**Meet the editor**

Born on June 11, 1951 in Tartu, Estonia. Education, PhD and current employer – University of Tartu. Position: Associate Professor of Differential and Integral Equations at the University of Tartu, Faculty of Mathematics and Computer Science, Institute of Mathematics. Administrative and organisational responsibilities: Estonian Operational Research Society, member of the Board;

Member of ECMI Educational Committee; member of Estonian Mathematical Society; former vice-dean of the Mathematical Faculty of the University of Tartu; editorial boards of several journals, member of different conferences organizing committees. Scientific interests: Mathematical Modelling, Numerical Methods, Differential Equations, Data Envelopment Analysis, GPS Tomography. Teaching: Differental Equations, Optimization, Models of Industrial Mathematics, Financial Mathematics, Modelling of Navigation Systems. Over 70 scientific publications. Current project: "Models of applied mathematics and mechanics".

Contents

**Preface IX** 

**Part 1 Fluid and Gas Dynamics 1** 

Chapter 2 **Simulation of the Scavenging** 

Chapter 3 **3D Multiphase Numerical** 

Chapter 1 **3D Numerical Modelling of Mould Filling of** 

**Process in Two-Stroke Engines 27** 

Chapter 4 **Numerical Simulation of the Unsteady** 

Chapter 5 **Numerical Modelling of Heavy Metals** 

**Part 2 Maxwell's Equations 157** 

Chapter 8 **Numerical Modelling and Design** 

Chapter 6 **Modelling Dynamics of Valley Glaciers 115**  Surendra Adhikari and Shawn J. Marshall

A.K. Ferouani, B. Liani and M. Lemerini

**of an Eddy Current Sensor 159**  Philip May and Erping Zhou

Rekha R. Rao, Lisa A. Mondy, David R. Noble,

**Modelling for Turbidity Current Flows 45** 

**Shock Interaction of Blunt Body Flows 73** 

**Transport Processes in Riverine Basins 91**  Seyed Mahmood Kashefipour and Ali Roshanfekr

Chapter 7 **Numerical Modelling of Dynamic Nitrogen at Atmospheric Pressure in a Negative DC Corona Discharge 143** 

**a Coat Hanger Distributer and Rectangular Cavity 3** 

María Isabel Lamas Galdo and Carlos G. Rodríguez Vidal

A. Georgoulas, P. Angelidis, K. Kopasakis and N. Kotsovinos

Leonid Bazyma, Vasyl Rashkovan and Vladimir Golovanevskiy

Matthew M. Hopkins, Carlton F. Brooks and Thomas A. Baer

### Contents

#### **Preface XI**

	- **Part 2 Maxwell's Equations 157**

X Contents


P. Esquivel, D. Cabuto, V. Sanchez and F. Chan

### Preface

Model in general sense is an analogue of the object under consideration, which replaces this object in human cognition. If the replacement action happens consciously, we call it Modelling. Science works by building models. Not cardboard or plasticine models, but models made out of symbols, special terms, theories, axioms, theorems etc. This helps to formulate ideas and identify underlying assumptions in formalized and abstracted form. Scientists can play with these symbolic models and adjust them until they start to behave in a way which resembles well enough to the things and objects they care about. When they have done this, we get an understanding of these things which is different and deeper than we could get if we would be limited to words and pictures. However, it is important to understand that models do not replace words and pictures, they sharpen them.

Models describe our beliefs about how the world functions. In mathematical modelling, those beliefs are translated into the language of mathematics. This has many advantages because the Mathematics is a very precise and concise language with well-defined rules for manipulations and the constructions in mathematical models are uniquely understandable. All the theoretical results that mathematicians have proved over hundreds of years are at our disposal legally, without law conflicts in any country. Mathematical Modelling is a fundamental and quantitative way to understand complex systems or phenomena is complementary to the traditional approaches of theory and experiment and deepens our knowledge about the world. Modelling is concerned with powerful methods of analysis designed to exploit high performance computing. The development of computers which can be used to perform numerical calculations and simulations is very important circumstance. The approach of Mathematical Modelling is becoming increasingly widespread in basic research and advanced technological applications, cross cutting the fields of physics, chemistry, mechanics, engineering, and technology. Mathematical Modelling is becoming an important subject as computers expand our ability to translate mathematical equations and formulations into concrete conclusions concerning the world, both natural and artificial, that we live in.

There is no such thing as a right model or a wrong model, particularly if we speak about mathematical models. Ideally we seek the simplest one that retains the essential features of the problem, object or phenomena. Every model should be compared with

#### XII Preface

available data and there should be as much input of available information about the process in question as possible. Often there is no alternative to a full numerical simulation of the process which means the solution of highly nonlinear equations using numerical approximations. Anyway, always it is necessary to understand how models are made. Numerical experiments and computer simulations are necessary parts of the whole process of contemporary Mathematical Modelling. For this, the existing commercial or free software is often used or adjusted. In the contributions of this book we find a large variety of computing environments and means mentioned and the information about these should be interesting for our readers.

Preface XI

Hopkins, Carlton F. Brooks and Thomas A. Baer from Sandia National Laboratories and Procter & Gamble Company, USA, we see an example of modelling of injection loading process of a ceramic paste into the mould which is a rectangular cavity. The process involves the complex interplay of extrusion of a viscous liquid into a mould where it displaces a gas phase. The inflow die should distribute the flow evenly across the mould. Numerical modelling is based on computational fluid dynamics and a finite element algorithm has been used to investigate filling behaviour for injection loading. Text is illustrated with several figures which explain the real experiments and comparison of these with numerical ones. In their Conclusion authors mention that the modelling has been successful in matching experimental data qualitatively. The simulations runs were made on 64 processors of Thunderbird (Sandia National Laboratories capacity computing platform) and ran in less than 3.5 hours. For future work, authors plan to investigate an advanced version of the level set method termed

The chapter "Simulation of the scavenging process in two-stroke engines" by María Isabel Lamas Galdo and Carlos G. Rodríguez Vidal, *Universidade da Coruña*, Spain, is focused on a numerical analysis to simulate the fluid flow inside the cylinder at the scavenging process. The Computational Fluid Dynamics is a very helpful tool to analyse the flow pattern inside the cylinder and these simulations can provide more detailed information than experimental studies due to the difficulties associated with the real measurement techniques. The contribution begins with introductory part about the performance of two-stroke engines which will facilitate the reader's understanding of the whole chapter. Then the Navier-Stokes governing equations of the flow inside the cylinder are introduced. In the fourth paragraph the generation of the mesh and other numerical details of computing experiment are described. In this work, a grid generation program, Gambit 2.4.6, was used to generate the mesh and for resolution of the equations the software ANSYS Fluent 6.3 was employed. A Computational Fluid Dynamics analysis carried out to study the scavenging process of two-stroke engines gave satisfactory results compared to experimental data. This study shows that Computational Fluid Dynamics predictions yield reasonably accurate results that allow improving the knowledge of the fluid flow characteristics.

"3D Multiphase Numerical Modelling for Turbidity Current Flows" of Anastasios Georgoulas, Kyriakos Kopasakis, Panagiotis Angelidis and Nikolaos Kotsovinos, Democritus University of Thrace, Xanthi, Greece, deals with modelling of gravity or density currents in river basin which constitute a large class of natural flows that are generated and driven by the density difference between two or even more fluids. In the case of floods, the suspended sediment concentration of river water rises to a great extent and the river plunges to the bottom of the receiving basin and forms so called hyperpycnal plume which is also known as turbidity current. Such flows are usually formed at river mouths in oceans, lakes or reservoirs, and can travel remarkable distances transferring, eroding and depositing large amounts of suspended sediments. These turbidity currents are very difficult to be observed and studied in the field due to their rare and unexpected occurrence nature, as they are usually formed during

the conformal decomposition finite element method.

In this book reader finds seventeen overviews of Mathematical modelling cases. All of these keep the presentation scheme where the real process itself is described at first, then governing rules in mathematical formulation are introduced and then the latter is numerically treated. The outputs of Numerical Modelling are always numbers even if these are represented graphically or visualized otherwise. Perhaps the most interest stage of the modelling process is interpretation of these numbers, translation of these into the real situation and making predictions and corrections in this. Here, in our book such kind of discussions are also given and these are very interesting, besides the explanations how to build mathematical models and how to use them.

The contributions are divided into three Sections according to the problems under consideration and corresponding mathematical models. The key word of the first Section, Fluid Dynamics, is turbulence. Anyone who has made more than a few airplane flights has almost surely had some bumpy ride when the airplane felt like a car on a rough road. That's turbulence. Mathematicians use vector fields to describe the motion of fluids. A fluid like air or water is made up of lots of tiny particles, molecules. One can think of each particle as pushing on its neighbours. In a gas like air, the particles hit each other and bounce off; in a liquid like water, there is a continual jostling, like people trying to get into a football ground. Each particle obeys Newton's laws of motion, the same laws that explain the motion of the planets but instead of quite simple planetary motion, one gets a rich variety of fluid behaviour, as it could be seen every time when to stir milk into coffee. To make a mathematical model of fluid motion, it is necessary to use the calculus. It was Daniel Bernoulli (1700 - 1782) who made a step, by discovering the equation named further by him. Bernoulli's work and that of his contemporaries was very important but had also many limitations, including the limitation to steady, frictionless flow. The limitation to steady flow was removed by Leonhard Euler (1707 - 1783) and the limitation to frictionless flow was removed thanks to the efforts of several mathematicians including Augustin Louis Cauchy (1789 - 1857), Claude-Louis Navier (1785 - 1836) and George Gabriel Stokes (1819 - 1903). The equation that resulted is more complicated than Bernoulli's, it uses vector calculus and it is the Navier-Stokes equation which lies at the foundations of modern fluid dynamics.

In "3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity" by Rekha R. Rao, Lisa A. Mondy, David R. Noble, Matthew M. Hopkins, Carlton F. Brooks and Thomas A. Baer from Sandia National Laboratories and Procter & Gamble Company, USA, we see an example of modelling of injection loading process of a ceramic paste into the mould which is a rectangular cavity. The process involves the complex interplay of extrusion of a viscous liquid into a mould where it displaces a gas phase. The inflow die should distribute the flow evenly across the mould. Numerical modelling is based on computational fluid dynamics and a finite element algorithm has been used to investigate filling behaviour for injection loading. Text is illustrated with several figures which explain the real experiments and comparison of these with numerical ones. In their Conclusion authors mention that the modelling has been successful in matching experimental data qualitatively. The simulations runs were made on 64 processors of Thunderbird (Sandia National Laboratories capacity computing platform) and ran in less than 3.5 hours. For future work, authors plan to investigate an advanced version of the level set method termed the conformal decomposition finite element method.

X Preface

available data and there should be as much input of available information about the process in question as possible. Often there is no alternative to a full numerical simulation of the process which means the solution of highly nonlinear equations using numerical approximations. Anyway, always it is necessary to understand how models are made. Numerical experiments and computer simulations are necessary parts of the whole process of contemporary Mathematical Modelling. For this, the existing commercial or free software is often used or adjusted. In the contributions of this book we find a large variety of computing environments and means mentioned

In this book reader finds seventeen overviews of Mathematical modelling cases. All of these keep the presentation scheme where the real process itself is described at first, then governing rules in mathematical formulation are introduced and then the latter is numerically treated. The outputs of Numerical Modelling are always numbers even if these are represented graphically or visualized otherwise. Perhaps the most interest stage of the modelling process is interpretation of these numbers, translation of these into the real situation and making predictions and corrections in this. Here, in our book such kind of discussions are also given and these are very interesting, besides the

The contributions are divided into three Sections according to the problems under consideration and corresponding mathematical models. The key word of the first Section, Fluid Dynamics, is turbulence. Anyone who has made more than a few airplane flights has almost surely had some bumpy ride when the airplane felt like a car on a rough road. That's turbulence. Mathematicians use vector fields to describe the motion of fluids. A fluid like air or water is made up of lots of tiny particles, molecules. One can think of each particle as pushing on its neighbours. In a gas like air, the particles hit each other and bounce off; in a liquid like water, there is a continual jostling, like people trying to get into a football ground. Each particle obeys Newton's laws of motion, the same laws that explain the motion of the planets but instead of quite simple planetary motion, one gets a rich variety of fluid behaviour, as it could be seen every time when to stir milk into coffee. To make a mathematical model of fluid motion, it is necessary to use the calculus. It was Daniel Bernoulli (1700 - 1782) who made a step, by discovering the equation named further by him. Bernoulli's work and that of his contemporaries was very important but had also many limitations, including the limitation to steady, frictionless flow. The limitation to steady flow was removed by Leonhard Euler (1707 - 1783) and the limitation to frictionless flow was removed thanks to the efforts of several mathematicians including Augustin Louis Cauchy (1789 - 1857), Claude-Louis Navier (1785 - 1836) and George Gabriel Stokes (1819 - 1903). The equation that resulted is more complicated than Bernoulli's, it uses vector calculus and it is the Navier-Stokes equation which lies

In "3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity" by Rekha R. Rao, Lisa A. Mondy, David R. Noble, Matthew M.

and the information about these should be interesting for our readers.

explanations how to build mathematical models and how to use them.

at the foundations of modern fluid dynamics.

The chapter "Simulation of the scavenging process in two-stroke engines" by María Isabel Lamas Galdo and Carlos G. Rodríguez Vidal, *Universidade da Coruña*, Spain, is focused on a numerical analysis to simulate the fluid flow inside the cylinder at the scavenging process. The Computational Fluid Dynamics is a very helpful tool to analyse the flow pattern inside the cylinder and these simulations can provide more detailed information than experimental studies due to the difficulties associated with the real measurement techniques. The contribution begins with introductory part about the performance of two-stroke engines which will facilitate the reader's understanding of the whole chapter. Then the Navier-Stokes governing equations of the flow inside the cylinder are introduced. In the fourth paragraph the generation of the mesh and other numerical details of computing experiment are described. In this work, a grid generation program, Gambit 2.4.6, was used to generate the mesh and for resolution of the equations the software ANSYS Fluent 6.3 was employed. A Computational Fluid Dynamics analysis carried out to study the scavenging process of two-stroke engines gave satisfactory results compared to experimental data. This study shows that Computational Fluid Dynamics predictions yield reasonably accurate results that allow improving the knowledge of the fluid flow characteristics.

"3D Multiphase Numerical Modelling for Turbidity Current Flows" of Anastasios Georgoulas, Kyriakos Kopasakis, Panagiotis Angelidis and Nikolaos Kotsovinos, Democritus University of Thrace, Xanthi, Greece, deals with modelling of gravity or density currents in river basin which constitute a large class of natural flows that are generated and driven by the density difference between two or even more fluids. In the case of floods, the suspended sediment concentration of river water rises to a great extent and the river plunges to the bottom of the receiving basin and forms so called hyperpycnal plume which is also known as turbidity current. Such flows are usually formed at river mouths in oceans, lakes or reservoirs, and can travel remarkable distances transferring, eroding and depositing large amounts of suspended sediments. These turbidity currents are very difficult to be observed and studied in the field due to their rare and unexpected occurrence nature, as they are usually formed during floods. So, mathematical and numerical models when properly designed and tested against field or laboratory data, can provide significant knowledge for turbidity current dynamics as well as for erosional and depositional characteristics. The model is based in a multiphase modification of the Reynolds Averaged Navier-Stokes Equations, the calculations of the model are performed using the robust Computational Fluid Dynamics solver ANSYS FLUENT. As authors mention in the Conclusion, the overall results of the laboratory scale application contribute considerably in the understanding of the dependence of the suspended sediment transport and deposition mechanism, from fundamental flow controlling parameters of natural, continuous, high-density turbidity currents that are usually formed during flood discharges at river outflows. Also it is shown that the proposed numerical approach can constitute a quite attractive alternative to laboratory experiments and field measurements since it allows the identification and the continuous monitoring of a wide range of flow parameters, with a relatively high accuracy.

Preface XIII

Iran, are also provided in this chapter. For numerical experiments the model FASTER - Flow and Solute Transport in Estuaries and Rivers - was used. The hydrodynamic module of FASTER model numerically solves the equations using Crank Nicolson

The next contribution, "Modelling Dynamics of Valley Glaciers" by Surendra Adhikari and Shawn J. Marshall from University of Calgary, Canada, the authors discuss the physics and mathematical models of ice flow in valley glacier and simulations of corresponding dynamics. They introduce ice rheology and brief summary of the history of numerical modelling in glaciology, describe the model physics and analyse various approximations associated with some models, provide an overview of numerical methods, concentrating on the finite element approach, and present a numerical comparison of several models. As we can read, glaciers and ice sheets presently occupy about 10% of the Earth's land surface in the annual mean, while valley glaciers make up only a small fraction of the global cryosphere ie. from the part of the Earth's surface where water is in solid form. However, the proper understanding of glacier dynamics is essential because valley glaciers are in close proximity to human settlement and any alteration in their dynamics affects society immediately. Over that, valley glaciers and ice caps are of significant concern for regional-scale fresh water resources and the dynamical response of glaciers have become proven indicators of climate change. For numerical experiments the open source FEM code Elmer was used. For each experiment considered in this study, the structured mesh was generated by using ElmerGrid, a two-dimensional mesh generator, capable of manipulating the mesh also in the third dimension. Numerical experiments require a large amount of memory and computation time, parallel runs were performed in a high-performance computing cluster provided by the Western

In the work "Numerical Simulation of Dynamic Nitrogen at Atmospheric Pressure in a Negative DC Corona Discharge" of A.K. Ferouani and M. Lemerini from the University Abou Bakr Belkaid, Tlemcen, Algéria, authors study the thermodynamics of the neutral gas subjected to energy injection as the result of electric discharge in the considered medium. The corona discharge is initiated when the electric field near the wire is sufficient to ionize the gaseous species. Numerous models of corona discharge have been proposed, the approach provided to the problem in this chapter allows considering the discharge only on its energetic aspect. The spatial-temporal evolution of the neutral gas particles is studied on the basis of hydrodynamic set of equations, i.e. equations of transport for mass, momentum and energy which is numerically

Second Section is dedicated to the Maxwell's equations. Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. From them one can develop most of the working relationships in the field. Because of their concise statement, they embody a high level of mathematical sophistication. The Maxwell equations are the set of four fundamental equations

schemes.

Canadian Research Grid (WestGrid).

solved by the Flux Corrected Transport method.

Next chapter is "Numerical Simulation of the Unsteady Shock Interaction of Blunt Body Flows", authors Leonid Bazyma from National Aerospace University "Kharkov Aviation Institute", Ukraine, Vasyl Rashkovan, National Polytechnic Institute, Mexico, and Vladimir Golovanevskiy from Western Australian School of Mines, Curtin University, Australia. We find there an example of modelling the head resistance of supersonic and hypersonic space vehicles which are extremely sensitive to aerodynamic resistance. For example, oblique shock waves, which are distributed from the bow part of an airplane or a rocket, can interact with a bow shock wave of any part of the fuselage construction and in certain cases the shock-wave interaction can result in significant negative and even catastrophic consequences for the aircraft. A cone or hemisphere serves as model of the head of the aircraft. The work presents the results of numerical simulation of the flow around a hemisphere at both the symmetric and asymmetric energy supply into the flow. The Godunov's difference scheme is realized for the system of non-stationary acoustic equations on a uniform rectangular grid.

"Numerical Modelling of Heavy Metals Transport Processes in Riverine Basins" by Seyed Mahmood Kashefipour from Shahid Chamran University, Ahwaz, Khuzestan, Iran and Ali Roshanfekr from Dalhousie University, Halifax, Canada. This chapter describes numerical modelling of heavy metals in a riverine basin and is important because the results of the work contribute to recognition and investigation of the heavy metals behaviours and different processes during their transportation along the rivers. Not only the sources and chemical and physical reactions, but also the environmental conditions affecting the rate of concentration variability of these substances are under consideration. The main purpose of this chapter is to describe the dissolved heavy metals modelling procedure and assess the impact of pH (the measure of the acidity) and electrical conductivity (EC) on the reaction coefficient used in dissolved lead and cadmium modelling. Details of the development of a modelling approach for predicting dissolved heavy metal fluxes and the application of the model to the field-measured data taken along the Karoon River, located in the south west of Iran, are also provided in this chapter. For numerical experiments the model FASTER - Flow and Solute Transport in Estuaries and Rivers - was used. The hydrodynamic module of FASTER model numerically solves the equations using Crank Nicolson schemes.

XII Preface

rectangular grid.

floods. So, mathematical and numerical models when properly designed and tested against field or laboratory data, can provide significant knowledge for turbidity current dynamics as well as for erosional and depositional characteristics. The model is based in a multiphase modification of the Reynolds Averaged Navier-Stokes Equations, the calculations of the model are performed using the robust Computational Fluid Dynamics solver ANSYS FLUENT. As authors mention in the Conclusion, the overall results of the laboratory scale application contribute considerably in the understanding of the dependence of the suspended sediment transport and deposition mechanism, from fundamental flow controlling parameters of natural, continuous, high-density turbidity currents that are usually formed during flood discharges at river outflows. Also it is shown that the proposed numerical approach can constitute a quite attractive alternative to laboratory experiments and field measurements since it allows the identification and the continuous monitoring of

Next chapter is "Numerical Simulation of the Unsteady Shock Interaction of Blunt Body Flows", authors Leonid Bazyma from National Aerospace University "Kharkov Aviation Institute", Ukraine, Vasyl Rashkovan, National Polytechnic Institute, Mexico, and Vladimir Golovanevskiy from Western Australian School of Mines, Curtin University, Australia. We find there an example of modelling the head resistance of supersonic and hypersonic space vehicles which are extremely sensitive to aerodynamic resistance. For example, oblique shock waves, which are distributed from the bow part of an airplane or a rocket, can interact with a bow shock wave of any part of the fuselage construction and in certain cases the shock-wave interaction can result in significant negative and even catastrophic consequences for the aircraft. A cone or hemisphere serves as model of the head of the aircraft. The work presents the results of numerical simulation of the flow around a hemisphere at both the symmetric and asymmetric energy supply into the flow. The Godunov's difference scheme is realized for the system of non-stationary acoustic equations on a uniform

"Numerical Modelling of Heavy Metals Transport Processes in Riverine Basins" by Seyed Mahmood Kashefipour from Shahid Chamran University, Ahwaz, Khuzestan, Iran and Ali Roshanfekr from Dalhousie University, Halifax, Canada. This chapter describes numerical modelling of heavy metals in a riverine basin and is important because the results of the work contribute to recognition and investigation of the heavy metals behaviours and different processes during their transportation along the rivers. Not only the sources and chemical and physical reactions, but also the environmental conditions affecting the rate of concentration variability of these substances are under consideration. The main purpose of this chapter is to describe the dissolved heavy metals modelling procedure and assess the impact of pH (the measure of the acidity) and electrical conductivity (EC) on the reaction coefficient used in dissolved lead and cadmium modelling. Details of the development of a modelling approach for predicting dissolved heavy metal fluxes and the application of the model to the field-measured data taken along the Karoon River, located in the south west of

a wide range of flow parameters, with a relatively high accuracy.

The next contribution, "Modelling Dynamics of Valley Glaciers" by Surendra Adhikari and Shawn J. Marshall from University of Calgary, Canada, the authors discuss the physics and mathematical models of ice flow in valley glacier and simulations of corresponding dynamics. They introduce ice rheology and brief summary of the history of numerical modelling in glaciology, describe the model physics and analyse various approximations associated with some models, provide an overview of numerical methods, concentrating on the finite element approach, and present a numerical comparison of several models. As we can read, glaciers and ice sheets presently occupy about 10% of the Earth's land surface in the annual mean, while valley glaciers make up only a small fraction of the global cryosphere ie. from the part of the Earth's surface where water is in solid form. However, the proper understanding of glacier dynamics is essential because valley glaciers are in close proximity to human settlement and any alteration in their dynamics affects society immediately. Over that, valley glaciers and ice caps are of significant concern for regional-scale fresh water resources and the dynamical response of glaciers have become proven indicators of climate change. For numerical experiments the open source FEM code Elmer was used. For each experiment considered in this study, the structured mesh was generated by using ElmerGrid, a two-dimensional mesh generator, capable of manipulating the mesh also in the third dimension. Numerical experiments require a large amount of memory and computation time, parallel runs were performed in a high-performance computing cluster provided by the Western Canadian Research Grid (WestGrid).

In the work "Numerical Simulation of Dynamic Nitrogen at Atmospheric Pressure in a Negative DC Corona Discharge" of A.K. Ferouani and M. Lemerini from the University Abou Bakr Belkaid, Tlemcen, Algéria, authors study the thermodynamics of the neutral gas subjected to energy injection as the result of electric discharge in the considered medium. The corona discharge is initiated when the electric field near the wire is sufficient to ionize the gaseous species. Numerous models of corona discharge have been proposed, the approach provided to the problem in this chapter allows considering the discharge only on its energetic aspect. The spatial-temporal evolution of the neutral gas particles is studied on the basis of hydrodynamic set of equations, i.e. equations of transport for mass, momentum and energy which is numerically solved by the Flux Corrected Transport method.

Second Section is dedicated to the Maxwell's equations. Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. From them one can develop most of the working relationships in the field. Because of their concise statement, they embody a high level of mathematical sophistication. The Maxwell equations are the set of four fundamental equations governing electromagnetism i.e., the behaviour of electric and magnetic fields. They were first written down in complete form by James Clerk Maxwell (1831 - 1879). Maxwell's equations have many very important implications in the life of a modern person, because people use devices that function on the base of the principles in Maxwell's equations every day without even knowing these. Maxwell's electromagnetic theory is one of the founding theories on which modern electrical science is based. The displacement-current concept introduced by James Clerk Maxwell is generally acknowledged as one of the most innovative concepts ever introduced in the development of physical science. Don't forget, it happened roughly one and half centuries ago. In the Section 2 reader find the contributions where the continuous analytical mathematical models are derived from the Maxwell's equations.

Preface XV

free-surface sub problem. The finite-element method is used for discretization of the magnetostatic sub problem, the Newton method is applied to find an element-wise distribution of the concentration and the finite-difference approach is used for the freesurface sub problem. Numerical results of different models are compared. All algorithms and the coupling of three sub problems were implemented in FORTRAN.

The last contribution of the second Section is "Finite Element Method applied to the modelling and Analysis of Induction Motors" by Rachek M'hemed and Merzouki Tarik from University Mouloud Mammeri of Tizi-ouzou, Algeria. Induction motor is an electromagnetic-mechanical actuator where interact several phenomena such as magnetic field, electrical circuits and mechanical motion. To model them, one must solve the system of nonlinear Partial Differential Equation derived from the Maxwell's equations combined to the materials properties, electric circuits, and mechanical motional equations. In this chapter an implementation of the finite element method for the modelling of rotating electrical machines, especially the squirrel cage three-phase induction motors, is presented. As a technical remark mention that the stator windings of the induction motors are usually in star or delta connection and that the squirrel rotor cage is formed by massive conductive bars short-circuited at their ends through massive and conductive end-rings. The finite element method is used to solve partial differential equation of electromagnetic phenomena occurred in induction motor. The resulting nonlinear time-dependent algebraic differential equations system obtained from the finite element formulation is solved using the Crank-Nicholson scheme, combined with the Newton-Raphson iteration method. The mechanical motional equation is solved by the fourth order Rung-Kutta method. It says in the Conclusion that the numerical results are in good agreement with corresponding results appearing

in the literature. Matlab and its PdeTool were used for numerical experiments.

which introduces us to the fractional ordinary differential equations.

Section three contains different case studies of mathematical modelling. We find examples about decision making and concurrent engineering problem, about modelling of behaviour of steels in casting and rolling technologies, about detecting masonry structures and ageing effects of old constructions, about earthquake information analysis. There is also a theoretical contribution from applied mathematics

The first chapter of third Section, "Numerical Evaluation of Product Development Processes" by Nadia Bhuiyan from Concordia University, Canada, is an exception in our book in the sense that the problem under consideration belongs to the field of operations research or decision science and the mathematical model is probabilistic. Concurrent engineering can broadly be defined as the integration of interrelated functions at the outset of the product development process in order to minimize risk and reduce effort downstream in the process, and to better meet customers' needs. In order to study and evaluate the performance of concurrent engineering and sequential new product development processes, an approach is provided which is based on an existing mathematical technique called the expected payoff method. The fundamental concept of the model is based on the premise that team members make decisions or

The Section begins with "Numerical Modelling and Design of an Eddy Current Sensor" by Philip May from Elcometer Instruments Ltd. and Erping Zhoub from University of Bolton, United Kingdom. The electrical impedance of the coil changes due to the influence of electrical eddy currents in the material. The data acquired from eddy current sensors however is affected by a large number of variables, which include sample conductivity, permeability, geometry of the objects, temperature and sensor lift off. This chapter focuses on the development and testing of a highly accurate and highly sensitive ferrite-cored sensor and a novel magnetic moment model of the sensor, which requires only the discretisation of the sensor core-air boundary interface. The first part of this chapter is the development of a set of partial differential equations to model the vector potential fields present in the regions bounding the sensor. Further the matrix method was developed in this chapter in order to calculate sensor coil impedance and induced voltage. The discretisation and approximation methods used in this are collocations, least square and Galerkin methods. Finally, a material profile equation for modelling the interaction between the sensor and test material is under consideration. Mathcad, version 11.0a and commercial Finite Element Method solver MagNet, version 6.25 are referred as software used for numerical experiments.

Next in this Section one finds "Numerical study of diffusion of interacting particles in a magnetic fluid layer" of authors Olga Lavrova and Viktor Polevikov from Belarusian State University, Belarus, and Lutz Tobiska from Otto-von-Guericke University, Germany. This study is devoted to the classical problem of ferrohydrostatics on stability, known as the normal field instability or the Rosensweig instability problem of a horizontal semi-infinite layer of magnetic fluid under the influence of gravity and a uniform magnetic field normal to the plane free surface of the layer. Magnetic fluids are stable colloidal suspensions of ferromagnetic nano-particles in a nonmagnetic liquid carrier. Motion of particles in magnetic fluids under the action of magnetic fields is of particular interest for contemporary mathematical and numerical modelling in ferrohydrodynamics. Mention that the particles are in Brownian motion inside the ferrofluid, when no magnetic field is applied and the gravity force has a negligible influence to the particles. The mathematical model introduced in this chapter consists of three parts: the magnetostatic sub problem, the concentration sub problem and the free-surface sub problem. The finite-element method is used for discretization of the magnetostatic sub problem, the Newton method is applied to find an element-wise distribution of the concentration and the finite-difference approach is used for the freesurface sub problem. Numerical results of different models are compared. All algorithms and the coupling of three sub problems were implemented in FORTRAN.

XIV Preface

numerical experiments.

governing electromagnetism i.e., the behaviour of electric and magnetic fields. They were first written down in complete form by James Clerk Maxwell (1831 - 1879). Maxwell's equations have many very important implications in the life of a modern person, because people use devices that function on the base of the principles in Maxwell's equations every day without even knowing these. Maxwell's electromagnetic theory is one of the founding theories on which modern electrical science is based. The displacement-current concept introduced by James Clerk Maxwell is generally acknowledged as one of the most innovative concepts ever introduced in the development of physical science. Don't forget, it happened roughly one and half centuries ago. In the Section 2 reader find the contributions where the continuous analytical mathematical models are derived from the Maxwell's equations.

The Section begins with "Numerical Modelling and Design of an Eddy Current Sensor" by Philip May from Elcometer Instruments Ltd. and Erping Zhoub from University of Bolton, United Kingdom. The electrical impedance of the coil changes due to the influence of electrical eddy currents in the material. The data acquired from eddy current sensors however is affected by a large number of variables, which include sample conductivity, permeability, geometry of the objects, temperature and sensor lift off. This chapter focuses on the development and testing of a highly accurate and highly sensitive ferrite-cored sensor and a novel magnetic moment model of the sensor, which requires only the discretisation of the sensor core-air boundary interface. The first part of this chapter is the development of a set of partial differential equations to model the vector potential fields present in the regions bounding the sensor. Further the matrix method was developed in this chapter in order to calculate sensor coil impedance and induced voltage. The discretisation and approximation methods used in this are collocations, least square and Galerkin methods. Finally, a material profile equation for modelling the interaction between the sensor and test material is under consideration. Mathcad, version 11.0a and commercial Finite Element Method solver MagNet, version 6.25 are referred as software used for

Next in this Section one finds "Numerical study of diffusion of interacting particles in a magnetic fluid layer" of authors Olga Lavrova and Viktor Polevikov from Belarusian State University, Belarus, and Lutz Tobiska from Otto-von-Guericke University, Germany. This study is devoted to the classical problem of ferrohydrostatics on stability, known as the normal field instability or the Rosensweig instability problem of a horizontal semi-infinite layer of magnetic fluid under the influence of gravity and a uniform magnetic field normal to the plane free surface of the layer. Magnetic fluids are stable colloidal suspensions of ferromagnetic nano-particles in a nonmagnetic liquid carrier. Motion of particles in magnetic fluids under the action of magnetic fields is of particular interest for contemporary mathematical and numerical modelling in ferrohydrodynamics. Mention that the particles are in Brownian motion inside the ferrofluid, when no magnetic field is applied and the gravity force has a negligible influence to the particles. The mathematical model introduced in this chapter consists of three parts: the magnetostatic sub problem, the concentration sub problem and the

The last contribution of the second Section is "Finite Element Method applied to the modelling and Analysis of Induction Motors" by Rachek M'hemed and Merzouki Tarik from University Mouloud Mammeri of Tizi-ouzou, Algeria. Induction motor is an electromagnetic-mechanical actuator where interact several phenomena such as magnetic field, electrical circuits and mechanical motion. To model them, one must solve the system of nonlinear Partial Differential Equation derived from the Maxwell's equations combined to the materials properties, electric circuits, and mechanical motional equations. In this chapter an implementation of the finite element method for the modelling of rotating electrical machines, especially the squirrel cage three-phase induction motors, is presented. As a technical remark mention that the stator windings of the induction motors are usually in star or delta connection and that the squirrel rotor cage is formed by massive conductive bars short-circuited at their ends through massive and conductive end-rings. The finite element method is used to solve partial differential equation of electromagnetic phenomena occurred in induction motor. The resulting nonlinear time-dependent algebraic differential equations system obtained from the finite element formulation is solved using the Crank-Nicholson scheme, combined with the Newton-Raphson iteration method. The mechanical motional equation is solved by the fourth order Rung-Kutta method. It says in the Conclusion that the numerical results are in good agreement with corresponding results appearing in the literature. Matlab and its PdeTool were used for numerical experiments.

Section three contains different case studies of mathematical modelling. We find examples about decision making and concurrent engineering problem, about modelling of behaviour of steels in casting and rolling technologies, about detecting masonry structures and ageing effects of old constructions, about earthquake information analysis. There is also a theoretical contribution from applied mathematics which introduces us to the fractional ordinary differential equations.

The first chapter of third Section, "Numerical Evaluation of Product Development Processes" by Nadia Bhuiyan from Concordia University, Canada, is an exception in our book in the sense that the problem under consideration belongs to the field of operations research or decision science and the mathematical model is probabilistic. Concurrent engineering can broadly be defined as the integration of interrelated functions at the outset of the product development process in order to minimize risk and reduce effort downstream in the process, and to better meet customers' needs. In order to study and evaluate the performance of concurrent engineering and sequential new product development processes, an approach is provided which is based on an existing mathematical technique called the expected payoff method. The fundamental concept of the model is based on the premise that team members make decisions or actions that maximize the payoff that these actions bring to the team. Team members must obtain, process, and communicate information to one another to make decisions that will optimize their performance. This leads to the network methods where network can be defined as a system of interconnected elements, all of which work together to produce a desired output. The principle of the expected payoff method has been applied mainly in the field of economics, management science, and in certain areas of artificial intelligence, with respect to decision-making.

Preface XVII

experimental curves obtained from the Gleeble machine are noisy and before the application of inverse analysis these were smoothed using Fast Fourier

Next contribution in this Section is "Distinct element method applied on old masonry structures", author Marwan Al-Heib Ineris – Ecole des Mines de Nancy, France. The analysis of old masonry constructions is a complicated task for several reasons: the characterization of the mechanical properties of the materials used is difficult and expensive, the mechanical properties of constructions are in large variability due to workmanship and use of natural materials, long construction and existing periods are caused significant changes in the core and constitution of structural elements, the sequence of construction is unknown, there can be unknown damages in the structure etc. However, several methods and computational tools are available for the assessment of the mechanical behaviour of old constructions. These have different levels of complexity from simple graphical methods and hand calculations to complex mathematical formulations and large systems of non-linear equations, different availability for the practitioner from readily available in any consulting engineer office to scarcely available in a few research-oriented institutions and large consulting offices, different time requirements and different costs. Three approaches are generally implemented to model the masonry elements: equivalent medium, discontinuous medium using continuous numerical methods as finite element or boundary element methods, and discontinuous medium using distinct element approach. In this chapter two case studies of the application of distinct element method are presented. The first concerns the simulation of the behaviour of an underground structure of old tunnel supported by masonry of stone elements. The second case study concerns the behaviour of a masonry wall under the effect of an underground excavation. The distinct element method is in use from 1971 and it considers the medium as an assembly of distinct rigid blocs that are linked together by joints. One can distinguish between rigid blocs and deformable blocs. Deformable blocs can be studied using difference element method. In this research UDEC, Universal Distinct Elements Code

"Phenomenological modelling of cyclic plasticity" by Radim Halama, Josef Sedlák and Michal Šofer, VŠB-Technical University of Ostrava, Czech Republic. Once more the stress-strain behaviour of metals under a cyclic loading is under consideration. This chapter is addressed to so called phenomenological models, which are based purely on the observed behaviour of materials. Background of the effects in cyclic plasticity of metals explained in the text helps to understand incremental theory of plasticity and main features of some cyclic plasticity models. The accumulation of axial plastic strain can occur cycle by cycle, this effect is called cyclic creep or ratcheting. The results of multiaxial ratcheting predictions are presented. It is concluded from the results of simulations that used combined hardening model can fairly well predict the trend of accumulation of plastic deformation in comparison with the experimental observations. The steel specimens were subjected to tension-compression and

Transformation algorithm.

was used.

"Numerical modelling of steel deformation at extra-high temperatures", contribution of Marcin Hojny and Miroslaw Glowacki AGH University of Science and Technology, Krakow, Poland, give us an example of new technology development which concentrate on energy preservation and environmental protection. New methods of steel strip manufacturing processes which could be characterized by very high temperature allowed at the mill entry, are under consideration. The mathematical and experimental modelling of mushy steel deformation is an innovative topic regarding the very high temperature range deformation processes. The model presented in this chapter allows the simulation of the deformation of material with mushy zone and consists of two main parts – mechanical and thermal and is written down as a system of partial differential equations. The mathematical model and numerical experiments are verified with the help of physical simulations realised in Institute for Ferrous Metallurgy in Gliwice, Poland using Gleeble 3800® thermo-mechanical simulator. The thermo-physical properties of the steel, necessary in calculations, were determined using commercial JMatPro software. The presented Def\_Semi\_Solid program is a unique tool, which can be very helpful and may enable the right interpretation of results of very high temperature tests. The Def\_Semi\_Solid system was used as a feedback unit with Gleeble 3800 simulator. Additionally, for numerical treatment of the mathematical model a Finite Element Method solver was used.

"Inverse analysis applied to mushy steel rheological properties testing using hybrid numerical-analytical model" by Miroslaw Glowacki, AGH University of Science and Technology, Poland. The application area is the same as in previous contribution – steel casting and rolling technologies, but the approach is based on the inverse analysis and providing a proposition of a hybrid numerical-analytical model of semi-solid steel deformation. Application of inverse analysis and the proposed model allows for the testing of rheological properties of steels at very high temperature. Mention that rheology means the study of the flow and deformation of matter, primarily in the liquid state. Steels testing experiments at temperature higher than 1400 °C are difficult due to deformation instability and risk of sample damage and cannot be interpreted using traditional methods. Appropriate interpretation is possible only with the help of a computer aided engineering system. This contribution reports a new model underlying such a system and code developed by the author's team which together with Gleeble 3800 physical simulator equipped with high temperature module allows for investigation of properties of semi-solid steel. The system Def\_Semi\_Solid is a result of theoretical research conducted in a team lead by the chapter author. The program is compatible with both Windows and Unix based platforms. The experimental curves obtained from the Gleeble machine are noisy and before the application of inverse analysis these were smoothed using Fast Fourier Transformation algorithm.

XVI Preface

actions that maximize the payoff that these actions bring to the team. Team members must obtain, process, and communicate information to one another to make decisions that will optimize their performance. This leads to the network methods where network can be defined as a system of interconnected elements, all of which work together to produce a desired output. The principle of the expected payoff method has been applied mainly in the field of economics, management science, and in certain

"Numerical modelling of steel deformation at extra-high temperatures", contribution of Marcin Hojny and Miroslaw Glowacki AGH University of Science and Technology, Krakow, Poland, give us an example of new technology development which concentrate on energy preservation and environmental protection. New methods of steel strip manufacturing processes which could be characterized by very high temperature allowed at the mill entry, are under consideration. The mathematical and experimental modelling of mushy steel deformation is an innovative topic regarding the very high temperature range deformation processes. The model presented in this chapter allows the simulation of the deformation of material with mushy zone and consists of two main parts – mechanical and thermal and is written down as a system of partial differential equations. The mathematical model and numerical experiments are verified with the help of physical simulations realised in Institute for Ferrous Metallurgy in Gliwice, Poland using Gleeble 3800® thermo-mechanical simulator. The thermo-physical properties of the steel, necessary in calculations, were determined using commercial JMatPro software. The presented Def\_Semi\_Solid program is a unique tool, which can be very helpful and may enable the right interpretation of results of very high temperature tests. The Def\_Semi\_Solid system was used as a feedback unit with Gleeble 3800 simulator. Additionally, for numerical treatment of

areas of artificial intelligence, with respect to decision-making.

the mathematical model a Finite Element Method solver was used.

"Inverse analysis applied to mushy steel rheological properties testing using hybrid numerical-analytical model" by Miroslaw Glowacki, AGH University of Science and Technology, Poland. The application area is the same as in previous contribution – steel casting and rolling technologies, but the approach is based on the inverse analysis and providing a proposition of a hybrid numerical-analytical model of semi-solid steel deformation. Application of inverse analysis and the proposed model allows for the testing of rheological properties of steels at very high temperature. Mention that rheology means the study of the flow and deformation of matter, primarily in the liquid state. Steels testing experiments at temperature higher than 1400 °C are difficult due to deformation instability and risk of sample damage and cannot be interpreted using traditional methods. Appropriate interpretation is possible only with the help of a computer aided engineering system. This contribution reports a new model underlying such a system and code developed by the author's team which together with Gleeble 3800 physical simulator equipped with high temperature module allows for investigation of properties of semi-solid steel. The system Def\_Semi\_Solid is a result of theoretical research conducted in a team lead by the chapter author. The program is compatible with both Windows and Unix based platforms. The

Next contribution in this Section is "Distinct element method applied on old masonry structures", author Marwan Al-Heib Ineris – Ecole des Mines de Nancy, France. The analysis of old masonry constructions is a complicated task for several reasons: the characterization of the mechanical properties of the materials used is difficult and expensive, the mechanical properties of constructions are in large variability due to workmanship and use of natural materials, long construction and existing periods are caused significant changes in the core and constitution of structural elements, the sequence of construction is unknown, there can be unknown damages in the structure etc. However, several methods and computational tools are available for the assessment of the mechanical behaviour of old constructions. These have different levels of complexity from simple graphical methods and hand calculations to complex mathematical formulations and large systems of non-linear equations, different availability for the practitioner from readily available in any consulting engineer office to scarcely available in a few research-oriented institutions and large consulting offices, different time requirements and different costs. Three approaches are generally implemented to model the masonry elements: equivalent medium, discontinuous medium using continuous numerical methods as finite element or boundary element methods, and discontinuous medium using distinct element approach. In this chapter two case studies of the application of distinct element method are presented. The first concerns the simulation of the behaviour of an underground structure of old tunnel supported by masonry of stone elements. The second case study concerns the behaviour of a masonry wall under the effect of an underground excavation. The distinct element method is in use from 1971 and it considers the medium as an assembly of distinct rigid blocs that are linked together by joints. One can distinguish between rigid blocs and deformable blocs. Deformable blocs can be studied using difference element method. In this research UDEC, Universal Distinct Elements Code was used.

"Phenomenological modelling of cyclic plasticity" by Radim Halama, Josef Sedlák and Michal Šofer, VŠB-Technical University of Ostrava, Czech Republic. Once more the stress-strain behaviour of metals under a cyclic loading is under consideration. This chapter is addressed to so called phenomenological models, which are based purely on the observed behaviour of materials. Background of the effects in cyclic plasticity of metals explained in the text helps to understand incremental theory of plasticity and main features of some cyclic plasticity models. The accumulation of axial plastic strain can occur cycle by cycle, this effect is called cyclic creep or ratcheting. The results of multiaxial ratcheting predictions are presented. It is concluded from the results of simulations that used combined hardening model can fairly well predict the trend of accumulation of plastic deformation in comparison with the experimental observations. The steel specimens were subjected to tension-compression and tension/torsion on the test machine MTS 858 MiniBionix. For implementing of numerical experiments commercial software based on finite element method, ANSYS, Abaqus, MSC.Nastran and MSC.Marc are compared in the work. The stress strain behaviour model was implemented to the ANSYS program via user subroutine.

Preface XIX

**Peep Miidla**

Estonia

Institute of Mathematics

Associate Professor of Differential and Integral Equations at the University of Tartu, Faculty of Mathematics and Computer Science,

This information is important to determine strategies for wide-area monitoring and

Mathematicians spend a great deal of time building, testing, comparing and revising models. More and more publications are dedicated to introducing, applying and interpreting these valuable tools. Mathematical models are and will be the principal

special protection systems.

instruments of modern science.

The chapter "Numerical schemes for fractional ordinary differential equations" of Weihua Deng and Can Li, Lanzhou University, People's Republic of China is mainly in theoretical interest. In recent years, fractional calculus and fractional differential equations become more and more popular because of their new powerful potential applications. A large number of new differential equations models that involve fractional calculus are developed. These models have been applied successfully in mechanics, biology, chemistry, electrical engineering etc. The fractional ordinary differential equations are introduced in terms of fractional derivative in the Caputo sense. The difficulties in solving fractional differential equations appear because fractional derivatives are non-local operators. This means that the next state of a system not only depends on its current state but also on its historical states starting from the initial time. This property is often close to reality and is the main reason why fractional calculus has become more and more useful and popular. In other words, this non-local property is good for modelling reality, but a challenge for numerical computations. In this chapter reader finds overviews of recent developments in the predictor-corrector approach for fractional dynamic systems, of the short memory principle and the nested meshes, of the predictor-corrector schemes and improved versions for initial value problem, of the convergence orders and arithmetic complexity. Collegial examples of applications of described ideas are also presented.

The last chapter "Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models" P. Esquivel from Technological institute of Tepic division of electrical and electronics engineering, Nayarit, V. Sanchez and F. Chan from University of Quintana Roo division of sciences and engineering, Quintana Roo, México deals with statistical techniques for the identification of dynamic systems, analysis of empirical orthogonal function. The analysis of empirical orthogonal functions is primarily a method of compressing of time and space variability of a data set into the lower possible number of spatial patterns. The conventional formulation of EOF analysis involves a set of optimal basis which is enforced to approach the original field with modes at infinite frequency. At local time 15:36:14.730, October 9, 1995 a submarine earthquake will occurred near the Mexican coast, Colima-Jalisco. This earthquake was recorded by sixteen stations of GPS-based multiple phase measurement units. The data obtained from this real event is used to study the practical applicability of the proposed method to characterize spatial-temporal behaviour and to assess oscillations patterns in wide-area dynamical systems. Additionally, the practical computation of mode shape identification in relation to the proposed decomposition from measurements data is discussed. Numerical results show that the proposed method can provide accurate estimation of dynamical effects, modal frequency, mode shapes, and time instants of intermittent transient responses. This information is important to determine strategies for wide-area monitoring and special protection systems.

XVIII Preface

tension/torsion on the test machine MTS 858 MiniBionix. For implementing of numerical experiments commercial software based on finite element method, ANSYS, Abaqus, MSC.Nastran and MSC.Marc are compared in the work. The stress strain behaviour model was implemented to the ANSYS program via user subroutine.

The chapter "Numerical schemes for fractional ordinary differential equations" of Weihua Deng and Can Li, Lanzhou University, People's Republic of China is mainly in theoretical interest. In recent years, fractional calculus and fractional differential equations become more and more popular because of their new powerful potential applications. A large number of new differential equations models that involve fractional calculus are developed. These models have been applied successfully in mechanics, biology, chemistry, electrical engineering etc. The fractional ordinary differential equations are introduced in terms of fractional derivative in the Caputo sense. The difficulties in solving fractional differential equations appear because fractional derivatives are non-local operators. This means that the next state of a system not only depends on its current state but also on its historical states starting from the initial time. This property is often close to reality and is the main reason why fractional calculus has become more and more useful and popular. In other words, this non-local property is good for modelling reality, but a challenge for numerical computations. In this chapter reader finds overviews of recent developments in the predictor-corrector approach for fractional dynamic systems, of the short memory principle and the nested meshes, of the predictor-corrector schemes and improved versions for initial value problem, of the convergence orders and arithmetic complexity. Collegial examples of applications of described ideas are also presented.

The last chapter "Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models" P. Esquivel from Technological institute of Tepic division of electrical and electronics engineering, Nayarit, V. Sanchez and F. Chan from University of Quintana Roo division of sciences and engineering, Quintana Roo, México deals with statistical techniques for the identification of dynamic systems, analysis of empirical orthogonal function. The analysis of empirical orthogonal functions is primarily a method of compressing of time and space variability of a data set into the lower possible number of spatial patterns. The conventional formulation of EOF analysis involves a set of optimal basis which is enforced to approach the original field with modes at infinite frequency. At local time 15:36:14.730, October 9, 1995 a submarine earthquake will occurred near the Mexican coast, Colima-Jalisco. This earthquake was recorded by sixteen stations of GPS-based multiple phase measurement units. The data obtained from this real event is used to study the practical applicability of the proposed method to characterize spatial-temporal behaviour and to assess oscillations patterns in wide-area dynamical systems. Additionally, the practical computation of mode shape identification in relation to the proposed decomposition from measurements data is discussed. Numerical results show that the proposed method can provide accurate estimation of dynamical effects, modal frequency, mode shapes, and time instants of intermittent transient responses.

Mathematicians spend a great deal of time building, testing, comparing and revising models. More and more publications are dedicated to introducing, applying and interpreting these valuable tools. Mathematical models are and will be the principal instruments of modern science.

#### **Peep Miidla**

Associate Professor of Differential and Integral Equations at the University of Tartu, Faculty of Mathematics and Computer Science, Institute of Mathematics Estonia

**Part 1** 

**Fluid and Gas Dynamics** 

## **Part 1**

**Fluid and Gas Dynamics** 

**1** 

*USA* 

**3D Numerical Modelling of Mould Filling of a** 

**Coat Hanger Distributer and Rectangular Cavity** 

Rekha R. Rao1, Lisa A. Mondy1, David R. Noble1, Matthew M. Hopkins1,

Filling processes occur in a wide range of industries, ranging from packaging of consumer products to manufacturing processes for making polymeric, metal and ceramic components. These processes involve the complex interplay of extrusion of a viscous liquid into a mould or container where it displaces a gas phase. Numerical modelling based on computational fluid dynamics can be useful for understanding the filling process. However, complexity arises in that the fluid dynamics in both the viscous liquid and gas phase must be resolved while concurrently determining the location of the fluid-gas interface and the interaction of this interface with the solid surface, *i.e.*, the wetting behaviour. Determining the free surface

Numerical methods have been applied to bottle and container filling for consumer products where the rheology can include shear thinning and viscoelastic effects and instabilities such as buckling and coiling may be prevalent [Tome, et al., 2001; Oishi et al., 2008; Roberts & Rao, 2011; Ville et al., 2011]. In metal casting simulations, the fluids are generally Newtonian, but complexity arises from the high injection rates leading to turbulent flow and temperature-dependent behaviour such as solidification [Ilinca & Hetu, 2000; Cross et al, 2006]. For injection moulding of polymers, time- and temperature-dependent effects are seen in conjunction with non-Newtonian rheology [Ilinca & Hetu, 2001; Kumar & Ghoshdastidar, 2002]. In powder injection moulding for ceramic and metal forming, a suspension of particles is injected into a mould to create a green part, which later sees further processing steps to produce the final part [Hwang & Kwon, 2002; Ilinca & Hetu, 2008]. Numerical methods for these problems range from finite difference, to finite volume,

General classes of algorithms for determining the location of the free surface include Eulerian, Lagrangian and arbitrary Lagrangian-Eulerian (ALE) descriptions. Eulerian methods use a fixed-grid with an interface capturing technique such as the volume of fluid [Hirt & Nichols, 1981] or level-set method [Sethian, 1999] to determine the location of the free surface. Traditional Lagrangian methods use a moving mesh as a material interface that advects with the fluid. These methods often require multiple remeshing steps to avoid mesh distortion and tangling [see for instance, Bach & Hassager, 1985; R. Radovitzky & M. Ortiz,

location and wetting behaviour is an integral part of the numerical method.

**1. Introduction** 

and finite element.

Carlton F. Brooks1 and Thomas A. Baer2

*1Sandia National Laboratories 2Procter and Gamble Company* 

## **3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity**

Rekha R. Rao1, Lisa A. Mondy1, David R. Noble1, Matthew M. Hopkins1, Carlton F. Brooks1 and Thomas A. Baer2

*1Sandia National Laboratories 2Procter and Gamble Company USA* 

#### **1. Introduction**

Filling processes occur in a wide range of industries, ranging from packaging of consumer products to manufacturing processes for making polymeric, metal and ceramic components. These processes involve the complex interplay of extrusion of a viscous liquid into a mould or container where it displaces a gas phase. Numerical modelling based on computational fluid dynamics can be useful for understanding the filling process. However, complexity arises in that the fluid dynamics in both the viscous liquid and gas phase must be resolved while concurrently determining the location of the fluid-gas interface and the interaction of this interface with the solid surface, *i.e.*, the wetting behaviour. Determining the free surface location and wetting behaviour is an integral part of the numerical method.

Numerical methods have been applied to bottle and container filling for consumer products where the rheology can include shear thinning and viscoelastic effects and instabilities such as buckling and coiling may be prevalent [Tome, et al., 2001; Oishi et al., 2008; Roberts & Rao, 2011; Ville et al., 2011]. In metal casting simulations, the fluids are generally Newtonian, but complexity arises from the high injection rates leading to turbulent flow and temperature-dependent behaviour such as solidification [Ilinca & Hetu, 2000; Cross et al, 2006]. For injection moulding of polymers, time- and temperature-dependent effects are seen in conjunction with non-Newtonian rheology [Ilinca & Hetu, 2001; Kumar & Ghoshdastidar, 2002]. In powder injection moulding for ceramic and metal forming, a suspension of particles is injected into a mould to create a green part, which later sees further processing steps to produce the final part [Hwang & Kwon, 2002; Ilinca & Hetu, 2008]. Numerical methods for these problems range from finite difference, to finite volume, and finite element.

General classes of algorithms for determining the location of the free surface include Eulerian, Lagrangian and arbitrary Lagrangian-Eulerian (ALE) descriptions. Eulerian methods use a fixed-grid with an interface capturing technique such as the volume of fluid [Hirt & Nichols, 1981] or level-set method [Sethian, 1999] to determine the location of the free surface. Traditional Lagrangian methods use a moving mesh as a material interface that advects with the fluid. These methods often require multiple remeshing steps to avoid mesh distortion and tangling [see for instance, Bach & Hassager, 1985; R. Radovitzky & M. Ortiz,

3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity 5

Understanding dynamic wetting, or the interaction between the free surface and the mould walls, has been the subject of numerous experimental and theoretical studies and is still an outstanding research topic [see for instance Blake, 2006; Ren et al, 2011]. The difficulty arises from the contradiction of a moving contact line at the fluid-gas interface and the no slip boundary conditions traditionally applied at solid surfaces. How can the contact line advance when the velocity vanishes at the solid surface? Highly viscous materials such as polymers and particle suspensions have large capillary numbers (a measure of the ratio of viscous forces to capillary forces) and are often hypothesized to obey a rolling motion condition with an 180o dynamic contact angle. Numerically, this approach is difficult to apply and other methods have been proposed. The simplest models use a Navier slip condition to allow either slip on the entire solid surface, slip for the gas phase only, or slip only at the dynamic contact line. These models are *ad hoc* and ignore any thermodynamics considerations such as the static contact angle and surface energy. More advanced wetting models generally give dynamic contact angle as a function of the local Ca, the static contact angle and other material properties of the fluid and the solid surface [see Schunk et al., 2006 for a brief review of this work]. Hoffman [1975] used experimental measurements to develop a universal correlation for the dynamic contact angle as a function of static contact angle and a local capillary number, while Cox [1986] developed a competing model using asymptotic analysis. Kistler [1983] used a linear model that was easy to implement in numerical computer codes. Shikhmurzaev [1994] used hydrodynamic theory and included a surface phase as part of the wetting model. Blake [1969] developed a molecular kinetic theory that reduces to a linear model for small contact angles. Blake noted that the advancing angle is a monotonically increasing function of Ca. The degree of velocity dependence will however increase steeply as viscosity increases or surface tension decreases. To model dynamic contact for the filling process, the dynamic angle is tied to the balance of forces at the advancing wetting line, namely the tangential wetting line force, liquid-gas surface tension force, and fluid viscous force. Here, a version of the Blake model is used that is straightforward to populate with experiments. The dynamic contact angle is measured in the laboratory as a function of the velocity of the wetting line for the fluids and surface used in our experiments as input to the wetting model. The Blake model is also easy to implement numerically, since the wetting speed can be written as a function of the dynamic and static contact angles. The performance of the Blake model at the high capillary number limit may be suspect, since it exhibits an unbounded dynamic contact angle while more physical models such as Cox and Hoffman reach a limit of 180o for large capillary numbers [Schunk et al, 2006].

Because the initial design of the mould exhibited pooling of the fluid in the centre of the cavity and this would lead to poor filling (see Figure 1), we tried two minor redesigns to the distributor to see if we could improve the flow into the mould and reduce pooling. Ideally, we would like to see a flat profile coming out of the distributor and a more one-dimensional front shape. Figure 2 shows the original mesh, Mesh 1, and a variation, Mesh 2, with a longer distributor, and Mesh 3 with a longer-taller distributor. The idea behind Mesh 2 was to give a longer length for the flow to develop a flat profile and fill up the distributor before entering the main cavity. Mesh 3 kept this longer distributor and made it wider on inflow to ease the fluid

**1.1 Dynamic wetting models and mould filling** 

**1.2 Mould design** 

1998; Zhang & Khayat, 2001]. To avoid these problems, Lagrangian mesh-free methods have been developed using smooth particle hydrodynamics or the material point method to avoid meshing issues [Kulasegaram et al, 2006; Kauzlaric et a;., 2011; Love & Sulsky, 2006]. These methods work well for being able to capture a moving interface with topological changes, but have difficulty in accurately solving the base physics, *e.g.* viscosity, and applying boundary conditions such as surface tension. ALE methods are hybrid techniques that seek to exploit the benefits of both the Eulerian and Lagrangian description in a hybrid manner, to determine the location of the interface [i.e. Sackinger et al, 1996; Lewis et al., 1998; Nithiarasu, 2005].

Injection loading of a ceramic paste is a high-rate process used to create green ceramic parts, which subsequently experience binder burnout and sintering to produce the final ceramic part. In the process of interest, the mould is a rectangular cavity, with an inflow from a coat hanger die that should distribute the flow evenly across the mould inflow. The cavity is small with dimensions of 1.3 cm by 3.6 cm in plane and a height of 0.4 cm. The inflow tube to the distributer has a diameter of 0.5cm. Figure 1 shows short shots (or incomplete filling of a mould) for the injection loading process, illustrating some of the defects that can occur when a fluid with complex shear and temperature-dependent rheology interacts with a high rate process.

Fig. 1. Short shots of injection loading of a ceramic paste in a part are shown [Rao et al, 2006]. The photo on the left is injection loading using a slow injection speed, while the photo on the right uses a higher injection speed. At low filling speed, the paste acts like a solid material. Even at high filling rates, when the paste begins to act as a fluid, pooling at the centre of the mould is seen and the desired mould shape is not achieved.

Temperature control and high-rate processing can limit the folding instability seen on the left of Figure 1. However, the pooling phenomenon shown on the right of Figure 1 occurs when optimal processing conditions are used, indicating that the design of the distributer may be responsible for material building up in the centre of the die. The complex shear-rate and temperature-dependent rheology of the ceramic paste was determined to follow a power-law dependence on shear rate and a Williams-Landau-Ferry temperature dependence [Rao et al., 2006]. The material shear thinned quickly to a constant viscosity value at moderate shear rates. Thus, because of the high shear rates in the injection loader, it was determined that the rheology was essentially Newtonian as long as the temperature remained constant at the processing temperature. Therefore, a study was undertaken to better understand the filling dynamics and reduce pooling by changing the distributer design using a Newtonian fluid.

#### **1.1 Dynamic wetting models and mould filling**

4 Numerical Modelling

1998; Zhang & Khayat, 2001]. To avoid these problems, Lagrangian mesh-free methods have been developed using smooth particle hydrodynamics or the material point method to avoid meshing issues [Kulasegaram et al, 2006; Kauzlaric et a;., 2011; Love & Sulsky, 2006]. These methods work well for being able to capture a moving interface with topological changes, but have difficulty in accurately solving the base physics, *e.g.* viscosity, and applying boundary conditions such as surface tension. ALE methods are hybrid techniques that seek to exploit the benefits of both the Eulerian and Lagrangian description in a hybrid manner, to determine the location of the interface [i.e. Sackinger et al, 1996; Lewis et al.,

Injection loading of a ceramic paste is a high-rate process used to create green ceramic parts, which subsequently experience binder burnout and sintering to produce the final ceramic part. In the process of interest, the mould is a rectangular cavity, with an inflow from a coat hanger die that should distribute the flow evenly across the mould inflow. The cavity is small with dimensions of 1.3 cm by 3.6 cm in plane and a height of 0.4 cm. The inflow tube to the distributer has a diameter of 0.5cm. Figure 1 shows short shots (or incomplete filling of a mould) for the injection loading process, illustrating some of the defects that can occur when a fluid with complex shear and temperature-dependent rheology interacts with a high

Fig. 1. Short shots of injection loading of a ceramic paste in a part are shown [Rao et al, 2006]. The photo on the left is injection loading using a slow injection speed, while the photo on the right uses a higher injection speed. At low filling speed, the paste acts like a solid material. Even at high filling rates, when the paste begins to act as a fluid, pooling at the

Temperature control and high-rate processing can limit the folding instability seen on the left of Figure 1. However, the pooling phenomenon shown on the right of Figure 1 occurs when optimal processing conditions are used, indicating that the design of the distributer may be responsible for material building up in the centre of the die. The complex shear-rate and temperature-dependent rheology of the ceramic paste was determined to follow a power-law dependence on shear rate and a Williams-Landau-Ferry temperature dependence [Rao et al., 2006]. The material shear thinned quickly to a constant viscosity value at moderate shear rates. Thus, because of the high shear rates in the injection loader, it was determined that the rheology was essentially Newtonian as long as the temperature remained constant at the processing temperature. Therefore, a study was undertaken to better understand the filling dynamics and reduce pooling by changing the distributer

centre of the mould is seen and the desired mould shape is not achieved.

1998; Nithiarasu, 2005].

rate process.

design using a Newtonian fluid.

Understanding dynamic wetting, or the interaction between the free surface and the mould walls, has been the subject of numerous experimental and theoretical studies and is still an outstanding research topic [see for instance Blake, 2006; Ren et al, 2011]. The difficulty arises from the contradiction of a moving contact line at the fluid-gas interface and the no slip boundary conditions traditionally applied at solid surfaces. How can the contact line advance when the velocity vanishes at the solid surface? Highly viscous materials such as polymers and particle suspensions have large capillary numbers (a measure of the ratio of viscous forces to capillary forces) and are often hypothesized to obey a rolling motion condition with an 180o dynamic contact angle. Numerically, this approach is difficult to apply and other methods have been proposed. The simplest models use a Navier slip condition to allow either slip on the entire solid surface, slip for the gas phase only, or slip only at the dynamic contact line. These models are *ad hoc* and ignore any thermodynamics considerations such as the static contact angle and surface energy. More advanced wetting models generally give dynamic contact angle as a function of the local Ca, the static contact angle and other material properties of the fluid and the solid surface [see Schunk et al., 2006 for a brief review of this work]. Hoffman [1975] used experimental measurements to develop a universal correlation for the dynamic contact angle as a function of static contact angle and a local capillary number, while Cox [1986] developed a competing model using asymptotic analysis. Kistler [1983] used a linear model that was easy to implement in numerical computer codes. Shikhmurzaev [1994] used hydrodynamic theory and included a surface phase as part of the wetting model. Blake [1969] developed a molecular kinetic theory that reduces to a linear model for small contact angles. Blake noted that the advancing angle is a monotonically increasing function of Ca. The degree of velocity dependence will however increase steeply as viscosity increases or surface tension decreases.

To model dynamic contact for the filling process, the dynamic angle is tied to the balance of forces at the advancing wetting line, namely the tangential wetting line force, liquid-gas surface tension force, and fluid viscous force. Here, a version of the Blake model is used that is straightforward to populate with experiments. The dynamic contact angle is measured in the laboratory as a function of the velocity of the wetting line for the fluids and surface used in our experiments as input to the wetting model. The Blake model is also easy to implement numerically, since the wetting speed can be written as a function of the dynamic and static contact angles. The performance of the Blake model at the high capillary number limit may be suspect, since it exhibits an unbounded dynamic contact angle while more physical models such as Cox and Hoffman reach a limit of 180o for large capillary numbers [Schunk et al, 2006].

#### **1.2 Mould design**

Because the initial design of the mould exhibited pooling of the fluid in the centre of the cavity and this would lead to poor filling (see Figure 1), we tried two minor redesigns to the distributor to see if we could improve the flow into the mould and reduce pooling. Ideally, we would like to see a flat profile coming out of the distributor and a more one-dimensional front shape. Figure 2 shows the original mesh, Mesh 1, and a variation, Mesh 2, with a longer distributor, and Mesh 3 with a longer-taller distributor. The idea behind Mesh 2 was to give a longer length for the flow to develop a flat profile and fill up the distributor before entering the main cavity. Mesh 3 kept this longer distributor and made it wider on inflow to ease the fluid

3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity 7

field, *u*, will be solenoidal and the continuity equation contains no density or pressure variables. (Note, this is a good assumption for the viscous liquid but a simplification for the

Conservation of momentum for a Newtonian fluid takes into account gradients in the fluid stress tensor, T, defined as the product of the viscosity and the shear rate tensor, ( )*<sup>t</sup> u u* , gradients in the pressure, p, as well as gravitational effects and inertial terms that can be dependent on time, t, and the fluid density, . Note that gravity, g, can be an important

<sup>2</sup> ( ) *<sup>u</sup> uu u p g*

Because we have a viscous fluid displacing a gas phase, the location of the interface between fluids is unknown *a priori*. To determine the location of the free surface as it evolves in time, we use an Eulerian interface-capturing scheme based on the level set method, the details of

We use the level set method of Sethian [1999] to determine the evolution of the interface with time. The level set is a scalar distance function, the zero of which coincides with the

We initialize this function to have a zero value at the fluid-gas interface, with negative distances residing in the fluid phase and positive distances in the gas phase. An advection

*v* 0

2

> 

  We use the equations of motion described in the previous section, but vary the material properties across the phase interface. This variation is handled using a smooth Heaviside

*t* 

Derivatives of the level set function can give us surface normal, *n*, and curvature,

1 1 ( ) (1 sin( )); <sup>2</sup>

 

function that modulates material properties to account for the change in phase.

() 1 ()

*fluid gas*

*H H*

 *n*

 

equation is then used to determine the location of the interface over time.

interface, which is useful for applying boundary conditions.

*gas*

*H*

 

(2)

(, ,) 0 *xyz* (3)

(4)

, at the

(5)

(6)

body force in filling processes and most filling processes fill counter to gravity.

*t* 

*u* 0 (1)

gas phase, which is actually compressible.)

which are included in the following section.

free surface or fluid-gas interface, *e.g.*

**2.2 Interface capturing** 

entering the cavity. These ideas were inspired by discussions in Sartor [1990] about die design. The meshes themselves all have the same cavity size and a similar amount of refinement, though Mesh 2 and Mesh 3 have more elements and unknowns due to the longer distributor.

Fig. 2. Geometries to be investigated: Mesh 1 is the original design, Mesh 2 incorporates a longer distributer, and Mesh 3 incorporates a longer distributer with a wider inflow to the cavity.

#### **1.3 Chapter organization**

In this chapter, we investigate the design of an injection loading mould using flow visualization experiments and numerical models. The finite element method is used to understand the interaction of the inflowing viscous liquid with the geometry and the displaced gas phase. Filling dynamics are determined with a diffuse-interface implementation of the level set method [Sethian 1999]. The Blake wetting model is used to represent the interaction of the free surface with the mould surface at the dynamic contact line [Blake & Haynes, 1969; Blake, 2006]. The flow visualization experiments are carried out under isothermal conditions using an acrylic mould and a viscous Newtonian fluid. Three different moulds are examined in two different orientations with gravity. Simulation results are given for these six cases.

The chapter is organized in the following manner. First, the equations and numerical method are presented. Next, the experimental methods used to provide input parameters for the models and flow visualization studies to better understand the filling dynamics and provide confidence in the numerical method are discussed. In the subsequent section, the results for injection moulding process are given for a Newtonian fluid into a coat hanger die distributer and a rectangular cavity, where the 3D level set simulations are compared to experiments. We conclude by summarizing the results and discussing future efforts.

#### **2. Equations and method**

#### **2.1 Equations of motion**

We can write the equations of motion for a single-phase fluid and then generalize them for our multiphase flow problem, where a viscous fluid displaces a gas. The fluids of interest are assumed to be incompressible and have a constant density, meaning that the velocity field, *u*, will be solenoidal and the continuity equation contains no density or pressure variables. (Note, this is a good assumption for the viscous liquid but a simplification for the gas phase, which is actually compressible.)

$$\nabla \cdot \boldsymbol{\mu} = 0 \tag{1}$$

Conservation of momentum for a Newtonian fluid takes into account gradients in the fluid stress tensor, T, defined as the product of the viscosity and the shear rate tensor, ( )*<sup>t</sup> u u* , gradients in the pressure, p, as well as gravitational effects and inertial terms that can be dependent on time, t, and the fluid density, . Note that gravity, g, can be an important body force in filling processes and most filling processes fill counter to gravity.

$$
\rho \rho (\frac{\partial u}{\partial t} + u \bullet \nabla u) = \mu \nabla^2 u - \nabla p + \rho \mathbf{g} \tag{2}
$$

Because we have a viscous fluid displacing a gas phase, the location of the interface between fluids is unknown *a priori*. To determine the location of the free surface as it evolves in time, we use an Eulerian interface-capturing scheme based on the level set method, the details of which are included in the following section.

#### **2.2 Interface capturing**

6 Numerical Modelling

entering the cavity. These ideas were inspired by discussions in Sartor [1990] about die design. The meshes themselves all have the same cavity size and a similar amount of refinement, though Mesh 2 and Mesh 3 have more elements and unknowns due to the longer distributor.

Mesh 1: Original Geometry Mesh 2: Longer Distributor Mesh 3: Longer-Taller Distributor

Fig. 2. Geometries to be investigated: Mesh 1 is the original design, Mesh 2 incorporates a longer distributer, and Mesh 3 incorporates a longer distributer with a wider inflow to the

In this chapter, we investigate the design of an injection loading mould using flow visualization experiments and numerical models. The finite element method is used to understand the interaction of the inflowing viscous liquid with the geometry and the displaced gas phase. Filling dynamics are determined with a diffuse-interface implementation of the level set method [Sethian 1999]. The Blake wetting model is used to represent the interaction of the free surface with the mould surface at the dynamic contact line [Blake & Haynes, 1969; Blake, 2006]. The flow visualization experiments are carried out under isothermal conditions using an acrylic mould and a viscous Newtonian fluid. Three different moulds are examined in two different orientations with gravity. Simulation results

The chapter is organized in the following manner. First, the equations and numerical method are presented. Next, the experimental methods used to provide input parameters for the models and flow visualization studies to better understand the filling dynamics and provide confidence in the numerical method are discussed. In the subsequent section, the results for injection moulding process are given for a Newtonian fluid into a coat hanger die distributer and a rectangular cavity, where the 3D level set simulations are compared to

We can write the equations of motion for a single-phase fluid and then generalize them for our multiphase flow problem, where a viscous fluid displaces a gas. The fluids of interest are assumed to be incompressible and have a constant density, meaning that the velocity

experiments. We conclude by summarizing the results and discussing future efforts.

cavity.

**1.3 Chapter organization** 

are given for these six cases.

**2. Equations and method** 

**2.1 Equations of motion** 

We use the level set method of Sethian [1999] to determine the evolution of the interface with time. The level set is a scalar distance function, the zero of which coincides with the free surface or fluid-gas interface, *e.g.*

$$
\phi(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \mathbf{0} \tag{3}
$$

We initialize this function to have a zero value at the fluid-gas interface, with negative distances residing in the fluid phase and positive distances in the gas phase. An advection equation is then used to determine the location of the interface over time.

$$\frac{\partial \hat{\phi}}{\partial t} + \nu \star \nabla \phi = 0 \tag{4}$$

Derivatives of the level set function can give us surface normal, *n*, and curvature, , at the interface, which is useful for applying boundary conditions.

$$n = \frac{\nabla \phi}{\|\nabla \phi\|}$$

$$\kappa = \frac{-\nabla^2 \phi}{\|\nabla \phi\|}$$

We use the equations of motion described in the previous section, but vary the material properties across the phase interface. This variation is handled using a smooth Heaviside function that modulates material properties to account for the change in phase.

$$\begin{aligned} H\_{gus}(\phi) &= \frac{1}{2} (1 + \frac{\phi}{\alpha} + \frac{1}{\pi} \sin(\frac{\pi \phi}{\alpha})); -\alpha \le \phi \le \alpha\\ H\_{gud}(\phi) &= 1 - H\_{gus}(\phi) \end{aligned} \tag{6}$$

3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity 9

The equations of motion (7) and the level set advection (4) were solved using a finite element method as implemented in ARIA [Notz et al., 2006]. Bilinear shape functions were used for the three velocity components, pressure, and level set. The LBB requirement on the velocity and pressure space was circumvented using Dohrmann-Bochev pressure stabilized pressure-projection (PSPP) [Dohrmann and Bochev, 2004] to allow for this equal order, bilinear, interpolation of all variables. (LBB compliant elements have the velocity space higher than the pressure space [Hughes, 2000].) The velocity vector and pressure unknowns were solved in the same matrix, while the level set equation was solved in a separate matrix, but at the same time step intervals. The level set equation was stabilized using a Taylor-Galerkin method [Donea, 1985]. The PSPP stabilization method greatly improved the condition number of the discretized matrix equations when compared to LBB elements or other stabilization methods, allowing for the use of an ILU preconditioner with a BiCGStab Krylov iterative solver. Further details of the modelling approach and equations, the numerical methods used

Boundary conditions for the dynamic contact line where the free surface and wall intersects are handled with a Blake wetting condition [Blake and Haynes, 1969; Blake, 2006]. Parameters for the model are informed by goniometer experiments that determine the wetting speed, vwet as a function of the dynamic contact angle, θ, and the static contact

sinh( (cos cos )) *wet*

  (11)

(12)

(13)

*<sup>v</sup> v v <sup>t</sup>* 

> cos / *<sup>w</sup>*

( ) 1 , ;0,

 

*<sup>v</sup> n T v f vt t*

 *n* 

When we integrate the stress, T, by parts in the finite element implementation of the momentum equation, a surface term is created as a natural boundary condition. We exploit the surface stress term and add on a Navier slip condition that includes the wetting speed

<sup>1</sup> ( () )

The value of the tangential wall velocity ramps from zero at a level set length scale away from the contact line to vwet from equation (11) at the contact line. Away from the contact line, we revert to no slip for the tangential wall velocity boundary condition. The normal velocity is enforced as no penetration everywhere. The shape of this ramp must be smooth in order to get a realistic wetting line that shows a smooth transition from the contact line to the bulk flow. For a sharp transition from slip to Blake, we ended up with unphysical looking cusps in the interface shape near the wall. The transient terms introduce dynamics

 

> 

*wet*

and the finite element implementation can be found in Rao *et al.* [2011].

angle, θs for the various fluids and surfaces of interest [Mondy *et al.,* 2007].

The dynamic contact angle can be calculated from the level set function.

*f*

*wet o s*

**2.3 Finite element implementation** 

**2.4 Contact-line wetting model** 

from the Blake model.

This is a diffuse interface implementation of the level set method, which allows for an interfacial zone of length 2. This zone is usually chosen to be four to six elements wide. Equation averaging is done using a Heaviside for the gas and viscous fluid equations. Because the properties are linear, this process results in Heaviside-averaged properties in a single momentum equation and an unchanged continuity equation:

$$\begin{aligned} \rho\_{\text{average}} (\frac{\hat{\mathbf{c}}u}{\hat{\mathbf{c}}t} + \boldsymbol{\mu} \bullet \mathbf{\nabla}u) &= \mu\_{\text{average}} \nabla^2 u - \nabla p + \rho\_{\text{average}} \mathbf{g} \\ \nabla \cdot \boldsymbol{\mu} &= 0 \end{aligned} \tag{7}$$

The properties have fluid properties in some regions and gas properties in other regions as modulated by the numerical Heaviside. In the diffuse interface region, the properties are averaged between fluid and gas values. Figure 3 shows a schematic representation of the Heaviside function.

$$\begin{aligned} \rho\_{\text{awraage}} &= H\_{\text{gas}} \rho\_{\text{gas}} + H\_{\text{fluid}} \rho\_{\text{fluid}}\\ \mu\_{\text{awraage}} &= H\_{\text{gas}} \mu\_{\text{gas}} + H\_{\text{fluid}} \mu\_{\text{fluid}} \end{aligned} \tag{8}$$

Fig. 3. Numerical Heaviside for averaging material properties.

The regularized Dirac delta function, which is defined as

$$\delta\_{\alpha}(\phi) = \frac{dH(\phi)}{d\phi} = \frac{|\phi|}{2\alpha} (1 + \cos(\frac{\pi\phi}{\alpha}))'\tag{9}$$

is used to apply surface tension and capillary boundary conditions via a continuous surface force approach [Brackbill et al, 1992]. This applies surface tension as a volumetric body force on the momentum equation, which is distributed throughout the interfacial zone region through the regularized Dirac delta function.

$$(\mu\_{gas} - \mu\_{fluid})n \bullet (\nabla u + (\nabla u)^\iota) = 2\sigma \delta\_a(\phi) \kappa \, n \tag{10}$$

#### **2.3 Finite element implementation**

8 Numerical Modelling

This is a diffuse interface implementation of the level set method, which allows for an interfacial zone of length 2. This zone is usually chosen to be four to six elements wide. Equation averaging is done using a Heaviside for the gas and viscous fluid equations. Because the properties are linear, this process results in Heaviside-averaged properties in a

> *average average average <sup>u</sup> uu u p g*

The properties have fluid properties in some regions and gas properties in other regions as modulated by the numerical Heaviside. In the diffuse interface region, the properties are averaged between fluid and gas values. Figure 3 shows a schematic representation of the

> *average gas gas fluid fluid average gas gas fluid fluid*

( ) ( ) (1 cos( )) <sup>2</sup>

is used to apply surface tension and capillary boundary conditions via a continuous surface force approach [Brackbill et al, 1992]. This applies surface tension as a volumetric body force on the momentum equation, which is distributed throughout the interfacial zone region

*gas fluid nu u n*

> 

(10)

, (9)

 

*dH d*

 *H H H H*

 

  (7)

(8)

<sup>2</sup> ( )

single momentum equation and an unchanged continuity equation:

*t*

Fig. 3. Numerical Heaviside for averaging material properties.

( ) ( ( )) 2 ( ) *<sup>t</sup>*

 

The regularized Dirac delta function, which is defined as

through the regularized Dirac delta function.

0

*u*

Heaviside function.

The equations of motion (7) and the level set advection (4) were solved using a finite element method as implemented in ARIA [Notz et al., 2006]. Bilinear shape functions were used for the three velocity components, pressure, and level set. The LBB requirement on the velocity and pressure space was circumvented using Dohrmann-Bochev pressure stabilized pressure-projection (PSPP) [Dohrmann and Bochev, 2004] to allow for this equal order, bilinear, interpolation of all variables. (LBB compliant elements have the velocity space higher than the pressure space [Hughes, 2000].) The velocity vector and pressure unknowns were solved in the same matrix, while the level set equation was solved in a separate matrix, but at the same time step intervals. The level set equation was stabilized using a Taylor-Galerkin method [Donea, 1985]. The PSPP stabilization method greatly improved the condition number of the discretized matrix equations when compared to LBB elements or other stabilization methods, allowing for the use of an ILU preconditioner with a BiCGStab Krylov iterative solver. Further details of the modelling approach and equations, the numerical methods used and the finite element implementation can be found in Rao *et al.* [2011].

#### **2.4 Contact-line wetting model**

Boundary conditions for the dynamic contact line where the free surface and wall intersects are handled with a Blake wetting condition [Blake and Haynes, 1969; Blake, 2006]. Parameters for the model are informed by goniometer experiments that determine the wetting speed, vwet as a function of the dynamic contact angle, θ, and the static contact angle, θs for the various fluids and surfaces of interest [Mondy *et al.,* 2007].

$$\nu\_{\rm var} = \nu\_o \sinh(\overline{\varphi}(\cos \theta\_s - \cos \theta)) - \tau \frac{\partial \nu\_{\rm var}}{\partial t} \tag{11}$$

The dynamic contact angle can be calculated from the level set function.

$$\cos \theta = n\_v \cdot \nabla \phi / \left| \nabla \phi \right| \tag{12}$$

When we integrate the stress, T, by parts in the finite element implementation of the momentum equation, a surface term is created as a natural boundary condition. We exploit the surface stress term and add on a Navier slip condition that includes the wetting speed from the Blake model.

$$\begin{aligned} \vec{n} \cdot T &= -\frac{1}{\beta} (-\tau \frac{\partial \vec{\boldsymbol{\nu}}}{\partial t} + \vec{\boldsymbol{\nu}} - f\_{\alpha}(\phi) \boldsymbol{\nu}\_{\text{var}} \vec{t}) \\ \boldsymbol{f}\_{a}(\phi) &= 1 - \frac{|\phi|}{\alpha}, -\alpha \le \phi < \alpha; 0, |\phi| > \alpha \end{aligned} \tag{13}$$

The value of the tangential wall velocity ramps from zero at a level set length scale away from the contact line to vwet from equation (11) at the contact line. Away from the contact line, we revert to no slip for the tangential wall velocity boundary condition. The normal velocity is enforced as no penetration everywhere. The shape of this ramp must be smooth in order to get a realistic wetting line that shows a smooth transition from the contact line to the bulk flow. For a sharp transition from slip to Blake, we ended up with unphysical looking cusps in the interface shape near the wall. The transient terms introduce dynamics

3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity 11

velocities, or the sessile drop can be allowed to relax to obtain "low" velocities. Figure 4 shows a schematic of the experimental apparatus and the results of this wetting test. During each experiment, the angle of contact and the location of the triple point of contact was recorded. The wetting line speed was then determined from the location of the triple point with time. The data fit to the Blake model (equation 9) gives a wetting constant vo of

Fig. 4. Sketch of apparatus for wetting parameter measurements (left). Dynamic wetting measurements of contact angle vs. velocity for 75-H-90000 UCON on acrylic (right).

Figures 5 through 7 are representative frames from the video recordings of the fill process using a vertical alignment of the mould as in the proposed ceramic injection process. The time it took to completely fill the original geometry, Mesh 1, (defined by the time when the liquid began to exit along the entire length of the vent) was 24.6 ± 1.2 s, to fill the geometry of Mesh 2 was 26.9 ± 2.0 s, and to fill the geometry of Mesh 3 was 24.6 ± 2.3 s. Because the fill rates varied, the time is shown in these figures in a nondimensional form. The initial time is taken to be when the front passes a line in the square entry channel 0.16 cm from the entrance of the distributor. Here, one can see the effects of changing the distributor geometry. In Figure 5 the liquid has just filled the distributor. All of the modified geometries help flatten the leading

front, especially Mesh 2. Mesh 2 fills the distributor with the fastest relative time.

**Mesh 2** 

Time/total time=0.13

Fig. 5. Comparison of the effect of distributor geometry on the shape of the fluid front

**Mesh 3** 

Time/total time=0.24

**3.3 Flow tests** 

**Mesh 1** 

Time/total time=0.32

entering the mould for vertical mould orientation.

0.00130cm/s and scale factor of 2.29 with a static contact angle θ of 37.3°.

into the wetting line motion and allow smooth movement of the contact line. The parameter is taken to be approximately the time step. The parameter is generally small so that equation (13) functions almost like a Dirichlet requirement on the fluid velocity.

#### **3. Experiments**

It was decided to visually record the flow of a simple, single phase, Newtonian liquid through transparent moulds to build confidence in the front tracking and wetting models used in the computations. Details of the geometries, experimental conditions, and properties of the materials used in each test will be given in the following sections.

#### **3.1 Geometries**

For these tests, we built transparent acrylic moulds identical to the three mesh geometries (Figure 2). The strength of the acrylic material making up the transparent moulds dictated that we inject at a lower injection pressure and lower operating temperature than the actual injection loading process. To mimic the actual injection process, we used a pressure driven syringe held at a constant pressure of 29.95 ± 0.10 psig during injection. The syringe of liquid was degassed in a vacuum chamber prior to the experiment. The syringe was modified to prevent leakage around the plunger and subsequent bubble formation. However, this resulted in more friction and a flow rate that varied somewhat from experiment to experiment. Hence, the time to fill the moulds varied from test to test, but was determined and recorded for each test. Reported below is the median and spread of the measured filling time for four repeated experiments in each geometry.

The tests were conducted at a room temperature that ranged from 23 to 24oC. To minimize the effect of temperature on the viscosity, the liquid was held in a water bath set to 23.5oC after degassing and before being used in an experiment. These conditions resulted in an injection rate that was approximately ten times slower than the actual process. However, the Reynolds number for the actual process and the validation experiments were similar and in the Stokes regime of much smaller than one. For an average fill time of 20 s, a cavity volume of 1.87 cm3, and characteristic length scale from the inflow port of 0.5cm, the Reynolds number was 0.0006 and the capillary number was 4.5.

#### **3.2 Materials and properties**

We chose a liquid, UCON 75-H-90,000 (oxyethylene/oxypropylenes from Dow Chemical) as our model fluid. This UCON was chosen since its viscosity is an order of magnitude lower than the ceramic paste at processing conditions. Combined with an order of magnitude decrease in the injection rate compared to the real system, this gives a similar Reynolds number. The liquid surface tension and wetting properties of the UCON lubricant were determined at 23.5°C. The viscosity of the lubricant was measured with a Rheologica™ constant stress rheometer and was equal to 390 Poise over a shear rate ranging from 0.1 to 10 sec-1, indicating Newtonian rheological behaviour. The density of the liquid was measured with a densitometer to be 1.09 g/cm3. The surface tension measured with a Du Noüy ring (mean circumference of 5.935 cm) was 42.40.1 dyne/cm.

The dynamic contact angle on acrylic was measured with a feed-through goniometer [Mondy *et al.*, 2007], in which liquid can be continuously injected to achieve "high" velocities, or the sessile drop can be allowed to relax to obtain "low" velocities. Figure 4 shows a schematic of the experimental apparatus and the results of this wetting test. During each experiment, the angle of contact and the location of the triple point of contact was recorded. The wetting line speed was then determined from the location of the triple point with time. The data fit to the Blake model (equation 9) gives a wetting constant vo of 0.00130cm/s and scale factor of 2.29 with a static contact angle θ of 37.3°.

Fig. 4. Sketch of apparatus for wetting parameter measurements (left). Dynamic wetting measurements of contact angle vs. velocity for 75-H-90000 UCON on acrylic (right).

#### **3.3 Flow tests**

10 Numerical Modelling

into the wetting line motion and allow smooth movement of the contact line. The parameter is taken to be approximately the time step. The parameter is generally small so

It was decided to visually record the flow of a simple, single phase, Newtonian liquid through transparent moulds to build confidence in the front tracking and wetting models used in the computations. Details of the geometries, experimental conditions, and properties

For these tests, we built transparent acrylic moulds identical to the three mesh geometries (Figure 2). The strength of the acrylic material making up the transparent moulds dictated that we inject at a lower injection pressure and lower operating temperature than the actual injection loading process. To mimic the actual injection process, we used a pressure driven syringe held at a constant pressure of 29.95 ± 0.10 psig during injection. The syringe of liquid was degassed in a vacuum chamber prior to the experiment. The syringe was modified to prevent leakage around the plunger and subsequent bubble formation. However, this resulted in more friction and a flow rate that varied somewhat from experiment to experiment. Hence, the time to fill the moulds varied from test to test, but was determined and recorded for each test. Reported below is the median and spread of the measured filling

The tests were conducted at a room temperature that ranged from 23 to 24oC. To minimize the effect of temperature on the viscosity, the liquid was held in a water bath set to 23.5oC after degassing and before being used in an experiment. These conditions resulted in an injection rate that was approximately ten times slower than the actual process. However, the Reynolds number for the actual process and the validation experiments were similar and in the Stokes regime of much smaller than one. For an average fill time of 20 s, a cavity volume of 1.87 cm3, and characteristic length scale from the inflow port of 0.5cm, the

We chose a liquid, UCON 75-H-90,000 (oxyethylene/oxypropylenes from Dow Chemical) as our model fluid. This UCON was chosen since its viscosity is an order of magnitude lower than the ceramic paste at processing conditions. Combined with an order of magnitude decrease in the injection rate compared to the real system, this gives a similar Reynolds number. The liquid surface tension and wetting properties of the UCON lubricant were determined at 23.5°C. The viscosity of the lubricant was measured with a Rheologica™ constant stress rheometer and was equal to 390 Poise over a shear rate ranging from 0.1 to 10 sec-1, indicating Newtonian rheological behaviour. The density of the liquid was measured with a densitometer to be 1.09 g/cm3. The surface tension measured with a Du Noüy ring

The dynamic contact angle on acrylic was measured with a feed-through goniometer [Mondy *et al.*, 2007], in which liquid can be continuously injected to achieve "high"

that equation (13) functions almost like a Dirichlet requirement on the fluid velocity.

of the materials used in each test will be given in the following sections.

time for four repeated experiments in each geometry.

**3.2 Materials and properties** 

Reynolds number was 0.0006 and the capillary number was 4.5.

(mean circumference of 5.935 cm) was 42.40.1 dyne/cm.

**3. Experiments** 

**3.1 Geometries** 

Figures 5 through 7 are representative frames from the video recordings of the fill process using a vertical alignment of the mould as in the proposed ceramic injection process. The time it took to completely fill the original geometry, Mesh 1, (defined by the time when the liquid began to exit along the entire length of the vent) was 24.6 ± 1.2 s, to fill the geometry of Mesh 2 was 26.9 ± 2.0 s, and to fill the geometry of Mesh 3 was 24.6 ± 2.3 s. Because the fill rates varied, the time is shown in these figures in a nondimensional form. The initial time is taken to be when the front passes a line in the square entry channel 0.16 cm from the entrance of the distributor. Here, one can see the effects of changing the distributor geometry. In Figure 5 the liquid has just filled the distributor. All of the modified geometries help flatten the leading front, especially Mesh 2. Mesh 2 fills the distributor with the fastest relative time.

Fig. 5. Comparison of the effect of distributor geometry on the shape of the fluid front entering the mould for vertical mould orientation.

3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity 13

Area Lower Corners (cm2)

Geometry Area Upper Corners

(cm2)

Table 1. Bubble sizes remaining in the corners for vertical mould orientation.

fill the geometry of Mesh 3 was 27.3 ± 1.8 s.

orientation.

Original 0.0153 ± 0.0012 0.0020 ± 0.0005 Mesh 2 0.0086 ± 0.0007 0.0012 ± 0.0002 Mesh 3 0.0128 ± 0.0005 0.0021 ± 0.0001

Next, we experimented with the orientation of the mould. Moulds were turned so that the distributor was perpendicular to gravity on the lower surface and the vent was moved to be on the upper surface. In other words, the flow direction and gravity are now perpendicular, with gravity acting in the thinnest cavity direction. Results for the original mesh showed that orientation with respect to gravity had a large impact on the likelihood of voids remaining in the corners of the mould. Figure 8 shows representative video frames at three stages of the fill: 1) when the distributor was completely filled, 2) when the front hit the far wall, and 3) when the liquid began to exit through the vent and the sides had completely wetted. Because the fill rates varied, the time is shown in a nondimensional form. The time it took to completely fill the original geometry in the horizontal position was 23.3 ± 0.9 s, to fill the geometry of Mesh 2 was 28.9 ± 1.5 s, and to

The top views of the horizontal orientation show that the liquid wets the top later than the bottom. The oval area of the middle image in Figure 8 indicates where the liquid has wetted the upper surface. The side views are more difficult to interpret because of the lighting challenges accompanying trying to see through the thickest section of the mould. However,

It is interesting to note that the horizontal alignment causes the front to hit the far wall before even wetting the sides at all. In other words, the front was less "flat" entering the mould from the distributor. Nevertheless, the results also show that the horizontal orientation, with the vent on top and the distributor entrance on bottom, resulted in bubbles only in the upper corners nearest the distributor. When oriented vertically, bubbles are trapped in corners both opposite the distributor and opposite the vent. The bubbles nearest the distributor with the horizontal orientation are approximately the same size as the bubbles trapped in the upper corners opposite the vent in the vertical

The effect of distributor geometry on the filling in the horizontal orientation is shown in Figure 9 and Figure 10, which correspond to Figure 5 and Figure 6 in the vertical orientation. Again, the geometry of Mesh 2 results in the flattest front entering the mould from the distributor (Figure 9). The front reaches the back wall at approximately the same time using the distributors of Mesh 2 and 3. By the time the front reaches the back wall both modified geometries result in more liquid filling the mould than with the original geometry;

the dark areas of the side view are where the liquid has wetted the sides.

however, Mesh 2 results in somewhat more than Mesh 3 (Figure 10).

Time/total time=0.82

Figure 6 shows the times at which the leading front hits the wall opposite the injection port. In this case, the front in Mesh 2 takes the longest relative time to reach this stage. The other geometries once again follow the same pattern as before, with Mesh 2 giving the flattest flow profile and the original mesh the displaying the most curvature of the interface. At this point, the fluid has wetted more of the side walls with the Mesh 2 geometry and the front is flatter. Because Mesh 2 has a flatter profile, it takes the longest time to reach the top wall compared to Mesh 1 and Mesh 3.

Fig. 6. Comparison of the effect of distributor geometry on the time it takes to reach the wall farthest from the injection port for vertical mould orientation.

Time/total time=0.86

Figure 7 shows the locations of the voids remaining in the mould once filled but without any over pressure (a relative time of 1). Small bubbles away from the corners are artefacts of the syringe loading process. The voids in geometries with redesigned distributors remain in the same locations as those seen in the original geometry. The relative areas of the bubbles on the images, which reflect the volume of air trapped, were determined and compared quantitatively in Table 1. One pixel resolution represents about 1×10-5 cm2. All redesigned distributors result in smaller bubbles in the upper corners than those in the original mesh. The lower bubbles of Mesh 2 are also smaller, whereas those in Mesh 3 are approximately the same as those in the original geometry.

Fig. 7. Voids remain in the front upper and lower corners of each geometry for the vertical mould orientation. Each frame consists of two views of the mould (top view and side view). These voids can be seen more easily from the side view.

Time/total time=0.80

Figure 6 shows the times at which the leading front hits the wall opposite the injection port. In this case, the front in Mesh 2 takes the longest relative time to reach this stage. The other geometries once again follow the same pattern as before, with Mesh 2 giving the flattest flow profile and the original mesh the displaying the most curvature of the interface. At this point, the fluid has wetted more of the side walls with the Mesh 2 geometry and the front is flatter. Because Mesh 2 has a flatter profile, it takes the longest time to reach the top wall

 **Mesh 2** 

**Mesh 1 Mesh 2 Mesh 3**

These voids can be seen more easily from the side view.

Fig. 7. Voids remain in the front upper and lower corners of each geometry for the vertical mould orientation. Each frame consists of two views of the mould (top view and side view).

farthest from the injection port for vertical mould orientation.

Time/total time=0.86

Fig. 6. Comparison of the effect of distributor geometry on the time it takes to reach the wall

Figure 7 shows the locations of the voids remaining in the mould once filled but without any over pressure (a relative time of 1). Small bubbles away from the corners are artefacts of the syringe loading process. The voids in geometries with redesigned distributors remain in the same locations as those seen in the original geometry. The relative areas of the bubbles on the images, which reflect the volume of air trapped, were determined and compared quantitatively in Table 1. One pixel resolution represents about 1×10-5 cm2. All redesigned distributors result in smaller bubbles in the upper corners than those in the original mesh. The lower bubbles of Mesh 2 are also smaller, whereas those in Mesh 3 are approximately

**Mesh 3** 

Time/total time=0.82

compared to Mesh 1 and Mesh 3.

 **Mesh 1** 

Time/total time=0.80

the same as those in the original geometry.


Table 1. Bubble sizes remaining in the corners for vertical mould orientation.

Next, we experimented with the orientation of the mould. Moulds were turned so that the distributor was perpendicular to gravity on the lower surface and the vent was moved to be on the upper surface. In other words, the flow direction and gravity are now perpendicular, with gravity acting in the thinnest cavity direction. Results for the original mesh showed that orientation with respect to gravity had a large impact on the likelihood of voids remaining in the corners of the mould. Figure 8 shows representative video frames at three stages of the fill: 1) when the distributor was completely filled, 2) when the front hit the far wall, and 3) when the liquid began to exit through the vent and the sides had completely wetted. Because the fill rates varied, the time is shown in a nondimensional form. The time it took to completely fill the original geometry in the horizontal position was 23.3 ± 0.9 s, to fill the geometry of Mesh 2 was 28.9 ± 1.5 s, and to fill the geometry of Mesh 3 was 27.3 ± 1.8 s.

The top views of the horizontal orientation show that the liquid wets the top later than the bottom. The oval area of the middle image in Figure 8 indicates where the liquid has wetted the upper surface. The side views are more difficult to interpret because of the lighting challenges accompanying trying to see through the thickest section of the mould. However, the dark areas of the side view are where the liquid has wetted the sides.

It is interesting to note that the horizontal alignment causes the front to hit the far wall before even wetting the sides at all. In other words, the front was less "flat" entering the mould from the distributor. Nevertheless, the results also show that the horizontal orientation, with the vent on top and the distributor entrance on bottom, resulted in bubbles only in the upper corners nearest the distributor. When oriented vertically, bubbles are trapped in corners both opposite the distributor and opposite the vent. The bubbles nearest the distributor with the horizontal orientation are approximately the same size as the bubbles trapped in the upper corners opposite the vent in the vertical orientation.

The effect of distributor geometry on the filling in the horizontal orientation is shown in Figure 9 and Figure 10, which correspond to Figure 5 and Figure 6 in the vertical orientation. Again, the geometry of Mesh 2 results in the flattest front entering the mould from the distributor (Figure 9). The front reaches the back wall at approximately the same time using the distributors of Mesh 2 and 3. By the time the front reaches the back wall both modified geometries result in more liquid filling the mould than with the original geometry; however, Mesh 2 results in somewhat more than Mesh 3 (Figure 10).

3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity 15

**Mesh 3** 

Time/total time=0.70

**Mesh 2** 

wall farthest from the injection port in a horizontal orientation.

Table 2. Bubble sizes remaining in horizontal orientation.

Time/total time=0.71

Fig. 10. Comparison of the effect of distributor geometry on the time it takes to reach the

Geometry Area (cm2) Original 0.0189 ± 0.0002 Mesh 2 0.0111 ± 0.0002 Mesh 3 0.0123 ± 0.0017

The simulations runs were made on 64 processors of Thunderbird (Sandia National Laboratories capacity computing platform) and ran in less than 3.5 hours. This allowed us to do real time design and sensitivity calculations for parameters such as wetting speed, second phase viscosity, level set length scale and inflow pressure. Properties for the liquid used for the validation simulations were the measured values discussed in the experimental section above. The density and viscosity of the displaced gas phase were taken as fictitious values of one thousand times smaller than the liquid phase density and viscosity. These

> **Material Property Value**  Density of liquid 1.09 g/cm3 Viscosity of liquid 390 Poise Density of gas 0.0011 g/cm3 Viscosity of gas 0.39 Poise Wetting speed, vo 0.0013 cm/s

Blake scale factor, 2.29 Static contact angle 37.3o

Table 3. Material properties used for validation simulations.

Surface tension 42.4 dyne/cm Inflow pressure 1.0x106 dyne/cm2

Sizes of the bubbles left in the various geometries in the horizontal orientation are listed in Table 2. Comparing these values with the measurements in Table 1, one can see that the amount of gas left in the vertical orientation is roughly the same as that in the horizontal orientation, although in the horizontal orientation there are fewer bubbles. Bubbles observed in other locations than the corners are almost always artefacts of the syringe loading process.

 **Mesh 1** 

**4. Simulations** 

values are summarized in Table 3.

Time/total time=0.82

Time/total time=0.26

Time/total time=0.33 Time/total time=0.81 Time/total time=1.0

Time/total time=0.22

Fig. 8. Original geometry (Mesh 1) oriented horizontally (A) and vertically (B). Images on the left show when the distributor fills completely, middle images show when the front first hits the back wall, and images on the right show when the part is filled to the point that the fluid leaves the vent area through the entire length of the vent. Bubbles left in the corners are circled in red.

Fig. 9. Comparison of the effect of distributor geometry on the shape of the fluid front entering the mould in a horizontal orientation. The flow direction and gravity are perpendicular, with gravity acting in the thinnest cavity direction.

Time/total time=0.13

Fig. 10. Comparison of the effect of distributor geometry on the time it takes to reach the wall farthest from the injection port in a horizontal orientation.

Sizes of the bubbles left in the various geometries in the horizontal orientation are listed in Table 2. Comparing these values with the measurements in Table 1, one can see that the amount of gas left in the vertical orientation is roughly the same as that in the horizontal orientation, although in the horizontal orientation there are fewer bubbles. Bubbles observed in other locations than the corners are almost always artefacts of the syringe loading process.


Table 2. Bubble sizes remaining in horizontal orientation.

#### **4. Simulations**

14 Numerical Modelling

Fig. 8. Original geometry (Mesh 1) oriented horizontally (A) and vertically (B). Images on the left show when the distributor fills completely, middle images show when the front first hits the back wall, and images on the right show when the part is filled to the point that the fluid leaves the vent area through the entire length of the vent. Bubbles left in the corners

Time/total time=0.33 Time/total time=0.81 Time/total time=1.0

Time/total time=0.26 Time/total time=0.82 Time/total time=1.0

Mesh 2 Time/total time=0.13

perpendicular, with gravity acting in the thinnest cavity direction.

Fig. 9. Comparison of the effect of distributor geometry on the shape of the fluid front entering the mould in a horizontal orientation. The flow direction and gravity are

Mesh 3 Time/total time=0.22

are circled in red.

A

B

Mesh 1 Time/total time=0.26 The simulations runs were made on 64 processors of Thunderbird (Sandia National Laboratories capacity computing platform) and ran in less than 3.5 hours. This allowed us to do real time design and sensitivity calculations for parameters such as wetting speed, second phase viscosity, level set length scale and inflow pressure. Properties for the liquid used for the validation simulations were the measured values discussed in the experimental section above. The density and viscosity of the displaced gas phase were taken as fictitious values of one thousand times smaller than the liquid phase density and viscosity. These values are summarized in Table 3.


Table 3. Material properties used for validation simulations.

3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity 17

The time to fill the mould for the vertical orientation for Mesh 1, Mesh 2, and Mesh 3 are 15.2s, 17.5s, and 13.2s, which was much faster than the experimental values of 24.6s, 26.9s, and 24.6s. We defined our fill time as when the vent had filled completely to a distance of the half the level-set length-scale or two elements. Unfortunately, the numerical fill times are difficult to obtain accurately as the gas phase viscosity makes a large difference in how fast a simulation will fill for the same value of pressure. For instance, when we reduced the second phase viscosity by a factor of 10 we got an increase in the inflow velocity when keeping all other parameters constant. Thus, the fact that the gas phase is harder to push out than it is for the experiments, adds a great deal of uncertainty to the fill times. However, we hope to be able to predict trends. Figure 12 shows the effect of the distributor geometry on

the meniscus shape for Mesh 1, Mesh 2, and Mesh 3 in a vertical orientation.

Mesh 2

which looks as if the front is pinned at the distributor.

Time/total time = 0.18

Fig. 12. Free surface profile after filling the distributor for Mesh 1, Mesh 2, and Mesh 3 for

Comparing Figure 12 and Figure 5, the numerical and experimental version of this profile, we can see that the simulations are exhibiting the physically correct trends. The original mesh takes the longest fractional time to fill the distributor, 42%, and gives the most bulging front shape. Mesh 2 is an improvement, taking 18% of the time to fill the distributor, while Mesh 3 is somewhere in between at 24%. The values for the experimental distributor dimensionless fill times are 32%, 13%, and 24%, so the simulations are also capturing the correct trends for fill time though they are not quantitative. The shape of the meniscus for Mesh 1 has more of a bulge at the edge of the distributor than the experimental meniscus,

We can also look at the profiles and dimensionless time to hit the back wall. These results

Mesh 3

Time/total time = 0.24

Mesh 1

Time/total time = 0.42

vertical mould orientation.

are given in Figure 13.

At 25oC, the viscosity of air is 2.0x10-4 Poise and the density of air is 0.0012 g/cm3, thus our second phase properties are very close for density, but three orders of magnitude too high for viscosity. The numerical method fails to converge for values of the liquid/gas viscosity ratio of more than 1000 for a diffuse interface implementation of the level set equations, so this is a necessary expedient requiring us to adhere to this fictitious viscosity value.

For the inflow condition, we used a constant inflow pressure of 1.0x106 dyne/cm2. This value was chosen to match the horizontal fill time of 23 seconds in the original mesh and then used for all other meshes and geometries. A shooting method was used, where different values of the inflow pressure were used and the solution was examined to see if it filled in the correct time. This required many simulations to be run. It is believed that the actual boundary condition for the experiments is somewhere between a constant velocity and constant pressure condition, but this is hard to replicate numerically. From the simulations, we found that the velocity changed quickly in the beginning for the pressure inflow boundary condition and subsequently reached a steady value.

The initial 3D mesh and boundary conditions are given Figure 11 for the mould filling simulations. We assume symmetry about the centreline and only solve half the problem to improve the computational efficiency. The mesh contains 6744 8-Node hexahedral elements giving 41300 total degrees of freedom for bilinear velocity/bilinear pressure interpolation. This mesh was shown to be adequate, as a more refined version of this mesh gave the same fill times and meniscus shapes [Rao et al., 2006].

Fig. 11. Initial mesh and boundary conditions for 3D level set simulations. The outflow vent is on the same side as the distributor for vertical simulations and opposite the distributor for horizontal simulations.

At 25oC, the viscosity of air is 2.0x10-4 Poise and the density of air is 0.0012 g/cm3, thus our second phase properties are very close for density, but three orders of magnitude too high for viscosity. The numerical method fails to converge for values of the liquid/gas viscosity ratio of more than 1000 for a diffuse interface implementation of the level set equations, so

For the inflow condition, we used a constant inflow pressure of 1.0x106 dyne/cm2. This value was chosen to match the horizontal fill time of 23 seconds in the original mesh and then used for all other meshes and geometries. A shooting method was used, where different values of the inflow pressure were used and the solution was examined to see if it filled in the correct time. This required many simulations to be run. It is believed that the actual boundary condition for the experiments is somewhere between a constant velocity and constant pressure condition, but this is hard to replicate numerically. From the simulations, we found that the velocity changed quickly in the beginning for the pressure

The initial 3D mesh and boundary conditions are given Figure 11 for the mould filling simulations. We assume symmetry about the centreline and only solve half the problem to improve the computational efficiency. The mesh contains 6744 8-Node hexahedral elements giving 41300 total degrees of freedom for bilinear velocity/bilinear pressure interpolation. This mesh was shown to be adequate, as a more refined version of this mesh gave the same

•Extra slides

Flow In

Z

Fig. 11. Initial mesh and boundary conditions for 3D level set simulations. The outflow vent is on the same side as the distributor for vertical simulations and opposite the distributor for

<sup>Y</sup> <sup>X</sup>

No penetration / no slip, except

near contact region

this is a necessary expedient requiring us to adhere to this fictitious viscosity value.

inflow boundary condition and subsequently reached a steady value.

fill times and meniscus shapes [Rao et al., 2006].

Outflow occurs at edges of

mold chamber

Centerline Symmetry

horizontal simulations.

The time to fill the mould for the vertical orientation for Mesh 1, Mesh 2, and Mesh 3 are 15.2s, 17.5s, and 13.2s, which was much faster than the experimental values of 24.6s, 26.9s, and 24.6s. We defined our fill time as when the vent had filled completely to a distance of the half the level-set length-scale or two elements. Unfortunately, the numerical fill times are difficult to obtain accurately as the gas phase viscosity makes a large difference in how fast a simulation will fill for the same value of pressure. For instance, when we reduced the second phase viscosity by a factor of 10 we got an increase in the inflow velocity when keeping all other parameters constant. Thus, the fact that the gas phase is harder to push out than it is for the experiments, adds a great deal of uncertainty to the fill times. However, we hope to be able to predict trends. Figure 12 shows the effect of the distributor geometry on the meniscus shape for Mesh 1, Mesh 2, and Mesh 3 in a vertical orientation.

Fig. 12. Free surface profile after filling the distributor for Mesh 1, Mesh 2, and Mesh 3 for vertical mould orientation.

Comparing Figure 12 and Figure 5, the numerical and experimental version of this profile, we can see that the simulations are exhibiting the physically correct trends. The original mesh takes the longest fractional time to fill the distributor, 42%, and gives the most bulging front shape. Mesh 2 is an improvement, taking 18% of the time to fill the distributor, while Mesh 3 is somewhere in between at 24%. The values for the experimental distributor dimensionless fill times are 32%, 13%, and 24%, so the simulations are also capturing the correct trends for fill time though they are not quantitative. The shape of the meniscus for Mesh 1 has more of a bulge at the edge of the distributor than the experimental meniscus, which looks as if the front is pinned at the distributor.

We can also look at the profiles and dimensionless time to hit the back wall. These results are given in Figure 13.

3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity 19

The void in the corner near the distributor does eventually fill in, since its size is less than the level-set length scale and we are using a diffuse interface method. The larger void at the vent never fills in as the viscous gas phase is trapped away from the vent by the fluid. For the numerical solutions it is hard to make any predictions about void size, though we can say that for similar values of the dimensionless time the voids for Mesh 2 will be smaller than Mesh 1,

We also examined the horizontal orientation numerically, which gave fill times for Mesh 1, Mesh 2, and Mesh 3 of 21.4s, 23.1s, and 22.2s compared to 23.3s, 28.9s, and 27.3s for the experiments. Again, we follow the trends of the experiment, but do not match quantitatively. Mesh 2 seems to take a longer time to fill for the experiment than one would predict numerically. Figure 15 shows a comparison of the profiles for Mesh 1 in a vertical

with Mesh 3 being somewhere in between, which does follow the experimental trend.

**Mesh 1**

**Mesh 1**

Time/total time = 0.60

Fig. 15. Mesh 1 oriented horizontally (A) and vertically (B). Leftmost pictures show profiles when the distributor is filled, middle shows profile when the fluid hits the back wall, and rightmost pictures show final profile. Both front and side views are given to highlight void

Time/total time = .41

**Mesh 1**

**Mesh 1**

Time/total time = 1.0

Time/total time = 1.0

and horizontal orientation.

A

**Mesh1** 

B

**Mesh 1** 

location.

Time/total time = 0.42

Time/total time = .29

Fig. 13. Free surface profile after hitting the back wall for Mesh 1, Mesh 2, and Mesh 3 for vertical mould orientation.

Comparing Figure 13 to Figure 6, for the simulation versus experiment, we can see differences in the meniscus shape. The numerical interface reaches the back wall for Mesh 1 and Mesh 3, before it wets the sidewall and Mesh 2 has a flatter profile in the experiments than the simulations. The percentage time to reach the back wall for the simulations on Mesh 1, Mesh 2, and Mesh 3 are 60%, 70%, and 55% compared to 80%, 86% and 82% for the experiments. Again, we capture the correct trends, but are still unable to match the data quantitatively. Figure 14 shows the full meniscus shape and void locations for the simulations on Mesh 1, Mesh 2, and Mesh 3.

Fig. 14. Final void location and front profile for Mesh 1, Mesh 2, and Mesh 3.

**Mesh 2** 

**Mesh 2** 

Fig. 14. Final void location and front profile for Mesh 1, Mesh 2, and Mesh 3.

Time/total time = 1.0

Time/total time = 0.70

Comparing Figure 13 to Figure 6, for the simulation versus experiment, we can see differences in the meniscus shape. The numerical interface reaches the back wall for Mesh 1 and Mesh 3, before it wets the sidewall and Mesh 2 has a flatter profile in the experiments than the simulations. The percentage time to reach the back wall for the simulations on Mesh 1, Mesh 2, and Mesh 3 are 60%, 70%, and 55% compared to 80%, 86% and 82% for the experiments. Again, we capture the correct trends, but are still unable to match the data quantitatively. Figure 14 shows the full meniscus shape and void locations for the

Fig. 13. Free surface profile after hitting the back wall for Mesh 1, Mesh 2, and Mesh 3 for

**Mesh 3** 

**Mesh 3** 

Time/total time = 1.0

Time/total time = 0.55

**Mesh 1** 

vertical mould orientation.

**Mesh 1** 

Time/total time = 1.0

Time/total time = .60

simulations on Mesh 1, Mesh 2, and Mesh 3.

The void in the corner near the distributor does eventually fill in, since its size is less than the level-set length scale and we are using a diffuse interface method. The larger void at the vent never fills in as the viscous gas phase is trapped away from the vent by the fluid. For the numerical solutions it is hard to make any predictions about void size, though we can say that for similar values of the dimensionless time the voids for Mesh 2 will be smaller than Mesh 1, with Mesh 3 being somewhere in between, which does follow the experimental trend.

We also examined the horizontal orientation numerically, which gave fill times for Mesh 1, Mesh 2, and Mesh 3 of 21.4s, 23.1s, and 22.2s compared to 23.3s, 28.9s, and 27.3s for the experiments. Again, we follow the trends of the experiment, but do not match quantitatively. Mesh 2 seems to take a longer time to fill for the experiment than one would predict numerically. Figure 15 shows a comparison of the profiles for Mesh 1 in a vertical and horizontal orientation.

Time/total time = 0.42 Time/total time = 0.60 Time/total time = 1.0 Fig. 15. Mesh 1 oriented horizontally (A) and vertically (B). Leftmost pictures show profiles when the distributor is filled, middle shows profile when the fluid hits the back wall, and rightmost pictures show final profile. Both front and side views are given to highlight void location.

3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity 21

Comparing Figure 16 and Figure 9, the numerical and experimental version of this profile, we can see that the simulations are again exhibiting the correct trends of the physical situation. Mesh 1 shows the most pooling at the centre of the mould, Mesh 2 has a flatter profile as does Mesh 3. The experiments predict filling times for Mesh 3 to be in between Mesh 1 and Mesh 2, and the simulations follow this trend. The dimensionless times to fill the distributor numerically for Mesh 1, Mesh 2, and Mesh 3 are 29%, 15%, and 21% compared to 26%, 13% and 22% for the experiments. In general, the simulations predict the vertical filling to be faster overall than the horizontal by several seconds for each of the geometries, whereas the experiments are faster in the vertical for Mesh 2 and 3, but slower for Mesh 1. This could have resulted from some experimental errors or from the uncertainty

Figure 17 shows the free surface profile as the fluid hits the back wall for Mesh 1, Mesh 2, and Mesh 3 in the horizontal orientation. The dimensionless times it takes to hit the back wall for Mesh 1, Mesh 2, and Mesh 3 are 41%, 55%, and 52% compared to values of 82%,

in injection rates and flow profiles from the experiments to the simulations.

**Mesh 2** 

Time/total time = .55

Fig. 17. Free surface profile after hitting the back wall for Mesh 1, Mesh 2, and Mesh 3 for

**Mesh 3** 

Time/total time = .52

71%, and 70% for the experiments seen in Figure 10.

**Mesh 1** 

horizontal mould orientation.

Time/total time = .41

Comparing Figure 15 to the experimental equivalent, Figure 8, we can see that we have quantitative differences but do match some trends. The numerical meniscus shape leaving the distributor looks similar to the experimental profile as it bulges more in the centre for the horizontal orientation, though the simulation is less dramatic. The numerical profiles when the fluid first hits the back wall are flatter for the vertical orientation than the horizontal, though the vertical should be even flatter to match the data. The numerical solutions predict two voids for each orientation, though the experiments do not show a second void for the horizontal orientation near the vent. However, this void may just be difficult to see experimentally. Conversely, the horizontal void at the outflow may be an artefact of the numerical method as we have a difficult balance at the outflow between wetting forces, gravity, the gas phase viscosity, and the material flowing out the vent. Also, our numerical vent is not identical to the experimental one and exhibits a slightly different area and shape.

Figure 16 shows the meniscus profiles for Mesh 1, Mesh 2, and Mesh 3 after filling the distributor for the horizontal orientation.

Fig. 16. Free surface profile after filling the distributor for Mesh 1, Mesh 2, and Mesh 3 for horizontal mould orientation.

Time/total time = .15

Time/total time = .21

Time/total time = .29

Comparing Figure 15 to the experimental equivalent, Figure 8, we can see that we have quantitative differences but do match some trends. The numerical meniscus shape leaving the distributor looks similar to the experimental profile as it bulges more in the centre for the horizontal orientation, though the simulation is less dramatic. The numerical profiles when the fluid first hits the back wall are flatter for the vertical orientation than the horizontal, though the vertical should be even flatter to match the data. The numerical solutions predict two voids for each orientation, though the experiments do not show a second void for the horizontal orientation near the vent. However, this void may just be difficult to see experimentally. Conversely, the horizontal void at the outflow may be an artefact of the numerical method as we have a difficult balance at the outflow between wetting forces, gravity, the gas phase viscosity, and the material flowing out the vent. Also, our numerical vent is not identical to the experimental one and exhibits a slightly different

Figure 16 shows the meniscus profiles for Mesh 1, Mesh 2, and Mesh 3 after filling the

area and shape.

**Mesh 1** 

horizontal mould orientation.

Time/total time = .29

**Mesh 2** 

Time/total time = .15

Fig. 16. Free surface profile after filling the distributor for Mesh 1, Mesh 2, and Mesh 3 for

**Mesh 3** 

Time/total time = .21

distributor for the horizontal orientation.

Comparing Figure 16 and Figure 9, the numerical and experimental version of this profile, we can see that the simulations are again exhibiting the correct trends of the physical situation. Mesh 1 shows the most pooling at the centre of the mould, Mesh 2 has a flatter profile as does Mesh 3. The experiments predict filling times for Mesh 3 to be in between Mesh 1 and Mesh 2, and the simulations follow this trend. The dimensionless times to fill the distributor numerically for Mesh 1, Mesh 2, and Mesh 3 are 29%, 15%, and 21% compared to 26%, 13% and 22% for the experiments. In general, the simulations predict the vertical filling to be faster overall than the horizontal by several seconds for each of the geometries, whereas the experiments are faster in the vertical for Mesh 2 and 3, but slower for Mesh 1. This could have resulted from some experimental errors or from the uncertainty in injection rates and flow profiles from the experiments to the simulations.

Figure 17 shows the free surface profile as the fluid hits the back wall for Mesh 1, Mesh 2, and Mesh 3 in the horizontal orientation. The dimensionless times it takes to hit the back wall for Mesh 1, Mesh 2, and Mesh 3 are 41%, 55%, and 52% compared to values of 82%, 71%, and 70% for the experiments seen in Figure 10.

Fig. 17. Free surface profile after hitting the back wall for Mesh 1, Mesh 2, and Mesh 3 for horizontal mould orientation.

Time/total time = .55

Time/total time = .52

Time/total time = .41

3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity 23

A diffuse interface finite element/level-set algorithm has been used to investigate filling behaviour for injection loading using a Blake wetting model. The modelling has been successful in matching experimental data qualitatively, but quantitative agreement is still lacking especially for the wetting dynamics and meniscus shape. For future work, we will investigate an advanced version of the level set method termed the conformal decomposition finite element method (CDFEM). CDFEM is a hybrid moving boundary algorithm, which uses a level set field to determine the location of the fluid-fluid interface and then dynamically adds mesh on the interface to facilitate the resolution of discontinuous material properties and fields, as well as the application of boundary conditions such as capillarity. This is a sharp interface method, where it is possible to apply jumps in material properties, material models, and field variables [Noble et al, 2010]. We believe this algorithm, which should be available soon, will lead to better agreement with experiments and should allow for straightforward inclusion of a

This work was supported by the Laboratory Directed Research and Development program at Sandia National Laboratories. We would like to thank Dr. Pin Yang, the project principal investigator, for his support of this study. We would also like to thank our Sandia reviewer Daniel Guildenbecher and P. Randall Schunk for their insightful

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-

Bach, P. & Hassager, O. (1985). "An algorithm for the use of the Lagrangian specification in

Blake, T. D. & Haynes, J. M. (1969). "Kinetics of liquid/liquid displacement," *J. Colloid* 

Blake, T. D. & De Coninck, J. (2002). "The Influence of Solid-Liquid Interactions on Dynamic

Brooks, C. F.; Grillet, A. M. & Emerson, J. A. (2006). "Experimental investigation of the

spontaneous wetting of polymers and polymer blends," *Langmuir,* 22, 9928-

Blake, T. D. (2006). "The physics of moving wetting lines," *J. Colloid Interface Sci.*, 299, 1-13. Brackbill, J. U.; Kothe, D. B. & Zemach, C. (1992). "A continuum method for modelling

Newtonian fluid mechanics and applications to free surface flow," *J. Fluid Mech*.,

**5. Conclusion** 

compressible gas phase.

**6. Acknowledgments** 

editorial comments.

AC04-94AL85000.

**7. References** 

152, 173-190.

9941.

*Interface Sci.*, 30, 421.

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surface-tension," *J. Comp. Phys.,* 100, 335-354.

For the simulations, Mesh 1 hits the back wall for the smallest dimensionless time while Mesh 2 and Mesh 3 take about the same time. For the experiments, Mesh 1 takes the longest time, while Mesh 2 and Mesh 3 do take about the same time. Thus for this case, we are capturing one trend, but not the differences between Mesh 1 and Mesh 2.

Table 4 summarizes the fill times to reach the distributor, back wall, and complete mould for the simulations and experiments for vertical and horizontal orientations on all three meshes.


Table 4. Summary of fill times for experiment and simulations to reach the distributor, the wall and completely full.

From Table 4 we can see that we capture some of the correct trends, especially for the vertical orientation, though some features elude us like the time to fill to the back wall for Mesh 1 in the horizontal orientation. Sources of uncertainty in the simulations include: 1) Lack of clarity of inflow conditions from the experiment to the simulation, since it is somewhere in between constant pressure and constant velocity, 2) Possible poor performance of the Blake wetting model at moderate capillary number and large dynamic contact angles, 3) Possible poor performance of the wetting model at wetting speed higher than experiments used to populate the model. Sources of error in the simulations include: 1) Numerically expedient of high gas phase viscosity, 2) Lack of compressibility for the gas dynamics, 3) The use of a diffuse interface model that smears out material property jumps, allowing viscous bleed through of the liquid phase into the gas phase. For future work, we will try to reduce these uncertainties and errors by using some of the advanced features in ARIA, which should be available soon, to allow for smaller values of the gas phase viscosity, such as sharp integration and a compressible gas phase. The optimal choice of wetting model for moderate to high capillary numbers continues to be an ongoing focus of our research.

### **5. Conclusion**

22 Numerical Modelling

For the simulations, Mesh 1 hits the back wall for the smallest dimensionless time while Mesh 2 and Mesh 3 take about the same time. For the experiments, Mesh 1 takes the longest time, while Mesh 2 and Mesh 3 do take about the same time. Thus for this case, we are

Table 4 summarizes the fill times to reach the distributor, back wall, and complete mould for the simulations and experiments for vertical and horizontal orientations on all three

> Expt. % Time - Wall

Sim. Time-Full

Sim. % Time - Dist

Sim. % Time - Wall

capturing one trend, but not the differences between Mesh 1 and Mesh 2.

Expt. % Time - Dist

1 Vertical 24.6s 32% 80% 15.2s 42% 60% 2 Vertical 26.9s 13% 86% 17.5s 18% 70% 3 Vertical 24.6s 24% 82% 13.2s 24% 55% 1 Horizontal 23.3s 26% 82% 21.4s 29% 41% 2 Horizontal 28.9s 13% 71% 23.1s 15% 55% 3 Horizontal 27.3s 22% 71% 15.6s 21% 52%

Table 4. Summary of fill times for experiment and simulations to reach the distributor, the

From Table 4 we can see that we capture some of the correct trends, especially for the vertical orientation, though some features elude us like the time to fill to the back wall for Mesh 1 in the horizontal orientation. Sources of uncertainty in the simulations include: 1) Lack of clarity of inflow conditions from the experiment to the simulation, since it is somewhere in between constant pressure and constant velocity, 2) Possible poor performance of the Blake wetting model at moderate capillary number and large dynamic contact angles, 3) Possible poor performance of the wetting model at wetting speed higher than experiments used to populate the model. Sources of error in the simulations include: 1) Numerically expedient of high gas phase viscosity, 2) Lack of compressibility for the gas dynamics, 3) The use of a diffuse interface model that smears out material property jumps, allowing viscous bleed through of the liquid phase into the gas phase. For future work, we will try to reduce these uncertainties and errors by using some of the advanced features in ARIA, which should be available soon, to allow for smaller values of the gas phase viscosity, such as sharp integration and a compressible gas phase. The optimal choice of wetting model for moderate to high capillary numbers continues to be an

Expt. Time - Full

meshes.

Mesh Orientation

wall and completely full.

ongoing focus of our research.

A diffuse interface finite element/level-set algorithm has been used to investigate filling behaviour for injection loading using a Blake wetting model. The modelling has been successful in matching experimental data qualitatively, but quantitative agreement is still lacking especially for the wetting dynamics and meniscus shape. For future work, we will investigate an advanced version of the level set method termed the conformal decomposition finite element method (CDFEM). CDFEM is a hybrid moving boundary algorithm, which uses a level set field to determine the location of the fluid-fluid interface and then dynamically adds mesh on the interface to facilitate the resolution of discontinuous material properties and fields, as well as the application of boundary conditions such as capillarity. This is a sharp interface method, where it is possible to apply jumps in material properties, material models, and field variables [Noble et al, 2010]. We believe this algorithm, which should be available soon, will lead to better agreement with experiments and should allow for straightforward inclusion of a compressible gas phase.

#### **6. Acknowledgments**

This work was supported by the Laboratory Directed Research and Development program at Sandia National Laboratories. We would like to thank Dr. Pin Yang, the project principal investigator, for his support of this study. We would also like to thank our Sandia reviewer Daniel Guildenbecher and P. Randall Schunk for their insightful editorial comments.

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

#### **7. References**


3D Numerical Modelling of Mould Filling of a Coat Hanger Distributer and Rectangular Cavity 25

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**2** 

*Spain* 

*Universidade da Coruña* 

**Simulation of the Scavenging** 

**Process in Two-Stroke Engines** 

María Isabel Lamas Galdo and Carlos G. Rodríguez Vidal

It is widely known that the scavenging process plays a very important role in the performance and efficiency of two-stroke engines. Briefly, scavenging is the process by which the fresh charge displaces the burnt gas from the cylinder. Due to the difficulties associated with the measurement techniques, CFD (Computational Fluid Dynamics) is a very helpful tool to analyze the flow pattern inside the cylinder. CFD simulations can provide more detailed information than experimental studies. For this reason, this chapter focuses on a numerical analysis to simulate the fluid flow and heat transfer inside the

This chapter is a continuation and extension of previous works (Lamas-Galdo *et al.,* 2011; Lamas & Rodriguez, 2012), in which CFD models were developed and validated with experimental results. The content is organized as follows. A brief description of two-stroke engines is given in Section 2. The mathematical model, i.e., the governing equations are presented in Section 3 and the numerical model is discussed in Section 4. After that, the results are shown in Section 5 and the conclusions of this chapter are discussed in Section 6.

Although the focus of this chapter is the numerical treatment of the scavenging process, it is important to introduce certain introductory aspects about the performance of two-stroke

A two-stroke engine is an internal combustion engine that completes the process cycle in one revolution of the crankshaft or two strokes of the piston: an up stroke and a down stroke. Both spark ignition and compression ignition engines exist today. Spark ignition engines are employed in light applications (chainsaws, motorcycles, outboard motors, etc) due to its low cost and simplicity. On the other hand, diesel compression ignition engines are mainly employed in large and weight applications, such as large industrial and marine engines, heavy machinery, locomotives, etc. Fig. 1 (a) shows a spark ignition engine installed on a motorbike and Fig. 1 (b) shows a large compression ignition engine, the MAN

engines. This will facilitate the reader's understanding of the chapter.

B&W 7S50MC, typically used in marine propulsion and industrial plants.

**1. Introduction** 

cylinder at the scavenging process.

**2.1 Mechanical aspects** 

**2. Introduction to the two-stroke engine** 

Zhang, J. & Khayat, R. E. (2001). "A Lagrangian boundary element approach to transient three-dimensional free surface flow in thin cavities, *Int. J. Num. Meth. Fluids*, 37, 399-418.

## **Simulation of the Scavenging Process in Two-Stroke Engines**

María Isabel Lamas Galdo and Carlos G. Rodríguez Vidal *Universidade da Coruña Spain* 

### **1. Introduction**

26 Numerical Modelling

Zhang, J. & Khayat, R. E. (2001). "A Lagrangian boundary element approach to transient

399-418.

three-dimensional free surface flow in thin cavities, *Int. J. Num. Meth. Fluids*, 37,

It is widely known that the scavenging process plays a very important role in the performance and efficiency of two-stroke engines. Briefly, scavenging is the process by which the fresh charge displaces the burnt gas from the cylinder. Due to the difficulties associated with the measurement techniques, CFD (Computational Fluid Dynamics) is a very helpful tool to analyze the flow pattern inside the cylinder. CFD simulations can provide more detailed information than experimental studies. For this reason, this chapter focuses on a numerical analysis to simulate the fluid flow and heat transfer inside the cylinder at the scavenging process.

This chapter is a continuation and extension of previous works (Lamas-Galdo *et al.,* 2011; Lamas & Rodriguez, 2012), in which CFD models were developed and validated with experimental results. The content is organized as follows. A brief description of two-stroke engines is given in Section 2. The mathematical model, i.e., the governing equations are presented in Section 3 and the numerical model is discussed in Section 4. After that, the results are shown in Section 5 and the conclusions of this chapter are discussed in Section 6.

#### **2. Introduction to the two-stroke engine**

Although the focus of this chapter is the numerical treatment of the scavenging process, it is important to introduce certain introductory aspects about the performance of two-stroke engines. This will facilitate the reader's understanding of the chapter.

#### **2.1 Mechanical aspects**

A two-stroke engine is an internal combustion engine that completes the process cycle in one revolution of the crankshaft or two strokes of the piston: an up stroke and a down stroke. Both spark ignition and compression ignition engines exist today. Spark ignition engines are employed in light applications (chainsaws, motorcycles, outboard motors, etc) due to its low cost and simplicity. On the other hand, diesel compression ignition engines are mainly employed in large and weight applications, such as large industrial and marine engines, heavy machinery, locomotives, etc. Fig. 1 (a) shows a spark ignition engine installed on a motorbike and Fig. 1 (b) shows a large compression ignition engine, the MAN B&W 7S50MC, typically used in marine propulsion and industrial plants.

Simulation of the Scavenging Process in Two-Stroke Engines 29

Before discussing the scavenging process, it is useful to describe the operation cycle of the two-stroke engine with direct injection. For this purpose, an engine with scavenge and exhaust ports instead valves will be considered. At the beginning of the cycle, when fuel injection and ignition have just taken place, the piston is at the TDC (top dead center). The temperature and pressure rise and consequently the piston is driven down, Fig. 3 (a) (note that the arrows indicate the direction of the piston). Along the power stroke, the exhaust ports are uncovered (opened) and, consequently, the burnt gases begin to flow out, Fig. 3 (b). The piston continues down. When the piston pasts over (and consequently opens) the scavenge ports, pressurized air enters and drives out the remaining exhaust gases, Fig. 3 (c). This process of introducing air and expelling burnt gases is called scavenging. The incoming air is used to clean out or scavenge the exhaust gases and then to fill or charge the space with fresh air. After reaching BDC (bottom dead center), the piston moves upward on its return stroke. The scavenge ports and then the exhaust ports are closed, Fig. 3 (d), and the air is then compressed as the piston moves to the top of its stroke. Soon before the piston reaches TDC, the injectors spray the fuel, the spark plug ignite the mixture and the cycle

SCAVENGE PORT

CONNECTING ROD

(a) (b) (c) (d)

Fig. 3. Basic engine operation. (a) Injection; (b) exhaust; (c) scavenge; (d) compression.

A drawback which has a decisive influence, not only on consumption but also on power and pollution, is the process of displacing the burnt gases from the cylinder and replacing them by the fresh-air charge, known as scavenging. In ideal scavenging, the entering scavenge air acts as a wedge in pushing the burnt gases out of the cylinder without mixing with them. Unfortunately, the real scavenging process is characterized by two problems common to two-stroke engines in general: *short-circuiting losses* and *mixing*. *Short-circuiting* consists on expelling some of the fresh-air charge directly to the exhaust and *mixing* consists on the fact

FUEL INJECTOR CYLINDER PISTON

**2.2 The scavenging process** 

starts again.

EXHAUST PORT

Gas Air

CRANKSHAFT

Fig. 1. (a) Spark ignition gasoline engine installed on a motorcycle. (b) Compression ignition diesel engine MAN B&W 7S50MC installed on a ship.

There are several mechanical details which vary from one engine to another. For example, Fig. 2 (a) shows a cross section of the spark ignition engine shown in Fig. 1 (a). In this engine, the charge is introduced to the cylinder by ports. The opening and closing of the ports is controlled by the sides of the piston covering and uncovering them as it moves up and down in the cylinder. As can be seen in the bottom part of Fig. 2 (a), this engine has a crankcase. This is a separate charging cylinder which employs the volume below the piston as a charging pump. On the other hand, Fig. 2 (b) shows a cross section of the compression ignition engine illustrated on Fig. 1 (a). This engine has one exhaust valve and several intake ports. In this case, the external air is introduced directly in the cylinder instead of being pumped from the crankcase.

Fig. 2. (a) Cross section of a spark ignition engine. (b) Cross section of a compression ignition engine, the MAN B&W 7S50MC.

#### **2.2 The scavenging process**

28 Numerical Modelling

(a) (b)

Fig. 1. (a) Spark ignition gasoline engine installed on a motorcycle. (b) Compression ignition

There are several mechanical details which vary from one engine to another. For example, Fig. 2 (a) shows a cross section of the spark ignition engine shown in Fig. 1 (a). In this engine, the charge is introduced to the cylinder by ports. The opening and closing of the ports is controlled by the sides of the piston covering and uncovering them as it moves up and down in the cylinder. As can be seen in the bottom part of Fig. 2 (a), this engine has a crankcase. This is a separate charging cylinder which employs the volume below the piston as a charging pump. On the other hand, Fig. 2 (b) shows a cross section of the compression ignition engine illustrated on Fig. 1 (a). This engine has one exhaust valve and several intake ports. In this case, the external air is introduced directly in the cylinder instead of being

(a) (b)

Fig. 2. (a) Cross section of a spark ignition engine. (b) Cross section of a compression

diesel engine MAN B&W 7S50MC installed on a ship.

pumped from the crankcase.

ignition engine, the MAN B&W 7S50MC.

Before discussing the scavenging process, it is useful to describe the operation cycle of the two-stroke engine with direct injection. For this purpose, an engine with scavenge and exhaust ports instead valves will be considered. At the beginning of the cycle, when fuel injection and ignition have just taken place, the piston is at the TDC (top dead center). The temperature and pressure rise and consequently the piston is driven down, Fig. 3 (a) (note that the arrows indicate the direction of the piston). Along the power stroke, the exhaust ports are uncovered (opened) and, consequently, the burnt gases begin to flow out, Fig. 3 (b). The piston continues down. When the piston pasts over (and consequently opens) the scavenge ports, pressurized air enters and drives out the remaining exhaust gases, Fig. 3 (c). This process of introducing air and expelling burnt gases is called scavenging. The incoming air is used to clean out or scavenge the exhaust gases and then to fill or charge the space with fresh air. After reaching BDC (bottom dead center), the piston moves upward on its return stroke. The scavenge ports and then the exhaust ports are closed, Fig. 3 (d), and the air is then compressed as the piston moves to the top of its stroke. Soon before the piston reaches TDC, the injectors spray the fuel, the spark plug ignite the mixture and the cycle starts again.

Fig. 3. Basic engine operation. (a) Injection; (b) exhaust; (c) scavenge; (d) compression.

A drawback which has a decisive influence, not only on consumption but also on power and pollution, is the process of displacing the burnt gases from the cylinder and replacing them by the fresh-air charge, known as scavenging. In ideal scavenging, the entering scavenge air acts as a wedge in pushing the burnt gases out of the cylinder without mixing with them. Unfortunately, the real scavenging process is characterized by two problems common to two-stroke engines in general: *short-circuiting losses* and *mixing*. *Short-circuiting* consists on expelling some of the fresh-air charge directly to the exhaust and *mixing* consists on the fact

Simulation of the Scavenging Process in Two-Stroke Engines 31

*<sup>p</sup> u uu t x xx*

where τij is the stress tensor. If the fluid is treated as Newtonian, the stress tensor

As only the scavenging process and not the combustion is treated on this chapter, only two components need to be computed: burnt gas and unburnt gas (air). In order to characterize

> ( )0 ( ) *Y V air air*

where *Yair* is the mass fraction of the air. The mass fraction of the burnt gases, *Ygas* , is given

Today's standard in engine simulation are Reynolds Averaged Navier-Stokes (RANS) methods. Another approach are Large Eddy Simulation (LES) techniques. LES and RANS techniques differ in the way they address the present impossibility to resolve all the scales present in engine flows. RANS simulations are based on a statistical averaging to solve only the mean flow. This implies that modelling concerns the whole spectrum of scales. In LES, a spatial or temporal filtering is used to represent the large turbulent scales of the flow, which are directly resolved, while the small scales are modeled. In LES, modeling thus concerns a much smaller part of the spectrum, which leads to an improvement of predictivity as compared to RANS. LES inherently allows to address large scale unsteady phenomena, and thus has a good potential to predict engine unsteadiness. The problem is that LES would lead to a CPU time that is way beyond reach of present supercomputers. Therefore, the use

In the field of RANS methods, the two-equation model standard k-ε is the most used to simulate engines. The RNG k-ε model is also widely employed, specially in the cases of

> *i i j i j j ijj <sup>p</sup> u uu u u*

(7)

*u ui <sup>j</sup>* , is the Boussinesq hypothesis to

*t x xxx*

The momentum conservation equation for a turbulent flow is given by:

' ' () ( ) ( ) *ij*

 

A common method to model the Reynolds stresses, ' '

relate the Reynolds stresses to the mean velocity gradients:

*Y t*

 

 

*ij ij*

*j i j*

2 3 *j i k*

 

*j i k u u u xx x*

  (3)

(4)

(5)

*Y Y* <sup>1</sup> *gas air* (6)

*i i j*

() ( ) *ij*

components are given by:

**3.2 Turbulence** 

o LES is not very common.

swirling flows.

the propagating interface, the following equation is solved:

by the restriction that the total mass fraction must sum to unity:

that there is a small amount of residual gases which remain trapped without being expelled, being mixed with some of the new air charge.

The main difficulty involved in designing an effective scavenging system is that there are too involving variables: piston chamber geometry, intake and exhaust ports design, opening and closing timings, compression ratio, fuel composition, inlet and exhaust pressures, etc, being necessary a detailed study to embrace all this factors. For years, the study of the fluid flow inside engines has been mainly supported by experimental tests such as PIV (Particle Image Velocity), LDA (Laser Doppler Anemometry), ICCD cameras, etc. However, these experimental tests are very laborious and expensive. As an alternative solution to experimental techniques, CFD has recently become a useful tool to study the fluid flow inside engines. In the field of engines, CFD is especially useful to design complex components such as combustion chambers, manifolds, injectors and other parameters. The first numerical simulations of engines appeared in the eighties (Sher, 1980; Carpenter & Ramos, 1986; Sweeny *et al.*, 1985; Ahmadi-Befrui *et al.*, 1989) but, unfortunately, these first numerical studies only provided, with poor accuracy, information about the general configuration of the flow field inside the cylinder. Besides, at that time it was very difficult to carry out a three-dimensional analysis. After these first numerical studies, a lot of works appeared in the nineties and in recent years. The number of CFD codes has also increased noticeably, appearing studies using KIVA (Epstein *et al.,* 1991; Amsden *et al.,* 1992), STAR-CD (Raghunathan & Kenny, 1997; Yu *et al.*  1997; Zahn *et al.*, 2000; Hariharan *et al.*, 2009), FIRE (Hori *et al.,* 1995; Laimböck *et al.*, 1998) Fluent (Pitta & Kuderu, 2008; Lamas-Galdo *et al.*, 2011), CFX (Albanesi *et al.*, 2009), etc.

#### **3. Mathematical model**

Once the basic performance of two-stroke engines was described, the methodology to simulate the scavenging process will be treated in this section.

#### **3.1 Governing equations**

The governing equations of the flow inside the cylinder are the Navier-Stokes ones. The energy equation is also needed to compute the thermal problem. Finally, as there are two components (air and burnt gases), one more equation must be added to characterize the propagating interface. These equations are briefly described in what follows.

In Cartesian tensor form, the continuity equation is given by:

$$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_i} (\rho u\_i) = 0 \tag{1}$$

where *ρ* is the density and u the velocity. It is very common to consider the flows as ideal gasses, so the density can be calculated as follows:

$$
\rho = \frac{p}{RT} \tag{2}
$$

where *p* is the pressure, *T* the temperature and *R* the ideal gas constant. The momentum conservation equation is given by:

$$\frac{\partial}{\partial t}(\rho u\_i) + \frac{\partial}{\partial \mathbf{x}\_j}(\rho u\_i u\_j) = -\frac{\partial p}{\partial \mathbf{x}\_i} + \frac{\partial \tau\_{ij}}{\partial \mathbf{x}\_j} \tag{3}$$

where τij is the stress tensor. If the fluid is treated as Newtonian, the stress tensor components are given by:

$$
\Delta \tau\_{ij} = \mu \left( \frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i} - \frac{2}{3} \mathcal{S}\_{ij} \frac{\partial u\_k}{\partial \mathbf{x}\_k} \right) \tag{4}
$$

As only the scavenging process and not the combustion is treated on this chapter, only two components need to be computed: burnt gas and unburnt gas (air). In order to characterize the propagating interface, the following equation is solved:

$$\frac{\partial(\rho Y\_{a\dot{a}r})}{\partial t} + \nabla \cdot (Y\_{a\dot{a}r}\rho \vec{V}) = 0 \tag{5}$$

where *Yair* is the mass fraction of the air. The mass fraction of the burnt gases, *Ygas* , is given by the restriction that the total mass fraction must sum to unity:

$$Y\_{\text{gas}} = 1 - Y\_{\text{air}} \tag{6}$$

#### **3.2 Turbulence**

30 Numerical Modelling

that there is a small amount of residual gases which remain trapped without being expelled,

The main difficulty involved in designing an effective scavenging system is that there are too involving variables: piston chamber geometry, intake and exhaust ports design, opening and closing timings, compression ratio, fuel composition, inlet and exhaust pressures, etc, being necessary a detailed study to embrace all this factors. For years, the study of the fluid flow inside engines has been mainly supported by experimental tests such as PIV (Particle Image Velocity), LDA (Laser Doppler Anemometry), ICCD cameras, etc. However, these experimental tests are very laborious and expensive. As an alternative solution to experimental techniques, CFD has recently become a useful tool to study the fluid flow inside engines. In the field of engines, CFD is especially useful to design complex components such as combustion chambers, manifolds, injectors and other parameters. The first numerical simulations of engines appeared in the eighties (Sher, 1980; Carpenter & Ramos, 1986; Sweeny *et al.*, 1985; Ahmadi-Befrui *et al.*, 1989) but, unfortunately, these first numerical studies only provided, with poor accuracy, information about the general configuration of the flow field inside the cylinder. Besides, at that time it was very difficult to carry out a three-dimensional analysis. After these first numerical studies, a lot of works appeared in the nineties and in recent years. The number of CFD codes has also increased noticeably, appearing studies using KIVA (Epstein *et al.,* 1991; Amsden *et al.,* 1992), STAR-CD (Raghunathan & Kenny, 1997; Yu *et al.*  1997; Zahn *et al.*, 2000; Hariharan *et al.*, 2009), FIRE (Hori *et al.,* 1995; Laimböck *et al.*, 1998)

Fluent (Pitta & Kuderu, 2008; Lamas-Galdo *et al.*, 2011), CFX (Albanesi *et al.*, 2009), etc.

simulate the scavenging process will be treated in this section.

In Cartesian tensor form, the continuity equation is given by:

gasses, so the density can be calculated as follows:

conservation equation is given by:

Once the basic performance of two-stroke engines was described, the methodology to

The governing equations of the flow inside the cylinder are the Navier-Stokes ones. The energy equation is also needed to compute the thermal problem. Finally, as there are two components (air and burnt gases), one more equation must be added to characterize the

0 *<sup>i</sup>*

(1)

(2)

*p RT*

where *p* is the pressure, *T* the temperature and *R* the ideal gas constant. The momentum

*i u*

where *ρ* is the density and u the velocity. It is very common to consider the flows as ideal

*t x* 

propagating interface. These equations are briefly described in what follows.

being mixed with some of the new air charge.

**3. Mathematical model** 

**3.1 Governing equations** 

Today's standard in engine simulation are Reynolds Averaged Navier-Stokes (RANS) methods. Another approach are Large Eddy Simulation (LES) techniques. LES and RANS techniques differ in the way they address the present impossibility to resolve all the scales present in engine flows. RANS simulations are based on a statistical averaging to solve only the mean flow. This implies that modelling concerns the whole spectrum of scales. In LES, a spatial or temporal filtering is used to represent the large turbulent scales of the flow, which are directly resolved, while the small scales are modeled. In LES, modeling thus concerns a much smaller part of the spectrum, which leads to an improvement of predictivity as compared to RANS. LES inherently allows to address large scale unsteady phenomena, and thus has a good potential to predict engine unsteadiness. The problem is that LES would lead to a CPU time that is way beyond reach of present supercomputers. Therefore, the use o LES is not very common.

In the field of RANS methods, the two-equation model standard k-ε is the most used to simulate engines. The RNG k-ε model is also widely employed, specially in the cases of swirling flows.

The momentum conservation equation for a turbulent flow is given by:

$$\frac{\partial}{\partial t}(\rho u\_i) + \frac{\partial}{\partial \mathbf{x}\_j}(\rho u\_i u\_j) = -\frac{\partial p}{\partial \mathbf{x}\_i} + \frac{\partial \tau\_{ij}}{\partial \mathbf{x}\_j} + \frac{\partial}{\partial \mathbf{x}\_j}(-\rho \overline{u\_i^{\prime} u\_j^{\prime}}) \tag{7}$$

A common method to model the Reynolds stresses, ' ' *u ui <sup>j</sup>* , is the Boussinesq hypothesis to relate the Reynolds stresses to the mean velocity gradients:

Simulation of the Scavenging Process in Two-Stroke Engines 33

AUXILIARY

(a) (b) Fig. 4. Cylinder geometry and distribution diagram. (a) Lateral view; (b) Plant view. Lamas-

> **Parameter Value**  Type of engine Two-stroke, Otto

Scavenging system Loop scavenge Fuel system Direct injection Power (W) 7500 Speed (rpm) 6000

Displacement (cm3) 127.3 Compression rate 9.86:1 Bore (mm) 53.8 Stroke (mm) 56

Connecting rod length (mm) 110

At maximum continuum rating, the in-cylinder, exhaust and intake pressures were measured experimentally. Piezoresistive sensors were employed to measure the exhaust and intake pressures, while a piezoelectric sensor was employed to measure the in-cylinder pressure. These sensors were connected to its corresponding charge amplifier and data acquisition system. The data were analyzed using the software *LabVIEW SignalExpress LE*. The in-cylinder pressure is shown in Fig. 5 and the intake and exhaust pressures are shown in Fig. 6. Note that, in this work, the crank angles were chosen with reference to TDC.

 MAIN SCAVENGE PORT

SCAVENGE PORT EXHAUST PORT

 MAIN SCAVENGE PORT

BDC

 MAIN SCAVENGE PORT

 AUXILIARY SCAVENGE PORT

Galdo *et al.* (2011).

Table 1. Technical specifications.

EXHAUST PORT

TDC

$$-\rho \overrightarrow{\dot{u\_i \dot{u\_j}}} = \mu\_t \left(\frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i}\right) - \frac{2}{3} \left(\rho \mathbf{k} + \mu\_t \frac{\partial u\_k}{\partial \mathbf{x}\_k}\right) \delta\_{ij} \tag{8}$$

where *δij* is the Kronecker delta (*δij*=1 if *i=j* and *δij*=0 if *i≠j*), which is included to make the formula applicable to the normal Reynolds stresses for which *i=j* (Versteeg and Malalasekera, 2007) and *μt* is the turbulent viscosity. The k-ε model includes two differential equations, corresponding to the turbulent kinetic energy (*k*), and its dissipation rate (*ε*), given by Ecs. (9) and (10) respectively.

$$\frac{\partial}{\partial t}(\rho k) + \frac{\partial}{\partial \mathbf{x}\_i}(\rho k u\_i) = \frac{\partial}{\partial \mathbf{x}\_j} \left| \alpha\_k \mu\_t \frac{\partial \mathbf{k}}{\partial \mathbf{x}\_j} \right| + \mathbf{G}\_k + \mathbf{G}\_b - \rho \mathbf{z} - \mathbf{Y}\_M \tag{9}$$

$$\frac{\partial}{\partial t}(\rho \varepsilon) + \frac{\partial}{\partial \boldsymbol{\chi}\_{i}}(\rho \varepsilon \boldsymbol{u}\_{i}) = \frac{\partial}{\partial \boldsymbol{\chi}\_{j}} \left| \boldsymbol{\alpha}\_{\varepsilon} \mu\_{t} \frac{\partial \varepsilon}{\partial \boldsymbol{\chi}\_{j}} \right| + \mathbf{C}\_{1\varepsilon} \frac{\varepsilon}{k} (\mathbf{G}\_{k} + \mathbf{G}\_{3\varepsilon} \mathbf{G}\_{b}) - \mathbf{C}\_{2\varepsilon} \rho \frac{\varepsilon^{2}}{k} \tag{10}$$

In the above equations, *Gk* represents the generation of turbulence kinetic energy due to the mean velocity gradients; *Gb* is the generation of turbulence kinetic energy due to buoyancy; *YM* represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. *Cμ*, *C1ε*, *C2ε*, *C3ε*, *σ<sup>k</sup>* and *σε* are constants and the terms *αk* and *αε* represent the inverse effective Prandtl numbers for *k* and *ε* respectively. These quantities were obtained by a RNG modified method which accounts for the effects of swirl or rotation. Details of the procedure are given elsewhere, (Fluent Inc., 2006).

The turbulent viscosity, *μt*, is computed by combining *k* and *ε* as follows:

$$
\mu\_t = \rho \mathbb{C}\_{\mu} \frac{k^2}{\varepsilon} \tag{11}
$$

Concerning the heat transfer problem, turbulent heat transport can be modeled using the concept of Reynolds' analogy to turbulent momentum transfer. The energy equation is thus given by the following:

$$\frac{\partial}{\partial t}(\rho \to) + \frac{\partial}{\partial \mathbf{x}\_i} [\mathbf{u}\_i(\rho \to p)] = \frac{\partial}{\partial \mathbf{x}\_j} \left| \left( \mathbf{k}\_t + \frac{\mathbf{C}\_p \mu\_t}{\mathbf{Pr}} \right) \frac{\partial \mathbf{T}}{\partial \mathbf{x}\_j} + \mathbf{u}\_i \boldsymbol{\tau}\_{ij} \right| \tag{12}$$

where *E* is the total energy.

#### **4. Numerical procedure**

In this section, the generation of the mesh and other numerical details will be described. Particularly, this section focuses on the engine studied in Lamas-Galdo *et al.* (2011), which is shown in Fig. 1 (a) and Fig. 2 (a). This is a single cylinder two-stroke engine. The geometry and distribution diagram are shown in Fig. 4, and other technical specifications are summarized in Table 1.

where *δij* is the Kronecker delta (*δij*=1 if *i=j* and *δij*=0 if *i≠j*), which is included to make the formula applicable to the normal Reynolds stresses for which *i=j* (Versteeg and Malalasekera, 2007) and *μt* is the turbulent viscosity. The k-ε model includes two differential equations, corresponding to the turbulent kinetic energy (*k*), and its dissipation rate (*ε*),

*<sup>k</sup> k ku GG Y*

*<sup>u</sup> C G GG C tx xxk <sup>k</sup>*

In the above equations, *Gk* represents the generation of turbulence kinetic energy due to the mean velocity gradients; *Gb* is the generation of turbulence kinetic energy due to buoyancy; *YM* represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. *Cμ*, *C1ε*, *C2ε*, *C3ε*, *σ<sup>k</sup>* and *σε* are constants and the terms *αk* and *αε* represent the inverse effective Prandtl numbers for *k* and *ε* respectively. These quantities were obtained by a RNG modified method which accounts for the effects of swirl or

(10)

2

*p t*

*i t i ij*

*i jj <sup>C</sup> <sup>T</sup> E u Ep k <sup>u</sup>*

 

In this section, the generation of the mesh and other numerical details will be described. Particularly, this section focuses on the engine studied in Lamas-Galdo *et al.* (2011), which is shown in Fig. 1 (a) and Fig. 2 (a). This is a single cylinder two-stroke engine. The geometry and distribution diagram are shown in Fig. 4, and other technical specifications are

*k*

 

*u u k*

() ( )*<sup>i</sup> kt k b M*

*tx xx*

*i jj*

 

*i jj*

 <sup>2</sup> 1 32 () ( )*<sup>i</sup> <sup>t</sup> k b*

 

 

rotation. Details of the procedure are given elsewhere, (Fluent Inc., 2006). The turbulent viscosity, *μt*, is computed by combining *k* and *ε* as follows:

() ( ) Pr

 *t*

*tx x x*

Concerning the heat transfer problem, turbulent heat transport can be modeled using the concept of Reynolds' analogy to turbulent momentum transfer. The energy equation is thus

 *C* 

3 *j i k i j t t ij j i k u u u*

 

(11)

 

(8)

 

(9)

(12)

*xx x*

' ' <sup>2</sup>

given by Ecs. (9) and (10) respectively.

given by the following:

where *E* is the total energy.

**4. Numerical procedure** 

summarized in Table 1.

 

Fig. 4. Cylinder geometry and distribution diagram. (a) Lateral view; (b) Plant view. Lamas-Galdo *et al.* (2011).


Table 1. Technical specifications.

At maximum continuum rating, the in-cylinder, exhaust and intake pressures were measured experimentally. Piezoresistive sensors were employed to measure the exhaust and intake pressures, while a piezoelectric sensor was employed to measure the in-cylinder pressure. These sensors were connected to its corresponding charge amplifier and data acquisition system. The data were analyzed using the software *LabVIEW SignalExpress LE*. The in-cylinder pressure is shown in Fig. 5 and the intake and exhaust pressures are shown in Fig. 6. Note that, in this work, the crank angles were chosen with reference to TDC.

Simulation of the Scavenging Process in Two-Stroke Engines 35

(a) (b)

(c) (d)

Hexahedral elements provide better accuracy and stability, so a structured hexahedral mesh was adopted. The numerical algorithm implemented automatically updates the mesh after each time step relative to the piston motion using a meshing tool called "dynamic layering", which consists on adding or removing layers of cells adjacent to a moving boundary based on the height of the layer adjacent to the moving surface. The procedure is shown in Fig. 8.

CYLINDER CYLINDER

PORT

Fig. 7. Computational mesh. (a) 92º crank angle; (b) 190º; (c) 215º; (d) 270º crank angle.

PORT PORT

Lamas-Galdo *et al.* (2011).

CYLINDER

Fig. 8. Layering procedure.

Concerning the temperatures, unfortunately, the in-cylinder temperatures can not be measured experimentally because a temperature sensor is not fast enough to accurate capture the in-cylinder temperature along the whole cycle.

Fig. 5. Evolution of the in-cylinder pressure.

Fig. 6. Evolution of the exhaust and intake pressures.

#### **4.1 Mesh generation**

The principle of operation of CFD codes is subdividing the domain into a number of smaller, non-overlapping sub-domains. The result is a grid (or mesh) of cells (or elements). In this work, a grid generation program, Gambit 2.4.6, was used to generate the mesh. In order to implement the movement of the piston, a moving mesh must be used. Figure 7 shows the mesh at several crankshaft angles. The computational domain includes the scavenge ports, exhaust port, cylinder and cylinder head.

Concerning the temperatures, unfortunately, the in-cylinder temperatures can not be measured experimentally because a temperature sensor is not fast enough to accurate

**Stroke** TDC BDC

Intake pressure Exhaust pressure

0 60 120 180 240 300 360

**Crankshaft angle (deg)**

The principle of operation of CFD codes is subdividing the domain into a number of smaller, non-overlapping sub-domains. The result is a grid (or mesh) of cells (or elements). In this work, a grid generation program, Gambit 2.4.6, was used to generate the mesh. In order to implement the movement of the piston, a moving mesh must be used. Figure 7 shows the mesh at several crankshaft angles. The computational domain includes the

capture the in-cylinder temperature along the whole cycle.

0

Fig. 5. Evolution of the in-cylinder pressure.


Fig. 6. Evolution of the exhaust and intake pressures.

scavenge ports, exhaust port, cylinder and cylinder head.

0

0.5

1

**Pressure (bar)**

**4.1 Mesh generation** 

1.5

2

2.5

5

10

**Pressure (bar)**

15

Fig. 7. Computational mesh. (a) 92º crank angle; (b) 190º; (c) 215º; (d) 270º crank angle. Lamas-Galdo *et al.* (2011).

Hexahedral elements provide better accuracy and stability, so a structured hexahedral mesh was adopted. The numerical algorithm implemented automatically updates the mesh after each time step relative to the piston motion using a meshing tool called "dynamic layering", which consists on adding or removing layers of cells adjacent to a moving boundary based on the height of the layer adjacent to the moving surface. The procedure is shown in Fig. 8.

Fig. 8. Layering procedure.

Simulation of the Scavenging Process in Two-Stroke Engines 37

It is very important to include the ports and ducts in the computational grid because they notably influence the movement of gases inside the cylinder and therefore the characteristics of the scavenging. For example, in the engine of Fig. 1 (b) and Fig. 2 (b), the intake ports and ducts are inclinated respect to the cylinder axis. Consequently, a swirling motion is promoted by the tangential velocities around the cilinder axis. This phenomena is shown in Fig. 10, which represents the velocity field in a tri-dimensional view, Fig. 10 (a), and in a

(a) (b)

Obviously, not all the engines are so sensible to the inlet ports and ducts geometry, but it is recommended to include them in the mesh instead a surface in which a boundary condition

All CFD models require initial and boundary conditions. Concerning the pressures, the experimentally values mentioned in the beginning of section 4 were employed as initial and

As the in-cylinder temperature can not be measured experimentally, the initial temperature must be estimated from an adaptation of the ideal Otto cycle, Fig. 11 (a) and 11 (b). Details of the procedure can be found in most undergraduate textbooks on internal combustion engines or thermodynamics, so they are not repeated here. As can be seen in Fig. 6 (b), the

Fig. 10. Velocity field [m/s] for 150º crankshaft angle. (a) Tri-dimensional view. (b) Transversal section A-A, at the base of the cylinder. Lamas & Rodríguez (2012).

is imposed.

boundary conditions.

**4.2 Boundary and initial conditions** 

temperature at 90º crankshaft angle is 1027 K.

transversal section at the base of the cylinder, Fig. 10 (b).

Sometimes it is not possible to employ hexahedral elements in the totality of the control volume. For example, the engine studied in Lamas & Rodríguez (2012), Fig. 1 (b) and Fig. 2 (b), has an exhaust valve in every cylinder. Due to the complex geometry of the valve and duct, tetrahedral elements were employed in that region. Besides, it was necessary to refine the region closed to the valve in order to capture the complex characteristics of the flow. The result is shown in Fig. 9.

Fig. 9. (a) Tri-dimensional mesh, 180º crankshaft angle. (b) Cross-section mesh, 180º crankshaft angle; (c) Cross-section mesh, 270º crankshaft angle. Lamas & Rodríguez (2012).

Sometimes it is not possible to employ hexahedral elements in the totality of the control volume. For example, the engine studied in Lamas & Rodríguez (2012), Fig. 1 (b) and Fig. 2 (b), has an exhaust valve in every cylinder. Due to the complex geometry of the valve and duct, tetrahedral elements were employed in that region. Besides, it was necessary to refine the region closed to the valve in order to capture the complex characteristics of the flow. The

(a) (b) (c)

Fig. 9. (a) Tri-dimensional mesh, 180º crankshaft angle. (b) Cross-section mesh, 180º crankshaft angle; (c) Cross-section mesh, 270º crankshaft angle. Lamas & Rodríguez (2012).

result is shown in Fig. 9.

It is very important to include the ports and ducts in the computational grid because they notably influence the movement of gases inside the cylinder and therefore the characteristics of the scavenging. For example, in the engine of Fig. 1 (b) and Fig. 2 (b), the intake ports and ducts are inclinated respect to the cylinder axis. Consequently, a swirling motion is promoted by the tangential velocities around the cilinder axis. This phenomena is shown in Fig. 10, which represents the velocity field in a tri-dimensional view, Fig. 10 (a), and in a transversal section at the base of the cylinder, Fig. 10 (b).

Fig. 10. Velocity field [m/s] for 150º crankshaft angle. (a) Tri-dimensional view. (b) Transversal section A-A, at the base of the cylinder. Lamas & Rodríguez (2012).

Obviously, not all the engines are so sensible to the inlet ports and ducts geometry, but it is recommended to include them in the mesh instead a surface in which a boundary condition is imposed.

#### **4.2 Boundary and initial conditions**

All CFD models require initial and boundary conditions. Concerning the pressures, the experimentally values mentioned in the beginning of section 4 were employed as initial and boundary conditions.

As the in-cylinder temperature can not be measured experimentally, the initial temperature must be estimated from an adaptation of the ideal Otto cycle, Fig. 11 (a) and 11 (b). Details of the procedure can be found in most undergraduate textbooks on internal combustion engines or thermodynamics, so they are not repeated here. As can be seen in Fig. 6 (b), the temperature at 90º crankshaft angle is 1027 K.

Simulation of the Scavenging Process in Two-Stroke Engines 39

In order to ensure that the CFD model is accurate enough, numerical results were compared to experimental ones. Particularly, the in-cylinder gauge pressure was validated. For the interval of time studied, from 90º to 270º crankshaft angles, the numerical and experimental results are shown in Fig. 12. Note that an acceptable concordance is obtained between CFD

Fig. 12. In-cylinder pressure numerically and experimentally obtained.

Figure 13 shows the gauge pressure field at several crank angles. As can be seen, the initial in-cylinder pressure, Fig. 13 (a), is 4.26 bar. As mentioned before, the intake and exhaust pressures are variable, imposed as boundary conditions at the intake and exhaust ports. At the beginning of the simulation, the pressure descends drastically due to the expansion of the piston (note that the arrows indicate the direction of the piston). When the ports are opened, Fig. 13 (b) and (c), the in-cylinder pressure is slightly superior to the exhaust pressure and slightly inferior to the intake pressure, therefore burnt gasses are expelled through the exhaust port and fresh air enters through the scavenge ports. Finally, when all ports are closed, Fig. 13 (d), the piston is ascending and the gasses are

**5. Results** 

and experimental results.

compressed, Fig. 13 (d).

**5.1 Pressure field and validation of the code** 

Fig. 11. (a) In-cylinder pressure experimentally measured and obtained from the ideal Otto cycle. (b) In-cylinder temperature obtained from the ideal Otto cycle.

#### **4.3 Resolution of the equations**

In this case, the software ANSYS Fluent 6.3 was employed. This is based on the finite volume method. Concerning the time discretization, an implicit method was chosen, with a constant timestep equivalent to 0.1º crankshaft angle. An explicit method could also have been chosen, but implicit methods are unconditionally stables and allow greater time steps. Concerning the pressure-velocity coupling, the PISO algorithm was employed because it is more recommended for transient calculations than the SIMPLE algorithm (Versteeg, 1995). A second order scheme was chosen for discretization of the continuity, momentum, energy and mass fraction equations.

Both the grid and time step sensibility were studied and it was verified that the size of the computational mesh and time increment are adequate to obtain results that are insensitive to further refinement of numerical parameters. In order to ensure this grid independence, several calculations with different mesh sizes and time step sizes were compared.

#### **5. Results**

38 Numerical Modelling

Fig. 11. (a) In-cylinder pressure experimentally measured and obtained from the ideal Otto

In this case, the software ANSYS Fluent 6.3 was employed. This is based on the finite volume method. Concerning the time discretization, an implicit method was chosen, with a constant timestep equivalent to 0.1º crankshaft angle. An explicit method could also have been chosen, but implicit methods are unconditionally stables and allow greater time steps. Concerning the pressure-velocity coupling, the PISO algorithm was employed because it is more recommended for transient calculations than the SIMPLE algorithm (Versteeg, 1995). A second order scheme was chosen for discretization of the continuity, momentum, energy

Both the grid and time step sensibility were studied and it was verified that the size of the computational mesh and time increment are adequate to obtain results that are insensitive to further refinement of numerical parameters. In order to ensure this grid independence,

several calculations with different mesh sizes and time step sizes were compared.

cycle. (b) In-cylinder temperature obtained from the ideal Otto cycle.

**4.3 Resolution of the equations** 

and mass fraction equations.

#### **5.1 Pressure field and validation of the code**

In order to ensure that the CFD model is accurate enough, numerical results were compared to experimental ones. Particularly, the in-cylinder gauge pressure was validated. For the interval of time studied, from 90º to 270º crankshaft angles, the numerical and experimental results are shown in Fig. 12. Note that an acceptable concordance is obtained between CFD and experimental results.

Fig. 12. In-cylinder pressure numerically and experimentally obtained.

Figure 13 shows the gauge pressure field at several crank angles. As can be seen, the initial in-cylinder pressure, Fig. 13 (a), is 4.26 bar. As mentioned before, the intake and exhaust pressures are variable, imposed as boundary conditions at the intake and exhaust ports. At the beginning of the simulation, the pressure descends drastically due to the expansion of the piston (note that the arrows indicate the direction of the piston). When the ports are opened, Fig. 13 (b) and (c), the in-cylinder pressure is slightly superior to the exhaust pressure and slightly inferior to the intake pressure, therefore burnt gasses are expelled through the exhaust port and fresh air enters through the scavenge ports. Finally, when all ports are closed, Fig. 13 (d), the piston is ascending and the gasses are compressed, Fig. 13 (d).

Simulation of the Scavenging Process in Two-Stroke Engines 41

The mass fraction field of air of the engine described in Fig. 1 (b) and Fig. 2 (b) is shown in Fig. 15. As can be seen, fresh air (red color) enters through the inlet ports situated at the bottom part of the cylinder and burnt gases (blue color) are expelled through the exhaust

 90º 130º 150º 180º 210º 230º 270º Fig. 15. Mass fraction field of air for several crankshaft positions. Lamas & Rodríguez (2012).

Fig. 16 shows the velocity field at 92.5º and 190ºcrankshaft angles. It is represented in a cross plane containing the auxiliary transfer port and the exhaust port. As the intake and exhaust ports are opened, fresh charge flows to the cylinder through the scavenge ports and exhaust

(a) (b)

Fig. 16. Velocity field (m/s). (a) 92º crankshaft angle; (b) 190º crankshaft angle.

valve situated at the top part of the cylinder.

gasses are expelled thought the exhaust port.

**5.3 Velocity field** 

Fig. 13. Pressure field [bar]. (a) 92º crank angle; (b) 190º crank angle; (c) 215º crank angle; (d) 270º crank angle. Lamas-Galdo *et al.* (2011).

#### **5.2 Mass fraction field**

The mass fraction field is shown in Fig. 14. Four positions were represented, 92.5º, 190º, 215º and 270º crank angles. Initially, the cylinder is full of burned gases (blue color), Fig. 14 (a). When the scavenging process begins, the fresh air charge (red color) throws away the burned gases out the cylinder, Fig. 14 (b) and (c). At the end of the process, Fig. 14 (d), the cylinder is full of fresh air charge.

Fig. 14. Mass fraction field [-]. (a) 92º crank angle; (b) 190º crank angle; (c) 215º crank angle; (d) 270º crank angle. Lamas-Galdo *et al.* (2011).

A very important advantage of CFD codes over experimental setups is that it is very easy to compute the portion of burnt gases which could not be expelled. In this work, it was quantified by means of the scavenging efficiency. This indicates the mass of delivered air that was trapped by comparison with the total mass of air and fresh charge that was retained at exhaust closure, Ec. (13), and its value was 82.5 for the parameters studied.

$$\eta = \frac{\text{mass of delivered air retained}}{\text{mass of mixture in the cylinder}} \tag{13}$$

The mass fraction field of air of the engine described in Fig. 1 (b) and Fig. 2 (b) is shown in Fig. 15. As can be seen, fresh air (red color) enters through the inlet ports situated at the bottom part of the cylinder and burnt gases (blue color) are expelled through the exhaust valve situated at the top part of the cylinder.

Fig. 15. Mass fraction field of air for several crankshaft positions. Lamas & Rodríguez (2012).

#### **5.3 Velocity field**

40 Numerical Modelling

 (a) (b) (c) (d) Fig. 13. Pressure field [bar]. (a) 92º crank angle; (b) 190º crank angle; (c) 215º crank angle; (d)

The mass fraction field is shown in Fig. 14. Four positions were represented, 92.5º, 190º, 215º and 270º crank angles. Initially, the cylinder is full of burned gases (blue color), Fig. 14 (a). When the scavenging process begins, the fresh air charge (red color) throws away the burned gases out the cylinder, Fig. 14 (b) and (c). At the end of the process, Fig. 14 (d), the

 (a) (b) (c) (d) Fig. 14. Mass fraction field [-]. (a) 92º crank angle; (b) 190º crank angle; (c) 215º crank angle;

A very important advantage of CFD codes over experimental setups is that it is very easy to compute the portion of burnt gases which could not be expelled. In this work, it was quantified by means of the scavenging efficiency. This indicates the mass of delivered air that was trapped by comparison with the total mass of air and fresh charge that was retained at exhaust closure, Ec. (13), and its value was 82.5 for the parameters studied.

> *mass of delivered air retained mass of mixture in the cylinder*

(13)

270º crank angle. Lamas-Galdo *et al.* (2011).

(d) 270º crank angle. Lamas-Galdo *et al.* (2011).

**5.2 Mass fraction field** 

cylinder is full of fresh air charge.

Fig. 16 shows the velocity field at 92.5º and 190ºcrankshaft angles. It is represented in a cross plane containing the auxiliary transfer port and the exhaust port. As the intake and exhaust ports are opened, fresh charge flows to the cylinder through the scavenge ports and exhaust gasses are expelled thought the exhaust port.

(a) (b)

Fig. 16. Velocity field (m/s). (a) 92º crankshaft angle; (b) 190º crankshaft angle.

Simulation of the Scavenging Process in Two-Stroke Engines 43

In the present chapter, a CFD analysis was carried out to study the scavenging process of two-stroke engines. The results were satisfactory compared to experimental data. In general, this study shows that CFD predictions yield reasonably accurate results that allow

This model is very useful to design the scavenging system of new two-stroke engines. The pressure field is useful for identifying areas where the gas flow is inefficient and should be corrected. The velocity field is useful for locating areas with too high, too low or inadequate orientation velocities. Finally, the mass fraction field is useful for checking the filling of fresh

Finally, it is very important to mention the disadvantages of CFD. First of all, a 3D CFD model is very tedious due to the large computational resources. Besides, the moving mesh required to simulate the movement of the piston is too computationally expensive to solve. Other disadvantage is that it must not be applied blindly as it has the capability to produce non-physical results due to erroneous modeling. The process of verification and validation of a CFD model is necessary to ensure the numerical model accurately captures the physical phenomena present. By comparing numerically obtained results with experimental results, confidence in the numerical model is achieved. Once thoroughly validated, a numerical model may be used to accurately predict the effect of design changes and experimentally

Ahmadi-Befrui, B.; Brandstatter, W.; Kratochwill, H. (1989). Multidimensional calculation of the flow processes in a loop-scavenged two-stroke cycle engine*. SAE Paper 890841*. Albanesi A., Destefanis C, Zanotti A. (2009) Intake port shape optimization in a four-valve high performance engine. *Mecánica Computacional.* Vol. 28, pp. 1355-1370. Amsden, A. A.; O´Rourke, P. J.; Butler, T. D.; Meintjes, K. and Fansler, T. D. Comparisons of

Blair G.P. (1996). *Design and Simulation of Two-Stroke Engines*. SAE International. ISBN 978-1-

Carpenter, M. H.; Ramos, J. I. (1986). Modelling a gasoline-injected two-stroke cycle engine.

Creaven J.P., Kenny K.G., Cunningham G. (2001). A computational and experimental study

Epstein, P. H.; Reitz, R. D. and Foster, D. E. (1991). Computations of two-stroke cylinder and

Hariharan Ramamoorthy, Mahalakshmi N. V., Krishnamoorthy Jeyachandran. (2009).

simulation. *International Journal of Applied Engineering Research*. Vol. 4-11. Hori, H.; Ogawa, T. and Toshihiko, K. (1985). CFD in-cylinder flow simulation of an engine

computed and measured three-dimensional velocity fields in a motored two-stroke

of the scavenging flow in the transfer duct of a motored two-stroke cycle engine.

Setting up a comprehensive CFD model of a small two stroke engine for

gases into the cylinder and detecting problems of short circuiting and gas drag.

improving the knowledge of the fluid flow characteristics.

**6. Conclusions** 

unobservable phenomena.

engine. SAE Paper 920418, 1992.

*Proc Instn Mech Engrs.* Vol.215-D.

port scavenging. *SAE Paper 919672*.

and flow visualization. *SAE Paper 950288*.

Fluent 6.3 Documentation, 2006. Fluent Inc.

56091-685-7, USA.

*SAE Paper 860167*.

**7. References** 

#### **5.4 Temperature field**

The temperature field at various crank angles is given in Fig 17. As mentioned above, the initial temperature, obtained from the ideal thermodynamic Otto cycle, was imposed as 1027 K, Fig. 17 (a). At the beginning of the simulation, the in-cylinder temperature descends due to the expansion of the piston. When the ports are opened, Fig. 17 (b) and (c), the temperature descends again because fresh air at 300 K enters through the scavenge ports and hot exhaust gases are expelled. At the end of the simulation all the ports are closed and the piston is rising. The compression of the piston makes the temperature increase. Finally, the in-cylinder average temperature at the end of the simulation, Fig. 17 (d), is 677 K.

Fig. 17. Temperature field. (a) 92.5º crank angle; (b) 190º crank angle; (c) 215º crank angle; (d) 270º crank angle.

The in-cylinder average temperature and heat transfer from 90º to 270º crankshaft angles is shown in Fig. 18.

Fig. 18. In-cylinder average temperatura and heat transfer.

#### **6. Conclusions**

42 Numerical Modelling

The temperature field at various crank angles is given in Fig 17. As mentioned above, the initial temperature, obtained from the ideal thermodynamic Otto cycle, was imposed as 1027 K, Fig. 17 (a). At the beginning of the simulation, the in-cylinder temperature descends due to the expansion of the piston. When the ports are opened, Fig. 17 (b) and (c), the temperature descends again because fresh air at 300 K enters through the scavenge ports and hot exhaust gases are expelled. At the end of the simulation all the ports are closed and the piston is rising. The compression of the piston makes the temperature increase. Finally,

the in-cylinder average temperature at the end of the simulation, Fig. 17 (d), is 677 K.

 (a) (b) (c) (d) Fig. 17. Temperature field. (a) 92.5º crank angle; (b) 190º crank angle; (c) 215º crank angle; (d)

The in-cylinder average temperature and heat transfer from 90º to 270º crankshaft angles is

90 120 150 180 210 240 270

**Crankshaft angle (deg)**

0.000

0.025

0.050

**Heat transfer (J/s)**

0.075

0.100

In-cylinder temperature

Heat transfer

**5.4 Temperature field** 

270º crank angle.

shown in Fig. 18.

300

Fig. 18. In-cylinder average temperatura and heat transfer.

500

700

**Temperature (K)**

900

1100

In the present chapter, a CFD analysis was carried out to study the scavenging process of two-stroke engines. The results were satisfactory compared to experimental data. In general, this study shows that CFD predictions yield reasonably accurate results that allow improving the knowledge of the fluid flow characteristics.

This model is very useful to design the scavenging system of new two-stroke engines. The pressure field is useful for identifying areas where the gas flow is inefficient and should be corrected. The velocity field is useful for locating areas with too high, too low or inadequate orientation velocities. Finally, the mass fraction field is useful for checking the filling of fresh gases into the cylinder and detecting problems of short circuiting and gas drag.

Finally, it is very important to mention the disadvantages of CFD. First of all, a 3D CFD model is very tedious due to the large computational resources. Besides, the moving mesh required to simulate the movement of the piston is too computationally expensive to solve. Other disadvantage is that it must not be applied blindly as it has the capability to produce non-physical results due to erroneous modeling. The process of verification and validation of a CFD model is necessary to ensure the numerical model accurately captures the physical phenomena present. By comparing numerically obtained results with experimental results, confidence in the numerical model is achieved. Once thoroughly validated, a numerical model may be used to accurately predict the effect of design changes and experimentally unobservable phenomena.

#### **7. References**


**3** 

*Greece* 

**3D Multiphase Numerical Modelling** 

A. Georgoulas, P. Angelidis, K. Kopasakis and N. Kotsovinos

Gravity or density currents constitute a large class of natural flows that are generated and driven by the density difference between two or even more fluids. The density difference between two fluids usually arises due to differences in temperature or salinity, but it can also arise due to the presence of suspended solid particles. These particulate currents, in the case of sediment laden water that enters a water basin, are classified according to the density difference with the ambient fluid into three major categories: a) hypopycnal currents, when the density of the sediment laden water is lower than that of the receiving water basin, b) homopycnal currents, when the density of the sediment laden water is almost equal to that of the receiving water basin, and c) hyperpycnal currents when their density is much greater than that of the receiving water body (Mulder & Alexander, 2001). In the case of floods, the suspended sediment concentration of river water rises to a great extent. Hence, the river plunges to the bottom of the receiving basin and forms a hyperpycnal plume which is also known as turbidity current. Such flows are usually formed at river mouths in oceans, lakes or reservoirs, and can travel remarkable distances transferring, eroding and depositing large amounts of suspended sediments (Mulder &

Turbidity currents are very difficult to be observed and studied in the field. This is due to their rare and unexpected occurrence nature, as they are usually formed during floods. Therefore, field investigations are usually limited to the study of the deposits originating from such currents. The anatomy of deposits originating from turbidity currents can be studied on a large scale, in order to identify the various depositional elements such as lobes, levees and submarine channels (Janbu et al., 2009). Furthermore, considerable research on the morphology of turbiditic systems and general deep-marine depositions is being increasingly

done with the use of 3D seismic sections (Posamentier & Kolla, 2003; Saller et al., 2006).

On the other hand, scaled laboratory experiments constitute an alternative and widely used method for simulating and studying the dynamics of turbidity currents. Many researchers have been focused in the study of the flow dynamics, depositional and erosional characteristics of laboratory turbidity currents, using scaled experimental models (Britter & Linden, 1980; Lovell, 1971; Garcia & Parker, 1989; Simpson & Britter, 1979). Advances in experimental technology in the last decades have increased the existing knowledge from

**1. Introduction** 

Alexander, 2001).

**for Turbidity Current Flows** 

*Democritus University of Thrace, Xanthi* 


### **3D Multiphase Numerical Modelling for Turbidity Current Flows**

A. Georgoulas, P. Angelidis, K. Kopasakis and N. Kotsovinos *Democritus University of Thrace, Xanthi Greece* 

#### **1. Introduction**

44 Numerical Modelling

Kato S., Nakagawa H., Kawahara Y., Adachi T., Nakashima M. (1991) Numerical analysis of

Laimböck, F. J.; Meist, G. and Grilc, S. (1998). CFD application in compact engine

Lamas-Galdo, M.; Rodríguez-Vidal, C.; Rodríguez-García, J.; Fernández-Quintás, M. (2011).

Payri F., Benajes J., Margot X. et al. (2004). CFD modeling of the in-cylinder flow in direct-

Pitta S. R., Kuderu R. (2008). A computational fluid dynamics analysis on stratified

Raghunathan, B. D. and Kenny, R. G. (1997). CFD simulation and validation of the flow

Rahman M.M., Hamada K.I., Noor M.M. et al. (2010) Heat transfer characteristics of intake

Sher, E. (1989). An improved gas dynamic model simulating the scavenging process in a

Sweeny, M. E. G.; Kenny, R. G.; Swann, G. B. G. and Blair, G. P. (1985). Computational fluid dynamics applied to two-stroke engine scavenging. *SAE Paper 851519*. Yu, L.; Campbell, T. and Pollock, W. (1997). A simulation model for direct-fuel-injection of

Zahn, W.; Rosskamp, H.; Raffenberg, M. and Klimmek, A. (2000). Analysis of a stratified

Zancanaro F.V., Vielmo H.A. (2010) Numerical analysis of the fluid flow in a high swirled

charging concept for high-performance two-stroke engine. *SAE Paper 2000-01-0900*.

diesel engine. *Proceedings of the 7th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics*. Antalya-Turkey, 19-21 July 2010, pp. 387-392.

injection diesel engines. *Computers & Fluids*. Vol.33 p.995-1021.

within a motored two-stroke engine. *SAE Paper 970359*.

*Journal*. Vol. 34-3, pp. 385-390.

Research.

development. *SAE Paper 982016*.

*Thermal Science*. Vol. 12-1, pp. 33-42.

two-stroke cycle engine. *SAE Paper 800037*.

two-stroke gasoline engines. *SAE Paper 970367*.

Vol.10-18, pp. 2019-2026.

the scavenging flow in a two stroke- cycle gasoline engine*. JSME International* 

Modelo de Mecánica de Fluidos Computacional para el proceso de barrido en un motor Otto de dos tiempos. DYNA Ingeniería e Industria, vol. 86-2, pp. 165-172. Lamas, M. I.; Rodríguez, C. G. (2012) CFD analysis of the scavenging process in the MAN

B&W 7S50MC two-stroke diesel marine engine. Submitted to Journal of Ship

scavenging system of medium capacity two-stroke internal combustion engines.

port for spark ignition engine: A comparative study. *Journal of applied sciences*.

Gravity or density currents constitute a large class of natural flows that are generated and driven by the density difference between two or even more fluids. The density difference between two fluids usually arises due to differences in temperature or salinity, but it can also arise due to the presence of suspended solid particles. These particulate currents, in the case of sediment laden water that enters a water basin, are classified according to the density difference with the ambient fluid into three major categories: a) hypopycnal currents, when the density of the sediment laden water is lower than that of the receiving water basin, b) homopycnal currents, when the density of the sediment laden water is almost equal to that of the receiving water basin, and c) hyperpycnal currents when their density is much greater than that of the receiving water body (Mulder & Alexander, 2001). In the case of floods, the suspended sediment concentration of river water rises to a great extent. Hence, the river plunges to the bottom of the receiving basin and forms a hyperpycnal plume which is also known as turbidity current. Such flows are usually formed at river mouths in oceans, lakes or reservoirs, and can travel remarkable distances transferring, eroding and depositing large amounts of suspended sediments (Mulder & Alexander, 2001).

Turbidity currents are very difficult to be observed and studied in the field. This is due to their rare and unexpected occurrence nature, as they are usually formed during floods. Therefore, field investigations are usually limited to the study of the deposits originating from such currents. The anatomy of deposits originating from turbidity currents can be studied on a large scale, in order to identify the various depositional elements such as lobes, levees and submarine channels (Janbu et al., 2009). Furthermore, considerable research on the morphology of turbiditic systems and general deep-marine depositions is being increasingly done with the use of 3D seismic sections (Posamentier & Kolla, 2003; Saller et al., 2006).

On the other hand, scaled laboratory experiments constitute an alternative and widely used method for simulating and studying the dynamics of turbidity currents. Many researchers have been focused in the study of the flow dynamics, depositional and erosional characteristics of laboratory turbidity currents, using scaled experimental models (Britter & Linden, 1980; Lovell, 1971; Garcia & Parker, 1989; Simpson & Britter, 1979). Advances in experimental technology in the last decades have increased the existing knowledge from

3D Multiphase Numerical Modelling for Turbidity Current Flows 47

Most of these previous CFD-based investigations treat turbidity currents with a quasisingle-phase approach, since the transport of sediment particles is taken into account through an advection-diffusion equation for sediment concentration. The present chapter aims to present the validity, usefulness and applicability of a three-dimensional, "uncommon", CFD-based, multiphase numerical approach for the simulation and study of the hydrodynamic and depositional characteristics of turbidity currents that are usually formed at river outflows in the sea, lakes and reservoirs. The numerical model is based in a multiphase modification of the Reynolds Averaged Navier-Stokes Equations (RANS). Turbulence closure is achieved through the application of the RNG (Renormalization-Group) k-ε turbulence model. The calculations of the model are performed using the robust CFD solver FLUENT. The proposed numerical model for the simulation of turbidity current

In the present section of the chapter (Section 1) a brief introduction on turbidity currents and a literature review on field, experimental and numerical studies are conducted while the main aim of the chapter is also stated. In Section 2 the theoretical background of the proposed numerical approach is presented and discussed in detail, while in Section 3 some main validation results are presented (Georgoulas et al., 2010). Section 4 presents the results of a laboratory-scale (Georgoulas, 2010) and a field scale (Georgoulas et al, 2009) application of the numerical approach. Finally, in Section 5 the main concussions that are withdrawn

Turbidity current flows can be characterized as multiphase flow systems, since they consist of a primary fluid phase (water) and secondary granular phases (suspended sediment classes) dispersed into the primary phase. Therefore, turbidity currents can be modeled through the application of suitable multiphase numerical models. Since, the particulate loading of turbidity currents may vary from small to considerably large values, an Eulerian-Eulerian multiphase numerical approach is considered to be more appropriate, as it can handle a wider range of particle volume fractions than an Eulerian-Lagrangian approach (maximum particles volume fraction of 10-12%). FLUENT provides various multiphase models that are based in the Eulerian-Eulerian approach. The "Eulerian" model that has been selected for the numerical approach that is presented in the present chapter, may require more computational effort, but it can handle a wider range of particulate loading values and is more accurate than the other available multiphase models in FLUENT. In this multiphase model, the different phases are treated mathematically as interpenetrating continua and therefore the concept of phasic volume fraction is introduced, where the volume fraction of each phase is assumed to be a continuous function of space and time. The sum of the volume fractions of the various phases is equal to unity. An accordingly modified set of momentum and continuity equations for each phase is solved. Pressure and inter-phase exchange coefficients are used in order to achieve coupling for these equations

The motion of the suspended sediment particles within a turbidity current as well as the motion generated in the ambient fluid are of highly turbulent nature. In order to account for

hydrodynamics was firstly introduced in the work of (Georgoulas et al., 2010).

from the present chapter are summarized.

**2. Numerical model description** 

**2.1 Overview** 

(Georgoulas et al., 2010).

macroscopic and qualitative descriptions of turbidity current behaviour and deposits, to detailed, quantitative results relating to the actual flow characteristics, such as the velocity, concentration as well as the turbulence structure of such flows (Baas et al., 2004; Garcia, 1994; Gladstone et al., 1998; Kneller et al., 1997).

Mathematical and numerical models when properly designed and tested against field or laboratory data, can provide significant knowledge for turbidity current dynamics as well as for erosional and depositional characteristics. Up to present, there are various numerical investigations dealing with turbidity current dynamics and flow characteristics, providing valuable results regarding these complex phenomena. The characteristics of a gravitycurrent head have been studied by (Hartel et al., 2000), using 3D Direct Numerical Simulations (DNS) of flow fronts in the lock-exchange configuration. (Kassem & Imran, 2001) present a 2D numerical approach for investigating the transformation of a plunging river flow into a turbidity current. In the work of (Heimsund et al., 2002) a computational, 3D, fluid-dynamics model for sediment transport, erosion and deposition by turbidity currents has been constructed using the CFD (Computational Fluid Dynamics) software Flow-3D. Another 3D numerical model using the CFX-4 code was developed, in order to simulate turbidity currents in Lake Lugano (Switzerland), in the work of (Lavelli et al., 2002). (Necker et al., 2002) presented 2D and 3D Direct Numerical Simulations of particledriven gravity currents, placing special emphasis on the sedimentation of particles, and the influence of particle settling on the flow dynamics. (Cantero et al., 2003) present two and three-dimensional CFD simulations of a discontinuous density current, using a stabilized equal-order finite element method. A comparative study on the convergence of CFD commercial codes, when simulating dense underflows is presented by (Bombardelli et al., 2004). Two codes are used for the proposed simulations: the first one is a comprehensive finite-element platform, whereas the other one is a commercial code. The lateral development of density-driven flow in a subaqueous channel is studied using a 3D numerical model, in the work of (Imran et al., 2004). The conditions under which turbidity currents may become self-sustaining through particle entrainment are investigated in the work of (Blanchette et al., 2005), using 2D Direct Numerical Simulations of resuspending gravity currents. A numerical model of turbidity currents with a deforming bottom boundary, that predicts the vertical structure of the flow velocity and concentration as well as the change in the bed level, due to erosion and deposition of suspended sediment, is developed in the work of (Huang et al., 2005). Lock-exchange gravity current flows, produced by the instantaneous release of a heavy fluid, are investigated by means of 2D Large-Eddy Simulation (LES) in the work of (Ooi et al., 2007). A numerical simulation of turbidity current using the *v <sup>f</sup>* <sup>2</sup> turbulence model is carried out in the work of (Mehdizadeh et al., 2008). (Cantero et al., 2008a), perform 2D Direct Numerical Simulations in order to investigate the effect of particle inertia on the dynamics of particulate gravity currents. They introduce an Eulerian-Eulerian formulation for gravity currents driven by inertial particles. 3D Direct Numerical Simulations of planar gravity currents have been conducted with the objective of identifying, visualizing and describing turbulent structures and their influence on flow dynamics, in the work of (Cantero et al., 2008b). The investigation of the effect of initial aspect ratio on the flow characteristics of suspension gravity currents as well as the diffusion of the turbidity under the presence of a turbidity fence is carried out in the work of (Singh, 2008), using 3D Large Eddy Simulations.

Most of these previous CFD-based investigations treat turbidity currents with a quasisingle-phase approach, since the transport of sediment particles is taken into account through an advection-diffusion equation for sediment concentration. The present chapter aims to present the validity, usefulness and applicability of a three-dimensional, "uncommon", CFD-based, multiphase numerical approach for the simulation and study of the hydrodynamic and depositional characteristics of turbidity currents that are usually formed at river outflows in the sea, lakes and reservoirs. The numerical model is based in a multiphase modification of the Reynolds Averaged Navier-Stokes Equations (RANS). Turbulence closure is achieved through the application of the RNG (Renormalization-Group) k-ε turbulence model. The calculations of the model are performed using the robust CFD solver FLUENT. The proposed numerical model for the simulation of turbidity current hydrodynamics was firstly introduced in the work of (Georgoulas et al., 2010).

In the present section of the chapter (Section 1) a brief introduction on turbidity currents and a literature review on field, experimental and numerical studies are conducted while the main aim of the chapter is also stated. In Section 2 the theoretical background of the proposed numerical approach is presented and discussed in detail, while in Section 3 some main validation results are presented (Georgoulas et al., 2010). Section 4 presents the results of a laboratory-scale (Georgoulas, 2010) and a field scale (Georgoulas et al, 2009) application of the numerical approach. Finally, in Section 5 the main concussions that are withdrawn from the present chapter are summarized.

#### **2. Numerical model description**

#### **2.1 Overview**

46 Numerical Modelling

macroscopic and qualitative descriptions of turbidity current behaviour and deposits, to detailed, quantitative results relating to the actual flow characteristics, such as the velocity, concentration as well as the turbulence structure of such flows (Baas et al., 2004; Garcia,

Mathematical and numerical models when properly designed and tested against field or laboratory data, can provide significant knowledge for turbidity current dynamics as well as for erosional and depositional characteristics. Up to present, there are various numerical investigations dealing with turbidity current dynamics and flow characteristics, providing valuable results regarding these complex phenomena. The characteristics of a gravitycurrent head have been studied by (Hartel et al., 2000), using 3D Direct Numerical Simulations (DNS) of flow fronts in the lock-exchange configuration. (Kassem & Imran, 2001) present a 2D numerical approach for investigating the transformation of a plunging river flow into a turbidity current. In the work of (Heimsund et al., 2002) a computational, 3D, fluid-dynamics model for sediment transport, erosion and deposition by turbidity currents has been constructed using the CFD (Computational Fluid Dynamics) software Flow-3D. Another 3D numerical model using the CFX-4 code was developed, in order to simulate turbidity currents in Lake Lugano (Switzerland), in the work of (Lavelli et al., 2002). (Necker et al., 2002) presented 2D and 3D Direct Numerical Simulations of particledriven gravity currents, placing special emphasis on the sedimentation of particles, and the influence of particle settling on the flow dynamics. (Cantero et al., 2003) present two and three-dimensional CFD simulations of a discontinuous density current, using a stabilized equal-order finite element method. A comparative study on the convergence of CFD commercial codes, when simulating dense underflows is presented by (Bombardelli et al., 2004). Two codes are used for the proposed simulations: the first one is a comprehensive finite-element platform, whereas the other one is a commercial code. The lateral development of density-driven flow in a subaqueous channel is studied using a 3D numerical model, in the work of (Imran et al., 2004). The conditions under which turbidity currents may become self-sustaining through particle entrainment are investigated in the work of (Blanchette et al., 2005), using 2D Direct Numerical Simulations of resuspending gravity currents. A numerical model of turbidity currents with a deforming bottom boundary, that predicts the vertical structure of the flow velocity and concentration as well as the change in the bed level, due to erosion and deposition of suspended sediment, is developed in the work of (Huang et al., 2005). Lock-exchange gravity current flows, produced by the instantaneous release of a heavy fluid, are investigated by means of 2D Large-Eddy Simulation (LES) in the work of (Ooi et al., 2007). A numerical simulation of turbidity current using the *v <sup>f</sup>* <sup>2</sup> turbulence model is carried out in the work of (Mehdizadeh et al., 2008). (Cantero et al., 2008a), perform 2D Direct Numerical Simulations in order to investigate the effect of particle inertia on the dynamics of particulate gravity currents. They introduce an Eulerian-Eulerian formulation for gravity currents driven by inertial particles. 3D Direct Numerical Simulations of planar gravity currents have been conducted with the objective of identifying, visualizing and describing turbulent structures and their influence on flow dynamics, in the work of (Cantero et al., 2008b). The investigation of the effect of initial aspect ratio on the flow characteristics of suspension gravity currents as well as the diffusion of the turbidity under the presence of a turbidity

fence is carried out in the work of (Singh, 2008), using 3D Large Eddy Simulations.

1994; Gladstone et al., 1998; Kneller et al., 1997).

Turbidity current flows can be characterized as multiphase flow systems, since they consist of a primary fluid phase (water) and secondary granular phases (suspended sediment classes) dispersed into the primary phase. Therefore, turbidity currents can be modeled through the application of suitable multiphase numerical models. Since, the particulate loading of turbidity currents may vary from small to considerably large values, an Eulerian-Eulerian multiphase numerical approach is considered to be more appropriate, as it can handle a wider range of particle volume fractions than an Eulerian-Lagrangian approach (maximum particles volume fraction of 10-12%). FLUENT provides various multiphase models that are based in the Eulerian-Eulerian approach. The "Eulerian" model that has been selected for the numerical approach that is presented in the present chapter, may require more computational effort, but it can handle a wider range of particulate loading values and is more accurate than the other available multiphase models in FLUENT. In this multiphase model, the different phases are treated mathematically as interpenetrating continua and therefore the concept of phasic volume fraction is introduced, where the volume fraction of each phase is assumed to be a continuous function of space and time. The sum of the volume fractions of the various phases is equal to unity. An accordingly modified set of momentum and continuity equations for each phase is solved. Pressure and inter-phase exchange coefficients are used in order to achieve coupling for these equations (Georgoulas et al., 2010).

The motion of the suspended sediment particles within a turbidity current as well as the motion generated in the ambient fluid are of highly turbulent nature. In order to account for

where,

and αq is the volume fraction of phase q.

where ρq is the physical density of phase q.

*n*

*p*

*N*

*l*

pressure shared by all phases,

phases. The stress-strain tensors

the solution domain,

force, *Flift*,*<sup>q</sup>*

1

1

*q* 

is a lift force and *Fvm*,*<sup>q</sup>*

 

 

The effective density of phase q is:

Documentation, 2010):

3D Multiphase Numerical Modelling for Turbidity Current Flows 49

(2)

(3)

(4)

 

is a virtual mass force. Kls = Ksl is the momentum

3

 

*<sup>s</sup>* are calculated by the following relationships:

 

(7)

(5)

(6)

is the gravitational

is an external body

is the velocity of phase p, p is the

 

*n q q* 1 1

*q qq* 

*rq t*

( )( )

( )( )

 

 

*p q q lift q vm q pq*

*K FF F* , ,

*l s s lift s vm s ls*

is the velocity of phase q,

*<sup>T</sup>*

 

*K FF F* , ,

acceleration, Kpq is the interphase momentum exchange coefficient, *Fq*

 *q* and 

> 

( ( )) ( )

( ( )) ( )

The continuity, the fluid-fluid, and fluid-solid momentum equations that are actually solved by the model are described by equations (4), (5) and (6) respectively, for the general case of a n-phase flow consisting of granular and non-granular secondary phases (ANSYS FLUENT

*q qq qq*

 

 

> 

<sup>1</sup> ( ) ( )0

*q q qqq qq q q qq*

*s s sss ss s s s ss*

where ρrq is the phase reference density, or the volume averaged density of the qth phase in

exchange coefficient between fluid phase l and solid phase s and N is the total number of

*q qq <sup>q</sup> q q qq q <sup>I</sup>* <sup>2</sup>

 

 

 *p* 

*<sup>q</sup>* is the qth phase stress-strain tensor, *g*

*<sup>p</sup> p g <sup>t</sup>*

*<sup>p</sup> <sup>g</sup> <sup>t</sup>*

the effect of turbulence in the numerical simulations of the present investigation, the instantaneous governing equations are not applied directly but they are ensemble-averaged, converting turbulent fluctuations into Reynolds stresses, which represent the effects of turbulence. This averaging procedure for the numerical simulation of turbulent flows is known as RANS (Reynolds-averaged Navier-Stokes equations). The averaged governing equations contain additional unknown variables, and turbulence models are needed to determine these variables in terms of known quantities. Therefore, with this averaging approach the turbulence is modeled and only the unsteady, mean flow structures that are primarily larger than the turbulent eddies are resolved. This is the main difference with the other two widely used numerical approaches for turbulent flows, known as DNS (Direct Numerical Simulation) and LES (Large Eddy Simulation). In DNS, the Navier-Stokes equations are applied and solved directly without the application of a turbulence model, resolving the whole range of turbulent eddies. In LES on the other hand, large eddies are resolved directly, while small eddies are modeled. DNS and LES may provide detailed information on turbidity current flows but their major disadvantage is that their application is limited due to large computational requirements. On the other hand, RANS may not provide detailed information from a microscopic point of view, but is quite accurate and attractive for modeling large scale, three dimensional flows of practical engineering interest due to the relatively low computational cost (Georgoulas et al., 2010).

In the numerical approach presented here, the Renormalization-group (RNG) k-ε model is applied for turbulence closure. This model was derived using a rigorous statistical technique, the renormalization group theory. The basic form of the RNG k-ε model is similar to the standard k-ε model, but it includes a number of refinements, rendering it more appropriate for the case of turbidity currents, as it is more accurate for swirling flows and rapidly strained flows and also accounts for low Reynolds number effects. Moreover, it provides an analytical formula for the calculation of the turbulent Prandtl numbers. At this point it should be mentioned that in the present numerical approach, the RNG k-ε model is also modified accordingly in order to simultaneously account for the primary (continuous) phase and the secondary (dispersed) phases of the simulated flows. This modification in FLUENT is based on a number of assumptions. In more detail, turbulent predictions for the continuous phase are obtained using the RNG k-ε model, supplemented with extra terms that include the interphase turbulent momentum transfer. Predictions for turbulence quantities for the dispersed phases are obtained using the Tchen theory of dispersion of discrete particles by homogeneous turbulence. Interphase turbulent momentum transfer is also assumed, in order to take into account the dispersion of the secondary phases transported by the turbulent fluid motion. Finally, a phase-weighted averaging process is assumed, so that no volume fraction fluctuations are introduced into the continuity equations (Georgoulas et al., 2010).

#### **2.2 Governing equations**

The volume of phase q, Vq is defined by the following relationship (ANSYS FLUENT Documentation, 2010):

$$\left.V\right|\_{q} = \int\_{V} a\_{q}dV\tag{1}$$

where,

48 Numerical Modelling

the effect of turbulence in the numerical simulations of the present investigation, the instantaneous governing equations are not applied directly but they are ensemble-averaged, converting turbulent fluctuations into Reynolds stresses, which represent the effects of turbulence. This averaging procedure for the numerical simulation of turbulent flows is known as RANS (Reynolds-averaged Navier-Stokes equations). The averaged governing equations contain additional unknown variables, and turbulence models are needed to determine these variables in terms of known quantities. Therefore, with this averaging approach the turbulence is modeled and only the unsteady, mean flow structures that are primarily larger than the turbulent eddies are resolved. This is the main difference with the other two widely used numerical approaches for turbulent flows, known as DNS (Direct Numerical Simulation) and LES (Large Eddy Simulation). In DNS, the Navier-Stokes equations are applied and solved directly without the application of a turbulence model, resolving the whole range of turbulent eddies. In LES on the other hand, large eddies are resolved directly, while small eddies are modeled. DNS and LES may provide detailed information on turbidity current flows but their major disadvantage is that their application is limited due to large computational requirements. On the other hand, RANS may not provide detailed information from a microscopic point of view, but is quite accurate and attractive for modeling large scale, three dimensional flows of practical engineering interest

In the numerical approach presented here, the Renormalization-group (RNG) k-ε model is applied for turbulence closure. This model was derived using a rigorous statistical technique, the renormalization group theory. The basic form of the RNG k-ε model is similar to the standard k-ε model, but it includes a number of refinements, rendering it more appropriate for the case of turbidity currents, as it is more accurate for swirling flows and rapidly strained flows and also accounts for low Reynolds number effects. Moreover, it provides an analytical formula for the calculation of the turbulent Prandtl numbers. At this point it should be mentioned that in the present numerical approach, the RNG k-ε model is also modified accordingly in order to simultaneously account for the primary (continuous) phase and the secondary (dispersed) phases of the simulated flows. This modification in FLUENT is based on a number of assumptions. In more detail, turbulent predictions for the continuous phase are obtained using the RNG k-ε model, supplemented with extra terms that include the interphase turbulent momentum transfer. Predictions for turbulence quantities for the dispersed phases are obtained using the Tchen theory of dispersion of discrete particles by homogeneous turbulence. Interphase turbulent momentum transfer is also assumed, in order to take into account the dispersion of the secondary phases transported by the turbulent fluid motion. Finally, a phase-weighted averaging process is assumed, so that no volume fraction fluctuations are introduced into the continuity

The volume of phase q, Vq is defined by the following relationship (ANSYS FLUENT

*q q V V*

*dV* (1)

due to the relatively low computational cost (Georgoulas et al., 2010).

equations (Georgoulas et al., 2010).

**2.2 Governing equations** 

Documentation, 2010):

$$\sum\_{q=1}^{n} \alpha\_q = 1 \tag{2}$$

and αq is the volume fraction of phase q.

The effective density of phase q is:

$$
\stackrel{\wedge}{\rho}\_q = \alpha\_q \rho\_q \tag{3}
$$

where ρq is the physical density of phase q.

The continuity, the fluid-fluid, and fluid-solid momentum equations that are actually solved by the model are described by equations (4), (5) and (6) respectively, for the general case of a n-phase flow consisting of granular and non-granular secondary phases (ANSYS FLUENT Documentation, 2010):

$$\frac{1}{\rho\_{\eta\eta}} \left( \frac{\partial}{\partial t} (\alpha\_q \rho\_q) + \nabla \cdot (\alpha\_q \rho\_q \stackrel{\rightarrow}{\nu}\_q) = 0 \right) \tag{4}$$

$$\begin{aligned} \stackrel{\circ}{\partial}(\stackrel{\rightarrow}{\alpha\_q}\rho\_q\stackrel{\rightarrow}{\upsilon\_q}) + \nabla \cdot (\stackrel{\rightarrow}{\alpha\_q}\rho\_q\stackrel{\rightarrow}{\upsilon\_q}\stackrel{\rightarrow}{\upsilon\_q}) &= -\alpha\_q\nabla p + \nabla \cdot \stackrel{\rightarrow}{\tau\_q} + \alpha\_q\rho\_q\stackrel{\rightarrow}{\varsigma} + \\ \stackrel{\circ}{\sum}(\stackrel{\rightarrow}{\mathcal{K}}(\stackrel{\rightarrow}{\upsilon\_{p\bar{q}}}\stackrel{\rightarrow}{\upsilon\_p}\stackrel{\rightarrow}{\upsilon\_q})) + \stackrel{\rightarrow}{\{F\_q + F\_{\bar{l}\bar{l}\bar{t},q} + \stackrel{\rightarrow}{F\_{\upsilon m,q}}\}} \end{aligned} \tag{5}$$

$$\begin{aligned} \stackrel{\circ}{\partial} \begin{matrix} \stackrel{\circ}{\partial} \begin{matrix} \stackrel{\circ}{\partial} \end{matrix} \stackrel{\rightarrow}{\partial} \begin{matrix} \stackrel{\circ}{\partial} \end{matrix} + \stackrel{\circ}{\nabla} \begin{matrix} \stackrel{\circ}{\partial} \end{matrix} \stackrel{\circ}{\partial} \begin{matrix} \stackrel{\circ}{\partial} \end{matrix} = \begin{matrix} \stackrel{\circ}{\partial} \end{matrix} \end{aligned} \end{aligned} \tag{6}$$

$$\begin{aligned} \stackrel{\circ}{\partial} \begin{matrix} \stackrel{\circ}{\partial} \end{matrix} \stackrel{\circ}{\partial} \begin{matrix} \stackrel{\circ}{\partial} \end{matrix} \stackrel{\circ}{\partial} \begin{matrix} \stackrel{\circ}{\partial} \end{matrix} \end{aligned} \right) = \begin{matrix} \stackrel{\circ}{\partial} \end{aligned} \tag{7}$$

$$\begin{aligned} \stackrel{\circ}{\partial} \begin{matrix} \stackrel{\circ}{\partial} \end{matrix} \stackrel{\circ}{\partial} \begin{matrix} \stackrel{\circ}{\partial} \end{matrix} \stackrel{\circ}{\partial} \begin{matrix} \stackrel{\circ}{\partial} \end{matrix} \end{aligned} \tag{6}$$

where ρrq is the phase reference density, or the volume averaged density of the qth phase in the solution domain, *q* is the velocity of phase q, *p* is the velocity of phase p, p is the pressure shared by all phases, *<sup>q</sup>* is the qth phase stress-strain tensor, *g* is the gravitational acceleration, Kpq is the interphase momentum exchange coefficient, *Fq* is an external body force, *Flift*,*<sup>q</sup>* is a lift force and *Fvm*,*<sup>q</sup>* is a virtual mass force. Kls = Ksl is the momentum exchange coefficient between fluid phase l and solid phase s and N is the total number of phases. The stress-strain tensors *q* and *<sup>s</sup>* are calculated by the following relationships:

$$
\stackrel{\blacksquare}{\tau}\_{\eta} = \alpha\_{q}\mu\_{q}\Big(\stackrel{\dashv}{\nabla\dot{\nu}\_{q}} + \stackrel{\dashv}{\nabla\dot{\nu}\_{q}}^{\top}\Big) + \alpha\_{q}\Big(\stackrel{\dashv}{\lambda\_{q}} - \frac{2}{3}\mu\_{q}\Big)\nabla\stackrel{\dashv}{\cdot\dot{\nu}\_{q}}\Big|\stackrel{\blacksquare}{\Gamma} \tag{7}
$$

3D Multiphase Numerical Modelling for Turbidity Current Flows 51

through water, as in this case the density of air is much smaller than the density of the ambient water and the added mass (by the surrounding water) in the air bubbles would be much larger than their own mass. In all of the turbidity current cases considered in the present chapter, the secondary phase density (solid particles) is larger than the primary phase density (fresh water)

The general transport equations for the turbulence kinetic energy k and the turbulence dissipation rate ε, of the RNG k-ε turbulence model, can be described by equations (11) and

> 

*i jj <sup>k</sup> k ku G G*

*u tx x x*

 

where u represents velocity, ρ is the local mixture density, Gk is the generation of turbulence kinetic energy due to mean velocity gradients, Gb is the generation of turbulence kinetic energy due to buoyancy, αk and αε are the inverse effective Prandtl numbers for k and ε respectively, μeff is the effective viscosity and C1ε, C2ε and C3ε are turbulence model constants. The term Rε in the ε equation accounts for the effects of rapid strain and

The governing equations in the proposed multiphase numerical approach are solved sequentially, using the control-volume method. Hence, the equations are integrated about each control-volume, yielding discrete equations for the conservation of each quantity. An implicit formulation is used, in order for the discretized equations to be converted to linear equations for the dependent variables in every computational cell. Further details regarding

At inlets, a velocity-inlet boundary condition is used. With this type of boundary condition, a uniform distribution of all the dependent variables is prescribed at the face representing the sediment laden water inflow. In more detail, the velocity magnitudes of the primary and secondary phases with directions normal to the inlet face are specified, assuming constant, uniform values. Moreover, the volume fractions of the secondary phases at the inlet are also

For the outlets, a pressure-outlet boundary condition is applied. Using this type of boundary condition, all flow quantities at the outlets are extrapolated from the flow in the interior domain. A set of "backflow'' conditions can be also specified, allowing reverse direction flow at the pressure outlet boundary during the solution process. In other words, this type of

the solution procedure can be found at (ANSYS FLUENT Documentation, 2010).

*C G CG C R k k*

*tx x x*

() ( )

*k b*

 

 

1 32

( )

*i k eff k b*

*i eff i jj*

 

 

2

  (11)

(12)

and therefore the virtual mass force is not taken into consideration.

(12) respectively (ANSYS FLUENT Documentation, 2010):

() ( )

streamline curvature (ANSYS FLUENT Documentation, 2010).

**2.3 Solution procedure** 

**2.4 Boundary conditions** 

specified.

$$
\overline{\dot{\boldsymbol{\tau}}}\_{s} = \boldsymbol{\alpha}\_{s} \boldsymbol{\mu}\_{s} \left( \boldsymbol{\nabla} \boldsymbol{\dot{\nu}}\_{s} + \boldsymbol{\nabla} \boldsymbol{\dot{\nu}}\_{s}^{\top} \right) + \boldsymbol{\alpha}\_{s} \left( \boldsymbol{\mathcal{A}}\_{s} - \frac{2}{3} \boldsymbol{\mu}\_{s} \right) \boldsymbol{\nabla} \cdot \boldsymbol{\vec{\mathcal{O}}}\_{s} \overline{\boldsymbol{I}} \tag{8}
$$

where, μq and μs are the shear viscosities of phases q and s, λq and λs are the bulk viscosities of phases q and s, and *I* is the identity tensor.

The momentum exchange between the various phases involved in a multiphase flow is based in the value of the interphase exchange coefficients. Therefore, these coefficients are very important for the simulation of granular multiphase flows, as turbidity currents. In the numerical approach presented here, the fluid-solid momentum exchange coefficient, between the ambient water (primary phase) and the suspended sediment particles (secondary phase) is calculated using the Syamlal-O'Brien model, which is based on measurements of the terminal velocities of particles in fluidized or settling beds. This model was selected, as a series of trial numerical runs indicated that this gives the best results, in comparison with corresponding experimental measurements, for the case of turbidity currents (Georgoulas et al., 2010). As it can be seen from Equation (6), in the case of granular flows, in the regime where the solids volume fraction is less than its maximum allowed value, a solids pressure is calculated independently and used for the pressure gradient term

( *<sup>s</sup> p* ), in the fluid-solid momentum equation. This solids pressure is composed of a kinetic term as well as a second term due to particle collisions and is calculated using the following relationship (ANSYS FLUENT Documentation, 2010):

$$p\_s = \alpha\_s \rho\_s \Theta\_s + 2\rho\_s (1 + e\_{ss}) \alpha\_s^2 g\_{0,ss} \Theta\_s \tag{9}$$

where ess is the coefficient of restitution for particle collisions, g0,ss is the radial distribution function, and Θs is the granular temperature which is proportional to the kinetic energy of the fluctuating particle motion. Trial numerical simulations indicated that the solids pressure is significant at various regions and stages of turbidity current flows (Georgoulas et al., 2010).

The effect of lift forces in the secondary phase solid particles is also taken into account. These lift forces act on particles mainly due to velocity gradients in the primary-phase flow field. The lift force will be more significant for larger particles. A main assumption is that the particle diameter is much smaller than the interparticle spacing. Hence, the inclusion of lift forces is not appropriate for closely packed particles or for very small particles. The lift force acting on a secondary phase p in a primary phase q is calculated in FLUENT, using the following equation (ANSYS FLUENT Documentation, 2010):

$$\vec{F}\_{\text{lift}} = -\mathbf{C}\_{L}\rho\_{q}\alpha\_{p}(\vec{\nu}\_{q} - \vec{\nu}\_{p}) \times (\nabla \times \vec{\nu}\_{q}) \tag{10}$$

where CL is the lift coefficient. For the turbidity current cases that are presented in the present chapter, values of the lift coefficient ranging from 0.1 to 0.5 give the best results in comparison with corresponding experimental measurements (Georgoulas et al., 2010).

The virtual mass force is usually significant, in cases where the secondary phase density is much smaller than the primary phase density (ANSYS FLUENT Documentation, 2010). For example, the virtual mass force would be significant in the case of air bubbles moving

*s ss <sup>s</sup> s s ss s <sup>I</sup>* <sup>2</sup>

where, μq and μs are the shear viscosities of phases q and s, λq and λs are the bulk viscosities

The momentum exchange between the various phases involved in a multiphase flow is based in the value of the interphase exchange coefficients. Therefore, these coefficients are very important for the simulation of granular multiphase flows, as turbidity currents. In the numerical approach presented here, the fluid-solid momentum exchange coefficient, between the ambient water (primary phase) and the suspended sediment particles (secondary phase) is calculated using the Syamlal-O'Brien model, which is based on measurements of the terminal velocities of particles in fluidized or settling beds. This model was selected, as a series of trial numerical runs indicated that this gives the best results, in comparison with corresponding experimental measurements, for the case of turbidity currents (Georgoulas et al., 2010). As it can be seen from Equation (6), in the case of granular flows, in the regime where the solids volume fraction is less than its maximum allowed value, a solids pressure is calculated independently and used for the pressure gradient term ( *<sup>s</sup> p* ), in the fluid-solid momentum equation. This solids pressure is composed of a kinetic term as well as a second term due to particle collisions and is calculated using the following

> *s s s s s ss s ss s p e g*<sup>2</sup> 0,

where ess is the coefficient of restitution for particle collisions, g0,ss is the radial distribution function, and Θs is the granular temperature which is proportional to the kinetic energy of the fluctuating particle motion. Trial numerical simulations indicated that the solids pressure is significant at various regions and stages of turbidity current flows (Georgoulas et

The effect of lift forces in the secondary phase solid particles is also taken into account. These lift forces act on particles mainly due to velocity gradients in the primary-phase flow field. The lift force will be more significant for larger particles. A main assumption is that the particle diameter is much smaller than the interparticle spacing. Hence, the inclusion of lift forces is not appropriate for closely packed particles or for very small particles. The lift force acting on a secondary phase p in a primary phase q is calculated in FLUENT, using the

> *F C lift Lqp*

where CL is the lift coefficient. For the turbidity current cases that are presented in the present chapter, values of the lift coefficient ranging from 0.1 to 0.5 give the best results in comparison with corresponding experimental measurements (Georgoulas et al., 2010).

The virtual mass force is usually significant, in cases where the secondary phase density is much smaller than the primary phase density (ANSYS FLUENT Documentation, 2010). For example, the virtual mass force would be significant in the case of air bubbles moving

 

( )( ) *qp q*

 

(10)

 

2 (1 ) (9)

 

 

3

   

(8)

*<sup>T</sup>*

 

 

of phases q and s, and *I* is the identity tensor.

relationship (ANSYS FLUENT Documentation, 2010):

following equation (ANSYS FLUENT Documentation, 2010):

al., 2010).

through water, as in this case the density of air is much smaller than the density of the ambient water and the added mass (by the surrounding water) in the air bubbles would be much larger than their own mass. In all of the turbidity current cases considered in the present chapter, the secondary phase density (solid particles) is larger than the primary phase density (fresh water) and therefore the virtual mass force is not taken into consideration.

The general transport equations for the turbulence kinetic energy k and the turbulence dissipation rate ε, of the RNG k-ε turbulence model, can be described by equations (11) and (12) respectively (ANSYS FLUENT Documentation, 2010):

$$\frac{\partial}{\partial t}(\rho k) + \frac{\partial}{\partial \mathbf{x}\_i}(\rho k u\_i) = \frac{\partial}{\partial \mathbf{x}\_j} \left( \alpha\_k \mu\_{\mathrm{eff}} \frac{\partial \mathbf{k}}{\partial \mathbf{x}\_j} \right) + \mathbf{G}\_k + \mathbf{G}\_b - \rho \varepsilon \tag{11}$$

$$\begin{aligned} \frac{\partial}{\partial t}(\rho \varepsilon) + \frac{\partial}{\partial \mathbf{x}\_i}(\rho \varepsilon \mathbf{u}\_i) &= \frac{\partial}{\partial \mathbf{x}\_j} \left( \mathbf{a}\_\varepsilon \mu\_{\varepsilon \| \mathbf{f}} \frac{\partial \varepsilon}{\partial \mathbf{x}\_j} \right) + \\ \mathbf{C}\_{1\varepsilon} \frac{\mathcal{E}}{k} (\mathbf{G}\_k + \mathbf{C}\_{3\varepsilon} \mathbf{G}\_b) &- \mathbf{C}\_{2\varepsilon} \rho \frac{\varepsilon^2}{k} - \mathbf{R}\_\varepsilon \end{aligned} \tag{12}$$

where u represents velocity, ρ is the local mixture density, Gk is the generation of turbulence kinetic energy due to mean velocity gradients, Gb is the generation of turbulence kinetic energy due to buoyancy, αk and αε are the inverse effective Prandtl numbers for k and ε respectively, μeff is the effective viscosity and C1ε, C2ε and C3ε are turbulence model constants. The term Rε in the ε equation accounts for the effects of rapid strain and streamline curvature (ANSYS FLUENT Documentation, 2010).

#### **2.3 Solution procedure**

The governing equations in the proposed multiphase numerical approach are solved sequentially, using the control-volume method. Hence, the equations are integrated about each control-volume, yielding discrete equations for the conservation of each quantity. An implicit formulation is used, in order for the discretized equations to be converted to linear equations for the dependent variables in every computational cell. Further details regarding the solution procedure can be found at (ANSYS FLUENT Documentation, 2010).

#### **2.4 Boundary conditions**

At inlets, a velocity-inlet boundary condition is used. With this type of boundary condition, a uniform distribution of all the dependent variables is prescribed at the face representing the sediment laden water inflow. In more detail, the velocity magnitudes of the primary and secondary phases with directions normal to the inlet face are specified, assuming constant, uniform values. Moreover, the volume fractions of the secondary phases at the inlet are also specified.

For the outlets, a pressure-outlet boundary condition is applied. Using this type of boundary condition, all flow quantities at the outlets are extrapolated from the flow in the interior domain. A set of "backflow'' conditions can be also specified, allowing reverse direction flow at the pressure outlet boundary during the solution process. In other words, this type of

3D Multiphase Numerical Modelling for Turbidity Current Flows 53

Details regarding the above mentioned laboratory experiments (experimental set-up, initial conditions) and their numerical reproduction (computational geometry, computational mesh, boundary conditions, etc.) can be found in the work of (Georgoulas et al., 2010). However, for the purposes of the present chapter, the key quantitative results that prove that the proposed numerical model predictions are realistic and reliable are presented and discussed in subsections 3.1 and 3.2 that follow, for the cases of the fixed-volume releases

Front speed is one of the most studied parameters for lock-exchange turbidity currents. Figure 1 compares the simulated and observed current front position versus time for all the lock-gate cases considered in the work of (Georgoulas et al., 2010). As it can be seen, in general the numerical simulations show a good match with the experimental data, adequately predicting the differences in the flow front advance among the generated currents with respect to the different relative proportions of coarse (%C in the legend) and fine particles (%F in the legend) that were used in the initial suspensions. The observed divergence between the experimental and the numerical curves at various times, might be partially attributed to possible overestimation or under-estimation of the flow front position in the particular laboratory runs, due to the difficulty in the visual definition of the exact flow front position, since these laboratory difficulties are stated in the work of (Gladstone et al., 1998). Another possible reason for the observed divergence might be the overall assumptions in the numerical simulations (e.g.

Fig. 1. Comparison of numerical (Georgoulas et al., 2010) and experimental (Gladstone et al.,

1998) results, of flow front advance with respect to time.

(Gladstone et al., 1998) and the steady-state releases (Baas et al., 2004), respectively.

**3.1 Fixed-volume releases** 

uniform grain size in each particle class).

outlet condition serves as an open flow boundary, allowing the flow to freely exit or enter the computational domain during the calculations.

At the free ambient water surfaces, a symmetry boundary condition is used, which is typically well above the generated turbidity currents. Thus, there are neither convective nor diffusive fluxes across the top surface. This type of free surface boundary condition has also been used by other researchers in literature (Imran et al., 2004; Huang et al., 2005) for the case of turbidity currents. (Farrell & Stefan, 1986) have found that for a plunging reservoir flow, the relative error that can be introduced by this approximation of the free surface, is of the order of 10-3 and does not influence the velocity field.

The solid boundaries are specified as stationary walls with a no-slip shear condition. Turbulent flows are significantly affected by the presence of walls. Very close to the wall, viscous damping and kinematic blocking reduce the tangential and normal velocity fluctuations respectively. However, in the outer part of the near wall region, the turbulence is rapidly augmented by the production of turbulent kinetic energy due to the relatively large gradients in mean velocity. In FLUENT, there are two different approaches for modeling the near wall region. In the first approach, the viscous sub-layer and the buffer sub-layer are not resolved. Instead, semi-empirical formulas known as "wall functions" (e.g. "standard wall functions") are used in order to link the viscosity affected sub-layers between the wall and the fully-turbulent region. In the second approach, known as "nearwall modeling" approach (e.g. "enhanced wall treatment), the turbulence models are modified in order for the viscosity affected near-wall regions to be resolved with a computational mesh all the way to the wall. However, the computational mesh must be significantly fine in these regions. This approach may require more computational effort, but it gives more accurate predictions at the near-wall region of the computational domain. Therefore, wall functions should only be used in cases where the complexity and size of the computational domain as well as the available computational resources, do not allow the construction of very fine meshes at the near-wall regions (ANSYS FLUENT Documentation, 2010).

#### **3. Numerical model validation**

A detailed verification of the proposed numerical model is conducted in the work of (Georgoulas et al., 2010), where two different series of published laboratory experiments on turbidity currents, conducted by (Gladstone et al., 1998) and (Baas et al., 2004) are reproduced numerically, and the results are compared aiming to evaluate how realistic and reliable the numerical simulations of the proposed model are. The first series of laboratory experiments (Gladstone et al., 1998) consist of fixed-volume lock-gate releases of dilute mixtures containing two different sizes of suspended silicon carbide particles, in various initial proportions, within a rectangular flume (Run A – Run G). The second series of laboratory experiments (Baas et al., 2004) consist of high-density sediment-water mixtures released with a steady rate, through a small inflow gate, into an inclined channel which is connected to a tank, were an expansion table covered with loose sediment is positioned. The mixtures consist of either fine sand, very fine sand or coarse silt. Apart from the suspended sediment grain size, the initial suspended sediment volume fraction, the water-sediment mixture discharge and the channel slope angle and bed roughness, are varied among the experimental runs (Run 1 – Run 14).

Details regarding the above mentioned laboratory experiments (experimental set-up, initial conditions) and their numerical reproduction (computational geometry, computational mesh, boundary conditions, etc.) can be found in the work of (Georgoulas et al., 2010). However, for the purposes of the present chapter, the key quantitative results that prove that the proposed numerical model predictions are realistic and reliable are presented and discussed in subsections 3.1 and 3.2 that follow, for the cases of the fixed-volume releases (Gladstone et al., 1998) and the steady-state releases (Baas et al., 2004), respectively.

#### **3.1 Fixed-volume releases**

52 Numerical Modelling

outlet condition serves as an open flow boundary, allowing the flow to freely exit or enter

At the free ambient water surfaces, a symmetry boundary condition is used, which is typically well above the generated turbidity currents. Thus, there are neither convective nor diffusive fluxes across the top surface. This type of free surface boundary condition has also been used by other researchers in literature (Imran et al., 2004; Huang et al., 2005) for the case of turbidity currents. (Farrell & Stefan, 1986) have found that for a plunging reservoir flow, the relative error that can be introduced by this approximation of the free surface, is of

The solid boundaries are specified as stationary walls with a no-slip shear condition. Turbulent flows are significantly affected by the presence of walls. Very close to the wall, viscous damping and kinematic blocking reduce the tangential and normal velocity fluctuations respectively. However, in the outer part of the near wall region, the turbulence is rapidly augmented by the production of turbulent kinetic energy due to the relatively large gradients in mean velocity. In FLUENT, there are two different approaches for modeling the near wall region. In the first approach, the viscous sub-layer and the buffer sub-layer are not resolved. Instead, semi-empirical formulas known as "wall functions" (e.g. "standard wall functions") are used in order to link the viscosity affected sub-layers between the wall and the fully-turbulent region. In the second approach, known as "nearwall modeling" approach (e.g. "enhanced wall treatment), the turbulence models are modified in order for the viscosity affected near-wall regions to be resolved with a computational mesh all the way to the wall. However, the computational mesh must be significantly fine in these regions. This approach may require more computational effort, but it gives more accurate predictions at the near-wall region of the computational domain. Therefore, wall functions should only be used in cases where the complexity and size of the computational domain as well as the available computational resources, do not allow the construction of very fine meshes at the near-wall regions (ANSYS FLUENT

A detailed verification of the proposed numerical model is conducted in the work of (Georgoulas et al., 2010), where two different series of published laboratory experiments on turbidity currents, conducted by (Gladstone et al., 1998) and (Baas et al., 2004) are reproduced numerically, and the results are compared aiming to evaluate how realistic and reliable the numerical simulations of the proposed model are. The first series of laboratory experiments (Gladstone et al., 1998) consist of fixed-volume lock-gate releases of dilute mixtures containing two different sizes of suspended silicon carbide particles, in various initial proportions, within a rectangular flume (Run A – Run G). The second series of laboratory experiments (Baas et al., 2004) consist of high-density sediment-water mixtures released with a steady rate, through a small inflow gate, into an inclined channel which is connected to a tank, were an expansion table covered with loose sediment is positioned. The mixtures consist of either fine sand, very fine sand or coarse silt. Apart from the suspended sediment grain size, the initial suspended sediment volume fraction, the water-sediment mixture discharge and the channel slope angle and bed roughness, are varied among the

the computational domain during the calculations.

the order of 10-3 and does not influence the velocity field.

Documentation, 2010).

**3. Numerical model validation** 

experimental runs (Run 1 – Run 14).

Front speed is one of the most studied parameters for lock-exchange turbidity currents. Figure 1 compares the simulated and observed current front position versus time for all the lock-gate cases considered in the work of (Georgoulas et al., 2010). As it can be seen, in general the numerical simulations show a good match with the experimental data, adequately predicting the differences in the flow front advance among the generated currents with respect to the different relative proportions of coarse (%C in the legend) and fine particles (%F in the legend) that were used in the initial suspensions. The observed divergence between the experimental and the numerical curves at various times, might be partially attributed to possible overestimation or under-estimation of the flow front position in the particular laboratory runs, due to the difficulty in the visual definition of the exact flow front position, since these laboratory difficulties are stated in the work of (Gladstone et al., 1998). Another possible reason for the observed divergence might be the overall assumptions in the numerical simulations (e.g. uniform grain size in each particle class).

Fig. 1. Comparison of numerical (Georgoulas et al., 2010) and experimental (Gladstone et al., 1998) results, of flow front advance with respect to time.

3D Multiphase Numerical Modelling for Turbidity Current Flows 55

In order to examine the validity of the vertical structure of the simulated steady-state releases, the non-dimensional vertical profiles of the streamwise velocity component for numerical runs 1, 7 and 14 (Georgoulas et al., 2010) are constructed and compared with corresponding dimensionless experimental data from the laboratory work of (Garcia, 1994). The numerical profiles and the corresponding experimental data are illustrated in Figure 4. As it can be seen, the numerically predicted dimensionless data fall within the scatter range of the dimensionless data for supercritical currents that resulted from the laboratory experiments of (Garcia, 1994). However, at the near-wall region of the numerical profiles, a sharp change is observed in relation to the experimental values. This sharp change at the near-wall region could be attributed to the 3cm mesh resolution that was used in the steadystate release runs and the application of the standard wall functions that do not resolve but instead link the viscosity affected near-wall region with the fully turbulent outer region, though the use of empirically derived formulas. Since, this sharp change is not presented in the lock-gate cases (Figure 2), it can be concluded that the application of the "enhanced wall treatment" that was used in the numerical reproduction of lock-gate releases should be preferable at the bottom wall boundaries, in cases that the complexity and size of the computational domain geometry as well as the available computational resources, allow the construction of high-resolution meshes at the near-wall regions, since this provides more

accurate and detailed predictions in the vicinity of the bottom wall boundaries.

Fig. 3. Head velocity variation with respect to the initial suspended sediment concentration, for turbidity currents laden with fine sand, very fine sand and coarse silt. Comparison of

numerical (Georgoulas et al., 2010) and experimental results (Baas et al., 2004).

In order to also examine the validity of the vertical structure of the simulated lock-gate cases, the non-dimensional vertical profiles of the stream-wise velocity component for numerical runs A and D are constructed and compared with analogous dimensionless experimental data from the laboratory work of (Garcia, 1994). The numerical profiles and the corresponding experimental data are compared in Figure 2. As it can be seen, the numerically predicted dimensionless profiles (Georgoulas et al., 2010) fall within the general scatter range of the dimensionless data for subcritical currents that resulted from the laboratory experiments of (Garcia, 1994). Therefore, it can be concluded that the proposed numerical model gives fairly reasonable predictions regarding the vertical structure of the simulated currents.

Fig. 2. Comparison of numerical dimensionless velocity profiles (Georgoulas et al., 2010) with analogous experimental data (Garcia, 1994), for numerical Runs A, and D that reproduce the experiments of (Gladstone et al., 1998).

#### **3.2 Steady-state releases**

The relationship between head velocity and initial suspended sediment concentration for fine-sand, very-fine sand and coarse silt laden turbidity currents is depicted in Figure 3, both for the numerical (Georgoulas et al., 2010) and the corresponding experimental runs (Runs 1, 3, 4, 7, 8, 13 and 14) (Bass et al., 2004). Once again, the numerical values are very close to the corresponding experimental values. Moreover, it is evident that the numerical model captures the same trend in the head velocity variation with respect to the increase of the initial suspended sediment concentration, in comply with the experimental runs.

In order to also examine the validity of the vertical structure of the simulated lock-gate cases, the non-dimensional vertical profiles of the stream-wise velocity component for numerical runs A and D are constructed and compared with analogous dimensionless experimental data from the laboratory work of (Garcia, 1994). The numerical profiles and the corresponding experimental data are compared in Figure 2. As it can be seen, the numerically predicted dimensionless profiles (Georgoulas et al., 2010) fall within the general scatter range of the dimensionless data for subcritical currents that resulted from the laboratory experiments of (Garcia, 1994). Therefore, it can be concluded that the proposed numerical model gives fairly reasonable predictions regarding the vertical structure of the

Fig. 2. Comparison of numerical dimensionless velocity profiles (Georgoulas et al., 2010) with analogous experimental data (Garcia, 1994), for numerical Runs A, and D that

The relationship between head velocity and initial suspended sediment concentration for fine-sand, very-fine sand and coarse silt laden turbidity currents is depicted in Figure 3, both for the numerical (Georgoulas et al., 2010) and the corresponding experimental runs (Runs 1, 3, 4, 7, 8, 13 and 14) (Bass et al., 2004). Once again, the numerical values are very close to the corresponding experimental values. Moreover, it is evident that the numerical model captures the same trend in the head velocity variation with respect to the increase of

the initial suspended sediment concentration, in comply with the experimental runs.

reproduce the experiments of (Gladstone et al., 1998).

**3.2 Steady-state releases** 

simulated currents.

In order to examine the validity of the vertical structure of the simulated steady-state releases, the non-dimensional vertical profiles of the streamwise velocity component for numerical runs 1, 7 and 14 (Georgoulas et al., 2010) are constructed and compared with corresponding dimensionless experimental data from the laboratory work of (Garcia, 1994). The numerical profiles and the corresponding experimental data are illustrated in Figure 4. As it can be seen, the numerically predicted dimensionless data fall within the scatter range of the dimensionless data for supercritical currents that resulted from the laboratory experiments of (Garcia, 1994). However, at the near-wall region of the numerical profiles, a sharp change is observed in relation to the experimental values. This sharp change at the near-wall region could be attributed to the 3cm mesh resolution that was used in the steadystate release runs and the application of the standard wall functions that do not resolve but instead link the viscosity affected near-wall region with the fully turbulent outer region, though the use of empirically derived formulas. Since, this sharp change is not presented in the lock-gate cases (Figure 2), it can be concluded that the application of the "enhanced wall treatment" that was used in the numerical reproduction of lock-gate releases should be preferable at the bottom wall boundaries, in cases that the complexity and size of the computational domain geometry as well as the available computational resources, allow the construction of high-resolution meshes at the near-wall regions, since this provides more accurate and detailed predictions in the vicinity of the bottom wall boundaries.

Fig. 3. Head velocity variation with respect to the initial suspended sediment concentration, for turbidity currents laden with fine sand, very fine sand and coarse silt. Comparison of numerical (Georgoulas et al., 2010) and experimental results (Baas et al., 2004).

3D Multiphase Numerical Modelling for Turbidity Current Flows 57

inclined channel into the tank, the turbidity current is free to expand in all directions (unconfined turbidity current). The proposed laboratory scale configuration, serves as a simplified experimental analog of natural, hyperpycnal turbidity currents that initially travel, laterally confined within a subaqueous canyon with a sloped bottom and then, after they exit from the downstream end of the canyon, they spread out laterally unconstrained in

the horizontal or mild sloped bottom of the receiving basin (sea, lake or reservoir).

Fig. 5. General configuration of investigated physical problem.

in all series of numerical experiments.

The symbols and the explanations of the controlling flow parameters that are investigated (varied) in each series of numerical experiments, in the present application, are summarized in Table 1. Each series of numerical experiments consists of four runs. The initial conditions of these runs are summarized in Table 2. The numerical experiments in each case are named accordingly to the varied parameter and its corresponding value in each numerical experiment. It should also be mentioned that in each series of numerical experiments (A, B, C and D) there is a common Reference Numerical Experiment (R.N.E.), which for ease purposes in the analysis of the results is named as S5, C25, D150 and R0 for Series A, B, C and D, respectively. Finally, it should be mentioned that the inflow discharge of the incoming fresh water – suspended sediment mixtures is continuous and steady, with a value of Qinflow=0.0078 m3/sec (that corresponds in an inflow velocity value of Vinflow=1.24 m/sec)

Fig. 4. Comparison of numerical dimensionless velocity profiles (Georgoulas et al., 2010) with analogous experimental data (Garcia, 1994), for numerical Runs 1, 7 and 14 that reproduce the experiments of (Baas et al., 2004).

#### **4. Applications of numerical model**

#### **4.1 Laboratory scale application**

The present subsection of the chapter describes a laboratory scale application of the proposed numerical model that aims to identify the effect of various flow controlling parameters (bed slope, bed roughness, initial suspended sediment concentration and suspended sediment diameter) in the hydrodynamic and depositional characteristics of continuous, high density turbidity currents (Georgoulas, 2010). For this purpose, four different series of parametric numerical experiments are conducted, using a laboratory scale experimental set-up, similar to the one used in the laboratory experiments of (Baas et al., 2004). In each series of numerical experiments, the initial value of only one of the above mentioned controlling parameters is varied, while the initial values of the rest parameters are kept constant.

The geometry and the general conditions of the physical problem under investigation are depicted in Figure 5. As it can be seen, the physical problem consists of turbidity currents that are generated during the continuous inflow of fresh water – suspended sediment mixtures (through an inflow gate, of height hgate=0.035 m, width wgate=0.18 m and crosssectional area of Agate=0.0063 m), into an inclined channel connected to a horizontal bottomed tank at its downstream end. The turbidity current flow within the inclined channel is laterally confined (confined turbidity current), while after its exit from the

Fig. 4. Comparison of numerical dimensionless velocity profiles (Georgoulas et al., 2010) with analogous experimental data (Garcia, 1994), for numerical Runs 1, 7 and 14 that

The present subsection of the chapter describes a laboratory scale application of the proposed numerical model that aims to identify the effect of various flow controlling parameters (bed slope, bed roughness, initial suspended sediment concentration and suspended sediment diameter) in the hydrodynamic and depositional characteristics of continuous, high density turbidity currents (Georgoulas, 2010). For this purpose, four different series of parametric numerical experiments are conducted, using a laboratory scale experimental set-up, similar to the one used in the laboratory experiments of (Baas et al., 2004). In each series of numerical experiments, the initial value of only one of the above mentioned controlling parameters is varied, while the initial values of the rest parameters

The geometry and the general conditions of the physical problem under investigation are depicted in Figure 5. As it can be seen, the physical problem consists of turbidity currents that are generated during the continuous inflow of fresh water – suspended sediment mixtures (through an inflow gate, of height hgate=0.035 m, width wgate=0.18 m and crosssectional area of Agate=0.0063 m), into an inclined channel connected to a horizontal bottomed tank at its downstream end. The turbidity current flow within the inclined channel is laterally confined (confined turbidity current), while after its exit from the

reproduce the experiments of (Baas et al., 2004).

**4. Applications of numerical model** 

**4.1 Laboratory scale application** 

are kept constant.

inclined channel into the tank, the turbidity current is free to expand in all directions (unconfined turbidity current). The proposed laboratory scale configuration, serves as a simplified experimental analog of natural, hyperpycnal turbidity currents that initially travel, laterally confined within a subaqueous canyon with a sloped bottom and then, after they exit from the downstream end of the canyon, they spread out laterally unconstrained in the horizontal or mild sloped bottom of the receiving basin (sea, lake or reservoir).

Fig. 5. General configuration of investigated physical problem.

The symbols and the explanations of the controlling flow parameters that are investigated (varied) in each series of numerical experiments, in the present application, are summarized in Table 1. Each series of numerical experiments consists of four runs. The initial conditions of these runs are summarized in Table 2. The numerical experiments in each case are named accordingly to the varied parameter and its corresponding value in each numerical experiment. It should also be mentioned that in each series of numerical experiments (A, B, C and D) there is a common Reference Numerical Experiment (R.N.E.), which for ease purposes in the analysis of the results is named as S5, C25, D150 and R0 for Series A, B, C and D, respectively. Finally, it should be mentioned that the inflow discharge of the incoming fresh water – suspended sediment mixtures is continuous and steady, with a value of Qinflow=0.0078 m3/sec (that corresponds in an inflow velocity value of Vinflow=1.24 m/sec) in all series of numerical experiments.

3D Multiphase Numerical Modelling for Turbidity Current Flows 59

69370 and 93487, respectively. In all situations the same mesh characteristics (cell size, cell clustering growth rates, cell layers in the vicinity of the bottom boundary etc.) are used. As it can be seen from Figure 6, the largest part of the computational mesh consists of tetrahedral cells of varying size, that are locally refined at regions where more computational accuracy is required (regions of sudden changes in the calculated quantities), such as the region in the

vicinity of the inflow boundary and the downstream end of the inclined channel.

Fig. 6. Computational geometry, mesh and boundary conditions (R.N.E.).

In order to ensure that the numerical solutions presented are mesh independent, sensitivity tests were performed with computational meshes of different total cell number. Figure 7 illustrates indicatively, the flow front position of the generated turbidity current with respect

Fig. 7. Mesh size sensitivity test, on turbidity current front position with respect to time

(R.N.E.).


Table 1. Investigated, fundamental controlling parameters, of turbidity current flows.


Table 2. Numerical experiments initial conditions.

As it can be seen from Table 2, the overall channel slope values that were used in the numerical experiments are 1º, 5º, 10º and 20º. Therefore, in order to conduct the numerical experiments of Series A, four different computational geometries, one for each channel slope, where constructed. In all the rest series of numerical experiments (B, C and D) the geometry with 5º channel slope is used. The computational geometry, computational mesh and boundary conditions, which were used in the numerical simulations are illustrated in Figure 6, for the case of the 5º channel slope that also corresponds to the R.N.E.. For the rest configurations these characteristics are similar and therefore are not illustrated schematically. In the computational geometries, that correspond to a channel slope of 1º, 5º, 10º and 20º, the computational meshes consist of a total number of cells (control volumes) of 51770, 58398,

**Parameter Symbol Explanation** 

"Initial, volumetric concentration of suspended sediment particles in the inflow mixture"

"Grain diameter of suspended sediment particles in the inflow mixture"

"Roughness of channel and tank bed expressed as equivalent roughness of uniformly distributed suspended sediment particles of specific grain size"

**Si (°) Ci (% vol.) Di (μm) Ri (μm)** 

**Α** Channel slope **Si** "inclination angle of channel bed"

Table 1. Investigated, fundamental controlling parameters, of turbidity current flows.

A S1 1 25 150 0 A S5 5 25 150 0 A S10 10 25 150 0 A S20 20 25 150 0 B C5 5 5 150 0 B C10 5 10 150 0 B C15 5 15 150 0 B C25 5 25 150 0 C D80 5 25 80 0 C D100 5 25 100 0 C D120 5 25 120 0 C D150 5 25 150 0 D R0 5 25 150 0 D R80 5 25 150 80 D R235 5 25 150 235 D R500 5 25 150 500

As it can be seen from Table 2, the overall channel slope values that were used in the numerical experiments are 1º, 5º, 10º and 20º. Therefore, in order to conduct the numerical experiments of Series A, four different computational geometries, one for each channel slope, where constructed. In all the rest series of numerical experiments (B, C and D) the geometry with 5º channel slope is used. The computational geometry, computational mesh and boundary conditions, which were used in the numerical simulations are illustrated in Figure 6, for the case of the 5º channel slope that also corresponds to the R.N.E.. For the rest configurations these characteristics are similar and therefore are not illustrated schematically. In the computational geometries, that correspond to a channel slope of 1º, 5º, 10º and 20º, the computational meshes consist of a total number of cells (control volumes) of 51770, 58398,

**Series of Numerical Experiments** 

**Series of Numerical Experiments**  **Investigated/Varied** 

concentration **Ci** 

**<sup>Β</sup>** Suspended sediment

**C** Grain diameter **Di**

**D** Bed roughness **Ri**

**Numerical Experiment Name** 

Table 2. Numerical experiments initial conditions.

69370 and 93487, respectively. In all situations the same mesh characteristics (cell size, cell clustering growth rates, cell layers in the vicinity of the bottom boundary etc.) are used. As it can be seen from Figure 6, the largest part of the computational mesh consists of tetrahedral cells of varying size, that are locally refined at regions where more computational accuracy is required (regions of sudden changes in the calculated quantities), such as the region in the vicinity of the inflow boundary and the downstream end of the inclined channel.

Fig. 6. Computational geometry, mesh and boundary conditions (R.N.E.).

In order to ensure that the numerical solutions presented are mesh independent, sensitivity tests were performed with computational meshes of different total cell number. Figure 7 illustrates indicatively, the flow front position of the generated turbidity current with respect

Fig. 7. Mesh size sensitivity test, on turbidity current front position with respect to time (R.N.E.).

3D Multiphase Numerical Modelling for Turbidity Current Flows 61

the inclined channel (b=0.22 m) and the flow time t is normalized with the time needed for the slowest of the generated turbidity currents (in each series of numerical experiments) to exit from the downstream boundary of the expansion tank (texit(S1)=12 sec for Series A, texit(C5)=18 sec for Series B, texit(D80)=10 sec for Series C and texit(R500)=13 sec for Series D). As it can be seen the resulting curves in each numerical experiment of Series A, B, C and D have a similar form, consisting of three distinct parts. In the first part the flow front velocity of the generated turbidity currents is almost constant, in the second part a gradual acceleration of the flow front is observed and in the third part, a gradual deceleration of the flow front is evident. In the first part, the flow of the generated turbidity currents is primarily controlled by their initial momentum, due to the continuous and steady discharge of the inflowing fresh water – suspended sediment mixtures from the inflow gate and therefore the flow front velocity remains constant. In the second part that the flow front of the currents has already traveled almost half the length of the inclined channel, the observed acceleration of the front is due to the continuous increase of the gravitational force effect, since the currents are flowing over an inclined bottom boundary. At the third part, the turbidity currents have already entered the expansion tank and their flow is laterally unconfined, expanding radically in all directions over the horizontal bottom boundary of the tank. Therefore, the continuous reduction of their excess density, due to the continuous entrainment of the ambient water of the tank and the consequent gradual deposition of suspended sediment particles,

causes a gradual dissipation and deceleration of the generated turbidity current flows.

(a) (b)

(c) (d)

Fig. 9. Dimensionless flow front position with respect to dimensionless time for, (a) Series A

numerical experiments, (b) Series B numerical experiments, (c) Series C numerical

experiments and (d) Series D numerical experiments.

to time, for three different computational meshes in the case of the R.N.E. The first computational mesh is the one used in the simulations (58,398 computational cells), the second one is a coarser mesh (36,133 computational cells) and the third one is a finer mesh (119,907 computational cells). It is obvious (Figure 7) that the resulting curves in each case show a good degree of convergence and therefore the solution can be considered to be mesh independent. In more detail, comparing the results of the coarser mesh with the corresponding results of the finer mesh, it is concluded that increasing the total number of cells by a factor of 3.33, the average differences of the flow front position values with respect to time is only 1.85%.

In order to visualize the flow of the generated turbidity currents in the simulations of the present investigation, the three-dimensional time evolution of the interface, between the generated turbidity current and the ambient water, for the case of the R.N.E., is depicted in Figure 8.

Fig. 8. Three dimensional time evolution of the interface (grey surface) between the generated turbidity current and the ambient water (R.N.E.).

It is obvious that 3sec after the inflow of the fresh water – suspended sediment mixture the generated turbidity current, flows within the inclined channel (laterally confined part of the flow). At t=5 sec, the turbidity current head has already exited from the downstream part of the channel and has started to expand radically in the horizontal bed of the tank (unconfined part of the flow). At t=10 sec the head of the current has just reached the downstream open boundary of the computational domain, while at t=20 sec it has already exited the computational domain from the downstream as well as the left and right side open boundaries. Finally, at t=40sec the evolution of the current within the computational domain has already reached a quasi-steady state.

In Figure 9 the resulting curves of the generated turbidity current flow front position with respect to time, are illustrated in dimensionless form, for the numerical experiments of Series A, B, C and D respectively. For comparison purposes, the varied parameter in each series of numerical experiments is normalized with its lowest value (S1=1° for Series A, C5=5% by vol. for Series B, D80=80 μm for Series C and R80=80 μm for Series D), the horizontal distance X of the flow front from the inflow gate is normalized with the width of

to time, for three different computational meshes in the case of the R.N.E. The first computational mesh is the one used in the simulations (58,398 computational cells), the second one is a coarser mesh (36,133 computational cells) and the third one is a finer mesh (119,907 computational cells). It is obvious (Figure 7) that the resulting curves in each case show a good degree of convergence and therefore the solution can be considered to be mesh independent. In more detail, comparing the results of the coarser mesh with the corresponding results of the finer mesh, it is concluded that increasing the total number of cells by a factor of 3.33, the average differences of the flow front position values with respect

In order to visualize the flow of the generated turbidity currents in the simulations of the present investigation, the three-dimensional time evolution of the interface, between the generated turbidity current and the ambient water, for the case of the R.N.E., is depicted in

Fig. 8. Three dimensional time evolution of the interface (grey surface) between the

It is obvious that 3sec after the inflow of the fresh water – suspended sediment mixture the generated turbidity current, flows within the inclined channel (laterally confined part of the flow). At t=5 sec, the turbidity current head has already exited from the downstream part of the channel and has started to expand radically in the horizontal bed of the tank (unconfined part of the flow). At t=10 sec the head of the current has just reached the downstream open boundary of the computational domain, while at t=20 sec it has already exited the computational domain from the downstream as well as the left and right side open boundaries. Finally, at t=40sec the evolution of the current within the computational

In Figure 9 the resulting curves of the generated turbidity current flow front position with respect to time, are illustrated in dimensionless form, for the numerical experiments of Series A, B, C and D respectively. For comparison purposes, the varied parameter in each series of numerical experiments is normalized with its lowest value (S1=1° for Series A, C5=5% by vol. for Series B, D80=80 μm for Series C and R80=80 μm for Series D), the horizontal distance X of the flow front from the inflow gate is normalized with the width of

generated turbidity current and the ambient water (R.N.E.).

domain has already reached a quasi-steady state.

to time is only 1.85%.

Figure 8.

the inclined channel (b=0.22 m) and the flow time t is normalized with the time needed for the slowest of the generated turbidity currents (in each series of numerical experiments) to exit from the downstream boundary of the expansion tank (texit(S1)=12 sec for Series A, texit(C5)=18 sec for Series B, texit(D80)=10 sec for Series C and texit(R500)=13 sec for Series D). As it can be seen the resulting curves in each numerical experiment of Series A, B, C and D have a similar form, consisting of three distinct parts. In the first part the flow front velocity of the generated turbidity currents is almost constant, in the second part a gradual acceleration of the flow front is observed and in the third part, a gradual deceleration of the flow front is evident. In the first part, the flow of the generated turbidity currents is primarily controlled by their initial momentum, due to the continuous and steady discharge of the inflowing fresh water – suspended sediment mixtures from the inflow gate and therefore the flow front velocity remains constant. In the second part that the flow front of the currents has already traveled almost half the length of the inclined channel, the observed acceleration of the front is due to the continuous increase of the gravitational force effect, since the currents are flowing over an inclined bottom boundary. At the third part, the turbidity currents have already entered the expansion tank and their flow is laterally unconfined, expanding radically in all directions over the horizontal bottom boundary of the tank. Therefore, the continuous reduction of their excess density, due to the continuous entrainment of the ambient water of the tank and the consequent gradual deposition of suspended sediment particles, causes a gradual dissipation and deceleration of the generated turbidity current flows.

Fig. 9. Dimensionless flow front position with respect to dimensionless time for, (a) Series A numerical experiments, (b) Series B numerical experiments, (c) Series C numerical experiments and (d) Series D numerical experiments.

3D Multiphase Numerical Modelling for Turbidity Current Flows 63

to a distance of X/b=1 and then follows a less rapid increase up to a maximum value, at a distance of X/b=11 that is close to the downstream end of the channel (X/b=13.6). The rapid increase of the volume fraction values in the vicinity of the inflow point (X/b=0 to 1) is probably due to the local increase of the volume fraction value of the inflowing mixtures, as a result of the resistance that is exerted from the ambient fluid. In the unconstrained and horizontal bottom part of the flow (tank), the suspended sediment volumetric concentration follows an irregular decrease with respect to the longitudinal distance, reaching an almost constant minimum value in the vicinity of the downstream boundary of the computational domain. The fact that in all cases, the maximum value of the suspended sediment volumetric concentration at the bottom boundary of the computational domain is found near the downstream end of the channel, is probably due to the sudden reduction in the velocity of the generated turbidity currents which is a result of the flow transition from the laterally constrained (channel) to the unconstrained (tank) part of the computational domain. This sudden drop of velocity is reasonable to cause intense particle deposition just

Examining separately the effect of each controlling parameter in the flow front advance velocity and in the deposit density of the current at the bottom boundary, it can be concluded that in general, the increase of the channel slope causes an increase in the flow front advance velocity and a reduction in the deposit density. The increase of the initial suspended sediment concentration causes an increase both in the flow front advance velocity and in the deposit density. The increase of the suspended sediment grain diameter causes an increase both in the flow front advance velocity and in the deposit density. Finally, the increase of the bed roughness causes a reduction in the flow front advance

From the presentation and the analysis of the above results so far, it is evident that the investigated controlling parameters affect with a different way and in a comparably different degree the dynamic and depositional characteristics of turbidity currents. Therefore in order to compare the relative percentage effect of the varied controlling parameters in the main flow characteristics of the generated turbidity currents, Figure 11 presents diagrams of the relative percentage change of the maximum flow front advance velocity (Figure 11 a) and the maximum value of suspended sediment volume fraction at the bottom boundary (Figure 11 b), in relation to the relative percentage change of the varied controlling parameters. It should be mentioned that for comparison purposes, the relative percentage change in each case is calculated using absolute differences. It should also be mentioned that in the case of Series D numerical experiments, only the experiments R80, R235, and R500 are taken into consideration, where the values of the bottom boundary roughness are greater than zero. It is obvious that the variation of the initial suspended sediment concentration as well as the suspended sediment grain diameter have the biggest effect in the flow of the generated turbidity currents. This can be probably attributed to the direct effect of the proposed controlling parameters in the main driving force of turbidity currents, which is the excess density of the current in relation to the ambient water density. The variation of the bed roughness has the smallest effect, while the variation of the channel slope causes a moderate effect in the turbidity current flows,

upstream of the channel exit to the expansion tank.

velocity and an increase in the deposit density.

in relation to the rest controlling parameters.

In order to investigate the exact quantitative effect of the varied controlling parameters to the depositional characteristics of the generated turbidity currents, in Figure 10 the suspended sediment volumetric concentration at the bottom boundary of the domain is plotted against the horizontal distance from the inflow gate, for flow time t=40 sec, where the flow of generated turbidity currents within the computational domain have reached a quasi-steady state. For comparison purposes, the varied parameter in each series of numerical experiments is normalized with its lowest value (S1=1° for Series A, C5=5% by vol. for Series B, D80=80 μm for Series C and R80=80 μm for Series D), the horizontal distance X of the flow front from the inflow gate is normalized with the width of the inclined channel (b=0.22 m) and the suspended sediment volume fraction at the bottom boundary Cvol is normalized with the values, CS1=0.25 for Series A, CC5=0.05 for Series B, CD80=0.25 for Series C and CR80=0.25 for Series D numerical experiments. It should be mentioned that the suspended sediment volume fraction values at the bottom boundary of the computational domain are taken at the central axis of the generated flows. It is obvious that in each case the resulting curves have a similar form. In more detail, in the laterally constrained and sloped bottom part of the flow (channel), the suspended sediment volumetric concentration at the bottom boundary increases rapidly with the longitudinal distance from the inflow gate, up

Fig. 10. Dimensionless suspended sediment volume fraction at the bottom boundary of the computational domain, with respect to the dimensionless horizontal distance from the inflow gate for, (a) Series A numerical experiments, (b) Series B numerical experiments, (c) Series C numerical experiments and (d) Series D numerical experiments, 40 sec after the beginning of the inflow of the fresh water – suspended sediment mixtures.

In order to investigate the exact quantitative effect of the varied controlling parameters to the depositional characteristics of the generated turbidity currents, in Figure 10 the suspended sediment volumetric concentration at the bottom boundary of the domain is plotted against the horizontal distance from the inflow gate, for flow time t=40 sec, where the flow of generated turbidity currents within the computational domain have reached a quasi-steady state. For comparison purposes, the varied parameter in each series of numerical experiments is normalized with its lowest value (S1=1° for Series A, C5=5% by vol. for Series B, D80=80 μm for Series C and R80=80 μm for Series D), the horizontal distance X of the flow front from the inflow gate is normalized with the width of the inclined channel (b=0.22 m) and the suspended sediment volume fraction at the bottom boundary Cvol is normalized with the values, CS1=0.25 for Series A, CC5=0.05 for Series B, CD80=0.25 for Series C and CR80=0.25 for Series D numerical experiments. It should be mentioned that the suspended sediment volume fraction values at the bottom boundary of the computational domain are taken at the central axis of the generated flows. It is obvious that in each case the resulting curves have a similar form. In more detail, in the laterally constrained and sloped bottom part of the flow (channel), the suspended sediment volumetric concentration at the bottom boundary increases rapidly with the longitudinal distance from the inflow gate, up

Fig. 10. Dimensionless suspended sediment volume fraction at the bottom boundary of the computational domain, with respect to the dimensionless horizontal distance from the inflow gate for, (a) Series A numerical experiments, (b) Series B numerical experiments, (c) Series C numerical experiments and (d) Series D numerical experiments, 40 sec after the

beginning of the inflow of the fresh water – suspended sediment mixtures.

(a) (b)

(c) (d)

to a distance of X/b=1 and then follows a less rapid increase up to a maximum value, at a distance of X/b=11 that is close to the downstream end of the channel (X/b=13.6). The rapid increase of the volume fraction values in the vicinity of the inflow point (X/b=0 to 1) is probably due to the local increase of the volume fraction value of the inflowing mixtures, as a result of the resistance that is exerted from the ambient fluid. In the unconstrained and horizontal bottom part of the flow (tank), the suspended sediment volumetric concentration follows an irregular decrease with respect to the longitudinal distance, reaching an almost constant minimum value in the vicinity of the downstream boundary of the computational domain. The fact that in all cases, the maximum value of the suspended sediment volumetric concentration at the bottom boundary of the computational domain is found near the downstream end of the channel, is probably due to the sudden reduction in the velocity of the generated turbidity currents which is a result of the flow transition from the laterally constrained (channel) to the unconstrained (tank) part of the computational domain. This sudden drop of velocity is reasonable to cause intense particle deposition just upstream of the channel exit to the expansion tank.

Examining separately the effect of each controlling parameter in the flow front advance velocity and in the deposit density of the current at the bottom boundary, it can be concluded that in general, the increase of the channel slope causes an increase in the flow front advance velocity and a reduction in the deposit density. The increase of the initial suspended sediment concentration causes an increase both in the flow front advance velocity and in the deposit density. The increase of the suspended sediment grain diameter causes an increase both in the flow front advance velocity and in the deposit density. Finally, the increase of the bed roughness causes a reduction in the flow front advance velocity and an increase in the deposit density.

From the presentation and the analysis of the above results so far, it is evident that the investigated controlling parameters affect with a different way and in a comparably different degree the dynamic and depositional characteristics of turbidity currents. Therefore in order to compare the relative percentage effect of the varied controlling parameters in the main flow characteristics of the generated turbidity currents, Figure 11 presents diagrams of the relative percentage change of the maximum flow front advance velocity (Figure 11 a) and the maximum value of suspended sediment volume fraction at the bottom boundary (Figure 11 b), in relation to the relative percentage change of the varied controlling parameters. It should be mentioned that for comparison purposes, the relative percentage change in each case is calculated using absolute differences. It should also be mentioned that in the case of Series D numerical experiments, only the experiments R80, R235, and R500 are taken into consideration, where the values of the bottom boundary roughness are greater than zero. It is obvious that the variation of the initial suspended sediment concentration as well as the suspended sediment grain diameter have the biggest effect in the flow of the generated turbidity currents. This can be probably attributed to the direct effect of the proposed controlling parameters in the main driving force of turbidity currents, which is the excess density of the current in relation to the ambient water density. The variation of the bed roughness has the smallest effect, while the variation of the channel slope causes a moderate effect in the turbidity current flows, in relation to the rest controlling parameters.

3D Multiphase Numerical Modelling for Turbidity Current Flows 65

Fig. 12. 3D digitized bottom relief model of the North Aegean Sea, region of Evros River Outflow (region 1) and region of numerical simulation (hatched region within Sub-region 2).

into account.

The resulting numerical geometry, the computational mesh and the boundary conditions that are used for the simulation of the present paper, are illustrated in Figure 13. The computational mesh consists of a total number of 56,720 hexahedral cells (Figure 13 a). For the inflow (Figure 13 b) a "velocity inlet" boundary condition is applied. For the shoreline east and west of Evros River outflow (Figure 13 c), a "wall" boundary condition is applied. For the open sea boundary of the flow field (Figure 13 d) a "pressure outlet" boundary condition is applied. For the bottom boundary a "wall" boundary condition is used (Figure 13 e), while for the free water surface of the ambient sea water a "symmetry" boundary condition is used (Figure 13 f). At this point, it should be mentioned that during the simulation, the entire flow field is rotated with respect to the Z-axis (vertical axis), with a rotational speed that corresponds to the rotational frequency (Coriolis parameter) of the North Aegean Sea region (latitude φ = 40°N), in order for the Coriolis force effect to be taken

The initial conditions applied for the simulation, are summarized in Table 3. As it can be seen the numerical simulation was conducted with relatively simplified conditions, in order to investigate more clearly the effects of the bottom topography and the Coriolis force, in the results of the studied flow. Therefore, the inflow discharge from Evros River was assumed to be steady, and the potential effects of other parallel to the shore, subaqueous and/ or surface currents were not taken into consideration. The value of the Evros river discharge, which was indicatively used in the simulation (4,555 m3/sec), corresponds in a big flood discharge of the proposed river. Since there are not any available data for the maximum sediment discharge at Evros River mouth, the initial suspended sediment concentration that was used for the numerical simulation, was estimated, taking into consideration the sediment discharge measurements, upstream of the river mouth, in the work of (Gergov, 1996). The initial condition for the ambient water was assumed to be constant for the entire flow field, with a salinity of 38.6 ppt and a temperature of 15.0 °C that correspond to a density value of 1028.75 (kg/m3). The width of Evros River at the inflow position into the North Aegean Sea

was assumed to be 450 m, while the corresponding depth was assumed to be 2.5 m.

Fig. 11. Dependence of the maximum flow front velocity (a) and the maximum suspended sediment volume fraction at the bottom boundary of the computational domain (b), from the investigated flow controlling parameters (expressed as relative percentage change).

Summarizing, the overall results of the present numerical investigation contribute considerably in the understanding of the dependence of the suspended sediment transport and deposition mechanism, from fundamental flow controlling parameters of natural, continuous, high-density turbidity currents that are usually formed during flood discharges at river outflows in the sea, lakes and reservoirs.

#### **4.2 Field scale application**

The present subsection of the chapter describes a field scale application of the proposed numerical model that aims to identify the dynamic behavior and the main flow characteristics of turbidity currents, which are potentially formed at Evros river mouth (Georgoulas et al., 2009). More specifically, the numerical model is applied at Evros river mouth (Greece), in order to simulate the river's suspended sediment transport and dispersal into the North Aegean Sea, in the case of a flood discharge, where the suspended sediment concentration of the river water is considerably high, in order for a turbidity current to be formed. It should be mentioned that the effects of the bed morphology and the Coriolis force are taken into account, during the numerical simulation.

The flow examined, is a flood discharge of Evros River that is based in existing flood data. It is treated numerically as a multiphase flow, with saline water (North Aegean Sea) being the primary phase and fresh water and suspended sediment particles (Evros River) being the secondary phases. For the present numerical application, two separate phases of suspended sediment particles are assumed. The first consists of fine sand particles of 0.235 mm diameter and the second consists of very fine sand particles of 0.069mm diameter.

The geometry used in the numerical simulation, has been extracted from a 3D digitized bottom relief model of the North Aegean Sea, which is illustrated in Figure 12 below. The region, denoted by number 1 in the digitized bottom relief model, is the wider region of Evros river mouth, while the hatched area in the sub-region denoted by number 2, is the part that was selected for the numerical simulation.

(a) (b)

Fig. 11. Dependence of the maximum flow front velocity (a) and the maximum suspended sediment volume fraction at the bottom boundary of the computational domain (b), from the investigated flow controlling parameters (expressed as relative percentage change).

Summarizing, the overall results of the present numerical investigation contribute considerably in the understanding of the dependence of the suspended sediment transport and deposition mechanism, from fundamental flow controlling parameters of natural, continuous, high-density turbidity currents that are usually formed during flood discharges

The present subsection of the chapter describes a field scale application of the proposed numerical model that aims to identify the dynamic behavior and the main flow characteristics of turbidity currents, which are potentially formed at Evros river mouth (Georgoulas et al., 2009). More specifically, the numerical model is applied at Evros river mouth (Greece), in order to simulate the river's suspended sediment transport and dispersal into the North Aegean Sea, in the case of a flood discharge, where the suspended sediment concentration of the river water is considerably high, in order for a turbidity current to be formed. It should be mentioned that the effects of the bed morphology and the Coriolis force

The flow examined, is a flood discharge of Evros River that is based in existing flood data. It is treated numerically as a multiphase flow, with saline water (North Aegean Sea) being the primary phase and fresh water and suspended sediment particles (Evros River) being the secondary phases. For the present numerical application, two separate phases of suspended sediment particles are assumed. The first consists of fine sand particles of 0.235 mm

The geometry used in the numerical simulation, has been extracted from a 3D digitized bottom relief model of the North Aegean Sea, which is illustrated in Figure 12 below. The region, denoted by number 1 in the digitized bottom relief model, is the wider region of Evros river mouth, while the hatched area in the sub-region denoted by number 2, is the

diameter and the second consists of very fine sand particles of 0.069mm diameter.

at river outflows in the sea, lakes and reservoirs.

are taken into account, during the numerical simulation.

part that was selected for the numerical simulation.

**4.2 Field scale application** 

Fig. 12. 3D digitized bottom relief model of the North Aegean Sea, region of Evros River Outflow (region 1) and region of numerical simulation (hatched region within Sub-region 2).

The resulting numerical geometry, the computational mesh and the boundary conditions that are used for the simulation of the present paper, are illustrated in Figure 13. The computational mesh consists of a total number of 56,720 hexahedral cells (Figure 13 a). For the inflow (Figure 13 b) a "velocity inlet" boundary condition is applied. For the shoreline east and west of Evros River outflow (Figure 13 c), a "wall" boundary condition is applied. For the open sea boundary of the flow field (Figure 13 d) a "pressure outlet" boundary condition is applied. For the bottom boundary a "wall" boundary condition is used (Figure 13 e), while for the free water surface of the ambient sea water a "symmetry" boundary condition is used (Figure 13 f). At this point, it should be mentioned that during the simulation, the entire flow field is rotated with respect to the Z-axis (vertical axis), with a rotational speed that corresponds to the rotational frequency (Coriolis parameter) of the North Aegean Sea region (latitude φ = 40°N), in order for the Coriolis force effect to be taken into account.

The initial conditions applied for the simulation, are summarized in Table 3. As it can be seen the numerical simulation was conducted with relatively simplified conditions, in order to investigate more clearly the effects of the bottom topography and the Coriolis force, in the results of the studied flow. Therefore, the inflow discharge from Evros River was assumed to be steady, and the potential effects of other parallel to the shore, subaqueous and/ or surface currents were not taken into consideration. The value of the Evros river discharge, which was indicatively used in the simulation (4,555 m3/sec), corresponds in a big flood discharge of the proposed river. Since there are not any available data for the maximum sediment discharge at Evros River mouth, the initial suspended sediment concentration that was used for the numerical simulation, was estimated, taking into consideration the sediment discharge measurements, upstream of the river mouth, in the work of (Gergov, 1996). The initial condition for the ambient water was assumed to be constant for the entire flow field, with a salinity of 38.6 ppt and a temperature of 15.0 °C that correspond to a density value of 1028.75 (kg/m3). The width of Evros River at the inflow position into the North Aegean Sea was assumed to be 450 m, while the corresponding depth was assumed to be 2.5 m.

3D Multiphase Numerical Modelling for Turbidity Current Flows 67

The 3D, time evolution of the root and dispersal of the suspended sediment - fresh water mixture that enters the flow field, is illustrated in Figure 14. It is observed that a part of the suspended sediment – fresh water mixture, that enters the flow field, is spreading at the free water surface, while most of the mixture plunges underneath the free sea water surface, forming a turbidity current, which continuous to flow along the bottom. This subaqueous current, initially spreads out radically in all directions, with an irregular shape, due to the mixing of the current with the ambient fluid. After the second hour of real flow, a major part of the turbidity current clearly deviates to the left of the main inflow axis, due to the general slope gradient of the bottom relief in this direction. However, it is obvious that a smaller part of the turbidity current deviates to the right of the main inflow axis, due to the effect of the Coriolis force, encountering even negative slope gradients. The effect of the Coriolis force is also evident in the part of the suspended sediment – fresh water mixture that is spreading in the free surface. The surface that is enclosed by the dashed line in Figure 14, constitutes the so called "plunge region" of the generated turbidity current. After this line the mixture plunges underneath the free water surface and continuous to flow along the

Fig. 14. Time evolution of the root and dispersal of the suspended sediment – fresh water

In order to investigate the different concentration distributions, for the two separate suspended sediment phases, Figure 16 is plotted, which illustrates the corresponding volume fraction contours for each of these phases, in two (perpendicular to each other) vertical sections within the flow field, 4 hours (real flow time) after the beginning of the simulation. It is reminded that at the river inflow into the sea, the initial concentration values for the fine sand and the very fine sand phases are 0.7% by volume (table 3). The position of each of the proposed sections, within the flow field, is shown in Figure 15.

bottom of the receiving basin.

mixture that enters the flow field.

Fig. 13. Numerical simulation geometry, computational mesh and boundary conditions.


Table 3. Numerical simulation initial conditions.

Fig. 13. Numerical simulation geometry, computational mesh and boundary conditions.

Inflow discharge (m3/sec) 4,555 Fine sand particle diameter (mm) 0.235 Very fine sand particle diameter (mm) 0.069 Fine sand concentration (vol. %) 0.7% Very fine sand concentration (vol. %) 0.7% Saline water density (kg/m3) 1028,75 Fresh water density (kg/m3) 998,2 Sand particle density (kg/m3) 2,650 Coriolis parameter f (Hz) 9.35x10-5

Table 3. Numerical simulation initial conditions.

The 3D, time evolution of the root and dispersal of the suspended sediment - fresh water mixture that enters the flow field, is illustrated in Figure 14. It is observed that a part of the suspended sediment – fresh water mixture, that enters the flow field, is spreading at the free water surface, while most of the mixture plunges underneath the free sea water surface, forming a turbidity current, which continuous to flow along the bottom. This subaqueous current, initially spreads out radically in all directions, with an irregular shape, due to the mixing of the current with the ambient fluid. After the second hour of real flow, a major part of the turbidity current clearly deviates to the left of the main inflow axis, due to the general slope gradient of the bottom relief in this direction. However, it is obvious that a smaller part of the turbidity current deviates to the right of the main inflow axis, due to the effect of the Coriolis force, encountering even negative slope gradients. The effect of the Coriolis force is also evident in the part of the suspended sediment – fresh water mixture that is spreading in the free surface. The surface that is enclosed by the dashed line in Figure 14, constitutes the so called "plunge region" of the generated turbidity current. After this line the mixture plunges underneath the free water surface and continuous to flow along the bottom of the receiving basin.

Fig. 14. Time evolution of the root and dispersal of the suspended sediment – fresh water mixture that enters the flow field.

In order to investigate the different concentration distributions, for the two separate suspended sediment phases, Figure 16 is plotted, which illustrates the corresponding volume fraction contours for each of these phases, in two (perpendicular to each other) vertical sections within the flow field, 4 hours (real flow time) after the beginning of the simulation. It is reminded that at the river inflow into the sea, the initial concentration values for the fine sand and the very fine sand phases are 0.7% by volume (table 3). The position of each of the proposed sections, within the flow field, is shown in Figure 15.

3D Multiphase Numerical Modelling for Turbidity Current Flows 69

In the present chapter an "uncommon", CFD-based, three-dimensional, multiphase numerical approach for the simulation and study of the hydrodynamic and depositional characteristics of turbidity currents is presented. The main advantages of the proposed multiphase numerical approach, in relation to previous quasi single-phase approaches are

 Separate velocity fields are calculated for each phase (ambient water, inflow/carrier water and various classes of suspended sediment), since the laws for the conservation of mass and momentum are accordingly modified, in order to be satisfied by each phase

 The use of the RNG k-ε turbulence model significantly increases the applicability of the proposed numerical approach, as it can also account for turbidity current flows with

 The total number of flow phases that can be simulated is only limited by the available memory of the computational resources. Hence, it can also be used for the simulation of polydisperse turbidity currents that contain many classes of suspended sediment

It can handle a wide range of particulate loading, and therefore is capable for the

 It is based on the finite volume method, and therefore it can be applied in situations with complex geometries, like in the case of turbidity currents that are formed at natural, water basin beds (sea, lakes, reservoirs), where morphological anomalies are

The method is tested against published laboratory data that are available in the literature and the comparison of the numerical and experimental results indicate that its predictions

The overall results of the laboratory scale application contribute considerably in the understanding of the dependence of the suspended sediment transport and deposition mechanism, from fundamental flow controlling parameters of natural, continuous, highdensity turbidity currents that are usually formed during flood discharges at river outflows. It is found that the investigated controlling parameters affect with a different way and in a comparably different degree the dynamic and depositional characteristics of turbidity currents. In more detail, the increase of the channel slope causes an increase in the flow front advance velocity and a reduction in the deposit density. The increase of the initial suspended sediment concentration causes an increase both in the flow front advance velocity and in the deposit density. The increase of the suspended sediment grain diameter causes an increase both in the flow front advance velocity and in the deposit density, while the increase of the bed roughness causes a reduction in the flow front advance velocity and an increase in the deposit density. Finally, from the comparison of the relative percentage effect of each of the examined controlling parameters in the main hydrodynamic and depositional characteristics of the generated turbidity currents, it is found that the variation of the initial suspended sediment concentration as well as the suspended sediment grain diameter have the biggest effect, the variation of the bed roughness has the smallest effect,

particles, which are more close to natural turbidity current flows.

simulation of both dilute and dense turbidity current flows.

**5. Conclusion** 

the following:

individually.

usually present.

are realistic and reliable.

low Reynolds numbers.

Fig. 15. Position of vertical sections within the flow field (Flow time, t = 4hr).

Fig. 16. Volume fractions of fine sand and very fine sand phases in vertical sections 1 and 2 (Flow time, t = 4 hr).

As it is observed, both of the suspended sediment phases have similar concentration distributions, with increasing values from the interface with the ambient fluid to the bottom of the receiving basin. However, the top and more dilute concentration layer of the very fine sand phase (volume fraction values 0.00 to 0.01), occupies more height within the flow field, than in the case of the fine sand phase. This fact is obviously due to the different settling velocities between the particles of the very fine sand and the fine sand. It is also evident that the part of the suspended sediment – fresh water mixture that spreads out along the free surface of the receiving basin, contains mainly fresh water and very fine sand particles. This can be seen in vertical section 1, where the plunge point of the very fine sand phase is traced in a considerably longer distance, from the inflow point (~1250 m), than in the case of the fine sand phase (~150 m).

Summarizing, from the results of the present subsection it is concluded that for the assumed initial conditions, the inflow of the suspended sediment – fresh water mixture from Evros River into the North Aegean Sea Basin, forms a turbidity current, which plunges to the bottom of the receiving basin. The effects of the bottom topography as well as the Coriolis force, in the root and dispersal of the studied turbidity current, are highly evident. Finally, different responses, of the different types of suspended sediment particles (fine sand and very fine sand) within the flow field, are also evident.

### **5. Conclusion**

68 Numerical Modelling

Fig. 15. Position of vertical sections within the flow field (Flow time, t = 4hr).

Fig. 16. Volume fractions of fine sand and very fine sand phases in vertical sections 1 and 2

As it is observed, both of the suspended sediment phases have similar concentration distributions, with increasing values from the interface with the ambient fluid to the bottom of the receiving basin. However, the top and more dilute concentration layer of the very fine sand phase (volume fraction values 0.00 to 0.01), occupies more height within the flow field, than in the case of the fine sand phase. This fact is obviously due to the different settling velocities between the particles of the very fine sand and the fine sand. It is also evident that the part of the suspended sediment – fresh water mixture that spreads out along the free surface of the receiving basin, contains mainly fresh water and very fine sand particles. This can be seen in vertical section 1, where the plunge point of the very fine sand phase is traced in a considerably longer distance, from the inflow point (~1250 m), than in the case of the

Summarizing, from the results of the present subsection it is concluded that for the assumed initial conditions, the inflow of the suspended sediment – fresh water mixture from Evros River into the North Aegean Sea Basin, forms a turbidity current, which plunges to the bottom of the receiving basin. The effects of the bottom topography as well as the Coriolis force, in the root and dispersal of the studied turbidity current, are highly evident. Finally, different responses, of the different types of suspended sediment particles (fine sand and

(Flow time, t = 4 hr).

fine sand phase (~150 m).

very fine sand) within the flow field, are also evident.

In the present chapter an "uncommon", CFD-based, three-dimensional, multiphase numerical approach for the simulation and study of the hydrodynamic and depositional characteristics of turbidity currents is presented. The main advantages of the proposed multiphase numerical approach, in relation to previous quasi single-phase approaches are the following:


The method is tested against published laboratory data that are available in the literature and the comparison of the numerical and experimental results indicate that its predictions are realistic and reliable.

The overall results of the laboratory scale application contribute considerably in the understanding of the dependence of the suspended sediment transport and deposition mechanism, from fundamental flow controlling parameters of natural, continuous, highdensity turbidity currents that are usually formed during flood discharges at river outflows. It is found that the investigated controlling parameters affect with a different way and in a comparably different degree the dynamic and depositional characteristics of turbidity currents. In more detail, the increase of the channel slope causes an increase in the flow front advance velocity and a reduction in the deposit density. The increase of the initial suspended sediment concentration causes an increase both in the flow front advance velocity and in the deposit density. The increase of the suspended sediment grain diameter causes an increase both in the flow front advance velocity and in the deposit density, while the increase of the bed roughness causes a reduction in the flow front advance velocity and an increase in the deposit density. Finally, from the comparison of the relative percentage effect of each of the examined controlling parameters in the main hydrodynamic and depositional characteristics of the generated turbidity currents, it is found that the variation of the initial suspended sediment concentration as well as the suspended sediment grain diameter have the biggest effect, the variation of the bed roughness has the smallest effect,

3D Multiphase Numerical Modelling for Turbidity Current Flows 71

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while the variation of the channel slope causes a moderate effect in the turbidity current flows, in relation to the rest controlling parameters.

From the field scale application it can be concluded that for the assumed initial conditions, the inflow of the suspended sediment – fresh water mixture from Evros River into the North Aegean Sea Basin, forms a turbidity current, which plunges to the bottom of the receiving basin. The effects of the bottom topography as well as the Coriolis force, in the root and dispersal of the studied turbidity current, are highly evident. More specifically, a big part of the turbidity current deviates to the left of the main inflow axis, due to the general slope gradient of the bottom relief in this direction. Another, smaller part of the turbidity current deviates to the right of the main inflow axis, due to the Coriolis force effect, encountering even negative slope gradients. The different responses, of the different types of suspended sediment particles (fine sand and very fine sand) within the flow field, are also characteristic. In more detail, in the concentration distributions, the upper, more dilute layer of the very fine sand concentration occupies more height within the flow field, than in the case of the fine sand case. This fact is obviously due to the different settling velocities, between the particles of the very fine sand and the fine sand.

Finally, the overall results presented in the present chapter indicate, the capabilities of the proposed numerical approach, as a possible and suitable tool for the further investigation of the hydrodynamic behavior of turbidity currents. It is shown that the proposed numerical approach can constitute a quite attractive alternative to laboratory experiments and field measurements since it allows the identification and the continuous monitoring of a wide range of flow parameters, with a relatively high accuracy.

#### **6. References**

ANSYS FLUENT Documentation, (2010). *User's Guide,* Version 13.0


while the variation of the channel slope causes a moderate effect in the turbidity current

From the field scale application it can be concluded that for the assumed initial conditions, the inflow of the suspended sediment – fresh water mixture from Evros River into the North Aegean Sea Basin, forms a turbidity current, which plunges to the bottom of the receiving basin. The effects of the bottom topography as well as the Coriolis force, in the root and dispersal of the studied turbidity current, are highly evident. More specifically, a big part of the turbidity current deviates to the left of the main inflow axis, due to the general slope gradient of the bottom relief in this direction. Another, smaller part of the turbidity current deviates to the right of the main inflow axis, due to the Coriolis force effect, encountering even negative slope gradients. The different responses, of the different types of suspended sediment particles (fine sand and very fine sand) within the flow field, are also characteristic. In more detail, in the concentration distributions, the upper, more dilute layer of the very fine sand concentration occupies more height within the flow field, than in the case of the fine sand case. This fact is obviously due to the different settling velocities,

Finally, the overall results presented in the present chapter indicate, the capabilities of the proposed numerical approach, as a possible and suitable tool for the further investigation of the hydrodynamic behavior of turbidity currents. It is shown that the proposed numerical approach can constitute a quite attractive alternative to laboratory experiments and field measurements since it allows the identification and the continuous monitoring of a wide

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between the particles of the very fine sand and the fine sand.

range of flow parameters, with a relatively high accuracy.

ANSYS FLUENT Documentation, (2010). *User's Guide,* Version 13.0

**6. References** 

3091

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15 pp., ISSN 2156–2202

Greece, August, 2003

Vol. 23, Bariloche, Argentina


**4** 

*1Ukraine 2Mexico 3Australia* 

**Numerical Simulation of the Unsteady** 

**Shock Interaction of Blunt Body Flows** 

*1National Aerospace University "Kharkov Aviation Institute"* 

*3Western Australian School of Mines, Curtin University* 

*2National Polytechnic Institute* 

Leonid Bazyma1, Vasyl Rashkovan2 and Vladimir Golovanevskiy3

Supersonic and hypersonic space vehicles are extremely sensitive to aerodynamic resistance. The combination of the two main rocket operation factors, low altitude and high velocity, produces considerable heat flows in the stagnation region of the nose. For this reason, passive heat-transfer analysis under such conditions is very important for understanding

The possibility of use of energy supply as the method of the overall control of the airflow is defined in the experimental and theoretical research (Adegren et al., 2001, 2005; Bazyma & Rashkovan, 2005; Tret'yakov et al., 1994, 1996). As an example, reduction of head resistance of a supersonic aircraft can be achieved with the introduction of energy into the contrary incoming flow. On the other hand, supply of energy may be used to minimize negative consequences of the shock-wave interaction when the streamlining of the aerodynamic configurations of the compound form occurs. For example, oblique shock waves, which are distributed from the bow part of an airplane or a rocket, can interact with a bow shock wave of any part of the fuselage construction (tail unit, suspension, hood, diffuser, etc.). In certain cases the shock-wave interaction can result in significant negative and even catastrophic

The use of a laser to supply energy has been experimentally shown to be a good approach for both general and local flow control. Experimental research (Tret'yakov et al., 1994, 1996) shows that an extensive region of energy supply is realized in the supersonic flow when a powerful optical pulsating discharge is applied, with a thermal wake developing behind the

A cone or hemisphere in the thermal wake, located from 1.0 to 4.0 diameters distance from the focal plane of irradiation from a CO2 laser, results in a reduction of aerodynamic drag of over a factor of two when a 100-kHz pulse frequency is applied (Tret'yakov et al., 1996). The thermal wake becomes continuous for 10–100-kHz radiation pulses (Tret'yakov et al., 1996).

**1. Introduction** 

and solving rocket operation problems.

consequences for the aircraft.

area where the energy is supplied.


Mehdizadeh, A.; Firoozabadi, B. & Farhanieh, B. (2008). Numerical Simulation of Turbidity

Current Using *<sup>2</sup> v f* Turbulence Model. *Journal of Applied Fluid Mechanics*, Vol.1, No. 2, pp. 45-55, ISSN 1735-3645


## **Numerical Simulation of the Unsteady Shock Interaction of Blunt Body Flows**

Leonid Bazyma1, Vasyl Rashkovan2 and Vladimir Golovanevskiy3

*1National Aerospace University "Kharkov Aviation Institute" 2National Polytechnic Institute 3Western Australian School of Mines, Curtin University 1Ukraine 2Mexico 3Australia* 

#### **1. Introduction**

72 Numerical Modelling

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Mulder, T. & Alexander, J. (2001). The Physical Character of Subaqueous Sedimentary

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Channelized Turbiditic System: The Eocene Kusuri Formation in the Sinop Basin, North-Central Turkey, In: *Sedimentary Processes, Environments and Basins: A Tribute to Peter Friend,* G. Nichols, E. Williams & C. Paola (Eds.), 457-517, Blackwell

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Experimental Results and Possible Geological Significance. *Sedimentology*, Vol.17,

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Density Flows and their Deposits. *Sedimentology*, Vol.48, No.2, (December 2001), pp.

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Supersonic and hypersonic space vehicles are extremely sensitive to aerodynamic resistance. The combination of the two main rocket operation factors, low altitude and high velocity, produces considerable heat flows in the stagnation region of the nose. For this reason, passive heat-transfer analysis under such conditions is very important for understanding and solving rocket operation problems.

The possibility of use of energy supply as the method of the overall control of the airflow is defined in the experimental and theoretical research (Adegren et al., 2001, 2005; Bazyma & Rashkovan, 2005; Tret'yakov et al., 1994, 1996). As an example, reduction of head resistance of a supersonic aircraft can be achieved with the introduction of energy into the contrary incoming flow. On the other hand, supply of energy may be used to minimize negative consequences of the shock-wave interaction when the streamlining of the aerodynamic configurations of the compound form occurs. For example, oblique shock waves, which are distributed from the bow part of an airplane or a rocket, can interact with a bow shock wave of any part of the fuselage construction (tail unit, suspension, hood, diffuser, etc.). In certain cases the shock-wave interaction can result in significant negative and even catastrophic consequences for the aircraft.

The use of a laser to supply energy has been experimentally shown to be a good approach for both general and local flow control. Experimental research (Tret'yakov et al., 1994, 1996) shows that an extensive region of energy supply is realized in the supersonic flow when a powerful optical pulsating discharge is applied, with a thermal wake developing behind the area where the energy is supplied.

A cone or hemisphere in the thermal wake, located from 1.0 to 4.0 diameters distance from the focal plane of irradiation from a CO2 laser, results in a reduction of aerodynamic drag of over a factor of two when a 100-kHz pulse frequency is applied (Tret'yakov et al., 1996). The thermal wake becomes continuous for 10–100-kHz radiation pulses (Tret'yakov et al., 1996).

Numerical Simulation of the Unsteady Shock Interaction of Blunt Body Flows 75

The equations of gas dynamics in the cylindrical coordinates, in contrast to (Bazyma &

<sup>0</sup> *r ur r*

 

> 

> >

 

 

> 

, ω - components of the velocity vector on *x*, *r* and *φ*

 

; (1)

; (2)

; (3)

; (4)

, (5)

*e* . (6)

, (7)

 

, (8)

 

<sup>2</sup> ( ) <sup>0</sup> *ur p ur ur u*

<sup>2</sup> *r ur* ( ) *p r <sup>p</sup> tx r* 

 

 

2 2 ( )( ) <sup>0</sup> *r ur p rp*

*er ue p r e p r e p* ( /) ( /) ( /) *qr tx r* 

respectively; *е* – total energy of the mass unit of the gas; *q* – energy supplied to the mass of the gas by the external source; *t* – time. The system is completed with the perfect gas

> *p* ( 1)

The energy supply is prescribed the same as in (Bazyma & Rashkovan, 2005; Guvernyuk &

1 <sup>1</sup> (,)*n <sup>n</sup> q Wxr t f f* 

where δ – is the Dirak's impulse function; *f* – pulse repetition rate; *W* – average mass density of the energy supply. Here, in contrast to (Bazyma & Rashkovan, 2005; Guvernyuk & Samoilov, 1997), *W* was taken in the form that permits modeling different shapes of heat

0 2

 

 3/2 2 2 <sup>2</sup> 1 2 03

<sup>1</sup> cos ( ) sin exp *<sup>p</sup> kr kx x kr*

 

where *W*0, *k*1, *k*2, *k*3 and *L* are constants defining deposited energy density and thermal spot

Similar to (Bazyma & Rashkovan, 2005), the solution of the system of equations (1-4) was conducted using the Godunov´s method (Godunov, 1976). The 11060 and the 1106032

*R L*

 

 

 

> 

 

 

> 

 

*tx r*

*tx r* 

 

 

 

 

Rashkovan, 2005) are in the form that includes the azimuthal component

 

 

   

*tx r*

 

 


spot at the asymmetric energy supply:

*W W*

**3. Numerical method** 

 

where *p* - pressure;

equation:

shape.

Samoilov, 1997):

Theoretical modeling results of the influence of a heat-release pulsating source on the supersonic flow around a hemisphere are presented by Guvernyuk & Samoilov (1997). The explicit total-variation-diminishing (TVD) method in Chakravarthy's) formulae (Chakravarthy & Osher, 1985; Chakravarthy, 1986) is applied in these calculations (Guvernyuk & Samoilov, 1997). In the case of *M* =3*,* =1*.*4 and a constant deposited energy per unit mass, the aerodynamic load upon the body exhibited a decrease. A pulse repetition rate corresponded to a minimum drag was determined and it was concluded that the use of the pulsating energy supply might be more effective than a constant energy source.

Other results (Georgievskii & Levin, 1988) show that pulse repetition rate, power supplied to the flow and the area into which this energy is supplied all greatly influence both the pressure distribution and the model surface and its flow regimes.

The energy supply parameters such as intensity and heat spot configuration influence the flow re-formation significantly as their combination determines the possibility of either airflow choking in the source (i.e. with the separated wave) or the choke-free flow. This considerably affects the spot properties behind the flow and consequently the stagnation pressure and configuration resistance.

This work presents the results of numerical simulation of the flow around a hemisphere at both the symmetric and asymmetric energy supply into the flow, when the energy supply is realized at 900 angle to the velocity vector of the incoming supersonic airflow.

The two types of the heat spot form considered were: the axis-symmetric spot (i.e. thin disk in the two-dimensional space) and the heat spot of the ellipsoidal form (in the threedimensional space) with its main axis perpendicular to the symmetry axis. The results of the numerical simulation correspond well with the experimental data (Adegren et al., 2001).

#### **2. Problem definition**

Let us consider the energy supply from above the sphere and along the airflow at Mach number *M*∞ = 3.45 and ratio of specific heats = 1.4 in the incoming supersonic gas flow, i.e. the same as in (Adegren et al., 2001). The energy supply scheme is illustrated in Figure 1. Assume that at the time *t* =0, a pulsating power supply source is initiated in front of the sphere.

Fig. 1. Schematic of energy supply.

Theoretical modeling results of the influence of a heat-release pulsating source on the supersonic flow around a hemisphere are presented by Guvernyuk & Samoilov (1997). The explicit total-variation-diminishing (TVD) method in Chakravarthy's) formulae (Chakravarthy & Osher, 1985; Chakravarthy, 1986) is applied in these calculations

per unit mass, the aerodynamic load upon the body exhibited a decrease. A pulse repetition rate corresponded to a minimum drag was determined and it was concluded that the use of

Other results (Georgievskii & Levin, 1988) show that pulse repetition rate, power supplied to the flow and the area into which this energy is supplied all greatly influence both the

The energy supply parameters such as intensity and heat spot configuration influence the flow re-formation significantly as their combination determines the possibility of either airflow choking in the source (i.e. with the separated wave) or the choke-free flow. This considerably affects the spot properties behind the flow and consequently the stagnation

This work presents the results of numerical simulation of the flow around a hemisphere at both the symmetric and asymmetric energy supply into the flow, when the energy supply is

The two types of the heat spot form considered were: the axis-symmetric spot (i.e. thin disk in the two-dimensional space) and the heat spot of the ellipsoidal form (in the threedimensional space) with its main axis perpendicular to the symmetry axis. The results of the numerical simulation correspond well with the experimental data (Adegren et al., 2001).

Let us consider the energy supply from above the sphere and along the airflow at Mach

the same as in (Adegren et al., 2001). The energy supply scheme is illustrated in Figure 1. Assume that at the time *t* =0, a pulsating power supply source is initiated in front of the

realized at 900 angle to the velocity vector of the incoming supersonic airflow.

the pulsating energy supply might be more effective than a constant energy source.

=1*.*4 and a constant deposited energy

= 1.4 in the incoming supersonic gas flow, i.e.

(Guvernyuk & Samoilov, 1997). In the case of *M* =3*,*

pressure and configuration resistance.

number *M*∞ = 3.45 and ratio of specific heats

Fig. 1. Schematic of energy supply.

**2. Problem definition** 

sphere.

pressure distribution and the model surface and its flow regimes.

The equations of gas dynamics in the cylindrical coordinates, in contrast to (Bazyma & Rashkovan, 2005) are in the form that includes the azimuthal component

$$\frac{\partial \rho r}{\partial t} + \frac{\partial \rho ur}{\partial \mathbf{x}} + \frac{\partial \rho vr}{\partial r} + \frac{\partial \rho o}{\partial \mathbf{p}} = \mathbf{0} \; ; \tag{1}$$

$$\frac{\partial \text{pur}}{\partial t} + \frac{\partial (p + \rho u^2)r}{\partial \mathbf{x}} + \frac{\partial \text{pur}r}{\partial r} + \frac{\partial \text{puo}\rho}{\partial \rho} = \mathbf{0} \; ; \tag{2}$$

$$\frac{\partial \rho \upsilon r}{\partial t} + \frac{\partial \rho \upsilon \upsilon r}{\partial \upsilon} + \frac{\partial (p + \rho \upsilon^2)r}{\partial r} + \frac{\partial \rho \upsilon \upsilon o}{\partial p} = p \ ; \tag{3}$$

$$\frac{\partial \text{por}}{\partial t} + \frac{\partial \text{puor}}{\partial \text{x}} + \frac{\partial (p + \rho \nu^2) r}{\partial r} + \frac{\partial (p + \rho \nu^2)}{\partial \text{p}} = 0 \; ; \tag{4}$$

$$\frac{\partial \rho \text{er}}{\partial t} + \frac{\partial \rho u (e + p \;/\; \rho) r}{\partial \mathbf{x}} + \frac{\partial \rho v (e + p \;/\; \rho) r}{\partial r} + \frac{\partial \rho o (e + p \;/\; \rho)}{\partial \mathbf{p}} = \rho q r \,, \tag{5}$$

where *p* - pressure; - density; *u*, , ω - components of the velocity vector on *x*, *r* and *φ* respectively; *е* – total energy of the mass unit of the gas; *q* – energy supplied to the mass of the gas by the external source; *t* – time. The system is completed with the perfect gas equation:

$$p = (\gamma - 1)\rho e\,,\tag{6}$$

The energy supply is prescribed the same as in (Bazyma & Rashkovan, 2005; Guvernyuk & Samoilov, 1997):

$$q = \mathcal{W}(\mathbf{x}, r) \sum\_{n=1}^{\infty} \frac{1}{f} \mathcal{S} \left( t - \frac{n}{f} \right),\tag{7}$$

where δ – is the Dirak's impulse function; *f* – pulse repetition rate; *W* – average mass density of the energy supply. Here, in contrast to (Bazyma & Rashkovan, 2005; Guvernyuk & Samoilov, 1997), *W* was taken in the form that permits modeling different shapes of heat spot at the asymmetric energy supply:

$$\mathcal{W} = \mathcal{W}\_0 \left(\frac{p\_o}{\rho\_o}\right)^{3/2} \frac{1}{R} \exp\left(-\frac{k\_1 \left(r \cos \varphi\right)^2 + k\_2 \left(\mathbf{x} - \mathbf{x}\_0\right)^2 + k\_3 \left(r \sin \varphi\right)^2}{L^2}\right),\tag{8}$$

where *W*0, *k*1, *k*2, *k*3 and *L* are constants defining deposited energy density and thermal spot shape.

#### **3. Numerical method**

Similar to (Bazyma & Rashkovan, 2005), the solution of the system of equations (1-4) was conducted using the Godunov´s method (Godunov, 1976). The 11060 and the 1106032

Numerical Simulation of the Unsteady Shock Interaction of Blunt Body Flows 77

 , max , max , *x y x y u aa u a a* 

The stability condition so given is extended to the quasi-linear equations of gas dynamics. Calculations show that this condition (9) provides the necessary stability. Nevertheless, this

In our work, the time step was chosen from cell to cell according to the stability condition as

1 1 1 1 , , 1 1 , 2 2 , 2 2 2 2 , min *x r n m n m n m x r n m* 

*k k* <sup>1</sup> *K*

, , / , /, , , , , , /,

*r rR x xR t tR a f fa R a aa u ua a a p p a W Wa R*

 

where *a∞* the velocity of the sound of the incident flow. In the following text, we have

The boundary conditions are defined similar to those in (Guvernyuk & Samoilov, 1997). On the surface of the body and along the symmetry axis, solid-wall inviscid boundary conditions were applied. Along the external inflow boundary, undisturbed freestream conditions were utilized. On the downstream outflow boundary, extrapolation of the flow

The initial data in calculations without energy deposition corresponded to the

, *u u* M ,

Subsequent solutions of the flow field about a hemisphere with energy deposition were initialized using the flow field about a hemisphere solution without energy deposition. The

Preliminary supersonic flow calculations around the hemisphere were conducted in (Bazyma & Rashkovan, 2005) to confirm the adequacy of our numerical scheme (see Fig. 3). Data reported in (Guvernyuk & Samoilov, 1997), where the explicit TVD method of

solutions were advanced in time until the average flow conditions were stabilized.

 

 

, , *p*, *W*.

> 0 ,

 , (10)

, (11)

, (12)

2 3

(13)

0 , (14)

 

 

dimensionless parameters of the incident stream:

*p p* 1 /

is the ratio of specific heats.

condition is usually used with right-hand side less than one.

 

In "*k*" space its value with respect to time is calculated using the spacing "*k*+1" as:

In order to use dimensionless values, we make use of the following equalities:

 

where *K* is a safety factor similar in meaning to the Courant number.

 

omitted bars above the dimensionless values *r*, *x*, *t*, *f*, *a*, *u*,

quantities from the adjacent internal boundary was performed.

 , 1 

where *a* is the velocity of the sound.

follows:

where

grids were used for the axis-symmetric and the three-dimensional problems respectivelly. In both cases the grid was designed with densening of the nodes near the body or in the areas of energy supply (i.e. the incoming flow disturbance areas).

The calculations were performed using the same finite difference scheme of the first-order approximation as that used by (Godunov, 1976). Computational grid used in calculations is shown in Figure 2.

Fig. 2. Computational grid.

The necessary and sufficient condition for stability is that the permissible spacing in time *τ* must satisfy the inequality

$$\frac{\tau}{\tau\_x} + \frac{\tau}{\tau\_y} \le 1 \tag{9}$$

resulting from a stability study of Godunov's difference scheme realized on the system of non-stationary acoustic equations on a uniform rectangular (or parallelogram) grid. Here, *x*and *y* are the time spacing of the one-dimensional scheme. Physically *x* and *y* are mean time intervals, in which waves appearing at the break decomposition on the cell boundary reach the neighboring boundaries:

$$
\tau\_x = \frac{\Delta x}{\max\left(u+a, a-u\right)},
\tau\_y = \frac{\Delta y}{\max\left(\upsilon+a, a-\upsilon\right)},\tag{10}
$$

where *a* is the velocity of the sound.

76 Numerical Modelling

grids were used for the axis-symmetric and the three-dimensional problems respectivelly. In both cases the grid was designed with densening of the nodes near the body or in the areas

The calculations were performed using the same finite difference scheme of the first-order approximation as that used by (Godunov, 1976). Computational grid used in calculations is

The necessary and sufficient condition for stability is that the permissible spacing in time *τ*

*x y* 

resulting from a stability study of Godunov's difference scheme realized on the system of non-stationary acoustic equations on a uniform rectangular (or parallelogram) grid. Here,

time intervals, in which waves appearing at the break decomposition on the cell boundary

 

  1

*y* are the time spacing of the one-dimensional scheme. Physically *x* and *y* are mean

(9)

of energy supply (i.e. the incoming flow disturbance areas).

shown in Figure 2.

Fig. 2. Computational grid.

must satisfy the inequality

reach the neighboring boundaries:

*x*and  The stability condition so given is extended to the quasi-linear equations of gas dynamics. Calculations show that this condition (9) provides the necessary stability. Nevertheless, this condition is usually used with right-hand side less than one.

In our work, the time step was chosen from cell to cell according to the stability condition as follows:

$$
\tau \underset{n - \frac{1}{2}, m - \frac{1}{2}}{1} = \left(\frac{\tau\_x \tau\_r}{\tau\_x + \tau\_r}\right)\_{n - \frac{1}{2}, m - \frac{1}{2}} \\
' \overline{\tau} = \underset{n, m}{\min} \,\tau \underset{n - \frac{1}{2}, m - \frac{1}{2}}{1} \\
\tag{11}
$$

In "*k*" space its value with respect to time is calculated using the spacing "*k*+1" as:

$$
\tau^{k+1} = K \overline{\tau}^k \, , \tag{12}
$$

where *K* is a safety factor similar in meaning to the Courant number.

In order to use dimensionless values, we make use of the following equalities:

$$r = \overline{r}\mathbb{R}, \quad \mathbf{x} = \overline{\mathbf{x}}\mathbb{R}, \quad t = \overline{t}\mathbb{R} \mid a\_{\circ}, \quad f = \overline{f} a\_{\circ} \text{ / } \mathbb{R}, \quad a = \overline{a} a\_{\circ}. \tag{13}$$

$$\mathbf{u} = \overline{\mathbf{u}} a\_{\circ}, \quad \mathbf{u} = \overline{\mathbf{u}} a\_{\circ}, \quad \alpha = \overline{\alpha} a\_{\circ}, \quad \rho = \overline{\rho} \rho\_{\circ}, \quad p = \overline{p} \rho\_{\circ} a\_{\circ}^{2}, \quad \mathcal{W} = \overline{\mathcal{W}} a\_{\circ}^{3} \text{ / } \mathbb{R},$$

where *a∞* the velocity of the sound of the incident flow. In the following text, we have omitted bars above the dimensionless values *r*, *x*, *t*, *f*, *a*, *u*, , , *p*, *W*.

The boundary conditions are defined similar to those in (Guvernyuk & Samoilov, 1997). On the surface of the body and along the symmetry axis, solid-wall inviscid boundary conditions were applied. Along the external inflow boundary, undisturbed freestream conditions were utilized. On the downstream outflow boundary, extrapolation of the flow quantities from the adjacent internal boundary was performed.

The initial data in calculations without energy deposition corresponded to the dimensionless parameters of the incident stream:

$$p = p\_{\circ} = 1 \;/\; \rho \; , \; \rho = \rho\_{\circ} = 1 \; , \; u = \mathfrak{u}\_{\circ} = \mathrm{M}\_{\circ} \; , \; \nu = 0 \; , \; \alpha = 0 \; , \tag{14}$$

whereis the ratio of specific heats.

Subsequent solutions of the flow field about a hemisphere with energy deposition were initialized using the flow field about a hemisphere solution without energy deposition. The solutions were advanced in time until the average flow conditions were stabilized.

Preliminary supersonic flow calculations around the hemisphere were conducted in (Bazyma & Rashkovan, 2005) to confirm the adequacy of our numerical scheme (see Fig. 3). Data reported in (Guvernyuk & Samoilov, 1997), where the explicit TVD method of

Numerical Simulation of the Unsteady Shock Interaction of Blunt Body Flows 79

standoff distance, formed recirculation zones and the flow in general reported in

Fig. 4. Mach number isolines while flowing around the hemisphere with supersonic gas flow at pulse repetition rate *f*=2 (*t*=11.2): a) results of (Guvernyuk & Samoilov, 1997); b)

The grid resolution study was also conducted, with test calculations for hemisphere and cavity hemisphere carried out using the 219×119 grid. The 219×119 grid was obtained through twice the 110×60 grid spacing reduction. Minimum surface cell spacing values were 0.024 for the 110×60 grid (reduced to the sphere radius) and 0.012 for the 219×119 grid.

A comparison of the results derived with the use of the 110×60 (continuous line) and 219×119 grids can be seen in Figure 3b above. As the resolution of the 219×119 (dotted line) grid is higher than that of the 110×60 grid, the solution derived with the use of the 219×119 grid marked out some peculiarities of the pressure change on the compression stage. These peculiarities correspond to the solution in (Guvernyuk & Samoilov, 1997) as well; however they were not seen through the grid applied. It is worth noting that solution difference obtained with the 110×60 and 219×119 grids is rather small for the hemisphere with energy

results of (Bazyma & Rashkovan, 2005), 110×60 grid.

deposition.

(Guvernyuk & Samoilov, 1997) and derived in our work are in good agreement.

Chakravarthy's formulation (Chakravarthy & Osher, 1985; Chakravarthy, 1986) was used, showed good correspondence in all the observed flow regimes.

Fig. 3. Pressure at the hemisphere stagnation point versus dimensionless time at the pulse repetition rate *f*=0.5 (spherical heat spot; *W*0 =20, *x*0 = -3.5, *L* = 0.5): a) results of (Guvernyuk & Samoilov, 1997); b) results of (Bazyma & Rashkovan, 2005) (── − 110×60 grid; - - - - − 219×119 grid).

Shown in Figure 3 is the dependence of the pressure at the stagnation point while flowing around the hemisphere (for M∞ = 3, = 1.4) on the dimensionless time at pulse repetition rate *f* = 0.5. Results are shown both from (Guvernyuk & Samoilov, 1997) (a) and from the our previous work (b). As can be seen from Figure 3, the main pulsation parameters (i.e. period and amplitude) and their character obtained in our work correspond well with the data of (Guvernyuk & Samoilov, 1997). However, while the resolution of the numerical scheme used in (Guvernyuk & Samoilov, 1997) is somewhat higher the scheme used in our previous work (Bazyma & Rashkovan, 2005) allows simulation of the quasi-stationary pulsation process.

Shown in Figure 4a is the flow visualization near the hemisphere under the influence of the pulsating thermal source (spherical heat spot; *W*0 =20, *x*0 = -3.5, *L* = 0.5). These results were obtained in (Guvernyuk & Samoilov, 1997), and compared with the analogous data of the (Bazyma & Rashkovan, 2005) (Figure 4b). As can be seen in Figure 4, bow shock wave

Chakravarthy's formulation (Chakravarthy & Osher, 1985; Chakravarthy, 1986) was used,

Fig. 3. Pressure at the hemisphere stagnation point versus dimensionless time at the pulse repetition rate *f*=0.5 (spherical heat spot; *W*0 =20, *x*0 = -3.5, *L* = 0.5): a) results of (Guvernyuk & Samoilov, 1997); b) results of (Bazyma & Rashkovan, 2005) (── − 110×60 grid; - - - - −

Shown in Figure 3 is the dependence of the pressure at the stagnation point while flowing

rate *f* = 0.5. Results are shown both from (Guvernyuk & Samoilov, 1997) (a) and from the our previous work (b). As can be seen from Figure 3, the main pulsation parameters (i.e. period and amplitude) and their character obtained in our work correspond well with the data of (Guvernyuk & Samoilov, 1997). However, while the resolution of the numerical scheme used in (Guvernyuk & Samoilov, 1997) is somewhat higher the scheme used in our previous work (Bazyma & Rashkovan, 2005) allows simulation of the quasi-stationary

Shown in Figure 4a is the flow visualization near the hemisphere under the influence of the pulsating thermal source (spherical heat spot; *W*0 =20, *x*0 = -3.5, *L* = 0.5). These results were obtained in (Guvernyuk & Samoilov, 1997), and compared with the analogous data of the (Bazyma & Rashkovan, 2005) (Figure 4b). As can be seen in Figure 4, bow shock wave

= 1.4) on the dimensionless time at pulse repetition

219×119 grid).

pulsation process.

around the hemisphere (for M∞ = 3,

showed good correspondence in all the observed flow regimes.

standoff distance, formed recirculation zones and the flow in general reported in (Guvernyuk & Samoilov, 1997) and derived in our work are in good agreement.

Fig. 4. Mach number isolines while flowing around the hemisphere with supersonic gas flow at pulse repetition rate *f*=2 (*t*=11.2): a) results of (Guvernyuk & Samoilov, 1997); b) results of (Bazyma & Rashkovan, 2005), 110×60 grid.

The grid resolution study was also conducted, with test calculations for hemisphere and cavity hemisphere carried out using the 219×119 grid. The 219×119 grid was obtained through twice the 110×60 grid spacing reduction. Minimum surface cell spacing values were 0.024 for the 110×60 grid (reduced to the sphere radius) and 0.012 for the 219×119 grid.

A comparison of the results derived with the use of the 110×60 (continuous line) and 219×119 grids can be seen in Figure 3b above. As the resolution of the 219×119 (dotted line) grid is higher than that of the 110×60 grid, the solution derived with the use of the 219×119 grid marked out some peculiarities of the pressure change on the compression stage. These peculiarities correspond to the solution in (Guvernyuk & Samoilov, 1997) as well; however they were not seen through the grid applied. It is worth noting that solution difference obtained with the 110×60 and 219×119 grids is rather small for the hemisphere with energy deposition.

Numerical Simulation of the Unsteady Shock Interaction of Blunt Body Flows 81

The bow shock stand-off distance for the undisturbed model at Mach 3.45 was calculated and compared to the Lobb (Lobb, 1964) approximation to Van Dyke's (Van Dyke, 2003) shock stand-off model. The model predicted the stand-off distances within 3 percent of the

> ( 1) 2 0.41 ( 1) *<sup>M</sup> <sup>D</sup>*

where, Δ is the stand-off distance, *D* is the sphere diameter, and *M*∞ = 3.45 is the freestream

Naturally, obtaining the heat spot form similar to that used in the experiment (Adegren et al., 2001) for the two-dimensional case is impossible. However, with the heat spot size small compared to the size of the streamlined body (i.e. sphere) and low values of the energy density supplied to the incoming flow obtaining some similarity can be expected. For the spheroidal heat spot form (i.e. its axis of rotation coincides with the axis of rotation of the streamlined body), the character of the pressure change in the critical point of the body is sufficiently close (at the stage of compression and the first phase of expansion) to that obtained in the experiment (Adegren et al., 2001) for the value of energy supplied in the

Fig. 5. Pressure variation at the critical point of the sphere versus time: *W*0 = 0.19, *f* = 0.00068,

Increasing the spot size and the energy supply density will not allow to obtain the conditions fully adequate to those of the physical experiment. At the same time, varying these two parameters (i.e. the size and configuration of the spot on the one hand and the energy supply density on the other hand) allow the character of the pressure change in the critical point of the body adequate to the experiment by the higher values of the energy supply density (127 mJ/pulse/1.3±0.7mm3, and 258 mJ/pulse/3±1mm3) to be obtained.

 

2 2

*M* 

, (16)

calculated distances. The model for shock stand-off distance is given as

Mach number.

**4.2 Symmetric energy supply** 

pulse in the order of 13mJ/1mm3 (see Figure 5).

ellipsoid heat point; *k*1 = *k*3 = 0.016, *k*2 =0.39.

The details of the numerical scheme, along with test examples, are given in (Godunov, 1976) and (Bazyma & Kholyavko, 1996; Bazyma & Rashkovan, 2006).

#### **4. Results and discussion**

When calculating the three-dimensional problem, the energy supply modeled was at 900 angle to the velocity vector of the incoming flow. The two types of the heat spot form considered were: the axis-symmetric spot (i.e. thin disk) and the heat spot of the ellipsoidal form with the main axis perpendicular to the symmetry axis.

The dimensionless parameters of the undisturbed contrary flow are assumed as the initial data in calculations without power supply.

The Table 1 shows the list of operational parameters for the facility used in the wind tunnel experiments (Adegren et al., 2001). This tunnel is a basic blowdown tunnel with an exhaust into atmospheric pressure.


Table 1. Operating Parameters for the Rutgers Mach 3.45 Supersonic Wind Tunnel.

#### **4.1 General calculation procedure**

The energy supplied to the mass unit of gas is prescribed in the form

$$q = \gamma^{-3/2} \mathcal{W}\_0 \exp\left(-\frac{k\_1 \left(r \cos \varphi\right)^2 + k\_2 \left(\mathbf{x} - \mathbf{x}\_0\right)^2 + k\_3 \left(r \sin \varphi\right)^2}{L^2}\right) t^\* \delta(t - t^\*)\,. \tag{15}$$

Here *x*0 = -3.0 (the energy is supplied at the distance of one diameter of the sphere from its surface, (Adegren et al., 2001)), *t*\* = *f*--1 at the pulse frequency *f* =0.00068 (that corresponds to the frequency 10 Hz, (Adegren et al., 2001)). The form of the heat spot is defined by the parameters *L*, *k*1, *k*2, *k*3. The value *L* =0.01 is fixed in all the calculations; the values *k*1, *k*2, *k*<sup>3</sup> are being variated, that permitted to obtain the heat spot dimensions characteristic for the experiment (Adegren et al., 2001) (the volume of the heat spot is evaluated approximately from 1 to 3 mm3; The sphere radius in the experiment is 12.75 mm). Thus, for example, at *k*<sup>1</sup> = *k*2 = *k*<sup>3</sup> one can obtain a spherical heat spot.

The parameter *W*<sup>0</sup> is being varied in the range 0.19 – 1.75 that in total with the selection of values *k*1, *k*2, *k*3 provides the change of the energy density in the impulse that is provided in the experiment (Adegren et al., 2001) (13 mJ/pulse/1±0.5mm3, 127 mJ/pulse/1.3±0.7mm3, and 258 mJ/pulse/3±1mm3).

The bow shock stand-off distance for the undisturbed model at Mach 3.45 was calculated and compared to the Lobb (Lobb, 1964) approximation to Van Dyke's (Van Dyke, 2003) shock stand-off model. The model predicted the stand-off distances within 3 percent of the calculated distances. The model for shock stand-off distance is given as

$$
\Delta = 0.41D \frac{\text{(\(\gamma - 1\))} \ M\_{\text{\(\gamma + 1\)}}^2 + 2}{\text{(\(\gamma + 1\))} \ M\_{\text{\(\gamma - 1\)}}^2} \text{ ,} \tag{16}
$$

where, Δ is the stand-off distance, *D* is the sphere diameter, and *M*∞ = 3.45 is the freestream Mach number.

#### **4.2 Symmetric energy supply**

80 Numerical Modelling

The details of the numerical scheme, along with test examples, are given in (Godunov, 1976)

When calculating the three-dimensional problem, the energy supply modeled was at 900 angle to the velocity vector of the incoming flow. The two types of the heat spot form considered were: the axis-symmetric spot (i.e. thin disk) and the heat spot of the ellipsoidal

The dimensionless parameters of the undisturbed contrary flow are assumed as the initial

The Table 1 shows the list of operational parameters for the facility used in the wind tunnel experiments (Adegren et al., 2001). This tunnel is a basic blowdown tunnel with an exhaust

and (Bazyma & Kholyavko, 1996; Bazyma & Rashkovan, 2006).

form with the main axis perpendicular to the symmetry axis.

Mach Number 3.45 Operating Stagnation Pressure 1.4 MPa Typical Stagnation Temperature 290 K Mass Flow Rate 9.8 Kg/s Total Run Time 1.8 minutes Test Area Cross Section 15 cm x 15 cm

Test Area Length 30 cm

The energy supplied to the mass unit of gas is prescribed in the form

0 2

Table 1. Operating Parameters for the Rutgers Mach 3.45 Supersonic Wind Tunnel.

 2 2 <sup>2</sup> 3/2 1 2 03 \* \*

*q W t tt L*

Here *x*0 = -3.0 (the energy is supplied at the distance of one diameter of the sphere from its surface, (Adegren et al., 2001)), *t*\* = *f*--1 at the pulse frequency *f* =0.00068 (that corresponds to the frequency 10 Hz, (Adegren et al., 2001)). The form of the heat spot is defined by the parameters *L*, *k*1, *k*2, *k*3. The value *L* =0.01 is fixed in all the calculations; the values *k*1, *k*2, *k*<sup>3</sup> are being variated, that permitted to obtain the heat spot dimensions characteristic for the experiment (Adegren et al., 2001) (the volume of the heat spot is evaluated approximately from 1 to 3 mm3; The sphere radius in the experiment is 12.75 mm). Thus, for example, at *k*<sup>1</sup>

The parameter *W*<sup>0</sup> is being varied in the range 0.19 – 1.75 that in total with the selection of values *k*1, *k*2, *k*3 provides the change of the energy density in the impulse that is provided in the experiment (Adegren et al., 2001) (13 mJ/pulse/1±0.5mm3, 127 mJ/pulse/1.3±0.7mm3,

cos ( ) sin exp ( ) *kr kx x kr*

 

 

. (15)

**4. Results and discussion** 

into atmospheric pressure.

**4.1 General calculation procedure** 

= *k*2 = *k*<sup>3</sup> one can obtain a spherical heat spot.

and 258 mJ/pulse/3±1mm3).

data in calculations without power supply.

Naturally, obtaining the heat spot form similar to that used in the experiment (Adegren et al., 2001) for the two-dimensional case is impossible. However, with the heat spot size small compared to the size of the streamlined body (i.e. sphere) and low values of the energy density supplied to the incoming flow obtaining some similarity can be expected. For the spheroidal heat spot form (i.e. its axis of rotation coincides with the axis of rotation of the streamlined body), the character of the pressure change in the critical point of the body is sufficiently close (at the stage of compression and the first phase of expansion) to that obtained in the experiment (Adegren et al., 2001) for the value of energy supplied in the pulse in the order of 13mJ/1mm3 (see Figure 5).

Fig. 5. Pressure variation at the critical point of the sphere versus time: *W*0 = 0.19, *f* = 0.00068, ellipsoid heat point; *k*1 = *k*3 = 0.016, *k*2 =0.39.

Increasing the spot size and the energy supply density will not allow to obtain the conditions fully adequate to those of the physical experiment. At the same time, varying these two parameters (i.e. the size and configuration of the spot on the one hand and the energy supply density on the other hand) allow the character of the pressure change in the critical point of the body adequate to the experiment by the higher values of the energy supply density (127 mJ/pulse/1.3±0.7mm3, and 258 mJ/pulse/3±1mm3) to be obtained.

Numerical Simulation of the Unsteady Shock Interaction of Blunt Body Flows 83

Before energy supply Start of energy supply 40 μs

50 μs 70 μs 80 μs

110 μs 150 μs 240 μs

Fig. 7. Time histories of pressure isolines:

*W*0 = 0.38, *f* = 0.00068, disk heat point; *k*1 = *k*3 = 0.00045, *k*2 =0.089.

Character of pressure variation in the critical point of the sphere versus time for other values of *k*1, *k*2, and *k*3 providing smaller volume of the heat spot (i.e. by the factor of 2 approximately) but twice the energy supply density (i.e. *W*0= 0.38) is shown in Figure 6, curve 1. It is worth noting that the character of the pressure variation in both cases is similar. Curves 2 and 3 in Figure 6, with a considerable difference in pressure amplitude at the compression stage, are obtained for the form of the heat spot that is the flattened axissymmetric disk with its radius comparable with the radius of the sphere.

Fig. 6. Pressure variation at the critical point of the sphere versus time: 1 – *W*0 = 0.38, *f* = 0.00068, ellipsoid heat point; *k*1 = *k*3 = 0.016, *k*2 =0.56; 2 – *W*0 = 0.38, *f* = 0.00068, disk heat point; *k*1 = *k*3 = 0.00045, *k*2 =0.089;3 – *W*0 = 0.38, *f* = 0.00068, disk heat point; *k*1 = *k*3 = 0.00023, *k*<sup>2</sup> =0.082.

As in the experiment (Adegren et al., 2001; refer figure 20) the time history of recorded pressure at centerline location of sphere surface for the three energy levels shows a common behavior comprised of an initial pressure rise, expansion, compression and transient decay. The expansion, compression and transient decay are similar to the ideal gas Euler simulations of (Georgievskii & Levin, 1993) for the interaction of a thermal spot with a sphere at Mach 3. The expansion lowers the surface pressure at the centerline by 40%. However, the initial compression phase observed in our research and in the experiment (Adegren et al., 2001) was not noted by (Georgievskii & Levin, 1993).

The interaction of the thermal spot, with the bow shock (Figure 7-9, t = 40-90 microseconds) causes a blooming of the bow shock (due to the lens effect of the thermal spot). This behavior is consistent with the simulations of Georgievski and Levin (Georgievskii & Levin, 1993).

Character of pressure variation in the critical point of the sphere versus time for other values of *k*1, *k*2, and *k*3 providing smaller volume of the heat spot (i.e. by the factor of 2 approximately) but twice the energy supply density (i.e. *W*0= 0.38) is shown in Figure 6, curve 1. It is worth noting that the character of the pressure variation in both cases is similar. Curves 2 and 3 in Figure 6, with a considerable difference in pressure amplitude at the compression stage, are obtained for the form of the heat spot that is the flattened axis-

Fig. 6. Pressure variation at the critical point of the sphere versus time: 1 – *W*0 = 0.38, *f* = 0.00068, ellipsoid heat point; *k*1 = *k*3 = 0.016, *k*2 =0.56; 2 – *W*0 = 0.38, *f* = 0.00068, disk heat point; *k*1 = *k*3 = 0.00045, *k*2 =0.089;3 – *W*0 = 0.38, *f* = 0.00068, disk heat point; *k*1 = *k*3 = 0.00023, *k*<sup>2</sup>

As in the experiment (Adegren et al., 2001; refer figure 20) the time history of recorded pressure at centerline location of sphere surface for the three energy levels shows a common behavior comprised of an initial pressure rise, expansion, compression and transient decay. The expansion, compression and transient decay are similar to the ideal gas Euler simulations of (Georgievskii & Levin, 1993) for the interaction of a thermal spot with a sphere at Mach 3. The expansion lowers the surface pressure at the centerline by 40%. However, the initial compression phase observed in our research and in the experiment

The interaction of the thermal spot, with the bow shock (Figure 7-9, t = 40-90 microseconds) causes a blooming of the bow shock (due to the lens effect of the thermal spot). This behavior is consistent with the simulations of Georgievski and Levin

(Adegren et al., 2001) was not noted by (Georgievskii & Levin, 1993).

=0.082.

(Georgievskii & Levin, 1993).

symmetric disk with its radius comparable with the radius of the sphere.

110 μs 150 μs 240 μs

Fig. 7. Time histories of pressure isolines: *W*0 = 0.38, *f* = 0.00068, disk heat point; *k*1 = *k*3 = 0.00045, *k*2 =0.089.

Numerical Simulation of the Unsteady Shock Interaction of Blunt Body Flows 85

Before energy supply Start of energy supply 50 μs

60 μs 70 μs 80 μs

110 μs 150 μs 240 μs

Fig. 9. Time histories of pressure isolines:

*W*0 = 0.38, *f* = 0.00068, disk heat point; *k*1 = *k*3 = 0.00023, *k*2 =0.082.

Fig. 8. Time histories of pressure isolines:

*W*0 = 0.38, *f* = 0.00068, disk heat point; *k*1 = *k*3 = 0.00045, *k*2 =0.089.

Before energy supply Start of energy supply 50 μs

60 μs 70 μs 80 μs

110 μs 150 μs 240 μs

Fig. 8. Time histories of pressure isolines:

*W*0 = 0.38, *f* = 0.00068, disk heat point; *k*1 = *k*3 = 0.00045, *k*2 =0.089.

Fig. 9. Time histories of pressure isolines: *W*0 = 0.38, *f* = 0.00068, disk heat point; *k*1 = *k*3 = 0.00023, *k*2 =0.082.

Numerical Simulation of the Unsteady Shock Interaction of Blunt Body Flows 87

Before energy supply Start of energy supply

10 μs 50 μs

80 μs 240 μs

Fig. 11. Time histories of pressure isolines:

*W*0 = 0.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 0.67, *k*1 =0.007.

Naturally, with these conditions satisfied the energy supply density was even less than in the experiment (Adegren et al., 2001). However, the character of the pressure variation at the stage of compression and expansion was similar to that obtained in the experiment for energy supply densities of 127 mJ/pulse/1.3±0.7mm3 and 258 mJ/pulse/3±1mm3.

#### **4.3 Asymmetric energy supply**

Pressure variation at the critical point of the sphere, relevant to the corresponding pressure obtained before the heat influence, versus time after the energy supply is shown in Figure 10.

Fig. 10. Pressure variation at the critical point of the sphere versus time: 1 – *W*0 = 0.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 0.67, *k*1 =0.007; 2 – *W*0 = 1.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 1, *k*1 =0.06; 3 – *W*0 = 1.75, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*<sup>3</sup> = 1, *k*1 =0.17.

Figure 11 shows the pressure history for the asymmetric energy deposition.

The Mach number in the contrary flow is M=3.45. The distance from the heat spot to the sphere was equal to one diameter of the sphere. The zone of the energy supply was an ellipsoid with the volume in the order of 1mm3 with its rotational axis perpendicular to the incoming flow velocity vector. The process of interaction of the heat track and the sphere contained two stages: short stage of compression and long stage of expansion. The obtained results are in good agreement with the experimental data (Adegren et al., 2001), both quantitatively and qualitatively.

Figure 12 shows the fields of the equal pressure on the sphere surface at 500 μs time after the heat influence (here the pressure is related to the contrary flow pressure). As can be seen, by this time the symmetrical vortex areas on the surface of the sphere still exist. It is worthwhile to point out that the flow rotation velocity is sufficiently high; with its maximum value reaching 0.3 of the contrary flow sound velocity (see Figure 13).

Naturally, with these conditions satisfied the energy supply density was even less than in the experiment (Adegren et al., 2001). However, the character of the pressure variation at the stage of compression and expansion was similar to that obtained in the experiment for

Pressure variation at the critical point of the sphere, relevant to the corresponding pressure obtained before the heat influence, versus time after the energy supply is shown

energy supply densities of 127 mJ/pulse/1.3±0.7mm3 and 258 mJ/pulse/3±1mm3.

Fig. 10. Pressure variation at the critical point of the sphere versus time:

Figure 11 shows the pressure history for the asymmetric energy deposition.

1 – *W*0 = 0.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 0.67, *k*1 =0.007; 2 – *W*0 = 1.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 1, *k*1 =0.06; 3 – *W*0 = 1.75, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*<sup>3</sup>

The Mach number in the contrary flow is M=3.45. The distance from the heat spot to the sphere was equal to one diameter of the sphere. The zone of the energy supply was an ellipsoid with the volume in the order of 1mm3 with its rotational axis perpendicular to the incoming flow velocity vector. The process of interaction of the heat track and the sphere contained two stages: short stage of compression and long stage of expansion. The obtained results are in good agreement with the experimental data (Adegren et al., 2001), both

Figure 12 shows the fields of the equal pressure on the sphere surface at 500 μs time after the heat influence (here the pressure is related to the contrary flow pressure). As can be seen, by this time the symmetrical vortex areas on the surface of the sphere still exist. It is worthwhile to point out that the flow rotation velocity is sufficiently high; with its

maximum value reaching 0.3 of the contrary flow sound velocity (see Figure 13).

**4.3 Asymmetric energy supply** 

in Figure 10.

= 1, *k*1 =0.17.

quantitatively and qualitatively.

Fig. 11. Time histories of pressure isolines: *W*0 = 0.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 0.67, *k*1 =0.007.

Numerical Simulation of the Unsteady Shock Interaction of Blunt Body Flows 89





The application of energy deposition for local flow control requires low power in terms of the energy deposition into the flow. This low power requirement could potentially translate into small, low weight energy generation systems utilizing optical lasers or electric arc units etc. for effective and efficient flow control. This warrants consideration of the use of energy supply as effective means for solution of local issues of supersonic flow of various aircraft.

The authors would like to express their gratitude to Professor Alexander V. Gaydachuk of the National Aerospace University "Kharkov Aviation Institute" for his input and guidance

Adegren, R.; Elliot, G.; Knight D.; Zheltovodov, A. & Beutner, T. (2001). Energy deposition

Adelgren, R.; Yan, H.; Elliott, G.; Knight D.; Beutner, T. & Zheltovodov, A. (2005). Control

Bazyma, L. & Rashkovan, V. (2005). Stabilization of Blunt Nose Cavity Flows Using Energy

Bazyma, L. & Kholyavko V.I. (1996). A modification of Godunov's finite difference scheme

Bazyma, L. & Rashkovan, V. (2006). Separation Flow Control by the Gas Injection Contrary Supersonic Stream, *AIAA J.,* Vol. 44, No. 12, pp. 2887-2895, ISSN 0001-1452

on Edney IV Interaction by Pulsed Laser Energy Deposition, *AIAA J.,* Vol. 43, No. 2,

Deposition. *Journal of Spacecraft and Rockets*, Vol.42, No.5, (September-October 2005),

on a mobile grid. *Computational Mathematics and Mathematical Physics*, Vol. 36, No.4,

in supersonic flows, *AIAA Paper* N2001–0885, 2001

The results of our research can be summarized as follows:

symmetric and asymmetric energy supply cases.

**5. Conclusion** 

magnitude;

stage of expansion;

**6. Acknowledgment** 

**7. References** 

during the manuscript preparation.

pp. 256-269, ISSN 0001-1452

pp. 790-794, ISSN 0022-4650

pp. 525-532, ISSN 0965-5425

energy supply conditions;

Fig. 12. Pressure on the sphere surface at 500 μs time after the energy supply: a *- W*0 = 0.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 0.67, *k*1 =0.007; b – *W*0 = 1.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 1, *k*1 =0.06; c - *W*0 = 1.75, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 1, *k*1 =0.17.

Fig. 13. The azimuthal velocity component (in the vector form) on the sphere surface at 500 μs time after the energy supply:

a *- W*0 = 0.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 0.67, *k*1 =0.007; b – *W*0 = 1.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 1, *k*1 =0.06;

c - *W*0 = 1.75, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 1, *k*1 =0.17.

The energy supply parameters, i.e. the heat spot configuration and the energy supply intensity, influence the flow reconstruction significantly as their combination defines the possibility of either flow choking in the source (i.e. with the separated wave) or the chokefree flow. This considerably affects the track properties behind the flow and consequently the braking pressure and the configuration resistance. Energy supply into the incoming airflow allows the loads caused by the shock-wave influence on the construction elements of aircraft to be minimised.

### **5. Conclusion**

88 Numerical Modelling

a b c

a b c

Fig. 13. The azimuthal velocity component (in the vector form) on the sphere surface at 500

The energy supply parameters, i.e. the heat spot configuration and the energy supply intensity, influence the flow reconstruction significantly as their combination defines the possibility of either flow choking in the source (i.e. with the separated wave) or the chokefree flow. This considerably affects the track properties behind the flow and consequently the braking pressure and the configuration resistance. Energy supply into the incoming airflow allows the loads caused by the shock-wave influence on the construction elements of

Fig. 12. Pressure on the sphere surface at 500 μs time after the energy supply:

a *- W*0 = 0.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 0.67, *k*1 =0.007; b – *W*0 = 1.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 1, *k*1 =0.06; c - *W*0 = 1.75, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 1, *k*1 =0.17.

a *- W*0 = 0.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 0.67, *k*1 =0.007; b – *W*0 = 1.5, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 1, *k*1 =0.06; c - *W*0 = 1.75, *f* = 0.00068, ellipsoid heat point, *k*2 = *k*3 = 1, *k*1 =0.17.

μs time after the energy supply:

aircraft to be minimised.

The results of our research can be summarized as follows:


The application of energy deposition for local flow control requires low power in terms of the energy deposition into the flow. This low power requirement could potentially translate into small, low weight energy generation systems utilizing optical lasers or electric arc units etc. for effective and efficient flow control. This warrants consideration of the use of energy supply as effective means for solution of local issues of supersonic flow of various aircraft.

### **6. Acknowledgment**

The authors would like to express their gratitude to Professor Alexander V. Gaydachuk of the National Aerospace University "Kharkov Aviation Institute" for his input and guidance during the manuscript preparation.

#### **7. References**


**5** 

*1Iran 2Canada* 

**Numerical Modelling of Heavy Metals** 

Seyed Mahmood Kashefipour1 and Ali Roshanfekr2

*1Department of Hydraulic Structures,* 

*Dalhousie University, Halifax, NS* 

*Shahid Chamran University, Ahwaz, Khuzestan, 2Department of Civil and Resource Engineering,* 

**Transport Processes in Riverine Basins** 

There has been a growing concern in the international community and an increased awareness of riverine pollution problems, particularly with regard to water pollution (Falconer & Lin, 2003). Human and aquatic life is often threatened by the transport of pollutants through riverine systems to coastal waters and it is therefore not surprising to find that, from a water quality point of view, rivers have been studied very extensively and for longer than any other water bodies (Thomann & Mueller, 1987). This is probably due to the fact that people live close to, or interact with, rivers and streams. Many rivers and estuaries have suffered environmental damage due to discharges from manufacturing processes and wastewater from centres of pollution over several decades. In recent years these environmental concerns have made the development of computer models that predict the dispersion of pollutants in natural water systems more urgent. The main attraction of such models, in contrast with physical models, is their low cost and the fact that they easily adapt to new situations. Thus the widespread popularity of mathematical modelling techniques for the hydrodynamic and pollutant transport in rivers justifies any attempt to develop new models based on novel and rigorous approaches (Nassehi & Bikangaga, 1993). This chapter describes numerical modelling of heavy metals in a riverine basin. It would be necessary to recognize and introduce the heavy metals behaviours and different processes during their transportation along the rivers; for example their sources, chemical and physical reactions, and also introducing the environmental conditions affecting the rate of concentration variability of these substances. The one-dimensional (1D) partial differential governing equations (PDE) of hydrodynamic and water quality will be fully described with the corresponding numerical solution methods. As a part of water quality PDE equations the 1D Advection-Dispersion Equation (ADE) will be described and it will be shown that how the dissolved heavy metals, *i.e.* lead and cadmium, may be numerically modelled through the source term of this equation. Details of the development of a modelling approach for predicting dissolved heavy metal fluxes and the application of the model to the Karoon River, located in the south west of Iran are also provided in this chapter. The

**1. Introduction** 


### **Numerical Modelling of Heavy Metals Transport Processes in Riverine Basins**

Seyed Mahmood Kashefipour1 and Ali Roshanfekr2

*1Department of Hydraulic Structures, Shahid Chamran University, Ahwaz, Khuzestan, 2Department of Civil and Resource Engineering, Dalhousie University, Halifax, NS 1Iran 2Canada* 

#### **1. Introduction**

90 Numerical Modelling

Chakravarthy, S. & Osher, S. (1985). New Class of High Accuracy TVD Schemes for

Chakravarthy, S. (1986). The Versality and Reliability of Euler Solvers Based on High-

Georgievskii, P. & Levin, B. (1988). Supersonic Flow Around Bodies in the Presence of

Georgievski, P. & Levin, V. (1993). Unsteady Interaction of a Sphere with Atmospheric

Godunov, S. Ed(s). (1976). *Numerical Solution of Multidimensional Problems in Gas Dynamics*,

Guvernyuk, S. & Samoilov, A. (1997). Control of supersonic flow around bodies by means of

Lobb, R.K. (1964). Experimental Measurement of Shock Detachment Distance on Spheres

Tret'yakov, P.; Grachev, G.; Ivanchenko, A.; Krainev, V.; Ponomarenko, A. & Tishenko, V.

Tret'yakov, P.; Garanin, A.; Grachev, G.; Krainev, V.; Ponomarenko, A.; Tishenko, V. &

Van Dyke, M.D. (2003). The Supersonic Blunt-Body Problem - Review and Extension. *AIAA* 

*Mekhanika Zhidkosti i Gaza*, No. 4, June, pp. 174-183, ISSN 0568-5281

Nelson, WC (ed.), pp. 519-527, Pergamon Press, New York, NY

*Doclady*, Vol.39, No.6, pp. 415-416, ISSN 1063-7753

*J.,* Vol. 41, No. 7, pp. 265-276, ISSN 0001-1452

External Heat Pulsed Sources, *Technical Physics Letters*, Vol. 14, No. 8, pp. 684–687

Temperature In homogeneity at Supersonic Speed, *Akademiya Nauk SSSR, Izvestiya,* 

a pulsed heat source. *Technical Physics Letters*, Vol.23, No.5, pp. 333-336, ISSN 1063-

Fired in Air at Hypervelocities, In *The High Temperature Aspects of Hypersonic Flow*,

(1994). Optical breakdown stabilization in the supersonic argon flow. *Physics-*

Yakovlev, V. (1996). Control of supersonic flow around bodies by means of highpower recurrent optical breakdown. *Physics-Doclady*, Vol.41, No.11, pp. 566-567,

Hyperbolic Conservation Laws, AIAA Paper 85-0363

Accuracy TVD Formulations," AIAA Paper 86-0243

(ISSN 1063-7850)

ISSN 1063-7753

7850

Nauka, Moscow (in Russian)

There has been a growing concern in the international community and an increased awareness of riverine pollution problems, particularly with regard to water pollution (Falconer & Lin, 2003). Human and aquatic life is often threatened by the transport of pollutants through riverine systems to coastal waters and it is therefore not surprising to find that, from a water quality point of view, rivers have been studied very extensively and for longer than any other water bodies (Thomann & Mueller, 1987). This is probably due to the fact that people live close to, or interact with, rivers and streams. Many rivers and estuaries have suffered environmental damage due to discharges from manufacturing processes and wastewater from centres of pollution over several decades. In recent years these environmental concerns have made the development of computer models that predict the dispersion of pollutants in natural water systems more urgent. The main attraction of such models, in contrast with physical models, is their low cost and the fact that they easily adapt to new situations. Thus the widespread popularity of mathematical modelling techniques for the hydrodynamic and pollutant transport in rivers justifies any attempt to develop new models based on novel and rigorous approaches (Nassehi & Bikangaga, 1993).

This chapter describes numerical modelling of heavy metals in a riverine basin. It would be necessary to recognize and introduce the heavy metals behaviours and different processes during their transportation along the rivers; for example their sources, chemical and physical reactions, and also introducing the environmental conditions affecting the rate of concentration variability of these substances. The one-dimensional (1D) partial differential governing equations (PDE) of hydrodynamic and water quality will be fully described with the corresponding numerical solution methods. As a part of water quality PDE equations the 1D Advection-Dispersion Equation (ADE) will be described and it will be shown that how the dissolved heavy metals, *i.e.* lead and cadmium, may be numerically modelled through the source term of this equation. Details of the development of a modelling approach for predicting dissolved heavy metal fluxes and the application of the model to the Karoon River, located in the south west of Iran are also provided in this chapter. The

Numerical Modelling of Heavy Metals Transport Processes in Riverine Basins 93

Fig. 1. Schematic illustration of dissolved heavy metal process in riverine waters.

metals by the particles in solution.

**3. Theoretical background** 

natural or modified river system.

(Ng *et al.,* 1996).

One way to numerically model heavy metals in riverine basins is to assume a reaction coefficient in the source term of the ADE (Advection-Dispersion Equation), which will be described later, showing the presence of the desired substance in the solution. In many studies (such as: Nassehi & Bikangaga, 1993; Shrestha & Orlob, 1996; Wu *et al.,* 2001 & 2005; etc.) the researchers assumed a constant reaction coefficient with time, whereas in the field this coefficient may vary according to the rate of pH, salinity, temperature or even other chemical substances and other hydraulic characteristics of the river. Roshanfekr *et al.* (2008a & 2008b) found that pH and EC play an essential role in adsorption and desorption of heavy

Numerical models provide a valuable tool for predicting the fate and transport of dissolved heavy metals in river environments and are increasingly used for such hydro-environmental management studies of river waters. However, computer-based tools used for predicting such heavy metal concentrations are still used infrequently, even though they can support decision-making by the regulatory authorities, marine environment agencies and industry

The use of computers has provided the opportunity to better understand and assess our water resources through comprehensive numerical model simulations and testing of various schemes or options. The numerical model allows the user to assess the hydraulic conditions in the river basin and thus, establish a better understanding of human impacts upon a

Any numerical model used to predict the flow and dissolved heavy metal transport processes in rivers depends primarily on solving the governing hydro-environmental equations. In most riverine systems, the basin is regarded as a 1D system, with longitudinal flow dominating throughout the system. Any type of hydro-environmental model

model was calibrated and verified against field-measured time series data for discharges, water levels and dissolved lead and cadmium heavy metal concentrations.

It is found that pH and EC play an essential role in adsorption and desorption of heavy metals by the particles in solution (Roshanfekr *et al.*, 2008a & 2008b). Based on the effect of different substances such as pH and salinity on dissolved heavy metal concentrations in rivers, the necessity of heavy metal modelling with more accuracy in predicting the concentration is inevitable.

This chapter provides a methodology predicting a varying reaction coefficient for dissolved lead and cadmium heavy metals using pH and EC (as a function of salinity) which affects the reaction coefficient in the ADE for improved accuracy. Also the procedure for dissolved heavy metal modelling and finding the best relationship between pH and EC with the reaction coefficient is described. Finally, the best relationships for dissolved lead and cadmium reaction coefficients were introduced and the results were successfully compared with the corresponding measured values.

#### **2. Heavy metals and their transport processes in a riverine basin**

Modelling dissolved heavy metal transport in rivers requires a good understanding of the phenomenon. Heavy metals generally exist in two phases in river waters, *i.e.* in the dissolved phase in the water column and in the particulate phase adsorbed on the sediments. The behaviour of heavy metals in the aquatic environment is strongly influenced by adsorption on organic and inorganic particles. The dissolved fraction of heavy metals may be transported via the process of advection-dispersion (Wu *et al.,* 2005). These pollutants are non-conservative in nature and their concentrations depend on salinity and pH, which may vary with time and along a river (Pafilippaki *et al.*, 2008). As a result, the dissolved metal may come out of solution or even re-dissolve, depending on conditions along the time or channel (Nassehi & Bikangaga, 1993).

Figure 1 illustrates the dissolved heavy metal transport process in a riverine basin. This process is very complicated, since presence and mobility of the heavy metal is highly depended on the environmental conditions, *e.g.* bed and suspended sediments. A quick review in the literature for the couple of recent years shows that the main attention was focused mostly on the measurements of heavy metals in alluvial rivers with or without sediments. For example, Rauf *et al.* (2009), Akan *et al.* (2010) and Kumar *et al.* (2011) investigated the effect of sediments on the transport of heavy metals; the seasonal variations of the heavy metals in rivers were investigated by Papafilippaki *et al.* (2008) and Sanayei *et al.* (2009). The effect of heavy metals on the river self purification process was studied by Mala & Maly (2009).

It should be noted that much attention should be given to the study of heavy metal transport dynamics. In fact, the main factors closely related to the heavy metal pollution transport-transformation in natural bodies are: water flow, sediment motion, pH value, water salinity, water temperature, sediment size distribution, sediment concentration, mineral composition for sediments, degree of mineralization of water and time. Thus theoretically the mathematical model of heavy metal transport dynamics should include equations describing all factors mentioned above (Haung *et al.*, 2007; Haung, 2010).

Fig. 1. Schematic illustration of dissolved heavy metal process in riverine waters.

One way to numerically model heavy metals in riverine basins is to assume a reaction coefficient in the source term of the ADE (Advection-Dispersion Equation), which will be described later, showing the presence of the desired substance in the solution. In many studies (such as: Nassehi & Bikangaga, 1993; Shrestha & Orlob, 1996; Wu *et al.,* 2001 & 2005; etc.) the researchers assumed a constant reaction coefficient with time, whereas in the field this coefficient may vary according to the rate of pH, salinity, temperature or even other chemical substances and other hydraulic characteristics of the river. Roshanfekr *et al.* (2008a & 2008b) found that pH and EC play an essential role in adsorption and desorption of heavy metals by the particles in solution.

### **3. Theoretical background**

92 Numerical Modelling

model was calibrated and verified against field-measured time series data for discharges,

It is found that pH and EC play an essential role in adsorption and desorption of heavy metals by the particles in solution (Roshanfekr *et al.*, 2008a & 2008b). Based on the effect of different substances such as pH and salinity on dissolved heavy metal concentrations in rivers, the necessity of heavy metal modelling with more accuracy in predicting the

This chapter provides a methodology predicting a varying reaction coefficient for dissolved lead and cadmium heavy metals using pH and EC (as a function of salinity) which affects the reaction coefficient in the ADE for improved accuracy. Also the procedure for dissolved heavy metal modelling and finding the best relationship between pH and EC with the reaction coefficient is described. Finally, the best relationships for dissolved lead and cadmium reaction coefficients were introduced and the results were successfully compared

Modelling dissolved heavy metal transport in rivers requires a good understanding of the phenomenon. Heavy metals generally exist in two phases in river waters, *i.e.* in the dissolved phase in the water column and in the particulate phase adsorbed on the sediments. The behaviour of heavy metals in the aquatic environment is strongly influenced by adsorption on organic and inorganic particles. The dissolved fraction of heavy metals may be transported via the process of advection-dispersion (Wu *et al.,* 2005). These pollutants are non-conservative in nature and their concentrations depend on salinity and pH, which may vary with time and along a river (Pafilippaki *et al.*, 2008). As a result, the dissolved metal may come out of solution or even re-dissolve, depending on conditions

Figure 1 illustrates the dissolved heavy metal transport process in a riverine basin. This process is very complicated, since presence and mobility of the heavy metal is highly depended on the environmental conditions, *e.g.* bed and suspended sediments. A quick review in the literature for the couple of recent years shows that the main attention was focused mostly on the measurements of heavy metals in alluvial rivers with or without sediments. For example, Rauf *et al.* (2009), Akan *et al.* (2010) and Kumar *et al.* (2011) investigated the effect of sediments on the transport of heavy metals; the seasonal variations of the heavy metals in rivers were investigated by Papafilippaki *et al.* (2008) and Sanayei *et al.* (2009). The effect of heavy metals on the river self purification process was studied by

It should be noted that much attention should be given to the study of heavy metal transport dynamics. In fact, the main factors closely related to the heavy metal pollution transport-transformation in natural bodies are: water flow, sediment motion, pH value, water salinity, water temperature, sediment size distribution, sediment concentration, mineral composition for sediments, degree of mineralization of water and time. Thus theoretically the mathematical model of heavy metal transport dynamics should include

equations describing all factors mentioned above (Haung *et al.*, 2007; Haung, 2010).

water levels and dissolved lead and cadmium heavy metal concentrations.

**2. Heavy metals and their transport processes in a riverine basin** 

concentration is inevitable.

Mala & Maly (2009).

with the corresponding measured values.

along the time or channel (Nassehi & Bikangaga, 1993).

Numerical models provide a valuable tool for predicting the fate and transport of dissolved heavy metals in river environments and are increasingly used for such hydro-environmental management studies of river waters. However, computer-based tools used for predicting such heavy metal concentrations are still used infrequently, even though they can support decision-making by the regulatory authorities, marine environment agencies and industry (Ng *et al.,* 1996).

The use of computers has provided the opportunity to better understand and assess our water resources through comprehensive numerical model simulations and testing of various schemes or options. The numerical model allows the user to assess the hydraulic conditions in the river basin and thus, establish a better understanding of human impacts upon a natural or modified river system.

Any numerical model used to predict the flow and dissolved heavy metal transport processes in rivers depends primarily on solving the governing hydro-environmental equations. In most riverine systems, the basin is regarded as a 1D system, with longitudinal flow dominating throughout the system. Any type of hydro-environmental model

Numerical Modelling of Heavy Metals Transport Processes in Riverine Basins 95

Water quality modelling involves the prediction of water pollution using mathematical simulation techniques. The following sections describe the governing equations and their numerical solution for heavy metals modelling in riverine basins. The model was then

The transport of heavy metals in the dissolved phase can be described by the following one-

The individual terms in the advection-dispersion equation refer to: (1) local effects, (2) transport by advection, (3) longitudinal dispersion and turbulent diffusion, (4) sources or sinks of dissolved heavy metals and (5) transformation term defining absorbed and

*<sup>d</sup> St* =transformation term defining absorbed and desorbed particulate fluxes to or from

*<sup>d</sup> Q SL L <sup>S</sup> x*

*x* =distance between two consecutive cross-sections which can be either constant or variable.

The longitudinal dispersion coefficient ( *Dx* ) in natural rivers is dependent upon many hydrodynamic parameters including: depth, width, velocity and shear velocity (Fischer *et al*. 1979). There are many empirical and/or semi-empirical equations describing this very important dynamic coefficient. Kashefipour & Falconer (2002) presented two empirical equations based on applying the dimensional analysis procedure to more than 80 data sets in 30 natural rivers, to estimate longitudinal dispersion coefficient in natural channels and showed that these equations performed relatively better than the other existing equations. In this chapter the Kashefipour & Falconer (2002) relationship has been used to estimate the

*TU U <sup>D</sup> HU*

 

*HU U*

0.620 0.572 \*

*SA SQ <sup>S</sup> AD S S t xx x* <sup>0</sup>

*d d*

(3)

4 5

<sup>0</sup> (4)

\*

(5)

*x t*

**3.2 Equations for water quality modelling** 

*Dx* =longitudinal dispersion coefficient; *<sup>d</sup> <sup>S</sup>*<sup>0</sup> =source or sink of dissolved heavy metal;

*QL* =lateral inflow or outflow discharge;

sediments (source term).

where:

applied to Karoon River for lead and cadmium modelling.

dimensional advection-dispersion equation (ADE) (Kashefipour, 2002):

desorbed particulate fluxes to or from sediments (source term), where:

*S* =cross-sectional averaged dissolved heavy metal concentration;

Sources or sinks of dissolved heavy metals can be defined as:

*SL* =lateral inflow or outflow dissolved heavy metal concentration;

longitudinal dispersion coefficient. This relationship is written as:

7.428 1.775

*x*

1 2 3

commonly used by environmental engineers and water managers to predict the dissolved heavy metals concentrations in rivers generally involves solving the hydrodynamic and water quality equations as described below.

#### **3.1 Equations for hydrodynamic modelling**

In order to dynamically model the heavy metals in riverine basins, the governing hydrodynamic partial deferential equations must be numerically solved. The governing equations and their numerical solutions for modelling the river hydrodynamics (*i.e.* velocity and water elevation at any point and time) are therefore presented in this section.

For unsteady flow and hydrodynamic modelling the velocity and the water-level at any point of the basin and any time is of interest. The velocity component in rivers is usually assumed as a one-dimensional vector. The one-dimensional governing hydrodynamic equations describing flow and water elevations in rivers are based on the well-known St. Venant equations, applicable to 1D unsteady open-channel flows. Various forms of the St. Venant equations have been formulated in the field for unsteady open-channel flows since the 1950s, when numerical model simulations were first developed. The most widely used form in practice is generally written as (Cunge *et al*., 1980; Wu, 2008):

$$T\frac{\partial \xi}{\partial t} + \frac{\partial Q}{\partial \mathbf{x}} = \frac{Q\_L}{\delta \mathbf{x}} \tag{1}$$

$$
\underbrace{\frac{\partial \mathbb{Q}}{\partial t}}\_{1} + \underbrace{\beta \frac{\partial}{\partial \mathbf{x}} \left(\frac{\mathbf{Q}^2}{A}\right)}\_{2} + \underbrace{gA \frac{\partial \xi}{\partial \mathbf{x}}}\_{3} + \underbrace{g \frac{\mathbf{Q} \left|\mathbf{Q}\right|}{C\_Z^2 AR}}\_{4} = 0\tag{2}
$$

The individual terms in the momentum equation can be defined as: (1) local acceleration, (2) advective acceleration, (3) pressure gradient and (4) bed resistance, where:


=momentum correction factor due to non uniform velocity over the cross-section;


Equations (1) and (2) are solved numerically to provide the varying values of discharge and water elevations.

#### **3.2 Equations for water quality modelling**

Water quality modelling involves the prediction of water pollution using mathematical simulation techniques. The following sections describe the governing equations and their numerical solution for heavy metals modelling in riverine basins. The model was then applied to Karoon River for lead and cadmium modelling.

The transport of heavy metals in the dissolved phase can be described by the following onedimensional advection-dispersion equation (ADE) (Kashefipour, 2002):

$$
\underbrace{\frac{\partial \text{SA}}{\partial t}}\_{1} + \underbrace{\frac{\partial \text{SQ}}{\partial \mathbf{x}}}\_{2} - \underbrace{\frac{\partial}{\partial \mathbf{x}} \left[ \text{AD}\_{\mathbf{x}} \frac{\partial \mathbf{S}}{\partial \mathbf{x}} \right]}\_{3} = \underbrace{\mathbf{S}\_{0}^{d}}\_{4} + \underbrace{\mathbf{S}\_{t}^{d}}\_{5} \tag{3}
$$

The individual terms in the advection-dispersion equation refer to: (1) local effects, (2) transport by advection, (3) longitudinal dispersion and turbulent diffusion, (4) sources or sinks of dissolved heavy metals and (5) transformation term defining absorbed and desorbed particulate fluxes to or from sediments (source term), where:

*S* =cross-sectional averaged dissolved heavy metal concentration;

*Dx* =longitudinal dispersion coefficient;

*<sup>d</sup> <sup>S</sup>*<sup>0</sup> =source or sink of dissolved heavy metal;

*<sup>d</sup> St* =transformation term defining absorbed and desorbed particulate fluxes to or from sediments (source term).

Sources or sinks of dissolved heavy metals can be defined as:

$$S\_0^d = \frac{Q\_L S\_L}{\Delta \mathbf{x}} \tag{4}$$

where:

94 Numerical Modelling

commonly used by environmental engineers and water managers to predict the dissolved heavy metals concentrations in rivers generally involves solving the hydrodynamic and

In order to dynamically model the heavy metals in riverine basins, the governing hydrodynamic partial deferential equations must be numerically solved. The governing equations and their numerical solutions for modelling the river hydrodynamics (*i.e.* velocity

For unsteady flow and hydrodynamic modelling the velocity and the water-level at any point of the basin and any time is of interest. The velocity component in rivers is usually assumed as a one-dimensional vector. The one-dimensional governing hydrodynamic equations describing flow and water elevations in rivers are based on the well-known St. Venant equations, applicable to 1D unsteady open-channel flows. Various forms of the St. Venant equations have been formulated in the field for unsteady open-channel flows since the 1950s, when numerical model simulations were first developed. The most widely used

> *Q QL <sup>T</sup> txx*

The individual terms in the momentum equation can be defined as: (1) local acceleration,

Equations (1) and (2) are solved numerically to provide the varying values of discharge and

2

(1)

(2)

0

 *<sup>Z</sup> Q Q Q Q gA g t xA x C AR* 2

=momentum correction factor due to non uniform velocity over the cross-section;

*QL* =lateral inflow or outflow (positive for inflow and negative for outflow);

*x* =longitudinal distance between two consecutive nodes;

(2) advective acceleration, (3) pressure gradient and (4) bed resistance, where:

<sup>1</sup> <sup>3</sup> <sup>2</sup> <sup>4</sup>

 

and water elevation at any point and time) are therefore presented in this section.

form in practice is generally written as (Cunge *et al*., 1980; Wu, 2008):

*T* =top width of the channel;

*A* =wetted cross-section area; *R AP* / =hydraulic radius;

*g* =acceleration due to gravity;

*CZ* =Chezy coefficient;

water elevations.

*P* =wetted parameter of the cross-section;

*x t*, =river flow direction and time respectively.

=water elevation above datum;

*Q* =discharge;

water quality equations as described below.

**3.1 Equations for hydrodynamic modelling** 

*QL* =lateral inflow or outflow discharge;

*SL* =lateral inflow or outflow dissolved heavy metal concentration;

*x* =distance between two consecutive cross-sections which can be either constant or variable.

The longitudinal dispersion coefficient ( *Dx* ) in natural rivers is dependent upon many hydrodynamic parameters including: depth, width, velocity and shear velocity (Fischer *et al*. 1979). There are many empirical and/or semi-empirical equations describing this very important dynamic coefficient. Kashefipour & Falconer (2002) presented two empirical equations based on applying the dimensional analysis procedure to more than 80 data sets in 30 natural rivers, to estimate longitudinal dispersion coefficient in natural channels and showed that these equations performed relatively better than the other existing equations. In this chapter the Kashefipour & Falconer (2002) relationship has been used to estimate the longitudinal dispersion coefficient. This relationship is written as:

$$D\_x = \left[7.428 + 1.775 \left(\frac{T}{H}\right)^{0.620} \left(\frac{U\_\*}{U}\right)^{0.572}\right] \text{H} \text{U} \left(\frac{U}{U\_\*}\right) \tag{5}$$

Numerical Modelling of Heavy Metals Transport Processes in Riverine Basins 97

accurate simulation. Therefore, the following sections describe the procedure for calculating varied reaction coefficients for dissolved lead and cadmium modelling using pH and EC changes in the water column. The key point is that the chemical characteristics of the flow, such as pH and EC, can affect the dissolved heavy metals from sorption and desorption, to or from the sediments, and these characteristics can have an important effect on the dissolved heavy metal concentrations. For more accurate heavy metal modelling, varied

Based on the different characteristics of each heavy metal (such as lead, cadmium and etc.) the varied reaction coefficient should be computed and the corresponding relation of the

Reaction coefficient, also known as the decay coefficient, is the ratio for the number of atoms that decay in a given period of time compared with the total number of atoms of the same kind present at the beginning of that period (Zhen-Gang, 2008). There are different environmental parameters, such as: temperature, pH, salinity and etc., which generally

value may be related with to temperature as given by the following equation (see

*TEMP O*( 20)

*f pH Salinity Temperature* (, , ,...) (7)

can be defined as:

(8)

reaction coefficients has been suggested linking pH and EC to the kinetic processes.

may not provide an

assuming zero value or a constant value for the reaction coefficient

reaction coefficient should be used separately for each metal.

affect the reaction coefficient in heavy metals. Therefore,

The 

where:

Orlob, 1983):

 <sup>20</sup> = reaction coefficient at 20oC; *TEMP* = temperature of water;

equilibrium conditions).

seems more reasonable.

 20

*O* = temperature coefficient which it can vary from 1.047 to 1.135 (Orlob, 1983).

Theoretically the best mathematical model for heavy metal reaction coefficient should consist of all factors affecting the heavy metals concentration. Roshanfekr *et al.* (2008a & 2008b) found that pH and EC play a more essential role in adsorption and desorption of heavy metals by the particles in solution. Therefore assuming a variable reaction coefficient

The first step to calculate a variable reaction coefficient is to select the parameters that are most likely affecting the dissolved heavy metal concentration. Then the efforts can be focused on finding suitable functions to represent the reaction coefficient rate for dissolved heavy metals (*e.g.* lead and cadmium) in rivers. In calibrating the model against measured dissolved lead and cadmium data, five approaches for each dissolved metal can be used:

1. No rate of reaction for dissolved heavy metal (used by some researchers and models for

2. A constant reaction coefficient for the rate of reaction during the whole simulation time (the general practice in dissolved heavy metals modelling used by many researchers). 3. A time varying reaction coefficient for the rate of the reaction using pH as a variable.

where:

*H* =averaged depth over the cross-section; *U* = cross-sectional average velocity; *U*\* =local shear velocity.

In addition Tavakolizadeh (2006) used this dispersion coefficient for water quality modelling in Karoon River and achieved acceptable results for different water quality parameters.

#### **3.3 Heavy metals modelling**

Heavy metals can exist in both the dissolved and adsorbed particulate phases in rivers. The distribution between these two phases may be expressed by a partition coefficient. In recent years much effort has been focused on correlating the partitioning rate of heavy metals in particulate and dissolved phases to several environmental factors and water properties. It would be possible to numerically model each type of heavy metals in the water column, separately (Wu *et al.*, 2001 & 2005). However, it is sometimes important to the environmental managers to have a good understanding of the ratio of these two types of heavy metal presence in water body. A few researchers assume a reaction coefficient in the transformation term, *i.e. <sup>d</sup> St* , in the form of Equation (6) to model either dissolved or particulate heavy metal in the water column.

Since in outfalls a proportion of a pollutant that is added to the water column generally decays, and settles according to the chemical and hydraulic characteristics of the flow, it can be concluded that the pollutant may also be added from or to the sediments. Therefore, for water bodies close to outfalls the conditions are not generally consistent with equilibrium conditions. For equilibrium conditions it can be assumed that the parameter *<sup>d</sup> St* in Equation (3) is equal zero. On the other hand, the transformation of heavy metal from dissolved phase to the particulate phase and vice versa is assumed to be equal. A review of the literature has shown that a number of researchers include this type of assumption in their models, such as Wu *et al.* (2005). However, another group of researches, for example Nassehi & Bikangaga (1993), assumed a decay term having a form of Equation (6) with a constant coefficient.

The fate and decay of toxic subtances can result from physical, chemical, and/or biological reaction. Transformation processes are those in which toxic subtances are essentialy irreversibly destroyed, changed, or removed from the water system. These transformation processes are often described by kinematic equations. Most decay processes are expressed as first-order reactions. Therefore, in this chapter the first-order chemical reaction was used as the transformation parameter in Equation (3) for dissolved heavy metal modelling and is written as follows (Zhen-Gang, 2008):

$$S\_t^d = -\kappa SA \tag{6}$$

 is a reaction coefficient rate, which may have a positive or negative value as the dissolved heavy metals disappears or accumulates in a given river section.

Since the exchange of the heavy metal substance between particulate and dissolved phases is a chemical process and is highly dependent on the environmental conditions, it seems that assuming zero value or a constant value for the reaction coefficient may not provide an accurate simulation. Therefore, the following sections describe the procedure for calculating varied reaction coefficients for dissolved lead and cadmium modelling using pH and EC changes in the water column. The key point is that the chemical characteristics of the flow, such as pH and EC, can affect the dissolved heavy metals from sorption and desorption, to or from the sediments, and these characteristics can have an important effect on the dissolved heavy metal concentrations. For more accurate heavy metal modelling, varied reaction coefficients has been suggested linking pH and EC to the kinetic processes.

Based on the different characteristics of each heavy metal (such as lead, cadmium and etc.) the varied reaction coefficient should be computed and the corresponding relation of the reaction coefficient should be used separately for each metal.

Reaction coefficient, also known as the decay coefficient, is the ratio for the number of atoms that decay in a given period of time compared with the total number of atoms of the same kind present at the beginning of that period (Zhen-Gang, 2008). There are different environmental parameters, such as: temperature, pH, salinity and etc., which generally affect the reaction coefficient in heavy metals. Therefore, can be defined as:

$$\kappa = f(pH, \text{Salinity}, Temperature, \dots) \tag{7}$$

The value may be related with to temperature as given by the following equation (see Orlob, 1983):

$$
\kappa = \kappa\_{20} \times O^{\text{(TEMP-20)}} \tag{8}
$$

where:

96 Numerical Modelling

In addition Tavakolizadeh (2006) used this dispersion coefficient for water quality modelling

Heavy metals can exist in both the dissolved and adsorbed particulate phases in rivers. The distribution between these two phases may be expressed by a partition coefficient. In recent years much effort has been focused on correlating the partitioning rate of heavy metals in particulate and dissolved phases to several environmental factors and water properties. It would be possible to numerically model each type of heavy metals in the water column, separately (Wu *et al.*, 2001 & 2005). However, it is sometimes important to the environmental managers to have a good understanding of the ratio of these two types of heavy metal presence in water body. A few researchers assume a reaction coefficient in the transformation term, *i.e. <sup>d</sup> St* , in the form of Equation (6) to model either dissolved or

Since in outfalls a proportion of a pollutant that is added to the water column generally decays, and settles according to the chemical and hydraulic characteristics of the flow, it can be concluded that the pollutant may also be added from or to the sediments. Therefore, for water bodies close to outfalls the conditions are not generally consistent with equilibrium conditions. For equilibrium conditions it can be assumed that the parameter *<sup>d</sup> St* in Equation (3) is equal zero. On the other hand, the transformation of heavy metal from dissolved phase to the particulate phase and vice versa is assumed to be equal. A review of the literature has shown that a number of researchers include this type of assumption in their models, such as Wu *et al.* (2005). However, another group of researches, for example Nassehi & Bikangaga (1993), assumed a decay term having a form of Equation (6) with a constant coefficient.

The fate and decay of toxic subtances can result from physical, chemical, and/or biological reaction. Transformation processes are those in which toxic subtances are essentialy irreversibly destroyed, changed, or removed from the water system. These transformation processes are often described by kinematic equations. Most decay processes are expressed as first-order reactions. Therefore, in this chapter the first-order chemical reaction was used as the transformation parameter in Equation (3) for dissolved heavy metal modelling and is

> *<sup>d</sup> S SA <sup>t</sup>*

is a reaction coefficient rate, which may have a positive or negative value as the dissolved

Since the exchange of the heavy metal substance between particulate and dissolved phases is a chemical process and is highly dependent on the environmental conditions, it seems that

heavy metals disappears or accumulates in a given river section.

(6)

in Karoon River and achieved acceptable results for different water quality parameters.

where:

*H* =averaged depth over the cross-section; *U* = cross-sectional average velocity;

particulate heavy metal in the water column.

written as follows (Zhen-Gang, 2008):

*U*\* =local shear velocity.

**3.3 Heavy metals modelling** 

<sup>20</sup> = reaction coefficient at 20oC;

*TEMP* = temperature of water;

*O* = temperature coefficient which it can vary from 1.047 to 1.135 (Orlob, 1983).

Theoretically the best mathematical model for heavy metal reaction coefficient should consist of all factors affecting the heavy metals concentration. Roshanfekr *et al.* (2008a & 2008b) found that pH and EC play a more essential role in adsorption and desorption of heavy metals by the particles in solution. Therefore assuming a variable reaction coefficient seems more reasonable.

The first step to calculate a variable reaction coefficient is to select the parameters that are most likely affecting the dissolved heavy metal concentration. Then the efforts can be focused on finding suitable functions to represent the reaction coefficient rate for dissolved heavy metals (*e.g.* lead and cadmium) in rivers. In calibrating the model against measured dissolved lead and cadmium data, five approaches for each dissolved metal can be used:


Numerical Modelling of Heavy Metals Transport Processes in Riverine Basins 99

A summary of numerical solution method is described here. The difference form of the continuity equation using the Crank-Nicholson central scheme around the node i (Figure 2)

*nn n n n nn n T xx QQ QQ Q Wi i i i i i i i i Li <sup>t</sup>*

=a weighting coefficient between 0 and 1 to split the spatial derivatives between the upper

The non-conservative form of the momentum equation may be discretised using the finite

*i Wi n n nn i i n n <sup>n</sup> i i ii i i <sup>n</sup> <sup>i</sup> i ii*

1 1 2 1

By rearranging Equations (9) and (10) the following algebraic linear equations may be

*n nn i i ii i i i aQ b cQ d* 1 11 1/2 1/2 

*nn n ii i i ii i a bQ c d* 11 1 1/2 1/2 1/2 1/2 1 1/2

The staggered varying grid size with the numerical scheme is alternatively applied to the continuity and momentum equations to produce a set of linear algebraic equations (*i.e.*

*Q T Q Q A A*

1 1 1 1/2 1/2 1/2 1/2 1/2

 

Fig. 2. Domain of the discretization of continuity and momentum equations.

1 2 <sup>1</sup>

 

*i ii n n n n*

*i i*

 

*i n n*

 

1/2 1 1 1

 

*n*

*gA x x*

written, respectively:

*i i*

1 1

difference central scheme around the node (i+1/2) as shown in Figure 2, yields:

*Q Q Q QQ <sup>t</sup> x xA*

1 1/2

 

*tA A xx*

1/2 1/2 1

1/2 1/2 1 1 1/2 1/2

1

 

*n n n nn i i n i i Li*

1 1 1 1 1/2 1/2 1/2 1/2 1/2 1/2

(11)

*n n*

*gQ Q*

1 1/2 1/2

*Zi i i*

1/2 1/2 1/2

(10)

*C AR*

 0 

1

 

*i i n n i i n nn*

1 2

 

(12)

1

(9)

can be written as:

 

and lower time levels ( 0 1

n and n+1=refer to time t and *t t* , respectively;

).

1

where:


For each one of these five cases a number of simulation calibration runs were carried out and the initial reaction coefficient was subsequently adjusted by comparing the predicted dissolved lead and cadmium concentrations with the corresponding measured values at sites and for the times of measured values. Final values of the reaction coefficients for each indicator were adopted when the best fit occurred between the series of data. The adjusted rate of reaction coefficients were then correlated with pH, EC and both to find the best relationships for as a function of pH and/or EC. These equations (*i.e.* Equations (20) to (25)) were added to the model as a part of the numerical solution of the ADE (see Equation (3)). The model was then validated using the corresponding measured data for different time series at the survey site.

#### **4. Numerical modelling**

Following sections describe the numerical methods used to solve the hydrodynamic and water quality partial differential equations for heavy metals modelling.

#### **4.1 Numerical methods for hydrodynamic equations solutions**

There are many implicit and explicit numerical methods used for solving 1D hydrodynamic equations (*i.e.* Equations (1) and (2)), in which the stability, accuracy and consistency of the numerical solution are important. Almost all implicit methods are unconditionally stable, however the accuracy of model predictions is highly depended on the Courant number (*i.e. C Ut x <sup>r</sup>* / ). Different methods for numerical solution of the above equations may be found in Abbott and Basco (1997).

In this study, the numerical model FASTER (Flow and Solute Transport in Estuaries and Rivers) (Kashefipour *et al*., 1999) was used. This model was first developed by Kashefipour (2002) and has since been extended and improved to predict the dissolved heavy metals concentrations for different reaction coefficients. The hydrodynamic module of FASTER model numerically solves the Saint Venant equations using Crank Nicolson with an implicit staggered scheme (Wu, 2008). This model uses the influenced line technique, enabling the model to remain implicit and thereby unconditionally stable and accurate over the whole domain, especially in river confluences. This model can be applied for complex channel networks with complex geometry and has been successfully applied to many research projects in Cardiff University, UK (Kashefipour *et al.*, 2002). In the numerical method used for this model, the hydrodynamic equations were formulated on a staggered grid to provide advantages in treating the typical hydrodynamic boundary conditions that are commonly used in such models. The implicit finite difference solution of the governing hydrodynamic equations is second order accurate in space and time and is unconditionally stable. However, where reasonable precision is required the Courant number, expressed in the form of *C tU x <sup>r</sup>* / , should be less than five. The scheme remains stable for higher Courant numbers, but the accuracy may reduce particularly at wave peaks.

A summary of numerical solution method is described here. The difference form of the continuity equation using the Crank-Nicholson central scheme around the node i (Figure 2) can be written as:

$$\frac{1}{\Delta t} T\_{\mathsf{VI}}^{n} \left( \xi\_{i}^{n+1} - \xi\_{i}^{n} \right) \left( \mathbf{x}\_{i+1/2} - \mathbf{x}\_{i-1/2} \right) + \theta \left( \mathbf{Q}\_{i+1/2}^{n+1} - \mathbf{Q}\_{i-1/2}^{n+1} \right) + (1 - \theta) \left( \mathbf{Q}\_{i+1/2}^{n} - \mathbf{Q}\_{i-1/2}^{n} \right) = \mathbf{Q}\_{\mathsf{li}}^{n+1} \tag{9}$$

where:

98 Numerical Modelling

For each one of these five cases a number of simulation calibration runs were carried out and the initial reaction coefficient was subsequently adjusted by comparing the predicted dissolved lead and cadmium concentrations with the corresponding measured values at sites and for the times of measured values. Final values of the reaction coefficients for each indicator were adopted when the best fit occurred between the series of data. The adjusted rate of reaction coefficients were then correlated with pH, EC and both to find the best

(25)) were added to the model as a part of the numerical solution of the ADE (see Equation (3)). The model was then validated using the corresponding measured data for different

Following sections describe the numerical methods used to solve the hydrodynamic and

There are many implicit and explicit numerical methods used for solving 1D hydrodynamic equations (*i.e.* Equations (1) and (2)), in which the stability, accuracy and consistency of the numerical solution are important. Almost all implicit methods are unconditionally stable, however the accuracy of model predictions is highly depended on the Courant number (*i.e. C Ut x <sup>r</sup>* / ). Different methods for numerical solution of the above equations may be

In this study, the numerical model FASTER (Flow and Solute Transport in Estuaries and Rivers) (Kashefipour *et al*., 1999) was used. This model was first developed by Kashefipour (2002) and has since been extended and improved to predict the dissolved heavy metals concentrations for different reaction coefficients. The hydrodynamic module of FASTER model numerically solves the Saint Venant equations using Crank Nicolson with an implicit staggered scheme (Wu, 2008). This model uses the influenced line technique, enabling the model to remain implicit and thereby unconditionally stable and accurate over the whole domain, especially in river confluences. This model can be applied for complex channel networks with complex geometry and has been successfully applied to many research projects in Cardiff University, UK (Kashefipour *et al.*, 2002). In the numerical method used for this model, the hydrodynamic equations were formulated on a staggered grid to provide advantages in treating the typical hydrodynamic boundary conditions that are commonly used in such models. The implicit finite difference solution of the governing hydrodynamic equations is second order accurate in space and time and is unconditionally stable. However, where reasonable precision is required the Courant number, expressed in the form of *C tU x <sup>r</sup>* / , should be less than five. The scheme remains stable for higher

water quality partial differential equations for heavy metals modelling.

Courant numbers, but the accuracy may reduce particularly at wave peaks.

**4.1 Numerical methods for hydrodynamic equations solutions** 

as a function of pH and/or EC. These equations (*i.e.* Equations (20) to

4. A time varying reaction coefficient for the rate of the reaction using EC as a variable. 5. A time varying reaction coefficient for the rate of the reaction using both pH and EC

variables.

relationships for

time series at the survey site.

**4. Numerical modelling** 

found in Abbott and Basco (1997).

n and n+1=refer to time t and *t t* , respectively;

 =a weighting coefficient between 0 and 1 to split the spatial derivatives between the upper and lower time levels ( 0 1 ).

Fig. 2. Domain of the discretization of continuity and momentum equations.

The non-conservative form of the momentum equation may be discretised using the finite difference central scheme around the node (i+1/2) as shown in Figure 2, yields:

 *n n n nn i i n i i Li i ii n n n n i Wi n n nn i i n n <sup>n</sup> i i ii i i <sup>n</sup> <sup>i</sup> i ii n i n n i i i i Q Q Q QQ <sup>t</sup> x xA Q T Q Q A A tA A xx gA x x* 1 1 1 1/2 1/2 1/2 1/2 1/2 1 1/2 1 1/2 1/2 1 1 1/2 1/2 1 1 2 1 1/2 1/2 1 1/2 1 1 1 1 1 1 2 <sup>1</sup> 1 *n n i i n n i i n nn Zi i i gQ Q C AR* 1 1/2 1/2 1 2 1/2 1/2 1/2 0 (10)

By rearranging Equations (9) and (10) the following algebraic linear equations may be written, respectively:

$$a\_i \mathbf{Q}\_{i-1/2}^{n+1} + b\_i \mathbf{Q}\_i^{n+1} + c\_i \mathbf{Q}\_{i+1/2}^{n+1} = d\_i \tag{11}$$

$$a\_{i+1/2}\xi\_i^{n+1} + b\_{i+1/2}\xi\_{i+1/2}^{n+1} + c\_{i+1/2}\xi\_{i+1}^{n+1} = d\_{i+1/2} \tag{12}$$

The staggered varying grid size with the numerical scheme is alternatively applied to the continuity and momentum equations to produce a set of linear algebraic equations (*i.e.*

Numerical Modelling of Heavy Metals Transport Processes in Riverine Basins 101

In Equation (14) the term (1) describes the local change of the solute concentration within the control volume from time (t) to time (t+t). Terms (2) to (5) refer to changes in the solute concentration due to: advection, diffusion, lateral inputs and transformation, respectively. The discrete forms of the terms in Equation (14) using finite volume method can be written

> *t t n n i i t V*

1 1 2 2 1 1

*n n n n i i i i l l i i i i i i*

*S S S S*

 

1/2 1/2 1 1

*x x x x*

2 2 1 1 1/2 1/2 1 1

*Q S dVdt Q S A t*

*S dVdt S V t V t*

*A D A D*

*L L t V*

1 4

More information regarding the FASTER model may be found in Kashefipour (2002) and

In the current chapter the dissolved heavy metal reaction coefficient comprises two parameters, including pH and EC. The coefficient was formulated using a linear regression relationship. These varied reaction coefficients were then added to the model for predicting dissolved lead and cadmium. The procedure of development and the equations added to the model can be followed in the modelling application and dissolved heavy metal results

The Karoon River is the largest and the only navigable river in south west of Iran (see Figure 3(a)). In this study the Mollasani-Farsiat reach of the Karoon River, a distance of 110Km was selected due to the high amount of heavy metal concentrations along this reach (see Figure

*x*

*t t d d i i t t t V n n*

*t t n n n n i i i i t V*

1 1

<sup>1</sup>

*t*

*S S QA S S QA <sup>t</sup>*

*<sup>l</sup> t V*

1

*t t*

*t*

*n n n n i i i i i i*

1 1 1 1 1 1 1/2 1/2

 

*n n n n i i i i i i*

*A D A D*

*t t L L*

1 1 1/2 1/2

*<sup>S</sup> AD dVdt x x*

*S S QA S S QA*

*SA dVdt SA SA V*

*SQ dVdt SQA SQA SQA SQA t*

 

 

*n n n n i i i i l l i i i i i i*

*S S S S*

1 1

*x x x x*

(19)

*AS AS*

*AS AS*

(18)

1 1 1/2 1/2

1/2 1/2

*n n*

 

*i i*

 

 

1

1/2 1/2 1/2 1/2

(15)

(16)

(17)

as follows:

2 1

*x*

Yang *et al.* (2002).

sections respectively.

**5. Case study** 

Equations (11) and (12)) for each three consecutive ξ and Q points. Applying Equations (11) and (12) simultaneously at all grid points in the discrete solution domain, from time *nt* to (n+1) t, yields a matrix system of linear algebraic equations based on *<sup>n</sup>* <sup>1</sup> and *<sup>n</sup> Q* <sup>1</sup> . A general equation, which contains all of the linear algebraic equations and may be solved by the Thomas algorithm or Gauss elimination procedure for the numerical solution of the governing partial differential equations, may be defined using the following equation:

$$\begin{bmatrix} B \end{bmatrix} \begin{bmatrix} X \end{bmatrix} = \begin{bmatrix} D \end{bmatrix} \tag{13}$$

where:

*B* = matrix of the coefficients;

*X* = matrix of the variables;

*D* = matrix of the constants of the linear equations.

Due to the staggered method, the matrix *B* is usually given as a tri-diagonal matrix and then Equation (13) can be generally solved using the well known Thomas algorithm. More information regarding the numerical solution and application of the influence line technique in FASTER model to keep the whole numerical solution in implicit form, specially in river confluences and junctions, can be found in Kashefipour (2002).

#### **4.2 Numerical methods for ADE solution**

In order to solve the ADE, an implicit algorithm has been developed and used in the FASTER model. This finite volume based solution procedure calculates the advection of a concentrate of solute, or suspended sediments at each face of any control volume, by means of a modified form of the highly accurate ULTIMATE QUICKEST1 scheme (Lin & Falconer, 1997). As before, a space staggered grid system is used to solve the finite volume form of the ADE, in which the variable S is located at the center of the control volume (Falconer *et al.*, 2005).

Double integration of the one-dimensional ADE, *i.e.* Equation (3), with respect to time and volume over the control volume, as shown in Figure 2 gives:

$$\underbrace{\int\_{t}^{t+\Delta t} \int\_{V} \frac{\partial SA}{\partial t} dV dt}\_{1} + \underbrace{\int\_{t}^{t+\Delta t} \int\_{V} \frac{\partial SQ}{\partial \mathbf{x}} dV dt}\_{2} - \underbrace{\int\_{t}^{t+\Delta t} \int\_{V} \frac{\partial}{\partial \mathbf{x}} AD\_{l} \frac{\partial S}{\partial \mathbf{x}} dV dt}\_{5} = \tag{14}$$
 
$$\underbrace{\int\_{t}^{t+\Delta t} \int\_{V} \frac{Q\_{l}S\_{L}}{\Delta \mathbf{x}} dV dt}\_{4} + \underbrace{\int\_{t}^{t+\Delta t} \int\_{V} S\_{t}^{d} dV dt}\_{5}$$

where:

V= volume.

*<sup>1</sup> Universal Limiter Transient Interpolation Modelling for Advection Term Equation - Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (Leonard, 1979 & 1991).* 

In Equation (14) the term (1) describes the local change of the solute concentration within the control volume from time (t) to time (t+t). Terms (2) to (5) refer to changes in the solute concentration due to: advection, diffusion, lateral inputs and transformation, respectively. The discrete forms of the terms in Equation (14) using finite volume method can be written as follows:

$$\int\_{t}^{t+\Delta t} \int\_{V} \frac{\partial SA}{\partial t} dV dt = \left[ \left( SA \right)\_{i}^{n+1} - \left( SA \right)\_{i}^{n} \right] \Delta V \tag{15}$$

$$\begin{aligned} &\int\_{t}^{t+\Delta t} \int\_{V} \frac{\partial SQ}{\partial \mathbf{x}} dV dt = \left[ \left. \nu \left( (SQA)\_{i+1/2}^{n+1} - (SQA)\_{i-1/2}^{n+1} \right) + \left( 1 - \nu \right) \left( (SQA)\_{i+1/2}^{n} - (SQA)\_{i-1/2}^{n} \right) \right] \right] \Delta t = \\ &\frac{\Delta t}{2} \left[ \nu \left( (S\_{i} + S\_{i+1}) \right)^{n+1} \left( QA \right)\_{i+1/2}^{n+1} - \left( S\_{i} + S\_{i-1} \right)^{n+1} \left( QA \right)\_{i-1/2}^{n+1} \right] + \\ &\overline{\frac{\Delta}{2}} \left[ \left( 1 - \nu \right) \left( \left( S\_{i} + S\_{i+1} \right)^{n} \left( QA \right)\_{i+1/2}^{n} - \left( S\_{i} + S\_{i-1} \right)^{n} \left( QA \right)\_{i-1/2}^{n} \right) \right]} \end{aligned} \tag{16}$$

$$\begin{aligned} \int\_{t}^{t+\Delta t} \int\_{V} \frac{\partial}{\partial \mathbf{x}} A D\_{l} \frac{\partial \mathbf{S}}{\partial \mathbf{x}} dV dt &= \\ \Delta t \Bigg[ \begin{aligned} & \left[ \Psi \left[ \left( A^{2} D\_{l} \right)\_{i+1/2}^{n+1} \frac{\left( S\_{i+1} - S\_{i} \right)^{n+1}}{\mathbf{x}\_{i+1} - \mathbf{x}\_{i}} - \left( A^{2} D\_{l} \right)\_{i-1/2}^{n+1} \frac{\left( S\_{i} - S\_{i-1} \right)^{n+1}}{\mathbf{x}\_{i} - \mathbf{x}\_{i-1}} \right] \\ &+ \left( 1 - \Psi \right) \Bigg[ \left( A^{2} D\_{l} \right)\_{i+1/2}^{n} \frac{\left( S\_{i+1} - S\_{i} \right)^{n}}{\mathbf{x}\_{i+1} - \mathbf{x}\_{i}} - \left( A^{2} D\_{l} \right)\_{i-1/2}^{n} \frac{\left( S\_{i} - S\_{i-1} \right)^{n}}{\mathbf{x}\_{i} - \mathbf{x}\_{i-1}} \Bigg] \end{aligned} \tag{17}$$

$$\int\_{t}^{t+\Delta t} \int\_{V} \frac{\underline{Q}\_{L} S\_{L}}{\Delta \mathbf{x}} dV dt = \overline{\underline{Q}\_{L} S\_{L}} A \Delta t \tag{18}$$

$$\int\_{t}^{t+\Lambda t} \int\_{V} -S\_t^d dV dt = \overline{-S\_t^d} \Delta V \Delta t = -\frac{1}{4} \begin{vmatrix} \left( \kappa A S \right)\_{i+1/2}^{n+1} + \left( \kappa A S \right)\_{i-1/2}^{n+1} \\ \left( \kappa A S \right)\_{i+1/2}^n + \left( \kappa A S \right)\_{i-1/2}^n \end{vmatrix} \Delta V \Delta t \tag{19}$$

More information regarding the FASTER model may be found in Kashefipour (2002) and Yang *et al.* (2002).

In the current chapter the dissolved heavy metal reaction coefficient comprises two parameters, including pH and EC. The coefficient was formulated using a linear regression relationship. These varied reaction coefficients were then added to the model for predicting dissolved lead and cadmium. The procedure of development and the equations added to the model can be followed in the modelling application and dissolved heavy metal results sections respectively.

#### **5. Case study**

100 Numerical Modelling

Equations (11) and (12)) for each three consecutive ξ and Q points. Applying Equations (11) and (12) simultaneously at all grid points in the discrete solution domain, from time *nt* to

general equation, which contains all of the linear algebraic equations and may be solved by the Thomas algorithm or Gauss elimination procedure for the numerical solution of the governing partial differential equations, may be defined using the following equation:

Due to the staggered method, the matrix *B* is usually given as a tri-diagonal matrix and then Equation (13) can be generally solved using the well known Thomas algorithm. More information regarding the numerical solution and application of the influence line technique in FASTER model to keep the whole numerical solution in implicit form, specially in river

In order to solve the ADE, an implicit algorithm has been developed and used in the FASTER model. This finite volume based solution procedure calculates the advection of a concentrate of solute, or suspended sediments at each face of any control volume, by means of a modified form of the highly accurate ULTIMATE QUICKEST1 scheme (Lin & Falconer, 1997). As before, a space staggered grid system is used to solve the finite volume form of the ADE, in which the

Double integration of the one-dimensional ADE, *i.e.* Equation (3), with respect to time and

*1 Universal Limiter Transient Interpolation Modelling for Advection Term Equation - Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (Leonard, 1979* 

*SA SQ <sup>S</sup> dVdt dVdt AD dVdt t xx x*

*BX D* (13)

and *<sup>n</sup> Q* <sup>1</sup> . A

(14)

(n+1) t, yields a matrix system of linear algebraic equations based on *<sup>n</sup>* <sup>1</sup>

where:

where:

*& 1991).* 

V= volume.

*B* = matrix of the coefficients; *X* = matrix of the variables;

*D* = matrix of the constants of the linear equations.

**4.2 Numerical methods for ADE solution** 

confluences and junctions, can be found in Kashefipour (2002).

volume over the control volume, as shown in Figure 2 gives:

*t t t t L L <sup>d</sup> <sup>t</sup> t V t V*

*x*

variable S is located at the center of the control volume (Falconer *et al.*, 2005).

*t t t t t t*

*Q S dVdt S dVdt*

5 4

*<sup>l</sup> t V t V t V*

12 3

The Karoon River is the largest and the only navigable river in south west of Iran (see Figure 3(a)). In this study the Mollasani-Farsiat reach of the Karoon River, a distance of 110Km was selected due to the high amount of heavy metal concentrations along this reach (see Figure

2000.

**5.1 Application of hydrodynamic modelling** 

statistical analysis of the model results is illustrated in Table 1.

Numerical Modelling of Heavy Metals Transport Processes in Riverine Basins 103

The 1D grid, covering the region from Mollasani to Farsiat, was represented using 113 segments, with extensive bathymetric data at each cross-section being collected during the most recent bathymetric survey conducted by Khuzestan Water and Power Authorities in

The time series water elevations recorded at the Farsiat hydrometric station were chosen as the downstream boundary and the measured discharges and heavy metal concentrations at the Mollasani station were used as the upstream boundary conditions for flow and water quality modules of the main model, respectively. Also concentrations of dissolved lead and cadmium were measured from more than fifteen outfalls and industrial locations along the Mollasani and Farsiat reach. Cross-sections No.1, 36, 49 and 113 corresponded to the crosssections at the gauging stations of Farsiat, Shekare, Ahwaz and Mollasani, respectively.

The hydrodynamic module of the FASTER model was calibrated against the data provided for the year 2004, starting from the month of March. The main hydrodynamic parameter used for calibration was the manning roughness coefficient. The river was separated into 4 parts, with the manning coefficient varying from 0.026 to 0.050. Good agreement was obtained between the predicted water levels and corresponding field data at the Ahwaz gauging station as the hydrometric survey site, with a difference in results being less than 3% (see Figure 4(a)) and also the model discharges agreed well with the field data obtained at the Ahwaz gauging station with the difference being less than 16% (see Figure 4(b)). The hydrodynamic module was then validated using another series of measured data (see Figures 5(a) and (b)). As can be seen from these figures the predicted data also gave relatively good correlation with the corresponding measured values. A summary of the

(a) Comparison of water levels with the corresponding measured data for model calibration

3(b)). The Karoon River basin has a network of gauging stations and there are several effluent inputs to the river between gauging stations at Mollasani and Farsiat, including industrial units such as: piping, steel, paint making, agriculture, paper mill, fish cultivation and power plant industries draining from wastewater works into the river (see Figure 3(c)) (Diagomanolin *et al*., 2004). Hydrodynamic and water quality data were acquired via Khuzestan Water and Power Authority (KWPA). A set of six field-measured data were available from March 2004, including discharge and water levels measurements at the Mollasani, Ahwaz and Farsiat gauging stations and pH, EC, dissolved lead and cadmium concentrations at the Mollasani and Shekare gauging stations (see Figure 3(c)).

Fig. 3. (a) Location of Karoon River, (b) Karoon river network and gauging stations and (c) Outfalls, gauging stations and cross-sections used in the model between Mollasani and Farsiat reach.

The 1D grid, covering the region from Mollasani to Farsiat, was represented using 113 segments, with extensive bathymetric data at each cross-section being collected during the most recent bathymetric survey conducted by Khuzestan Water and Power Authorities in 2000.

The time series water elevations recorded at the Farsiat hydrometric station were chosen as the downstream boundary and the measured discharges and heavy metal concentrations at the Mollasani station were used as the upstream boundary conditions for flow and water quality modules of the main model, respectively. Also concentrations of dissolved lead and cadmium were measured from more than fifteen outfalls and industrial locations along the Mollasani and Farsiat reach. Cross-sections No.1, 36, 49 and 113 corresponded to the crosssections at the gauging stations of Farsiat, Shekare, Ahwaz and Mollasani, respectively.

#### **5.1 Application of hydrodynamic modelling**

102 Numerical Modelling

3(b)). The Karoon River basin has a network of gauging stations and there are several effluent inputs to the river between gauging stations at Mollasani and Farsiat, including industrial units such as: piping, steel, paint making, agriculture, paper mill, fish cultivation and power plant industries draining from wastewater works into the river (see Figure 3(c)) (Diagomanolin *et al*., 2004). Hydrodynamic and water quality data were acquired via Khuzestan Water and Power Authority (KWPA). A set of six field-measured data were available from March 2004, including discharge and water levels measurements at the Mollasani, Ahwaz and Farsiat gauging stations and pH, EC, dissolved lead and cadmium concentrations at the Mollasani and Shekare gauging stations (see Figure 3(c)).

Fig. 3. (a) Location of Karoon River, (b) Karoon river network and gauging stations and (c) Outfalls, gauging stations and cross-sections used in the model between Mollasani and

Farsiat reach.

The hydrodynamic module of the FASTER model was calibrated against the data provided for the year 2004, starting from the month of March. The main hydrodynamic parameter used for calibration was the manning roughness coefficient. The river was separated into 4 parts, with the manning coefficient varying from 0.026 to 0.050. Good agreement was obtained between the predicted water levels and corresponding field data at the Ahwaz gauging station as the hydrometric survey site, with a difference in results being less than 3% (see Figure 4(a)) and also the model discharges agreed well with the field data obtained at the Ahwaz gauging station with the difference being less than 16% (see Figure 4(b)). The hydrodynamic module was then validated using another series of measured data (see Figures 5(a) and (b)). As can be seen from these figures the predicted data also gave relatively good correlation with the corresponding measured values. A summary of the statistical analysis of the model results is illustrated in Table 1.

(a) Comparison of water levels with the corresponding measured data for model calibration

Numerical Modelling of Heavy Metals Transport Processes in Riverine Basins 105

(b) Comparison of discharges with the corresponding measured data for model verification

*Water Elevation* 0.350 0.935 2.17 1.013 0.869 2.98 *Discharge* 1.580 0.960 15.20 1.870 0.930 13.21

*X X*

*R*

1

*i ip im n i im*

1 % 100 

*Xip* =Predicted Data; *Xim* = Measured Data and *n* =Number of Data (Azmathullah

*X X*

 

*X*

*n*

1 2

*i*

1

**CALIBRATION VERIFICATION**  *RMSE (a) R2 (b) %Error (c) RMSE R2 %Error* 

0.5 <sup>2</sup>

*i ip im n n i i ip im*

*X X*

2

*X X*

2 2 1 1

Fig. 5. Results of hydrodynamic model verification.

*(a) Root Mean Square Error <sup>n</sup> ip im*

*(c) Average Absolute Error <sup>n</sup>*

Table 1. A summary of the hydrodynamic model results.

where:

*et al*., 2005).

*RMSE*

*(b) Coefficient of Determination (r-Square) <sup>n</sup>*

*Error*

(b) Comparison of discharges with the corresponding measured data for model calibration Fig. 4. Results of hydrodynamic model calibration.

(a) Comparison of water levels with the corresponding measured data for model verification

(b) Comparison of discharges with the corresponding measured data for model calibration

(a) Comparison of water levels with the corresponding measured data for model verification

Fig. 4. Results of hydrodynamic model calibration.

(b) Comparison of discharges with the corresponding measured data for model verification Fig. 5. Results of hydrodynamic model verification.


Table 1. A summary of the hydrodynamic model results.

Numerical Modelling of Heavy Metals Transport Processes in Riverine Basins 107

verification in Figures 6(a) and (b), respectively. The comparison showed that the error of simulation had reduced to 1.9% and 15% for calibration and verification of the model,

Based on the fact that the reaction coefficient relates to the EC value, a number of simulations were also carried out to find a suitable formulation for describing the reaction coefficient with the EC value. Using the measured dissolved lead concentration, it was found that the most suitable relationship between the reaction coefficient for dissolved lead

*EC* = (Electrical Conductivity), the mean EC of the river at the site for each time (micro

The predicted lead concentration, for which the reaction coefficients were calculated using Equation (21) in the model, were compared with the corresponding measured values for calibration and verification in Figures 6(a) and (b), respectively. This showed that the error of simulation had also declined to 0.8% and 10.8% for calibration and verification of the

For the last run of the dissolved lead, a number of simulations were carried out to find a formulation for describing the relationship between the reaction coefficient and both the pH and EC variables. Using the measured dissolved lead concentrations, it was found that the most suitable relationship between the reaction coefficient for dissolved lead and pH and

The predicted results, for which the reaction coefficient were calculated using Equation (22) in the model, were then compared with the corresponding measured values for calibration and verification in Figures 6(a) and (b) with the errors of 0.4% and 8.3% respectively. These

A summary of the statistical analysis for the different model results is shown in Table 2. As it is clear from this table the predicted dissolved lead concentrations improved, giving lower

0 2.9544 25.24 4.2072 33.30

*Const*. 0.4064 3.35 2.6855 17.13

0.1646 1.4934 *pH* 0.2633 1.89 2.3984 14.97

0.00023 0.581 *EC* 0.1396 0.84 1.7807 10.77

0.160 0.000402 0.401 *pH EC* 0.0671 0.35 1.5329 8.29

results showed another improvement in the predicted dissolved lead concentrations.

errors when varying reaction coefficients were applied as a part of ADE.

0.160 0.000402 0.401 *pH EC* ( *R*<sup>2</sup> 1.000 ) (22)

**CALIBRATION VERIFICATION**  *RMSE %Error RMSE %Error* 

0.00023 0.581 *EC* ( *R*<sup>2</sup> 0.924 ) (21)

respectively.

where:

mhos/cm).

model, respectively.

and the EC of the river was of the following form:

EC as variables for the river was of the following form:

Table 2. A summary of the dissolved lead model results.

#### **5.2 Application of dissolved heavy metals modelling**

As discussed above, the rate of reaction plays an important role in predicting the concentration distribution of the dissolved heavy metals for river, estuarine and coastal waters. In the current section effort was made to find suitable functions to represent the reaction coefficient rate for dissolved lead and cadmium metal modelling in rivers. These functions were established using a comparison of the predicted heavy metal concentrations with the corresponding measured values at the Shekare gauging station (see Figure 3). In calibrating the model against measured dissolved lead and cadmium levels, five approaches for each dissolved metal were used. The model with the adjusted rate of reaction was then validated using the corresponding measured data for different time series at the survey site (Kashefipour *et al.*, 2006). In the following sections the equations and the results of the modelling for dissolved lead and cadmium, with the derived equations for the reaction coefficients are illustrated.

#### **5.2.1 Results of dissolved lead modelling**

For the first run a conservative dissolved lead was assumed, leading to a zero value for the rate of reaction coefficient. The fit between the predicted and measured data showed 25.2% and 33.3% errors for calibration and verification of the model respectively. As can be seen from Figures 6(a) and (b) the predicted dissolved lead in this case did not agree well with the corresponding measured data at the survey site (*i.e.* Shekare gauging station).

In the second run for predicting the dissolved lead concentration, the dissolved metal concentration was assumed to be non-conservative with the reaction coefficient in Equation (6) being constant. The best fit between the predicted and measured dissolved lead concentrations occurred for a reaction coefficient of 0.12 day-1. This assumption led to a prediction error of 3.4% and 17.1% for calibration and verification of the model, respectively (see Figures 6(a) and (b)). However, some research results suggest that the reaction coefficients for different pH and salinity conditions were not constant. A more detailed investigation is being planned to determine the rate of reaction coefficient for different pH and EC (*i.e.* as a function of salinity).

According to the above findings, it seems that using a variable reaction coefficient, which can be adjusted automatically within a numerical model, depending on the pH, EC or pH and EC values may give better calibration results. A number of simulations were carried out to find a formulation for describing the relationship between the reaction coefficient and the pH value. Using the measured dissolved lead concentrations, it was found that the most suitable relationship between the reaction coefficient for dissolved lead and pH of the river was of the following form:

$$
\kappa = -0.1646 \times pH + 1.4934 \qquad \left(R^2 = 0.643\text{ }\right) \tag{20}
$$

where:

*pH* =the mean pH of the river at the site for each time.

The predicted results, for which the reaction coefficient was calculated using Equation (20) in the model, were compared with the corresponding measured values for calibration and verification in Figures 6(a) and (b), respectively. The comparison showed that the error of simulation had reduced to 1.9% and 15% for calibration and verification of the model, respectively.

Based on the fact that the reaction coefficient relates to the EC value, a number of simulations were also carried out to find a suitable formulation for describing the reaction coefficient with the EC value. Using the measured dissolved lead concentration, it was found that the most suitable relationship between the reaction coefficient for dissolved lead and the EC of the river was of the following form:

$$
\kappa = -0.00023 \times EC + 0.581 \qquad \left( R^2 = 0.924 \right) \tag{21}
$$

where:

106 Numerical Modelling

As discussed above, the rate of reaction plays an important role in predicting the concentration distribution of the dissolved heavy metals for river, estuarine and coastal waters. In the current section effort was made to find suitable functions to represent the reaction coefficient rate for dissolved lead and cadmium metal modelling in rivers. These functions were established using a comparison of the predicted heavy metal concentrations with the corresponding measured values at the Shekare gauging station (see Figure 3). In calibrating the model against measured dissolved lead and cadmium levels, five approaches for each dissolved metal were used. The model with the adjusted rate of reaction was then validated using the corresponding measured data for different time series at the survey site (Kashefipour *et al.*, 2006). In the following sections the equations and the results of the modelling for dissolved lead and cadmium, with the derived equations for the reaction

For the first run a conservative dissolved lead was assumed, leading to a zero value for the rate of reaction coefficient. The fit between the predicted and measured data showed 25.2% and 33.3% errors for calibration and verification of the model respectively. As can be seen from Figures 6(a) and (b) the predicted dissolved lead in this case did not agree well with

In the second run for predicting the dissolved lead concentration, the dissolved metal concentration was assumed to be non-conservative with the reaction coefficient in Equation (6) being constant. The best fit between the predicted and measured dissolved lead concentrations occurred for a reaction coefficient of 0.12 day-1. This assumption led to a prediction error of 3.4% and 17.1% for calibration and verification of the model, respectively (see Figures 6(a) and (b)). However, some research results suggest that the reaction coefficients for different pH and salinity conditions were not constant. A more detailed investigation is being planned to determine the rate of reaction coefficient for different pH

According to the above findings, it seems that using a variable reaction coefficient, which can be adjusted automatically within a numerical model, depending on the pH, EC or pH and EC values may give better calibration results. A number of simulations were carried out to find a formulation for describing the relationship between the reaction coefficient and the pH value. Using the measured dissolved lead concentrations, it was found that the most suitable relationship between the reaction coefficient for dissolved lead and pH of the river

The predicted results, for which the reaction coefficient was calculated using Equation (20) in the model, were compared with the corresponding measured values for calibration and

0.1646 1.4934 *pH* ( *R*<sup>2</sup> 0.643 ) (20)

the corresponding measured data at the survey site (*i.e.* Shekare gauging station).

**5.2 Application of dissolved heavy metals modelling** 

coefficients are illustrated.

**5.2.1 Results of dissolved lead modelling** 

and EC (*i.e.* as a function of salinity).

*pH* =the mean pH of the river at the site for each time.

was of the following form:

where:

*EC* = (Electrical Conductivity), the mean EC of the river at the site for each time (micro mhos/cm).

The predicted lead concentration, for which the reaction coefficients were calculated using Equation (21) in the model, were compared with the corresponding measured values for calibration and verification in Figures 6(a) and (b), respectively. This showed that the error of simulation had also declined to 0.8% and 10.8% for calibration and verification of the model, respectively.

For the last run of the dissolved lead, a number of simulations were carried out to find a formulation for describing the relationship between the reaction coefficient and both the pH and EC variables. Using the measured dissolved lead concentrations, it was found that the most suitable relationship between the reaction coefficient for dissolved lead and pH and EC as variables for the river was of the following form:

$$\kappa = 0.160 \times pH - 0.000402 \times EC - 0.401 \qquad \left(R^2 = 1.000\right) \tag{22}$$

The predicted results, for which the reaction coefficient were calculated using Equation (22) in the model, were then compared with the corresponding measured values for calibration and verification in Figures 6(a) and (b) with the errors of 0.4% and 8.3% respectively. These results showed another improvement in the predicted dissolved lead concentrations.

A summary of the statistical analysis for the different model results is shown in Table 2. As it is clear from this table the predicted dissolved lead concentrations improved, giving lower errors when varying reaction coefficients were applied as a part of ADE.


Table 2. A summary of the dissolved lead model results.

Numerical Modelling of Heavy Metals Transport Processes in Riverine Basins 109

reaction coefficient of 0.38 day-1 was used in the model. This assumption significantly reduce the error to 7.7% and 8.5% for calibration and verification of the model, respectively

In the third run the reaction coefficient was assumed to be varying with pH. The most suitable relationship between the reaction coefficient for dissolved cadmium and pH in the

The predicted dissolved cadmium for which the reaction coefficients were calculated using Equation (23) in the model, were compared with the corresponding measured values for calibration and verification in Figures 7(a) and (b), respectively. This comparison showed that the error of simulation had declined to 1.8% and 4.2% for calibration and verification of

The fourth model run was carried out using the EC as a variable in computing the reaction coefficient. For the measured data of dissolved cadmium, the most suitable function for

This function was added to the model for predicting the results of dissolved cadmium with EC. The results showed that the error of simulation was 2.5% and 2.6% for calibration and

For the last run for dissolved cadmium a number of simulations were carried out to find a formulation for time varying reaction coefficients for the rate of reaction using both the pH and EC as variables. With using the measured data of dissolved cadmium, the most suitable function for relating the reaction coefficient with pH and EC in river was found to be:

The predicted dissolved cadmium for which the reaction coefficients were calculated using Equation (25) in the model, were compared with the corresponding measured values for calibration and verification in Figures 7(a) and (b), respectively. This showed that the error of simulation had declined to 2.2% and 2.3% for calibration and verification of the model, respectively. A summary of the statistical analysis for the different model

0 0.1086 71.11 0.1483 76.42

*Const*. 0.0126 7.67 0.0171 8.47

0.2462 2.3738 *pH* 0.0028 1.78 0.0114 4.16

0.000201 0.7286 *EC* 0.0041 2.54 0.0050 2.56

0.1231 0.0001 1.5512 *pH EC* 0.0035 2.18 0.0046 2.29

Table 3. A summary of the dissolved cadmium model results.

relating the reaction coefficient with EC in the river was found to be:

0.2462 2.3738 *pH* ( *R*<sup>2</sup> 0.703 ) (23)

0.000201 0.7286 *EC* ( *R*<sup>2</sup> 0.350 ) (24)

0.1231 0.0001 1.5512 *pH EC* ( *R*<sup>2</sup> 0.560 ) (25)

**CALIBRATION VERIFICATION**  *RMSE %Error RMSE %Error* 

(see Figures 7(a) and (b)).

the model, respectively.

river was found to be of following form:

verification, respectively (see Figures 7(a) and (b)).

results is shown in Table 3.

Fig. 6. Comparison of predicted dissolved Lead with the corresponding measured values.

#### **5.2.2 Results of dissolved cadmium modelling**

The same procedure was carried out for dissolved cadmium modelling. For the first run cadmium was assumed to be conservative for which the predicted data did not show reasonable agreement with the measured data and the error was estimated to be 71.1% and 76.4% for calibration and verification of the model respectively (see Figures 7(a) and (b)).

For the second run cadmium was assumed to be non-conservative, with a constant reaction coefficient and the best fit between the predicted and measured data occurred when a

(a) Calibration

(b) Verification Fig. 6. Comparison of predicted dissolved Lead with the corresponding measured values.

The same procedure was carried out for dissolved cadmium modelling. For the first run cadmium was assumed to be conservative for which the predicted data did not show reasonable agreement with the measured data and the error was estimated to be 71.1% and 76.4% for calibration and verification of the model respectively (see Figures 7(a) and (b)).

For the second run cadmium was assumed to be non-conservative, with a constant reaction coefficient and the best fit between the predicted and measured data occurred when a

**5.2.2 Results of dissolved cadmium modelling** 

reaction coefficient of 0.38 day-1 was used in the model. This assumption significantly reduce the error to 7.7% and 8.5% for calibration and verification of the model, respectively (see Figures 7(a) and (b)).

In the third run the reaction coefficient was assumed to be varying with pH. The most suitable relationship between the reaction coefficient for dissolved cadmium and pH in the river was found to be of following form:

$$
\kappa = -0.2462 \times pH + 2.3738 \qquad \qquad \left(R^2 = 0.703\right) \tag{23}
$$

The predicted dissolved cadmium for which the reaction coefficients were calculated using Equation (23) in the model, were compared with the corresponding measured values for calibration and verification in Figures 7(a) and (b), respectively. This comparison showed that the error of simulation had declined to 1.8% and 4.2% for calibration and verification of the model, respectively.

The fourth model run was carried out using the EC as a variable in computing the reaction coefficient. For the measured data of dissolved cadmium, the most suitable function for relating the reaction coefficient with EC in the river was found to be:

$$
\kappa = -0.000201 \times EC + 0.7286 \qquad \left(R^2 = 0.350\right) \tag{24}
$$

This function was added to the model for predicting the results of dissolved cadmium with EC. The results showed that the error of simulation was 2.5% and 2.6% for calibration and verification, respectively (see Figures 7(a) and (b)).

For the last run for dissolved cadmium a number of simulations were carried out to find a formulation for time varying reaction coefficients for the rate of reaction using both the pH and EC as variables. With using the measured data of dissolved cadmium, the most suitable function for relating the reaction coefficient with pH and EC in river was found to be:

$$\kappa = -0.1231 \times pH - 0.0001 \times EC + 1.5512 \qquad \left(R^2 = 0.560\right) \tag{25}$$

The predicted dissolved cadmium for which the reaction coefficients were calculated using Equation (25) in the model, were compared with the corresponding measured values for calibration and verification in Figures 7(a) and (b), respectively. This showed that the error of simulation had declined to 2.2% and 2.3% for calibration and verification of the model, respectively. A summary of the statistical analysis for the different model results is shown in Table 3.


Table 3. A summary of the dissolved cadmium model results.

Numerical Modelling of Heavy Metals Transport Processes in Riverine Basins 111

coefficient in estuarine waters is primarily a function of salinity. Nassehi & Bikangaga (1993) calculated the value of the reaction coefficient for dissolved zinc in different elements of a river. Wu *et al.* (2005) used salinity for modelling the partitioning coefficient of heavy metals in the Mersey estuary and concluded that the modelling results agreed well with the

It should be noted that the proposed method in this chapter is valid for rivers with large variations in salinity and pH. Therefore, this method could be used for rivers either close to the coastal waters, and thus affected by tides, or such rivers that have many agricultural inputs from saline soils draining into them. The chosen reach of the Karoon River in this research was an example of the second type of river. The average minimum and maximum EC for three years data collection (2002 - 2004) at the Ahwaz hydrometric station (see Figure 3) were 707 and 2254 µΩ-1/cm, respectively. The pH values also ranged from a minimum of

In deriving Equations (20) to (22) for lead and the similar ones for cadmium (Table 3), it was assumed that the environmental factors and water properties remained constant during the whole simulation period. Since the model was calibrated using measured dissolved lead and cadmium at the site this assumption was thought to be valid. However, there are some limitations in using these equations. Firstly, simultaneous measurements of dissolved lead and cadmium were only made at one site and for six months. More field-measured data are needed to validate and improve the formulae, which relate the pH and EC values to the reaction coefficient for dissolved lead and cadmium. Secondly, a one-dimensional model was used. Although one-dimensional models have been successfully used in riverine hydrodynamic and water quality studies, it seems that applying a two- or three-dimensional model may improve the derived equations. However, using two- or three-dimensional models needs extensive field-measured data. The importance of the models is to estimate the desirable variables as accurately as possible. Measuring some special environmental variables, such as heavy metals, in the field is sensitive and ideally needs extensive laboratory studies with sophisticated instruments and with large investments. Measuring pH and EC in riverine systems is relatively straightforward and can be done with even portable instruments. The main idea from this research work is therefore to introduce a procedure that relates the pH and EC values to reaction coefficients of heavy metal substances, such as lead and cadmium, for model predictions. Hence, for heavy metals modelling studies, measurements of pH and EC would be a suitable tool for relatively

The results show an average improvement of 25% and 71.5% in error estimations of lead and cadmium, respectively, when using pH and EC as two variables affecting the dynamic

Details are given of the hydro-environmental model to predict the dissolved heavy metals concentration along rivers using a varied reaction coefficient approach to the source term of the Advection-Dispersion Equation (ADE). The main purpose of this chapter was to describe the dissolved heavy metals modelling procedure and assess the impact of pH and EC on the reaction coefficient used in dissolved lead and cadmium modelling. The

measured data.

7.3 to a maximum of 8.5 at this station.

accurate estimation of these substances.

processes of these heavy metals.

**7. Summary and conclusions** 

Fig. 7. Comparison of predicted dissolved Cadmium with the corresponding measured values.

#### **6. Discussion**

Salinity has been found by many investigators to be more influential on the reaction coefficient than any other environmental or water properties in riverine and estuarine waters. The results published by Turner *et al.* (2002) showed that the trace metal distribution

(a) Calibration

(b) Verification

Salinity has been found by many investigators to be more influential on the reaction coefficient than any other environmental or water properties in riverine and estuarine waters. The results published by Turner *et al.* (2002) showed that the trace metal distribution

Fig. 7. Comparison of predicted dissolved Cadmium with the corresponding measured

values.

**6. Discussion** 

coefficient in estuarine waters is primarily a function of salinity. Nassehi & Bikangaga (1993) calculated the value of the reaction coefficient for dissolved zinc in different elements of a river. Wu *et al.* (2005) used salinity for modelling the partitioning coefficient of heavy metals in the Mersey estuary and concluded that the modelling results agreed well with the measured data.

It should be noted that the proposed method in this chapter is valid for rivers with large variations in salinity and pH. Therefore, this method could be used for rivers either close to the coastal waters, and thus affected by tides, or such rivers that have many agricultural inputs from saline soils draining into them. The chosen reach of the Karoon River in this research was an example of the second type of river. The average minimum and maximum EC for three years data collection (2002 - 2004) at the Ahwaz hydrometric station (see Figure 3) were 707 and 2254 µΩ-1/cm, respectively. The pH values also ranged from a minimum of 7.3 to a maximum of 8.5 at this station.

In deriving Equations (20) to (22) for lead and the similar ones for cadmium (Table 3), it was assumed that the environmental factors and water properties remained constant during the whole simulation period. Since the model was calibrated using measured dissolved lead and cadmium at the site this assumption was thought to be valid. However, there are some limitations in using these equations. Firstly, simultaneous measurements of dissolved lead and cadmium were only made at one site and for six months. More field-measured data are needed to validate and improve the formulae, which relate the pH and EC values to the reaction coefficient for dissolved lead and cadmium. Secondly, a one-dimensional model was used. Although one-dimensional models have been successfully used in riverine hydrodynamic and water quality studies, it seems that applying a two- or three-dimensional model may improve the derived equations. However, using two- or three-dimensional models needs extensive field-measured data. The importance of the models is to estimate the desirable variables as accurately as possible. Measuring some special environmental variables, such as heavy metals, in the field is sensitive and ideally needs extensive laboratory studies with sophisticated instruments and with large investments. Measuring pH and EC in riverine systems is relatively straightforward and can be done with even portable instruments. The main idea from this research work is therefore to introduce a procedure that relates the pH and EC values to reaction coefficients of heavy metal substances, such as lead and cadmium, for model predictions. Hence, for heavy metals modelling studies, measurements of pH and EC would be a suitable tool for relatively accurate estimation of these substances.

The results show an average improvement of 25% and 71.5% in error estimations of lead and cadmium, respectively, when using pH and EC as two variables affecting the dynamic processes of these heavy metals.

#### **7. Summary and conclusions**

Details are given of the hydro-environmental model to predict the dissolved heavy metals concentration along rivers using a varied reaction coefficient approach to the source term of the Advection-Dispersion Equation (ADE). The main purpose of this chapter was to describe the dissolved heavy metals modelling procedure and assess the impact of pH and EC on the reaction coefficient used in dissolved lead and cadmium modelling. The

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hydrodynamic module was first calibrated and validated using the field-measured data taken at a site located along the Karoon River, the largest river in the south west of Iran. In order to find the best equation between pH and EC with the reaction coefficient used in the ADE too, many model runs were carried out and the water quality module was subsequently calibrated by adjusting the reaction coefficient. For each measured lead or cadmium value at any time the most appropriate reaction coefficient was specified and from there for the considered heavy metals a few equation between pH and EC with the reaction coefficient were proposed and added to the water quality module of the model. The main findings from the model simulations can be summarized as follows:


#### **8. References**


hydrodynamic module was first calibrated and validated using the field-measured data taken at a site located along the Karoon River, the largest river in the south west of Iran. In order to find the best equation between pH and EC with the reaction coefficient used in the ADE too, many model runs were carried out and the water quality module was subsequently calibrated by adjusting the reaction coefficient. For each measured lead or cadmium value at any time the most appropriate reaction coefficient was specified and from there for the considered heavy metals a few equation between pH and EC with the reaction coefficient were proposed and added to the water quality module of the model. The main

1. Five different procedures were used for estimating the rate of reaction coefficient for dissolved lead and cadmium, including: a zero reaction coefficient, a constant reaction coefficient, a varying reaction coefficient with pH, a varying reaction coefficient with

2. Improvements were achieved in the predicted dissolved lead and cadmium

3. The best fit between the predicted and measured values for simulation with a constant reaction coefficient was obtained when the coefficient was set to 0.12 and 0.38 day-1 for

4. According to Equations (20) to (22) for lead and the similar ones in Table 3 for cadmium and the measured pH and EC values, the ranges of reaction coefficients were calculated to be: (0.11-0.18, 0.10-0.29, 0.10-0.43) and (0.31-0.40, 0.31-0.48, 0.31-0.44) for lead and cadmium for the three suggested procedures, respectively. The error estimation was decreased from an average of 30% to 4% for lead and 74% to 2.2% for cadmium when

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*engineers,* Longman Singapore Publishers (Pte) Ltd., 425pp.

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EC and a varying reaction coefficient with both pH and EC.

dissolved lead and cadmium, respectively.

Vol.131, No.10, pp.898-908

*Management,* Vol.1, No.1, pp.81-89

Ltd., pp.305-328

*hydraulics*, Pitman Publishing Limited, Boston

**8. References** 


**1. Introduction**

of the planet.

(e.g. Christensen et al., 2007; Gillett et al., 2011).

**0**

**6**

*Canada*

**Modelling Dynamics of Valley Glaciers**

Ever since the Paleoproterozoic snowball Earth era, ca. 2.4 billion years ago (e.g. Hoffman & Schrag, 2000; Kirschvink, 1992), and beyond, the landscape of the planet Earth has been shaped up by the tremendous amount of scouring due to the repeated waxing and waning of ice masses. Over time, the dynamics of ice masses – a major part of Earth's cryosphere – has played a crucial role in global climate through complex interactions and feedbacks between the atmosphere, biosphere, and oceans. The cryosphere remains as one of the major dynamical components of the Earth system, participating in the geomorphologic and climatic evolution

Presently, glaciers and ice sheets occupy ca. 10% of the Earth's land surface in the annual mean (Lemke et al., 2007). If it were to melt out completely, the mean sea level would rise by more than 64 m. The majority of this contribution comes from the large ice sheets of Antarctica, 56.6 m (Lythe et al., 2001), and Greenland, 7.3 m (Bamber et al., 2001). Glaciers and ice caps outside of Greenland and Antarctica contribute in a range between 0.15 m (Ohmura, 2004) and 0.37 m (Dyurgerov & Meier, 2005). In the ongoing warm epoch of climate since the little ice age, beginning in the late 19th century, glaciers and ice sheets have been retreating in most regions of the world (e.g. Cook et al., 2005; Krabill et al., 1999; Zemp et al., 2006). Such a response of the cryosphere creates a high-degree of disequilibrium, with positive feedbacks on the Earth's climate system, whereby the planet is likely to face ongoing and accelerated ice loss. Giving proper attention to the cryospheric component of climate system, most climate models forecast continued warming and glacier retreat at least until the end of 21st century

On this premise, glaciological studies bear a tremendous importance; they are useful, for instance, (1) to understand the complex interaction between the ice and climate (e.g. Goelzer et al., 2011; Kaser, 2001), (2) to trace out the past climatic signals (e.g. Oerlemans, 2005; Thompson et al., 2003), (3) to assess the glacier-related hazards (e.g. Allen et al., 2009), and (4) to estimate glacial contributions to sea level rise (e.g. Leclercq et al., 2011; Meier, 1984; Raper & Braithwaite, 2006). To make future projections and to understand the intrinsic dynamical phenomena underlying glacier-climate interactions, such as the thermomechanical evolution

In this chapter, we discuss the physics and numerics of ice flow models with various degrees of complexity and we simulate the corresponding dynamics of a valley glacier. While valley glaciers make up only a tiny fraction (< 1.0%) of the global cryosphere, proper understanding of glacier dynamics is essential for several reasons. First, valley glaciers are in close proximity

of ice masses, numerical modelling, supplemented by field data, is the only option.

Surendra Adhikari and Shawn J. Marshall *Department of Geography, University of Calgary*


### **Modelling Dynamics of Valley Glaciers**

Surendra Adhikari and Shawn J. Marshall

*Department of Geography, University of Calgary Canada*

#### **1. Introduction**

114 Numerical Modelling

Rauf, A.; Javed, M.; Ubaidyllah, M. & Abdullah, S. (2009). Assessment of heavy metals in

Roshanfekr, A.; Kashefipour, S.M. & Jafarzadeh, N. (2008a). A new approach for modeling

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Sanayei, Y.; Ismail, N. & Talebi, S.M. (2009). Determination of heavy metals in Zayandeh Rood river, Isfahan, Iran. *World Applied Sciences Journal,* Vo.6, No.9, pp.1209-1214 Shrestha, P.L. & Orlob, G.T. (1996). Multiphase distribution of cohesive sediments and

Tavakolizadeh, A.A. (2006). *Modelling hydrodynamics and water quality in riverine systems*.

Thomann, A. & Mueller, J.A. (1987). *Principles of surface water quality modelling control.* Harper

Turner, A.; Martino, M. & Le roux, S.M. (2002). Trace metal distribution coefficients in the

Wu, Y.; Falconer, R.A. & Lin, B. (2005). Modelling trace metal concentration distributions in estuarine waters. *Journal of Estuarine, Costal and Shelf Science,* Vol. 64, pp. 699-709 Yang, L.; Lin, B.; Kashefipour, S.M. & Falconer, R.A. (2002). Integration of a 1-D river model

Zhen-Gang, J. (2008). *Hydrodynamics and Water Quality: Modeling Rivers, Lakes, and Estuaries.*

Wu, W. (2008). *Computational River Dynamics*, Taylor & Francis Group, London, 494pp. Wu, Y.; Falconer, R.A. & Lin, B. (2001). Hydro-environmental modelling of heavy metal

*IWA Journal of Hydroinformatics,* Vol.10, No.3, pp.245-255

*Science and Technology Journal,* Vol.36, No.21, pp.4578-4584

*Hydraulics,* Beijing, China, pp.732-739

John Wiley & Sons, Inc., Hoboken, NJ, 676pp.

Vol.11, No.2, pp.197-200

Vol.8, No.12, pp.2242-2249

No.8, pp.730-740

Iran. 136 pp. (In Farsi)

Collins, New York

pp.693-701

sediments of river Ravi, Pakistan. *International Journal of Agriculture & Biology,* 

dissolved lead using an integrated 1D and 2D model, *Journal of Applied Sciences,* 

metals for riverine systems using a new approach to the source term in the ADE,

heavy metals in estuarine systems. *Journal of Environmental Engineering,* Vol.122,

MSc. Thesis, Department of Hydraulic Structures, Shahid Chamran University,

Mersey estuary, UK: Evidence got salting out of metal complexes. *Environmental* 

fluxes in an estuary. *Proceedings of XXIX IAHR Congress, Theme b: Environmental* 

with object-oriented methodology. *Environmental Modelling and Software,* Vol.17,

Ever since the Paleoproterozoic snowball Earth era, ca. 2.4 billion years ago (e.g. Hoffman & Schrag, 2000; Kirschvink, 1992), and beyond, the landscape of the planet Earth has been shaped up by the tremendous amount of scouring due to the repeated waxing and waning of ice masses. Over time, the dynamics of ice masses – a major part of Earth's cryosphere – has played a crucial role in global climate through complex interactions and feedbacks between the atmosphere, biosphere, and oceans. The cryosphere remains as one of the major dynamical components of the Earth system, participating in the geomorphologic and climatic evolution of the planet.

Presently, glaciers and ice sheets occupy ca. 10% of the Earth's land surface in the annual mean (Lemke et al., 2007). If it were to melt out completely, the mean sea level would rise by more than 64 m. The majority of this contribution comes from the large ice sheets of Antarctica, 56.6 m (Lythe et al., 2001), and Greenland, 7.3 m (Bamber et al., 2001). Glaciers and ice caps outside of Greenland and Antarctica contribute in a range between 0.15 m (Ohmura, 2004) and 0.37 m (Dyurgerov & Meier, 2005). In the ongoing warm epoch of climate since the little ice age, beginning in the late 19th century, glaciers and ice sheets have been retreating in most regions of the world (e.g. Cook et al., 2005; Krabill et al., 1999; Zemp et al., 2006). Such a response of the cryosphere creates a high-degree of disequilibrium, with positive feedbacks on the Earth's climate system, whereby the planet is likely to face ongoing and accelerated ice loss. Giving proper attention to the cryospheric component of climate system, most climate models forecast continued warming and glacier retreat at least until the end of 21st century (e.g. Christensen et al., 2007; Gillett et al., 2011).

On this premise, glaciological studies bear a tremendous importance; they are useful, for instance, (1) to understand the complex interaction between the ice and climate (e.g. Goelzer et al., 2011; Kaser, 2001), (2) to trace out the past climatic signals (e.g. Oerlemans, 2005; Thompson et al., 2003), (3) to assess the glacier-related hazards (e.g. Allen et al., 2009), and (4) to estimate glacial contributions to sea level rise (e.g. Leclercq et al., 2011; Meier, 1984; Raper & Braithwaite, 2006). To make future projections and to understand the intrinsic dynamical phenomena underlying glacier-climate interactions, such as the thermomechanical evolution of ice masses, numerical modelling, supplemented by field data, is the only option.

In this chapter, we discuss the physics and numerics of ice flow models with various degrees of complexity and we simulate the corresponding dynamics of a valley glacier. While valley glaciers make up only a tiny fraction (< 1.0%) of the global cryosphere, proper understanding of glacier dynamics is essential for several reasons. First, valley glaciers are in close proximity

thought to depend primarily on ice temperature, as well as on crystal size and orientation (anisotropy), water and impurity content, ice density and pressure, and perhaps on other several factors. Only the thermal dependence of *A* has been parameterized, following an Arrhenius relation (e.g. Hooke, 1981; Paterson & Budd, 1982), and coupled successfully with dynamical ice-flow models (e.g. Huybrechts & Oerlemans, 1988; Marshall & Clarke, 1997). Several attempts have also been made to account for anisotropic effects through the introduction of a "flow enhancement factor", both empirically (e.g. Wang & Warner, 1999) and through physically-based parameterizations (e.g. Gillet-Chaulet et al., 2005; Morland &

Modelling Dynamics of Valley Glaciers 117

Similarly, the choice of flow law exponent, *n*, is also not obvious, as it varies in a range 1.5− 4.2 under different stress regimes (Weertman, 1973). For the realistic scenarios, i.e. *τ* ≈ 50 − 200 kPa, *n* = 3 is representative (e.g. Cuffey & Paterson, 2010). Assuming isothermal and isotropic ice masses for purposes here, we use *n* = 3 and *A* = 10−<sup>16</sup> Pa−<sup>3</sup> a−<sup>1</sup> as in, e.g. Pattyn et al.

In the early 1950s, the power-law relation between *τ* and *�*˙ (Eqs. 1–3) was formulated based on laboratory experiments (Glen, 1952) and field observations on the closure of boreholes (Nye, 1953). Subsequently, the Glen-Nye law, commonly known as the Glen's law after Glen (1955), emerged to describe the glacier ice as a quasi-viscous fluid with non-Newtonian flow behaviour. This not only discarded the then-prevailing theory of "extrusion flow" (see Waddington, 2010), but also opened the door for investigating the theoretical and mathematical foundations of modern glaciology. Nye's works (e.g. Nye, 1952; 1959) mark the beginning of such investigations, particularly focusing on the motion of glacier ice. Robin (1955) was the first to calculate ice temperature by considering glacial thermodynamics. Due to the lack of computational power, these early works were primarily based on the semi-analytical methods used in contemporary fluid mechanics. A nice summary of these

early works that form the foundation of physical glaciology is given in Clarke (1987).

are presented respectively by Payne et al. (1996) and Huybrechts et al. (1996).

SIA theory is strictly valid only where horizontal gradients in ice thickness and velocity are negligibly small and bedrock slopes are sufficiently gentle. These criteria are clearly violated in valley glaciers (e.g. Le Meur et al., 2004; Leysinger Vieli & Gudmundsson, 2004),

With the dawn of digital computing, model-based studies of glacier dynamics started in late 1960s (e.g. Campbell & Rasmussen, 1969). Soon after, several numerical models (e.g. Budd & Jenssen, 1975; Mahaffy, 1976; Oerlemans, 1982) were developed, including the ones with thermomechanical coupling (e.g. Jenssen, 1977). These pioneer models were based on the "shallow-ice" theory of glacier mechanics (e.g. Nye, 1959), which assumes that ice thickness is much less than the horizontal length scale over which a domain is discretized. This theory was later developed rigorously by Hutter (1983) and Morland (1984), which is now known formally as the shallow-ice approximation (SIA). SIA models have been used extensively for simulating large ice sheets (e.g. Calov & Hutter, 1996; Huybrechts & Oerlemans, 1988), as well as valley glaciers (e.g. Adhikari & Huybrechts, 2009; Oerlemans et al., 1998). In general, ice sheet models need to thermomechanically coupled, since the polar ice sheets span a range of temperature from the melting point to ca. –50◦C, whereas models for temperate valley glaciers are commonly isothermal. This is reasonable outside of the polar regions, as most of the world's valley glaciers are temperate: at the pressure-melting point throughout. For SIA models with and without the coupled thermodynamics, benchmark numerical experiments

Staroszczyk, 2003).

(2008) and Sargent & Fastook (2010).

to human settlement; any alteration in their dynamics affects society immediately. Second, valley glaciers and ice caps are of significant concern for watershed- and regional-scale water resources (e.g. Jansson et al., 2003; Viviroli et al., 2003); they, for instance, provide fresh water supply for municipal, agricultural, and industrial purposes. Third, the dynamical response of glaciers leaves footprints of past climate in their moraines (e.g. Beedle et al., 2009); they have hence become proven indicators of climate change. More importantly, the ice flow in valley glaciers and icefields comprises a high degree of complexity, primarily due to the irregular valley geometry. This demands a high-order treatment of glacier dynamics, thereby posing a challenge to numerical modellers. Finally, the fundamental physics of glaciers (i.e. mechanisms of ice flow) do not differ from those of larger ice sheets; experience in modelling valley-glacier dynamics can be directly extended to modelling of continental-scale ice sheets.

This chapter is hence designed to focus on the dynamics of valley glacier and its modelling. We (1) introduce ice rheology and briefly summarize the history of numerical modelling in glaciology, (2) describe the model physics and analyze the various approximations associated with the low-order (reduced) models, (3) provide an overview of numerical methods, concentrating on the finite element approach, and (4) present a numerical comparison of several models with various degrees of sophistication.

#### **2. Ice rheology and glacier modelling**

The rheological properties of glacier ice are practically independent of the isotropic pressure (e.g. Rigsby, 1958), and are therefore commonly described using deviatoric stresses rather than Cauchy stresses. The constitutive equation that relates deviatoric stresses to strain-rates in randomly oriented polycrystalline ice (under secondary creep) is given by the linearized inversion of Glen's flow law (Glen, 1955), i.e.

$$
\pi\_{\dot{i}\dot{j}} = 2\eta \dot{\epsilon}\_{\dot{i}\dot{j}\prime} \tag{1}
$$

where *τ* is the deviatoric stress tensor, *�*˙ is the strain-rate tensor, and *η* is the effective viscosity. The viscosity of glacier ice is strain-rate dependent and is given by,

$$\eta = \frac{1}{2} A^{-\frac{1}{n}} \dot{\varepsilon}\_{\varepsilon}^{\left(\frac{1-n}{n}\right)} \, \, \, \, \, \, \, \tag{2}$$

where *A* is the flow law rate factor, *n* is the flow law exponent, and *�*˙*e* is the effective strain-rate that can be understood from the second invariant of *�*˙, i.e.

$$
\mathcal{D}\dot{\epsilon}^2\_{\epsilon} = \dot{\epsilon}\_{\text{ij}}\dot{\epsilon}\_{\text{ji}}.\tag{3}
$$

By defining

$$
\dot{\epsilon}\_{ij} = \frac{1}{2} (u\_{i,j} + u\_{j,i})\_\prime \tag{4}
$$

deviatoric stresses in Equation (1) can easily be expressed in terms of ice velocity, *u*, – the readily observable glaciological field variable.

Hypotheses and experimental foundations of this theory of ice rheology are given by Glen (1958) and are reviewed in detail by, e.g. Alley (1992), Budd & Jacka (1989), Cuffey & Paterson (2010), Hooke (1981), and Marshall (2005). Since the form of the constitutive relation (Eq. 1) is well-established and can be explained in terms of dislocation theory, these discussions revolve around the suitable parameterizations of *A* and *n*. The flow law rate factor, *A*, is 2 Will-be-set-by-IN-TECH

to human settlement; any alteration in their dynamics affects society immediately. Second, valley glaciers and ice caps are of significant concern for watershed- and regional-scale water resources (e.g. Jansson et al., 2003; Viviroli et al., 2003); they, for instance, provide fresh water supply for municipal, agricultural, and industrial purposes. Third, the dynamical response of glaciers leaves footprints of past climate in their moraines (e.g. Beedle et al., 2009); they have hence become proven indicators of climate change. More importantly, the ice flow in valley glaciers and icefields comprises a high degree of complexity, primarily due to the irregular valley geometry. This demands a high-order treatment of glacier dynamics, thereby posing a challenge to numerical modellers. Finally, the fundamental physics of glaciers (i.e. mechanisms of ice flow) do not differ from those of larger ice sheets; experience in modelling valley-glacier dynamics can be directly extended to modelling of continental-scale ice sheets. This chapter is hence designed to focus on the dynamics of valley glacier and its modelling. We (1) introduce ice rheology and briefly summarize the history of numerical modelling in glaciology, (2) describe the model physics and analyze the various approximations associated with the low-order (reduced) models, (3) provide an overview of numerical methods, concentrating on the finite element approach, and (4) present a numerical comparison of

The rheological properties of glacier ice are practically independent of the isotropic pressure (e.g. Rigsby, 1958), and are therefore commonly described using deviatoric stresses rather than Cauchy stresses. The constitutive equation that relates deviatoric stresses to strain-rates in randomly oriented polycrystalline ice (under secondary creep) is given by the linearized

where *τ* is the deviatoric stress tensor, *�*˙ is the strain-rate tensor, and *η* is the effective viscosity.

where *A* is the flow law rate factor, *n* is the flow law exponent, and *�*˙*e* is the effective strain-rate

deviatoric stresses in Equation (1) can easily be expressed in terms of ice velocity, *u*, – the

Hypotheses and experimental foundations of this theory of ice rheology are given by Glen (1958) and are reviewed in detail by, e.g. Alley (1992), Budd & Jacka (1989), Cuffey & Paterson (2010), Hooke (1981), and Marshall (2005). Since the form of the constitutive relation (Eq. 1) is well-established and can be explained in terms of dislocation theory, these discussions revolve around the suitable parameterizations of *A* and *n*. The flow law rate factor, *A*, is

*<sup>η</sup>* <sup>=</sup> <sup>1</sup> 2 *A*<sup>−</sup> <sup>1</sup> *n �*˙ ( <sup>1</sup>−*<sup>n</sup>*

> 2*�*˙ 2

*�*˙*ij* <sup>=</sup> <sup>1</sup> 2

*τij* = 2*η�*˙*ij*, (1)

*<sup>n</sup>* ) *<sup>e</sup>* , (2)

*<sup>e</sup>* = *�*˙*ij�*˙ *ji*. (3)

(*ui*,*<sup>j</sup>* + *uj*,*i*), (4)

several models with various degrees of sophistication.

The viscosity of glacier ice is strain-rate dependent and is given by,

that can be understood from the second invariant of *�*˙, i.e.

readily observable glaciological field variable.

**2. Ice rheology and glacier modelling**

inversion of Glen's flow law (Glen, 1955), i.e.

By defining

thought to depend primarily on ice temperature, as well as on crystal size and orientation (anisotropy), water and impurity content, ice density and pressure, and perhaps on other several factors. Only the thermal dependence of *A* has been parameterized, following an Arrhenius relation (e.g. Hooke, 1981; Paterson & Budd, 1982), and coupled successfully with dynamical ice-flow models (e.g. Huybrechts & Oerlemans, 1988; Marshall & Clarke, 1997). Several attempts have also been made to account for anisotropic effects through the introduction of a "flow enhancement factor", both empirically (e.g. Wang & Warner, 1999) and through physically-based parameterizations (e.g. Gillet-Chaulet et al., 2005; Morland & Staroszczyk, 2003).

Similarly, the choice of flow law exponent, *n*, is also not obvious, as it varies in a range 1.5− 4.2 under different stress regimes (Weertman, 1973). For the realistic scenarios, i.e. *τ* ≈ 50 − 200 kPa, *n* = 3 is representative (e.g. Cuffey & Paterson, 2010). Assuming isothermal and isotropic ice masses for purposes here, we use *n* = 3 and *A* = 10−<sup>16</sup> Pa−<sup>3</sup> a−<sup>1</sup> as in, e.g. Pattyn et al. (2008) and Sargent & Fastook (2010).

In the early 1950s, the power-law relation between *τ* and *�*˙ (Eqs. 1–3) was formulated based on laboratory experiments (Glen, 1952) and field observations on the closure of boreholes (Nye, 1953). Subsequently, the Glen-Nye law, commonly known as the Glen's law after Glen (1955), emerged to describe the glacier ice as a quasi-viscous fluid with non-Newtonian flow behaviour. This not only discarded the then-prevailing theory of "extrusion flow" (see Waddington, 2010), but also opened the door for investigating the theoretical and mathematical foundations of modern glaciology. Nye's works (e.g. Nye, 1952; 1959) mark the beginning of such investigations, particularly focusing on the motion of glacier ice. Robin (1955) was the first to calculate ice temperature by considering glacial thermodynamics. Due to the lack of computational power, these early works were primarily based on the semi-analytical methods used in contemporary fluid mechanics. A nice summary of these early works that form the foundation of physical glaciology is given in Clarke (1987).

With the dawn of digital computing, model-based studies of glacier dynamics started in late 1960s (e.g. Campbell & Rasmussen, 1969). Soon after, several numerical models (e.g. Budd & Jenssen, 1975; Mahaffy, 1976; Oerlemans, 1982) were developed, including the ones with thermomechanical coupling (e.g. Jenssen, 1977). These pioneer models were based on the "shallow-ice" theory of glacier mechanics (e.g. Nye, 1959), which assumes that ice thickness is much less than the horizontal length scale over which a domain is discretized. This theory was later developed rigorously by Hutter (1983) and Morland (1984), which is now known formally as the shallow-ice approximation (SIA). SIA models have been used extensively for simulating large ice sheets (e.g. Calov & Hutter, 1996; Huybrechts & Oerlemans, 1988), as well as valley glaciers (e.g. Adhikari & Huybrechts, 2009; Oerlemans et al., 1998). In general, ice sheet models need to thermomechanically coupled, since the polar ice sheets span a range of temperature from the melting point to ca. –50◦C, whereas models for temperate valley glaciers are commonly isothermal. This is reasonable outside of the polar regions, as most of the world's valley glaciers are temperate: at the pressure-melting point throughout. For SIA models with and without the coupled thermodynamics, benchmark numerical experiments are presented respectively by Payne et al. (1996) and Huybrechts et al. (1996).

SIA theory is strictly valid only where horizontal gradients in ice thickness and velocity are negligibly small and bedrock slopes are sufficiently gentle. These criteria are clearly violated in valley glaciers (e.g. Le Meur et al., 2004; Leysinger Vieli & Gudmundsson, 2004),

*t = t +* Δ*t*

Modelling Dynamics of Valley Glaciers 119

*t*

*ui*,*<sup>i</sup>* = 0, (5) *σij*,*<sup>j</sup>* + *ρgi* = 0, (6)

*σij* = *τij* + *pδij*, (7)

Fig. 1. Flowchart of an ice flow model. The processes associated with the diagnostic and prognostic simulations are listed in the LHS and RHS boxes, respectively; black and red colors are used for clarity. Dotted boxes enclose the processes that are not considered in this

where *u* is the velocity vector, *σ* is the Cauchy stress tensor, *ρ* is ice density, and *g* is the gravity

whereby the momentum balance equation (Eq. 6) can be expressed in terms of the velocity vector, as explained in Section 2. The isotropic pressure is dependent on the trace of Cauchy stress tensor, i.e. *p* = *σii*/*k*, and is activated via the Kronecker delta, *δij*, only when normal

Intuitively, three-dimensional (3D) Stokes models, which solve a complete set of Stokes equations (Eqs. 5–6), describe the most sophisticated treatment of glacier dynamics. Virtually all models developed to date (e.g. SIA or high-order) can be considered as approximations of a Stokes model. Hindmarsh (2004) compares the numerical solutions of various approximations to the Stokes equations. Apart from the standard SIA model, he considers an 'L' family of models that include some of most common models, such as those of Blatter (1995), MacAyeal (1989) and Pattyn (2003). These L-models differ from each other in: (1) how they reduce the definitions of momentum balance (Eq. 6), strain-rate tensor (Eq. 4) and its second invariant (Eq. 3), and (2) how they obtain the approximate solutions of ice velocities from the previous iteration step to calculate effective viscosity (Eq. 2). Since such approximations are made primarily to optimize the solution accuracy and computational efficiency, it is not always obvious how to choose a particular model for a given glaciological scenario. Here, we present

study. Mid-arrows are used to depict the corresponding boundary conditions.

vector. We split *σ* into its deviatoric part, *τ*, and an isotropic pressure, *p*, i.e.

stresses are being considered (*δij* = 1 for *i* = *j*, and *δij* = 0 otherwise).

space, *<sup>k</sup>*, the Stokes problem can be stated as,

fast-flowing ice streams (e.g. Whillans & Van der Veen, 1997), and at the ice divides and grounding zones of ice-sheet/ice-shelf systems (e.g. Baral et al., 2001). Several attempts have therefore been made to capture high-order dynamics in ice flow models; effects of longitudinal stress gradients (e.g. Adhikari & Marshall, 2011; Shoemaker & Morland, 1984; Souˇcek & Martinec, 2008) and lateral drag (e.g. Adhikari & Marshall, in preperation; Nye, 1965) are particularly accounted via physically-based or numerical/empirical parameterizations. More complete representations of glacier dynamics are provided by high-order (e.g. Blatter, 1995; Pattyn, 2003) and Stokes (e.g. Jarosch, 2008; Jouvet et al., 2008; Zwinger et al., 2007) models. The development history of such models is nicely summarized by Blatter et al. (2010); corresponding benchmark experiments are presented by Pattyn et al. (2008).

With the material nonlinearity (see Eqs. 1–2), even SIA models are not analytically tractable; the coupled evolution of glacier temperatures, rheology, and high-order velocities therefore requires a numerical solution. A few attempts have been made to obtain analytical solutions (e.g. Bueler et al., 2007; Sargent & Fastook, 2010), however, at least as a tool for verification of numerical models in simple geometric and climatic settings.

#### **3. Model physics, approximations, and boundary conditions**

For full simulations of Earth's climate system, the dynamical models of glaciers and ice sheets are coupled with those of other climatic components, namely the atmosphere, the biosphere, and the ocean (see, for example, Fig. 1 in Huybrechts et al., 2011). The models of glaciers and ice sheets are usually accompanied by, for instance, (1) a mass balance model that describes the physics of mass exchange at the ice/atmosphere and ice/ocean interface, (2) a model of glacial isostatic processes whereby the underlying bed deforms due to the load of ice, and (3) a model of subglacial till deformation that yields the associated basal motion of ice. The intrinsic processes of ice flow can be described by a combination of gravitational creep deformation and decoupled basal sliding. To simulate creep deformation (i.e. effective viscosity) and to predict the regions where ice masses are warm-based (permitting basal sliding), a proper account of energy balance is essential. Along with the companion models listed above, a full three-dimensional ice flow model, equipped with thermomechanical coupling, is therefore required for the realistic simulations of glacier dynamics.

We outline a simple flowchart (Fig. 1) depicting the major components of a typical ice flow model. Given the boundary conditions and some description of mass budget, we accomplish ice flow modelling in a two-step simulation: (1) diagnostic simulation of a set of steady-state problems in order to obtain the quasi-stationary englacial velocity/stress and temperature fields at time *t*, and (2) prognostic simulation satisfying kinematic boundary conditions to update the glacier geometry at a subsequent time, *t* + Δ*t*. Below, we describe the physics and associated low-order approximations of several ice flow models.

#### **3.1 Diagnostic equations**

Dynamical models for ice flow are based on the fundamental physics of conservation of mass, momentum, and energy. Glacier velocities are so small that we can remove the acceleration term from the momentum balance equation; the dynamical problem in glaciology therefore reduces to a Stokes problem. For isothermal glacier domains in *k*(≥ 2)-dimensional Euclidean 4 Will-be-set-by-IN-TECH

fast-flowing ice streams (e.g. Whillans & Van der Veen, 1997), and at the ice divides and grounding zones of ice-sheet/ice-shelf systems (e.g. Baral et al., 2001). Several attempts have therefore been made to capture high-order dynamics in ice flow models; effects of longitudinal stress gradients (e.g. Adhikari & Marshall, 2011; Shoemaker & Morland, 1984; Souˇcek & Martinec, 2008) and lateral drag (e.g. Adhikari & Marshall, in preperation; Nye, 1965) are particularly accounted via physically-based or numerical/empirical parameterizations. More complete representations of glacier dynamics are provided by high-order (e.g. Blatter, 1995; Pattyn, 2003) and Stokes (e.g. Jarosch, 2008; Jouvet et al., 2008; Zwinger et al., 2007) models. The development history of such models is nicely summarized by Blatter et al. (2010);

With the material nonlinearity (see Eqs. 1–2), even SIA models are not analytically tractable; the coupled evolution of glacier temperatures, rheology, and high-order velocities therefore requires a numerical solution. A few attempts have been made to obtain analytical solutions (e.g. Bueler et al., 2007; Sargent & Fastook, 2010), however, at least as a tool for verification of

For full simulations of Earth's climate system, the dynamical models of glaciers and ice sheets are coupled with those of other climatic components, namely the atmosphere, the biosphere, and the ocean (see, for example, Fig. 1 in Huybrechts et al., 2011). The models of glaciers and ice sheets are usually accompanied by, for instance, (1) a mass balance model that describes the physics of mass exchange at the ice/atmosphere and ice/ocean interface, (2) a model of glacial isostatic processes whereby the underlying bed deforms due to the load of ice, and (3) a model of subglacial till deformation that yields the associated basal motion of ice. The intrinsic processes of ice flow can be described by a combination of gravitational creep deformation and decoupled basal sliding. To simulate creep deformation (i.e. effective viscosity) and to predict the regions where ice masses are warm-based (permitting basal sliding), a proper account of energy balance is essential. Along with the companion models listed above, a full three-dimensional ice flow model, equipped with thermomechanical coupling, is therefore

We outline a simple flowchart (Fig. 1) depicting the major components of a typical ice flow model. Given the boundary conditions and some description of mass budget, we accomplish ice flow modelling in a two-step simulation: (1) diagnostic simulation of a set of steady-state problems in order to obtain the quasi-stationary englacial velocity/stress and temperature fields at time *t*, and (2) prognostic simulation satisfying kinematic boundary conditions to update the glacier geometry at a subsequent time, *t* + Δ*t*. Below, we describe the physics and

Dynamical models for ice flow are based on the fundamental physics of conservation of mass, momentum, and energy. Glacier velocities are so small that we can remove the acceleration term from the momentum balance equation; the dynamical problem in glaciology therefore reduces to a Stokes problem. For isothermal glacier domains in *k*(≥ 2)-dimensional Euclidean

corresponding benchmark experiments are presented by Pattyn et al. (2008).

numerical models in simple geometric and climatic settings.

required for the realistic simulations of glacier dynamics.

associated low-order approximations of several ice flow models.

**3.1 Diagnostic equations**

**3. Model physics, approximations, and boundary conditions**

Fig. 1. Flowchart of an ice flow model. The processes associated with the diagnostic and prognostic simulations are listed in the LHS and RHS boxes, respectively; black and red colors are used for clarity. Dotted boxes enclose the processes that are not considered in this study. Mid-arrows are used to depict the corresponding boundary conditions.

space, *<sup>k</sup>*, the Stokes problem can be stated as,

$$
u\_{i,i} = 0,\tag{5}$$

$$
\sigma\_{\dot{i}\dot{j},\dot{l}} + \rho g\_{\dot{i}} = 0,\tag{6}
$$

where *u* is the velocity vector, *σ* is the Cauchy stress tensor, *ρ* is ice density, and *g* is the gravity vector. We split *σ* into its deviatoric part, *τ*, and an isotropic pressure, *p*, i.e.

$$
\sigma\_{\rm ij} = \tau\_{\rm ij} + p \delta\_{\rm ij} \tag{7}
$$

whereby the momentum balance equation (Eq. 6) can be expressed in terms of the velocity vector, as explained in Section 2. The isotropic pressure is dependent on the trace of Cauchy stress tensor, i.e. *p* = *σii*/*k*, and is activated via the Kronecker delta, *δij*, only when normal stresses are being considered (*δij* = 1 for *i* = *j*, and *δij* = 0 otherwise).

Intuitively, three-dimensional (3D) Stokes models, which solve a complete set of Stokes equations (Eqs. 5–6), describe the most sophisticated treatment of glacier dynamics. Virtually all models developed to date (e.g. SIA or high-order) can be considered as approximations of a Stokes model. Hindmarsh (2004) compares the numerical solutions of various approximations to the Stokes equations. Apart from the standard SIA model, he considers an 'L' family of models that include some of most common models, such as those of Blatter (1995), MacAyeal (1989) and Pattyn (2003). These L-models differ from each other in: (1) how they reduce the definitions of momentum balance (Eq. 6), strain-rate tensor (Eq. 4) and its second invariant (Eq. 3), and (2) how they obtain the approximate solutions of ice velocities from the previous iteration step to calculate effective viscosity (Eq. 2). Since such approximations are made primarily to optimize the solution accuracy and computational efficiency, it is not always obvious how to choose a particular model for a given glaciological scenario. Here, we present

where *ux*(*x*, *z*) and *ux*(*x*, *b*) are respectively the velocities at any depth *z* and at the bedrock *z* = *b*. Similarly, *gz* is the vertical component of the gravity vector, *h*(*x*) is the ice thickness, *s*

Modelling Dynamics of Valley Glaciers 121

In prognostic simulations of each model, the glacier surface, *z* = *s*, evolves satisfying the

where subscript *t* represents time, index *i* refers to the horizontal coordinates, i.e. *i* = (*x*, *y*) ⊂ �3, and *<sup>i</sup>* <sup>=</sup> *<sup>x</sup>* ⊂ �2, *uz*(*s*) is the vertical velocity at the glacier surface, and *<sup>m</sup>* is the mass balance function. To compute the unknown *s* at a new time *t* + Δ*t*, we obtain glacier surface velocities from the diagnostic simulation of domain at an antecedent time *t*, and prescribe

In addition to the kinematic boundary condition (Eq. 9), the upper ice surface satisfies the stress free criterion, i.e. *σij*(*s*) ≈ 0. This involves an assumption that the role of atmospheric pressure on the overall dynamics of glacier ice is negligibly small. The kinematic boundary condition (similar to Eq. 9) is also applied at the ice/bedrock interface. However, we impose a no-slip basal criterion, i.e. *ui*(*b*) = 0. In doing so, we assume no mass exchange due to ice melting/refreezing, i.e. *m*(*b*,*t*) = 0, thus restricting the evolution of basal ice, i.e. *b*,*<sup>t</sup>* = 0.

The lateral boundary condition (glacier margin) is typically free in glacier simulations, with ice free to advance or retreat within a domain. Domain extent is designed such that the ice mass does not reach the boundary; it then freely evolves within the domain, with a zone with ice thickness *h* = 0 around the periphery. If the combined snow accumulation rates and ice flux into an empty grid cell exceed local mass loss (ablation), the grid cell becomes glacierized

The existence of numerical solutions to strongly nonlinear Stokes problem discussed in Section 3.1 is proven in Colinge & Rappaz (1999). Various approaches have been used to obtain solutions in 3D space; numerical schemes such as finite difference (e.g. Colinge & Blatter, 1998; Huybrechts et al., 1996; Marshall & Clarke, 1997; Pattyn, 2003), finite element (e.g. Gudmundsson, 1999; Hanson, 1995; Jarosch, 2008; Jouvet et al., 2008; Picasso et al., 2008; Zwinger et al., 2007), control volume (e.g. Price et al., 2007), and spectral (e.g. Hindmarsh, 2004) methods have all been employed. Finite difference (FD) and finite element (FE) methods are more common; we use the latter one. Along with a brief overview of the FD method, below

Most "classical" models of glacier and ice sheet dynamics are based on FDM, where the 3D domain is split into a series of regular grid cells on a Cartesian or spherical (i.e. Earth) grid. Staggered grids are used, solving ice thickness, stress and temperature in the cell centre and

*s*,*<sup>t</sup>* + *ui*(*s*)*s*,*<sup>i</sup>* = *uz*(*s*) + *m*(*s*, *t*), (9)

is the glacier surface, and *αs*(*x*) is the surface slope.

*m*(*s*, *t*) as a vertical flux with units m ice eq. a–1.

and is included in the overall glacier continuum.

**4.1 Finite difference method (FDM)**

**4. Numerical methods applied to glacier dynamics**

we provide theoretical and numerical details of the FE method.

**3.2 Prognostic equations**

**3.3 Boundary conditions**

kinematic boundary condition,

new brands of models, whose associated approximations clearly define the distinct physical mechanisms of glacier dynamics; the scope of each model therefore becomes apparent.

In a 3D domain of land-based glacier, key physical processes that act to balance a gravity-driven ice-flow consist of basal drag, *τb*, resistance associated with longitudinal stress gradients, *τlon*, and lateral drag, *τlat*. Mathematical details of these resistances can be found, for example, in Van der Veen (1999) and Whillans (1987). Based on the physical mechanisms associated with each resistance, we define three families of models. Ice flow in the first family of models is controlled collectively by (*τ<sup>b</sup>* + *τlon* + *τlat*), in the second one by (*τ<sup>b</sup>* + *τlon*) only, and in the third one by *τ<sup>b</sup>* alone. For a fairly wide glacier (such that *τlat* ≈ 0) resting on steep and undulating bedrock (such that *τlon* is significant), for example, the second family of models should be optimal to yield realistic simulations. Members of a given model family differ from each other mainly in that they deal with different spatial dimensions. Below, we describe each of them; relevant governing equations are given in Appendix A.

#### **3.1.1 Full-system Stokes (FS) model**

If a model solves the Stokes equations (Eqs. 5–6) in 3D space, �3, we call it the full-system Stokes model (FS). This is the only member of the first model family. Here, the indices (*i*, *j*) refer to Cartesian coordinates (*x*, *y*, *z*); *x* is the horizontal coordinate along the principal flow direction, *y* is the second horizontal coordinate along the lateral direction, and *z* is the vertical coordinate opposite to gravity.

#### **3.1.2 Plane-strain Stokes (PS) model**

The 3D plane-strain Stokes model (PS3) does not strictly follow the plane-strain approximations as its name suggests. It rather excludes the lateral gradients of stress deviators, i.e. *τij*,*<sup>y</sup>* = 0, in the momentum balance equation (Eq. 6), and those of ice velocities, i.e. *ui*,*<sup>y</sup>* = 0, in the strain-rate definition (Eq. 4). The flowline version of this model (PS2) solves the Stokes equations in a two-dimensional (2D) space, �2, and hence follows the plane-strain approximations. In a flowline model, the indices (*i*, *j*) refer to Cartesian coordinates (*x*, *z*), where *x* and *z* are once again the horizontal and vertical coordinates, respectively.

#### **3.1.3 Shear-deformational (SD) model**

In 3D shear-deformational model (SD3), the vertical shear stresses, i.e. *τiz* with *i* = (*x*, *y*), are the only non-zero stress components. As for the PS3 model, it also excludes the lateral gradients of stress deviators and ice velocities. In its flowline counterpart (SD2), *τxz* is the only non-zero stress component; no further assumption is needed.

The standard zeroth-order SIA models (Hutter, 1983) also belong to the SD family. The three-dimensional (SIA3) and flowline (SIA2) shallow-ice models can be derived respectively from the SD3 and SD2 models, by further assuming that the horizontal gradients in vertical shear stresses and ice velocities are negligible, i.e. *τiz*,*<sup>x</sup>* = 0 and *ui*,*<sup>x</sup>* = 0. The laminar-flow model (LF) is the simplest of SD models. There exist analytical solutions for ice velocities in isothermal, laminar flow (e.g. Cuffey & Paterson, 2010; Van der Veen, 1999); the horizontal velocity, *ux*, at any point on the flowline plane (*x*, *z*), for example, is given by

$$u\_{\mathbf{x}}(\mathbf{x}, \mathbf{z}) = u\_{\mathbf{x}}(\mathbf{x}, \mathbf{b}) + \frac{2A}{n+1} \left[ \rho g\_{\mathbf{z}} a\_{\mathbf{s}}(\mathbf{x}) \right]^n h(\mathbf{x})^{n+1} \left[ 1 - \left( \frac{\mathbf{s} - \mathbf{z}}{h(\mathbf{x})} \right)^{n+1} \right],\tag{8}$$

where *ux*(*x*, *z*) and *ux*(*x*, *b*) are respectively the velocities at any depth *z* and at the bedrock *z* = *b*. Similarly, *gz* is the vertical component of the gravity vector, *h*(*x*) is the ice thickness, *s* is the glacier surface, and *αs*(*x*) is the surface slope.

#### **3.2 Prognostic equations**

6 Will-be-set-by-IN-TECH

new brands of models, whose associated approximations clearly define the distinct physical mechanisms of glacier dynamics; the scope of each model therefore becomes apparent.

In a 3D domain of land-based glacier, key physical processes that act to balance a gravity-driven ice-flow consist of basal drag, *τb*, resistance associated with longitudinal stress gradients, *τlon*, and lateral drag, *τlat*. Mathematical details of these resistances can be found, for example, in Van der Veen (1999) and Whillans (1987). Based on the physical mechanisms associated with each resistance, we define three families of models. Ice flow in the first family of models is controlled collectively by (*τ<sup>b</sup>* + *τlon* + *τlat*), in the second one by (*τ<sup>b</sup>* + *τlon*) only, and in the third one by *τ<sup>b</sup>* alone. For a fairly wide glacier (such that *τlat* ≈ 0) resting on steep and undulating bedrock (such that *τlon* is significant), for example, the second family of models should be optimal to yield realistic simulations. Members of a given model family differ from each other mainly in that they deal with different spatial dimensions. Below, we

If a model solves the Stokes equations (Eqs. 5–6) in 3D space, �3, we call it the full-system Stokes model (FS). This is the only member of the first model family. Here, the indices (*i*, *j*) refer to Cartesian coordinates (*x*, *y*, *z*); *x* is the horizontal coordinate along the principal flow direction, *y* is the second horizontal coordinate along the lateral direction, and *z* is the vertical

The 3D plane-strain Stokes model (PS3) does not strictly follow the plane-strain approximations as its name suggests. It rather excludes the lateral gradients of stress deviators, i.e. *τij*,*<sup>y</sup>* = 0, in the momentum balance equation (Eq. 6), and those of ice velocities, i.e. *ui*,*<sup>y</sup>* = 0, in the strain-rate definition (Eq. 4). The flowline version of this model (PS2) solves the Stokes equations in a two-dimensional (2D) space, �2, and hence follows the plane-strain approximations. In a flowline model, the indices (*i*, *j*) refer to Cartesian coordinates (*x*, *z*),

In 3D shear-deformational model (SD3), the vertical shear stresses, i.e. *τiz* with *i* = (*x*, *y*), are the only non-zero stress components. As for the PS3 model, it also excludes the lateral gradients of stress deviators and ice velocities. In its flowline counterpart (SD2), *τxz* is the

The standard zeroth-order SIA models (Hutter, 1983) also belong to the SD family. The three-dimensional (SIA3) and flowline (SIA2) shallow-ice models can be derived respectively from the SD3 and SD2 models, by further assuming that the horizontal gradients in vertical shear stresses and ice velocities are negligible, i.e. *τiz*,*<sup>x</sup>* = 0 and *ui*,*<sup>x</sup>* = 0. The laminar-flow model (LF) is the simplest of SD models. There exist analytical solutions for ice velocities in isothermal, laminar flow (e.g. Cuffey & Paterson, 2010; Van der Veen, 1999); the horizontal

where *x* and *z* are once again the horizontal and vertical coordinates, respectively.

only non-zero stress component; no further assumption is needed.

*ux*(*x*, *<sup>z</sup>*) = *ux*(*x*, *<sup>b</sup>*) + <sup>2</sup>*<sup>A</sup>*

velocity, *ux*, at any point on the flowline plane (*x*, *z*), for example, is given by

*<sup>n</sup>* <sup>+</sup> <sup>1</sup> [*ρgzαs*(*x*)]

*<sup>n</sup> h*(*x*)*n*+<sup>1</sup>

 1 −

 *<sup>s</sup>* <sup>−</sup> *<sup>z</sup> h*(*x*)

*<sup>n</sup>*+<sup>1</sup> 

, (8)

describe each of them; relevant governing equations are given in Appendix A.

**3.1.1 Full-system Stokes (FS) model**

coordinate opposite to gravity.

**3.1.2 Plane-strain Stokes (PS) model**

**3.1.3 Shear-deformational (SD) model**

In prognostic simulations of each model, the glacier surface, *z* = *s*, evolves satisfying the kinematic boundary condition,

$$\mathbf{s}\_{,l} + \mathfrak{u}\_{l}(\mathbf{s})\mathbf{s}\_{,l} = \mathfrak{u}\_{z}(\mathbf{s}) + m(\mathbf{s},t),\tag{9}$$

where subscript *t* represents time, index *i* refers to the horizontal coordinates, i.e. *i* = (*x*, *y*) ⊂ �3, and *<sup>i</sup>* <sup>=</sup> *<sup>x</sup>* ⊂ �2, *uz*(*s*) is the vertical velocity at the glacier surface, and *<sup>m</sup>* is the mass balance function. To compute the unknown *s* at a new time *t* + Δ*t*, we obtain glacier surface velocities from the diagnostic simulation of domain at an antecedent time *t*, and prescribe *m*(*s*, *t*) as a vertical flux with units m ice eq. a–1.

#### **3.3 Boundary conditions**

In addition to the kinematic boundary condition (Eq. 9), the upper ice surface satisfies the stress free criterion, i.e. *σij*(*s*) ≈ 0. This involves an assumption that the role of atmospheric pressure on the overall dynamics of glacier ice is negligibly small. The kinematic boundary condition (similar to Eq. 9) is also applied at the ice/bedrock interface. However, we impose a no-slip basal criterion, i.e. *ui*(*b*) = 0. In doing so, we assume no mass exchange due to ice melting/refreezing, i.e. *m*(*b*,*t*) = 0, thus restricting the evolution of basal ice, i.e. *b*,*<sup>t</sup>* = 0.

The lateral boundary condition (glacier margin) is typically free in glacier simulations, with ice free to advance or retreat within a domain. Domain extent is designed such that the ice mass does not reach the boundary; it then freely evolves within the domain, with a zone with ice thickness *h* = 0 around the periphery. If the combined snow accumulation rates and ice flux into an empty grid cell exceed local mass loss (ablation), the grid cell becomes glacierized and is included in the overall glacier continuum.

#### **4. Numerical methods applied to glacier dynamics**

The existence of numerical solutions to strongly nonlinear Stokes problem discussed in Section 3.1 is proven in Colinge & Rappaz (1999). Various approaches have been used to obtain solutions in 3D space; numerical schemes such as finite difference (e.g. Colinge & Blatter, 1998; Huybrechts et al., 1996; Marshall & Clarke, 1997; Pattyn, 2003), finite element (e.g. Gudmundsson, 1999; Hanson, 1995; Jarosch, 2008; Jouvet et al., 2008; Picasso et al., 2008; Zwinger et al., 2007), control volume (e.g. Price et al., 2007), and spectral (e.g. Hindmarsh, 2004) methods have all been employed. Finite difference (FD) and finite element (FE) methods are more common; we use the latter one. Along with a brief overview of the FD method, below we provide theoretical and numerical details of the FE method.

#### **4.1 Finite difference method (FDM)**

Most "classical" models of glacier and ice sheet dynamics are based on FDM, where the 3D domain is split into a series of regular grid cells on a Cartesian or spherical (i.e. Earth) grid. Staggered grids are used, solving ice thickness, stress and temperature in the cell centre and

according to the standard Galerkin method, is then set to zero, i.e.

 Ω*<sup>e</sup> ψu* 

system of these equations in 3D Cartesian coordinates.

the stabilized finite elements (Franca & Frey, 1992).

*σij*,*j*, *δ*1*σij*,*<sup>j</sup>*

residuals. The stability parameters are chosen following (Franca & Frey, 1992),

each relevant Γ*e*, *s*˜ varies according to the chosen interpolation function *ψ<sup>s</sup>*

*s*˜(*x*, *y*, *t*) = *ψ<sup>s</sup>*

stabilization (for the FS model) are given in Appendix C.

 Γ*e ψ<sup>s</sup>*

**4.2.2 Prognostic equations**

(Eq. 9), whose weighted sum is set to zero,

*<sup>δ</sup>*<sup>1</sup> <sup>=</sup> *mkh*<sup>2</sup> *k* <sup>4</sup>*<sup>η</sup>* , *<sup>δ</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup>*<sup>η</sup>*

 Ω*<sup>e</sup> ψ<sup>p</sup> u*˜*i*,*<sup>i</sup>* 

*σij*,*j*(*u*˜*i*, *p*˜) + *ρgi*

Here the weights are chosen to be the assumed interpolation functions; this is unique to the Galerkin method (e.g. Rao, 2005). See Appendix B for the details of construction of a linear

Modelling Dynamics of Valley Glaciers 123

As the governing equations comprise a one-degree high-order of derivative for the velocity vector than that for the isotropic pressure (see Appendix A), a typical Taylor-Hood element (Hood & Taylor, 1974) with quadratic interpolation function for velocities and linear one for pressure is recommended. For simplicity, however, we use the same order of interpolation function, so that *<sup>ψ</sup><sup>u</sup>* <sup>=</sup> *<sup>ψ</sup><sup>p</sup>* <sup>≡</sup> *<sup>ψ</sup>*. Any instability arising as a result is accommodated by using

Stabilization involves addition of mesh-dependent terms to the Galerkin formulation. These additional terms are the Euler-Lagrange equations evaluated elementwise, so that exact solutions satisfy both the Galerkin and these additional terms. The additional terms are,

> +

*ρgi*, *δ*1*σij*,*<sup>j</sup>*

Equation (14) is added to the elemental coefficient matrix and Equation (15) is added to the RHS force vector. In Equation (14), the first term inside the first inner product is the residual of momentum balance equation (Eq. 13), excluding the force term, and the first term inside the second inner product is the residual of continuity equation (Eq. 12). The second terms associated with stability parameters *δ*<sup>1</sup> and *δ*<sup>2</sup> are the stabilization contributions to the weight functions. Here, these contributions are assumed to be the same as the respective system

where *mk* depends on the type of the element and *hk* on its size. Details of diagnostic system

In prognostic simulations, we seek the approximate solution, *s*˜, of *s* along the ice surface. Over

*<sup>i</sup>*(*x*, *<sup>y</sup>*, *<sup>t</sup>*)*si* <sup>=</sup> [*ψ<sup>s</sup>*

With this approximation of field variables, we obtain the residuals of the prognostic equation

*s*˜,*<sup>t</sup>* + *uis*˜,*<sup>i</sup>* − (*uz* + *m*)

*ui*,*i*, *δ*2*ui*,*<sup>i</sup>*

*mk*

dΩ*<sup>e</sup>* = 0, (12)

dΩ*<sup>e</sup>* = 0. (13)

, (14)

. (15)

, (16)

(*x*, *y*, *t*)] {*s*} . (17)

dΓ*<sup>e</sup>* = 0. (18)

, such that

3D velocity fields at the cell interfaces, with discretization of the governing equations through standard second-order FD approximations (e.g. Huybrechts & Oerlemans, 1988). Vertical resolution is usually fine compared to horizontal resolution, with 10-40 layers in the vertical, on an adaptive, stretched grid that is updated each time step as the glacier thins or thickens. Sometimes a nonlinear vertical grid transformation is also introduced in order to increase resolution near the bed, where velocity gradients are strongest.

For SIA models, the system of FD equations is particularly efficient to solve, as the governing equation at a point is dependent on only local conditions (ice thickness, surface slope). In this case the solution depends only on the nearest neighbours to a grid cell, presenting a banded matrix system that is amenable to sparse matrix techniques. With a more complete representation of the physics, i.e. the Stokes and high-order models, solutions are non-local and computational costs increase by at an order of magnitude or more (Blatter, 1995).

FD discretizations are also limited in their ability to describe complex geometries, as found in valley glaciers, ice shelves, and fjords: many of the most interesting glaciological situations. Much of the most interesting dynamics is at the glacier or ice sheet margin, where increased resolution is desirable. We therefore turn to FE method for glaciological simulations for the remainder of this chapter.

#### **4.2 Finite element method (FEM)**

The existence and convergence of FE solutions for the glaciological (Stokes) problem have been proven in Chow et al. (2004) and Glowinski & Rappaz (2003). Let <sup>Ω</sup> ∈ �*k*|*<sup>k</sup>* <sup>≥</sup> 2 be a continuous glacier domain, enclosed by a boundary Γ. In FE schemes, we decompose Ω into a finite number of elemental domains Ω*e*, thereby generating a finite number of boundary domains Γ*e*. Satisfying the relevant boundary conditions, we first seek the approximate solutions of field variables within each sub-domain, Ω*e*. We then assemble them over the continuous domain, Ω, to obtain the required solutions.

There are several methods, such as variational and weighted residuals, to formulate the FE counterparts of the governing equations. We use the standard Galerkin method – a weighted residual approach in which the weighted sum of system residuals arising from the FE approximations of a continuous domain is set to zero.

#### **4.2.1 Diagnostic equations**

Let *u*˜ and *p*˜ be the approximate solutions of field variables that vary within Ω*e*, according to the respective interpolation functions *ψ<sup>u</sup>* and *ψp*, such that

$$\vec{u}(\mathbf{x}, y, z) = \psi\_i^{\mu}(\mathbf{x}, y, z)u\_i = \left[\psi^{\mu}(\mathbf{x}, y, z)\right] \left\{u\right\},\tag{10}$$

$$\vec{p}(\mathbf{x}, y, z) = \psi\_{\mathbf{i}}^{p}(\mathbf{x}, y, z) p\_{\mathbf{i}} = \left[\psi^{p}(\mathbf{x}, y, z)\right] \left\{p\right\},\tag{11}$$

where index *i* refers to the elemental degrees of freedom associated with the velocity vector and pressure, respectively. Hence, {*u*} and {*p*} denote the nodal velocities and pressure.

By plugging in the approximations of field variables (Eqs. 10–11), we obtain the residuals of the diagnostic equations presented in Section 3.1. The weighted sum of these residuals, 8 Will-be-set-by-IN-TECH

3D velocity fields at the cell interfaces, with discretization of the governing equations through standard second-order FD approximations (e.g. Huybrechts & Oerlemans, 1988). Vertical resolution is usually fine compared to horizontal resolution, with 10-40 layers in the vertical, on an adaptive, stretched grid that is updated each time step as the glacier thins or thickens. Sometimes a nonlinear vertical grid transformation is also introduced in order to increase

For SIA models, the system of FD equations is particularly efficient to solve, as the governing equation at a point is dependent on only local conditions (ice thickness, surface slope). In this case the solution depends only on the nearest neighbours to a grid cell, presenting a banded matrix system that is amenable to sparse matrix techniques. With a more complete representation of the physics, i.e. the Stokes and high-order models, solutions are non-local

FD discretizations are also limited in their ability to describe complex geometries, as found in valley glaciers, ice shelves, and fjords: many of the most interesting glaciological situations. Much of the most interesting dynamics is at the glacier or ice sheet margin, where increased resolution is desirable. We therefore turn to FE method for glaciological simulations for the

The existence and convergence of FE solutions for the glaciological (Stokes) problem have been proven in Chow et al. (2004) and Glowinski & Rappaz (2003). Let <sup>Ω</sup> ∈ �*k*|*<sup>k</sup>* <sup>≥</sup> 2 be a continuous glacier domain, enclosed by a boundary Γ. In FE schemes, we decompose Ω into a finite number of elemental domains Ω*e*, thereby generating a finite number of boundary domains Γ*e*. Satisfying the relevant boundary conditions, we first seek the approximate solutions of field variables within each sub-domain, Ω*e*. We then assemble them over the

There are several methods, such as variational and weighted residuals, to formulate the FE counterparts of the governing equations. We use the standard Galerkin method – a weighted residual approach in which the weighted sum of system residuals arising from the

Let *u*˜ and *p*˜ be the approximate solutions of field variables that vary within Ω*e*, according to

where index *i* refers to the elemental degrees of freedom associated with the velocity vector and pressure, respectively. Hence, {*u*} and {*p*} denote the nodal velocities and pressure. By plugging in the approximations of field variables (Eqs. 10–11), we obtain the residuals of the diagnostic equations presented in Section 3.1. The weighted sum of these residuals,

*<sup>i</sup>* (*x*, *<sup>y</sup>*, *<sup>z</sup>*)*ui* <sup>=</sup> [*ψu*(*x*, *<sup>y</sup>*, *<sup>z</sup>*)] {*u*} , (10)

*<sup>i</sup>* (*x*, *<sup>y</sup>*, *<sup>z</sup>*)*pi* <sup>=</sup> [*ψp*(*x*, *<sup>y</sup>*, *<sup>z</sup>*)] {*p*} , (11)

and computational costs increase by at an order of magnitude or more (Blatter, 1995).

resolution near the bed, where velocity gradients are strongest.

remainder of this chapter.

**4.2.1 Diagnostic equations**

**4.2 Finite element method (FEM)**

continuous domain, Ω, to obtain the required solutions.

FE approximations of a continuous domain is set to zero.

the respective interpolation functions *ψ<sup>u</sup>* and *ψp*, such that

*u*˜(*x*, *y*, *z*) = *ψ<sup>u</sup>*

*<sup>p</sup>*˜(*x*, *<sup>y</sup>*, *<sup>z</sup>*) = *<sup>ψ</sup><sup>p</sup>*

according to the standard Galerkin method, is then set to zero, i.e.

$$\int\_{\Omega\_{\varepsilon}} \psi^p \left[ \vec{u}\_{i,i} \right] \, \mathrm{d}\Omega\_{\varepsilon} = 0,\tag{12}$$

$$\int\_{\Omega\_{\varepsilon}} \psi^{u} \left[ \sigma\_{\vec{ij},\vec{j}}(\vec{u}\_{i\prime}\vec{p}) + \rho g\_{i} \right] \,\mathrm{d}\Omega\_{\varepsilon} = 0. \tag{13}$$

Here the weights are chosen to be the assumed interpolation functions; this is unique to the Galerkin method (e.g. Rao, 2005). See Appendix B for the details of construction of a linear system of these equations in 3D Cartesian coordinates.

As the governing equations comprise a one-degree high-order of derivative for the velocity vector than that for the isotropic pressure (see Appendix A), a typical Taylor-Hood element (Hood & Taylor, 1974) with quadratic interpolation function for velocities and linear one for pressure is recommended. For simplicity, however, we use the same order of interpolation function, so that *<sup>ψ</sup><sup>u</sup>* <sup>=</sup> *<sup>ψ</sup><sup>p</sup>* <sup>≡</sup> *<sup>ψ</sup>*. Any instability arising as a result is accommodated by using the stabilized finite elements (Franca & Frey, 1992).

Stabilization involves addition of mesh-dependent terms to the Galerkin formulation. These additional terms are the Euler-Lagrange equations evaluated elementwise, so that exact solutions satisfy both the Galerkin and these additional terms. The additional terms are,

$$
\left<\sigma\_{\text{ij},\text{j}}\,\delta\_1\sigma\_{\text{ij},\text{j}}\right> + \left,\tag{14}
$$

$$
\left<\rho g\_{i\prime} \delta\_1 \sigma\_{i\dot{\jmath},\dot{\jmath}}\right>.\tag{15}
$$

Equation (14) is added to the elemental coefficient matrix and Equation (15) is added to the RHS force vector. In Equation (14), the first term inside the first inner product is the residual of momentum balance equation (Eq. 13), excluding the force term, and the first term inside the second inner product is the residual of continuity equation (Eq. 12). The second terms associated with stability parameters *δ*<sup>1</sup> and *δ*<sup>2</sup> are the stabilization contributions to the weight functions. Here, these contributions are assumed to be the same as the respective system residuals. The stability parameters are chosen following (Franca & Frey, 1992),

$$
\delta\_1 = \frac{m\_k h\_k^2}{4\eta}, \; \delta\_2 = \frac{2\eta}{m\_k}.\tag{16}
$$

where *mk* depends on the type of the element and *hk* on its size. Details of diagnostic system stabilization (for the FS model) are given in Appendix C.

#### **4.2.2 Prognostic equations**

In prognostic simulations, we seek the approximate solution, *s*˜, of *s* along the ice surface. Over each relevant Γ*e*, *s*˜ varies according to the chosen interpolation function *ψ<sup>s</sup>* , such that

$$\left[\mathbf{s}(\mathbf{x},\mathbf{y},t)=\psi\_{i}^{s}(\mathbf{x},\mathbf{y},t)\mathbf{s}\_{i}=\left[\psi^{s}(\mathbf{x},\mathbf{y},t)\right]\{\mathbf{s}\}\right.\tag{17}$$

With this approximation of field variables, we obtain the residuals of the prognostic equation (Eq. 9), whose weighted sum is set to zero,

$$\int\_{\Gamma\_{\ell}} \psi^{s} \left[ \mathfrak{s}\_{,t} + \mathfrak{u}\_{i} \mathfrak{s}\_{,i} - (\mathfrak{u}\_{2} + m) \right] \, \mathrm{d}\Gamma\_{\ell} = 0. \tag{18}$$

until the steady-state criterion, *es*, is reached; we use implicit scheme (first-order backward differentiation formula, BDF, scheme) for such time-dependent integrations. We advise maintaining *el* < *enl* < *es* for good convergence. Other aspects of Elmer (e.g. effects of mesh density on solution accuracy and computational efficiency, and parallel simulations) are

Modelling Dynamics of Valley Glaciers 125

For each experiment considered in this study, we generate the structured mesh by using ElmerGrid. ElmerGrid is basically a 2D mesh generator, but is also capable of extruding and manipulating the mesh in the third dimension. Since 3D experiments require a large amount of memory and computation time, we perform parallel runs in a high-performance computing

We consider a 10 <sup>×</sup> 2.5 km<sup>2</sup> glacial valley. To mimic a typical real-world glacier scenario, we

where *α<sup>b</sup>* is the mean bedrock slope in radians, *L* and *W* are the longitudinal and lateral extents of the valley, *ax* and *ay* are the amplitudes of the topographical variation in *x* and *y* directions, and *θ* is the sinusoidal offset of the flowline. Here, we use *α<sup>b</sup>* = 12◦, *L* = 10 km, *W* = 2.5 km, *ax* = 200 m, *ay* = (500 + 0.05*x*) m, and *θ* = 500 sin (2*πx*/*L*) m. The plan view of basal

Next, we define the climatic regime. For *z* ≤ 4.6 km and over a 500-m wide corridor around

where *β* is the linear mass balance gradient, *E* is the equilibrium line altitude (ELA), and Δ*m*(*t*) is the time-dependent mass balance perturbation; *m* (*s*, *t*) = 0 elsewhere. For now, we choose *β* = 0.01 m ice eq. a−<sup>1</sup> m−1, *E* = 3.7 km, and Δ*m*(*t*) = 0 m ice eq. a−1. Since *s*(*x*, *y*) evolves in prognostic simulations (Eq. 9), the parameterization of *m*(*s*, *t*) (Eq. 20) ensures that our models

Under these geometric and climatic settings, we grow glaciers to steady state using several models. In order to illustrate the importance of each physical mechanism of ice flow, we consider three different models (one from each model family), namely the FS, PS3 and SIA3 models. We denote them respectively by FS, PS and SD, unless otherwise specified. Each model domain consists of 50 k bilinear quadrilateral elements, with average horizontal dimensions 50 <sup>×</sup> 50 m2. The vertical dimension of element varies according to the ice thickness; we use five vertical layers. Below, we carry out numerical comparison of these 3D models in terms of steady state geometry, surface velocity, basal shear stress, and response timescales. Considering pragmatic flowline models (PS2 and SIA2), we also present a brief tutorial on modelling valley glacier dynamics, which involves (1) sensitivity tests for a glacier, (2) reconstruction of past climate or glacier extent, and (3) projection of a glacier's future.

0.5*π* +

4*πx L* 

<sup>−</sup> *ay* sin

*m*(*s*, *t*) = *β* [*s*(*x*, *y*) − *E*] + Δ*m*(*t*), (20)

(*<sup>y</sup>* <sup>+</sup> *<sup>θ</sup>*) *<sup>π</sup> W* 

, (19)

include meanders and bumps in the subglacial topography, *b* (*x*, *y*), as defined below,

cluster provided by the Western Canadian Research Grid (WestGrid).

topography is shown in Figure 3a; the central flowline is also depicted.

the central flowline (see Fig. 3a), the mass balance function, *m* (*s*, *t*), is chosen as,

**5. Numerical comparison of physical approximations**

*<sup>b</sup>* (*x*, *<sup>y</sup>*) <sup>=</sup> <sup>5000</sup> <sup>−</sup> *<sup>x</sup>* tan *<sup>α</sup><sup>b</sup>* <sup>+</sup> *ax* sin

capture the height/mass-balance feedback inclusively.

given by Gagliardini & Zwinger (2008).

Fig. 2. Flowchart of the solution scheme. Here also, red color is used to distinguish the prognostic simulations from the diagnostic ones

This hyperbolic equation is stabilized by adding the element-wise terms (Donea & Huerta, 2003) to the mass and coefficient matrices, as well as to the force vector. Mathematical details of FE formulation and stabilization of the prognostic equation are given in Appendix D.

#### **4.2.3 Elmer and model numerics**

We use the open source FEM code Elmer (http://www.csc.fi/elmer), adapted for Glen's flow law for ice (Glen, 1955). Elmer gives approximate (numerical) solutions for both the FS and reduced models by solving the weak forms of the respective governing equations. The solutions from the FS and PS2 models are tested by Gagliardini & Zwinger (2008), against the ISMIP-HOM (Ice Sheet Model Intercomparison Project for Higher-Order Models; Pattyn et al., 2008) benchmark experiments. We solve additional subroutines to obtain FE solutions for the PS3 and SD family of models. We validate SIA2 model by comparing results with the corresponding analytical solutions (Eq. 8; see Adhikari & Marshall, 2011).

We sketch a flowchart of the solution scheme employed in Elmer (Fig. 2). The linear system of equations obtained from the Galerkin formulation (Eq. B16) is in the core of the solver. This can be solved by using either direct or iterative methods. The direct method yields an exact solution up to the machine precision; this, however, is not feasible for large problems. We therefore use an iterative method, the Krylov subspace method method (biconjugate gradient stabilized method, BiCGStab) with with an incomplete lower-upper factorization (ILU4) as the system pre-conditioner, and obtain the approximate solutions. Given the mesh density and element type, the accuracy of such solutions relies on the chosen convergence criterion, *el*; the smaller the value of *el*, the more accurate the solutions. However, too small a choice of *el* makes the job computationally inefficient.

We then solve the material nonlinearity associated with the constitutive relation. We apply a fixed point iteration scheme (the Picard linearization) to linearize the system by expressing *η* in terms of *ui* from the previous iteration step. Here also, a suitable convergence criterion, *enl*, should be satisfied. For a given transient domain, we integrate the prognostic equation 10 Will-be-set-by-IN-TECH

[*K*]{φ}*=*[*F*]

*K*=*K*(φ), *enl*

Fig. 2. Flowchart of the solution scheme. Here also, red color is used to distinguish the

This hyperbolic equation is stabilized by adding the element-wise terms (Donea & Huerta, 2003) to the mass and coefficient matrices, as well as to the force vector. Mathematical details of FE formulation and stabilization of the prognostic equation are given in Appendix D.

We use the open source FEM code Elmer (http://www.csc.fi/elmer), adapted for Glen's flow law for ice (Glen, 1955). Elmer gives approximate (numerical) solutions for both the FS and reduced models by solving the weak forms of the respective governing equations. The solutions from the FS and PS2 models are tested by Gagliardini & Zwinger (2008), against the ISMIP-HOM (Ice Sheet Model Intercomparison Project for Higher-Order Models; Pattyn et al., 2008) benchmark experiments. We solve additional subroutines to obtain FE solutions for the PS3 and SD family of models. We validate SIA2 model by comparing results with the

We sketch a flowchart of the solution scheme employed in Elmer (Fig. 2). The linear system of equations obtained from the Galerkin formulation (Eq. B16) is in the core of the solver. This can be solved by using either direct or iterative methods. The direct method yields an exact solution up to the machine precision; this, however, is not feasible for large problems. We therefore use an iterative method, the Krylov subspace method method (biconjugate gradient stabilized method, BiCGStab) with with an incomplete lower-upper factorization (ILU4) as the system pre-conditioner, and obtain the approximate solutions. Given the mesh density and element type, the accuracy of such solutions relies on the chosen convergence criterion, *el*; the smaller the value of *el*, the more accurate the solutions. However, too small a choice of

We then solve the material nonlinearity associated with the constitutive relation. We apply a fixed point iteration scheme (the Picard linearization) to linearize the system by expressing *η* in terms of *ui* from the previous iteration step. Here also, a suitable convergence criterion, *enl*, should be satisfied. For a given transient domain, we integrate the prognostic equation

corresponding analytical solutions (Eq. 8; see Adhikari & Marshall, 2011).

prognostic simulations from the diagnostic ones

*el* makes the job computationally inefficient.

**4.2.3 Elmer and model numerics**

*es*

*t* = *t +* Δ*t*

*el*

until the steady-state criterion, *es*, is reached; we use implicit scheme (first-order backward differentiation formula, BDF, scheme) for such time-dependent integrations. We advise maintaining *el* < *enl* < *es* for good convergence. Other aspects of Elmer (e.g. effects of mesh density on solution accuracy and computational efficiency, and parallel simulations) are given by Gagliardini & Zwinger (2008).

For each experiment considered in this study, we generate the structured mesh by using ElmerGrid. ElmerGrid is basically a 2D mesh generator, but is also capable of extruding and manipulating the mesh in the third dimension. Since 3D experiments require a large amount of memory and computation time, we perform parallel runs in a high-performance computing cluster provided by the Western Canadian Research Grid (WestGrid).

#### **5. Numerical comparison of physical approximations**

We consider a 10 <sup>×</sup> 2.5 km<sup>2</sup> glacial valley. To mimic a typical real-world glacier scenario, we include meanders and bumps in the subglacial topography, *b* (*x*, *y*), as defined below,

$$b\left(\mathbf{x}, y\right) = 5000 - \mathbf{x} \tan a\_{\theta} + a\_{\mathrm{x}} \sin \left(0.5\pi + \frac{4\pi\mathbf{x}}{L}\right) - a\_{\mathrm{y}} \sin \left[\left(y + \theta\right) \frac{\pi}{W}\right],\tag{19}$$

where *α<sup>b</sup>* is the mean bedrock slope in radians, *L* and *W* are the longitudinal and lateral extents of the valley, *ax* and *ay* are the amplitudes of the topographical variation in *x* and *y* directions, and *θ* is the sinusoidal offset of the flowline. Here, we use *α<sup>b</sup>* = 12◦, *L* = 10 km, *W* = 2.5 km, *ax* = 200 m, *ay* = (500 + 0.05*x*) m, and *θ* = 500 sin (2*πx*/*L*) m. The plan view of basal topography is shown in Figure 3a; the central flowline is also depicted.

Next, we define the climatic regime. For *z* ≤ 4.6 km and over a 500-m wide corridor around the central flowline (see Fig. 3a), the mass balance function, *m* (*s*, *t*), is chosen as,

$$m(\mathbf{s}, t) = \beta \left[ \mathbf{s}(\mathbf{x}, \mathbf{y}) - E \right] + \Delta m(t), \tag{20}$$

where *β* is the linear mass balance gradient, *E* is the equilibrium line altitude (ELA), and Δ*m*(*t*) is the time-dependent mass balance perturbation; *m* (*s*, *t*) = 0 elsewhere. For now, we choose *β* = 0.01 m ice eq. a−<sup>1</sup> m−1, *E* = 3.7 km, and Δ*m*(*t*) = 0 m ice eq. a−1. Since *s*(*x*, *y*) evolves in prognostic simulations (Eq. 9), the parameterization of *m*(*s*, *t*) (Eq. 20) ensures that our models capture the height/mass-balance feedback inclusively.

Under these geometric and climatic settings, we grow glaciers to steady state using several models. In order to illustrate the importance of each physical mechanism of ice flow, we consider three different models (one from each model family), namely the FS, PS3 and SIA3 models. We denote them respectively by FS, PS and SD, unless otherwise specified. Each model domain consists of 50 k bilinear quadrilateral elements, with average horizontal dimensions 50 <sup>×</sup> 50 m2. The vertical dimension of element varies according to the ice thickness; we use five vertical layers. Below, we carry out numerical comparison of these 3D models in terms of steady state geometry, surface velocity, basal shear stress, and response timescales. Considering pragmatic flowline models (PS2 and SIA2), we also present a brief tutorial on modelling valley glacier dynamics, which involves (1) sensitivity tests for a glacier, (2) reconstruction of past climate or glacier extent, and (3) projection of a glacier's future.

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>0</sup>

SD model PS model FS model

> SD model PS model FS model

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>0</sup>

SD model PS model FS model

Shear stress along the flowline

Horizontal distance, x (km)

Length response to climate

SD model PS model FS model

<sup>0</sup> <sup>40</sup> <sup>80</sup> <sup>120</sup> <sup>160</sup> −1

Time after imposing Δm, t (years)

70

−0.5

0

Length change, ΔL (km)

Fig. 4. (a) Surface velocity and (b) basal shear stress along the central flowline. Evolutions of (c) ice volume and (d) glacier length in response to step changes in climate, imposed on the

and FS models, respectively. This indicates the significance of resistances associated with the

The steady state horizontal velocity at the upper ice surface, *ux*(*x*, *y*,*s*), along the central flowline is plotted in Figure 4a. In each model case, we find smaller velocities around the basal depression at *x* ≈ 3 km; while ice flows faster at places with steeper basal slopes (see Fig. 3d). Due to resistive effects of high-order dynamics, the FS and PS models yield relatively smaller ice velocities. The mean surface velocity along the central flowline in the SD model is

We also calculate the deviatoric stresses from the englacial velocity field, using Equations (1–4). The vertical shear stress at the ice/bedrock interface, *τxz*(*x*, *y*, *b*), in a steady state longitudinal profile along the central flowline is shown in Figure 4b. As expected, basal shear stress is smaller in the PS and FS models, where other stress components are also active (i.e. *τxz* is not the only non-zero component) to control the glacier ice flow. In the SD model, *τxz*(*x*, *y*, *b*) characterizes *τ<sup>b</sup>* (e.g. Adhikari & Marshall, 2011), which is the sole resistance to the gravitational driving stress, *τd*; it follows that *τxz*(*x*, *y*, *b*) ≈ *τd*. Therefore, we calculate the errors as the difference between the SD and FS/PS models with respect to the former one. This gives a rough idea about the fractional contributions of high-order resistances, i.e. other than *τb*, to balance the gravity-driven ice flow; we find *τlon* ≈ *eSD*.*PS* = 6.1% and

0.5

1

140

Basal stress, τxz (kPa)

Modelling Dynamics of Valley Glaciers 127

210

280

Surface velocity along the flowline

a b

c d

Horizontal distance, x (km)

Volume response to climate

<sup>0</sup> <sup>40</sup> <sup>80</sup> <sup>120</sup> <sup>160</sup> −120

Time after imposing Δm, t (years)

steady-state glaciers whose geometries are shown in Fig. 3.

high-order dynamics; *τlat* once again appears to be more crucial than *τlon*.

higher by 5.9% and 23.6% than in the PS and FS models, respectively (Table 1).

35

−60

0

Volume change, ΔV (106 m3

)

60

120

70

Surface velocity, ux (m a−1)

105

140

Fig. 3. (a) Basal topography, showing the central flowline. The mass balance function (Eq. 20) is applied only over a passage enclosed by dotted lines. (b) Evolution of ice volume. (c) Steady state ice thickness obtained from the FS model. (d) Longitudinal profiles of steady state geometry along the central flowline; the corresponding ELA is shown with a dotted line.

#### **5.1 Geometry and field variables**

The evolution of a glacier from zero ice volume to steady state is shown for each model case (Fig. 3b). By accounting for the high-order physical mechanisms, FS and PS models hold more ice mass than does the SD model. To assess the importance of high-order dynamics, i.e. the role of *τlat* and/or *τlon*, we compute errors between the models. We denote the error, for example, by *ePS*.*FS* to explain a difference between the PS and FS models with respect to the latter one. The errors *ePS*.*FS* and *eSD*.*PS* therefore illustrate the sole role of *τlat* and *τlon*, respectively; while *eSD*.*FS* explains their collective effects. The steady state ice volume obtained from each model and the associated errors are listed in Table 1. For the chosen geometric setting, the role of *τlat* (*ePS*.*FS* = −10.6%) is relatively more pronounced than that of *τlon* (*eSD*.*PS* = −7.6%).

The plan view of the steady state ice thickness obtained from the FS model is shown in Figure 3c. The maximum ice thickness is observed along the central flowline, and specifically around the basal depression at *x* ≈ 3 km. This is true for each model case, as shown in Figure 3d. Although the longitudinal profiles of surface elevation appear to superimpose on each other, there is a considerable difference in both the mean (Table 1) and maximum values of ice thickness. The SD model generates a glacier that is 6.4% and 15.9% thinner than the PS 12 Will-be-set-by-IN-TECH

3

4

Elevation, z (km)

Fig. 3. (a) Basal topography, showing the central flowline. The mass balance function (Eq. 20) is applied only over a passage enclosed by dotted lines. (b) Evolution of ice volume. (c) Steady state ice thickness obtained from the FS model. (d) Longitudinal profiles of steady state geometry along the central flowline; the corresponding ELA is shown with a dotted line.

The evolution of a glacier from zero ice volume to steady state is shown for each model case (Fig. 3b). By accounting for the high-order physical mechanisms, FS and PS models hold more ice mass than does the SD model. To assess the importance of high-order dynamics, i.e. the role of *τlat* and/or *τlon*, we compute errors between the models. We denote the error, for example, by *ePS*.*FS* to explain a difference between the PS and FS models with respect to the latter one. The errors *ePS*.*FS* and *eSD*.*PS* therefore illustrate the sole role of *τlat* and *τlon*, respectively; while *eSD*.*FS* explains their collective effects. The steady state ice volume obtained from each model and the associated errors are listed in Table 1. For the chosen geometric setting, the role of *τlat* (*ePS*.*FS* = −10.6%) is relatively more pronounced than that

The plan view of the steady state ice thickness obtained from the FS model is shown in Figure 3c. The maximum ice thickness is observed along the central flowline, and specifically around the basal depression at *x* ≈ 3 km. This is true for each model case, as shown in Figure 3d. Although the longitudinal profiles of surface elevation appear to superimpose on each other, there is a considerable difference in both the mean (Table 1) and maximum values of ice thickness. The SD model generates a glacier that is 6.4% and 15.9% thinner than the PS

5

d

200

400

Ice volume, V (106 m3

)

600

800

b

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>2</sup>

SD model PS model FS model

Central flowline geometry

<sup>0</sup> <sup>150</sup> <sup>300</sup> <sup>450</sup> <sup>600</sup> <sup>0</sup>

SD model PS model FS model

Evolution of glacier

Time, t (years)

Horizontal distance, x (km)

Horizontal distance, x (km)

2080 2860 3640 4420 5200

Ice thickness, h(x,y) (m)

Horizontal distance, x (km)

0 65 130 195 260

**5.1 Geometry and field variables**

of *τlon* (*eSD*.*PS* = −7.6%).

<sup>0</sup> 2.5 <sup>5</sup> 7.5 <sup>10</sup> <sup>0</sup>

<sup>0</sup> 2.5 <sup>5</sup> 7.5 <sup>10</sup> <sup>0</sup>

Basal topography, b(x,y) (m)

Lateral distance, y (km)

Lateral distance, y (km)

0.5 1 1.5 2 2.5

0.5 1 1.5 2 2.5

Fig. 4. (a) Surface velocity and (b) basal shear stress along the central flowline. Evolutions of (c) ice volume and (d) glacier length in response to step changes in climate, imposed on the steady-state glaciers whose geometries are shown in Fig. 3.

and FS models, respectively. This indicates the significance of resistances associated with the high-order dynamics; *τlat* once again appears to be more crucial than *τlon*.

The steady state horizontal velocity at the upper ice surface, *ux*(*x*, *y*,*s*), along the central flowline is plotted in Figure 4a. In each model case, we find smaller velocities around the basal depression at *x* ≈ 3 km; while ice flows faster at places with steeper basal slopes (see Fig. 3d). Due to resistive effects of high-order dynamics, the FS and PS models yield relatively smaller ice velocities. The mean surface velocity along the central flowline in the SD model is higher by 5.9% and 23.6% than in the PS and FS models, respectively (Table 1).

We also calculate the deviatoric stresses from the englacial velocity field, using Equations (1–4). The vertical shear stress at the ice/bedrock interface, *τxz*(*x*, *y*, *b*), in a steady state longitudinal profile along the central flowline is shown in Figure 4b. As expected, basal shear stress is smaller in the PS and FS models, where other stress components are also active (i.e. *τxz* is not the only non-zero component) to control the glacier ice flow. In the SD model, *τxz*(*x*, *y*, *b*) characterizes *τ<sup>b</sup>* (e.g. Adhikari & Marshall, 2011), which is the sole resistance to the gravitational driving stress, *τd*; it follows that *τxz*(*x*, *y*, *b*) ≈ *τd*. Therefore, we calculate the errors as the difference between the SD and FS/PS models with respect to the former one. This gives a rough idea about the fractional contributions of high-order resistances, i.e. other than *τb*, to balance the gravity-driven ice flow; we find *τlon* ≈ *eSD*.*PS* = 6.1% and

−300 −200 −100 <sup>0</sup> <sup>100</sup> −2

SIA2 model PS2 model FS model

−200 −150 −100 −50 <sup>0</sup> <sup>7</sup>

Time, t (year)

Projection of glacier length

<sup>0</sup> <sup>25</sup> <sup>50</sup> <sup>75</sup> <sup>100</sup> 5.5

Time, t (year)

SIA2 model PS2 model FS model

SIA2 model PS2 model FS model

(Reconstruction of) glacier length

7.6

6.5

7.5

Glacier length, L (km)

Fig. 5. Reconstructions of the past (a) climate and (b) glacier extent. A pre-defined Δ*m*(*t*) is imposed on the FS model to induce the evolution of glacier length. Using each of the PS2 and SIA2 models, the past Δ*m*(*t*) is then reconstructed so that the FS glacier extent is properly simulated. Future projections of (c) flowline ice volume and (d) glacier extent. Negative and

the model simulates such variations in the glacier length properly. The reconstructed glacier lengths and associated past climate are shown in Figure 5b and 5a, respectively. Due to the lack of high-order resistances, the PS2 and SIA2 models require a larger Δ*m*(*t*) in order to generate the (flowline) mass flux large enough to maintain the FS glacier lengths. With the corresponding climate history, the PS2 and SIA2 models are now ready to project the glacier's

With respect to the present climate, we consider two distinct scenarios for the future. The present climate is obtained by averaging the data of past 30 years. In addition to this, the future scenarios are defined as the linear changes in climate with <sup>Δ</sup>*m*(*t*) = <sup>±</sup>1.5 m ice eq. a−<sup>1</sup> by the end of *t* = 100 years (Fig. 5a). Under such climatic scenarios, we simulate each model to project the future dynamics of glacier. The evolutions of ice volume along the central flowline in a longitudinal band of unit width and glacier length are shown in Figures 5c and 5d, respectively. It is clear that the present time (*t* = 0 years) glacier lengths are similar in each model case; the corresponding ice volumes differ considerably, however. Short of tuning the ice dynamics (i.e. altering the ice viscosity or basal sliding rates), it is not possible to

8.5

8.2

Glacier length, L (km)

8.8

Modelling Dynamics of Valley Glaciers 129

(Reconstruction of) past climate

a b

c d

Time, t (year)

Projection of glacier volume

<sup>0</sup> <sup>25</sup> <sup>50</sup> <sup>75</sup> <sup>100</sup> <sup>400</sup>

Time, t (year)

SIA2 model PS2 model FS model

positive times respectively indicate the past and future.

0

600

future.

Ice volume, V (103 m3

)

800

1000

2

Δm (m ice eq. a−1)

4

(*τlon* + *τlat*) ≈ *eSD*.*FS* = 17.6% (Table 1). These figures represent lower-limit estimates, as *τ<sup>d</sup>* = *ρghα<sup>s</sup>* (e.g. Van der Veen, 1999) should be larger for the high-order models, which hold thicker ice masses.

#### **5.2 Response of the glacier to climate change**

Before simulating the past and future dynamics of a glacier, it is useful to conduct a simple sensitivity test by imposing a step change in climate, i.e. mass balance, on the steady state geometry. This yields the characteristic timescales of a glacier, specifically the response times, which explain the length of time over which the glacier carries in its memory the mass balance history. On the corresponding steady state geometry of each model (Section 5.1), we impose <sup>Δ</sup>*m*(*t*) = <sup>±</sup>1 m ice eq. a−<sup>1</sup> in turn and we let the glacier respond until it attains a new steady state. The volume and length response of a glacier in each model case are plotted respectively in Figure 4c and 4d. Based on the e-folding concept (e.g. Jóhannesson et al., 1989), we calculate both the volume, *tv*, and length response time, *tl*, as the time required for a glacier to adjust <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−<sup>1</sup> ≈ 63% of total change in volume, Δ*V*, and length, Δ*L*, respectively.

Response times, *tv* and *tl*, are listed in Table 1; values are given for 2D flowline models as well. Based on these values, we note a few important points. First, a glacier takes less time, by ca. 17 (3D) and ca. 8 years (2D), to adjust its ice volume vs. its length. The relatively shorter *tv* is primarily due to the instantaneous response of ice thickness to the climatic perturbation. Secondly, all models with a given spatial dimension yield nearly the same response times, i.e. *tv* ≈ 25 and *tl* ≈ 42 years (3D), and *tv* ≈ 12 and *tl* ≈ 21 years (2D). Leysinger Vieli & Gudmundsson (2004) find the same for 2D models, and they suggest that simpler models are sufficient for the purpose of estimating response times. This however does not imply that 2D models yield representative timescales for 3D cases. For the chosen geometry, 3D models appear to take twice as long to respond as the flowline models. This is mainly because the flowline models only capture the maximum velocity, along the central flowline, and hence adjust its geometry more quickly; whereas 3D models have an integrated ice flux across the glacier, which gives a slower average velocity, and consequently take longer time to adjust its geometry. Therefore, the flowline models, which lack the proper account of effects of varying glacier width, do not yield realistic estimates of response times for valley glaciers.

#### **5.3 Projecting the glacier's future using flowline models**

One of the key reasons for using dynamical ice flow models is to simulate the future states of a glacier under several possible climatic scenarios. To be able to obtain realistic predictions of ice volume, ice flow models should be constrained properly, ensuring that they truly represent the dynamics of the glacier at hand. Depending upon the availability of field data, this is usually accomplished through simulations of the ice surface velocities and/or historical front variations (e.g. Adhikari & Huybrechts, 2009). As we have considered synthetic glaciers, we take advantage to assume that the FS dynamics represents a real-world scenario and we constrain the reduced models accordingly. The majority of valley glacier simulations are based on flowline dynamics (e.g. Oerlemans et al., 1998); we choose PS2 and SIA2 models to reconstruct the past and project the future of a glacier.

By mimicking a real-world climatic history with inter-decadal variability, we impose a pre-defined Δ*m*(*t*) upon the FS model (Fig. 5a), and record the corresponding changes in the terminus position (Fig. 5b). For each of the PS2 and SIA2 models, we tune Δ*m*(*t*) so that 14 Will-be-set-by-IN-TECH

(*τlon* + *τlat*) ≈ *eSD*.*FS* = 17.6% (Table 1). These figures represent lower-limit estimates, as *τ<sup>d</sup>* = *ρghα<sup>s</sup>* (e.g. Van der Veen, 1999) should be larger for the high-order models, which hold

Before simulating the past and future dynamics of a glacier, it is useful to conduct a simple sensitivity test by imposing a step change in climate, i.e. mass balance, on the steady state geometry. This yields the characteristic timescales of a glacier, specifically the response times, which explain the length of time over which the glacier carries in its memory the mass balance history. On the corresponding steady state geometry of each model (Section 5.1), we impose <sup>Δ</sup>*m*(*t*) = <sup>±</sup>1 m ice eq. a−<sup>1</sup> in turn and we let the glacier respond until it attains a new steady state. The volume and length response of a glacier in each model case are plotted respectively in Figure 4c and 4d. Based on the e-folding concept (e.g. Jóhannesson et al., 1989), we calculate both the volume, *tv*, and length response time, *tl*, as the time required for a glacier to adjust

≈ 63% of total change in volume, Δ*V*, and length, Δ*L*, respectively.

glacier width, do not yield realistic estimates of response times for valley glaciers.

One of the key reasons for using dynamical ice flow models is to simulate the future states of a glacier under several possible climatic scenarios. To be able to obtain realistic predictions of ice volume, ice flow models should be constrained properly, ensuring that they truly represent the dynamics of the glacier at hand. Depending upon the availability of field data, this is usually accomplished through simulations of the ice surface velocities and/or historical front variations (e.g. Adhikari & Huybrechts, 2009). As we have considered synthetic glaciers, we take advantage to assume that the FS dynamics represents a real-world scenario and we constrain the reduced models accordingly. The majority of valley glacier simulations are based on flowline dynamics (e.g. Oerlemans et al., 1998); we choose PS2 and SIA2 models

By mimicking a real-world climatic history with inter-decadal variability, we impose a pre-defined Δ*m*(*t*) upon the FS model (Fig. 5a), and record the corresponding changes in the terminus position (Fig. 5b). For each of the PS2 and SIA2 models, we tune Δ*m*(*t*) so that

**5.3 Projecting the glacier's future using flowline models**

to reconstruct the past and project the future of a glacier.

Response times, *tv* and *tl*, are listed in Table 1; values are given for 2D flowline models as well. Based on these values, we note a few important points. First, a glacier takes less time, by ca. 17 (3D) and ca. 8 years (2D), to adjust its ice volume vs. its length. The relatively shorter *tv* is primarily due to the instantaneous response of ice thickness to the climatic perturbation. Secondly, all models with a given spatial dimension yield nearly the same response times, i.e. *tv* ≈ 25 and *tl* ≈ 42 years (3D), and *tv* ≈ 12 and *tl* ≈ 21 years (2D). Leysinger Vieli & Gudmundsson (2004) find the same for 2D models, and they suggest that simpler models are sufficient for the purpose of estimating response times. This however does not imply that 2D models yield representative timescales for 3D cases. For the chosen geometry, 3D models appear to take twice as long to respond as the flowline models. This is mainly because the flowline models only capture the maximum velocity, along the central flowline, and hence adjust its geometry more quickly; whereas 3D models have an integrated ice flux across the glacier, which gives a slower average velocity, and consequently take longer time to adjust its geometry. Therefore, the flowline models, which lack the proper account of effects of varying

thicker ice masses.

<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−<sup>1</sup>

**5.2 Response of the glacier to climate change**

Fig. 5. Reconstructions of the past (a) climate and (b) glacier extent. A pre-defined Δ*m*(*t*) is imposed on the FS model to induce the evolution of glacier length. Using each of the PS2 and SIA2 models, the past Δ*m*(*t*) is then reconstructed so that the FS glacier extent is properly simulated. Future projections of (c) flowline ice volume and (d) glacier extent. Negative and positive times respectively indicate the past and future.

the model simulates such variations in the glacier length properly. The reconstructed glacier lengths and associated past climate are shown in Figure 5b and 5a, respectively. Due to the lack of high-order resistances, the PS2 and SIA2 models require a larger Δ*m*(*t*) in order to generate the (flowline) mass flux large enough to maintain the FS glacier lengths. With the corresponding climate history, the PS2 and SIA2 models are now ready to project the glacier's future.

With respect to the present climate, we consider two distinct scenarios for the future. The present climate is obtained by averaging the data of past 30 years. In addition to this, the future scenarios are defined as the linear changes in climate with <sup>Δ</sup>*m*(*t*) = <sup>±</sup>1.5 m ice eq. a−<sup>1</sup> by the end of *t* = 100 years (Fig. 5a). Under such climatic scenarios, we simulate each model to project the future dynamics of glacier. The evolutions of ice volume along the central flowline in a longitudinal band of unit width and glacier length are shown in Figures 5c and 5d, respectively. It is clear that the present time (*t* = 0 years) glacier lengths are similar in each model case; the corresponding ice volumes differ considerably, however. Short of tuning the ice dynamics (i.e. altering the ice viscosity or basal sliding rates), it is not possible to

**FS model PS model SD model** *ePS*.*FS eSD*.*FS eSD*.*PS* Steady state geometry and field variables

Response timescales

Future projections (after *t* = 100 years)

**Volume,** *<sup>V</sup>* (106 <sup>m</sup>3) 679.5 607.5 561.3 <sup>−</sup>10.6 <sup>−</sup>17.4 <sup>−</sup>7.6 **Thickness,** *h* (m) 133.0 119.4 111.8 −10.2 −15.9 −6.4 **Velocity,** *ux* (m a−1) 69.2 80.8 85.5 16.7 23.6 5.9 **Stress,** *τxz* (kPa) 154.6 176.3 187.7 - 17.7 6.1

Modelling Dynamics of Valley Glaciers 131

**Timescale,** *tv* (a) 26.0 25.0 23.5 −3.8 −9.6 −6.0 \$ - 12.5 12.0 −51.9 −53.8 - **Timescale,** *tl* (a) 42.0 42.0 41.0 0.0 −2.4 −2.4 \$ - 21.5 20.5 −48.8 −51.2 -

**Volume,** *<sup>V</sup>* (103 <sup>m</sup>3)\* 871.8 821.1 762.9 <sup>−</sup>5.8 <sup>−</sup>12.5 <sup>−</sup>7.1

**Length,** *L* (km)\* 7.9 8.0 8.0 1.8 1.8 0.0

There are more than 200 k small glaciers and ice caps on Earth; it is not feasible to use the numerically intensive FS model to simulate every valley glacier. Therefore, we mostly encounter simple flowline models, e.g. the SIA2 and PS2 models, being used in the valley glacier applications. Without adding the numerical complexity, the dynamical reach of such models can be extended through the introduction of parameterized correction factors. The effects of longitudinal stress gradients can be accounted for by embedding *L*-factors (Adhikari & Marshall, 2011); the lateral drag associated with the valley walls (Nye, 1965) and stick/slip basal interface (Adhikari & Marshall, in preperation) can also be captured via analogous correction factors. This offers a pragmatic middle ground for simulating glacier response to

Undoubtedly, the biggest challenge in glacier modelling is the lack of sufficient field data. Geometric and climatic data (e.g. basal and surface topographies, glacier length records, and mass balance fields), as well as observations of ice velocities, are not available in most cases. They are essential to justify the cost of using complex, 3D, FS models. Furthermore, the lack of proper theories and associated data to describe basal processes, e.g. basal sliding, is also a subject of concern that we have not discussed in the text; in many cases, poor characterization

We acknowledge support from the Western Canadian Cryospheric Network (WC2N), funded by the Canadian Foundation for Climate and Atmospheric Sciences (CFCAS), and the Natural

of basal flow is the limiting factor in modelling glacier dynamics.

Sciences and Engineering Research Council (NSERC) of Canada.

Table 1. Numerical comparison of several models, one from each model family. Errors between the models, e.g. *ePS*.*FS*, are given in percentage. Response timescales are also calculated for flowline models (\$). The future projection results represent for the central flowline; values are listed for both the positive (\*) and negative (#) mass balance (Fig. 5a)

climate change.

**7. Acknowledgements**
