Computational Particle Hydrodynamics

Chapter 1

Abstract

the coupled simulation.

mesh interpolation

1. Introduction

3

probability between two energy states.

A Coupling Algorithm of

Dynamics (CFPD)

Albert S. Kim and Hyeon-Ju Kim

Computational Fluid and Particle

Computational fluid dynamics (CFD) and particle hydrodynamics (PHD) have been developed almost independently. CFD is classified into Eulerian and Lagrangian. The Eulerian approach observes fluid motion at specific locations in the space, and the Lagrangian approach looks at fluid motion where the observer follows an individual fluid parcel moving through space and time. In classical mechanics, particle dynamic simulations include molecular dynamics, Brownian dynamics, dissipated particle dynamics, Stokesian dynamics, and granular dynamics (often called discrete element method). Dissipative hydrodynamic method unifies these dynamic simulation algorithms and provides a general view of how to mimic particle motion in gas and liquid. Studies on an accurate and rigorous coupling of CFD and PHD are in literature still in a growing stage. This chapter shortly reviews the past development of CFD and PHD and proposes a general algorithm to couple the two dynamic simulations without losing theoretical rigor and numerical accuracy of

Keywords: computational fluid dynamics (CFD), computational fluid and particle

The first simulations of a liquid were conducted at the Los Alamos National Laboratory in the early 1950s, using the Los Alamos computer, Mathematical Analyzer, Numerical Integrator, and Computer (MANIAC). This computer was developed under the direction of Nicholas Metropolis, who is the pioneer of the modern (Metropolis) Monte Carlo simulation [1, 2]. The first MC simulation was conducted using the Lennard-Jones potential to investigate the material properties of liquid argon [3]. In the MC simulations, the phase space was searched to find more probable thermodynamic states and calculate macroscopic material properties (i.e., experimentally observable) using averages of the same physical quantity over the micro-thermodynamic states. In principle, the MC method assumes that the system of interest is in a static equilibrium state, and therefore, the time evolution is replaced by the phase-space averaging. The Boltzmann factor is used as a transition

dynamics (CFPD), dissipative hydrodynamics, tetrahedron mesh,

#### Chapter 1

## A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD)

Albert S. Kim and Hyeon-Ju Kim

#### Abstract

Computational fluid dynamics (CFD) and particle hydrodynamics (PHD) have been developed almost independently. CFD is classified into Eulerian and Lagrangian. The Eulerian approach observes fluid motion at specific locations in the space, and the Lagrangian approach looks at fluid motion where the observer follows an individual fluid parcel moving through space and time. In classical mechanics, particle dynamic simulations include molecular dynamics, Brownian dynamics, dissipated particle dynamics, Stokesian dynamics, and granular dynamics (often called discrete element method). Dissipative hydrodynamic method unifies these dynamic simulation algorithms and provides a general view of how to mimic particle motion in gas and liquid. Studies on an accurate and rigorous coupling of CFD and PHD are in literature still in a growing stage. This chapter shortly reviews the past development of CFD and PHD and proposes a general algorithm to couple the two dynamic simulations without losing theoretical rigor and numerical accuracy of the coupled simulation.

Keywords: computational fluid dynamics (CFD), computational fluid and particle dynamics (CFPD), dissipative hydrodynamics, tetrahedron mesh, mesh interpolation

#### 1. Introduction

The first simulations of a liquid were conducted at the Los Alamos National Laboratory in the early 1950s, using the Los Alamos computer, Mathematical Analyzer, Numerical Integrator, and Computer (MANIAC). This computer was developed under the direction of Nicholas Metropolis, who is the pioneer of the modern (Metropolis) Monte Carlo simulation [1, 2]. The first MC simulation was conducted using the Lennard-Jones potential to investigate the material properties of liquid argon [3]. In the MC simulations, the phase space was searched to find more probable thermodynamic states and calculate macroscopic material properties (i.e., experimentally observable) using averages of the same physical quantity over the micro-thermodynamic states. In principle, the MC method assumes that the system of interest is in a static equilibrium state, and therefore, the time evolution is replaced by the phase-space averaging. The Boltzmann factor is used as a transition probability between two energy states.

The first CFD paper was published by Hess and Smith [4]. Their method is known as panel method as the surface was discretized with many panels. More accurate CFD work of advanced panel method can be found elsewhere [5, 6]. In general, CFD research is categorized by Eulerian and Lagrangian approaches, which are grid-dependent and meshfree, respectively. In the Eulerian category, the CFD performance regarding numerical accuracy and computational speed depends on how to discretize the computational domain. Popular methods include finite volume method (FVM) [7], finite element method (FEM), finite difference method (FDM), spectral element method (SEM), boundary element (BEM), and highresolution discretization schemes.

formalism of near-field hydrodynamic contribution to the global hydrodynamic resistance (i.e., mobility inverted) often overestimates the real hydrodynamic repulsion forces. One of the primary reasons is that the lubrication theory assumes that particle surfaces are perfectly smooth, while in reality particle surfaces are rough and therefore surface friction also plays an essential role during collision events. Stokesian dynamics includes the most accurate estimation of the hydrodynamic tensors using the grand mobility matrix and pair-wise lubrication interactions. The fluctuation-dissipation theorem is better satisfied in SD by using manybody hydrodynamic interactions. In SD simulations, the fluid flow is given at any point in the computational domain as a combination of the unidirectional velocity, vortex velocity, and the rate of strain that is traceless and symmetric. The feedback interaction between a particle and fluid is already included in the expansion of Faxen rule. Due to the logarithmic characteristics of the lubrication forces, if two particles touch each other, they experience infinite repulsive forces. This characteristic of the lubrication force indicates the hard-sphere behavior of the colliding particles conceptually, implicitly assuming a perfectly elastic interparticle collision

A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD)

DOI: http://dx.doi.org/10.5772/intechopen.86895

Dissipative hydrodynamics (DHD) overcomes significant limitations of the par-

ticle dynamic method discussed above. DHD is a generalized method of which special cases converge to MD, BD, DPD, and SD by turning on or off specific force mechanisms. Details of DHD can be found elsewhere [14–16]. CFD and PHD were developed and applied without strong mutual influences. Particle tracking method can be viewed as a reasonable way to investigate the hydrodynamic motion of particles under the influence of ambient fluid flow. But, it has several fundamental limitations by neglecting particle density, particle shapes and sizes, and particlefluid interactions. More importantly, the basic two-body interactions due to collisions, viscous flow, and electromagnetic properties are not included, and dispersion forces were dropped. In this light, the particle tracking method does not track particles but fluid elements moving along streamlines. While theories of particle hydrodynamics are not rigorously applied to engineering processes, this chapter includes a possible method to couple CFD and DHD in a seamless, robust way.

2. Dissipative hydrodynamics as unified particle dynamics

discussed.

5

2.1 Overview

simulation algorithms [16, 17].

In this section, a unified view of preexisting particle dynamic method is

In the deterministic simulations, particle dynamics can be classified based on sizes of objects of interests. The purposes of particle dynamics are to provide the exact solution of a complex problem, bridging the theory and experiments. Note that the fundamental principles often provide governing equations, which were proven to be valid. It is difficult to solve a governing equation of a problem, if it has complex geometry and coupled boundary conditions. Experimental observations show natural processes using quantified information. Dissipative hydrodynamics is a generalized algorithm that unifies most of the preexisting particle dynamic

At microscale of an order of nanometers, molecular dynamics deals with the motion of many molecules in various phases such as gas, liquid, and solid. Suppose

i.e., ε ¼ 1 where ε is the coefficient of restitution.

The deterministic molecular dynamic (MD) method treats a particle as a point mass, which has a finite mass and other physical properties but does not have any volume and shape. Given external and interparticle forces, positions and velocities in linear motion are predicted using Newton's second law for N particles. When the motion of solute molecules in the solvent fluid is of concern, then the deterministic forces form solvent to solute molecules can be mathematically replaced by random fluctuating forces. These random forces have a zero mean over time, and its magnitude is determined by the dissipation fluctuation theorem [8].

Langevin's equation includes the hydrodynamic drag balanced by the random forces due to thermal fluctuation [9]. Solving N body Langevin equation is called Brownian dynamics (BD) [10]. Although BD can include effects of particle sizes in the dynamic simulations, it is fundamentally limited to the low concentration of solutes due to the Oseen tensor (see Appendix for details). Brownian dynamics (BD) was initially developed to reduce computational load by replacing deterministic interactions between a solute molecule and many solvent molecules by randomly fluctuating, probabilistic interaction as a net driving force. BD presumes a dilute solution, which means that a mean distance between solutes is much longer than the size of the molecule. A large number of solvent molecules exist around solutes, which is enough to exert random forces due to a tremendous number of collisions. When BD is applied to the dynamic motion of multiple particles, the lack of hydrodynamic interactions may provide erroneous results because the fluctuation-dissipation principle is not quantitatively well balanced. This limitation of BD to a single particle or a dilute solution is at the equivalent level of Stokes' drag coefficient, used to calculate the particle diffusivity.

Dissipative particle dynamics (DPD) is an updated version of BD, which specifically includes the interparticle hydrodynamic forces [11, 12]. Two functions, often denoted as wR and wD, are proposed to quantify the presumed relationship of pair-wise hydrodynamic forces/torques, determined by the dissipation fluctuation theorem. The proposed forms of the hydrodynamic interaction are vector wise as the real viscous forces are tensor-wise. Dissipative particle dynamics (DPD) updates BD by including an approximate form of tensor-wise hydrodynamic interactions as a pair-wise vector form. A force exerted on a particle has three types such as conservative, dissipative, and random forces, and DPD satisfies the fluctuation-dissipation theorem by balancing the dissipative and random forces.

Stokesian dynamics (SD) uses the grand mobility matrix to include the far-field hydrodynamic forces/torques [13]. In-depth comparative analyses of present particle dynamic methods can be found elsewhere [14]. This mobility matrix is inverted and updated by adding differences between near-field lubrication forces and farfield two-body interactions. As the lubrication force is proportional to the logarithm of the surface-to-surface distance between two particles, it diverges during events of particle collisions. Besides, if two particles are close to each other, i.e., the surface-to-surface distance is much smaller than the particle diameter, the SD

#### A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD) DOI: http://dx.doi.org/10.5772/intechopen.86895

formalism of near-field hydrodynamic contribution to the global hydrodynamic resistance (i.e., mobility inverted) often overestimates the real hydrodynamic repulsion forces. One of the primary reasons is that the lubrication theory assumes that particle surfaces are perfectly smooth, while in reality particle surfaces are rough and therefore surface friction also plays an essential role during collision events. Stokesian dynamics includes the most accurate estimation of the hydrodynamic tensors using the grand mobility matrix and pair-wise lubrication interactions. The fluctuation-dissipation theorem is better satisfied in SD by using manybody hydrodynamic interactions. In SD simulations, the fluid flow is given at any point in the computational domain as a combination of the unidirectional velocity, vortex velocity, and the rate of strain that is traceless and symmetric. The feedback interaction between a particle and fluid is already included in the expansion of Faxen rule. Due to the logarithmic characteristics of the lubrication forces, if two particles touch each other, they experience infinite repulsive forces. This characteristic of the lubrication force indicates the hard-sphere behavior of the colliding particles conceptually, implicitly assuming a perfectly elastic interparticle collision i.e., ε ¼ 1 where ε is the coefficient of restitution.

Dissipative hydrodynamics (DHD) overcomes significant limitations of the particle dynamic method discussed above. DHD is a generalized method of which special cases converge to MD, BD, DPD, and SD by turning on or off specific force mechanisms. Details of DHD can be found elsewhere [14–16]. CFD and PHD were developed and applied without strong mutual influences. Particle tracking method can be viewed as a reasonable way to investigate the hydrodynamic motion of particles under the influence of ambient fluid flow. But, it has several fundamental limitations by neglecting particle density, particle shapes and sizes, and particlefluid interactions. More importantly, the basic two-body interactions due to collisions, viscous flow, and electromagnetic properties are not included, and dispersion forces were dropped. In this light, the particle tracking method does not track particles but fluid elements moving along streamlines. While theories of particle hydrodynamics are not rigorously applied to engineering processes, this chapter includes a possible method to couple CFD and DHD in a seamless, robust way.

#### 2. Dissipative hydrodynamics as unified particle dynamics

In this section, a unified view of preexisting particle dynamic method is discussed.

#### 2.1 Overview

The first CFD paper was published by Hess and Smith [4]. Their method is known as panel method as the surface was discretized with many panels. More accurate CFD work of advanced panel method can be found elsewhere [5, 6]. In general, CFD research is categorized by Eulerian and Lagrangian approaches, which are grid-dependent and meshfree, respectively. In the Eulerian category, the CFD performance regarding numerical accuracy and computational speed depends on how to discretize the computational domain. Popular methods include finite volume method (FVM) [7], finite element method (FEM), finite difference method (FDM), spectral element method (SEM), boundary element (BEM), and high-

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

The deterministic molecular dynamic (MD) method treats a particle as a point mass, which has a finite mass and other physical properties but does not have any volume and shape. Given external and interparticle forces, positions and velocities in linear motion are predicted using Newton's second law for N particles. When the motion of solute molecules in the solvent fluid is of concern, then the deterministic forces form solvent to solute molecules can be mathematically replaced by random fluctuating forces. These random forces have a zero mean over time, and its mag-

Langevin's equation includes the hydrodynamic drag balanced by the random forces due to thermal fluctuation [9]. Solving N body Langevin equation is called Brownian dynamics (BD) [10]. Although BD can include effects of particle sizes in the dynamic simulations, it is fundamentally limited to the low concentration of solutes due to the Oseen tensor (see Appendix for details). Brownian dynamics (BD) was initially developed to reduce computational load by replacing deterministic interactions between a solute molecule and many solvent molecules by randomly fluctuating, probabilistic interaction as a net driving force. BD presumes a dilute solution, which means that a mean distance between solutes is much longer than the size of the molecule. A large number of solvent molecules exist around solutes, which is enough to exert random forces due to a tremendous number of collisions. When BD is applied to the dynamic motion of multiple particles, the lack

nitude is determined by the dissipation fluctuation theorem [8].

of hydrodynamic interactions may provide erroneous results because the

coefficient, used to calculate the particle diffusivity.

4

fluctuation-dissipation principle is not quantitatively well balanced. This limitation of BD to a single particle or a dilute solution is at the equivalent level of Stokes' drag

Dissipative particle dynamics (DPD) is an updated version of BD, which specifically includes the interparticle hydrodynamic forces [11, 12]. Two functions, often denoted as wR and wD, are proposed to quantify the presumed relationship of pair-wise hydrodynamic forces/torques, determined by the dissipation fluctuation theorem. The proposed forms of the hydrodynamic interaction are vector wise as the real viscous forces are tensor-wise. Dissipative particle dynamics (DPD) updates BD by including an approximate form of tensor-wise hydrodynamic interactions as a pair-wise vector form. A force exerted on a particle has three types

such as conservative, dissipative, and random forces, and DPD satisfies the fluctuation-dissipation theorem by balancing the dissipative and random forces. Stokesian dynamics (SD) uses the grand mobility matrix to include the far-field hydrodynamic forces/torques [13]. In-depth comparative analyses of present particle dynamic methods can be found elsewhere [14]. This mobility matrix is inverted and updated by adding differences between near-field lubrication forces and farfield two-body interactions. As the lubrication force is proportional to the logarithm of the surface-to-surface distance between two particles, it diverges during events of particle collisions. Besides, if two particles are close to each other, i.e., the surface-to-surface distance is much smaller than the particle diameter, the SD

resolution discretization schemes.

In the deterministic simulations, particle dynamics can be classified based on sizes of objects of interests. The purposes of particle dynamics are to provide the exact solution of a complex problem, bridging the theory and experiments. Note that the fundamental principles often provide governing equations, which were proven to be valid. It is difficult to solve a governing equation of a problem, if it has complex geometry and coupled boundary conditions. Experimental observations show natural processes using quantified information. Dissipative hydrodynamics is a generalized algorithm that unifies most of the preexisting particle dynamic simulation algorithms [16, 17].

At microscale of an order of nanometers, molecular dynamics deals with the motion of many molecules in various phases such as gas, liquid, and solid. Suppose that a system contains N molecules in volume V at temperature T. Newton's second laws of motion for this system is

$$m\_j \mathfrak{a}\_j = m\_j \frac{\mathrm{d}^2 r\_j}{\mathrm{d}t^2} = \sum\_{i=1,\ i \neq j}^N \mathbf{F}\_{ij} \tag{1}$$

where m is the particle mass at position x, β is the drag coefficient, and f

f 0

dv dt

where T is the absolute temperature, kB is Boltzmann's constant, and δð Þt is Dirac's delta function. Discarding the random force, one divides both sides of

¼ � <sup>v</sup>

which indicates that m=β has a dimension of time. This time scale is called the

<sup>τ</sup><sup>p</sup> <sup>¼</sup> <sup>m</sup>

which is a time scale that after a particle noticeably slows down after it starts moving with an initial velocity under the drag force. Stokes derived the drag

where μ is the absolute fluid viscosity and ap is the radius of the primary

Governing equation. A governing equation of DHD simulation is as follows:

where M is a diagonal matrix of mass and moment of inertia; v and U are translational/rotational velocities of the particle and the fluid, respectively; Q<sup>p</sup> is the generalized interparticle and conservative force/torque vector; R is the grand resistance matrix; and B is the Brownian matrix of zero mean and finite variance:

where δð Þt is the Dirac-Delta function. And, dW is the Ito-Wiener process [23, 24] having the following mathematical properties: Wk ¼ 0 at t ¼ 0, Wkð Þt is continuous, dWk ð Þ � Wkð Þ� t þ δt Wkð Þt follows the normal distribution, and finally dW � dW ¼ dt. The Brownian matrix B can be calculated by decomposing

where A is the decomposed matrix to be obtained and I is the identity matrix.

where C is a vector with zero mean and unit variance, i.e., h i C ¼ 0 and

f 0 ð Þ 0 f 0

A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD)

ð Þ<sup>t</sup> <sup>¼</sup> <sup>0</sup> (3)

<sup>m</sup>=<sup>β</sup> (5)

<sup>β</sup> (6)

β ¼ 6πμap (7)

<sup>M</sup> � <sup>d</sup><sup>v</sup> <sup>¼</sup> <sup>Q</sup><sup>p</sup> ½ � � <sup>R</sup> � ð Þ <sup>v</sup> � <sup>U</sup> <sup>d</sup><sup>t</sup> <sup>þ</sup> <sup>B</sup> � <sup>d</sup><sup>W</sup> (8)

h i <sup>B</sup> <sup>¼</sup> 0 and <sup>B</sup>tr h i � <sup>B</sup> <sup>¼</sup> <sup>2</sup>kBT<sup>R</sup> (9)

<sup>R</sup> <sup>¼</sup> <sup>A</sup>tr � <sup>I</sup> � <sup>A</sup> (10)

<sup>I</sup> <sup>¼</sup> <sup>C</sup>tr h i � <sup>C</sup> (11)

ð Þ<sup>t</sup> <sup>¼</sup> <sup>2</sup>kBTβδð Þ<sup>t</sup> (4)

random fluctuating force of zero mean:

DOI: http://dx.doi.org/10.5772/intechopen.86895

Eq. (2) by m to have

coefficient

particle [22].

7

particle relaxation time, define as

the grand resistance matrix such as

Statistically, the the identity matrix can be expressed as

<sup>C</sup><sup>2</sup> <sup>¼</sup> 1, respectively. The Brownian matrix is defined as

<sup>0</sup> is a

where Fij is a force exerted on particle j of mass mj from particle i of mass mi at time t. Position r<sup>i</sup> and velocity v<sup>i</sup> of molecule i are updated from time t to t þ δt, i.e., from its initial values of <sup>r</sup>ið Þ¼ <sup>t</sup> <sup>¼</sup> <sup>0</sup> <sup>r</sup><sup>0</sup> <sup>i</sup> and <sup>v</sup>ið Þ¼ <sup>t</sup> <sup>¼</sup> <sup>0</sup> <sup>v</sup><sup>0</sup> <sup>i</sup> , respectively, using the acceleration a<sup>i</sup> determined using Eq. (1). Numerical evolution of Eq. (1) requires a specific algorithm for double integration [18–21].

A macroscale of order of millimeters, granular dynamics often called the discrete element method (DEM) includes specifically collision rules using the restitution and friction coefficients. During inelastic collisions of nonrotating particles, the particle kinetic energy is continuously lost, and their motion is decelerated. For a collision of rotating spheres, the surface friction provides an effective torque (as action and reaction) of the same magnitude and opposite directions to two colliding spheres. Rotational motion of a non-touching particle in a fluid medium generates angular dissipation of kinetic energy. Considering granules, i.e., non-Brownian particles, in a gas phase often neglect solute molecules and approximate the system as multiple particles undergoing the gravitational force field in a vacuum phase. As granular particle mass is much higher than that of colloids or nanoparticles in an aqueous solution, in a stationary phase the gravitational force is often balanced by normal forces developed at interfaces of particles to touching neighbors or a rigid wall. Implementation of the hydrodynamic lubrication interactions to granular particles is a difficult task, which requires an in-depth understanding of microscopic surfacedeformation phenomena, linked to macroscopic particle motion.

A universal simulation method that can seamlessly include forces/torques exerted on arbitrary particles is therefore of great necessity. The method, first of all, should be able to:


Dissipative hydrodynamics (DHD) has all the features required to be a universal simulation method for particle dynamics by taking specific advantages from MD, BD, SD, DPD, and DEM. A detailed review can be found elsewhere [14].

#### 2.2 DHD formalism

Particle relaxation time. For a single particle motion in a viscous fluid, the governing equation can be in 1D space for simplicity:

$$m\ddot{x} = -\beta v + f'(t) \tag{2}$$

A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD) DOI: http://dx.doi.org/10.5772/intechopen.86895

where m is the particle mass at position x, β is the drag coefficient, and f <sup>0</sup> is a random fluctuating force of zero mean:

$$
\left< f'(t) \right> = \mathbf{0} \tag{3}
$$

$$
\left< f'(\mathbf{0}) f'(t) \right> = 2k\_B T \beta \delta(t) \tag{4}
$$

where T is the absolute temperature, kB is Boltzmann's constant, and δð Þt is Dirac's delta function. Discarding the random force, one divides both sides of Eq. (2) by m to have

$$\frac{\text{d}v}{\text{d}t} = -\frac{v}{m/\beta} \tag{5}$$

which indicates that m=β has a dimension of time. This time scale is called the particle relaxation time, define as

$$
\tau\_p = \frac{m}{\beta} \tag{6}
$$

which is a time scale that after a particle noticeably slows down after it starts moving with an initial velocity under the drag force. Stokes derived the drag coefficient

$$
\beta = \mathsf{G} \pi \mu \mathfrak{a}\_p \tag{7}
$$

where μ is the absolute fluid viscosity and ap is the radius of the primary particle [22].

Governing equation. A governing equation of DHD simulation is as follows:

$$\mathbf{M} \cdot \mathbf{d}\boldsymbol{\sigma} = [\mathbf{Q}^p - \mathbf{R} \cdot (\boldsymbol{\nu} - \mathbf{U})] \mathbf{d}t + \mathbf{B} \cdot \mathbf{d}W \tag{8}$$

where M is a diagonal matrix of mass and moment of inertia; v and U are translational/rotational velocities of the particle and the fluid, respectively; Q<sup>p</sup> is the generalized interparticle and conservative force/torque vector; R is the grand resistance matrix; and B is the Brownian matrix of zero mean and finite variance:

$$
\langle \mathbf{B} \rangle = \mathbf{0} \text{ and } \langle \mathbf{B}^{\text{tr}} \cdot \mathbf{B} \rangle = 2k\_B T \mathbf{R} \tag{9}
$$

where δð Þt is the Dirac-Delta function. And, dW is the Ito-Wiener process [23, 24] having the following mathematical properties: Wk ¼ 0 at t ¼ 0, Wkð Þt is continuous, dWk ð Þ � Wkð Þ� t þ δt Wkð Þt follows the normal distribution, and finally dW � dW ¼ dt. The Brownian matrix B can be calculated by decomposing the grand resistance matrix such as

$$\mathbf{R} = \mathbf{A}^{\text{tr}} \cdot \mathbf{I} \cdot \mathbf{A} \tag{10}$$

where A is the decomposed matrix to be obtained and I is the identity matrix. Statistically, the the identity matrix can be expressed as

$$I = \langle \mathbf{C}^{\text{tr}} \cdot \mathbf{C} \rangle \tag{11}$$

where C is a vector with zero mean and unit variance, i.e., h i C ¼ 0 and <sup>C</sup><sup>2</sup> <sup>¼</sup> 1, respectively. The Brownian matrix is defined as

that a system contains N molecules in volume V at temperature T. Newton's second

d2 rj <sup>d</sup>t<sup>2</sup> <sup>¼</sup> <sup>∑</sup>

where Fij is a force exerted on particle j of mass mj from particle i of mass mi at time t. Position r<sup>i</sup> and velocity v<sup>i</sup> of molecule i are updated from time t to t þ δt, i.e.,

acceleration a<sup>i</sup> determined using Eq. (1). Numerical evolution of Eq. (1) requires a

A macroscale of order of millimeters, granular dynamics often called the discrete element method (DEM) includes specifically collision rules using the restitution and friction coefficients. During inelastic collisions of nonrotating particles, the particle kinetic energy is continuously lost, and their motion is decelerated. For a collision of rotating spheres, the surface friction provides an effective torque (as action and reaction) of the same magnitude and opposite directions to two colliding spheres. Rotational motion of a non-touching particle in a fluid medium generates angular dissipation of kinetic energy. Considering granules, i.e., non-Brownian particles, in a gas phase often neglect solute molecules and approximate the system as multiple particles undergoing the gravitational force field in a vacuum phase. As granular particle mass is much higher than that of colloids or nanoparticles in an aqueous solution, in a stationary phase the gravitational force is often balanced by normal forces developed at interfaces of particles to touching neighbors or a rigid wall. Implementation of the hydrodynamic lubrication interactions to granular particles is a difficult task, which requires an in-depth understanding of microscopic surface-

N <sup>i</sup>¼1,i6¼<sup>j</sup>

<sup>i</sup> and <sup>v</sup>ið Þ¼ <sup>t</sup> <sup>¼</sup> <sup>0</sup> <sup>v</sup><sup>0</sup>

Fij (1)

<sup>i</sup> , respectively, using the

mja<sup>j</sup> ¼ mj

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

deformation phenomena, linked to macroscopic particle motion.

1. Investigate the accelerating/decelerating motion of particles.

3. Include many-body hydrodynamic interactions.

governing equation can be in 1D space for simplicity:

4.Mimic inelastic collisions between spherical particles.

2. Satisfy the fluctuation-dissipation theorem for Brownian particles.

BD, SD, DPD, and DEM. A detailed review can be found elsewhere [14].

5. Apply constraint forces to form a nonspherical rigid body consisting of unequal

Dissipative hydrodynamics (DHD) has all the features required to be a universal simulation method for particle dynamics by taking specific advantages from MD,

Particle relaxation time. For a single particle motion in a viscous fluid, the

mx€ ¼ �βv þ f

0

ð Þt (2)

A universal simulation method that can seamlessly include forces/torques exerted on arbitrary particles is therefore of great necessity. The method, first of all,

laws of motion for this system is

from its initial values of <sup>r</sup>ið Þ¼ <sup>t</sup> <sup>¼</sup> <sup>0</sup> <sup>r</sup><sup>0</sup>

should be able to:

spherical particles.

2.2 DHD formalism

6

specific algorithm for double integration [18–21].

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

$$\mathbf{B} = \sqrt{2k\_B T} \mathbf{C} \cdot \mathbf{A} \tag{12}$$

The disturbance velocity field at the particle surface Si is

A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD)

to the ambient flow field have then 11 degrees of freedom such as

<sup>u</sup><sup>i</sup> � <sup>U</sup><sup>∞</sup>; <sup>ω</sup><sup>i</sup> � <sup>Ω</sup><sup>∞</sup>; �E<sup>∞</sup> <sup>ð</sup> Þ ¼ vix � <sup>V</sup><sup>∞</sup>

β, replaced by the grand resistant matrix R.

DOI: http://dx.doi.org/10.5772/intechopen.86895

The generalized relative velocity is

<sup>i</sup> ð Þ¼ <sup>r</sup> <sup>u</sup><sup>i</sup> � <sup>U</sup><sup>∞</sup> <sup>þ</sup> <sup>ω</sup><sup>i</sup> � <sup>Ω</sup><sup>∞</sup> ð Þ� ð Þ� <sup>r</sup> � <sup>r</sup><sup>i</sup> <sup>E</sup><sup>∞</sup> : ð Þ <sup>r</sup> � <sup>r</sup><sup>i</sup> , <sup>r</sup><sup>∈</sup> Si (23)

<sup>x</sup> ; viy � <sup>V</sup><sup>∞</sup>

� �

<sup>x</sup> ;ωiy � <sup>ω</sup><sup>∞</sup>

� �

<sup>2</sup> ; �E<sup>∞</sup>

� �

<sup>y</sup> ; viy � <sup>V</sup><sup>∞</sup> y

<sup>3</sup> ; �E<sup>∞</sup>

<sup>y</sup> ;ωiz � <sup>Ω</sup><sup>∞</sup>

<sup>4</sup> ; �E<sup>∞</sup> 5

z

(24)

(25)

<sup>i</sup> ; S<sup>H</sup> i h itr

(26)

)

where u<sup>i</sup> and ω<sup>i</sup> are the translational and angular velocities of particle i, respectively. The translational/angular velocities and the rate of strain of particle i relative

<sup>þ</sup> <sup>ω</sup>ix � <sup>Ω</sup><sup>∞</sup>

For non-Brownian particles, the governing Eq. (8) is reduced back to that of Stokesian dynamics, which is Langevin's equation with the constant drag coefficient

When particle j is moving with linear and angular velocities of u<sup>j</sup> and ω<sup>j</sup> under the influences of the ambient flow field characterized using U<sup>∞</sup>, Ω<sup>∞</sup>, and E<sup>∞</sup>, it experiences the hydrodynamic force F<sup>H</sup> and torque T<sup>H</sup>. The stresslet S<sup>H</sup> can be obtained but does not directly contribute to the particle acceleration. The generalized velocity and force are related through the grand mobility matrix μ<sup>∞</sup>. Here, we use q, q\_, and F~ for generalized coordinates, velocities, and forces, respectively:

<sup>q</sup> <sup>¼</sup> ð Þ <sup>r</sup>; <sup>θ</sup> , <sup>v</sup> <sup>¼</sup> ð Þ <sup>u</sup>; <sup>ω</sup> , <sup>F</sup><sup>~</sup> <sup>¼</sup> ð Þ <sup>F</sup>; <sup>T</sup>

<sup>Δ</sup><sup>v</sup> <sup>¼</sup> <sup>u</sup> � <sup>U</sup>∞; <sup>ω</sup> � <sup>Ω</sup><sup>∞</sup> ð Þ

¼ � <sup>μ</sup><sup>∞</sup>

where S<sup>H</sup> is the hydrodynamic stresslet. The matrix μ<sup>∞</sup> (multiplied to F~ <sup>H</sup>

R<sup>∞</sup> Sv R<sup>∞</sup> SE � � Δv<sup>j</sup>

is called far-field grand mobility matrix. An inverse relationship of Eq. (25) is

¼ � <sup>R</sup><sup>∞</sup>

resistance matrix as an inverse of μ<sup>∞</sup>, having the mathematical identity as

� <sup>R</sup><sup>∞</sup>

R<sup>∞</sup> Sv R<sup>∞</sup> SE

i.e., forces and torques exerted on N bodies, can be expressed as

Δv<sup>j</sup> �E<sup>∞</sup> j

" #

F~ H i SH i

where the matrix <sup>R</sup><sup>∞</sup> (multiplied to <sup>Δ</sup>vj; �E<sup>∞</sup>

μ<sup>∞</sup> vF μ<sup>∞</sup> vS

� �

μ<sup>∞</sup> EF μ<sup>∞</sup> ES

Fv � ��<sup>1</sup> 6¼ <sup>μ</sup><sup>∞</sup>

matrices but R<sup>∞</sup>

9

" #

for both translational and angular motion. Then, the hydrodynamic interactions,

μ<sup>∞</sup> EF μ<sup>∞</sup> ES � � F~ <sup>H</sup>

Fv R<sup>∞</sup> FE

Fv R<sup>∞</sup> FE

where <sup>I</sup> is the identity matrix. Note that <sup>μ</sup><sup>∞</sup> ð Þ�<sup>1</sup> <sup>¼</sup> <sup>R</sup><sup>∞</sup> and <sup>R</sup><sup>∞</sup> ð Þ�<sup>1</sup> <sup>¼</sup> <sup>μ</sup><sup>∞</sup> for grand

� �

vF μ<sup>∞</sup> vS

i SH i

�E<sup>∞</sup> j

<sup>¼</sup> <sup>I</sup> <sup>0</sup> 0 I

vF for sub-matrices. The grand resistance matrix R in

� �tr) is the far-field grand

� � (27)

i

" #

" #

<sup>1</sup> ; �E<sup>∞</sup>

þ �E<sup>∞</sup>

vD

and substituted into (10) to provide

$$\mathbf{R} = (\mathbf{A}^{\mathrm{tr}} \cdot \mathbf{C}^{\mathrm{tr}}) \cdot (\mathbf{C} \cdot \mathbf{A}) = (\mathbf{C} \cdot \mathbf{A})^{\mathrm{tr}} \cdot (\mathbf{C} \cdot \mathbf{A}) = \frac{\mathbf{B}^{\mathrm{tr}} \cdot \mathbf{B}}{2k\_B T} \tag{13}$$

Therefore, B is obtained by calculating a square root of R matrix, which is equal to A matrix of Eq. (10). The identity relationship of Eq. (11) is not satisfied at specific time <sup>t</sup> but statistically by taking an average of <sup>C</sup>tr � <sup>C</sup> for a much longer period than the particle relaxation time τp. The effective force acting on a particle of a swarm of many particles in a viscous fluid is then represented as

$$\mathbf{Q}^p - \mathbf{R} \cdot (\boldsymbol{\nu} - \mathbf{U}) + \sqrt{2k\_B T} \mathbf{C} \cdot \mathbf{A} \cdot \mathbf{W}' \tag{14}$$

where W<sup>0</sup> ¼ dW=dt. Although A is deterministically calculated to satisfy Eq. (10), <sup>C</sup>tr � <sup>C</sup> <sup>¼</sup> I is not valid at an instance but statistically. In the same sense, Eq. (11) is satisfied statistically.

Hydrodynamic tensors In Eq. (14), the generalized force requires a calculation of the grand resistance matrix R, which will allow to generate A. Consider particle i among Np particles in a given volume V, translating with a linear velocity v<sup>i</sup> and rotating with an angular velocity ω<sup>i</sup> at an instantaneous position rið Þt . In the absence of particles, the fluid flow at the center of particle <sup>i</sup> can be represented as <sup>U</sup><sup>∞</sup>ð Þ <sup>r</sup><sup>i</sup> . At a point r ¼ ð Þ x; y; z ∈ Si on surface Si of particle i from the particle center ri, the flow field is described as

$$\mathbf{V}(\mathbf{r}) = \mathbf{U}^{\infty}(\mathbf{r}\_i) + \mathbf{Q}^{\infty} \times (\mathbf{r} - \mathbf{r}\_i) + \mathbf{E}^{\infty} : (\mathbf{r} - \mathbf{r}\_i) \tag{15}$$

where U<sup>∞</sup> is the unidirectional velocity, Ω<sup>∞</sup> is the vorticity,

$$
\boldsymbol{\Omega}^{\infty} = \frac{1}{2} \boldsymbol{\nabla} \times \mathbf{V} \tag{16}
$$

and E<sup>∞</sup> is the rate of strain

$$E\_{ij}^{\infty} = \frac{1}{2} \left( \frac{\partial V\_i}{\partial \mathbf{x}\_j} + \frac{\partial V\_j}{\partial \mathbf{x}\_i} \right) \tag{17}$$

which are evaluated at ri. Because the rate-of-strain matrix is symmetric and traceless, the original nine components are reduced to

$$E\_1^{\infty} = E\_{\infty}^{\infty} - E\_{\infty}^{\infty} = \frac{\partial V\_x}{\partial \mathbf{x}} - \frac{\partial V\_x}{\partial \mathbf{z}} \tag{18}$$

$$E\_2^{\infty} = 2E\_{xy}^{\infty} = \frac{\partial V\_x}{\partial y} + \frac{\partial V\_y}{\partial \mathbf{x}} \tag{19}$$

$$E\_3^{\infty} = 2E\_{\infty}^{\infty} = \frac{\partial V\_x}{\partial \mathbf{z}} + \frac{\partial V\_x}{\partial \mathbf{x}} \tag{20}$$

$$E\_4^{\infty} = 2E\_{yx}^{\infty} = \frac{\partial V\_y}{\partial x} + \frac{\partial V\_x}{\partial y} \tag{21}$$

$$E\_5^{\infty} = E\_{\gamma y}^{\infty} - E\_{xx}^{\infty} = \frac{\partial V\_y}{\partial y} - \frac{\partial V\_x}{\partial x} \tag{22}$$

A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD) DOI: http://dx.doi.org/10.5772/intechopen.86895

The disturbance velocity field at the particle surface Si is

$$\boldsymbol{\sigma}\_{i}^{D}(\boldsymbol{r}) = \boldsymbol{\mu}\_{i} - \boldsymbol{U}^{\infty} + (\boldsymbol{\mu}\_{i} - \boldsymbol{\Omega}^{\infty}) \times (\boldsymbol{r} - \boldsymbol{r}\_{i}) - \boldsymbol{E}^{\infty} : (\boldsymbol{r} - \boldsymbol{r}\_{i}), \quad \boldsymbol{r} \in \mathbb{S}\_{i} \tag{23}$$

where u<sup>i</sup> and ω<sup>i</sup> are the translational and angular velocities of particle i, respectively. The translational/angular velocities and the rate of strain of particle i relative to the ambient flow field have then 11 degrees of freedom such as

$$\begin{aligned} \left(\boldsymbol{\mu}\_{i} - \mathbf{U}^{\infty}, \boldsymbol{\alpha}\_{i} - \boldsymbol{\Omega}^{\infty}, -\mathbf{E}^{\infty}\right) &= \left(v\_{i\text{ir}} - V\_{\text{x}}^{\infty}, v\_{i\text{j}} - V\_{\text{y}}^{\infty}, v\_{i\text{j}} - V\_{\text{y}}^{\infty}\right) \\ &+ \left(\boldsymbol{\alpha}\_{i\text{ir}} - \boldsymbol{\Omega}\_{\text{x}}^{\infty}, \boldsymbol{\alpha}\_{i\text{j}} - \boldsymbol{\alpha}\_{\text{y}}^{\infty}, \boldsymbol{\alpha}\_{i\text{i}} - \boldsymbol{\Omega}\_{\text{x}}^{\infty}\right) \\ &+ \left(-E\_{1}^{\infty}, -E\_{2}^{\infty}, -E\_{3}^{\infty}, -E\_{4}^{\infty}, -E\_{5}^{\infty}\right) \end{aligned} \tag{24}$$

For non-Brownian particles, the governing Eq. (8) is reduced back to that of Stokesian dynamics, which is Langevin's equation with the constant drag coefficient β, replaced by the grand resistant matrix R.

When particle j is moving with linear and angular velocities of u<sup>j</sup> and ω<sup>j</sup> under the influences of the ambient flow field characterized using U<sup>∞</sup>, Ω<sup>∞</sup>, and E<sup>∞</sup>, it experiences the hydrodynamic force F<sup>H</sup> and torque T<sup>H</sup>. The stresslet S<sup>H</sup> can be obtained but does not directly contribute to the particle acceleration. The generalized velocity and force are related through the grand mobility matrix μ<sup>∞</sup>. Here, we use q, q\_, and F~ for generalized coordinates, velocities, and forces, respectively:

$$\boldsymbol{q} = (\boldsymbol{r}, \boldsymbol{\theta}), \quad \boldsymbol{\nu} = (\boldsymbol{\mu}, \boldsymbol{\alpha}), \quad \boldsymbol{\tilde{F}} = (\boldsymbol{F}, \boldsymbol{T}).$$

The generalized relative velocity is

$$\Delta \boldsymbol{\sigma} = (\boldsymbol{\mu} - \boldsymbol{U}^{\infty}, \boldsymbol{\mu} - \boldsymbol{\Omega}^{\infty})$$

for both translational and angular motion. Then, the hydrodynamic interactions, i.e., forces and torques exerted on N bodies, can be expressed as

$$
\begin{bmatrix}
\Delta \boldsymbol{\sigma}\_{j} \\
\end{bmatrix} = -\begin{bmatrix}
\mu\_{v\boldsymbol{F}}^{\infty} & \mu\_{v\boldsymbol{S}}^{\infty} \\
\mu\_{EF}^{\infty} & \mu\_{ES}^{\infty}
\end{bmatrix} \begin{bmatrix}
\boldsymbol{\tilde{F}}\_{i}^{H} \\
\boldsymbol{\mathsf{S}}\_{i}^{H}
\end{bmatrix} \tag{25}
$$

where S<sup>H</sup> is the hydrodynamic stresslet. The matrix μ<sup>∞</sup> (multiplied to F~ <sup>H</sup> <sup>i</sup> ; S<sup>H</sup> i h itr ) is called far-field grand mobility matrix. An inverse relationship of Eq. (25) is

$$
\begin{bmatrix} \tilde{\boldsymbol{F}}\_i^H \\ \mathbf{S}\_i^H \end{bmatrix} = - \begin{bmatrix} R\_{\mathcal{V}\nu}^{\infty} & R\_{\mathcal{FE}}^{\infty} \\ R\_{\mathcal{S}\nu}^{\infty} & R\_{\mathcal{SE}}^{\infty} \end{bmatrix} \begin{bmatrix} \Delta \boldsymbol{\mathcal{w}}\_j \\ -\mathbf{E}\_j^{\infty} \end{bmatrix} \tag{26}
$$

where the matrix <sup>R</sup><sup>∞</sup> (multiplied to <sup>Δ</sup>vj; �E<sup>∞</sup> i � �tr) is the far-field grand resistance matrix as an inverse of μ<sup>∞</sup>, having the mathematical identity as

$$
\begin{bmatrix}
\mu\_{\nu F}^{\infty} & \mu\_{\nu S}^{\infty} \\
\mu\_{EF}^{\infty} & \mu\_{ES}^{\infty}
\end{bmatrix} \cdot \begin{bmatrix}
R\_{Fv}^{\infty} & R\_{FE}^{\infty} \\
R\_{Sv}^{\infty} & R\_{SE}^{\infty}
\end{bmatrix} = \begin{bmatrix}
I & \mathbf{0} \\
\mathbf{0} & I
\end{bmatrix} \tag{27}
$$

where <sup>I</sup> is the identity matrix. Note that <sup>μ</sup><sup>∞</sup> ð Þ�<sup>1</sup> <sup>¼</sup> <sup>R</sup><sup>∞</sup> and <sup>R</sup><sup>∞</sup> ð Þ�<sup>1</sup> <sup>¼</sup> <sup>μ</sup><sup>∞</sup> for grand matrices but R<sup>∞</sup> Fv � ��<sup>1</sup> 6¼ <sup>μ</sup><sup>∞</sup> vF for sub-matrices. The grand resistance matrix R in

<sup>B</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

to A matrix of Eq. (10). The identity relationship of Eq. (11) is not satisfied at specific time <sup>t</sup> but statistically by taking an average of <sup>C</sup>tr � <sup>C</sup> for a much longer period than the particle relaxation time τp. The effective force acting on a particle of

a swarm of many particles in a viscous fluid is then represented as

where U<sup>∞</sup> is the unidirectional velocity, Ω<sup>∞</sup> is the vorticity,

E<sup>∞</sup> ij <sup>¼</sup> <sup>1</sup> 2

traceless, the original nine components are reduced to

E<sup>∞</sup> <sup>1</sup> <sup>¼</sup> <sup>E</sup><sup>∞</sup>

> E<sup>∞</sup> <sup>2</sup> <sup>¼</sup> <sup>2</sup>E<sup>∞</sup>

> E<sup>∞</sup> <sup>3</sup> <sup>¼</sup> <sup>2</sup>E<sup>∞</sup>

E<sup>∞</sup> <sup>4</sup> <sup>¼</sup> <sup>2</sup>E<sup>∞</sup>

E<sup>∞</sup> <sup>5</sup> <sup>¼</sup> <sup>E</sup><sup>∞</sup>

<sup>Ω</sup><sup>∞</sup> <sup>¼</sup> <sup>1</sup> 2

> ∂Vi ∂xj þ ∂Vj ∂xi � �

which are evaluated at ri. Because the rate-of-strain matrix is symmetric and

zz <sup>¼</sup> <sup>∂</sup>Vx

xy <sup>¼</sup> <sup>∂</sup>Vx ∂y þ ∂Vy ∂x

xz <sup>¼</sup> <sup>∂</sup>Vx ∂z þ ∂Vz ∂x

yz <sup>¼</sup> <sup>∂</sup>Vy ∂z þ ∂Vz ∂y

zz <sup>¼</sup> <sup>∂</sup>Vy

yy � <sup>E</sup><sup>∞</sup>

<sup>∂</sup><sup>x</sup> � <sup>∂</sup>Vz ∂z

<sup>∂</sup><sup>y</sup> � <sup>∂</sup>Vz ∂z

xx � <sup>E</sup><sup>∞</sup>

<sup>Q</sup><sup>p</sup> � <sup>R</sup> � ð Þþ <sup>v</sup> � <sup>U</sup> ffiffiffiffiffiffiffiffiffiffi

where W<sup>0</sup> ¼ dW=dt. Although A is deterministically calculated to satisfy Eq. (10), <sup>C</sup>tr � <sup>C</sup> <sup>¼</sup> I is not valid at an instance but statistically. In the same sense,

Hydrodynamic tensors In Eq. (14), the generalized force requires a calculation of the grand resistance matrix R, which will allow to generate A. Consider particle i among Np particles in a given volume V, translating with a linear velocity v<sup>i</sup> and rotating with an angular velocity ω<sup>i</sup> at an instantaneous position rið Þt . In the absence of particles, the fluid flow at the center of particle <sup>i</sup> can be represented as <sup>U</sup><sup>∞</sup>ð Þ <sup>r</sup><sup>i</sup> . At a point r ¼ ð Þ x; y; z ∈ Si on surface Si of particle i from the particle center ri, the flow

V rð Þ¼ <sup>U</sup><sup>∞</sup>ð Þþ <sup>r</sup><sup>i</sup> <sup>Ω</sup><sup>∞</sup> � ð Þþ <sup>r</sup> � <sup>r</sup><sup>i</sup> <sup>E</sup><sup>∞</sup> : ð Þ <sup>r</sup> � <sup>r</sup><sup>i</sup> (15)

<sup>R</sup> <sup>¼</sup> <sup>A</sup>tr � <sup>C</sup>tr ð Þ� ð Þ¼ <sup>C</sup> � <sup>A</sup> ð Þ <sup>C</sup> � <sup>A</sup> tr � ð Þ¼ <sup>C</sup> � <sup>A</sup> <sup>B</sup>tr � <sup>B</sup>

Therefore, B is obtained by calculating a square root of R matrix, which is equal

and substituted into (10) to provide

Eq. (11) is satisfied statistically.

and E<sup>∞</sup> is the rate of strain

field is described as

8

<sup>2</sup>kBT <sup>p</sup> <sup>C</sup> � <sup>A</sup> (12)

<sup>2</sup>kBT <sup>p</sup> <sup>C</sup> � <sup>A</sup> � <sup>W</sup><sup>0</sup> (14)

∇ � V (16)

(17)

(18)

(19)

(20)

(21)

(22)

<sup>2</sup>kBT (13)

Eq. (8) refers to R<sup>∞</sup> Fv of Eq. (26), and R<sup>∞</sup> FE � <sup>E</sup><sup>∞</sup> <sup>j</sup> is an extra forcing term due to the rate of strain.

3.1 Geometric calculations

DOI: http://dx.doi.org/10.5772/intechopen.86895

3.1.1 How to determine a cell containing the centroid position of my particle

A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD)

located at position r<sup>i</sup> for i ¼ 1 � 4, where r<sup>i</sup> ¼ xi; yi

where J is the Jacobian matrix given as

<sup>V</sup> <sup>¼</sup> <sup>1</sup>

Using this mathematical relationship, one can easily test whether a particle

tetrahedron, then the total volume of the tetrahedron can be divided by four small

Jð Þ¼ 1; 2; 3; 4

� �tr replaced by 1; xp; yp; zp

tetrahedron can be calculated as

position r<sup>p</sup> ¼ xp; yp; zp

Vp <sup>¼</sup> <sup>1</sup>

first column 1; x1; y1; z<sup>1</sup>

the tetrahedron, we have an inequality of

using

11

pieces and their sum is equal to V:

Computational grid cells have various structures such as hexahedron, wedge, prism, pyramid, tetrahedron, and tetrahedron wedge. Among them, hexahedron followed by tetrahedron structures is widely used to generate mesh structure of bulk (internal) spaces. Cubic and rectangular shapes are representative structures of hexahedrons, consisting of eight vertices (points) and six rectangular (or square) surfaces. On the other hand, a tetrahedron cell has only four vertices (as compared to eight in hexahedron) and three triangular surfaces. Each of these two cell structures has its advantages and disadvantages in CFD simulations. Formation of hexahedron meshes is straightforward, and numerical solutions are well converged to provide accurate results within a tolerable error, especially if edge lines are well aligned to the flow directions. However, if a computational domain includes complex and curved surface structures such as human faces or globes, the hexahedron meshes often provide unrealistic small exuberances instead of well-curved surfaces. As three triangular surfaces surround the tetrahedron volume, it can form well-fitted boundary layers of arbitrary shapes. As edges of tetrahedrons cannot be fully aligned on a straight line, the numerical convergence of tetrahedron meshes is often more sensitive to the fineness of generated mesh structures than that of hexahedron meshes. To overcome this limitation, one can make tetrahedron meshes often finer than that of hexahedron meshes to solve the same problem with similar accuracy. Nevertheless, there are unique mathematical advantages of using tetrahedron meshes for coupled simulations of CFD and particle hydrodynamics. Location test using volume calculation. A tetrahedron have four points, pi

; zi

1111 x<sup>1</sup> x<sup>2</sup> x<sup>3</sup> x<sup>4</sup> y<sup>1</sup> y<sup>2</sup> y<sup>3</sup> y<sup>4</sup> z<sup>1</sup> z<sup>2</sup> z<sup>3</sup> z<sup>4</sup>

� � is inside or outside the tetrahedron. If <sup>r</sup><sup>p</sup> is inside of the

<sup>6</sup> ½ � J pð Þþ ; <sup>2</sup>; <sup>3</sup>; <sup>4</sup> <sup>J</sup>ð Þþ <sup>1</sup>; <sup>p</sup>; <sup>3</sup>; <sup>4</sup> <sup>J</sup>ð Þþ <sup>1</sup>; <sup>2</sup>; <sup>p</sup>; <sup>4</sup> <sup>J</sup>ð Þ <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; <sup>p</sup> (32)

where, for example, J pð Þ ; 2; 3; 4 means the Jacobian matrix of Eq. (30) with the

� �tr

� �. The volume of the

<sup>6</sup> detð Þ<sup>J</sup> (29)

V ¼ Vp (31)

. Conversely, if r<sup>p</sup> is outside of

,

(30)

Note that at time t, the right-hand side of Eq. (25) is known, and one can calculate the generalized hydrodynamic force F~ <sup>H</sup> by using an appropriate solver in numerical linear algebra [25, 26]. As noted in Eq. (15), the required information to evolve the motion of particle j, located at r<sup>j</sup> with linear and angular velocities of v<sup>j</sup> and <sup>ω</sup>i, respectively, is the ambient flow field consisting of <sup>U</sup><sup>∞</sup>; ;Ω<sup>∞</sup>; �E<sup>∞</sup> ð Þ at the particle location. Unlike MD and BD, DHD explicitly includes the sizes of individual spherical particles. In this case, r<sup>j</sup> indicates the position of the particle centroid or center of mass, and the fluid field is calculated at r<sup>j</sup> without considering the presence of the particles. Therefore, the calculation of the ambient flow field is highly dependent on a mesh structure used for CFD simulations.

#### 3. Computational fluid dynamics coupled with dissipative hydrodynamics

A general governing equation for fluid dynamics is Navier–Stokes equation, of which most general form for incompressible fluid is

$$\frac{\partial \mathbf{U}}{\partial t} + \mathbf{U} \cdot \nabla \mathbf{U} = -\frac{1}{\rho} \nabla P + \frac{\eta}{\rho} \nabla^2 \mathbf{U} \tag{28}$$

where U is the flow velocity, P is pressure, η and ρ are the viscosity and density of the fluid, respectively. In the adjacent space of particle j at rj, U can be expanded as expressed in Eq. (15). Most fluid dynamic problems have at least three boundaries, which are the inlet, outlet, and side walls. For inlet and outlet surfaces, Neuman and Dirichlet or mixed boundary conditions are often used to set values or conditions of U and P. On the wall, zero velocity and zero-gradient pressure are usually assigned. The former condition assumes that there are strong adhesion forces between solvent molecules and wall surfaces. The solvent molecules are fixed on the wall. The velocities of the wall-adsorbed solvent molecules are equal to those of the solid walls, which is zero for non-moving walls. Values of ð Þ U; P are calculated at grid points in internal spaces surrounded by the boundary surfaces. The simulation accuracy and numerical convergence highly depend on the mesh structures. When CFD is coupled with particle dynamics, which is in this study, DHD, there are additional requirements that should be satisfied to evolve the motion of multi-particles moving in a fluid flow:


Possible methods to satisfy the three requirements are dependent on available CFD solvers and flexibility of applying customized modifications. Here we suggest fundamental approaches to meet the requirement numerically.

A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD) DOI: http://dx.doi.org/10.5772/intechopen.86895

#### 3.1 Geometric calculations

Eq. (8) refers to R<sup>∞</sup>

hydrodynamics

of strain.

Fv of Eq. (26), and R<sup>∞</sup>

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

dependent on a mesh structure used for CFD simulations.

which most general form for incompressible fluid is

motion of multi-particles moving in a fluid flow:

position, rk, for k ¼ 1 � Np

wall boundaries

kth particle

10

∂U ∂t

3. Computational fluid dynamics coupled with dissipative

A general governing equation for fluid dynamics is Navier–Stokes equation, of

ρ

where U is the flow velocity, P is pressure, η and ρ are the viscosity and density of the fluid, respectively. In the adjacent space of particle j at rj, U can be expanded as expressed in Eq. (15). Most fluid dynamic problems have at least three boundaries, which are the inlet, outlet, and side walls. For inlet and outlet surfaces, Neuman and Dirichlet or mixed boundary conditions are often used to set values or conditions of U and P. On the wall, zero velocity and zero-gradient pressure are usually assigned. The former condition assumes that there are strong adhesion forces between solvent molecules and wall surfaces. The solvent molecules are fixed on the wall. The velocities of the wall-adsorbed solvent molecules are equal to those of the solid walls, which is zero for non-moving walls. Values of ð Þ U; P are calculated at grid points in internal spaces surrounded by the boundary surfaces. The simulation accuracy and numerical convergence highly depend on the mesh structures. When CFD is coupled with particle dynamics, which is in this study, DHD, there are additional requirements that should be satisfied to evolve the

1. To identify a cell of the constructed mesh grid, which contains the kth particle's

2. To calculate the distance between r<sup>k</sup> and wall surfaces, if the particle is close to

3. To interpolate the flow field <sup>U</sup><sup>∞</sup>; ;Ω<sup>∞</sup>; �E<sup>∞</sup> ð Þ at <sup>r</sup><sup>k</sup> within a cell that contains

Possible methods to satisfy the three requirements are dependent on available CFD solvers and flexibility of applying customized modifications. Here we suggest

fundamental approaches to meet the requirement numerically.

<sup>∇</sup><sup>P</sup> <sup>þ</sup> <sup>η</sup> ρ ∇2

<sup>þ</sup> <sup>U</sup> � <sup>∇</sup><sup>U</sup> ¼ � <sup>1</sup>

FE � <sup>E</sup><sup>∞</sup>

Note that at time t, the right-hand side of Eq. (25) is known, and one can calculate the generalized hydrodynamic force F~ <sup>H</sup> by using an appropriate solver in numerical linear algebra [25, 26]. As noted in Eq. (15), the required information to evolve the motion of particle j, located at r<sup>j</sup> with linear and angular velocities of v<sup>j</sup> and <sup>ω</sup>i, respectively, is the ambient flow field consisting of <sup>U</sup><sup>∞</sup>; ;Ω<sup>∞</sup>; �E<sup>∞</sup> ð Þ at the particle location. Unlike MD and BD, DHD explicitly includes the sizes of individual spherical particles. In this case, r<sup>j</sup> indicates the position of the particle centroid or center of mass, and the fluid field is calculated at r<sup>j</sup> without considering the presence of the particles. Therefore, the calculation of the ambient flow field is highly

<sup>j</sup> is an extra forcing term due to the rate

U (28)

#### 3.1.1 How to determine a cell containing the centroid position of my particle

Computational grid cells have various structures such as hexahedron, wedge, prism, pyramid, tetrahedron, and tetrahedron wedge. Among them, hexahedron followed by tetrahedron structures is widely used to generate mesh structure of bulk (internal) spaces. Cubic and rectangular shapes are representative structures of hexahedrons, consisting of eight vertices (points) and six rectangular (or square) surfaces. On the other hand, a tetrahedron cell has only four vertices (as compared to eight in hexahedron) and three triangular surfaces. Each of these two cell structures has its advantages and disadvantages in CFD simulations. Formation of hexahedron meshes is straightforward, and numerical solutions are well converged to provide accurate results within a tolerable error, especially if edge lines are well aligned to the flow directions. However, if a computational domain includes complex and curved surface structures such as human faces or globes, the hexahedron meshes often provide unrealistic small exuberances instead of well-curved surfaces. As three triangular surfaces surround the tetrahedron volume, it can form well-fitted boundary layers of arbitrary shapes. As edges of tetrahedrons cannot be fully aligned on a straight line, the numerical convergence of tetrahedron meshes is often more sensitive to the fineness of generated mesh structures than that of hexahedron meshes. To overcome this limitation, one can make tetrahedron meshes often finer than that of hexahedron meshes to solve the same problem with similar accuracy. Nevertheless, there are unique mathematical advantages of using tetrahedron meshes for coupled simulations of CFD and particle hydrodynamics.

Location test using volume calculation. A tetrahedron have four points, pi , located at position r<sup>i</sup> for i ¼ 1 � 4, where r<sup>i</sup> ¼ xi; yi ; zi � �. The volume of the tetrahedron can be calculated as

$$V = \frac{1}{6} \det(I) \tag{29}$$

where J is the Jacobian matrix given as

$$J(1,2,3,4) = \begin{bmatrix} 1 & 1 & 1 & 1 \\ \varkappa\_1 & \varkappa\_2 & \varkappa\_3 & \varkappa\_4 \\ \varkappa\_1 & \varkappa\_2 & \varkappa\_3 & \varkappa\_4 \\ \varkappa\_1 & \varkappa\_2 & \varkappa\_3 & \varkappa\_4 \end{bmatrix} \tag{30}$$

Using this mathematical relationship, one can easily test whether a particle position r<sup>p</sup> ¼ xp; yp; zp � � is inside or outside the tetrahedron. If <sup>r</sup><sup>p</sup> is inside of the tetrahedron, then the total volume of the tetrahedron can be divided by four small pieces and their sum is equal to V:

$$V = V\_p \tag{31}$$

using

$$V\_p = \frac{1}{6}[J(p,2,3,4) + J(1,p,3,4) + J(1,2,p,4) + J(1,2,3,p)]\tag{32}$$

where, for example, J pð Þ ; 2; 3; 4 means the Jacobian matrix of Eq. (30) with the first column 1; x1; y1; z<sup>1</sup> � �tr replaced by 1; xp; yp; zp � �tr . Conversely, if r<sup>p</sup> is outside of the tetrahedron, we have an inequality of

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

$$\mathbf{V} \prec \mathbf{V}\_p \tag{33}$$

boundary, so the distance d cannot be smaller than the particle radius ap. After a single-step time evolution, if a particle is found overlapped with a wall surface, then collision rule will be applied to return the particle to the tetrahedron interior.

If a particle k is found inside a tetrahedron (of index l), the flow field consisting

<sup>i</sup> and <sup>∂</sup>jUi � <sup>∂</sup>Ui=∂xj for <sup>i</sup> and <sup>j</sup> <sup>¼</sup> <sup>1</sup> � 3. (Note that pressure value is not

A scalar quantity of interest S at each vertex point can be denoted as Sl. Then, the

Sp ¼ S � ξ ¼ ∑

U<sup>∞</sup>, vortex Ω<sup>∞</sup>, and the rate of strain E<sup>∞</sup>, which are calculated using U<sup>∞</sup>

located at r<sup>i</sup> and rj, translating with v<sup>i</sup> and v<sup>j</sup> and rotating with ω<sup>i</sup> and ωj, respectively. Relative position of particle i with respect to particle j is defined as rij ¼ r<sup>i</sup> � rj, and similarly the relative velocity of i to j is vij ¼ v<sup>i</sup> � vj. A normal

After an instantaneous collision, the two particle have the following

� � <sup>1</sup> <sup>þ</sup> <sup>ε</sup><sup>n</sup> ð Þg<sup>n</sup>

� � <sup>1</sup> <sup>þ</sup> <sup>ε</sup><sup>n</sup> ð Þg<sup>n</sup>

μij miai

μij mjaj

� � <sup>1</sup> � <sup>ε</sup><sup>t</sup>

� � <sup>1</sup> � <sup>ε</sup><sup>t</sup>

ij þ

ij þ

<sup>1</sup> <sup>þ</sup>~<sup>J</sup> � �nij � <sup>g</sup><sup>t</sup>

<sup>1</sup> <sup>þ</sup>~<sup>J</sup> � �nij � <sup>g</sup><sup>t</sup>

<sup>1</sup> � <sup>ε</sup><sup>t</sup> <sup>1</sup> <sup>þ</sup> <sup>1</sup>=~Jg<sup>t</sup> ij " # (43)

<sup>1</sup> � <sup>ε</sup><sup>t</sup> <sup>1</sup> <sup>þ</sup> <sup>1</sup>=~Jg<sup>t</sup> ij " # (44)

mi

μij mj

For DHD simulations, Sp represents each element of the unidirectional velocity

Suppose there are two particles colliding each other, which are particle i and j

<sup>n</sup>ij <sup>¼</sup> <sup>r</sup>ij

4 l¼1 ; zl

� � for <sup>l</sup> <sup>¼</sup> <sup>1</sup> � 4.

� � can be calculated as a

Slξ<sup>l</sup> (41)

<sup>∣</sup>rij<sup>∣</sup> (42)

ij (45)

ij (46)

<sup>i</sup> or its

of <sup>U</sup><sup>∞</sup>; ;Ω<sup>∞</sup>; �E<sup>∞</sup> ð Þ needs to be interpolated using values calculated adjacent locations. A proper choice of a set of these locations are the vertex points of the containing cell. Based on the definition of Ω<sup>∞</sup> and E<sup>∞</sup>, the basic quantities needed

3.1.3 How to interpolate the flow field at the position of my particle

A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD)

DOI: http://dx.doi.org/10.5772/intechopen.86895

required to calculate the generalized hydrodynamic forces <sup>F</sup>~H.) Four vertices of a cell can be represented as r<sup>l</sup> ¼ xl; yl

value of S at an arbitrary internal position r<sup>p</sup> ¼ xp; yp; zp

linear superposition of Sl and ξl:

are U<sup>∞</sup>

gradient ∂jUi.

velocities [14]:

13

3.2 Collision rules

vector from j to i is denoted as

v0

v0 <sup>j</sup> ¼ v<sup>j</sup> þ

ω0

ω0

<sup>i</sup> ¼ ω<sup>i</sup> þ

<sup>j</sup> ¼ ω<sup>j</sup> þ

<sup>i</sup> <sup>¼</sup> <sup>v</sup><sup>i</sup> � <sup>μ</sup>ij

If a particle is very close to one of the triangular surfaces of the tetrahedron, then the inequality check of Eq. (33) may not be done accurately. In this case, this location check can be extended to the nearest neighbor cells, especially one that shares the triangular surface that the particle is closely located.

For r<sup>p</sup> within the tetrahedron, dividing both sides of Eq. (32) by V gives

$$\mathbf{1} = \xi\_1 + \xi\_2 + \xi\_3 + \xi\_4 \tag{34}$$

where, for i ¼ 1 � 4,

$$
\xi\_i = \frac{J(\text{without } i)}{\text{6V}} \tag{35}
$$

which is solely determined by the internal position rp. A relationship between r<sup>p</sup> and ξ is

$$
\begin{bmatrix} \mathbf{1} \\ \mathbf{x}\_p \\ \mathbf{y}\_p \\ \mathbf{z}\_p \end{bmatrix} = \begin{bmatrix} \mathbf{1} & \mathbf{1} & \mathbf{1} & \mathbf{1} \\ \mathbf{x}\_1 & \mathbf{x}\_2 & \mathbf{x}\_3 & \mathbf{x}\_4 \\ \mathbf{y}\_1 & \mathbf{y}\_2 & \mathbf{y}\_3 & \mathbf{y}\_4 \\ \mathbf{z}\_1 & \mathbf{z}\_2 & \mathbf{z}\_3 & \mathbf{z}\_4 \end{bmatrix} \begin{bmatrix} \xi\_1 \\ \xi\_2 \\ \xi\_3 \\ \xi\_4 \end{bmatrix} \tag{36}
$$

where the inverse of the 4 � 4 matrix can be analytically available. Eq. (36) assumes that r<sup>p</sup> is located within the tetrahedron, but it can also use the particle location check as a better alternative to Eq. (33) because Eq. (36) requires one time solving of a 4 � 4 linear system, but Eq. (33) requires four times calculation of 4 � 4 Jacobian matrices.

#### 3.1.2 How to calculate a distance between wall surfaces to the position of my particle

Using the four vertices of ð Þ r1; r2; r3; r<sup>4</sup> of a tetrahedron cell, one can identify four triangular surfaces of S1ð Þ r2; r3; r<sup>4</sup> , S2ð Þ r1; r3; r<sup>4</sup> , S3ð Þ r1; r2; r<sup>4</sup> , and S4ð Þ r1; r2; r<sup>3</sup> . For example, let us consider S<sup>4</sup> having vertices of ð Þ r1; r2; r<sup>3</sup> . If we calculate relative position of vertices 2 and 3 with respect to vertex 1, denoted as

$$
\mathbf{r}\_{2/1} = \mathbf{r}\_2 - \mathbf{r}\_1 \tag{37}
$$

$$r\_{3/1} = r\_3 - r\_1 \tag{38}$$

their cross product allows us to calculate a unit vector normal to surface S4:

$$m\_4 = \frac{r\_{2/1} \times r\_{3/1}}{\left| r\_{2/1} \times r\_{3/1} \right|} \tag{39}$$

The particle position relative to r<sup>1</sup> is r<sup>p</sup>=<sup>1</sup> ¼ r<sup>p</sup> � r1. Then, the distance between surface S<sup>4</sup> and position r<sup>p</sup> is simply

$$d = \left| \mathfrak{n}\_4 \cdot \mathfrak{r}\_p \right| \tag{40}$$

where the absolute value is necessary because n can direct inside or outside the tetrahedron volume, depending on choice of the reference position, r1. This wallparticle distance calculation is necessary when the surface S<sup>4</sup> is known as a wall

A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD) DOI: http://dx.doi.org/10.5772/intechopen.86895

boundary, so the distance d cannot be smaller than the particle radius ap. After a single-step time evolution, if a particle is found overlapped with a wall surface, then collision rule will be applied to return the particle to the tetrahedron interior.

#### 3.1.3 How to interpolate the flow field at the position of my particle

If a particle k is found inside a tetrahedron (of index l), the flow field consisting of <sup>U</sup><sup>∞</sup>; ;Ω<sup>∞</sup>; �E<sup>∞</sup> ð Þ needs to be interpolated using values calculated adjacent locations. A proper choice of a set of these locations are the vertex points of the containing cell. Based on the definition of Ω<sup>∞</sup> and E<sup>∞</sup>, the basic quantities needed are U<sup>∞</sup> <sup>i</sup> and <sup>∂</sup>jUi � <sup>∂</sup>Ui=∂xj for <sup>i</sup> and <sup>j</sup> <sup>¼</sup> <sup>1</sup> � 3. (Note that pressure value is not required to calculate the generalized hydrodynamic forces <sup>F</sup>~H.)

Four vertices of a cell can be represented as r<sup>l</sup> ¼ xl; yl ; zl � � for <sup>l</sup> <sup>¼</sup> <sup>1</sup> � 4. A scalar quantity of interest S at each vertex point can be denoted as Sl. Then, the value of S at an arbitrary internal position r<sup>p</sup> ¼ xp; yp; zp � � can be calculated as a linear superposition of Sl and ξl:

$$\mathbf{S}\_p = \mathbf{S} \cdot \mathbf{f} = \sum\_{l=1}^{4} \mathbf{S}\_l \mathbf{f}\_l \tag{41}$$

For DHD simulations, Sp represents each element of the unidirectional velocity U<sup>∞</sup>, vortex Ω<sup>∞</sup>, and the rate of strain E<sup>∞</sup>, which are calculated using U<sup>∞</sup> <sup>i</sup> or its gradient ∂jUi.

#### 3.2 Collision rules

V <Vp (33)

1 ¼ ξ<sup>1</sup> þ ξ<sup>2</sup> þ ξ<sup>3</sup> þ ξ<sup>4</sup> (34)

<sup>6</sup><sup>V</sup> (35)

(36)

(39)

If a particle is very close to one of the triangular surfaces of the tetrahedron, then

the inequality check of Eq. (33) may not be done accurately. In this case, this location check can be extended to the nearest neighbor cells, especially one that

For r<sup>p</sup> within the tetrahedron, dividing both sides of Eq. (32) by V gives

<sup>ξ</sup><sup>i</sup> <sup>¼</sup> <sup>J</sup>ð Þ without <sup>i</sup>

which is solely determined by the internal position rp. A relationship between r<sup>p</sup>

1111 x<sup>1</sup> x<sup>2</sup> x<sup>3</sup> x<sup>4</sup> y<sup>1</sup> y<sup>2</sup> y<sup>3</sup> y<sup>4</sup> z<sup>1</sup> z<sup>2</sup> z<sup>3</sup> z<sup>4</sup>

where the inverse of the 4 � 4 matrix can be analytically available. Eq. (36) assumes that r<sup>p</sup> is located within the tetrahedron, but it can also use the particle location check as a better alternative to Eq. (33) because Eq. (36) requires one time solving of a 4 � 4 linear system, but Eq. (33) requires four times calculation of 4 � 4

3.1.2 How to calculate a distance between wall surfaces to the position of my particle

position of vertices 2 and 3 with respect to vertex 1, denoted as

Using the four vertices of ð Þ r1; r2; r3; r<sup>4</sup> of a tetrahedron cell, one can identify four triangular surfaces of S1ð Þ r2; r3; r<sup>4</sup> , S2ð Þ r1; r3; r<sup>4</sup> , S3ð Þ r1; r2; r<sup>4</sup> , and S4ð Þ r1; r2; r<sup>3</sup> . For example, let us consider S<sup>4</sup> having vertices of ð Þ r1; r2; r<sup>3</sup> . If we calculate relative

their cross product allows us to calculate a unit vector normal to surface S4:

<sup>n</sup><sup>4</sup> <sup>¼</sup> <sup>r</sup>2=<sup>1</sup> � <sup>r</sup>3=<sup>1</sup> r2=<sup>1</sup> � r3=<sup>1</sup>

� �

The particle position relative to r<sup>1</sup> is r<sup>p</sup>=<sup>1</sup> ¼ r<sup>p</sup> � r1. Then, the distance between

d ¼ n<sup>4</sup> � r<sup>p</sup> � � �

where the absolute value is necessary because n can direct inside or outside the tetrahedron volume, depending on choice of the reference position, r1. This wallparticle distance calculation is necessary when the surface S<sup>4</sup> is known as a wall

�

�

ξ1 ξ2 ξ3 ξ4

r2=<sup>1</sup> ¼ r<sup>2</sup> � r<sup>1</sup> (37) r3=<sup>1</sup> ¼ r<sup>3</sup> � r<sup>1</sup> (38)

� (40)

shares the triangular surface that the particle is closely located.

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

1 xp yp zp

where, for i ¼ 1 � 4,

and ξ is

Jacobian matrices.

surface S<sup>4</sup> and position r<sup>p</sup> is simply

12

Suppose there are two particles colliding each other, which are particle i and j located at r<sup>i</sup> and rj, translating with v<sup>i</sup> and v<sup>j</sup> and rotating with ω<sup>i</sup> and ωj, respectively. Relative position of particle i with respect to particle j is defined as rij ¼ r<sup>i</sup> � rj, and similarly the relative velocity of i to j is vij ¼ v<sup>i</sup> � vj. A normal vector from j to i is denoted as

$$m\_{ij} = \frac{r\_{ij}}{|r\_{ij}|} \tag{42}$$

After an instantaneous collision, the two particle have the following velocities [14]:

$$\boldsymbol{\sigma}'\_{i} = \boldsymbol{\sigma}\_{i} - \left(\frac{\mu\_{\vec{\boldsymbol{y}}}}{m\_{i}}\right) \left[ (\mathbf{1} + \boldsymbol{\varepsilon}^{n}) \mathbf{g}^{n}\_{\vec{\boldsymbol{y}}} + \frac{\mathbf{1} - \boldsymbol{\varepsilon}^{t}}{\mathbf{1} + \boldsymbol{1}/\tilde{\boldsymbol{f}} \mathbf{g}^{t}\_{\vec{\boldsymbol{y}}}} \right] \tag{43}$$

$$\mathbf{v}'\_{j} = \mathbf{v}\_{j} + \left(\frac{\mu\_{ij}}{m\_{j}}\right) \left[ (\mathbf{1} + \boldsymbol{\varepsilon}^{n}) \mathbf{g}^{n}\_{ij} + \frac{\mathbf{1} - \boldsymbol{\varepsilon}^{t}}{\mathbf{1} + \boldsymbol{1}/\boldsymbol{\tilde{J}} \mathbf{g}^{t}\_{ij}} \right] \tag{44}$$

$$
\alpha \boldsymbol{\nu}\_i' = \boldsymbol{\alpha}\_i + \left(\frac{\mu\_{ij}}{m\_i a\_i}\right) \left(\frac{\mathbf{1} - \boldsymbol{\epsilon}^t}{\mathbf{1} + \boldsymbol{\tilde{J}}}\right) \boldsymbol{n}\_{ij} \times \mathbf{g}\_{ij}^t \tag{45}
$$

$$
\mu a'\_j = \alpha\_{\hat{j}} + \left(\frac{\mu\_{ij}}{m\_j a\_j}\right) \left(\frac{1 - \varepsilon^t}{1 + \hat{J}}\right) \mathbf{n}\_{\hat{j}} \times \mathbf{g}^t\_{\hat{i}j} \tag{46}
$$

where 0 <sup>≤</sup>ε<sup>n</sup> <sup>≤</sup>1 and �1≤ε<sup>t</sup> <sup>≤</sup>1 are restitution coefficients in the normal and tangential directions, respectively, μij is a reduced mass defined as

$$
\mu\_{\vec{v}\,} = \frac{m\_i m\_j}{m\_i + m\_j},\tag{47}
$$

method to simulate the coupled fluid-particle motion within a reasonable time duration. A tetrahedron-based mesh is proposed to take the mathematical advantages of tetrahedron structure, consisting of four vertices and four triangular surfaces. Advantages of tetrahedron meshes include the following features: first, to test a location of particle within or outside a tetrahedron mesh-cell; second, to calculate a distance between a particle surface and a wall surface; third, to interpolate a fluid velocity and its gradient using those of values given at vertex locations; and finally to track each particle from one cell to the other by using a pre-built list of nearest neighbor cells. As DHD uses the SD algorithm for hydrodynamics of non-Brownian particles and Ito-Weiner process for random fluctuating forces, it can be used as a general particle hydrodynamic simulation method when it is coupled with CFD using specific mesh structures. Hexahedron-based meshes can be used for the same purpose with the intrinsic advantages of aligning grid edges to estimated streamline directions. When particles move in a channel of complex geometry, boundary surfaces can be better constructed using tetrahedron meshes. Open-sourced meshes include gmsh, tetgen, and netgen, which can import structure files, generate meshes, and export them to a CFD solver package. The current coupling algorithm of CFD and DHD is limited to cases that particle Reynolds number does not exceed 1.0, but this restriction can be avoided by considering dominant forces/torques exerted on particles and simulation time intervals as compared to the particle relaxation time. The new coupling method covered in this chapter may provide a new foundation in a coupled simulation of CFD and DHD including DEM.

A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD)

DOI: http://dx.doi.org/10.5772/intechopen.86895

This work was financially supported by the National R&D project of "Development of 1MW OTEC demonstration plant (4/5)" (PMS4080) funded by

The Navier–Stokes equation for a laminar flow is from Eq. (28):

�∇<sup>p</sup> <sup>þ</sup> <sup>η</sup>∇<sup>2</sup>

where δð Þr is the Dirac-Delta function, which indicates

ð V

where V<sup>0</sup> is a volume enclosing the origin r ¼ 0. The fundamental solution for v

Gð Þ x 8πη

U xð Þ¼ F �

U ¼ �Fδð Þ x (57) ∇ � U ¼ 0 (58)

δð Þ¼ x 6¼ 0 0 (59)

� Fδð Þr dV<sup>0</sup> ¼ �F (60)

(61)

the Ministry of Oceans and Fisheries of the Republic of Korea.

A.1 The Oseen tensor and Faxen law

Acknowledgements

A. Appendix

and p are

15

and gij is the relative velocity at the point of contact defined as

$$\mathbf{g}\_{\vec{\text{ij}}} = \boldsymbol{\nu}\_{\vec{\text{ij}}} - \left(a\_i \boldsymbol{a}\boldsymbol{\nu}\_i + a\_j \boldsymbol{a}\boldsymbol{\nu}\_j\right) \times \mathbf{n}\_{\vec{\text{ij}}} \tag{48}$$

whose normal and tangential components are

$$\mathbf{g}\_{\dot{\imath}}^{n} = \left(\mathbf{g}\_{\dot{\imath}\dot{\jmath}} \cdot \mathbf{n}\_{\dot{\imath}}\right) \mathbf{n}\_{\dot{\imath}}\tag{49}$$

$$\mathbf{g}\_{\vec{\imath}\vec{\jmath}}^{\vec{\imath}} = -\mathbf{n}\_{\vec{\imath}\vec{\jmath}} \times \left(\mathbf{n}\_{\vec{\imath}\vec{\jmath}} \times \mathbf{g}\_{\vec{\imath}\vec{\jmath}}\right) \tag{50}$$

If a collision between a wall surface and particle i occurs, then the wall can be represented as a stationary spherical particle j having infinite mass and radius, i.e., mj ! ∞, vj ! 0, and aj ! ∞:

$$\boldsymbol{\sigma}\_{i}^{\prime} = \boldsymbol{\sigma}\_{i} - \left[ (\mathbf{1} + \boldsymbol{\varepsilon}^{n}) \mathbf{g}\_{ij}^{n} + \left( \frac{\mathbf{1} - \boldsymbol{\varepsilon}^{t}}{\mathbf{1} + \boldsymbol{\tilde{J}} - \mathbf{1}} \right) \mathbf{g}\_{ij}^{t} \right] \tag{51}$$

$$
\boldsymbol{\alpha}\_{i}^{\prime} = \boldsymbol{\alpha}\_{i} + \left(\frac{\mathbf{1} - \boldsymbol{\varepsilon}^{t}}{\mathbf{1} + \boldsymbol{\tilde{J}}}\right) \frac{\boldsymbol{\mathfrak{n}}\_{\circ j} \times \mathbf{g}\_{i\circ}^{t}}{\boldsymbol{a}\_{i}} \tag{52}
$$

$$
\boldsymbol{\upsilon}'\_{\circ} = \boldsymbol{\upsilon}\_{\circ} = \mathbf{0} \tag{53}
$$

$$
\alpha\_j' = \alpha\_j = 0 \tag{54}
$$

where

$$\mathbf{g}\_{i\dot{j}} \rightarrow \mathbf{v}\_i - a\_i \left(\boldsymbol{\alpha}\_i \times \boldsymbol{n}\_{i\dot{j}}\right) \tag{55}$$

$$
\mathfrak{n}\_{\vec{\imath}} \to \mathfrak{n} \text{ to particle } i \tag{56}
$$

and n is the normal vector of the colliding wall surface inward to the liquid volume. During the collision of particle i with the wall, the wall is not moving so that ω<sup>0</sup> <sup>j</sup> ¼ ω<sup>j</sup> ¼ 0 and v<sup>0</sup> <sup>j</sup> ¼ v<sup>j</sup> ¼ 0.

#### 4. Concluding remarks

Each of computational fluid dynamics and particle hydrodynamics is a challenging research topic, as applied to real engineering problems. Movements of particles (viewed as small solid pieces) in a moving fluid require rigorous interfaces to couple the two strongly correlated dynamic events. When the ambient flow pushes suspended particles in a liquid, dynamic responses of particles to exerting fluid change the fluid motion at the next time step, which returns to the particles with modified magnitude and direction. This fluid-particle (or fluid–solid) interaction is under the regime of Newton's third law, i.e., action and reaction. Multi-body simulations including the fluid-particle interaction are generally a difficult task. In this chapter, we briefly reviewed the CFD and PHD literature and discussed a feasible

A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD) DOI: http://dx.doi.org/10.5772/intechopen.86895

method to simulate the coupled fluid-particle motion within a reasonable time duration. A tetrahedron-based mesh is proposed to take the mathematical advantages of tetrahedron structure, consisting of four vertices and four triangular surfaces. Advantages of tetrahedron meshes include the following features: first, to test a location of particle within or outside a tetrahedron mesh-cell; second, to calculate a distance between a particle surface and a wall surface; third, to interpolate a fluid velocity and its gradient using those of values given at vertex locations; and finally to track each particle from one cell to the other by using a pre-built list of nearest neighbor cells. As DHD uses the SD algorithm for hydrodynamics of non-Brownian particles and Ito-Weiner process for random fluctuating forces, it can be used as a general particle hydrodynamic simulation method when it is coupled with CFD using specific mesh structures. Hexahedron-based meshes can be used for the same purpose with the intrinsic advantages of aligning grid edges to estimated streamline directions. When particles move in a channel of complex geometry, boundary surfaces can be better constructed using tetrahedron meshes. Open-sourced meshes include gmsh, tetgen, and netgen, which can import structure files, generate meshes, and export them to a CFD solver package. The current coupling algorithm of CFD and DHD is limited to cases that particle Reynolds number does not exceed 1.0, but this restriction can be avoided by considering dominant forces/torques exerted on particles and simulation time intervals as compared to the particle relaxation time. The new coupling method covered in this chapter may provide a new foundation in a coupled simulation of CFD and DHD including DEM.

#### Acknowledgements

where 0 <sup>≤</sup>ε<sup>n</sup> <sup>≤</sup>1 and �1≤ε<sup>t</sup> <sup>≤</sup>1 are restitution coefficients in the normal and

<sup>μ</sup>ij <sup>¼</sup> mimj mi þ mj

gij ¼ vij � aiω<sup>i</sup> þ ajω<sup>j</sup>

ij ¼ gij � nij 

ij ¼ �nij � nij � gij

If a collision between a wall surface and particle i occurs, then the wall can be represented as a stationary spherical particle j having infinite mass and radius, i.e.,

ij þ

<sup>1</sup> � <sup>ε</sup><sup>t</sup> <sup>1</sup> <sup>þ</sup> <sup>~</sup><sup>J</sup>

gij ! v<sup>i</sup> � ai ω<sup>i</sup> � nij

and n is the normal vector of the colliding wall surface inward to the liquid volume. During the collision of particle i with the wall, the wall is not moving so

Each of computational fluid dynamics and particle hydrodynamics is a challenging research topic, as applied to real engineering problems. Movements of particles (viewed as small solid pieces) in a moving fluid require rigorous interfaces to couple the two strongly correlated dynamic events. When the ambient flow pushes suspended particles in a liquid, dynamic responses of particles to exerting fluid change the fluid motion at the next time step, which returns to the particles with modified magnitude and direction. This fluid-particle (or fluid–solid) interaction is under the regime of Newton's third law, i.e., action and reaction. Multi-body simulations including the fluid-particle interaction are generally a difficult task. In this chapter, we briefly reviewed the CFD and PHD literature and discussed a feasible

<sup>n</sup>ij � <sup>g</sup><sup>t</sup>

<sup>1</sup> � <sup>ε</sup><sup>t</sup> <sup>1</sup> <sup>þ</sup>~J�<sup>1</sup>

Þgt ij

<sup>j</sup> ¼ v<sup>j</sup> ¼ 0 (53)

<sup>j</sup> ¼ ω<sup>j</sup> ¼ 0 (54)

nij ! n to particle i (56)

(55)

ij ai

, (47)

nij (49)

(50)

(51)

(52)

� <sup>n</sup>ij (48)

tangential directions, respectively, μij is a reduced mass defined as

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

and gij is the relative velocity at the point of contact defined as

gn

<sup>i</sup> <sup>¼</sup> <sup>v</sup><sup>i</sup> � <sup>1</sup> <sup>þ</sup> <sup>ε</sup><sup>n</sup> ð Þg<sup>n</sup>

<sup>i</sup> ¼ ω<sup>i</sup> þ

v0

ω0

ω0

<sup>j</sup> ¼ v<sup>j</sup> ¼ 0.

gt

whose normal and tangential components are

v0

mj ! ∞, vj ! 0, and aj ! ∞:

<sup>j</sup> ¼ ω<sup>j</sup> ¼ 0 and v<sup>0</sup>

4. Concluding remarks

where

that ω<sup>0</sup>

14

This work was financially supported by the National R&D project of "Development of 1MW OTEC demonstration plant (4/5)" (PMS4080) funded by the Ministry of Oceans and Fisheries of the Republic of Korea.

#### A. Appendix

#### A.1 The Oseen tensor and Faxen law

The Navier–Stokes equation for a laminar flow is from Eq. (28):

$$-\nabla p + \eta \nabla^2 \mathbf{U} = -\mathbf{F}\delta(\mathbf{x})\tag{57}$$

$$\nabla \cdot \mathbf{U} = \mathbf{0} \tag{58}$$

where δð Þr is the Dirac-Delta function, which indicates

$$\delta(\mathfrak{x} \neq \mathbf{0}) = \mathbf{0} \tag{59}$$

$$\int\_{V} -F\delta(r) \mathbf{d}V' = -F \tag{60}$$

where V<sup>0</sup> is a volume enclosing the origin r ¼ 0. The fundamental solution for v and p are

$$U(\mathfrak{x}) = \mathbf{F} \cdot \frac{\mathcal{G}(\mathfrak{x})}{8\pi\eta} \tag{61}$$

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

$$p(\mathfrak{x}) = \mathbf{F} \cdot \frac{\mathcal{P}(\mathfrak{x})}{8\pi\eta} \tag{62}$$

References

335-341

27(3):720-733

1-138

[1] Metropolis N, Ulam S. The Monte Carlo method. Journal of the American Statistical Association. 1949;44(247):

DOI: http://dx.doi.org/10.5772/intechopen.86895

A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD)

[10] Ermak DL, McCammon JA.

[11] Hoogerbrugge P, Koelman J. Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhysics Letters. 1992;19:

[12] Koelman J, Hoogerbrugge P. Dynamic simulations of hard-sphere suspensions under steady shear. Europhysics Letters. 1993;21:363

[13] Brady JF, Bossis G. Stokesian dynamics. Annual Review of Fluid Mechanics. 1988;20:111-157

[14] Kim AS, Kim H-J. dissipative dynamics of granular materials. In: Granular Materials. Rijeka: InTechOpen;

[15] Kim AS. Constraint dissipative hydrodynamics (HydroRattle) algorithm for aggregate dynamics. Chemistry Letters. 2012;41(10):

[16] Kim AS. Dissipative hydrodynamics of rigid spherical particles. Chemistry Letters. 2012;41(10):1128-1130

Microhydrodynamics simulation of single-collector granular filtration. Chemistry Letters. 2012;41(10):

[18] Verlet L. Computer 'experiments' on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Physics Review. 1967;159(1):98-103

[19] Martys NS, Mountain RD. Velocity Verlet algorithm for dissipativeparticle-dynamics-based models of suspensions. Physical Review E. 1999;

2017. pp. 9-42

1285-1287

1288-1290

[17] Kim AS, Kang S-T.

59(3):3733 LP-3733736

155

Brownian dynamics with hydrodynamic interactions. The Journal of Chemical Physics. 1978;69(4):1352-1360

[2] Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. Equation of state calculations by fast computing machines. The Journal of Chemical Physics. 1953;21(6):1087-1092

[3] Wood WW, Parker FR. Monte carlo

equation of state of molecules interacting with the lennard-jones potential. i. a supercritical isotherm at about twice the critical temperature. The Journal of Chemical Physics. 1957;

[4] Hess J, Smith A. Calculation of potential flow about arbitrary bodies. Progress in Aerospace Sciences. 1967;8:

[5] Rubbert P, Saaris G. Review and evaluation of a three-dimensional lifting potential flow computational method for arbitrary configurations. In: 10th Aerospace Sciences Meeting. Reston, VA: American Institute of Aeronautics

[6] Maskew B. Prediction of subsonic aerodynamic characteristics: A case for low-order panel methods. Journal of

[7] Hyman JM, Knapp RJ, Scovel JC. High order finite volume approximations of differential operators on nonuniform grids. Physica D: Nonlinear Phenomena.

[8] Kubo R. The fluctuation-dissipation theorem. Reports on Progress in Physics. 1966;29:255-284

mouvement brownien. Comptes Rendus de l'Académie des Sciences (Paris).

[9] Langevin P. Sur la the'orie du

and Astronautics; 1972

Aircraft. 1982;19(2):157-163

1992;60(1–4):112-138

1908;146:530-533

17

The Oseen tensor Gð Þ x for the fluid velocity is given by

$$\mathcal{G}\_{\vec{v}\vec{j}} = \frac{1}{r}\delta\_{\vec{v}} + \frac{1}{r^3}\mathfrak{x}\_i\mathfrak{x}\_{\vec{j}} \tag{63}$$

which is independent of fluid properties. The Oseen tensor for the pressure p is

$$\mathcal{P}\_j(\mathfrak{x}) = 2\eta \frac{\mathfrak{x}\_j}{r^3} + \mathcal{P}\_j^{\infty} \tag{64}$$

where P<sup>∞</sup> <sup>j</sup> is a constant at the ambient condition. The Faxen laws determine the hydrodynamic force and torque, especially, exerted on a sphere of radius a, moving with the linear and angular velocities of u and ω:

$$\mathbf{F} = 6\pi\eta a \left[ \left( \mathbf{1} + \frac{a^2}{6} \nabla^2 \right) \mathbf{U}(\mathbf{x}) \right]\_{\mathbf{x}=0} - 6\pi\eta a \mathbf{u} \tag{65}$$

$$T = 8\pi\eta a^{\frac{3}{4}}[\Omega(\mathfrak{x}) - \mathfrak{o}]\_{\mathfrak{x}=\mathfrak{o}}\tag{66}$$

which indicates that the Stokes flow requires a quadrupole a<sup>2</sup>F∇<sup>2</sup> δð Þ x in addition to a monopole of �6πηau, which is a drag on a sphere undergoing steady translation.

#### Author details

Albert S. Kim<sup>1</sup> \* and Hyeon-Ju Kim<sup>2</sup>

1 Civil and Environmental Engineering, University of Hawai'i at Manoa, USA

2 Offshore Plant and Marine Energy Research Division, Korea Research Institute of Ships and Ocean Engineering, Korea

\*Address all correspondence to: albertsk@hawaii.edu

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A Coupling Algorithm of Computational Fluid and Particle Dynamics (CFPD) DOI: http://dx.doi.org/10.5772/intechopen.86895

#### References

pð Þ¼ x F �

which is independent of fluid properties. The Oseen tensor for the pressure p is

hydrodynamic force and torque, especially, exerted on a sphere of radius a, moving

xj <sup>r</sup><sup>3</sup> <sup>þ</sup> <sup>P</sup><sup>∞</sup>

<sup>j</sup> is a constant at the ambient condition. The Faxen laws determine the

U xð Þ

x¼0

<sup>G</sup>ij <sup>¼</sup> <sup>1</sup> r δij þ 1

Pjð Þ¼ x 2η

a2 <sup>6</sup> <sup>∇</sup><sup>2</sup> 

<sup>T</sup> <sup>¼</sup> <sup>8</sup>πηa<sup>3</sup>

which indicates that the Stokes flow requires a quadrupole a<sup>2</sup>F∇<sup>2</sup>

to a monopole of �6πηau, which is a drag on a sphere undergoing steady

1 Civil and Environmental Engineering, University of Hawai'i at Manoa, USA

2 Offshore Plant and Marine Energy Research Division, Korea Research Institute of

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

The Oseen tensor Gð Þ x for the fluid velocity is given by

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

with the linear and angular velocities of u and ω:

\* and Hyeon-Ju Kim<sup>2</sup>

\*Address all correspondence to: albertsk@hawaii.edu

Ships and Ocean Engineering, Korea

provided the original work is properly cited.

F ¼ 6πηa 1 þ

where P<sup>∞</sup>

translation.

Author details

Albert S. Kim<sup>1</sup>

16

Pð Þ x 8πη

(62)

<sup>r</sup><sup>3</sup> xixj (63)

<sup>j</sup> (64)

� 6πηau (65)

δð Þ x in addition

½ � <sup>Ω</sup>ð Þ� <sup>x</sup> <sup>ω</sup> <sup>x</sup>¼<sup>0</sup> (66)

[1] Metropolis N, Ulam S. The Monte Carlo method. Journal of the American Statistical Association. 1949;44(247): 335-341

[2] Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. Equation of state calculations by fast computing machines. The Journal of Chemical Physics. 1953;21(6):1087-1092

[3] Wood WW, Parker FR. Monte carlo equation of state of molecules interacting with the lennard-jones potential. i. a supercritical isotherm at about twice the critical temperature. The Journal of Chemical Physics. 1957; 27(3):720-733

[4] Hess J, Smith A. Calculation of potential flow about arbitrary bodies. Progress in Aerospace Sciences. 1967;8: 1-138

[5] Rubbert P, Saaris G. Review and evaluation of a three-dimensional lifting potential flow computational method for arbitrary configurations. In: 10th Aerospace Sciences Meeting. Reston, VA: American Institute of Aeronautics and Astronautics; 1972

[6] Maskew B. Prediction of subsonic aerodynamic characteristics: A case for low-order panel methods. Journal of Aircraft. 1982;19(2):157-163

[7] Hyman JM, Knapp RJ, Scovel JC. High order finite volume approximations of differential operators on nonuniform grids. Physica D: Nonlinear Phenomena. 1992;60(1–4):112-138

[8] Kubo R. The fluctuation-dissipation theorem. Reports on Progress in Physics. 1966;29:255-284

[9] Langevin P. Sur la the'orie du mouvement brownien. Comptes Rendus de l'Académie des Sciences (Paris). 1908;146:530-533

[10] Ermak DL, McCammon JA. Brownian dynamics with hydrodynamic interactions. The Journal of Chemical Physics. 1978;69(4):1352-1360

[11] Hoogerbrugge P, Koelman J. Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhysics Letters. 1992;19: 155

[12] Koelman J, Hoogerbrugge P. Dynamic simulations of hard-sphere suspensions under steady shear. Europhysics Letters. 1993;21:363

[13] Brady JF, Bossis G. Stokesian dynamics. Annual Review of Fluid Mechanics. 1988;20:111-157

[14] Kim AS, Kim H-J. dissipative dynamics of granular materials. In: Granular Materials. Rijeka: InTechOpen; 2017. pp. 9-42

[15] Kim AS. Constraint dissipative hydrodynamics (HydroRattle) algorithm for aggregate dynamics. Chemistry Letters. 2012;41(10): 1285-1287

[16] Kim AS. Dissipative hydrodynamics of rigid spherical particles. Chemistry Letters. 2012;41(10):1128-1130

[17] Kim AS, Kang S-T. Microhydrodynamics simulation of single-collector granular filtration. Chemistry Letters. 2012;41(10): 1288-1290

[18] Verlet L. Computer 'experiments' on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Physics Review. 1967;159(1):98-103

[19] Martys NS, Mountain RD. Velocity Verlet algorithm for dissipativeparticle-dynamics-based models of suspensions. Physical Review E. 1999; 59(3):3733 LP-3733736

Chapter 2

Flows

Abstract

Santiago Laín

Response Behavior of

Nonspherical Particles in

tions that maximize the cross section exposed to the flow.

model to describe the turbulent dynamics of the carrier phase.

response behavior, preferential orientation

1. Introduction

19

Keywords: kinematic simulations, Lagrangian tracking, nonspherical particles,

Nowadays, the use of numerical simulation techniques to assist the development and optimization of industrial processes dealing with turbulent multiphase flow has been included as one more step in their layout. Examples of them include pneumatic conveying, fluidized bed reactors, cyclones, classifiers, or flow mixers. Industrial sectors where such processes are important are the chemical, food, or paper industries as well as electric energy production. Due to the complexity of the involved flow, a great majority of simulations are carried out under Reynoldsaveraged Navier-Stokes (RANS) in connection with an appropriate turbulence

Two main frames are employed for the description of complex multiphase flows: the two-fluid model or Euler-Euler and the discrete particle models or Euler-Lagrange. In both of them, particles are approximated as point masses being

Homogeneous Isotropic Turbulent

In this study, the responsiveness of nonspherical particles, specifically ellipsoids and cylinders, in homogeneous and isotropic turbulence is investigated through kinematic simulations of the fluid velocity field. Particle tracking in such flow field includes not only the translational and rotational components but also the orientation through the Euler angles and parameters. Correlations for the flow coefficients, forces and torques, of the nonspherical particles in the range of intermediate Reynolds number are obtained from the literature. The Lagrangian time autocorrelation function, the translational and rotational particle response, and preferential orientation of the nonspherical particles in the turbulent flow are studied as function of their shape and inertia. As a result, particle autocorrelation functions, translational and rotational, decrease with aspect ratio, and particle linear root mean square velocity increases with aspect ratio, while rotational root mean square velocity first increases, reaches a maximum around aspect ratio 2, and then decreases again. Finally, cylinders do not present any preferential orientation in homogeneous isotropic turbulence, but ellipsoids do, resulting in preferred orienta-

[20] Hockney RW, Eastwood JW. Computer Simulation Using Particles. New York: Adam Hilger; 1988

[21] Hockney RW. The potential calculation and some applications. In: Methods in Computational Physics. Vol. 9. New York: Academic Press; 1970

[22] Stokes GG. On the effect of internal friction of fluids on the motion of pendulums. Transactions of the Cambridge Philosophical Society. 1851;9:1-106

[23] Wiener N. Differential space. Journal of Mathematical Physics. 1923;58:31-174

[24] Ito M. An extension of nonlinear evolution equations of the K-dV (mKdV) type to higher orders. Journal of the Physical Society of Japan. 1980;49(2): 771-778

[25] Anderson E. LAPACK Users' Guide. Philadelphia, PA: Society for Industrial and Applied Mathematics; 1987

[26] Blackford LS, Choi J, Cleary A, D'Azevedo E, Demmel J, Chillon I, et al. ScaLAPACK User's Guide. Philadelphia, PA: Society for Industrial and Applied Mathematics; 1997

#### Chapter 2

[20] Hockney RW, Eastwood JW. Computer Simulation Using Particles.

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

New York: Adam Hilger; 1988

[21] Hockney RW. The potential calculation and some applications. In: Methods in Computational Physics. Vol. 9. New York: Academic Press; 1970

1851;9:1-106

1923;58:31-174

771-778

18

[22] Stokes GG. On the effect of internal friction of fluids on the motion of pendulums. Transactions of the Cambridge Philosophical Society.

[23] Wiener N. Differential space. Journal of Mathematical Physics.

[24] Ito M. An extension of nonlinear evolution equations of the K-dV (mKdV) type to higher orders. Journal of the Physical Society of Japan. 1980;49(2):

[25] Anderson E. LAPACK Users' Guide. Philadelphia, PA: Society for Industrial and Applied Mathematics; 1987

[26] Blackford LS, Choi J, Cleary A, D'Azevedo E, Demmel J, Chillon I, et al. ScaLAPACK User's Guide. Philadelphia, PA: Society for Industrial and Applied

Mathematics; 1997

## Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows

Santiago Laín

### Abstract

In this study, the responsiveness of nonspherical particles, specifically ellipsoids and cylinders, in homogeneous and isotropic turbulence is investigated through kinematic simulations of the fluid velocity field. Particle tracking in such flow field includes not only the translational and rotational components but also the orientation through the Euler angles and parameters. Correlations for the flow coefficients, forces and torques, of the nonspherical particles in the range of intermediate Reynolds number are obtained from the literature. The Lagrangian time autocorrelation function, the translational and rotational particle response, and preferential orientation of the nonspherical particles in the turbulent flow are studied as function of their shape and inertia. As a result, particle autocorrelation functions, translational and rotational, decrease with aspect ratio, and particle linear root mean square velocity increases with aspect ratio, while rotational root mean square velocity first increases, reaches a maximum around aspect ratio 2, and then decreases again. Finally, cylinders do not present any preferential orientation in homogeneous isotropic turbulence, but ellipsoids do, resulting in preferred orientations that maximize the cross section exposed to the flow.

Keywords: kinematic simulations, Lagrangian tracking, nonspherical particles, response behavior, preferential orientation

#### 1. Introduction

Nowadays, the use of numerical simulation techniques to assist the development and optimization of industrial processes dealing with turbulent multiphase flow has been included as one more step in their layout. Examples of them include pneumatic conveying, fluidized bed reactors, cyclones, classifiers, or flow mixers. Industrial sectors where such processes are important are the chemical, food, or paper industries as well as electric energy production. Due to the complexity of the involved flow, a great majority of simulations are carried out under Reynoldsaveraged Navier-Stokes (RANS) in connection with an appropriate turbulence model to describe the turbulent dynamics of the carrier phase.

Two main frames are employed for the description of complex multiphase flows: the two-fluid model or Euler-Euler and the discrete particle models or Euler-Lagrange. In both of them, particles are approximated as point masses being transported in the carrier phase flow field; the solution of the flow around individual elements is usually too expensive and cannot be afforded. In the two-fluid model, both phases are conceived as two interpenetrating continua [1] whose properties are described by sets of partial differential equations. In the Euler-Lagrange approach, the discrete elements are considered as individual objects whose dynamics is governed by a Lagrangian motion equation. Therefore, to obtain the discrete phase variables in the computational domain, a large enough number of discrete element trajectories must be computed. On each particle, appropriate forces act reflecting the various microprocesses taking place at the element scale such as fluid-particle turbulent interaction, particle-(rough) wall interactions, and interparticle collisions [2]. Such technique is especially appropriate for the description of disperse multiphase flow, where usually particles have a size distribution, in confined domains where particle-wall collisions play a predominant role as pneumatic conveying, separation, and classification processes. Both techniques, twofluid model and Euler-Lagrange, have been applied mainly considering spherical particles. This means that the forces due to the flow (drag and lift) as well as the microprocesses modeling, wall-particle and inter-particle interactions, are assumed to be for spherical-shaped elements [3]. In practical situations, however, nonspherical particles are encountered, either of irregular shape, either with welldefined shapes (fibers or granulates). For example, the paper industry uses large amounts of turbulent liquid to handle and transport the fibers that compose the paper pulp. Besides, such particles in the flow experience particle Reynolds numbers larger than one, Re > 1. For such particles, the most relevant transport mechanisms such as aerodynamic transport, wall-particle interactions, and interparticle collisions are substantially different than those for spherical particles.

options appear to be dominant: the spherical particle equivalent diameter dp, and the sphericity, defined as the ratio between the surface of the spherical particle equivalent diameter and the actual surface of the nonspherical particle. None of such correlations takes into account the dependence of the drag coefficient with the particle orientation in the flow. There exist some correlations that consider such dependence [15–17] and, therefore, they are appropriated to be implemented in a Lagrangian computation scheme. Nevertheless, corresponding results for lift and pitching torque coefficients are still not equally available. One of the first results for

Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows

DOI: http://dx.doi.org/10.5772/intechopen.81045

the different coefficients in terms of the orientation of elliptic particles was obtained by Hölzer and Sommerfeld [18] using the lattice Boltzmann method (LBM). Vakil and Green [19] used DNS to study the drag and lift coefficients of cylindrical particles depending on their orientation and aspect ratio, for Reynolds number up to 40, providing a correlation for them. In such work, the underlying flow field was assumed to be uniform and the flow around the particle was

number of 300.

21

completely resolved. More recently, Zastawny et al. [20] applied DNS in the frame of the implicit mirroring immersed boundary (MIB) method to obtain the flow coefficients for four different ellipsoids. The authors provide specific correlations for the drag, lift, pitching torque, and rotational torque coefficients depending on the particle Reynolds number and orientation but without including the effect of the aspect ratio. The covered Reynolds number range was up to 300. Ouchene et al. [21] determined with DNS the drag, lift, and pitching torque coefficients for prolate ellipsoids with aspect ratio up to 32 and adjusted their results to proper correlations that include the effects of particle orientation and aspect ratio up to a Reynolds

The first numerical computations of very small nonspherical particles in pseudoturbulent flow were performed by Fan and Ahmadi [22] and Olson [23]. The hydrodynamic forces and torques were computed by the theoretical coefficients of the Stokes regime. Olson [23] estimated the time step for the translation and rotation motions in function of the fiber length, obtaining the corresponding dispersion coefficients. Fan and Ahmadi [22] showed that the dispersion of both, translation and rotation, was reduced with the fiber length. However, Olson [23] found a different result in the case of ellipsoidal particles. Lin et al. [24] investigated numerically the distribution of the orientation of the fibers in a developing mixing layer, comparing the obtained results with experiments. The fiber length was smaller than the Kolmogorov scale, so they employed the forces due to the flow of the Stokes regime. Zhang et al. [25], Mortensen et al. [26], and Marchioli et al. [27] studied the transport and deposition of ellipsoidal particles in a turbulent channel flow using direct numerical simulation (DNS). Again the hydrodynamic forces and torques were computed with Stokes regime expressions. Beyond the Stokes regime, van Wachem et al. [28] and Ouchene et al. [29] studied a turbulent channel flow laden with ellipsoidal particles using LES and DNS, respectively, employing the

flow coefficients developed by themselves in previous works.

Rosendahl group developed a model for the numerical computation of cylindrical and superellipsoidal particles in laminar and turbulent flows in the intermediate Reynolds numbers regime [30–32]. Particle angular velocity and orientation were computed by means of the Euler parameters. Using a linear relationship between the drag coefficient and the ellipsoid parameters, it was possible to establish a correlation valid up to Reynolds numbers of 1000. To estimate the influence of the orientation, a correlation between the maximum (90°) and minimum (0°) drag was employed. Drag force was calculated using the projected area perpendicularly to the flow. The lift force was expressed in function of the drag coefficient and particle orientation. Other lift forces, such as those due to the fluid velocity gradients or particle rotation, were not considered. The study case was a combustion chamber

With the objective of performing the numerical simulation of turbulent flows laden with nonspherical particles, additional information about the forces and torques due to flow (drag and lift forces and pitching and rotational torques due to the shear flow and particle rotation) is needed. It is known that for regular nonspherical particles, that is, ellipsoids or fibers, such forces depend on particle orientation with respect to the flow. For instance, fiber orientation plays a major role in chemical processes as injection, compression molding, or extrusion in which the mechanical properties of the suspensions are determined by the orientation distribution.

For the Stokes regime, particle Reynolds number much lower than 1, the behavior of the nonspherical particles can be determined by analytical methods. Forces and torques acting on an ellipsoidal particle were analytically computed by Jeffery [4]. In a series of papers, Brenner determined the forces due to the flow acting on arbitrary-shaped nonspherical particles in the Stokes regime under different flow configuration by means of theoretical methods [5]. In the creeping flow regime, also with particle Reynolds number much lower than 1, Bläser [6] computed the forces acting of the surface on an ellipsoid in free motion for different flow situations, which allow him to suggest a simple criterion for particle breakup.

The drag coefficient for particle Reynolds number higher than 1 must be obtained by experiments, physical or numerical, as the analytical methods are not applicable any more.

The experimental studies to determine the drag coefficients for nonspherical particles employ wind tunnels or sedimentation vessels. For a moderately wide particle Reynolds number range, there exist results for thin discs [7], isometric irregular particles [8], cylinders and plates [9], discs [10], and discs and cylinders [11]. Drag coefficients were developed in all cases only for certain particle orientations. Compiling such results, different correlations have been developed in terms of particle shape [12–14]. As representative parameter of the particle shape, two

#### Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows DOI: http://dx.doi.org/10.5772/intechopen.81045

options appear to be dominant: the spherical particle equivalent diameter dp, and the sphericity, defined as the ratio between the surface of the spherical particle equivalent diameter and the actual surface of the nonspherical particle. None of such correlations takes into account the dependence of the drag coefficient with the particle orientation in the flow. There exist some correlations that consider such dependence [15–17] and, therefore, they are appropriated to be implemented in a Lagrangian computation scheme. Nevertheless, corresponding results for lift and pitching torque coefficients are still not equally available. One of the first results for the different coefficients in terms of the orientation of elliptic particles was obtained by Hölzer and Sommerfeld [18] using the lattice Boltzmann method (LBM). Vakil and Green [19] used DNS to study the drag and lift coefficients of cylindrical particles depending on their orientation and aspect ratio, for Reynolds number up to 40, providing a correlation for them. In such work, the underlying flow field was assumed to be uniform and the flow around the particle was completely resolved. More recently, Zastawny et al. [20] applied DNS in the frame of the implicit mirroring immersed boundary (MIB) method to obtain the flow coefficients for four different ellipsoids. The authors provide specific correlations for the drag, lift, pitching torque, and rotational torque coefficients depending on the particle Reynolds number and orientation but without including the effect of the aspect ratio. The covered Reynolds number range was up to 300. Ouchene et al. [21] determined with DNS the drag, lift, and pitching torque coefficients for prolate ellipsoids with aspect ratio up to 32 and adjusted their results to proper correlations that include the effects of particle orientation and aspect ratio up to a Reynolds number of 300.

The first numerical computations of very small nonspherical particles in pseudoturbulent flow were performed by Fan and Ahmadi [22] and Olson [23]. The hydrodynamic forces and torques were computed by the theoretical coefficients of the Stokes regime. Olson [23] estimated the time step for the translation and rotation motions in function of the fiber length, obtaining the corresponding dispersion coefficients. Fan and Ahmadi [22] showed that the dispersion of both, translation and rotation, was reduced with the fiber length. However, Olson [23] found a different result in the case of ellipsoidal particles. Lin et al. [24] investigated numerically the distribution of the orientation of the fibers in a developing mixing layer, comparing the obtained results with experiments. The fiber length was smaller than the Kolmogorov scale, so they employed the forces due to the flow of the Stokes regime. Zhang et al. [25], Mortensen et al. [26], and Marchioli et al. [27] studied the transport and deposition of ellipsoidal particles in a turbulent channel flow using direct numerical simulation (DNS). Again the hydrodynamic forces and torques were computed with Stokes regime expressions. Beyond the Stokes regime, van Wachem et al. [28] and Ouchene et al. [29] studied a turbulent channel flow laden with ellipsoidal particles using LES and DNS, respectively, employing the flow coefficients developed by themselves in previous works.

Rosendahl group developed a model for the numerical computation of cylindrical and superellipsoidal particles in laminar and turbulent flows in the intermediate Reynolds numbers regime [30–32]. Particle angular velocity and orientation were computed by means of the Euler parameters. Using a linear relationship between the drag coefficient and the ellipsoid parameters, it was possible to establish a correlation valid up to Reynolds numbers of 1000. To estimate the influence of the orientation, a correlation between the maximum (90°) and minimum (0°) drag was employed. Drag force was calculated using the projected area perpendicularly to the flow. The lift force was expressed in function of the drag coefficient and particle orientation. Other lift forces, such as those due to the fluid velocity gradients or particle rotation, were not considered. The study case was a combustion chamber

transported in the carrier phase flow field; the solution of the flow around individual elements is usually too expensive and cannot be afforded. In the two-fluid model, both phases are conceived as two interpenetrating continua [1] whose properties are described by sets of partial differential equations. In the Euler-Lagrange approach, the discrete elements are considered as individual objects whose dynamics is governed by a Lagrangian motion equation. Therefore, to obtain the discrete phase variables in the computational domain, a large enough number of discrete element trajectories must be computed. On each particle, appropriate forces act reflecting the various microprocesses taking place at the element scale such as fluid-particle turbulent interaction, particle-(rough) wall interactions, and interparticle collisions [2]. Such technique is especially appropriate for the description of disperse multiphase flow, where usually particles have a size distribution, in confined domains where particle-wall collisions play a predominant role as pneumatic conveying, separation, and classification processes. Both techniques, twofluid model and Euler-Lagrange, have been applied mainly considering spherical particles. This means that the forces due to the flow (drag and lift) as well as the microprocesses modeling, wall-particle and inter-particle interactions, are assumed

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

to be for spherical-shaped elements [3]. In practical situations, however,

collisions are substantially different than those for spherical particles.

which allow him to suggest a simple criterion for particle breakup.

The drag coefficient for particle Reynolds number higher than 1 must be obtained by experiments, physical or numerical, as the analytical methods are not

The experimental studies to determine the drag coefficients for nonspherical particles employ wind tunnels or sedimentation vessels. For a moderately wide particle Reynolds number range, there exist results for thin discs [7], isometric irregular particles [8], cylinders and plates [9], discs [10], and discs and cylinders [11]. Drag coefficients were developed in all cases only for certain particle orientations. Compiling such results, different correlations have been developed in terms of particle shape [12–14]. As representative parameter of the particle shape, two

distribution.

applicable any more.

20

the shear flow and particle rotation) is needed. It is known that for regular nonspherical particles, that is, ellipsoids or fibers, such forces depend on particle orientation with respect to the flow. For instance, fiber orientation plays a major role in chemical processes as injection, compression molding, or extrusion in which the mechanical properties of the suspensions are determined by the orientation

nonspherical particles are encountered, either of irregular shape, either with welldefined shapes (fibers or granulates). For example, the paper industry uses large amounts of turbulent liquid to handle and transport the fibers that compose the paper pulp. Besides, such particles in the flow experience particle Reynolds numbers larger than one, Re > 1. For such particles, the most relevant transport mechanisms such as aerodynamic transport, wall-particle interactions, and interparticle

With the objective of performing the numerical simulation of turbulent flows laden with nonspherical particles, additional information about the forces and torques due to flow (drag and lift forces and pitching and rotational torques due to

For the Stokes regime, particle Reynolds number much lower than 1, the behavior of the nonspherical particles can be determined by analytical methods. Forces and torques acting on an ellipsoidal particle were analytically computed by Jeffery [4]. In a series of papers, Brenner determined the forces due to the flow acting on arbitrary-shaped nonspherical particles in the Stokes regime under different flow configuration by means of theoretical methods [5]. In the creeping flow regime, also with particle Reynolds number much lower than 1, Bläser [6] computed the forces acting of the surface on an ellipsoid in free motion for different flow situations,

with straw particles, which were quite well approximated by cylinders. It was found that straw particles were better dispersed than spheres [30], a fact that properly illustrates the importance of the correct modeling of nonspherical particles motion. In a later work [32], other forces such as added mass and pressure force were also included. Drag coefficient was computed using the Ganser [15] correlation, making it possible to numerically compute the biomass combustion chamber.

frame, the z<sup>0</sup> axis coincides with the particle symmetry axis and its position with

Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows

Goldstein [33] gives the transformation between the comoving and particle coordinate systems, which is frequently employed in regular nonspherical particle

A is the orthogonal matrix that performs the transformation. Its components are the direction cosines of the particle axes in the comoving frame, written in function of Euler angles ð Þ θ; ϕ; ψ . Such Euler angles are defined according to the

cos ψ cos ϕ � cos θ sin ϕ sin ψ cos ψ sin ϕ � cos θ cos ϕ sin ψ sin ψ sin θ � sin ψ cos ϕ � cos θ sin ϕ cos ψ � sin ψ sin ϕ þ cos θ cos ϕ cos ψ cos ψ sin θ

The time evolution of such Euler angles depends on the particle angular velocity regarding the particle frame axes. However, there is a difficulty in the sense that such time evolution equations present an unavoidable singularity. Therefore, instead of the Euler angles, the Euler parameters ð Þ ε1; ε2; ε3; η are used instead:

sin θ sin ϕ � sin θ cos ϕ cos θ

; <sup>ε</sup><sup>3</sup> <sup>¼</sup> sin <sup>ϕ</sup> <sup>þ</sup> <sup>ψ</sup>

� � <sup>2</sup>ð Þ <sup>ε</sup>1ε<sup>2</sup> <sup>þ</sup> <sup>ε</sup>3<sup>η</sup> <sup>2</sup>ð Þ <sup>ε</sup>1ε<sup>3</sup> � <sup>ε</sup>2<sup>η</sup>

<sup>2</sup>ð Þ <sup>ε</sup>1ε<sup>3</sup> <sup>þ</sup> <sup>ε</sup>2<sup>η</sup> <sup>2</sup>ð Þ <sup>ε</sup>3ε<sup>2</sup> � <sup>ε</sup>1<sup>η</sup> <sup>1</sup> � <sup>2</sup> <sup>ε</sup><sup>2</sup>

In the present study, the initial particle orientations are assigned by means of the Euler angles. From them, the corresponding Euler parameters are computed by Eq. (3), and with them, the initial transformation matrix is evaluated using Eq. (4). The Euler parameters evolve in time following Eq. (5), where the particle angular

<sup>1</sup> <sup>þ</sup> <sup>ε</sup><sup>2</sup> 3

ηω<sup>x</sup><sup>0</sup> � ε3ω<sup>y</sup><sup>0</sup> þ ε2ω<sup>z</sup><sup>0</sup> ε3ω<sup>x</sup><sup>0</sup> þ ηω<sup>y</sup><sup>0</sup> � ε1ω<sup>z</sup><sup>0</sup> �ε2ω<sup>x</sup><sup>0</sup> þ ε1ω<sup>y</sup><sup>0</sup> þ ηω<sup>z</sup><sup>0</sup> �ε1ω<sup>x</sup><sup>0</sup> � ε2ω<sup>y</sup><sup>0</sup> � ε3ω<sup>z</sup><sup>0</sup>

<sup>2</sup> cos <sup>θ</sup> 2

� � <sup>2</sup>ð Þ <sup>ε</sup>3ε<sup>2</sup> <sup>þ</sup> <sup>ε</sup>1<sup>η</sup>

<sup>1</sup> <sup>þ</sup> <sup>ε</sup><sup>2</sup> 2 � �

; <sup>η</sup> <sup>¼</sup> cos <sup>ϕ</sup> <sup>þ</sup> <sup>ψ</sup>

3 7

<sup>5</sup> (4)

x<sup>0</sup> ¼ A � x<sup>00</sup> (1)

3 7 5

(2)

<sup>2</sup> cos <sup>θ</sup> 2 (3)

(5)

respect to the comoving frame determines particle orientation.

DOI: http://dx.doi.org/10.5772/intechopen.81045

tracking [31].

A ¼

2 6 4

<sup>ε</sup><sup>1</sup> <sup>¼</sup> cos <sup>ϕ</sup> � <sup>ψ</sup>

<sup>2</sup> sin <sup>θ</sup> 2

A ¼

2.2 Particle motion equations

written as:

23

2 6 4

; <sup>ε</sup><sup>2</sup> <sup>¼</sup> sin <sup>ϕ</sup> � <sup>ψ</sup>

And the transformation matrix A is written as [33]:

<sup>2</sup> <sup>þ</sup> <sup>ε</sup><sup>2</sup> 3

<sup>2</sup>ð Þ <sup>ε</sup>1ε<sup>2</sup> � <sup>ε</sup>3<sup>η</sup> <sup>1</sup> � <sup>2</sup> <sup>ε</sup><sup>2</sup>

velocities are expressed in the particle frame of reference x<sup>0</sup> ¼ x<sup>0</sup> y<sup>0</sup> z<sup>0</sup> ½ �.

The nonspherical particle motion equations in a general fluid flow [34] are

<sup>1</sup> � <sup>2</sup> <sup>ε</sup><sup>2</sup>

dε<sup>1</sup> dt dε<sup>2</sup> dt dε<sup>3</sup> dt dη dt

¼ 1 2

<sup>2</sup> sin <sup>θ</sup> 2

x-convention of [33]:

This contribution aims to study the motion of nonspherical particles immersed in homogeneous isotropic turbulent (HIT) velocity fields built from kinematic simulation at moderate Reynolds numbers. Computations were performed in a tailored in-house code. Properties analyzed include the Lagrangian time autocorrelation function, the translational and rotational particle response, and preferential orientation of the nonspherical particles in the turbulent flow, all of them in terms of particle aspect ratio and inertia.

#### 2. Governing equations

#### 2.1 Coordinate systems

To build the trajectory of a regular nonspherical particle, it is necessary to solve for its translational as well as rotational motion. However, whereas translation is solved in an inertial frame, rotation is solved referred to the so-called particle frame. Thus, the relevant coordinate frames and the transformations between them have to be introduced.

Figure 1 illustrates, in the case of a cylindrical particle, the employed coordinate systems: x ¼ ½ � xyz is the inertial frame; x<sup>0</sup> ¼ x<sup>0</sup> y<sup>0</sup> z<sup>0</sup> ½ � is the particle frame, whose origin is in the particle center of mass and its axes are the particle principal axes; and x 00 ¼ x 00 y 00 z <sup>00</sup> is the comoving frame, which has its origin at the same point than particle frame but its axes are parallel to the inertial frame axes. In the particle

Figure 1. Illustration of a cylindrical particle and the employed coordinate systems.

Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows DOI: http://dx.doi.org/10.5772/intechopen.81045

frame, the z<sup>0</sup> axis coincides with the particle symmetry axis and its position with respect to the comoving frame determines particle orientation.

Goldstein [33] gives the transformation between the comoving and particle coordinate systems, which is frequently employed in regular nonspherical particle tracking [31].

$$\mathbf{x}' = \mathbf{A} \cdot \mathbf{x}'' \tag{1}$$

A is the orthogonal matrix that performs the transformation. Its components are the direction cosines of the particle axes in the comoving frame, written in function of Euler angles ð Þ θ; ϕ; ψ . Such Euler angles are defined according to the x-convention of [33]:

$$\mathbf{A} = \begin{bmatrix} \cos\psi\cos\phi - \cos\theta\sin\phi\sin\psi & \cos\psi\sin\phi - \cos\theta\cos\phi\sin\psi & \sin\psi\sin\theta \\ -\sin\psi\cos\phi - \cos\theta\sin\phi\cos\psi & -\sin\psi\sin\phi + \cos\theta\cos\phi\cos\psi & \cos\psi\sin\theta \\ \sin\theta\sin\phi & -\sin\theta\cos\phi & \cos\theta \end{bmatrix} \tag{2}$$

The time evolution of such Euler angles depends on the particle angular velocity regarding the particle frame axes. However, there is a difficulty in the sense that such time evolution equations present an unavoidable singularity. Therefore, instead of the Euler angles, the Euler parameters ð Þ ε1; ε2; ε3; η are used instead:

$$\varepsilon\_1 = \cos\frac{\phi - \psi}{2}\sin\frac{\theta}{2}; \varepsilon\_2 = \sin\frac{\phi - \psi}{2}\sin\frac{\theta}{2}; \varepsilon\_3 = \sin\frac{\phi + \psi}{2}\cos\frac{\theta}{2}; \eta = \cos\frac{\phi + \psi}{2}\cos\frac{\theta}{2} \tag{3}$$

And the transformation matrix A is written as [33]:

$$\mathbf{A} = \begin{bmatrix} \mathbf{1} - \mathbf{2} \begin{pmatrix} \varepsilon\_2^2 + \varepsilon\_3^2 \end{pmatrix} & \mathbf{2} \begin{pmatrix} \varepsilon\_1 \varepsilon\_2 + \varepsilon\_3 \eta \end{pmatrix} & \mathbf{2} \begin{pmatrix} \varepsilon\_1 \varepsilon\_3 - \varepsilon\_2 \eta \end{pmatrix} \\\ \mathbf{2} \begin{pmatrix} \varepsilon\_1 \varepsilon\_2 - \varepsilon\_3 \eta \end{pmatrix} & \mathbf{1} - \mathbf{2} \begin{pmatrix} \varepsilon\_1^2 + \varepsilon\_3^2 \end{pmatrix} & \mathbf{2} \begin{pmatrix} \varepsilon\_3 \varepsilon\_2 + \varepsilon\_1 \eta \end{pmatrix} \\\ \mathbf{2} \begin{pmatrix} \varepsilon\_1 \varepsilon\_3 + \varepsilon\_2 \eta \end{pmatrix} & \mathbf{2} \begin{pmatrix} \varepsilon\_3 \varepsilon\_2 - \varepsilon\_1 \eta \end{pmatrix} & \mathbf{1} - \mathbf{2} \begin{pmatrix} \varepsilon\_1^2 + \varepsilon\_2^2 \end{pmatrix} \end{bmatrix} \tag{4}$$

In the present study, the initial particle orientations are assigned by means of the Euler angles. From them, the corresponding Euler parameters are computed by Eq. (3), and with them, the initial transformation matrix is evaluated using Eq. (4). The Euler parameters evolve in time following Eq. (5), where the particle angular velocities are expressed in the particle frame of reference x<sup>0</sup> ¼ x<sup>0</sup> y<sup>0</sup> z<sup>0</sup> ½ �.

$$\begin{bmatrix} \frac{d\varepsilon\_1}{dt} \\\\ \frac{d\varepsilon\_2}{dt} \\\\ \frac{d\varepsilon\_3}{dt} \\\\ \frac{d\varepsilon\_4}{dt} \\\\ \frac{d\eta}{dt} \end{bmatrix} = \frac{1}{2} \begin{bmatrix} \eta a\_{\mathbf{x'}} - \varepsilon\_3 a\_{\mathbf{y'}} + \varepsilon\_2 a\_{\mathbf{z'}} \\\\ \varepsilon\_3 a\_{\mathbf{x'}} + \eta a\_{\mathbf{y'}} - \varepsilon\_1 a\_{\mathbf{z'}} \\\\ -\varepsilon\_2 a\_{\mathbf{x'}} + \varepsilon\_1 a\_{\mathbf{y'}} + \eta a\_{\mathbf{z'}} \\\\ -\varepsilon\_1 a\_{\mathbf{x'}} - \varepsilon\_2 a\_{\mathbf{y'}} - \varepsilon\_3 a\_{\mathbf{z'}} \end{bmatrix} \tag{5}$$

#### 2.2 Particle motion equations

The nonspherical particle motion equations in a general fluid flow [34] are written as:

with straw particles, which were quite well approximated by cylinders. It was found that straw particles were better dispersed than spheres [30], a fact that properly illustrates the importance of the correct modeling of nonspherical particles motion. In a later work [32], other forces such as added mass and pressure force were also included. Drag coefficient was computed using the Ganser [15] correlation, making

This contribution aims to study the motion of nonspherical particles immersed in homogeneous isotropic turbulent (HIT) velocity fields built from kinematic simulation at moderate Reynolds numbers. Computations were performed in a tailored in-house code. Properties analyzed include the Lagrangian time autocorrelation function, the translational and rotational particle response, and preferential orientation of the nonspherical particles in the turbulent flow, all of them in terms

To build the trajectory of a regular nonspherical particle, it is necessary to solve for its translational as well as rotational motion. However, whereas translation is solved in an inertial frame, rotation is solved referred to the so-called particle frame. Thus, the relevant coordinate frames and the transformations between them have

Figure 1 illustrates, in the case of a cylindrical particle, the employed coordinate systems: x ¼ ½ � xyz is the inertial frame; x<sup>0</sup> ¼ x<sup>0</sup> y<sup>0</sup> z<sup>0</sup> ½ � is the particle frame, whose origin is in the particle center of mass and its axes are the particle principal axes;

than particle frame but its axes are parallel to the inertial frame axes. In the particle

is the comoving frame, which has its origin at the same point

it possible to numerically compute the biomass combustion chamber.

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

of particle aspect ratio and inertia.

2. Governing equations

2.1 Coordinate systems

to be introduced.

and x 00 ¼ x 00 y 00 z <sup>00</sup>

Figure 1.

22

Illustration of a cylindrical particle and the employed coordinate systems.

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

Translational motion:

$$m\_p \frac{d\mathbf{u}\_p}{dt} = \mathbf{F} \tag{6}$$

CD <sup>¼</sup> CD,<sup>0</sup> <sup>þ</sup> ð Þ CD,<sup>90</sup> � CD,<sup>0</sup> sin <sup>a</sup>0φ; CD,<sup>0</sup> <sup>¼</sup> <sup>a</sup><sup>1</sup>

DOI: http://dx.doi.org/10.5772/intechopen.81045

CL <sup>¼</sup> <sup>b</sup><sup>1</sup> Reb<sup>2</sup> þ b3 Reb<sup>4</sup>

CT <sup>¼</sup> <sup>c</sup><sup>1</sup>

terms of cylinder length L and diameter D:

The correlations are expressed as:

can be found in Vakil and Green [19].

CD,cyl CD<sup>⊥</sup>

the particle frame.

25

a values are listed in [20] as also coefficients b, c, r:

Rec<sup>2</sup> þ

CD,cyl <sup>¼</sup> FD 1 2 ρeu 2 LD

Coefficients κ and those γ in functions Ai, i ¼ 0, 1, 2, 4:

in a cylinder, lCP, in terms of AR and φ, was proposed to be:

lCP ¼ 0:25

� �

c3 Rec<sup>4</sup> � �ð Þ sin <sup>φ</sup> <sup>c</sup>5þc6Rec<sup>7</sup>

Rea<sup>2</sup> þ

<sup>b</sup>5þb6Reb<sup>7</sup>

r3 Rer<sup>4</sup> R

; CL,cyl <sup>¼</sup> FL 1

CL,cylð Þ¼ φ; AR; ReD A2ð Þ AR; ReD sin 2φ þ A4ð Þ AR; ReD sin 4φ (14)

ð Þ¼ <sup>φ</sup>; AR; ReD <sup>A</sup>1ð Þ AR; ReD cos 2<sup>φ</sup> <sup>þ</sup> <sup>A</sup>0ð Þ AR; ReD ; CD⊥ð Þ¼ AR; ReD <sup>κ</sup><sup>1</sup> <sup>þ</sup> <sup>κ</sup><sup>2</sup>

Aið Þ¼ AR; ReD <sup>β</sup><sup>i</sup>1ð Þ AR ln ReD <sup>þ</sup> <sup>β</sup><sup>i</sup>2ð Þ AR ; <sup>β</sup>ijð Þ¼ AR <sup>γ</sup>ij<sup>1</sup> <sup>þ</sup> <sup>γ</sup>ij<sup>2</sup> exp <sup>γ</sup>ij3AR � � <sup>j</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>

However, expressions for the pitching and rotational torque coefficients are not

provided in [19]. Therefore, for the cylinders, the approach of [31] has been assumed. In [31], the distance between the center of mass and the center or pressure

1 � e

Then, the pitching torque T<sup>P</sup> is just the cross-product between the particle orientation unitary vector and the resultant force acting on it times lCP. Nevertheless, this torque is computed in the inertial frame of reference, so it should be transformed to the particle frame before being included in Eq. (7) to calculate the particle angular velocity. The approach to compute the viscous rotational torque T<sup>R</sup> is to integrate along the particle length the torque due to the drag force with respect the particle center of mass and it is described in [31]. This torque is given directly in

Particle motion equations and correlations for cylinders and ellipsoids presented in this section have been implemented in an in-house code. The numerical integration of the ordinary differential equations that govern the motion of nonspherical particles has been performed by a fourth-order Runge-Kutta method, with small

3 1ð Þ �AR � � cos <sup>3</sup>

φ � � �

L 2 R þ

Moreover, for the cylinders, Vakil and Green [19] developed correlations for drag and lift coefficients depending on orientation, Reynolds number based on its diameter ReD, and aspect ratio AR. In this case, such coefficients are expressed in

ð Þ sin φ

Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows

CR <sup>¼</sup> <sup>r</sup>1Rer<sup>2</sup>

a3

ð Þ cos φ

Rea<sup>4</sup> ; CD,<sup>90</sup> <sup>¼</sup> <sup>a</sup><sup>5</sup>

Rea<sup>6</sup> þ

<sup>b</sup>8þb9Reb<sup>10</sup> (10)

ð Þ cos <sup>φ</sup> <sup>c</sup>8þc9Rec<sup>10</sup> (11)

<sup>2</sup> <sup>ρ</sup>eu2LD (13)

AR � �Re <sup>κ</sup>3þκ<sup>4</sup> ð Þ AR D (15)

� (17)

(16)

a7 Rea<sup>8</sup> (9)

(12)

Rotational motion:

$$\begin{aligned} I\_{\mathbf{x'}} \frac{d\alpha\_{\mathbf{x'}}}{dt} - \alpha\_{\mathbf{y'}} \alpha\_{\mathbf{z'}} (I\_{\mathbf{y'}} - I\_{\mathbf{z'}}) &= T\_{\mathbf{x'}}\\ I\_{\mathbf{y'}} \frac{d\alpha\_{\mathbf{y'}}}{dt} - \alpha\_{\mathbf{x'}} \alpha\_{\mathbf{z'}} (I\_{\mathbf{z'}} - I\_{\mathbf{x'}}) &= T\_{\mathbf{y'}}\\ I\_{\mathbf{z'}} \frac{d\alpha\_{\mathbf{z'}}}{dt} - \alpha\_{\mathbf{y'}} \alpha\_{\mathbf{x'}} (I\_{\mathbf{x'}} - I\_{\mathbf{y'}}) &= T\_{\mathbf{z'}} \end{aligned} \tag{7}$$

Here, mp is the mass of the particle, <sup>u</sup><sup>p</sup> <sup>¼</sup> upx upy upz � � is the translational velocity of the particle center of mass, referred to the inertial frame, and F ¼ Fx Fy Fz � � is the external forces acting on the particle. The moments of inertia with respect to the particle frame axes are Ix<sup>0</sup> Iy<sup>0</sup> Iz<sup>0</sup> � �, and Tx<sup>0</sup> Ty<sup>0</sup> Tz<sup>0</sup> � � are the torques experienced by the particle. It should be remarked that the equations for the translation motion are computed in the inertial frame but those of the rotation motion are expressed in the particle frame. In case of the torque experienced by the particle, it has two contributions: the pitching torque, due to the noncoincidence of the particle center of mass and center of pressure (same fact that happens in an airfoil), and the rotational torque, due to the viscous resistance experienced by a rotating body inside a fluid, generated by the differences between fluid and particle rotational velocities.

In addition to a sphere, the four ellipsoids of Zawstawny et al. [20] have been chosen. They have different sphericities and aspect ratio (see Table 1). In Table 1, a denotes the major semiaxis and b the minor semiaxis.

Using DNS for ellipsoidal particles immersed in a uniform flow, Zastawny et al. [20] determined correlations for the flow coefficients (drag CD, lift CL, pitching torque CT, and rotational torque CR). Such coefficients are written as [20]:

$$\mathbf{C}\_{D} = \frac{F\_{D}}{\frac{1}{2}\rho\tilde{u}^{2}\frac{\pi}{4}d\_{p}^{2}}; \mathbf{C}\_{L} = \frac{F\_{L}}{\frac{1}{2}\rho\tilde{u}^{2}\frac{\pi}{4}d\_{p}^{2}}; \mathbf{C}\_{T} = \frac{F\_{T}}{\frac{1}{2}\rho\tilde{u}^{2}\frac{\pi}{8}d\_{p}^{3}}; \mathbf{C}\_{R} = \frac{F\_{R}}{\frac{1}{2}\rho\left(\frac{d\_{p}}{2}\right)^{5}|\mathbf{\tilde{M}}|^{2}}\tag{8}$$

Here, dp is the volume equivalent particle diameter or the diameter of a sphere with the same volume as the considered particle. The relative fluid velocity with respect to the particle is <sup>u</sup><sup>e</sup> <sup>¼</sup> <sup>u</sup> � <sup>u</sup><sup>p</sup> and <sup>Ω</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> � u � ω<sup>p</sup> is the fluid relative rotation with ω<sup>p</sup> being the particle angular velocity.

The developed correlations depend not only on particle Reynolds number Re <sup>¼</sup> <sup>ρ</sup>dpeu=<sup>μ</sup> and particle rotation number ReR <sup>¼</sup> <sup>ρ</sup>d<sup>2</sup> <sup>p</sup>j j Ω =μ, but also on orientation φ. They are written as [20]:


Table 1.

Ellipsoids evaluated by Zastawny et al. [20]. a and b are the major and minor semiaxis, respectively.

Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows DOI: http://dx.doi.org/10.5772/intechopen.81045

$$\mathbf{C}\_{\rm D} = \mathbf{C}\_{\rm D,0} + (\mathbf{C}\_{\rm D,90} - \mathbf{C}\_{\rm D,0}) \sin^{a\_0} \boldsymbol{\wp}; \mathbf{C}\_{\rm D,0} = \frac{a\_1}{\mathbf{R}^{a\_1}} + \frac{a\_3}{\mathbf{R}^{a\_4}}; \mathbf{C}\_{\rm D,90} = \frac{a\_5}{\mathbf{R}^{a\_6}} + \frac{a\_7}{\mathbf{R}^{a\_8}} \tag{9}$$

a values are listed in [20] as also coefficients b, c, r:

Translational motion:

Rotational motion:

particle frame axes are Ix<sup>0</sup> Iy<sup>0</sup> Iz<sup>0</sup>

CD <sup>¼</sup> FD 1 2 ρeu 2 π 4 d2 p

They are written as [20]:

Table 1.

24

mp du<sup>p</sup>

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

dt � <sup>ω</sup>y0ωz<sup>0</sup> Iy<sup>0</sup> � Iz<sup>0</sup>

dt � <sup>ω</sup>y0ωx<sup>0</sup> Ix<sup>0</sup> � Iy<sup>0</sup>

ity of the particle center of mass, referred to the inertial frame, and F ¼ Fx Fy Fz

� �, and Tx<sup>0</sup> Ty<sup>0</sup> Tz<sup>0</sup>

the external forces acting on the particle. The moments of inertia with respect to the

the particle. It should be remarked that the equations for the translation motion are computed in the inertial frame but those of the rotation motion are expressed in the particle frame. In case of the torque experienced by the particle, it has two contributions: the pitching torque, due to the noncoincidence of the particle center of mass and center of pressure (same fact that happens in an airfoil), and the rotational torque, due to the viscous resistance experienced by a rotating body inside a fluid, generated by the differences between fluid and particle rotational velocities. In addition to a sphere, the four ellipsoids of Zawstawny et al. [20] have been chosen. They have different sphericities and aspect ratio (see Table 1). In Table 1, a

Using DNS for ellipsoidal particles immersed in a uniform flow, Zastawny et al. [20] determined correlations for the flow coefficients (drag CD, lift CL, pitching torque CT, and rotational torque CR). Such coefficients are written as [20]:

> ; CT <sup>¼</sup> FT 1 2 ρeu 2 π 8 d3 p

Here, dp is the volume equivalent particle diameter or the diameter of a sphere with the same volume as the considered particle. The relative fluid velocity with

The developed correlations depend not only on particle Reynolds number

Shape Aspect ratio Sphericity

Ellipsoids evaluated by Zastawny et al. [20]. a and b are the major and minor semiaxis, respectively.

<sup>b</sup> <sup>¼</sup> <sup>5</sup>

<sup>b</sup> <sup>¼</sup> <sup>5</sup>

<sup>b</sup> <sup>¼</sup> <sup>5</sup>

<sup>b</sup> <sup>¼</sup> <sup>5</sup>

dt � <sup>ω</sup>x0ωz<sup>0</sup> Iz<sup>0</sup> � Ix ð Þ¼0 Ty<sup>0</sup>

� � <sup>¼</sup> Tx<sup>0</sup>

� � <sup>¼</sup> Tz<sup>0</sup>

Ix0 dωx<sup>0</sup>

Iy0 dωy<sup>0</sup>

Iz0 dωz<sup>0</sup>

denotes the major semiaxis and b the minor semiaxis.

; CL <sup>¼</sup> FL 1 2 ρeu 2 π 4 d2 p

respect to the particle is <sup>u</sup><sup>e</sup> <sup>¼</sup> <sup>u</sup> � <sup>u</sup><sup>p</sup> and <sup>Ω</sup> <sup>¼</sup> <sup>1</sup>

rotation with ω<sup>p</sup> being the particle angular velocity.

Re <sup>¼</sup> <sup>ρ</sup>dpeu=<sup>μ</sup> and particle rotation number ReR <sup>¼</sup> <sup>ρ</sup>d<sup>2</sup>

Ellipsoid 1 (prolate) <sup>a</sup>

Ellipsoid 2 (prolate) <sup>a</sup>

Disc (oblate) <sup>a</sup>

Fiber (prolate) <sup>a</sup>

Here, mp is the mass of the particle, u<sup>p</sup> ¼ upx upy upz

dt <sup>¼</sup> <sup>F</sup> (6)

� � is the translational veloc-

� � are the torques experienced by

; CR <sup>¼</sup> FR 1 2 ρ dp 2 � �<sup>5</sup>

<sup>2</sup> � u � ω<sup>p</sup> is the fluid relative

<sup>2</sup> 0.88

<sup>4</sup> 0.99

<sup>1</sup> 0.62

<sup>1</sup> 0.73

j j <sup>Ω</sup> <sup>2</sup>

<sup>p</sup>j j Ω =μ, but also on orientation φ.

(8)

(7)

� � is

$$\mathbf{C}\_{L} = \left(\frac{b\_{1}}{Re^{b\_{1}}} + \frac{b\_{3}}{Re^{b\_{4}}}\right) (\sin \varphi)^{b\_{5} + b\_{6}Re^{b\_{7}}} (\cos \varphi)^{b\_{8} + b\_{9}Re^{b\_{10}}} \tag{10}$$

$$C\_T = \left(\frac{c\_1}{Re^{\varepsilon\_1}} + \frac{c\_3}{Re^{\varepsilon\_4}}\right) (\sin \rho)^{c\_5 + c\_6 Re^{\varepsilon\_1}} (\cos \rho)^{c\_8 + c\_9 Re^{\varepsilon\_1}} \tag{11}$$

$$C\_R = r\_1 Re\_R^{r\_2} + \frac{r\_3}{Re\_R^{r\_4}}\tag{12}$$

Moreover, for the cylinders, Vakil and Green [19] developed correlations for drag and lift coefficients depending on orientation, Reynolds number based on its diameter ReD, and aspect ratio AR. In this case, such coefficients are expressed in terms of cylinder length L and diameter D:

$$\mathbf{C}\_{D,cyl} = \frac{F\_D}{\frac{1}{2}\rho \tilde{u}^2 LD}; \mathbf{C}\_{L,cyl} = \frac{F\_L}{\frac{1}{2}\rho \tilde{u} 2LD} \tag{13}$$

The correlations are expressed as:

$$\mathcal{L}\_{\text{L},\text{cyl}}(\rho, AR, Re\_D) = A\_2(AR, Re\_D)\sin 2\rho + A\_4(AR, Re\_D)\sin 4\rho \tag{14}$$

$$\frac{\mathbf{C\_{D,pl}}}{\mathbf{C\_{D\perp}}}(\boldsymbol{\wp}, \boldsymbol{AR}, \mathbf{Re\_{D}}) = \mathbf{A\_{1}}(\boldsymbol{AR}, \mathbf{Re\_{D}}) \cos 2\boldsymbol{\wp} + \mathbf{A\_{0}}(\boldsymbol{AR}, \mathbf{Re\_{D}}); \mathbf{C\_{D\perp}}(\boldsymbol{AR}, \mathbf{Re\_{D}}) = \left(\boldsymbol{\kappa\_{1}} + \frac{\boldsymbol{\kappa\_{2}}}{\boldsymbol{AR}}\right) \mathbf{Re\_{D}^{\left(\boldsymbol{x}\_{j} + \frac{\boldsymbol{\kappa\_{3}}}{2\mathbf{M}}\right)}}\tag{15}$$

Coefficients κ and those γ in functions Ai, i ¼ 0, 1, 2, 4:

$$A\_i(AR, \mathcal{Re}\_D) = \beta\_{i1}(AR) \ln \mathcal{Re}\_D + \beta\_{i2}(AR); \beta\_{ij}(AR) = \gamma\_{i1} + \gamma\_{i2} \exp\left(\gamma\_{i3} AR\right) \quad j = \textbf{1, 2} \tag{16}$$

can be found in Vakil and Green [19].

However, expressions for the pitching and rotational torque coefficients are not provided in [19]. Therefore, for the cylinders, the approach of [31] has been assumed. In [31], the distance between the center of mass and the center or pressure in a cylinder, lCP, in terms of AR and φ, was proposed to be:

$$d\_{\rm CP} = 0.25 \frac{L}{2} \left( 1 - e^{3(1 - AR)} \right) \left| \cos^3 \varphi \right| \tag{17}$$

Then, the pitching torque T<sup>P</sup> is just the cross-product between the particle orientation unitary vector and the resultant force acting on it times lCP. Nevertheless, this torque is computed in the inertial frame of reference, so it should be transformed to the particle frame before being included in Eq. (7) to calculate the particle angular velocity. The approach to compute the viscous rotational torque T<sup>R</sup> is to integrate along the particle length the torque due to the drag force with respect the particle center of mass and it is described in [31]. This torque is given directly in the particle frame.

Particle motion equations and correlations for cylinders and ellipsoids presented in this section have been implemented in an in-house code. The numerical integration of the ordinary differential equations that govern the motion of nonspherical particles has been performed by a fourth-order Runge-Kutta method, with small

enough time steps to avoid numerical instabilities [35, 36]. The fluid velocity field in which particles are immersed has been built by the kinematic simulation technique described in the next section. It is known that Runge-Kutta methods do not satisfy the time-reversal property, a fact that makes such methods inappropriate for integrating energy-conserving systems, for instance. However, particle equations are dissipative systems (as they include viscous drag forces) and, for them, Runge-Kutta algorithms can be used [37] provided that the time step is small enough to keep the errors bounded.

#### 3. Kinematic simulation

There exist different options to calculate the Lagrangian properties in a turbulent flow. The starting point is the trajectory equation in which the position x xð Þ <sup>0</sup>; t of a particle released at point x<sup>0</sup> at time t = 0 is calculated solving:

$$\frac{d\mathbf{x}}{dt} = \mathbf{u}(\mathbf{x}, t) \tag{18}$$

statistical characteristics of two-particle diffusion are independent of λ. In particular, in this work, two values of the unsteadiness parameter of 0 and 0.5 have been tested, without significant differences in the computed statistical properties. Therefore, following the suggestions of [39], the value 0.5 has been adopted in the

Comparison of spherical particle Reynolds stresses, obtained with KS versus theoretical values for i ¼ j.

Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows

DOI: http://dx.doi.org/10.5772/intechopen.81045

kn∙A<sup>n</sup> ¼ kn∙B<sup>n</sup> ¼ 0, it is solenoidal trajectory by trajectory. Moreover, as shown in [40], such field includes in each realization turbulent-like patterns as eddying,

To validate the spherical particles tracking in the KS velocity field, the values of particle Reynolds stresses (RS) in HIT have been selected. In this configuration, Hyland et al. [41] demonstrated that, as the fluid turbulence is homogeneous,

Computations were performed in a tailored in-house code. The turbulent velocity field generated with KS resembles one of the fields worked in [39]. Such velocity field is characterized by a fluctuating velocity u' = 1 m/s, a fluid Reynolds number of 10<sup>4</sup> resulting in a Kolmogorov length scale η<sup>K</sup> ≈ 6:286 mm, associated Kolmogorov time scale τ<sup>K</sup> ≈ 10 ms, and a fluid integral Lagrangian time scale of turbulence

The regular nonspherical particles studied have been the ellipsoids in [20] and the cylinders in [19]. In all cases, particles have the same particle volume equivalent diameter dp ¼ 200 μm, hence much smaller than ηK. Therefore, such particles can be thought as immersed in a uniform flow field. The Stokes number has been modified by adjusting the material density of particles being the Stokesian particle

q tð Þδij, that is, they are an isotropic tensor. Moreover, in the asymptotic limit, q tð Þ¼ ! ∞ βTL=ð Þ 1 þ βTL , where β is the inverse of particle relaxation time (see Eq. (21) below) and TL is the Lagrangian time scale of fluid turbulence. Figure 2 presents the numerical results for particle RS computed with KS and the asymptotic expression q tð Þ ! ∞ . As Figure 2 readily shows, the asymptotic particle RS are very well reproduced by the numerical particle tracking in the KS velocity field in the

piu<sup>0</sup> pjð Þ<sup>t</sup> <sup>=</sup> <sup>u</sup><sup>0</sup>

iu0 j <sup>¼</sup>

Because of the construction of the velocity field given by Eq. (19),

particle RS only depend on time and they can be written as u<sup>0</sup>

TL ¼ 0:56 s. Those values are matched by the present KS.

present computations.

Figure 2.

straining, and streaming regions.

range of two decades for βTL.

4. Numerical simulation

relaxation time defined as:

27

Here, u xð Þ ; t is the Eulerian velocity field. If it is known, it is possible to solve Eq. (18); however, finding u xð Þ ; t is not an easy task. One possibility is to work with Lagrangian statistics but then it would be needed to close the relevant Lagrangian correlations. Another option to solve Eq. (18) is to use DNS to obtain u xð Þ ; t ; however, this is computationally very expensive. A much more economical alternative is the use of kinematic simulation (KS) to compute the Lagrangian characteristics of turbulent flow fields. In this technique, stochastic fluid velocity fields are constructed in such a way that their statistical properties are in agreement with those extracted from experiments or reliable DNS. The main advantage of KS is that it employs an explicit continuous formula for computing u xð Þ ; t , so it is not needed to perform interpolation of the fluid velocity field. Moreover, KS results of two particle statistics in HIT have been validated versus DNS showing good agreement [38].

The three-dimensional Eulerian velocity field to be employed in Eq. (18) is built as a series of random Fourier modes. The velocity field is solenoidal at each realization by construction. Moreover, the energy spectrum of the Fourier modes is prescribed, for example, by a power law, so the effects of small flow scales on Lagrangian statistics are directly included. Such KS velocity field is written as [39]:

$$\mathbf{u}(\mathbf{x},t) = \sum\_{n=1}^{N} \mathbf{A}\_n \cos \left(\mathbf{k}\_n \bullet \mathbf{x} + \alpha\_n t\right) + \mathbf{B}\_n \sin \left(\mathbf{k}\_n \bullet \mathbf{x} + \alpha\_n t\right) \tag{19}$$

k<sup>n</sup> represents the n-th wave number; coefficients An, B<sup>n</sup> are random, uncorrelated vectors perpendicular to kn, whose amplitudes are chosen according to the prescribed energy spectrum E(k) [39]. Here, the energy spectrum has been the Kolmogorov decay law of �5/3.

ω<sup>n</sup> is the n-th frequency, which determines the unsteadiness of the corresponding mode; it is written proportional to the eddy-turnover time of the n-th mode:

$$
\alpha\_n = \lambda \sqrt{k\_n^3 E(k\_n)} \tag{20}
$$

Here, λ is a parameter of order 1 that governs the unsteadiness of the velocity field. In three-dimensional HIT flows, it has been demonstrated [38] that the

Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows DOI: http://dx.doi.org/10.5772/intechopen.81045

statistical characteristics of two-particle diffusion are independent of λ. In particular, in this work, two values of the unsteadiness parameter of 0 and 0.5 have been tested, without significant differences in the computed statistical properties. Therefore, following the suggestions of [39], the value 0.5 has been adopted in the present computations.

Because of the construction of the velocity field given by Eq. (19), kn∙A<sup>n</sup> ¼ kn∙B<sup>n</sup> ¼ 0, it is solenoidal trajectory by trajectory. Moreover, as shown in [40], such field includes in each realization turbulent-like patterns as eddying, straining, and streaming regions.

To validate the spherical particles tracking in the KS velocity field, the values of particle Reynolds stresses (RS) in HIT have been selected. In this configuration, Hyland et al. [41] demonstrated that, as the fluid turbulence is homogeneous, particle RS only depend on time and they can be written as u<sup>0</sup> piu<sup>0</sup> pjð Þ<sup>t</sup> <sup>=</sup> <sup>u</sup><sup>0</sup> iu0 j <sup>¼</sup> q tð Þδij, that is, they are an isotropic tensor. Moreover, in the asymptotic limit, q tð Þ¼ ! ∞ βTL=ð Þ 1 þ βTL , where β is the inverse of particle relaxation time (see Eq. (21) below) and TL is the Lagrangian time scale of fluid turbulence. Figure 2 presents the numerical results for particle RS computed with KS and the asymptotic expression q tð Þ ! ∞ . As Figure 2 readily shows, the asymptotic particle RS are very well reproduced by the numerical particle tracking in the KS velocity field in the range of two decades for βTL.

#### 4. Numerical simulation

Computations were performed in a tailored in-house code. The turbulent velocity field generated with KS resembles one of the fields worked in [39]. Such velocity field is characterized by a fluctuating velocity u' = 1 m/s, a fluid Reynolds number of 10<sup>4</sup> resulting in a Kolmogorov length scale η<sup>K</sup> ≈ 6:286 mm, associated Kolmogorov time scale τ<sup>K</sup> ≈ 10 ms, and a fluid integral Lagrangian time scale of turbulence TL ¼ 0:56 s. Those values are matched by the present KS.

The regular nonspherical particles studied have been the ellipsoids in [20] and the cylinders in [19]. In all cases, particles have the same particle volume equivalent diameter dp ¼ 200 μm, hence much smaller than ηK. Therefore, such particles can be thought as immersed in a uniform flow field. The Stokes number has been modified by adjusting the material density of particles being the Stokesian particle relaxation time defined as:

enough time steps to avoid numerical instabilities [35, 36]. The fluid velocity field in which particles are immersed has been built by the kinematic simulation technique described in the next section. It is known that Runge-Kutta methods do not satisfy the time-reversal property, a fact that makes such methods inappropriate for integrating energy-conserving systems, for instance. However, particle equations are dissipative systems (as they include viscous drag forces) and, for them, Runge-Kutta algorithms can be used [37] provided that the time step is small enough to

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

There exist different options to calculate the Lagrangian properties in a turbulent flow. The starting point is the trajectory equation in which the position x xð Þ <sup>0</sup>; t

Here, u xð Þ ; t is the Eulerian velocity field. If it is known, it is possible to solve Eq. (18); however, finding u xð Þ ; t is not an easy task. One possibility is to work with Lagrangian statistics but then it would be needed to close the relevant Lagrangian correlations. Another option to solve Eq. (18) is to use DNS to obtain u xð Þ ; t ; however, this is computationally very expensive. A much more economical alternative is the use of kinematic simulation (KS) to compute the Lagrangian characteristics of turbulent flow fields. In this technique, stochastic fluid velocity fields are constructed in such a way that their statistical properties are in agreement with those extracted from experiments or reliable DNS. The main advantage of KS is that it employs an explicit continuous formula for computing u xð Þ ; t , so it is not needed to perform interpolation of the fluid velocity field. Moreover, KS results of two particle statistics in HIT have been validated versus DNS showing good

The three-dimensional Eulerian velocity field to be employed in Eq. (18) is built as a series of random Fourier modes. The velocity field is solenoidal at each realization by construction. Moreover, the energy spectrum of the Fourier modes is prescribed, for example, by a power law, so the effects of small flow scales on

Lagrangian statistics are directly included. Such KS velocity field is written as [39]:

uncorrelated vectors perpendicular to kn, whose amplitudes are chosen according to the prescribed energy spectrum E(k) [39]. Here, the energy spectrum has been the

q

Here, λ is a parameter of order 1 that governs the unsteadiness of the velocity field. In three-dimensional HIT flows, it has been demonstrated [38] that the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k3 nE kð Þ<sup>n</sup>

k<sup>n</sup> represents the n-th wave number; coefficients An, B<sup>n</sup> are random,

ω<sup>n</sup> is the n-th frequency, which determines the unsteadiness of the corresponding mode; it is written proportional to the eddy-turnover time of the

ω<sup>n</sup> ¼ λ

A<sup>n</sup> cosð Þþ kn∙x þ ωnt B<sup>n</sup> sin ð Þ kn∙x þ ωnt (19)

(20)

dt <sup>¼</sup> u xð Þ ; <sup>t</sup> (18)

of a particle released at point x<sup>0</sup> at time t = 0 is calculated solving:

dx

keep the errors bounded.

3. Kinematic simulation

agreement [38].

n-th mode:

26

u xð Þ¼ ; t ∑

Kolmogorov decay law of �5/3.

N n¼1 Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

$$
\pi\_p = \beta^{-1} = \frac{\rho\_p d\_p^2}{18\mu} \tag{21}
$$

heaviest particles, where the various shapes present differences in their curves. It is interesting to realize that ILTSs of higher aspect ratio (AR) are below those of smaller AR, for both ellipsoids and cylinders. This effect is a Reynolds number effect due to the dependence of drag coefficient on shape and AR: an interaction between translation and rotation motions occurs that results in a spreading of the particle effective Stokes number. As a consequence, particles with higher Reynolds numbers also have larger effective inertia (reflected on an increased Stokes number) and, therefore, their LAF decreases slower, implying a higher ILTS. In the ellipsoids case, it happens that those of typ. 2 present a RL,tð Þτ curve slightly over that of the spherical particle, as they have lower effective Stokes number. Also, the LAF curve for the disc-like particles in this case is very similar to that of the fiber. In an analogous way to translational LAF, a rotational autocorrelation function

Computed RL,tð Þτ curves for ellipsoidal particles [20] (left) and cylindrical particles [19] (right). Fluid

Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows

RL,rð Þ¼ <sup>τ</sup> <sup>ω</sup>pð Þ <sup>0</sup> <sup>∙</sup> <sup>ω</sup>pð Þ<sup>τ</sup>

The obtained results for the Lagrangian rotational autocorrelation function of the ellipsoids of Zastawny et al. [20] are shown in the left part of Figure 4, whereas those of the cylinders of Vakil and Green [19] are in the right side of such figure. Again, horizontal axis is the nondimensional time delay, τ=TL. Similar to the case of translational motion, the angular velocities of heavy particles keep correlated for longer times than those of lighter particles. Such correlation time for ellipsoidal particles is much shorter than that of the translational motion. Also, for all inertia cases, the RAF of disc-like particles drops quicker than for the prolate ellipsoids. As mentioned for the translational correlations, the RAF curves for the prolate ellipsoids collapse for the lighter particles, but they show noticeable differences for the intermediate and large inertia particles demonstrating an effect of the aspect ratio on RL,rð Þτ . Ellipsoid 2, with the smaller AR, has the higher RAF curve of all prolate ellipsoids, while Ellipsoid 1 and

In the case of cylinders (Figure 4, right), the RAF curves for all AR and inertias are different. For the smallest inertia particles, RAF decreases with increasing AR, similar to what was found for LAFs in the translational motion. Moreover, the RL,rð Þτ curve presents negative values for the two largest aspect ratios of 10 and 20. For the intermediate particles, the RAF curves keep the same decreasing trend with increasing aspect ratio as the light particles; however, in this case, correlation times

<sup>ω</sup><sup>p</sup> <sup>∙</sup> <sup>ω</sup>pð Þ <sup>0</sup> (23)

(RAF) RL,rð Þτ can be defined in terms of the time delay τ as:

Lagrangian and Eulerian curves are included for comparison.

DOI: http://dx.doi.org/10.5772/intechopen.81045

Figure 3.

29

where the particle angular velocity is denoted by ωp.

the fiber have very similar rotational correlation functions.

If the Kolmogorov time scale τ<sup>K</sup> is taken as the fluid time scale, particle Stokes number is defined as St ¼ τp=τK. According to this nondimensional number, three particle inertia classes are considered: light (St ≈ 0:5), intermediate (St ≈ 10), and heavy (St ≈ 100). However, as cases with Re > 1 are considered, an effective particle relaxation time is introduced as τp,eff ¼ τp=ReCD, allowing the introduction of an effective Stokes number Steff ¼ τp,eff =τK. Therefore, the values of such effective Stokes number are Steff ≈ 0.3 (light), 5 (intermediate), and 40 (heavy).

Simulations proceed in the following way: for each KS realization of HIT fluid velocity field, a particle is located in the center of the domain with zero initial velocity; particle translational and rotational motion is computed from Eqs. (6) and (7), its orientation is calculated from Eq. (1), and its trajectory is built; particle tracking lasts for around 10 fluid integral time scales; and particle properties are stored every second for evaluation. Such process is carried out a sufficient number of times to reach significant statistical results. In this study, statistics has been performed based on 105 KS realizations.

In the following section, the results of the particle Lagrangian time autocorrelation function, the translational and rotational particle response, and preferential orientation of the nonspherical particles in the turbulent flow are analyzed as function of their shape and effective Stokes number.

#### 5. Results and discussion

The Lagrangian autocorrelation function RL,tð Þτ for translational motion is expressed as:

$$R\_{L,t}(\tau) = \frac{\left<\mathbf{u}\_p(\mathbf{0}) \bullet \mathbf{u}\_p(\tau)\right>}{\left<\mathbf{u}\_p \bullet \mathbf{u}\_p(\mathbf{0})\right>}\tag{22}$$

τ represents the time delay. With the objective of making results independent of particle injection conditions, statistics are started to be collected after 2 s. The obtained results for the Lagrangian autocorrelation function (LAF) of the ellipsoids of Zastawny et al. [20] are presented in the left part of Figure 3, including the results for spherical particles, whereas those of the cylinders of Vakil and Green [19] are in the right side of such figure. Horizontal axis is the nondimensional time delay, τ=TL. As the lighter particles have an autocorrelation function nearly equal to that of the fluid (tracer limit), the corresponding curves are not shown in Figure 3. Therefore, only the curves for intermediate and high inertia particles are presented. Moreover, in Figure 3 also the fluid Lagrangian (brown curve) and Eulerian (cyan curve) RL,tð Þτ 's are included for comparison.

As it can be readily seen from Figure 3, higher inertia particles are characterized by larger integral Lagrangian time scales (ILTSs) (defined as the integral up to infinity of RL,tð Þτ ), as a result of their smaller responsiveness to the turbulent fluctuations. Same as in [23], all particle LAFs are mainly in between the fluid LAF and Eulerian autocorrelation function (EAF). As a consequence of inertia, for the smallest values of τ, the heavy RL,tð Þτ overcomes the fluid EAF, differently from [23] who considered only noninertial particles.

Moreover, for intermediate inertia, the curves for all particle shapes nearly collapse in a single curve. On the other hand, a shape effect is noticeable for the Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows DOI: http://dx.doi.org/10.5772/intechopen.81045

Figure 3.

<sup>τ</sup><sup>p</sup> <sup>¼</sup> <sup>β</sup>�<sup>1</sup> <sup>¼</sup> <sup>ρ</sup>pd<sup>2</sup>

If the Kolmogorov time scale τ<sup>K</sup> is taken as the fluid time scale, particle Stokes number is defined as St ¼ τp=τK. According to this nondimensional number, three particle inertia classes are considered: light (St ≈ 0:5), intermediate (St ≈ 10), and heavy (St ≈ 100). However, as cases with Re > 1 are considered, an effective particle relaxation time is introduced as τp,eff ¼ τp=ReCD, allowing the introduction of an effective Stokes number Steff ¼ τp,eff =τK. Therefore, the values of such effective

Simulations proceed in the following way: for each KS realization of HIT fluid velocity field, a particle is located in the center of the domain with zero initial velocity; particle translational and rotational motion is computed from Eqs. (6) and (7), its orientation is calculated from Eq. (1), and its trajectory is built; particle tracking lasts for around 10 fluid integral time scales; and particle properties are stored every second for evaluation. Such process is carried out a sufficient number of times to reach significant statistical results. In this study, statistics has been

In the following section, the results of the particle Lagrangian time autocorrelation function, the translational and rotational particle response, and preferential orientation of the nonspherical particles in the turbulent flow are analyzed as

The Lagrangian autocorrelation function RL,tð Þτ for translational motion is

RL,tð Þ¼ <sup>τ</sup> <sup>u</sup>pð Þ <sup>0</sup> <sup>∙</sup> <sup>u</sup>pð Þ<sup>τ</sup>

particle injection conditions, statistics are started to be collected after 2 s. The obtained results for the Lagrangian autocorrelation function (LAF) of the ellipsoids of Zastawny et al. [20] are presented in the left part of Figure 3, including the results for spherical particles, whereas those of the cylinders of Vakil and Green [19] are in the right side of such figure. Horizontal axis is the nondimensional time delay, τ=TL. As the lighter particles have an autocorrelation function nearly equal to that of the fluid (tracer limit), the corresponding curves are not shown in Figure 3. Therefore, only the curves for intermediate and high inertia particles are presented. Moreover, in Figure 3 also the fluid Lagrangian (brown curve) and Eulerian (cyan

τ represents the time delay. With the objective of making results independent of

As it can be readily seen from Figure 3, higher inertia particles are characterized

by larger integral Lagrangian time scales (ILTSs) (defined as the integral up to infinity of RL,tð Þτ ), as a result of their smaller responsiveness to the turbulent fluctuations. Same as in [23], all particle LAFs are mainly in between the fluid LAF and Eulerian autocorrelation function (EAF). As a consequence of inertia, for the smallest values of τ, the heavy RL,tð Þτ overcomes the fluid EAF, differently from

Moreover, for intermediate inertia, the curves for all particle shapes nearly collapse in a single curve. On the other hand, a shape effect is noticeable for the

<sup>u</sup><sup>p</sup> <sup>∙</sup> <sup>u</sup>pð Þ <sup>0</sup> (22)

Stokes number are Steff ≈ 0.3 (light), 5 (intermediate), and 40 (heavy).

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

performed based on 105 KS realizations.

5. Results and discussion

expressed as:

28

function of their shape and effective Stokes number.

curve) RL,tð Þτ 's are included for comparison.

[23] who considered only noninertial particles.

P 18μ

(21)

Computed RL,tð Þτ curves for ellipsoidal particles [20] (left) and cylindrical particles [19] (right). Fluid Lagrangian and Eulerian curves are included for comparison.

heaviest particles, where the various shapes present differences in their curves. It is interesting to realize that ILTSs of higher aspect ratio (AR) are below those of smaller AR, for both ellipsoids and cylinders. This effect is a Reynolds number effect due to the dependence of drag coefficient on shape and AR: an interaction between translation and rotation motions occurs that results in a spreading of the particle effective Stokes number. As a consequence, particles with higher Reynolds numbers also have larger effective inertia (reflected on an increased Stokes number) and, therefore, their LAF decreases slower, implying a higher ILTS. In the ellipsoids case, it happens that those of typ. 2 present a RL,tð Þτ curve slightly over that of the spherical particle, as they have lower effective Stokes number. Also, the LAF curve for the disc-like particles in this case is very similar to that of the fiber.

In an analogous way to translational LAF, a rotational autocorrelation function (RAF) RL,rð Þτ can be defined in terms of the time delay τ as:

$$R\_{L,r}(\tau) = \frac{\left<\mathbf{o}\_p(\mathbf{0}) \bullet \mathbf{o}\_p(\tau)\right>}{\left<\mathbf{o}\_p \bullet \mathbf{o}\_p(\mathbf{0})\right>}\tag{23}$$

where the particle angular velocity is denoted by ωp.

The obtained results for the Lagrangian rotational autocorrelation function of the ellipsoids of Zastawny et al. [20] are shown in the left part of Figure 4, whereas those of the cylinders of Vakil and Green [19] are in the right side of such figure. Again, horizontal axis is the nondimensional time delay, τ=TL. Similar to the case of translational motion, the angular velocities of heavy particles keep correlated for longer times than those of lighter particles. Such correlation time for ellipsoidal particles is much shorter than that of the translational motion. Also, for all inertia cases, the RAF of disc-like particles drops quicker than for the prolate ellipsoids. As mentioned for the translational correlations, the RAF curves for the prolate ellipsoids collapse for the lighter particles, but they show noticeable differences for the intermediate and large inertia particles demonstrating an effect of the aspect ratio on RL,rð Þτ . Ellipsoid 2, with the smaller AR, has the higher RAF curve of all prolate ellipsoids, while Ellipsoid 1 and the fiber have very similar rotational correlation functions.

In the case of cylinders (Figure 4, right), the RAF curves for all AR and inertias are different. For the smallest inertia particles, RAF decreases with increasing AR, similar to what was found for LAFs in the translational motion. Moreover, the RL,rð Þτ curve presents negative values for the two largest aspect ratios of 10 and 20. For the intermediate particles, the RAF curves keep the same decreasing trend with increasing aspect ratio as the light particles; however, in this case, correlation times

ellipsoid with AR = 1.25. The trend of increasing particle fluctuating velocities with aspect ratio is consistent with the aforementioned fact that effective Stokes number tends to reduce with growing AR; therefore, particles with lower AR respond less to the fluid fluctuations than the more elongated ones. This result has been obtained for particles much smaller than Kolmogorov length scale; therefore, such particles can be considered to be immersed in a uniform flow field, just in the same conditions as the flow coefficient correlations were developed. In the study of Hölzer and Sommerfeld [42], nevertheless, a different result was found. Using a DNS based on the lattice Boltzmann method (LBM), Hölzer and Sommerfeld [42] obtained that relative particle fluctuating velocity reduced with increasing AR. The main difference with the present work is that Hölzer and Sommerfeld [42] employed particles with size well above ηK. The authors explained the fact arguing that particles averaged the fluid fluctuations on their surface, which augmented with increasing AR. Let us remark that both results are not conflicting as the size range of the employed particles in the two studies is very different. Further work combining DNS with nonspherical point particles smaller than η<sup>K</sup> is necessary to explain this point. Figure 5(b) presents the behavior of the particle angular rms velocity, ω<sup>0</sup>

Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows

DOI: http://dx.doi.org/10.5772/intechopen.81045

homogeneous and isotropic turbulence, the angular velocity of spherical particles is zero because of viscous damping and the absence of pitching torque. The situation is different in nonspherical particles because in them the geometrical and pressure centers do not coincide, so a net pitching torque is produced that promotes nonzero

up to a maximum of AR ≈ 2 and decreases with higher values of aspect ratio. The shape of the curve is the same for ellipsoids and cylinders. Such behavior was also found in [42], and it was explained observing that, with increasing AR, the moment of inertia along the major axis reduces and along the minor axis increases, which

In the following, the correlation relative velocity direction-particle orientation is analyzed depending on inertia and aspect ratio. A well-known fact is that regular nonspherical particles falling through a still liquid at intermediate Reynolds numbers tend to be oriented in a determined direction. Cylinders and prolate ellipsoids are prone to keep their symmetry axis (z' in Figure 1) perpendicular to the flow, thus maximizing drag. Differently, discs and oblate ellipsoids tend to move with the symmetry axis aligned with the flow, also maximizing drag [43]. However, spheroidal Stokes particles only show a preferential orientation if a persistent velocity gradient exists [27]. Therefore, in HIT flow where there are no mean velocity

<sup>p</sup>, for inertial particles, ω<sup>0</sup>

gradients, a Stokes particle will not have any preferred orientation.

and pressure do not coincide, this situation is unstable.

Newsom and Bruce [44] analyzed the influence of turbulence on preferential alignment of quite elongated fibers with Re ≈ 1. As explained by [44], preferential orientation of such fibers falling through a still fluid can only be clarified if fluid inertial effects are considered. In the Stokes regime, Khayat and Cox [43] demonstrate that the force distribution on the fiber is symmetrically distributed along its axis, independent of its orientation regarding the flow and, as a result, the fiber experiences a zero net torque. Beyond the Stokes regime, Re > 1, when the fiber has an oblique orientation with respect the flow, such distribution of the force is not any more symmetric and it experiences a net pitching torque. Such torque will promote a rotation that drives the fiber to adopt an orientation where its symmetry axis is orthogonal to the relative flow. Interestingly, if the fiber is oriented orthogonal or parallel to the flow, the net experienced torque is zero, due to the symmetry of the force distribution; however, in the first case, this situation is stable, while in the second case, where the centers of gravity

angular velocities. As illustrated in Figure 5(b), ω<sup>0</sup>

would lead to higher and lower ω<sup>0</sup>

what happened with u<sup>0</sup>

31

<sup>p</sup>. In

<sup>p</sup> increases with AR and reaches

<sup>p</sup> decreases as inertia augments.

<sup>p</sup>, respectively. On the other hand, and similar to

Figure 4. Computed RL,rð Þτ curves for ellipsoidal particles [20] (left) and cylindrical particles [19] (right).

Figure 5. Relative particle rms of particle linear velocity (a) and angular rms velocity (b). In all cases, the dependence on aspect ratio and inertia is considered. Closed symbols refer to prolate ellipsoids and open symbols to cylinders.

for angular velocities are very much increased and they are significantly higher than for ellipsoids. The previous trend is reversed for the heavy particles as the RAF curves augment with increasing AR; however, as Figure 4 (right) suggests, an asymptotic value for L/D seems to exist because the curves for AR = 10 and 20 are very close to each other. The change of behavior of the RAF with increasing inertia could be due to the fact that for the small and intermediate inertia, the particle relaxation time for rotation reduces with growing AR, whereas for heavy particles, such relaxation time behaves in the opposite way. Nevertheless, this fact must be further investigated, possibly using fully resolved simulations.

Next, the response of the nonspherical particles to the fluid fluctuating velocities is analyzed for both translation and rotation motions. Figure 5(a) shows the behavior of the particle's relative linear root mean square (rms) velocity, that is, u0 p=u<sup>0</sup> , where u<sup>0</sup> is the fluid rms fluctuating translational velocity and u<sup>0</sup> <sup>p</sup> denotes the same quantity but for the particles. The aspect ratio is in the horizontal axis, whereas the different curves correspond to the various inertia cases. On the one side, as it could be anticipated, u<sup>0</sup> <sup>p</sup> reduces with increasing particle inertia because the more inertial particles are not able to follow all fluid velocity fluctuations. In fact, as it was found for the LAF, the less inertial particles present, for both ellipsoids and cylinders and for all values of AR, the same fluctuating velocities as the fluid, indicating that they behave as fluid tracers.

As inertia increases, u<sup>0</sup> <sup>p</sup>=u<sup>0</sup> decreases monotonically, as expected. However, it increases with growing aspect ratio for both cylinders and ellipsoids. There is one exception that spherical particles have values of u<sup>0</sup> <sup>p</sup>=u<sup>0</sup> slightly above those of the

#### Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows DOI: http://dx.doi.org/10.5772/intechopen.81045

ellipsoid with AR = 1.25. The trend of increasing particle fluctuating velocities with aspect ratio is consistent with the aforementioned fact that effective Stokes number tends to reduce with growing AR; therefore, particles with lower AR respond less to the fluid fluctuations than the more elongated ones. This result has been obtained for particles much smaller than Kolmogorov length scale; therefore, such particles can be considered to be immersed in a uniform flow field, just in the same conditions as the flow coefficient correlations were developed. In the study of Hölzer and Sommerfeld [42], nevertheless, a different result was found. Using a DNS based on the lattice Boltzmann method (LBM), Hölzer and Sommerfeld [42] obtained that relative particle fluctuating velocity reduced with increasing AR. The main difference with the present work is that Hölzer and Sommerfeld [42] employed particles with size well above ηK. The authors explained the fact arguing that particles averaged the fluid fluctuations on their surface, which augmented with increasing AR. Let us remark that both results are not conflicting as the size range of the employed particles in the two studies is very different. Further work combining DNS with nonspherical point particles smaller than η<sup>K</sup> is necessary to explain this point.

Figure 5(b) presents the behavior of the particle angular rms velocity, ω<sup>0</sup> <sup>p</sup>. In homogeneous and isotropic turbulence, the angular velocity of spherical particles is zero because of viscous damping and the absence of pitching torque. The situation is different in nonspherical particles because in them the geometrical and pressure centers do not coincide, so a net pitching torque is produced that promotes nonzero angular velocities. As illustrated in Figure 5(b), ω<sup>0</sup> <sup>p</sup> increases with AR and reaches up to a maximum of AR ≈ 2 and decreases with higher values of aspect ratio. The shape of the curve is the same for ellipsoids and cylinders. Such behavior was also found in [42], and it was explained observing that, with increasing AR, the moment of inertia along the major axis reduces and along the minor axis increases, which would lead to higher and lower ω<sup>0</sup> <sup>p</sup>, respectively. On the other hand, and similar to what happened with u<sup>0</sup> <sup>p</sup>, for inertial particles, ω<sup>0</sup> <sup>p</sup> decreases as inertia augments.

In the following, the correlation relative velocity direction-particle orientation is analyzed depending on inertia and aspect ratio. A well-known fact is that regular nonspherical particles falling through a still liquid at intermediate Reynolds numbers tend to be oriented in a determined direction. Cylinders and prolate ellipsoids are prone to keep their symmetry axis (z' in Figure 1) perpendicular to the flow, thus maximizing drag. Differently, discs and oblate ellipsoids tend to move with the symmetry axis aligned with the flow, also maximizing drag [43]. However, spheroidal Stokes particles only show a preferential orientation if a persistent velocity gradient exists [27]. Therefore, in HIT flow where there are no mean velocity gradients, a Stokes particle will not have any preferred orientation.

Newsom and Bruce [44] analyzed the influence of turbulence on preferential alignment of quite elongated fibers with Re ≈ 1. As explained by [44], preferential orientation of such fibers falling through a still fluid can only be clarified if fluid inertial effects are considered. In the Stokes regime, Khayat and Cox [43] demonstrate that the force distribution on the fiber is symmetrically distributed along its axis, independent of its orientation regarding the flow and, as a result, the fiber experiences a zero net torque. Beyond the Stokes regime, Re > 1, when the fiber has an oblique orientation with respect the flow, such distribution of the force is not any more symmetric and it experiences a net pitching torque. Such torque will promote a rotation that drives the fiber to adopt an orientation where its symmetry axis is orthogonal to the relative flow. Interestingly, if the fiber is oriented orthogonal or parallel to the flow, the net experienced torque is zero, due to the symmetry of the force distribution; however, in the first case, this situation is stable, while in the second case, where the centers of gravity and pressure do not coincide, this situation is unstable.

for angular velocities are very much increased and they are significantly higher than for ellipsoids. The previous trend is reversed for the heavy particles as the RAF curves augment with increasing AR; however, as Figure 4 (right) suggests, an asymptotic value for L/D seems to exist because the curves for AR = 10 and 20 are very close to each other. The change of behavior of the RAF with increasing inertia could be due to the fact that for the small and intermediate inertia, the particle relaxation time for rotation reduces with growing AR, whereas for heavy particles, such relaxation time behaves in the opposite way. Nevertheless, this fact must be

Relative particle rms of particle linear velocity (a) and angular rms velocity (b). In all cases, the dependence on aspect ratio and inertia is considered. Closed symbols refer to prolate ellipsoids and open symbols to cylinders.

Computed RL,rð Þτ curves for ellipsoidal particles [20] (left) and cylindrical particles [19] (right).

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

Next, the response of the nonspherical particles to the fluid fluctuating velocities

<sup>p</sup> reduces with increasing particle inertia because

<sup>p</sup>=u<sup>0</sup> slightly above those of the

<sup>p</sup>=u<sup>0</sup> decreases monotonically, as expected. However, it

<sup>p</sup> denotes the

is analyzed for both translation and rotation motions. Figure 5(a) shows the behavior of the particle's relative linear root mean square (rms) velocity, that is,

, where u<sup>0</sup> is the fluid rms fluctuating translational velocity and u<sup>0</sup>

same quantity but for the particles. The aspect ratio is in the horizontal axis, whereas the different curves correspond to the various inertia cases. On the one

the more inertial particles are not able to follow all fluid velocity fluctuations. In fact, as it was found for the LAF, the less inertial particles present, for both ellipsoids and cylinders and for all values of AR, the same fluctuating velocities as the

increases with growing aspect ratio for both cylinders and ellipsoids. There is one

further investigated, possibly using fully resolved simulations.

u0 p=u<sup>0</sup>

30

Figure 4.

Figure 5.

side, as it could be anticipated, u<sup>0</sup>

As inertia increases, u<sup>0</sup>

fluid, indicating that they behave as fluid tracers.

exception that spherical particles have values of u<sup>0</sup>

Previous reasoning is valid too for another kind of nonspherical shapes as disclike, cylindrical, or ellipsoidal [45]. For high Reynolds numbers, that is, Re > 100 appears a secondary motion overimposed to the particles predominant movement direction. Such secondary motion is promoted by a wake instability and vortex detachment from the rear surface of the particles. Two main kinds of secondary motion can be observed: large quasi-periodic swings along the main path, and a more or less chaotic tumbling forming a definite angle with the main motion direction. There is a coupling between this kind of oscillatory motion and the wake instability [10]: a vortex detachment follows at the end of a particle swing. Nevertheless, in the present study, such secondary motions do not appear as the considered particle Reynolds number is not large enough, that is, Re < 40.

Let now θ be the angle formed by the relative velocity, u � up, and the particle symmetry axis, z'. Therefore, cos θ can be used to determine the orientation of the nonspherical particle with respect to the relative flow. In this work, particle preferential orientation is determined computing cos j j <sup>θ</sup> along the trajectories of 10<sup>5</sup> particles. Computed values of cos j j θ are sorted in equally distributed bins between 0 (particle axis orthogonal to relative velocity) and 1 (alignment between particle axis and relative velocity), and the corresponding probability density functions (Pdfs) are determined. Such Pdfs are shown in Figure 6 in terms of cos j j θ . Figure 6(a) shows the results for the prolate ellipsoids in terms of particle inertia,

> and Figure 6(b) presents the curves for the cylinders also depending on their inertia. Each inertia class is plotted in a separated frame. For the discs case, results

Orientation probability density functions (Pdfs) of disc-like particles regarding the relative flow direction.

Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows

DOI: http://dx.doi.org/10.5772/intechopen.81045

As it is observed in Figure 6(a), it is found that prolate ellipsoids do manifest preferential orientation with respect to the relative velocity. Of course, spherical particles do not have a preferred orientation and the corresponding Pdf is a horizontal line (black color). Prolate ellipsoids have a preference for orientating its symmetry axis orthogonal to the relative flow, tending to maximize the drag, similar to what occurs in particle sedimentation studies. The orientation preference increases with inertia, which is quite similar for all aspect ratios considered in this

On the other hand, as it is presented in Figure 6(b), cylinders seem not to have any preferred orientation in the HIT KS velocity field, being all the curves pretty flat. Only for the case of higher AR and lowest inertia, the curves show a trend to be slightly higher for values of cos j j θ closer to one than to zero. Such result is qualitatively similar to the DNS computations of [27] in the central region of the channel. For the discs, Figure 7 shows that there is a clear trend of the particle symmetry

In this study, regular nonspherical particle responsiveness to HIT flows has been

axis to be aligned with the relative flow, again maximizing drag, similar to the results obtained for sedimenting particles in stagnant fluid. Such trend is more

investigated in combination with KS of fluid velocity field. The main results obtained are the following: the particle LAF reduces when particle AR is augmented, because effective particle inertia decreases if aspect ratio increases; this is true for both translational and rotational time autocorrelation functions. In the case of cylinders, RL,rð Þτ is much higher than for ellipsoids, a fact that requires further clarification through particle resolved simulations. Additionally, the fluctuating particle velocity increases for growing AR in the considered case of particles much

are presented in Figure 7.

marked when particle inertia increases.

study.

Figure 7.

6. Conclusions

33

#### Figure 6.

Orientations probability density functions (Pdfs) of prolate ellipsoids (a) and cylinders (b) regarding the relative flow direction.

Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows DOI: http://dx.doi.org/10.5772/intechopen.81045

Figure 7. Orientation probability density functions (Pdfs) of disc-like particles regarding the relative flow direction.

and Figure 6(b) presents the curves for the cylinders also depending on their inertia. Each inertia class is plotted in a separated frame. For the discs case, results are presented in Figure 7.

As it is observed in Figure 6(a), it is found that prolate ellipsoids do manifest preferential orientation with respect to the relative velocity. Of course, spherical particles do not have a preferred orientation and the corresponding Pdf is a horizontal line (black color). Prolate ellipsoids have a preference for orientating its symmetry axis orthogonal to the relative flow, tending to maximize the drag, similar to what occurs in particle sedimentation studies. The orientation preference increases with inertia, which is quite similar for all aspect ratios considered in this study.

On the other hand, as it is presented in Figure 6(b), cylinders seem not to have any preferred orientation in the HIT KS velocity field, being all the curves pretty flat. Only for the case of higher AR and lowest inertia, the curves show a trend to be slightly higher for values of cos j j θ closer to one than to zero. Such result is qualitatively similar to the DNS computations of [27] in the central region of the channel.

For the discs, Figure 7 shows that there is a clear trend of the particle symmetry axis to be aligned with the relative flow, again maximizing drag, similar to the results obtained for sedimenting particles in stagnant fluid. Such trend is more marked when particle inertia increases.

#### 6. Conclusions

In this study, regular nonspherical particle responsiveness to HIT flows has been investigated in combination with KS of fluid velocity field. The main results obtained are the following: the particle LAF reduces when particle AR is augmented, because effective particle inertia decreases if aspect ratio increases; this is true for both translational and rotational time autocorrelation functions. In the case of cylinders, RL,rð Þτ is much higher than for ellipsoids, a fact that requires further clarification through particle resolved simulations. Additionally, the fluctuating particle velocity increases for growing AR in the considered case of particles much

Previous reasoning is valid too for another kind of nonspherical shapes as disclike, cylindrical, or ellipsoidal [45]. For high Reynolds numbers, that is, Re > 100 appears a secondary motion overimposed to the particles predominant movement direction. Such secondary motion is promoted by a wake instability and vortex detachment from the rear surface of the particles. Two main kinds of secondary motion can be observed: large quasi-periodic swings along the main path, and a more or less chaotic tumbling forming a definite angle with the main motion direction. There is a coupling between this kind of oscillatory motion and the wake instability [10]: a vortex detachment follows at the end of a particle swing. Nevertheless, in the present study, such secondary motions do not appear as the consid-

Let now θ be the angle formed by the relative velocity, u � up, and the particle symmetry axis, z'. Therefore, cos θ can be used to determine the orientation of the nonspherical particle with respect to the relative flow. In this work, particle preferential orientation is determined computing cos j j <sup>θ</sup> along the trajectories of 10<sup>5</sup> particles. Computed values of cos j j θ are sorted in equally distributed bins between 0 (particle axis orthogonal to relative velocity) and 1 (alignment between particle axis and relative velocity), and the corresponding probability density functions (Pdfs) are determined. Such Pdfs are shown in Figure 6 in terms of cos j j θ . Figure 6(a) shows the results for the prolate ellipsoids in terms of particle inertia,

Orientations probability density functions (Pdfs) of prolate ellipsoids (a) and cylinders (b) regarding the

ered particle Reynolds number is not large enough, that is, Re < 40.

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

Figure 6.

32

relative flow direction.

smaller than Kolmogorov length scale; such behavior is contrary, although not conflicting, to the findings of [42] for fully resolved particles with sizes larger than the Kolmogorov length scale. For both ellipsoids and cylinders, the particle angular rms velocity first increases with aspect ratio, reaches a maximum of AR ≈ 2, and then decreases again, which is explained because with increasing aspect ratio, the moment of inertia around the longitudinal axis decreases and around the radial axis increases, which would lead to higher and lower rms angular velocities, respectively. Finally, in agreement with Marchioli et al. [27], cylinders seem not to prefer any specific orientation in the KS HIT velocity field; however, prolate ellipsoids tend to be oriented with its symmetry axis orthogonal to the relative flow, maximizing the drag. Oblate ellipsoids and disc-like particles also show a preferential orientation, tending to align their symmetry axis with the relative flow velocity.

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DOI: http://dx.doi.org/10.5772/intechopen.81045

Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows

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35

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### Acknowledgements

The financial support of Universidad Autónoma de Occidente is gratefully acknowledged.

### Author details

Santiago Laín Department of Energetics and Mechanics, PAI+ Group, Universidad Autónoma de Occidente, Cali, Colombia

\*Address all correspondence to: slain@uao.edu.co

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Response Behavior of Nonspherical Particles in Homogeneous Isotropic Turbulent Flows DOI: http://dx.doi.org/10.5772/intechopen.81045

#### References

smaller than Kolmogorov length scale; such behavior is contrary, although not conflicting, to the findings of [42] for fully resolved particles with sizes larger than the Kolmogorov length scale. For both ellipsoids and cylinders, the particle angular rms velocity first increases with aspect ratio, reaches a maximum of AR ≈ 2, and then decreases again, which is explained because with increasing aspect ratio, the moment of inertia around the longitudinal axis decreases and around the radial axis increases, which would lead to higher and lower rms angular velocities, respectively. Finally, in agreement with Marchioli et al. [27], cylinders seem not to prefer any specific orientation in the KS HIT velocity field; however, prolate ellipsoids tend to be oriented with its symmetry axis orthogonal to the relative flow, maximizing the drag. Oblate ellipsoids and disc-like particles also show a preferential orientation, tending to align their symmetry axis with the relative flow velocity.

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

The financial support of Universidad Autónoma de Occidente is gratefully

Department of Energetics and Mechanics, PAI+ Group, Universidad Autónoma de

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Acknowledgements

acknowledged.

Author details

Occidente, Cali, Colombia

\*Address all correspondence to: slain@uao.edu.co

provided the original work is properly cited.

Santiago Laín

34

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[31] Yin C, Rosendahl L, Kaer SK, Sorensen H. Modelling the motion of cylindrical particles in a nonuniform flow. Chemical Engineering Science.

[32] Yin C, Rosendahl L, Kaer SK, Condra TJ. Use of numerical modelling in design for co-firing biomass in wallfired burners. Chemical Engineering

[33] Goldstein H. Classical Mechanics. 2nd ed. Vol. 793. New York: Addison-

[34] Gallily I, Cohen AH. On the orderly nature of the motion of nonspherical aerosol particles II. Inertial collision between a spherical large droplet and an axially symmetrical elongated particle. Journal of Colloid and Interface Science.

[35] Göz MF, Lain S, Sommerfeld M. Study of the numerical instabilities in Lagrangian tracking of bubbles and particles in two-phase flow. Computers and Chemical Engineering. 2004;28:

[36] Göz MF, Sommerfeld M, Lain S. Instabilities in Lagrangian tracking of

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2003;58:3489-3498

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2727-2733

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[20] Zastawny M, Mallouppas G, Zhao F, van Wachem B. Derivation of drag and lift force and torque coefficients for non-spherical particles in flows. International Journal of Multiphase

[21] Ouchene R, Khalij M, Arcen B, Tanière A. A new set of correlations of drag, lift and torque coefficients for non-spherical particles and large

Reynolds numbers. Powder Technology.

[22] Fan FG, Ahmadi G. Dispersion of ellipsoidal particles in an isotropic pseudo-turbulent flow field.

Transactions of the ASME, Journal of Fluids Engineering. 1995;117:154-161

[23] Olson JA. The motion of fibres in turbulent flow, stochastic simulation of isotropic homogeneous turbulence. International Journal of Multiphase

[24] Lin J, Shi X, Yu Z. The motion of fibers in an evolving mixing layer. International Journal of Multiphase

[25] Zhang H, Ahmadi G, Fan FG, McLaughlin JB. Ellipsoidal particles transport and deposition in turbulent channel flows. International Journal of Multiphase Flow. 2001;27:971-1009

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Chapter 3

Abstract

An Eulerian-Lagrangian Coupled

In this chapter, an Euler-Lagrangian double-way coupled model is presented for simulating the liquid particle dispersion ejected from a high-pressure nozzle. The Eulerian code is advanced regional prediction system (ARPS), developed by Center of Analysis and Prediction of Storm (CAPS) and Oklahoma University, USA, which is specialized in weather simulation. This code is the double way coupled with a Lagrangian one-particle model. The theoretical remarks of the double-way coupling, the simulation of the liquid droplet trajectory, and, finally, the droplet collision in the spray cloud using a binary collision model are descripts. The results of droplet velocities and diameters are compared with experimental laboratory measurements. Finally, agrochemical spraying over a cultivated field in weak wind and

Model for Droplets Dispersion

Carlos G. Sedano, César Augusto Aguirre and

from Nozzle Spray

high air temperature conditions is showed.

Lagrangian stochastic model

1. Introduction

39

Keywords: droplets, spray, multiphase flow, large-eddy simulation,

Numerous engineering applications are focused on solving the problems of dispersion of a sprayed droplet jet from a high-pressure nozzle into a gaseous medium. Spraying is used in internal combustion engines, application of agrochemicals over cultivated fields and greenhouses, irrigation systems, among others. Some questions raised by this topic of engineering can be studied using the computational simulation of multiphase flows. There are currently different ways to implement these tools. In the Eulerian approach, the physical domain is subdivided into cells of a grid space. Each cell has a portion of its volume filled by the liquid phase and the other part by the air. Continuity, momentum, energy and species, for a single-fluid mixture conservation equations, are solved in all pass time of the simulation [1, 2]. On the other hand, the Lagrangian approach at one particle proposes the velocities and positions of particles simulation (solids, liquid, vapor, or scalar species) by solving a stochastic equation following the Markov chains. The deterministic term is obtained from the average air velocity values, while the random term is like a white noise following a Brownian motion. The coupled Eulerian large-eddy simulation (LES) with Lagrangian one-particle stochastic method (STO) has been proposed in order to obtain more details on the turbulent properties of the

Armando B. Brizuela

#### Chapter 3

## An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

Carlos G. Sedano, César Augusto Aguirre and Armando B. Brizuela

#### Abstract

In this chapter, an Euler-Lagrangian double-way coupled model is presented for simulating the liquid particle dispersion ejected from a high-pressure nozzle. The Eulerian code is advanced regional prediction system (ARPS), developed by Center of Analysis and Prediction of Storm (CAPS) and Oklahoma University, USA, which is specialized in weather simulation. This code is the double way coupled with a Lagrangian one-particle model. The theoretical remarks of the double-way coupling, the simulation of the liquid droplet trajectory, and, finally, the droplet collision in the spray cloud using a binary collision model are descripts. The results of droplet velocities and diameters are compared with experimental laboratory measurements. Finally, agrochemical spraying over a cultivated field in weak wind and high air temperature conditions is showed.

Keywords: droplets, spray, multiphase flow, large-eddy simulation, Lagrangian stochastic model

#### 1. Introduction

Numerous engineering applications are focused on solving the problems of dispersion of a sprayed droplet jet from a high-pressure nozzle into a gaseous medium. Spraying is used in internal combustion engines, application of agrochemicals over cultivated fields and greenhouses, irrigation systems, among others. Some questions raised by this topic of engineering can be studied using the computational simulation of multiphase flows. There are currently different ways to implement these tools. In the Eulerian approach, the physical domain is subdivided into cells of a grid space. Each cell has a portion of its volume filled by the liquid phase and the other part by the air. Continuity, momentum, energy and species, for a single-fluid mixture conservation equations, are solved in all pass time of the simulation [1, 2]. On the other hand, the Lagrangian approach at one particle proposes the velocities and positions of particles simulation (solids, liquid, vapor, or scalar species) by solving a stochastic equation following the Markov chains. The deterministic term is obtained from the average air velocity values, while the random term is like a white noise following a Brownian motion. The coupled Eulerian large-eddy simulation (LES) with Lagrangian one-particle stochastic method (STO) has been proposed in order to obtain more details on the turbulent properties of the fluid carrying the particles. Several studies using this coupling methodology (LES-STO) can be found [3–11]. In this chapter, we focus on the ejection of droplets in air environment from a spray nozzle. The sprayed liquid is a water at 20°C temperature, and the ejection pressure reaches 3 bar. The atmosphere temperature is like the water ejected, but the air pressure is 1.013 bar. These conditions are like as Nuyttens experience [12]. The author carries out paired measurements of droplet diameters and velocities at 25 cm below the spray nozzle using phase Doppler particle analyzer (PDPA) instrument. The particle's Euler-Lagrangian double-way coupling code LES-STO is proposed for to simulate the trajectory of these particles in their liquid phase. The original finite-difference Eulerian LES code named advanced regional prediction systems (ARPS) developed by the University of Oklahoma's Center for Analysis and Forecasting of Storms (CAPS) [13] has been adapted by Aguirre [14] for the simulation of fluid particles in order to validate it with measurements of concentration of a passive gas made in a wind tunnel over flat ground [15] and in the presence of a gentle sloping hill [16]. First time, we present a random ejection algorithm of droplet diameters whose probability density function replies to the two-parameter Weibull distribution. These parameters are previously obtained using laboratory experimental data. Second time, we present the theoretical approach for obtaining the results of collision droplets into the spray. The binary collision droplet model [17–19] has been performed in the LES-STO code. This model uses the concept of symmetric weber number [20] to consider the relationship between the kinetic and surface energy of the two colliding droplets. Finally, an agrochemical spraying over a cultivated field in low wind velocity and high air temperature conditions is showed.

• The minor semiaxis of A ellipse (transverse direction y) is dy0 = 0.0046 m.

• The initial vertical velocity of liquid particles is adopted from the confined

<sup>U</sup><sup>0</sup> <sup>¼</sup> xc Lc W0,

8 ><

>:

xc ¼ 2dxoð Þ χ � 0:5 ,

<sup>o</sup> � <sup>x</sup><sup>2</sup> c � � dx<sup>2</sup>

where χ is a continuous uniform random variable in the [0, 1] interval whose

All particles are located at the z=h � Lc height within the A ellipse at the initial

The Rosin-Rammler (R-R) distribution function is a cumulative function of continuous random variable whose probability density function (p.d.f.) is a twoparameter Weibull. This distribution function is used [22, 23] to adjust experimental data of droplet diameter measurements as a function of liquid-sprayed fraction volume in order to obtain the shape m and scale k Weibull parameters. The experimental data require very precise measurements of the diameters of liquid droplets. The Doppler phase particle analyzer (PDPA) meets the necessary requirements and has the advantage of obtaining paired velocity and diameter data of very small droplets, which are ejected from the nozzle. Nuyttens [12] presents laboratory

<sup>V</sup><sup>0</sup> <sup>¼</sup> yc Lc W0,

.

(1)

(2)

• The initial horizontal component velocity of each liquid particle will depend on

where (xc, yc) are the relative horizontal positions of the liquid particle at center of the A ellipse. So, the horizontal velocity of droplets at initial time of the simulation depends directly on these relative distances. To determine the initial horizontal

> o dy<sup>2</sup> o � � � � <sup>1</sup>=<sup>2</sup> ð Þ χ � 0:5 ,

> > <sup>3</sup> <sup>p</sup> <sup>=</sup>6.

• The major semiaxis of A ellipse (longitudinal direction x) is

Spray cone HARDI ™ ISO F 110-O3 nozzle follows Nuyttens laboratory experience [12].

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

dx<sup>0</sup> ¼ tg ð Þ α=2 Lc ¼ 0:0328 m.

DOI: http://dx.doi.org/10.5772/intechopen.81110

Figure 1.

fluid simulation [21] W0 = 16.356 m s�<sup>1</sup>

the initial position within the A ellipse (Figure 1):

position (xc, yc), an algorithm of random variable is used:

8 < :

time of the simulation.

41

yc <sup>¼</sup> <sup>2</sup> dx<sup>2</sup>

average value is μχ <sup>¼</sup> <sup>0</sup>:5 and standard deviation σχ <sup>¼</sup> ffiffiffi

2.1.2 Initial distribution function of the liquid particles' diameters

### 2. Theoretical framework and techniques of numerical simulation

#### 2.1 Conditions and simulation of liquid particle ejection

In this section, we present a random ejection algorithm for simulating different diameters of droplets whose probability density function matches a Weibull distribution. The scale and shape parameters of Weibull distribution are previously obtained from laboratory experimental data using a phase Doppler particle analyzer (PDPA) performed by Nuyttens [12] from an HARDI™ spray nozzle. The sprayed liquid in laboratory experience has been water at 20°C temperature, and the ejection pressure reaches 3 bar. The atmosphere temperature is like the water ejected, but the air pressure is 1.013 bar and a calm wind.

#### 2.1.1 Initial conditions of the droplet positions and velocities

The initial conditions of ejection droplets are as follows:


An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray DOI: http://dx.doi.org/10.5772/intechopen.81110

#### Figure 1.

fluid carrying the particles. Several studies using this coupling methodology

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

2. Theoretical framework and techniques of numerical simulation

In this section, we present a random ejection algorithm for simulating different diameters of droplets whose probability density function matches a Weibull distribution. The scale and shape parameters of Weibull distribution are previously obtained from laboratory experimental data using a phase Doppler particle analyzer (PDPA) performed by Nuyttens [12] from an HARDI™ spray nozzle. The sprayed liquid in laboratory experience has been water at 20°C temperature, and the ejection pressure reaches 3 bar. The atmosphere temperature is like the water ejected,

2.1 Conditions and simulation of liquid particle ejection

but the air pressure is 1.013 bar and a calm wind.

liquid film.

40

2.1.1 Initial conditions of the droplet positions and velocities

technical specifications of the HARDI™ nozzle.

The initial conditions of ejection droplets are as follows:

• The nozzle height was located at h = 0.75 m over the ground.

• The elliptical shape A (Figure 1) for exit droplets. It was located at

Lc = 0.023 m below the nozzle because, according to the measurements of Nuyttens [12], it is the region of detachment of droplets from the sheet of the

• The angle of the spray in the transverse direction is α = 110° according to the

high air temperature conditions is showed.

(LES-STO) can be found [3–11]. In this chapter, we focus on the ejection of droplets in air environment from a spray nozzle. The sprayed liquid is a water at 20°C temperature, and the ejection pressure reaches 3 bar. The atmosphere temperature is like the water ejected, but the air pressure is 1.013 bar. These conditions are like as Nuyttens experience [12]. The author carries out paired measurements of droplet diameters and velocities at 25 cm below the spray nozzle using phase Doppler particle analyzer (PDPA) instrument. The particle's Euler-Lagrangian double-way coupling code LES-STO is proposed for to simulate the trajectory of these particles in their liquid phase. The original finite-difference Eulerian LES code named advanced regional prediction systems (ARPS) developed by the University of Oklahoma's Center for Analysis and Forecasting of Storms (CAPS) [13] has been adapted by Aguirre [14] for the simulation of fluid particles in order to validate it with measurements of concentration of a passive gas made in a wind tunnel over flat ground [15] and in the presence of a gentle sloping hill [16]. First time, we present a random ejection algorithm of droplet diameters whose probability density function replies to the two-parameter Weibull distribution. These parameters are previously obtained using laboratory experimental data. Second time, we present the theoretical approach for obtaining the results of collision droplets into the spray. The binary collision droplet model [17–19] has been performed in the LES-STO code. This model uses the concept of symmetric weber number [20] to consider the relationship between the kinetic and surface energy of the two colliding droplets. Finally, an agrochemical spraying over a cultivated field in low wind velocity and

Spray cone HARDI ™ ISO F 110-O3 nozzle follows Nuyttens laboratory experience [12].


$$\begin{cases} U\_0 = \frac{\mathcal{X}\_\mathcal{c}}{L\_\mathcal{c}} W\_{0\prime} \\ V\_0 = \frac{\mathcal{Y}\_\mathcal{c}}{L\_\mathcal{c}} W\_{0\prime} \end{cases} \tag{1}$$

where (xc, yc) are the relative horizontal positions of the liquid particle at center of the A ellipse. So, the horizontal velocity of droplets at initial time of the simulation depends directly on these relative distances. To determine the initial horizontal position (xc, yc), an algorithm of random variable is used:

$$\begin{cases} \varkappa\_c = 2d\varkappa\_o(\chi - 0.5), \\ \jmath\_c = 2\left( (d\varkappa\_o^2 - \varkappa\_c^2) \left( \frac{d\varkappa\_o^2}{d\jmath\_o^2} \right) \right)^{1/2} \end{cases} \tag{2}$$

where χ is a continuous uniform random variable in the [0, 1] interval whose average value is μχ <sup>¼</sup> <sup>0</sup>:5 and standard deviation σχ <sup>¼</sup> ffiffiffi <sup>3</sup> <sup>p</sup> <sup>=</sup>6.

All particles are located at the z=h � Lc height within the A ellipse at the initial time of the simulation.

#### 2.1.2 Initial distribution function of the liquid particles' diameters

The Rosin-Rammler (R-R) distribution function is a cumulative function of continuous random variable whose probability density function (p.d.f.) is a twoparameter Weibull. This distribution function is used [22, 23] to adjust experimental data of droplet diameter measurements as a function of liquid-sprayed fraction volume in order to obtain the shape m and scale k Weibull parameters. The experimental data require very precise measurements of the diameters of liquid droplets. The Doppler phase particle analyzer (PDPA) meets the necessary requirements and has the advantage of obtaining paired velocity and diameter data of very small droplets, which are ejected from the nozzle. Nuyttens [12] presents laboratory

measurements with this device for different working pressures with an HARDI ™ ISO F 110-O3 nozzle, among others, in calm air conditions. The values obtained in the Nuyttens [12] measurement experiences are as follows: average droplet diameters ϕ<sup>0</sup> ¼ 267:6 μm, standard deviation of droplet diameters σϕ ¼ 110:3 μm, and a modal value Mo ¼ 250:2 μm. These values allow finding the parameters of the Weibull p.d.f. for the initial conditions of ejected droplet diameter. In this work, we obtained the m and k parameters following the methodology presented [24] for the droplet diameters that are ejected with an internal pressure of 3 bar. The properties of the Weibull distribution and the fit with the experimental measurement data are described in Appendix A.

#### 2.1.3 Initial randomization of the droplet diameters

Once the shape and scale parameters of the Weibull p.d.f. are obtained, which characterize the diameters of drops ejected from the spray nozzle, it is necessary to carry out a temporal sequence for the simulation of these diameters. An algorithm based on the function of the random variable χ, already used, is proposed for the initial position of droplets in the A ellipse (Figure 1). The randomization algorithm should allow the diameters to be assigned to each ejected droplets such that the mean and standard deviation of p.d.f. droplets simulated over a long period time are close to those corresponding to the Weibull p.d.f. Using the central limit theorem for a set of values corresponding to the random variable χ, a normalized random variable Z of mean value Z ¼ 0 and standard deviation σ<sup>Z</sup> ¼ 1 can be obtained

$$Z = \frac{\overline{\chi} - \mu\_{\chi}}{\frac{\sigma\_{\chi}}{\sqrt{n}}},\tag{3}$$

Incorporating this value into Eq. (4), the random variable ϕ can be written as:

ϕ<sup>0</sup> ¼ Mo þ σϕ:Z, ϕ<sup>0</sup> ¼ Mo, symmetric normal p ð Þ :d:f: ,

Eqs. (1), (2), and (7) provide the initial conditions of velocities, positions, and diameters of the ejected liquid particles from the HARDI™ ISO F110-O3 nozzle.

Several simplifications are imposed at ejection and trajectory simulation of liquid

• Particles are considered to have a constant spherical shape in their trajectory.

Assuming these simplifications, the force per unit mass to which the liquid particles are submitted is based on a balance between gravity and drag forces per unit mass:

where Fli is the force actuating over l liquid particle in i direction i = 1, 2, 3 (x, y, z) on Cartesian coordinate system (Figure 1), ml is the mass of liquid particle, Ui is the air velocity, Vi is the liquid particle velocity, and τ is the characteristic time response or relaxation time of liquid particle, which represents the time required for the liquid adapt to sudden changes in air velocity. This last parameter can be estimated [26] as:

where CD is the dynamic drag coefficient due to the air viscosity, ρ<sup>l</sup> is the liquid particle density, and ρ is the air density. It is important to note that, if the spray injection pressure is increased, the relative velocity between the droplets and the air will be higher. This implies that the relaxation time will be decreased. In addition, if the diameter of the liquid particle decreases, the relaxation time will also be shorter. As the droplet is considered spherical, the drag coefficient CD depends on both, the diameter of droplet and the air viscosity, which is used for the calculation of the

<sup>ℜ</sup>el <sup>¼</sup> <sup>ϕ</sup> Ui � Vi j j

Several drag coefficient expressions have been analyzed [27]. The authors showed that Turton expression [28] has given better results in this simulation case:

1

� gi δi3 |ffl{zffl} Gravity acceleration

<sup>2</sup> <sup>ϕ</sup><sup>0</sup> � Mo � � !, <sup>ϕ</sup><sup>0</sup> 6¼ Mo, asymmetric Weibull p ð Þ :d:f: :

(7)

, (8)

Ui � Vi j j , (9)

<sup>ν</sup> : (10)

Z

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

• The rotating motion of the particle is not considered.

Fl i ml

• The ratio between droplet and air densities is very large.

<sup>¼</sup> Ui � Vi τ |fflfflfflffl{zfflfflfflffl} Drag acceleration

<sup>τ</sup> <sup>¼</sup> <sup>4</sup> 3 φ CD ρl ρ

Reynolds number referred to the droplet ℜel:

43

ϕ<sup>0</sup> ¼ Mo þ σϕ:Z: exp

particle phenomena:

2.1.4 Dynamic parameters of liquid particles

DOI: http://dx.doi.org/10.5772/intechopen.81110

8 ><

>:

where n is the sample size. Michelot [25] performed several tests with different n values using 1 million particles to find an acceptable number from computational cost time. The author concludes that n = 50 is a good value to obtain the standard normal random variable for generating random numbers with χ whose μχ ¼ 0:5 and σχ <sup>¼</sup> ffiffiffi <sup>3</sup> <sup>p</sup> <sup>=</sup>6.

With this method, it is possible to simulate diameters of liquid particles that follow the normal distribution from random number generation

$$
\phi\_0 = \overline{\phi\_0} + \sigma\_\phi Z,\tag{4}
$$

where ϕ<sup>0</sup> and σϕ are the mean and standard deviation values of the Weibull p.d.f., respectively, whose expressions are given as:

$$\begin{cases} \overline{\phi\_0} = k \, \Gamma \Big( 1 + \frac{1}{m} \Big), \\\\ \sigma\_{\phi} = k \, \Big\{ \Gamma \big( 1 + \frac{2}{m} \big) - \left[ \Gamma \left( 1 + \frac{1}{m} \right) \right]^2 \Big\}^{\prime \frac{1}{2}} \end{cases} . \tag{5}$$

However, in Eq. (4), we do not consider the asymmetry of the Weibull p.d.f. For this, it is necessary to incorporate the mode (Mo) of Weibull p.d.f. whose expression is given as:

$$\mathbf{M}\mathbf{o} = k \left(\frac{m-1}{m}\right)^{1/m}.\tag{6}$$

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray DOI: http://dx.doi.org/10.5772/intechopen.81110

Incorporating this value into Eq. (4), the random variable ϕ can be written as:

$$\begin{cases} \phi\_0 = \mathcal{M}\boldsymbol{\sigma} + \sigma\_{\phi} \mathcal{Z}, & \overline{\phi\_0} = \mathcal{M}\boldsymbol{\sigma}, \text{ symmetric } (\text{normal p.d.f.}), \\\\ \phi\_0 = \mathcal{M}\boldsymbol{\sigma} + \sigma\_{\phi} \mathcal{Z}. \exp\left(\frac{\mathcal{Z}}{2(\overline{\phi\_0} - \mathcal{M}\boldsymbol{\sigma})}\right), & \overline{\phi\_0} \neq \mathcal{M}\boldsymbol{\sigma}, \text{ asymmetric } (\text{Weibull p.d.f.}). \end{cases} \tag{7}$$

Eqs. (1), (2), and (7) provide the initial conditions of velocities, positions, and diameters of the ejected liquid particles from the HARDI™ ISO F110-O3 nozzle.

#### 2.1.4 Dynamic parameters of liquid particles

measurements with this device for different working pressures with an HARDI ™ ISO F 110-O3 nozzle, among others, in calm air conditions. The values obtained in the Nuyttens [12] measurement experiences are as follows: average droplet diameters ϕ<sup>0</sup> ¼ 267:6 μm, standard deviation of droplet diameters σϕ ¼ 110:3 μm, and a modal value Mo ¼ 250:2 μm. These values allow finding the parameters of the Weibull p.d.f. for the initial conditions of ejected droplet diameter. In this work, we obtained the m and k parameters following the methodology presented [24] for the droplet diameters that are ejected with an internal pressure of 3 bar. The properties of the Weibull distribution and the fit with the experimental measurement data are

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

Once the shape and scale parameters of the Weibull p.d.f. are obtained, which characterize the diameters of drops ejected from the spray nozzle, it is necessary to carry out a temporal sequence for the simulation of these diameters. An algorithm based on the function of the random variable χ, already used, is proposed for the initial position of droplets in the A ellipse (Figure 1). The randomization algorithm should allow the diameters to be assigned to each ejected droplets such that the mean and standard deviation of p.d.f. droplets simulated over a long period time are close to those corresponding to the Weibull p.d.f. Using the central limit theorem for a set of values corresponding to the random variable χ, a normalized random variable Z of mean value Z ¼ 0 and standard deviation σ<sup>Z</sup> ¼ 1 can be obtained

> <sup>Z</sup> <sup>¼</sup> <sup>χ</sup> � μχ σχ ffiffi

where n is the sample size. Michelot [25] performed several tests with different n values using 1 million particles to find an acceptable number from computational cost time. The author concludes that n = 50 is a good value to obtain the standard normal random variable for generating random numbers with χ whose μχ ¼ 0:5

With this method, it is possible to simulate diameters of liquid particles that

where ϕ<sup>0</sup> and σϕ are the mean and standard deviation values of the Weibull p.d.f.,

,

However, in Eq. (4), we do not consider the asymmetry of the Weibull p.d.f. For this, it is necessary to incorporate the mode (Mo) of Weibull p.d.f. whose expres-

> m � �1=<sup>m</sup>

� � � � <sup>2</sup> n o1=<sup>2</sup>

1 m � �

> m � � � <sup>Γ</sup> <sup>1</sup> <sup>þ</sup> <sup>1</sup>

Mo <sup>¼</sup> <sup>k</sup> <sup>m</sup> � <sup>1</sup>

follow the normal distribution from random number generation

ϕ<sup>0</sup> ¼ k:Γ 1 þ

σϕ <sup>¼</sup> <sup>k</sup>: <sup>Γ</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>

respectively, whose expressions are given as:

8 >>><

>>>:

<sup>n</sup> <sup>p</sup> , (3)

ϕ<sup>0</sup> ¼ ϕ<sup>0</sup> þ σϕZ, (4)

:

: (6)

(5)

m

described in Appendix A.

and σχ <sup>¼</sup> ffiffiffi

sion is given as:

42

<sup>3</sup> <sup>p</sup> <sup>=</sup>6.

2.1.3 Initial randomization of the droplet diameters

Several simplifications are imposed at ejection and trajectory simulation of liquid particle phenomena:


Assuming these simplifications, the force per unit mass to which the liquid particles are submitted is based on a balance between gravity and drag forces per unit mass:

$$\frac{F\_{l\_i}}{m\_l} = \underbrace{\frac{U\_i - V\_i}{\tau}}\_{\text{Drag}} - \underbrace{g\_i \delta\_{i3}}\_{\text{Gravity}},\tag{8}$$

where Fli is the force actuating over l liquid particle in i direction i = 1, 2, 3 (x, y, z) on Cartesian coordinate system (Figure 1), ml is the mass of liquid particle, Ui is the air velocity, Vi is the liquid particle velocity, and τ is the characteristic time response or relaxation time of liquid particle, which represents the time required for the liquid adapt to sudden changes in air velocity. This last parameter can be estimated [26] as:

$$\tau = \frac{4}{3} \frac{\varrho}{C\_D} \frac{\rho\_l}{\rho} \frac{1}{|U\_i - V\_i|},\tag{9}$$

where CD is the dynamic drag coefficient due to the air viscosity, ρ<sup>l</sup> is the liquid particle density, and ρ is the air density. It is important to note that, if the spray injection pressure is increased, the relative velocity between the droplets and the air will be higher. This implies that the relaxation time will be decreased. In addition, if the diameter of the liquid particle decreases, the relaxation time will also be shorter. As the droplet is considered spherical, the drag coefficient CD depends on both, the diameter of droplet and the air viscosity, which is used for the calculation of the Reynolds number referred to the droplet ℜel:

$$\Re e\_l = \phi \frac{|U\_i - V\_i|}{\nu}. \tag{10}$$

Several drag coefficient expressions have been analyzed [27]. The authors showed that Turton expression [28] has given better results in this simulation case: Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

$$\begin{cases} \text{C}\_{D} = \frac{24}{\Re e\_{l}} (1 + 0.173 \Re e\_{l}^{0.657}) + \frac{0.413}{1 + 16300 \Re e\_{l}^{-1.09}}, & \text{if } \Re e\_{l} < 2 \times 10^{5} \\\\ \text{C}\_{D} = 0.465, & \text{if } \Re e\_{l} \ge 2 \times 10^{5} \end{cases}. \tag{11}$$

When the drag and gravity forces are balanced, the liquid particles reach the sedimentation regime. In this case of free fall, the droplets have only vertical velocity component. This velocity is named sedimentation velocity Ui � Vi j jδi<sup>3</sup> ¼ Vs. From Eqs. (8) and (9):

$$V\_s = \left(\frac{4}{3} \frac{\rho\_l \phi \mathbf{g}\_i \delta\_{i3}}{\rho \mathbf{C}\_{D,s}}\right)^{1/2}.\tag{12}$$

The tensors hij Ui ð Þ ; t and qij Ui ð Þ ; t are determined dynamically according to the

It is necessary to simulate the air velocity at liquid particle position Ui. The LES

<sup>i</sup> þ u�

The LES code advanced regional prediction system (ARPS) developed by Center of Analysis and Prediction of Storm (CAPS) and Oklahoma University [13] numerically integrates the time-dependent equations of mass balance, forces and energy of the largest turbulent scales. Filtered continuity as in Eq. (17), filtered momentum of fluid velocity as in Eq. (18), and filtered momentum of scalars as in Eq. (19) are

<sup>0</sup> � t

<sup>00</sup> ð Þ: (15)

<sup>i</sup> : (16)

¼ 0, (17)

� Ii, (18)

, (19)

∂~ S a⊕ ij ∂xj

ij is the anisotropic deformation tensor, ν is the

Fli (20)

characteristics of the turbulence at each position of the simulation and at each instant time. This requires that the LES equations are coupling with Lagrangian stochastic model. In Eq. (14), ηð Þt denotes the random characteristic variable of zero

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

η<sup>i</sup> t <sup>0</sup> ð Þη<sup>j</sup> t <sup>00</sup> ð Þ D E <sup>¼</sup> <sup>δ</sup>ij<sup>δ</sup> <sup>t</sup>

method decomposes in a resolved component and a fluctuation :

Ui <sup>¼</sup> <sup>u</sup><sup>⊕</sup>

∂u~<sup>⊕</sup> i ∂xi

<sup>B</sup><sup>⊕</sup> � <sup>∂</sup>p<sup>0</sup>

molecular viscosity, and Φ<sup>ψ</sup> represents the sink and sources of the scalars variables ψ. The variables with tilde indicate that they have been weighted by the density of air

� �, which is only dependent of z (vertical) height. The pressure equation is obtained using the material derivative of the state equation for moist air and replacing the time derivative of density by velocity divergence using the continuity equation. The correlation terms containing unsolved scales ~τij and ~τi<sup>ψ</sup> are modeled using the

From Lagrangian stochastic equation to the Eulerian LES model taken into account, the number of liquid particles is very large near the nozzle. The coupling has been computed by adding term at filtered momentum of fluid velocity Eq. (18), which expresses the additional momentum due to the presence of liquid

> <sup>I</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> <sup>Δ</sup><sup>V</sup> <sup>∑</sup> nΔ l¼1

⊕ ∂xi

� <sup>∂</sup>~τij ∂xj þ 2ν

<sup>¼</sup> <sup>Φ</sup><sup>ψ</sup> � <sup>∂</sup>~τi<sup>ψ</sup>

∂xj

¼ ρgi

S a

∂ u~<sup>⊕</sup> <sup>j</sup> ψ<sup>⊕</sup> � �

∂xj

mean and covariance:

described as follows:

u~⊕ <sup>i</sup> <sup>¼</sup> <sup>ρ</sup>u<sup>⊕</sup> i

45

∂u~<sup>⊕</sup> i ∂t þ

where B<sup>⊕</sup> is a buoyancy force, ~

dynamic Smagorinsky formulation [29].

2.2.3 Lagrangian to the Eulerian coupling

particles per volume of carrier fluid:

∂ u~<sup>⊕</sup> <sup>i</sup> u<sup>⊕</sup> j � �

∂xj

∂ψ~ <sup>⊕</sup> ∂t þ

2.2.2 The Eulerian flow model

DOI: http://dx.doi.org/10.5772/intechopen.81110

Note that CD,s is the drag coefficient at sedimentation regime. It depends on ℜel (Eq. (11)) and therefore on the sedimentation velocity itself (Eq. (10)). So, Vs can only be calculated iteratively.

The time elapsed until the particle reaches the sedimentation velocity can be written as:

$$
\pi\_s = \frac{V\_s}{\mathcal{g}}.\tag{13}
$$

This is an important parameter of liquid particles because if the time elapsed until the liquid particle reaches the ground is longer than the sedimentation time, it will be exposed to drift.

#### 2.2 Euler-Lagrangian double-way coupled model

The double-way coupled model presents a bidirectional coupling between the Eulerian and Lagrangian equation systems. Based on the Eulerian approach, the large-eddy simulation (LES) technique is proposed to obtain a detailed turbulent flow. The turbulent intensity of the fluid that transports the liquid particles is taken into account in the simulation of its trajectories. In this approach, it is not possible to obtain a full description of all eddies, so the LES technique is applied for resolving the larges scales of turbulence. The small scales are modeled by subgrid eddy viscosity model (SGS). A dynamic SGS model proposed by Germano [29] is implemented in ARPS by Aguirre [14]. On the other hand, the Lagrangian form is proposed to simulate the trajectories of the liquid particles. In the double-way-coupled LES-STO model, it is considered that the intensity of turbulent flow is taken into account in Lagrangian stochastic equation, and the presence of the liquid particles is taken into account in the momentum equation for LES.

#### 2.2.1 The Lagrangian stochastic model

The governing equations of the liquid particles trajectories are based on a Lagrangian stochastic model at a one-particle and one-time scale following the classical equation of Langevin. The air velocity model at liquid particle position Ui has a deterministic term and a random term:

$$\frac{d\mathbf{U}\_i}{dt} = \underbrace{h\_{\vec{\eta}}(U\_i, t)}\_{\text{Deterministic}} + \underbrace{q\_{\vec{\eta}}(U\_i, t)\eta\_j(t)}\_{\text{Random}}.\tag{14}$$

#### An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray DOI: http://dx.doi.org/10.5772/intechopen.81110

The tensors hij Ui ð Þ ; t and qij Ui ð Þ ; t are determined dynamically according to the characteristics of the turbulence at each position of the simulation and at each instant time. This requires that the LES equations are coupling with Lagrangian stochastic model. In Eq. (14), ηð Þt denotes the random characteristic variable of zero mean and covariance:

$$
\left\langle \eta\_i(t')\eta\_j(t'') \right\rangle = \delta\_{\vec{\eta}}\delta(t'-t''). \tag{15}
$$

#### 2.2.2 The Eulerian flow model

CD <sup>¼</sup> <sup>24</sup> ℜel

8 < :

written as:

for LES.

44

2.2.1 The Lagrangian stochastic model

has a deterministic term and a random term:

dUi

dt <sup>¼</sup> hij Ui ð Þ ; <sup>t</sup> |fflfflfflffl{zfflfflfflffl} Deterministic

1 þ 0:173ℜe

Ui � Vi j jδi<sup>3</sup> ¼ Vs. From Eqs. (8) and (9):

only be calculated iteratively.

will be exposed to drift.

2.2 Euler-Lagrangian double-way coupled model

0:657 l � � <sup>þ</sup>

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

velocity component. This velocity is named sedimentation velocity

Vs <sup>¼</sup> <sup>4</sup> 3 ρlϕgi δi3 ρCD,s � �1=<sup>2</sup>

CD <sup>¼</sup> <sup>0</sup>:465, if <sup>ℜ</sup>el <sup>≥</sup> <sup>2</sup> � <sup>10</sup><sup>5</sup>

When the drag and gravity forces are balanced, the liquid particles reach the sedimentation regime. In this case of free fall, the droplets have only vertical

Note that CD,s is the drag coefficient at sedimentation regime. It depends on ℜel (Eq. (11)) and therefore on the sedimentation velocity itself (Eq. (10)). So, Vs can

The time elapsed until the particle reaches the sedimentation velocity can be

<sup>τ</sup><sup>s</sup> <sup>¼</sup> Vs

This is an important parameter of liquid particles because if the time elapsed until the liquid particle reaches the ground is longer than the sedimentation time, it

The double-way coupled model presents a bidirectional coupling between the Eulerian and Lagrangian equation systems. Based on the Eulerian approach, the large-eddy simulation (LES) technique is proposed to obtain a detailed turbulent flow. The turbulent intensity of the fluid that transports the liquid particles is taken into account in the simulation of its trajectories. In this approach, it is not possible to obtain a full description of all eddies, so the LES technique is applied for resolving the larges scales of turbulence. The small scales are modeled by subgrid eddy viscosity model (SGS). A dynamic SGS model proposed by Germano [29] is implemented in ARPS by Aguirre [14]. On the other hand, the Lagrangian form is proposed to simulate the trajectories of the liquid particles. In the double-way-coupled LES-STO model, it is considered that the intensity of turbulent flow is taken into account in Lagrangian stochastic equation, and the presence of the liquid particles is taken into account in the momentum equation

The governing equations of the liquid particles trajectories are based on a Lagrangian stochastic model at a one-particle and one-time scale following the classical equation of Langevin. The air velocity model at liquid particle position Ui

þ qij Ui ð Þ ; t η<sup>j</sup>


ð Þt

: (14)

0:413 <sup>1</sup> <sup>þ</sup> <sup>16300</sup>ℜe�1:<sup>09</sup> l

, if <sup>ℜ</sup>el < 2 � <sup>10</sup><sup>5</sup>

: (12)

<sup>g</sup> : (13)

:

(11)

It is necessary to simulate the air velocity at liquid particle position Ui. The LES method decomposes in a resolved component and a fluctuation :

$$U\_i = \mathfrak{u}\_i^{\Theta} + \mathfrak{u}\_i^{-}. \tag{16}$$

The LES code advanced regional prediction system (ARPS) developed by Center of Analysis and Prediction of Storm (CAPS) and Oklahoma University [13] numerically integrates the time-dependent equations of mass balance, forces and energy of the largest turbulent scales. Filtered continuity as in Eq. (17), filtered momentum of fluid velocity as in Eq. (18), and filtered momentum of scalars as in Eq. (19) are described as follows:

$$\frac{\partial \tilde{u}\_i^{\Theta}}{\partial \mathbf{x}\_i} = \mathbf{0},\tag{17}$$

$$\frac{\partial \ddot{u}\_i^{\oplus}}{\partial t} + \frac{\partial \left(\ddot{u}\_i^{\oplus} u\_j^{\oplus}\right)}{\partial \mathbf{x}\_j} = \overline{\rho} \mathbf{g}\_i B^{\oplus} - \frac{\partial p^{\prime \oplus}}{\partial \mathbf{x}\_i} - \frac{\partial \ddot{\mathbf{r}}\_{ij}}{\partial \mathbf{x}\_j} + 2\nu \frac{\partial \ddot{S}\_{ij}^{a \oplus}}{\partial \mathbf{x}\_j} - \Im\_{i\nu} \tag{18}$$

$$\frac{\partial \tilde{\boldsymbol{\mu}}^{\oplus}}{\partial t} + \frac{\partial \left(\tilde{\boldsymbol{u}}\_{j}^{\oplus} \boldsymbol{\mu}^{\oplus}\right)}{\partial \boldsymbol{\alpha}\_{j}} = \boldsymbol{\Phi}\_{\boldsymbol{\mu}} - \frac{\partial \tilde{\boldsymbol{\tau}}\_{i\boldsymbol{\mu}}}{\partial \boldsymbol{\alpha}\_{j}},\tag{19}$$

where B<sup>⊕</sup> is a buoyancy force, ~ S a ij is the anisotropic deformation tensor, ν is the molecular viscosity, and Φ<sup>ψ</sup> represents the sink and sources of the scalars variables ψ. The variables with tilde indicate that they have been weighted by the density of air u~⊕ <sup>i</sup> <sup>¼</sup> <sup>ρ</sup>u<sup>⊕</sup> i � �, which is only dependent of z (vertical) height. The pressure equation is obtained using the material derivative of the state equation for moist air and replacing the time derivative of density by velocity divergence using the continuity equation. The correlation terms containing unsolved scales ~τij and ~τi<sup>ψ</sup> are modeled using the dynamic Smagorinsky formulation [29].

#### 2.2.3 Lagrangian to the Eulerian coupling

From Lagrangian stochastic equation to the Eulerian LES model taken into account, the number of liquid particles is very large near the nozzle. The coupling has been computed by adding term at filtered momentum of fluid velocity Eq. (18), which expresses the additional momentum due to the presence of liquid particles per volume of carrier fluid:

$$\mathcal{O}\_i = \frac{1}{\Delta V} \sum\_{l=1}^{n\_{\Delta}} F\_{li} \tag{20}$$

where is the force of the liquid particle l in i direction, ΔV = ΔxΔyΔz is the grid cell volume, and n<sup>Δ</sup> is the number of liquid particles within the grid cell. Using Eq. (8), the additional momentum is given by:

$$\Im \Im\_i = \frac{1}{\Delta V} \sum\_{l=1}^{n\_{\Delta}} \left[ m\_l \left( \frac{U\_i - V\_i}{\tau} - \mathbf{g}\_j \delta\_{\bar{\beta}} \right) \right]. \tag{21}$$

It is necessary to note that in the first Eq. (26), the rate Δt=τ must be greater than one for convergence of the numerical solution. Cases in which the liquid particle diameter is very small, the second Eq. (26) must be used considering that the sedimentation velocity Vs has been reached before Δt time step has elapsed. For this reason, the simulation time step Δt must be chosen less than the relaxation time of the smallest possible liquid particle. According to the experimental measurements by Nuyttens [12], the liquid particles whose diameters are smaller to 50 μm are exposed to drift before reaching the ground. These particles have relaxation times less than 7.6 ms. So, a simulation time step Δt = 0.2 ms has been chosen due to little size of cell grid ΔV, whereby liquid particles whose diameters are smaller than 7 μm

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

Once the droplets ejected from the spray nozzle and having simulated their positions, velocities and diameters along their trajectory, it is necessary to consider the collision. Binary droplet collision models are a widely used theoretical approximation to obtain the outcome of the interaction droplets [17–20, 33–39]. This model consists of estimating the positions, velocities, and diameters of droplets after the collision. In addition, satellite droplets can be created from the ligament breakup as

The binary droplet collision is simulated using three important parameters. The ratio of the droplet diameters Δ (Eq. (27)), the dimensionless symmetric Weber number (Wes) [20] relating kinetic energy vs. surface energy (Eq. (28)), and the dimensionless impact parameter (Imp) takes into account the way in which the two

> <sup>Δ</sup> <sup>¼</sup> <sup>ϕ</sup><sup>S</sup> ϕL

> > � � � 2 þ V ! mL � � �

ϕ<sup>L</sup> þ ϕ<sup>S</sup>

� � � 2

,

mR is the velocity of mass center. If the droplets have the same

ρlϕSΔ<sup>3</sup> V ! mS � � �

Imp <sup>¼</sup> <sup>2</sup><sup>X</sup>

!

V !

8 < :

V ! mS ¼ V ! <sup>S</sup> � V ! mR

mass center of the incoming droplets, and X is the projection of the distance between the droplet centers in the normal direction to the relative velocity

> mL ¼ V ! <sup>L</sup> � V ! mR,

where the subscripts S and L indicate the smaller and larger droplet, respec-

mS and V !

Wes ¼

, (27)

, (29)

mL are the relative velocities to the

(30)

<sup>12</sup>σ<sup>Δ</sup> <sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>2</sup> � � , (28)

are being considered for calculation with the second Eq. (26).

2.3 Binary collision droplet model

DOI: http://dx.doi.org/10.5772/intechopen.81110

2.3.1 Parameters of binary collision

droplets impact (Eq. (29)):

tively, σ denote the surface tension, V

<sup>L</sup> as shown in Figure 2.

V ! <sup>R</sup> ¼ V ! <sup>S</sup> � V !

In Eq. (30), V

density:

47

!

a consequence of it.

#### 2.2.4 Eulerian to Lagrangian coupling

Aguirre and Brizuela [11] show that the coupling LES-STO model allows to find the expressions of the deterministic and random terms of Eq. (14) using the velocity-filtered density function (VFDF) proposed by Gicquel et al. [30]:

$$\begin{cases} h\_{\vec{\eta}}(U\_i, t) = \frac{d u\_j^{\oplus}}{dt} + a\_{\vec{\eta}} u\_j^{-}, \\ q\_{\vec{\eta}}(U\_i, t) = \sqrt{\mathcal{C}\_0 \varepsilon} \,\delta\_{\vec{\eta}}. \end{cases} \tag{22}$$

The material derivatives of the velocity-filtered air flow, subgrid turbulent kinetic energy K<sup>¬</sup> , and energy molecular dissipation ε are calculated using the ARPS code at each position and time step of the simulation. The Kolmogorov constant value is C0 = 2.1. Therefore, we only need to evaluate the αij tensor. Aguirre and Brizuela [11] propose the expression of αij for inhomogeneous and anisotropic turbulence:

$$a\_{\vec{\eta}} = \frac{1}{2K^-} \frac{dK^-}{dt} \delta\_{\vec{\eta}} - \left(\frac{3}{4} C\_0\right) \frac{\varepsilon}{K^-} \delta\_{\vec{\eta}} + \left(\frac{R\_{\vec{\eta}}}{2K^-} - \frac{\delta\_{\vec{\eta}}}{3}\right) \frac{\varepsilon}{K^-}.\tag{23}$$

The subgrid turbulent kinetic energy is solved by 1.5 order transport equation [31] and Rij ¼ u� <sup>i</sup> u� j � �<sup>⊕</sup> is the Reynolds SGS stress tensor.

The unresolved velocity component u� <sup>j</sup> in Eq. (16) is obtained in discrete form using the Markov chains:

$$u\_{j^{(n)}}^{-} = u\_{j^{(n-1)}}^{-} + a\_{\vec{\eta}^{\,}(n)} u\_{j^{(n-1)}}^{-} \Delta t + \sqrt{C\_0 \varepsilon \Delta t} \, \chi\_{(n)},\tag{24}$$

where the subscript (n) denotes the value at present time of the simulation, while (n�1) is the value in the previous time step. The first pass time, subscript (0), is considered as isotropic homogeneous turbulence [32]:

$$
\mu\_{j(0)}^- = \sqrt{\frac{2}{3} K\_{(0)}^-} \varkappa\_{(0)}.\tag{25}
$$

With Eqs. (16) and (22–25), it is possible to calculate the air velocity at the liquid particle position. The equations describing the motion of the liquid particle in its discrete form are:

$$\begin{cases} V\_{i(n+1)} = V\_{i(n)} + \frac{\Delta t}{\tau} \left( U\_{i(n)} - V\_{i(n)} \right) - g\_i \Delta t \delta\_{i3}, & \text{if} \quad \tau > \Delta t, \\\\ V\_{i(n+1)} = U\_{i(n)} - V\_{i} \delta\_{i3}, & \text{if} \quad \tau \le \Delta t, \\\\ X\_{i(n+1)} = \frac{V\_{i(n+1)} - V\_{i(n)}}{2} \Delta t. \end{cases} \tag{26}$$

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray DOI: http://dx.doi.org/10.5772/intechopen.81110

It is necessary to note that in the first Eq. (26), the rate Δt=τ must be greater than one for convergence of the numerical solution. Cases in which the liquid particle diameter is very small, the second Eq. (26) must be used considering that the sedimentation velocity Vs has been reached before Δt time step has elapsed. For this reason, the simulation time step Δt must be chosen less than the relaxation time of the smallest possible liquid particle. According to the experimental measurements by Nuyttens [12], the liquid particles whose diameters are smaller to 50 μm are exposed to drift before reaching the ground. These particles have relaxation times less than 7.6 ms. So, a simulation time step Δt = 0.2 ms has been chosen due to little size of cell grid ΔV, whereby liquid particles whose diameters are smaller than 7 μm are being considered for calculation with the second Eq. (26).

#### 2.3 Binary collision droplet model

where is the force of the liquid particle l in i direction, ΔV = ΔxΔyΔz is the grid cell volume, and n<sup>Δ</sup> is the number of liquid particles within the grid cell. Using

> Ui � Vi <sup>τ</sup> � gj

Aguirre and Brizuela [11] show that the coupling LES-STO model allows to find

du<sup>⊕</sup> j dt <sup>þ</sup> <sup>α</sup>iju�

<sup>C</sup>0<sup>ε</sup> <sup>p</sup> <sup>δ</sup>ij:

<sup>K</sup>� <sup>δ</sup>ij <sup>þ</sup>

j nð Þ �<sup>1</sup> <sup>Δ</sup><sup>t</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

The subgrid turbulent kinetic energy is solved by 1.5 order transport equation

is the Reynolds SGS stress tensor.

where the subscript (n) denotes the value at present time of the simulation, while (n�1) is the value in the previous time step. The first pass time, subscript (0),

r

ffiffiffiffiffiffiffiffiffiffiffiffi 2 3 K� ð Þ 0

With Eqs. (16) and (22–25), it is possible to calculate the air velocity at the liquid particle position. The equations describing the motion of the liquid particle in its

� � � �

δj3

j ,

Rij <sup>2</sup>K� � <sup>δ</sup>ij 3 � � ε

<sup>j</sup> in Eq. (16) is obtained in discrete form

<sup>C</sup>0εΔ<sup>t</sup> <sup>p</sup> <sup>χ</sup>ð Þ <sup>n</sup> , (24)

χð Þ <sup>0</sup> : (25)

Δtδ<sup>i</sup>3, if τ > Δt,

, and energy molecular dissipation ε are calculated using the ARPS

: (21)

(22)

(26)

<sup>K</sup>� : (23)

ml

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

the expressions of the deterministic and random terms of Eq. (14) using the velocity-filtered density function (VFDF) proposed by Gicquel et al. [30]:

hij Ui ð Þ¼ ; t

qij Ui ð Þ¼ ; <sup>t</sup> ffiffiffiffiffiffiffiffi

The material derivatives of the velocity-filtered air flow, subgrid turbulent

code at each position and time step of the simulation. The Kolmogorov constant value is C0 = 2.1. Therefore, we only need to evaluate the αij tensor. Aguirre and Brizuela [11] propose the expression of αij for inhomogeneous and anisotropic

Eq. (8), the additional momentum is given by:

2.2.4 Eulerian to Lagrangian coupling

<sup>α</sup>ij <sup>¼</sup> <sup>1</sup> 2K�

<sup>i</sup> u� j � �<sup>⊕</sup>

> u� j nð Þ ¼ u�

Vi nð Þ <sup>þ</sup><sup>1</sup> ¼ Vi nð Þ þ

Xi nð Þ <sup>þ</sup><sup>1</sup> <sup>¼</sup> Vi nð Þ <sup>þ</sup><sup>1</sup> � Vi nð Þ

The unresolved velocity component u�

kinetic energy K<sup>¬</sup>

[31] and Rij ¼ u�

discrete form are:

8 ><

>:

�

46

using the Markov chains:

turbulence:

<sup>I</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> <sup>Δ</sup><sup>V</sup> <sup>∑</sup> nΔ l¼1

> 8 ><

> >:

dK�

is considered as isotropic homogeneous turbulence [32]:

Δt

2

dt <sup>δ</sup>ij � <sup>3</sup>

4 C<sup>0</sup> � � ε

j nð Þ �<sup>1</sup> <sup>þ</sup> <sup>α</sup>ij nð Þu�

u� <sup>j</sup>ð Þ <sup>0</sup> ¼

<sup>τ</sup> Ui nð Þ � Vi nð Þ � � � gi

Δt:

Vi nð Þ <sup>þ</sup><sup>1</sup> ¼ Ui nð Þ � Vsδ<sup>i</sup>3, if τ ≤ Δt,

Once the droplets ejected from the spray nozzle and having simulated their positions, velocities and diameters along their trajectory, it is necessary to consider the collision. Binary droplet collision models are a widely used theoretical approximation to obtain the outcome of the interaction droplets [17–20, 33–39]. This model consists of estimating the positions, velocities, and diameters of droplets after the collision. In addition, satellite droplets can be created from the ligament breakup as a consequence of it.

#### 2.3.1 Parameters of binary collision

The binary droplet collision is simulated using three important parameters. The ratio of the droplet diameters Δ (Eq. (27)), the dimensionless symmetric Weber number (Wes) [20] relating kinetic energy vs. surface energy (Eq. (28)), and the dimensionless impact parameter (Imp) takes into account the way in which the two droplets impact (Eq. (29)):

$$
\Delta = \frac{\phi\_S}{\phi\_L},
\tag{27}
$$

$$\text{Wes} = \frac{\rho\_l \phi\_{\text{S}} \Delta^3 \left| \overrightarrow{V}\_{m\text{S}} \right|^2 + \left| \overrightarrow{V}\_{mL} \right|^2}{12 \sigma \Delta \left( 1 + \Delta^2 \right)},\tag{28}$$

$$\text{Imp} = \frac{2\text{X}}{\phi\_L + \phi\_S},\tag{29}$$

where the subscripts S and L indicate the smaller and larger droplet, respectively, σ denote the surface tension, V ! mS and V ! mL are the relative velocities to the mass center of the incoming droplets, and X is the projection of the distance between the droplet centers in the normal direction to the relative velocity V ! <sup>R</sup> ¼ V ! <sup>S</sup> � V ! <sup>L</sup> as shown in Figure 2.

$$\begin{cases} \overrightarrow{V}\_{mL} = \overrightarrow{V}\_{L} - \overrightarrow{V}\_{mRr} \\ \overrightarrow{V}\_{mS} = \overrightarrow{V}\_{S} - \overrightarrow{V}\_{mR} \end{cases},\tag{30}$$

In Eq. (30), V ! mR is the velocity of mass center. If the droplets have the same density:

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

Figure 2. Scheme of small droplet S and large droplet L before collision (dashed line) and at contact instant (solid line).

$$
\overrightarrow{V}\_{mR} = \frac{\overrightarrow{V}\_S \phi\_S^3 + \overrightarrow{V}\_L \phi\_L^3}{\phi\_S^3 + \phi\_L^3} \tag{31}
$$

It is necessary to obtain Dp. It is the distance from large droplet center

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

!

The plane equation passing through the small drop center is uRxS þ vRyS þ wRzSþ

constant of plane equation. This constant is obtained as: D ¼ �uRxS � vRyS � wRzR.

Dp <sup>¼</sup> uRxL <sup>þ</sup> vRyL <sup>þ</sup> wRzL <sup>þ</sup> <sup>D</sup> uR<sup>2</sup> <sup>þ</sup> vR<sup>2</sup> <sup>þ</sup> wR<sup>2</sup> ð Þ<sup>1</sup>

In each time of numerical simulation, it checks whether the collision between two droplets occurs. For obtaining a more optimize algorithm, collision boxes are placed around and inside the liquid particle ejection spray. The sizes of grid boxes vary dynamically, adjusting to the boundaries of the particle domain as shown in Figure 3a. The size of the boxes is the same as the Eulerian calculation grid in horizontal direction Δb = Δx = Δy = 0.1 m. In this way, every drop inside this box will be questioned about whether it collided with the other drops that are in the same box. When a binary collision is successfully found (e.g., 5–6; 3–7 in

Figure 3b), the pairs are marked and removed from the next iteration of detection. This technique was proposed by Michelot [25] and used by Aguirre [4, 14] to consider the diffusion of chemical species that are carried by fluid particles. It is evident that due to the temporal discretization of the numerical solution of droplet motion equations, it is almost impossible that for an instant of discretized time t(n) = t(n-1) + Δt, the contact between two drops can be concurrent. Most likely, by that instant time t(n-1), the contact is about to occur and at instant time t(n), it has already occurred. When the distance between the centers of the two drops inside a collision box is less than the sum of their radii, then the drops collided. In this case, the time Δt" elapsed from the collision to the computation instant time t(n) to be calculated. The particles are repositioned at the moment of collision t" = t(n) � Δt", and the impact factor is calculated according to the positions of their centers as shown in Figure 4. The lapse time Δt" for repositioning the droplets at instant of

(a) Collision boxes around and inside the spray ejection of droplets and (b) droplets inside the collision box at

. This plane P-P is shown in green color in Figure 2.

!

2

 

<sup>R</sup> velocity passing through the small

<sup>R</sup>. In this expression, D is a

: (36)

xL; yL; zL

droplet center xS; yS; zS

to the perpendicular plane at V

DOI: http://dx.doi.org/10.5772/intechopen.81110

So, the Dp distance can be obtained as:

collision is computed in resolving:

Figure 3.

49

t(n) instant time of the simulation.

D ¼ 0, where ð Þ uR; vR; wR are the components of V

2.3.2 Numerical resolution of the impact coefficient

 

The relative velocity droplets of the mass center can be resumed using Eq. (27):

$$\begin{cases} \overrightarrow{V}\_{mL} = +\frac{\Delta^3 \overrightarrow{V}\_{mR}}{\Delta^3 + \mathbf{1}}, \\\\ \overrightarrow{V}\_{mS} = -\frac{\overrightarrow{V}\_{mR}}{\Delta^3 + \mathbf{1}}. \end{cases} \tag{32}$$

For the impact parameter in Eq. (29), it is necessary to compute X variable. This variable is the projection of segment b ¼ 0:5 ϕ<sup>S</sup> þ ϕ<sup>L</sup> ð Þ on the plane perpendicular to the relative velocity V ! <sup>R</sup>. The impact factor will be equal to the cosine of γ angle:

$$\text{Imp} = \cos \chi = \frac{X}{b},\tag{33}$$

$$
\sin \chi = \frac{D\_p}{b},
\tag{34}
$$

So, inserting Eq. (34) into Eq. (33) results in:

$$\text{Imp} = \cos \chi = \left[ \mathbf{1} - \left( \frac{\mathbf{2}D\_p}{\phi\_S + \phi\_L} \right)^2 \right]^{\frac{1}{2}}. \tag{35}$$

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray DOI: http://dx.doi.org/10.5772/intechopen.81110

It is necessary to obtain Dp. It is the distance from large droplet center xL; yL; zL to the perpendicular plane at V ! <sup>R</sup> velocity passing through the small droplet center xS; yS; zS . This plane P-P is shown in green color in Figure 2. The plane equation passing through the small drop center is uRxS þ vRyS þ wRzSþ D ¼ 0, where ð Þ uR; vR; wR are the components of V ! <sup>R</sup>. In this expression, D is a constant of plane equation. This constant is obtained as: D ¼ �uRxS � vRyS � wRzR. So, the Dp distance can be obtained as:

$$D\_p = \left| \frac{u\_R \mathbf{x}\_L + v\_R \mathbf{y}\_L + w\_R \mathbf{z}\_L + D}{(u\_R^2 + v\_R^2 + w\_R^2)^{\frac{1}{2}}} \right|. \tag{36}$$

#### 2.3.2 Numerical resolution of the impact coefficient

In each time of numerical simulation, it checks whether the collision between two droplets occurs. For obtaining a more optimize algorithm, collision boxes are placed around and inside the liquid particle ejection spray. The sizes of grid boxes vary dynamically, adjusting to the boundaries of the particle domain as shown in Figure 3a. The size of the boxes is the same as the Eulerian calculation grid in horizontal direction Δb = Δx = Δy = 0.1 m. In this way, every drop inside this box will be questioned about whether it collided with the other drops that are in the same box. When a binary collision is successfully found (e.g., 5–6; 3–7 in Figure 3b), the pairs are marked and removed from the next iteration of detection. This technique was proposed by Michelot [25] and used by Aguirre [4, 14] to consider the diffusion of chemical species that are carried by fluid particles. It is evident that due to the temporal discretization of the numerical solution of droplet motion equations, it is almost impossible that for an instant of discretized time t(n) = t(n-1) + Δt, the contact between two drops can be concurrent. Most likely, by that instant time t(n-1), the contact is about to occur and at instant time t(n), it has already occurred. When the distance between the centers of the two drops inside a collision box is less than the sum of their radii, then the drops collided. In this case, the time Δt" elapsed from the collision to the computation instant time t(n) to be calculated. The particles are repositioned at the moment of collision t" = t(n) � Δt", and the impact factor is calculated according to the positions of their centers as shown in Figure 4. The lapse time Δt" for repositioning the droplets at instant of collision is computed in resolving:

#### Figure 3.

(a) Collision boxes around and inside the spray ejection of droplets and (b) droplets inside the collision box at t(n) instant time of the simulation.

V ! mR <sup>¼</sup> <sup>V</sup> ! Sϕ<sup>3</sup> <sup>S</sup> þ V ! Lϕ<sup>3</sup> L

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

8 >>>><

>>>>:

the relative velocity V

48

Figure 2.

!

So, inserting Eq. (34) into Eq. (33) results in:

V !

V ! mL ¼ þ

mS ¼ � <sup>V</sup>

Imp <sup>¼</sup> cos <sup>γ</sup> <sup>¼</sup> <sup>X</sup>

sin <sup>γ</sup> <sup>¼</sup> Dp

Imp <sup>¼</sup> cos <sup>γ</sup> <sup>¼</sup> <sup>1</sup> � <sup>2</sup>Dp

ϕ3 <sup>S</sup> <sup>þ</sup> <sup>ϕ</sup><sup>3</sup> L

Scheme of small droplet S and large droplet L before collision (dashed line) and at contact instant (solid line).

The relative velocity droplets of the mass center can be resumed using Eq. (27):

Δ3 V ! mR <sup>Δ</sup><sup>3</sup> <sup>þ</sup> <sup>1</sup> ,

! mR <sup>Δ</sup><sup>3</sup> <sup>þ</sup> <sup>1</sup> :

For the impact parameter in Eq. (29), it is necessary to compute X variable. This variable is the projection of segment b ¼ 0:5 ϕ<sup>S</sup> þ ϕ<sup>L</sup> ð Þ on the plane perpendicular to

<sup>R</sup>. The impact factor will be equal to the cosine of γ angle:

ϕ<sup>S</sup> þ ϕ<sup>L</sup> � �<sup>2</sup> " #<sup>1</sup>

<sup>b</sup> , (33)

: (35)

<sup>b</sup> , (34)

2

(31)

(32)

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

#### Figure 4.

Positions of droplets before and after the collision for a time lapse Δt. Droplets before collision at t(n�1) instant time. Droplets after collision at t(n) = t(n�1) + Δt instant time. Repositioning of droplets at the instant of collision (t <sup>00</sup> = t(n) – Δt <sup>00</sup>).

not only on the velocity of both drops but also on their relative size and impact coefficient. Two of the four possible outcomes of the binary collision are susceptible to generating satellite droplets. These droplets are usually much smaller in size than the parent-drops and are, therefore, more prone to drift and evaporation. If these droplets are composed of a phosphonate-acid solution (such as glyphosate), then after evaporation, the solute will drift away from the airflow very quickly.

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

DOI: http://dx.doi.org/10.5772/intechopen.81110

Impc‐<sup>s</sup> <sup>¼</sup>

Time sequence diagram of the binary droplet collision and its outcomes.

[20] as follows:

Figure 5.

(Impc-r) is:

51

The outcomes of collision droplets are defined using a map collision. This map is the graphic representations between the Wes vs. Imp (Wes-Imp) frontier curves among the different outcomes of binary collision that are displayed on this map. Several researchers proposed equations for frontier curves. The transition impact factor between coalescence and stretching separation (Impc-s) is according to Rabe

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>0</sup>:53<sup>2</sup> <sup>þ</sup> <sup>4</sup>:24Wes <sup>p</sup> � <sup>0</sup>:<sup>53</sup>

The transition impact factor between reflexive and stretching separation appears

It should be noted that the boundary curve between coalescence and reflexive separation Impc-r increases with the increase of Wes to the value of Imp = 0.28. This behavior indicates that for low Imp values (on-head collision) and relatively low droplet velocities before collision, surface energy is greater than kinetic energy and the result of the collision is stable coalescence. However, for the same Imp values but with higher velocities, the kinetic energy is predominant; the droplets have an unstable coalescence and then separate. This separation can generate satellite droplets. On the other hand, if the Imp is higher (tangential collision of the droplets), then coalescence as a result of the collision is more improbable since only a fraction

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>0</sup>:<sup>45</sup> Wes <sup>r</sup>

The transition impact factor between coalescence and reflexive separation

Impc‐<sup>r</sup> <sup>¼</sup> <sup>0</sup>:<sup>3059</sup>

when the Wes > 2.5 and can be considered a constant value Impr–<sup>s</sup> = 0.28.

4Wes : (38)

: (39)

$$\left(\frac{\phi\_{\rm S} + \phi\_{\rm L}}{2}\right)^2 = \left(\mathbf{x}\_{\rm S} - \mathbf{x}\_{\rm L} - \boldsymbol{\mu}\_{\rm R}\boldsymbol{\Delta t}^{\prime}\right)^2 + \left(\mathbf{y}\_{\rm S} - \mathbf{y}\_{\rm L} - \boldsymbol{\nu}\_{\rm R}\boldsymbol{\Delta t}^{\prime}\right)^2 + \left(\mathbf{z}\_{\rm S} - \mathbf{z}\_{\rm L} - \boldsymbol{\nu}\_{\rm R}\boldsymbol{\Delta t}^{\prime}\right)^2. \tag{37}$$

The outcomes of collision droplets are computed using the map collision theory. Once the droplets collided and the effects of collision are into account on the droplets, they are repositioned by advancing the same pass time Δt".

#### 2.3.3 Binary droplet collision map

The outcomes of the binary droplet collision model propose different scenarios:


Figure 5 shows a time sequence of the binary droplet collision for each outcome described above. It is important to note that the result of the binary collision depends An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray DOI: http://dx.doi.org/10.5772/intechopen.81110

Figure 5. Time sequence diagram of the binary droplet collision and its outcomes.

not only on the velocity of both drops but also on their relative size and impact coefficient. Two of the four possible outcomes of the binary collision are susceptible to generating satellite droplets. These droplets are usually much smaller in size than the parent-drops and are, therefore, more prone to drift and evaporation. If these droplets are composed of a phosphonate-acid solution (such as glyphosate), then after evaporation, the solute will drift away from the airflow very quickly.

The outcomes of collision droplets are defined using a map collision. This map is the graphic representations between the Wes vs. Imp (Wes-Imp) frontier curves among the different outcomes of binary collision that are displayed on this map. Several researchers proposed equations for frontier curves. The transition impact factor between coalescence and stretching separation (Impc-s) is according to Rabe [20] as follows:

$$\text{Imp}\_{\text{c-s}} = \frac{\sqrt{0.53^2 + 4.24 \text{Wes}} - 0.53}{4 \text{Wes}}.\tag{38}$$

The transition impact factor between coalescence and reflexive separation (Impc-r) is:

$$\text{Imp}\_{\text{c-r}} = 0.3059 \sqrt{1 - \frac{0.45}{\text{Wes}}}.\tag{39}$$

The transition impact factor between reflexive and stretching separation appears when the Wes > 2.5 and can be considered a constant value Impr–<sup>s</sup> = 0.28.

It should be noted that the boundary curve between coalescence and reflexive separation Impc-r increases with the increase of Wes to the value of Imp = 0.28. This behavior indicates that for low Imp values (on-head collision) and relatively low droplet velocities before collision, surface energy is greater than kinetic energy and the result of the collision is stable coalescence. However, for the same Imp values but with higher velocities, the kinetic energy is predominant; the droplets have an unstable coalescence and then separate. This separation can generate satellite droplets. On the other hand, if the Imp is higher (tangential collision of the droplets), then coalescence as a result of the collision is more improbable since only a fraction

ϕ<sup>S</sup> þ ϕ<sup>L</sup> 2 <sup>2</sup>

at the instant of collision (t

Figure 4.

¼ xS � xL � uRΔt <sup>00</sup> <sup>2</sup>

<sup>00</sup> = t(n) – Δt

<sup>00</sup>).

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

2.3.3 Binary droplet collision map

kinetic energy.

satellite droplets.

50

generate satellite droplets.

exchanging mass between them.

þ yS � yL � vRΔt <sup>00</sup> <sup>2</sup>

The outcomes of collision droplets are computed using the map collision theory.

Positions of droplets before and after the collision for a time lapse Δt. Droplets before collision at t(n�1) instant time. Droplets after collision at t(n) = t(n�1) + Δt instant time. Repositioning of droplets

The outcomes of the binary droplet collision model propose different scenarios:

the collision. In this case, the surface energy is relatively greater than the

Stretching: the two drops collide tangentially, so they separate and can

Bouncing: the two colliding drops remain separated after collision without

Figure 5 shows a time sequence of the binary droplet collision for each outcome described above. It is important to note that the result of the binary collision depends

Coalescence: the two droplets that collide to form a single drop as a result of

Reflexive: the two colliding droplets almost head-on, so they join together as one, but the kinetic energy is large enough to separate again and can generate

Once the droplets collided and the effects of collision are into account on the

droplets, they are repositioned by advancing the same pass time Δt".

þ zS � zL � wRΔt <sup>00</sup> <sup>2</sup>

:

(37)

of the volume of the drops interacts during the collision. The contact surface of both drops is smaller and therefore the surface energy as well. This reduces the likelihood of stable coalescence as a consequence of the collision. This behavior is evident in the Impc-s frontier curve, which decreases the coalescence area as the Imp increases.

For bounce, the model proposed by Estrade [35] calculates the number of transition Weber Web according to the Imp, Δ and a shape parameter, φ:

$$\text{We}\_{\text{b}} = \frac{\Delta \left( \mathbf{1} + \Delta^2 \right) (4\rho - \mathbf{12})}{\xi \left( \mathbf{1} - \text{Imp}^2 \right)},\tag{40}$$

where ξ is computed as:

$$\xi = \begin{cases} 1 - \frac{\left(2 - \lambda\right)^2 (1 + \lambda)}{4} & \text{if } \lambda > 1, \\\frac{\lambda^2 (3 - \lambda)}{4} & \text{if } \lambda \le 1, \end{cases} \tag{41}$$

and λ = (1 � Imp)(1 + Δ). The shape parameter φ can be computed as Zhang [19]:

$$\rho = \text{3.351} \left( \frac{\rho\_l}{\text{1.16}} \right)^{\frac{2}{3}}. \tag{42}$$

(or not) to satellite droplets. This volume is computed and taken into account the magnitude of the opposing surface (Esurten), stretching (Estrtch), and viscous dissi-

Map collision droplets with areas of outcomes collision. Coalescence, reflexive separation, stretching separation, and bounce. Bounce frontiers: Δ = 1, Δ = 0.75, Δ = 0.5, and

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

DOI: http://dx.doi.org/10.5772/intechopen.81110

CVS <sup>¼</sup> Estrtch � Esurten � Edissip Estrtch þ Esurten þ Edissip

Case CVS ≤ 0: if this coefficient is carried at negative value, it is assumed that fragmentation of droplets does not occur. The drops only lose kinetic energy. The center mass of droplet velocities is affected by Z coefficient indicating the fraction of energy that is dissipated during collision. For this case, Kim [18] proposes:

<sup>Z</sup> <sup>¼</sup> Imp � ffiffiffiffiffiffiffiffi

The relative velocities of mass center after collision can be written by using

where ecoal <sup>¼</sup> min 1:0; <sup>2</sup>:<sup>4</sup> <sup>f</sup> We�<sup>1</sup> � � is the coalescence efficiency,

, and We <sup>¼</sup> <sup>ρ</sup>lϕSΔ<sup>3</sup>

<sup>f</sup> <sup>¼</sup> <sup>Δ</sup>�<sup>3</sup> � <sup>2</sup>:4Δ�<sup>2</sup> <sup>þ</sup> <sup>2</sup>:7Δ�<sup>1</sup>

53

Figure 6.

Δ = 0.1.

which follow O'Rourke model [40].

momentum conservation equation:

1 � ffiffiffiffiffiffiffiffi

ecoal p

σ�<sup>1</sup> V ! R � � � � � � 2 , (44)

ecoal <sup>p</sup> , (45)

, is the Weber number,

pation energies (Edissip) by using a separation coefficient (CVS):

The transition bounces into Wes-Imp map collision droplets, and the Weber symmetric bounce frontier Wesb is used. So, it is obtained from Web (Eq. (40)) as:

$$\text{Wes}\_{\text{b}} = \text{We}\_{\text{b}} \frac{\Delta^2}{12\left(1 + \Delta^3\right)\left(1 + \Delta^2\right)},\tag{43}$$

The Wes-Imp map collision droplets define areas , , and where the outcomes of the binary droplet collision are represented. These areas are bounded by frontiers curves as proposed in Eqs. (38)–(43). The areas with frontier curves are shown in Figure 6. The frontier curves between and change their position as a function of Δ.

#### 2.3.4 Numerical models of the binary droplet collision

The binary droplet collision model allows obtaining the diameters and velocities of the droplets after the collision. The values of these variables are obtained according to the proposed models [17–19, 34] (coalescence, reflexive, and stretching separations) and [35] (bounce outcome).

#### 2.3.4.1 Coalescence

For coalescence outcome, the two droplets coalesce into one. This occurs preferably at low Weber numbers as surface tensions exceed kinetic energy. The new droplet velocity is the velocity of mass center before the collision V ! ð Þ new ¼ V ! mR (Eq. (31)). For droplets of the same density, its diameter is <sup>ϕ</sup>ð Þ new <sup>¼</sup> <sup>ϕ</sup><sup>3</sup> <sup>L</sup> <sup>þ</sup> <sup>ϕ</sup><sup>3</sup> S � �<sup>1</sup>=<sup>3</sup> .

#### 2.3.4.2 Stretching

Munnannur and Reitz [17] calculate the interaction volume between the droplets. This volume is released from both drops creating a ligament that gives rise

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray DOI: http://dx.doi.org/10.5772/intechopen.81110

#### Figure 6.

of the volume of the drops interacts during the collision. The contact surface of both drops is smaller and therefore the surface energy as well. This reduces the likelihood of stable coalescence as a consequence of the collision. This behavior is evident in the Impc-s frontier curve, which decreases the coalescence area as the Imp increases. For bounce, the model proposed by Estrade [35] calculates the number of tran-

Web <sup>¼</sup> <sup>Δ</sup> <sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>2</sup> � �ð Þ <sup>4</sup><sup>φ</sup> � <sup>12</sup>

4

and λ = (1 � Imp)(1 + Δ). The shape parameter φ can be computed as Zhang [19]:

1:16 � �<sup>2</sup> 3

Δ2

<sup>φ</sup> <sup>¼</sup> <sup>3</sup>:<sup>351</sup> <sup>ρ</sup><sup>l</sup>

The transition bounces into Wes-Imp map collision droplets, and the Weber symmetric bounce frontier Wesb is used. So, it is obtained from Web (Eq. (40)) as:

The Wes-Imp map collision droplets define areas , , and where the outcomes of the binary droplet collision are represented. These areas are bounded by frontiers curves as proposed in Eqs. (38)–(43). The areas with frontier curves are shown in Figure 6. The frontier curves between and change their

The binary droplet collision model allows obtaining the diameters and velocities

For coalescence outcome, the two droplets coalesce into one. This occurs preferably at low Weber numbers as surface tensions exceed kinetic energy. The new

Munnannur and Reitz [17] calculate the interaction volume between the droplets. This volume is released from both drops creating a ligament that gives rise

of the droplets after the collision. The values of these variables are obtained according to the proposed models [17–19, 34] (coalescence, reflexive, and

droplet velocity is the velocity of mass center before the collision V

(Eq. (31)). For droplets of the same density, its diameter is <sup>ϕ</sup>ð Þ new <sup>¼</sup> <sup>ϕ</sup><sup>3</sup>

ð Þ 1 þ λ

<sup>ξ</sup> <sup>1</sup> � Imp<sup>2</sup> � � , (40)

: (42)

!

ð Þ new ¼ V ! mR

<sup>L</sup> <sup>þ</sup> <sup>ϕ</sup><sup>3</sup> S � �<sup>1</sup>=<sup>3</sup>

.

12 1 <sup>þ</sup> <sup>Δ</sup><sup>3</sup> � � <sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>2</sup> � � , (43)

(41)

if λ > 1,

if λ≤1,

sition Weber Web according to the Imp, Δ and a shape parameter, φ:

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

<sup>1</sup> � ð Þ <sup>2</sup> � <sup>λ</sup> <sup>2</sup>

λ2 ð Þ 3 � λ 4

Wesb ¼ Web

2.3.4 Numerical models of the binary droplet collision

stretching separations) and [35] (bounce outcome).

8 >><

>>:

where ξ is computed as:

position as a function of Δ.

2.3.4.1 Coalescence

2.3.4.2 Stretching

52

ξ ¼

Map collision droplets with areas of outcomes collision. Coalescence, reflexive separation, stretching separation, and bounce. Bounce frontiers: Δ = 1, Δ = 0.75, Δ = 0.5, and Δ = 0.1.

(or not) to satellite droplets. This volume is computed and taken into account the magnitude of the opposing surface (Esurten), stretching (Estrtch), and viscous dissipation energies (Edissip) by using a separation coefficient (CVS):

$$\mathbf{C}\_{\rm VS} = \frac{\mathbf{E}\_{\rm strch} - \mathbf{E}\_{\rm surten} - \mathbf{E}\_{\rm dispip}}{\mathbf{E}\_{\rm strch} + \mathbf{E}\_{\rm surten} + \mathbf{E}\_{\rm dispip}},\tag{44}$$

Case CVS ≤ 0: if this coefficient is carried at negative value, it is assumed that fragmentation of droplets does not occur. The drops only lose kinetic energy. The center mass of droplet velocities is affected by Z coefficient indicating the fraction of energy that is dissipated during collision. For this case, Kim [18] proposes:

$$Z = \frac{\text{Imp} - \sqrt{\mathcal{e}\_{coal}}}{1 - \sqrt{\mathcal{e}\_{coal}}},\tag{45}$$

where ecoal <sup>¼</sup> min 1:0; <sup>2</sup>:<sup>4</sup> <sup>f</sup> We�<sup>1</sup> � � is the coalescence efficiency, <sup>f</sup> <sup>¼</sup> <sup>Δ</sup>�<sup>3</sup> � <sup>2</sup>:4Δ�<sup>2</sup> <sup>þ</sup> <sup>2</sup>:7Δ�<sup>1</sup> , and We <sup>¼</sup> <sup>ρ</sup>lϕSΔ<sup>3</sup> σ�<sup>1</sup> V ! R � � � � � � 2 , is the Weber number, which follow O'Rourke model [40].

The relative velocities of mass center after collision can be written by using momentum conservation equation:

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

$$\begin{cases} \stackrel{\rightarrow}{V}\_{mS \ (new)} = Z \stackrel{\rightarrow}{V}\_{mS} \\ \stackrel{\rightarrow}{V}\_{mL \ (new)} = Z \stackrel{\rightarrow}{V}\_{mL} \end{cases} \tag{46}$$

where the diameters of parent droplets after collision are:

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

8 ><

DOI: http://dx.doi.org/10.5772/intechopen.81110

>:

2.3.4.3 Reflexive

identical diameters:

2.3.4.4 Bounce

follows:

55

initial radius of ligaments is r<sup>3</sup>

ϕL new ð Þ ¼ ð Þ 1 � Ψ<sup>L</sup>

ϕ<sup>S</sup> ð Þ new ¼ ð Þ 1 � Ψ<sup>S</sup>

The volume of ligament is the entire temporarily merged mass of two droplets. The model of satellite droplet formation is similar at stretching outcome, but the

<sup>0</sup> <sup>¼</sup> ð Þ <sup>ϕ</sup>L=<sup>2</sup> <sup>3</sup> <sup>þ</sup> ð Þ <sup>ϕ</sup>S=<sup>2</sup> <sup>3</sup>

Munnannur [17] for reflexive outcome uses the time scale of temporal evolution ligament. When T ≤ 3, a single satellite droplet is formed and the three droplets (considering the ligament breaks up into two end-droplets and one-satellite droplet) have same size. However, according to the experimental studies of Ashrgiz [34], Kim [18] affirms that no uniform droplet sizes are obtained for the enddroplets and a single satellite droplet after reflexive collision. We have adopted the last criteria, so the satellite droplet diameters are computed with Eqs. (48) and (49),

but the number of satellite droplets is Nsat <sup>¼</sup> <sup>6</sup> <sup>r</sup>0=ϕsat ð Þ<sup>3</sup> � 2. If Nsat <sup>≤</sup> 0, it is assumed that the ligament breaks up without satellite droplet and the two enddroplets have their own radius. If 0 < Nsat ≤ 1, it is assumed that a single satellite droplet is formed that is smaller than the two end-droplets after collision. The diameter of single satellite droplets is ϕsat and the end-droplets after collision have

<sup>ϕ</sup>L new ð Þ <sup>¼</sup> <sup>ϕ</sup>S new ð Þ <sup>¼</sup> <sup>8</sup>r<sup>3</sup>

droplets are computed with Eqs. (46) and (50).

When Nsat > 1, the ligament breaks up into uniform droplets with identical diameters ϕL new ð Þ ¼ ϕS new ð Þ ¼ ϕsat. The velocities of end-droplets and satellite

In this case, the droplets bounce maintaining their diameters after the impact. In

the general case, oblique collision between droplets is considered. The droplet velocities after collision must be decomposed into a normal component and a tangential component to the plane of impact. The tangential component after impact remains unchanged, but the normal component is affected by a soft inelastic rebound assuming a restitution coefficient en,p = 0.97 by following Almohammed [42]. This restitution coefficient takes into account the dissipation of kinetic energy during the impact. The normal velocities of droplets at instant of collision are as

Vn L ¼ Vi L

8 >><

>>:

velocities after collision are given as:

Vn S ¼ Vi S

where xi R ¼ xi S � xi L is the relative position between the droplets in i = 1, 2, 3 (x, y, z) and b is the center droplet distance (Figure 2). The normal component

j j xi R b

j j xi R b

,

1 <sup>6</sup> πϕ<sup>3</sup> L

1 <sup>6</sup> πϕ<sup>3</sup> S :

<sup>0</sup> � <sup>ϕ</sup><sup>3</sup> sat 2 � �<sup>1</sup>=<sup>3</sup>

. The model proposed by

: (52)

(53)

(51)

The velocities after collision can be obtained by using Eqs. (30) and (31). The diameters of droplets after collision are unaltered.

Case CVS > 0: the separation volumes from droplets determine the evolution of the temporary fluid ligament that would form between them. In this model, it is assumed that the ligament has a uniform cylindrical shape, and the radius ro of ligament at initial instant time of the temporal evolution with a momentum balance equation can be obtained:

$$\frac{1}{6}\pi \left(\Psi\_S \phi\_S^3 + \Psi\_L \phi\_L^3\right) = \pi r\_0^2 \eta,\tag{47}$$

where Ψ<sup>S</sup> and Ψ<sup>L</sup> are the fraction of volumes lost from the smaller and large droplets to form the ligament [17, 18], and η is its initial time instantaneous length (Figure 7). Another assumption is η = ro. In this model, a time scale of temporal evolution ligament is proposed: T ¼ 0:75k<sup>2</sup> ffiffiffiffiffiffiffiffiffiffi We0 <sup>p</sup> . If T <sup>≤</sup> 2, the ligament contracts in a single satellite whose radius is ro. k2 = 0.45 and We0 ¼ 2r0ð Þ ρl=σ V ! R � � � � � � 2 . Otherwise, it is necessary to compute the evolution time of ligament radius equation for obtaining the final value rbu:

$$\frac{3}{4\sqrt{2}}k\_1k\_2\sqrt{\mathrm{We}\_0}\left(\frac{r\_{bu}}{r\_0}\right)^{\gamma\_2} + \left(\frac{r\_{bu}}{r\_0}\right)^2 - 1 = 0,\tag{48}$$

k<sup>1</sup> = 11.5 following Kim [18]. Eq. (48) can be solved by iteration with an initial value ð Þ¼ rbu=r<sup>0</sup> 1 and <sup>Δ</sup>t = <sup>1</sup> � <sup>10</sup>�<sup>2</sup> .

The diameter of satellite droplets can be determined by following Georjon [41]:

$$
\phi\_{\rm sat} = \mathbf{3.78} r\_{bu}. \tag{49}
$$

The number of satellite droplets is calculated from the mass conservation by assuming uniform satellites size Nsat <sup>¼</sup> <sup>6</sup> <sup>r</sup>0=ϕsat ð Þ<sup>3</sup> . The velocities of satellite droplets can be obtained from momentum equation where the velocities of parent droplets after collision are computed by Eq. (46).

$$
\overrightarrow{\mathbf{V}}\_{\text{sat}} = \frac{\left(\phi\_L^3 \overrightarrow{\mathbf{V}}\_L - \phi\_L^3 \underbrace{}\_{\text{(new)}} \overrightarrow{\mathbf{V}}\_L \underbrace{}\_{\text{(new)}}\right) + \left(\phi\_S^3 \overrightarrow{\mathbf{V}}\_S - \phi\_S^3 \underbrace{}\_{\text{(new)}} \overrightarrow{\mathbf{V}}\_{\text{S (new)}}\right)}{N\_{\text{sat}} \phi\_{\text{sat}}^3},\tag{50}
$$

Figure 7.

Collision model for the stretching outcome. (a) Formation instant time of ligament and (b) temporal evolution ligament.

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray DOI: http://dx.doi.org/10.5772/intechopen.81110

where the diameters of parent droplets after collision are:

$$\begin{cases} \phi\_{L\ (new)} = (1 - \Psi\_L) \frac{1}{6} \pi \phi\_L^3\\ \phi\_{\mathcal{S}\ (new)} = (1 - \Psi\_{\mathcal{S}}) \frac{1}{6} \pi \phi\_{\mathcal{S}}^3 \end{cases} \tag{51}$$

#### 2.3.4.3 Reflexive

V !

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

8 < :

diameters of droplets after collision are unaltered.

evolution ligament is proposed: T ¼ 0:75k<sup>2</sup>

3 4 ffiffi 2 p k1k<sup>2</sup>

assuming uniform satellites size Nsat <sup>¼</sup> <sup>6</sup> <sup>r</sup>0=ϕsat ð Þ<sup>3</sup>

droplets after collision are computed by Eq. (46).

L new ð Þ<sup>V</sup> ! L new ð Þ

� �

value ð Þ¼ rbu=r<sup>0</sup> 1 and <sup>Δ</sup>t = <sup>1</sup> � <sup>10</sup>�<sup>2</sup>

ϕ3 LV ! <sup>L</sup> � <sup>ϕ</sup><sup>3</sup>

1 <sup>6</sup> <sup>π</sup> <sup>Ψ</sup>Sϕ<sup>3</sup>

a single satellite whose radius is ro. k2 = 0.45 and We0 ¼ 2r0ð Þ ρl=σ V

ffiffiffiffiffiffiffiffiffiffi We0 p rbu

it is necessary to compute the evolution time of ligament radius equation for

r0 � �7=<sup>2</sup>

.

equation can be obtained:

obtaining the final value rbu:

V ! sat ¼

Figure 7.

ligament.

54

V ! mS new ð Þ ¼ ZV

mL new ð Þ ¼ ZV

The velocities after collision can be obtained by using Eqs. (30) and (31). The

Case CVS > 0: the separation volumes from droplets determine the evolution of the temporary fluid ligament that would form between them. In this model, it is assumed that the ligament has a uniform cylindrical shape, and the radius ro of ligament at initial instant time of the temporal evolution with a momentum balance

> <sup>S</sup> <sup>þ</sup> <sup>Ψ</sup>Lϕ<sup>3</sup> L � � <sup>¼</sup> <sup>π</sup><sup>r</sup>

where Ψ<sup>S</sup> and Ψ<sup>L</sup> are the fraction of volumes lost from the smaller and large droplets to form the ligament [17, 18], and η is its initial time instantaneous length (Figure 7). Another assumption is η = ro. In this model, a time scale of temporal

ffiffiffiffiffiffiffiffiffiffi We0

þ

k<sup>1</sup> = 11.5 following Kim [18]. Eq. (48) can be solved by iteration with an initial

The diameter of satellite droplets can be determined by following Georjon [41]:

The number of satellite droplets is calculated from the mass conservation by

<sup>þ</sup> <sup>ϕ</sup><sup>3</sup> SV ! <sup>S</sup> � <sup>ϕ</sup><sup>3</sup>

Nsatϕ<sup>3</sup> sat

Collision model for the stretching outcome. (a) Formation instant time of ligament and (b) temporal evolution

lets can be obtained from momentum equation where the velocities of parent

rbu r0 � �<sup>2</sup>

ϕsat ¼ 3:78rbu: (49)

<sup>S</sup> ð Þ new <sup>V</sup> ! S ð Þ new

� �

! mS

! mL :

2

<sup>0</sup>η, (47)

! R � � � � � � 2

� 1 ¼ 0, (48)

. The velocities of satellite drop-

, (50)

. Otherwise,

<sup>p</sup> . If T <sup>≤</sup> 2, the ligament contracts in

(46)

The volume of ligament is the entire temporarily merged mass of two droplets. The model of satellite droplet formation is similar at stretching outcome, but the initial radius of ligaments is r<sup>3</sup> <sup>0</sup> <sup>¼</sup> ð Þ <sup>ϕ</sup>L=<sup>2</sup> <sup>3</sup> <sup>þ</sup> ð Þ <sup>ϕ</sup>S=<sup>2</sup> <sup>3</sup> . The model proposed by Munnannur [17] for reflexive outcome uses the time scale of temporal evolution ligament. When T ≤ 3, a single satellite droplet is formed and the three droplets (considering the ligament breaks up into two end-droplets and one-satellite droplet) have same size. However, according to the experimental studies of Ashrgiz [34], Kim [18] affirms that no uniform droplet sizes are obtained for the enddroplets and a single satellite droplet after reflexive collision. We have adopted the last criteria, so the satellite droplet diameters are computed with Eqs. (48) and (49), but the number of satellite droplets is Nsat <sup>¼</sup> <sup>6</sup> <sup>r</sup>0=ϕsat ð Þ<sup>3</sup> � 2. If Nsat <sup>≤</sup> 0, it is assumed that the ligament breaks up without satellite droplet and the two enddroplets have their own radius. If 0 < Nsat ≤ 1, it is assumed that a single satellite droplet is formed that is smaller than the two end-droplets after collision. The diameter of single satellite droplets is ϕsat and the end-droplets after collision have identical diameters:

$$
\phi\_{L\ (new)} = \phi\_{S\ (new)} = \left(\frac{8r\_0^3 - \phi\_{sat}^3}{2}\right)^{1/3}.\tag{52}
$$

When Nsat > 1, the ligament breaks up into uniform droplets with identical diameters ϕL new ð Þ ¼ ϕS new ð Þ ¼ ϕsat. The velocities of end-droplets and satellite droplets are computed with Eqs. (46) and (50).

#### 2.3.4.4 Bounce

In this case, the droplets bounce maintaining their diameters after the impact. In the general case, oblique collision between droplets is considered. The droplet velocities after collision must be decomposed into a normal component and a tangential component to the plane of impact. The tangential component after impact remains unchanged, but the normal component is affected by a soft inelastic rebound assuming a restitution coefficient en,p = 0.97 by following Almohammed [42]. This restitution coefficient takes into account the dissipation of kinetic energy during the impact. The normal velocities of droplets at instant of collision are as follows:

$$\begin{cases} V\_{n\,\,L} = V\_{i\,\,L} \frac{|\boldsymbol{\omega}\_{i\,\,R}|}{b} \\ V\_{n\,\,S} = V\_{i\,\,S} \frac{|\boldsymbol{\omega}\_{i\,\,R}|}{b} \end{cases} \tag{53}$$

where xi R ¼ xi S � xi L is the relative position between the droplets in i = 1, 2, 3 (x, y, z) and b is the center droplet distance (Figure 2). The normal component velocities after collision are given as:

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

$$\begin{cases} V\_{n\ L\ (new)} = V\_{n\ L} - \frac{\Delta^3}{\mathbf{1} + \Delta^3} f\_n^{dd} \\ V\_{n\ S\ (new)} = V\_{n\ S} + \frac{\mathbf{1}}{\mathbf{1} + \Delta^3} f\_n^{dd} \end{cases} \tag{54}$$

where f dd <sup>n</sup> is the normal impulse of a droplet-droplet collision: f dd <sup>n</sup> ¼ � 1 þ en,p � �ð Þ Vn S � Vn L .

#### 3. Results and discussion

#### 3.1 Ejection of liquid particle simulation

In order to obtain the Weibull p.d.f. corresponding to the droplet diameters as described in Section 2.1, the scale k = 301.228 and shape m = 2.606 parameters were obtained with R-R method. Figure 8 shows the minimum square adjusted of regression line with a correlation coefficient R2 = 0.9965.

#### 3.2 Effects of the Eulerian-Lagrangian double-way coupling

The trajectories of liquid particles are simulated with an Euler-Lagrangian doubleway coupled model descript in Section 2.2. The influence of droplets to air velocity is shown in Figure 9 (a) for t = 1 s instant time and (b) for t = 20 s instant time of the simulation. The vertical velocities of air W are shown in color scale. Eddies around the spray plume are formed and extend up to 3 m from the center of the spray (c). It is observed that eddies are formed by influence of jet droplets. This effect should be taken into account as droplets of very small diameters are captured by eddies and are prone to contribute to drift. This effect can be seen in Figure 10 where the small droplets follow the streamlines of eddies on both sides of the sprayer.

The results of the vertical droplet velocities distribution as a function of the droplet diameters obtained at 0.35 m below the nozzle are shown in Figure 11. These are compared with the laboratory measurements of Nuyttens [12]. It is

Figure 9.

spray.

57

Influence of droplets to air velocity at different instant times of the simulation. (a) t = 1 s, (b) t = 20 s, (c) zoom of eddy formed at 2.5 m from center spray at 20 s, and W is a vertical component of air velocity around the

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

DOI: http://dx.doi.org/10.5772/intechopen.81110

#### Figure 8.

Regression line of R-R distribution function and data measurement of droplet diameters.

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray DOI: http://dx.doi.org/10.5772/intechopen.81110

#### Figure 9.

Vn L new ð Þ <sup>¼</sup> Vn L � <sup>Δ</sup><sup>3</sup>

Vn S new ð Þ ¼ Vn S þ

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

<sup>n</sup> is the normal impulse of a droplet-droplet collision:

In order to obtain the Weibull p.d.f. corresponding to the droplet diameters as described in Section 2.1, the scale k = 301.228 and shape m = 2.606 parameters were obtained with R-R method. Figure 8 shows the minimum square adjusted of

The trajectories of liquid particles are simulated with an Euler-Lagrangian doubleway coupled model descript in Section 2.2. The influence of droplets to air velocity is shown in Figure 9 (a) for t = 1 s instant time and (b) for t = 20 s instant time of the simulation. The vertical velocities of air W are shown in color scale. Eddies around the spray plume are formed and extend up to 3 m from the center of the spray (c). It is observed that eddies are formed by influence of jet droplets. This effect should be taken into account as droplets of very small diameters are captured by eddies and are prone to contribute to drift. This effect can be seen in Figure 10 where the small

The results of the vertical droplet velocities distribution as a function of the droplet diameters obtained at 0.35 m below the nozzle are shown in Figure 11. These are compared with the laboratory measurements of Nuyttens [12]. It is

8 >><

>>:

regression line with a correlation coefficient R2 = 0.9965.

3.2 Effects of the Eulerian-Lagrangian double-way coupling

droplets follow the streamlines of eddies on both sides of the sprayer.

Regression line of R-R distribution function and data measurement of droplet diameters.

where f

<sup>n</sup> ¼ � 1 þ en,p

f dd

Figure 8.

56

dd

3. Results and discussion

� �ð Þ Vn S � Vn L .

3.1 Ejection of liquid particle simulation

<sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>3</sup> <sup>f</sup>

1 <sup>1</sup> <sup>þ</sup> <sup>Δ</sup><sup>3</sup> <sup>f</sup> dd n

,

(54)

dd n

Influence of droplets to air velocity at different instant times of the simulation. (a) t = 1 s, (b) t = 20 s, (c) zoom of eddy formed at 2.5 m from center spray at 20 s, and W is a vertical component of air velocity around the spray.

Figure 10. Position of droplets at t = 20 s of the simulation classified by their diameters.

and stretching separation. When considering the total number of droplet binary collision events, 21.1% corresponds to coalescence, 0.6% to reflexive separation, 8.8% to stretching separation, and 69.5% to bounce. The amount of satellite droplets arising from the separation by reflexive and stretching is displayed with numbers. It is noted that the number of satellite drops increases with the number of symmetrical Weber for both separately. This behavior indicates that the greater velocity the droplets are ejected from the spray nozzle, the more likely it is that satellite droplets will appear as a result of reflexive and stretching separation. As mentioned above,

numbers next to the symbols indicate the number of satellite droplets formed.

Map outcomes from binary droplet collision model. Coalescence, reflexive, stretching, and bounce. The

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

DOI: http://dx.doi.org/10.5772/intechopen.81110

Figure 12.

Figure 13.

59

Drift of spraying droplets from a nozzle at 0.75 m over ground.

Figure 11.

Distribution of vertical droplet velocities in (m/s) as a function of the diameters (μm). Droplet simulation. Mean and extreme range values measured by Nuyttens [12].

observed that the dispersion of velocity values for each diameter class is greater in laboratory measurements than in simulation. In addition, for diameters less than 200 μm, the model slightly underestimates the vertical velocity values relative to the laboratory results.

#### 3.3 Binary collision droplet map

The collision map for binary droplet model descripted in Section 2.3 is shown in Figure 12. The map allows showing the events of coalescence, bounce, reflexive,

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray DOI: http://dx.doi.org/10.5772/intechopen.81110

#### Figure 12.

Map outcomes from binary droplet collision model. Coalescence, reflexive, stretching, and bounce. The numbers next to the symbols indicate the number of satellite droplets formed.

and stretching separation. When considering the total number of droplet binary collision events, 21.1% corresponds to coalescence, 0.6% to reflexive separation, 8.8% to stretching separation, and 69.5% to bounce. The amount of satellite droplets arising from the separation by reflexive and stretching is displayed with numbers. It is noted that the number of satellite drops increases with the number of symmetrical Weber for both separately. This behavior indicates that the greater velocity the droplets are ejected from the spray nozzle, the more likely it is that satellite droplets will appear as a result of reflexive and stretching separation. As mentioned above,

Figure 13. Drift of spraying droplets from a nozzle at 0.75 m over ground.

observed that the dispersion of velocity values for each diameter class is greater in laboratory measurements than in simulation. In addition, for diameters less than 200 μm, the model slightly underestimates the vertical velocity values relative to

Distribution of vertical droplet velocities in (m/s) as a function of the diameters (μm). Droplet simulation.

Mean and extreme range values measured by Nuyttens [12].

The collision map for binary droplet model descripted in Section 2.3 is shown in Figure 12. The map allows showing the events of coalescence, bounce, reflexive,

the laboratory results.

Figure 11.

58

Figure 10.

Position of droplets at t = 20 s of the simulation classified by their diameters.

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

3.3 Binary collision droplet map

this can cause an increase in the proportion of sprayed product not reaching its destination, leaving it adrift.

equipment." We are also grateful to the Laboratory of Prototyping of Electronics and 3D Printing of the School of Engineering, UNER, for the work of printing the

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

The Weibull probability density function (Weibull p.d.f.) can be used for describing a lot of technical applications for which the distribution of ground material, particles dispersion, or droplet diameters in spray jet normally in the μm-band have behaviors with a random characteristic. In this case, the diameters of droplets ejected from a spray nozzle are simulated using a

Let us name the random variable ϕ<sup>0</sup> that represents the initial diameter of the liquid particles, and the expression of Weibull p.d.f. for this random

<sup>0</sup> exp � <sup>ϕ</sup><sup>0</sup>

f ϕ<sup>0</sup> ð Þ¼ 0, if ϕ<sup>0</sup> < 0:

In this p.d.f., m > 0 is the shape parameter and k > 0 is the scale parameter. The form of the density function of the Weibull distribution changes drastically with the value of m parameters. The k parameter does not change the shape of the distribution, but it extends along the random variable ϕ0. In this way, if the parameters m and k are chosen correctly, it is possible to obtain the shape and stretch of the

To take these data into account, the cumulative function of the Weibull distri-

k � � � �<sup>m</sup>

Ln �Ln 1 � F ϕ<sup>0</sup> f g ½ � ð Þ ¼ m:Ln ϕ<sup>0</sup> ð Þ� m:Lnð Þk : (57)

<sup>F</sup> <sup>ϕ</sup><sup>0</sup> ð Þ¼ <sup>1</sup> � exp � <sup>ϕ</sup><sup>0</sup>

If we associate Eq. (A.3) with a linear equation f(X) = mX + a where the dependent variable is f Xð Þ¼ Ln �Ln 1 � F ϕ<sup>0</sup> f g ½ � ð Þ and the independent variable is X ¼ Ln ϕ<sup>0</sup> ð Þ, m is the slope and a = �mLn(k) is intercept, in which the line cuts on the Y=f(X) axis. The method to obtain the Weibull p.d.f. parameters consists of plotting over logarithmic scale; the results of the experimental measurements Ln �Ln 1 � F ϕ<sup>0</sup> f g ½ � ð Þ vs. Ln ϕ<sup>0</sup> ð Þand approximate the point cloud to a linear regression by the least squares to obtain the slope (shape parameter) m and the constant

With these two parameters (m, k), it is possible to obtain the Weibull p.d.f. that describes the droplet diameter's distribution corresponding to the experiment

k � � � �<sup>m</sup>

, if ϕ<sup>0</sup> ≥ 0,

: (56)

(55)

<sup>f</sup> <sup>ϕ</sup><sup>0</sup> ð Þ¼ <sup>m</sup>

bution, named Rosin-Rammler (R-R), is used.

Eq. (A.2) can be written as:

measurements.

61

8 < :

DOI: http://dx.doi.org/10.5772/intechopen.81110

<sup>k</sup><sup>m</sup> <sup>ϕ</sup><sup>m</sup>�<sup>1</sup>

Weibull p.d.f. that fits the experimentally measured diameter data.

The R-R distribution function F ϕ<sup>0</sup> ð Þ is expressed as:

term a. The k scale parameter is obtained by k ¼ exp ð Þ �a=m .

nozzle.

A. Appendix A

Weibull p.d.f.

variable is:

#### 3.4 Simulation of droplet dispersion from a nozzle in a cultivated field

Figure 13 shows the drift simulation of droplets spraying over cultivate field with a nozzle at 0.75 m above the ground.

The meteorological conditions of air temperature at nozzle level are 30°C with 2ms<sup>1</sup> velocity wind. The simulation time shown in Figure 13 is 20 s after the start of spraying. The drift of small droplets (less than 50 μm in diameter) exceeds 8 m in the area of application. Of the total liquid sprayed, 0.43% corresponds to droplets smaller than 50 μm measured by Nuyttens [12] in wind tunnel at 50 cm below the spray nozzle. In the simulation shown in Figure 13, this percentage does not change because the satellite droplets generated are greater than 80 μm. The number of satellite droplets generated by stretching and reflective separation in these conditions was obtained. Of the 120 satellite droplets analyzed, 35.3% have diameters less than 150 μm, 61.3% have diameters between 150 and 250 μm, and 3.4% have diameters between 250 and 350 μm. There were no satellite droplets with diameters larger than 350 μm.

#### 4. Conclusions

In the present work, it was possible to simulate and validate the ejection velocity of the liquid particles from an HARDI™ ISO F110 03 nozzle placed at 0.75 m over ground. The diameters of the drops were randomized to the volume applied following a procedure of Rosin-Rammler distribution function for obtaining the parameters of Weibull probability density function with a correlation coefficient R2 = 0.997. The double-way coupled Euler-Lagrangian model has been used for obtaining the trajectory of droplet spraying. Eddies at both sides of spraying have been captured by the model. These extend up to 3 m from the center of the spray. The vertical component droplet velocity was simulated and validated with laboratory measurements. The velocity of droplets smaller than 200 μm slightly underestimates with respect to laboratory data. The binary collision models have been implemented into the code to consider particle collision events. A collision detection algorithm using collision boxes was presented and used to optimize computation times. The drift of droplets with air temperature of 30°C and wind speed value 2ms<sup>1</sup> has been simulated in cultivated fields. The drift of droplets smaller than 50 μm diameter exceeds 8 m of the application area. No satellite droplets smaller than 80 μm are generated under field simulation conditions. The largest proportion of satellite droplets generated as a result of the droplet collision has a diameter between 80 and 250 μm.

#### Acknowledgements

The present work is funded through the PIO CONICET-UNER 2015–2016 project and the FCyT-UADER research project "Development of a simulation model for the study of the drift of agricultural sprayings, using a flat fan nozzle, from trailing

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray DOI: http://dx.doi.org/10.5772/intechopen.81110

equipment." We are also grateful to the Laboratory of Prototyping of Electronics and 3D Printing of the School of Engineering, UNER, for the work of printing the nozzle.

#### A. Appendix A

this can cause an increase in the proportion of sprayed product not reaching its

Figure 13 shows the drift simulation of droplets spraying over cultivate field

The meteorological conditions of air temperature at nozzle level are 30°C with 2ms<sup>1</sup> velocity wind. The simulation time shown in Figure 13 is 20 s after the start of spraying. The drift of small droplets (less than 50 μm in diameter) exceeds 8 m in the area of application. Of the total liquid sprayed, 0.43% corresponds to droplets smaller than 50 μm measured by Nuyttens [12] in wind tunnel at 50 cm below the spray nozzle. In the simulation shown in Figure 13, this percentage does not change because the satellite droplets generated are greater than 80 μm. The number of satellite droplets generated by stretching and reflective separation in these conditions was obtained. Of the 120 satellite droplets analyzed, 35.3% have diameters less than 150 μm, 61.3% have diameters between 150 and 250 μm, and 3.4% have diameters between 250 and 350 μm. There were no satellite droplets with diameters

In the present work, it was possible to simulate and validate the ejection velocity of the liquid particles from an HARDI™ ISO F110 03 nozzle placed at 0.75 m over ground. The diameters of the drops were randomized to the volume applied following a procedure of Rosin-Rammler distribution function for obtaining the parameters of Weibull probability density function with a correlation coefficient R2 = 0.997. The double-way coupled Euler-Lagrangian model has been used for obtaining the trajectory of droplet spraying. Eddies at both sides of spraying have been captured by the model. These extend up to 3 m from the center of the spray. The vertical component droplet velocity was simulated and validated with laboratory measurements. The velocity of droplets smaller than 200 μm slightly underestimates with respect to laboratory data. The binary collision models have been implemented into the code to consider particle collision events. A collision detection algorithm using collision boxes was presented and used to optimize computation times. The drift of droplets with air temperature of 30°C and wind speed value 2ms<sup>1</sup> has been simulated in cultivated fields. The drift of droplets smaller than 50 μm diameter exceeds 8 m of the application area. No satellite droplets smaller than 80 μm are generated under field simulation conditions. The largest proportion of satellite droplets generated as a result of the droplet collision has a diameter

The present work is funded through the PIO CONICET-UNER 2015–2016 project and the FCyT-UADER research project "Development of a simulation model for the study of the drift of agricultural sprayings, using a flat fan nozzle, from trailing

3.4 Simulation of droplet dispersion from a nozzle in a cultivated field

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

destination, leaving it adrift.

larger than 350 μm.

4. Conclusions

between 80 and 250 μm.

Acknowledgements

60

with a nozzle at 0.75 m above the ground.

The Weibull probability density function (Weibull p.d.f.) can be used for describing a lot of technical applications for which the distribution of ground material, particles dispersion, or droplet diameters in spray jet normally in the μm-band have behaviors with a random characteristic. In this case, the diameters of droplets ejected from a spray nozzle are simulated using a Weibull p.d.f.

Let us name the random variable ϕ<sup>0</sup> that represents the initial diameter of the liquid particles, and the expression of Weibull p.d.f. for this random variable is:

$$\begin{cases} f(\phi\_0) = \frac{m}{k^m} \phi\_0^{m-1} \exp\left[-\left(\frac{\phi\_0}{k}\right)^m\right], & \text{if } \phi\_0 \ge 0, \\ f(\phi\_0) = 0, & \text{if } \phi\_0 < 0. \end{cases} \tag{55}$$

In this p.d.f., m > 0 is the shape parameter and k > 0 is the scale parameter. The form of the density function of the Weibull distribution changes drastically with the value of m parameters. The k parameter does not change the shape of the distribution, but it extends along the random variable ϕ0. In this way, if the parameters m and k are chosen correctly, it is possible to obtain the shape and stretch of the Weibull p.d.f. that fits the experimentally measured diameter data.

To take these data into account, the cumulative function of the Weibull distribution, named Rosin-Rammler (R-R), is used.

The R-R distribution function F ϕ<sup>0</sup> ð Þ is expressed as:

$$F(\phi\_0) = 1 - \exp\left[-\left(\frac{\phi\_0}{k}\right)^m\right].\tag{56}$$

Eq. (A.2) can be written as:

$$\operatorname{Ln}\{-\operatorname{Ln}[\mathbf{1} - F(\phi\_0)]\} = m \operatorname{Ln}(\phi\_0) - m \operatorname{Ln}(k). \tag{57}$$

If we associate Eq. (A.3) with a linear equation f(X) = mX + a where the dependent variable is f Xð Þ¼ Ln �Ln 1 � F ϕ<sup>0</sup> f g ½ � ð Þ and the independent variable is X ¼ Ln ϕ<sup>0</sup> ð Þ, m is the slope and a = �mLn(k) is intercept, in which the line cuts on the Y=f(X) axis. The method to obtain the Weibull p.d.f. parameters consists of plotting over logarithmic scale; the results of the experimental measurements Ln �Ln 1 � F ϕ<sup>0</sup> f g ½ � ð Þ vs. Ln ϕ<sup>0</sup> ð Þand approximate the point cloud to a linear regression by the least squares to obtain the slope (shape parameter) m and the constant term a. The k scale parameter is obtained by k ¼ exp ð Þ �a=m .

With these two parameters (m, k), it is possible to obtain the Weibull p.d.f. that describes the droplet diameter's distribution corresponding to the experiment measurements.

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

References

10.2516/ogst/2015024

[2] Sabelnikov V, Lipatnikov A,

Chakraborty N, Nishiki S, Hasegawa T. A transport equation for reaction rate in turbulent flow. Physics of Fluids. 2016; 28(8):081701. DOI: 10.1063/1.4960390

[3] Wei G, Vinkovic I, Shao L, Simoëns S. Scalar dispersion by a large-eddy simulation and a Lagrangian stochastic subgrid model. Physics of Fluids. 2006; 18:095101. DOI: 10.1063/1.2337329

[4] Aguirre C, Brizuela A, Vinkovic I, Simoëns S. A subgrid Lagrangian stochastic model for turbulent passive

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[5] Vinkovic I, Aguirre C, Simoëns S. Large-eddy simulation and Lagrangian stochastic modelling of passive scalar dispersion in a turbulent boundary layers. Journal of Turbulence. 2006;

[6] Vinkovic I, Aguirre C, Simoëns S, Gorokhovski M. Large-eddy simulation

inhomogeneous turbulent wall flow. International Journal of Multiphase

[7] Vinkovic I, Aguirre C, Ayrault M, Simoëns S. Large-eddy simulation of the dispersion of solid particles in a turbulent boundary layers. Journal of Boundary Layers-Meteorology. 2006:1472-1573. DOI: 10.1007/s10546-006-9072-6

[8] Stephens DW, Keough S, Sideroff C. Euler-Lagrange large eddy simulation of

and reactive scalar dispersion.

7(30):1468-5248

63

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Flow. 2006;32(3):344-364

[1] Battistoni M, Xue Q, Som S. Largeeddy simulation (LES) of spray transients: Start and end of injection phenomena. Oil & Gas Science and Technology—Rev IFP Energies Nouvelles. 2015;2015:1-24. DOI:

DOI: http://dx.doi.org/10.5772/intechopen.81110

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

a square cross-sectioned bubble column.

incorporating Chemeca; 27 Sept–01 Oct 2015; Melbourne, Victoria; Paper no.

turbulent channel flow: Universalities in statistic, subgrid stochastic models and

[10] Orcellet E, Berri J, Aguirre C, Müller G. Atmospheric dispersion study of TRS compounds emitted from a pulp mill plant in coastal regions of the Uruguay river, South America. Aerosol and Air

Computational tools for the simulation of atmospheric pollution transport during a severe wind event in

Argentina. In: Atmospheric Hazards—

Communication, and Societal Impacts. Chapter 6. InTech Open Science, Open Mind.; 2016. pp. 111-136. DOI: 10.5772/

[12] Nuyttens D. Drift from field crops prayers: The influence of spray application technology determined using indirect and direct drift

assessment means [thesis]. Germany:

[13] Xue M, Droegemeier K, Wong V. The advanced regional prediction system (ARPS). A multi-scale

nonhydrostatic atmospheric simulation and prediction model. Part I: Model dynamics and verification. Meteorology Atmospheric Physics. 2000;75:161-193

wetenschappen, Katholieke Universiteit

In: APCChE 2015 Congress

[9] Zamansky R, Vinkovic I, Gorokhovsky M. Acceleration in

an application. Journal of Fluid Mechanics. 2013;721:627-668. DOI:

Quality Research. 2016;16(6): 1473-1482. DOI: 10.4209/aaqr.2015.

[11] Aguirre C, Brizuela A.

Case Studies in Modeling,

Faculteit Bio-ingenieurs

Leuven; 2007

3134665. 2015. pp. 1-12

10.1017/jfm.2013.48

02.0112

63552

#### Author details

Carlos G. Sedano<sup>1</sup> , César Augusto Aguirre2,3\* and Armando B. Brizuela1,2,3

1 School of Science and Technology, Autonomous University of Entre Ríos (FCyT-UADER), Entre Ríos, Argentina

2 School of Agricultural Sciences, National University of Entre Ríos (FCA-UNER), Entre Ríos, Argentina

3 National Research Council of Science and Technology (CONICET), Buenos Aires, Argentina

\*Address all correspondence to: cesaraguirredalotto@gmail.com

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray DOI: http://dx.doi.org/10.5772/intechopen.81110

#### References

[1] Battistoni M, Xue Q, Som S. Largeeddy simulation (LES) of spray transients: Start and end of injection phenomena. Oil & Gas Science and Technology—Rev IFP Energies Nouvelles. 2015;2015:1-24. DOI: 10.2516/ogst/2015024

[2] Sabelnikov V, Lipatnikov A, Chakraborty N, Nishiki S, Hasegawa T. A transport equation for reaction rate in turbulent flow. Physics of Fluids. 2016; 28(8):081701. DOI: 10.1063/1.4960390

[3] Wei G, Vinkovic I, Shao L, Simoëns S. Scalar dispersion by a large-eddy simulation and a Lagrangian stochastic subgrid model. Physics of Fluids. 2006; 18:095101. DOI: 10.1063/1.2337329

[4] Aguirre C, Brizuela A, Vinkovic I, Simoëns S. A subgrid Lagrangian stochastic model for turbulent passive and reactive scalar dispersion. International Journal of Heat and Fluid Flow. 2006;27(4):627-635. DOI: 10.1016/j.ijheatfluidflow.2006.02.011

[5] Vinkovic I, Aguirre C, Simoëns S. Large-eddy simulation and Lagrangian stochastic modelling of passive scalar dispersion in a turbulent boundary layers. Journal of Turbulence. 2006; 7(30):1468-5248

[6] Vinkovic I, Aguirre C, Simoëns S, Gorokhovski M. Large-eddy simulation of droplet dispersion for inhomogeneous turbulent wall flow. International Journal of Multiphase Flow. 2006;32(3):344-364

[7] Vinkovic I, Aguirre C, Ayrault M, Simoëns S. Large-eddy simulation of the dispersion of solid particles in a turbulent boundary layers. Journal of Boundary Layers-Meteorology. 2006:1472-1573. DOI: 10.1007/s10546-006-9072-6

[8] Stephens DW, Keough S, Sideroff C. Euler-Lagrange large eddy simulation of a square cross-sectioned bubble column. In: APCChE 2015 Congress incorporating Chemeca; 27 Sept–01 Oct 2015; Melbourne, Victoria; Paper no. 3134665. 2015. pp. 1-12

[9] Zamansky R, Vinkovic I, Gorokhovsky M. Acceleration in turbulent channel flow: Universalities in statistic, subgrid stochastic models and an application. Journal of Fluid Mechanics. 2013;721:627-668. DOI: 10.1017/jfm.2013.48

[10] Orcellet E, Berri J, Aguirre C, Müller G. Atmospheric dispersion study of TRS compounds emitted from a pulp mill plant in coastal regions of the Uruguay river, South America. Aerosol and Air Quality Research. 2016;16(6): 1473-1482. DOI: 10.4209/aaqr.2015. 02.0112

[11] Aguirre C, Brizuela A. Computational tools for the simulation of atmospheric pollution transport during a severe wind event in Argentina. In: Atmospheric Hazards— Case Studies in Modeling, Communication, and Societal Impacts. Chapter 6. InTech Open Science, Open Mind.; 2016. pp. 111-136. DOI: 10.5772/ 63552

[12] Nuyttens D. Drift from field crops prayers: The influence of spray application technology determined using indirect and direct drift assessment means [thesis]. Germany: Faculteit Bio-ingenieurs wetenschappen, Katholieke Universiteit Leuven; 2007

[13] Xue M, Droegemeier K, Wong V. The advanced regional prediction system (ARPS). A multi-scale nonhydrostatic atmospheric simulation and prediction model. Part I: Model dynamics and verification. Meteorology Atmospheric Physics. 2000;75:161-193

Author details

Carlos G. Sedano<sup>1</sup>

Entre Ríos, Argentina

Argentina

62

(FCyT-UADER), Entre Ríos, Argentina

provided the original work is properly cited.

, César Augusto Aguirre2,3\* and Armando B. Brizuela1,2,3

1 School of Science and Technology, Autonomous University of Entre Ríos

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

\*Address all correspondence to: cesaraguirredalotto@gmail.com

2 School of Agricultural Sciences, National University of Entre Ríos (FCA-UNER),

3 National Research Council of Science and Technology (CONICET), Buenos Aires,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

[14] Aguirre C. Dispersión et Mélange Atmosphérique Euléro-Lagrangien de Particules Fluides Réactives. Application à des cas simples et complexes [thesis]. Lyon, France: Université Claude Bernard; 2005. 337 p

[15] Fackrell J, Robins A. Concentration fluctuation and fluxes in plumes from point sources in a turbulent boundary layers. Journal of Fluid Mechanics. 1982; 117:1-26

[16] Gong W. A wind tunnel study of turbulent dispersion over two- and three-dimensional gentle Hills from upwind point sources in neutral flow. Boundary Layers Meteorology. 1991;54: 211-230

[17] Munnannur A, Reitz R. A new predictive model for fragmenting and non-fragmenting binary droplet collisions. International Journal of Multiphase Flow. 2007;33(8):873-896. DOI: 10.1016/j.ijmultiphaseflow. 2007.03.003

[18] Kim S, Lee D, Lee C. Modeling of binary droplet collisions for application to inter-impingement sprays. International Journal of Multiphase Flow. 2009;35:533-549. DOI: 10.1016/j. ijmultiphaseflow.2009.02.010

[19] Zhang H, Li Y, Li J, Liu Q. Study on separation abilities of moisture separators based on droplet collision models. Nuclear Engineering and Design. 2017;325:135-148. DOI: 10.1016/ j.nucengdes.2017.09.030

[20] Rabe C, Malet J, Feuillebois F. Experimental investigation of water droplet binary collisions and description of outcomes with a symmetric Weber number. Physics of Fluids. 2010;22: 047101. DOI: 10.1063/1.3392768

[21] Sedano C, Aguirre C, Brizuela A. Simulación de la Eyección de Spray Líquido desde un Pico de Pulverizadora para Aplicación de Herbicidas.

Asociación Argentina de Mecánica Computacional AMCA - Mecánica Computacional. 2017;VXXIII: 1049-1068

[30] Gicquel LYM, Givi P, Jaberi FA, Pope SB. Velocity filtered density function for large-eddy simulation of turbulent flow. Physics of Fluids. 2002;

DOI: http://dx.doi.org/10.5772/intechopen.81110

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray

[38] Ko GH, Ryou HS. Modeling of droplet collision-induced breakup process. International Journal of

02.004. ISSN 0301-9322

2017.05.094

University; 1981

Multiphase Flow. 2005;31:723-738. DOI: 10.1016/j.ijmultiphaseflow.2005.

[39] Hu C, Xia S, Li C, Wu G. Threedimensional numerical investigation and modeling of binary alumina droplet collisions. International Journal of Heat and Mass Transfer. 2017;113:569-588. DOI: 10.1016/j.ijheatmasstransfer.

[40] O'Rourke P. Collective drop effects in vaporizing liquid sprays [PhD dissertation]. Princeton, NJ: Dept. Mech. Aerospace Engg., Princeton

[41] Georjon TL, Reitz RD. A dropshattering collision model for

multidimensional spray computations. Atomization and Sprays. 1999;9(3)

[42] Almohammed N. Modelling and simulation of particle agglomeration droplet coalescence and particle-wall adhesion in turbulent multiphase flow [thesis]. Hamburg, Germany: Helmet-Schmidt University; 2018; 402 p

[31] Deardorff JW. Stratocumuluscapped mixed layer derived from a three dimensional model. Boundary Layer Meteorology. 1980;18:495-527

[32] Pope SB. Lagrangian PDF methods for turbulent flows. Annual Review of Fluid Mechanics. 1994;26:23-63

[33] Brazier-Smith PR, Jennings SG, Latham J. The interaction of falling water drops: Coalescence. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 1972;326(1566):393-408. DOI: 10.1098/

[34] Ashrgiz N, Poo JY. Coalescence and separation in binary collisions of liquid drops. Journal of Fluid Mechanics. 1990;

[35] Estrade JP, Carentz H, Lavergne G, Biscos Y. Experimental investigation of dynamic binary collision of ethanol droplets—A model for droplet

coalescence and bouncing. International Journal of Heat and Fluid Flow. 1999;

[36] Brenn G, Valkovska D, Danov K. The formation of satellite droplets by unstable binary drop collisions. Physics of Fluids. 1994;13(9):2463, 2001-2477.

[37] Kollár LE, Farzaneh M, Karev AR.

coalescence in an icing wind tunnel and the influence of these processes on droplet size distribution. International Journal of Multiphase Flow. 2005;31(1): 69-92. DOI: 10.1016/j.ijmultiphaseflow.

Modeling droplet collision and

2004.08.007. ISSN 0301-9322

65

20(5):486-491. DOI: 10.1016/ S0142-727X(99)00036-3

DOI: 10.1063/1.1384892

14(3):1196-1213

rspa.1972.0016

221:183-204

[22] Ayres D, Caldas M, Semião V, da Graça Carvalho M. Prediction of the droplet size and velocity joint distribution for sprays. Fuel. 2001;80: 383-394

[23] Baetens K. Development and application of drift prediction models in fields praying [thesis]. Leuven: Faculteit Bio-ingenieurs wetenschappen, Katholieke Universiteit; 2009

[24] Macías-García A, Cuerda-Correa E, Díaz-Díez M. Application of the Rossin-Rammler and Gates-Gaudin-Schuhmann models to the particle size analysis of agglomerated cork. Material Characterization. 2004;52:159-164. DOI: 10.1016/j.matchar.2004.04.007

[25] Michelot C. Développement d'un modèle stochastique lagrangien. Application à la dispersion et à la chimie de l'atmosphère [thesis]. Lyon, France: Université Claude Bernard; 1996. 180 p

[26] Holterman H. Kinetic and evaporation of waterdrops in air, Wageningen: IMAG. Report 2003-12; Wageningen UR, InstituutvoorvMilieu en Agritechniek; 2003. 67 p

[27] Sedano C, Aguirre A, Brizuela B. Numerical simulation of spray ejection from nozzle for herbicide application: Comparison of drag coefficient expressions. Computers and Electronics in Agriculture. Elsevier. Forthcoming

[28] Turton R, Levenspiel O. A short note on the drag correlation for sphere. Powder Technology. 1986;47:83-86. DOI: 10.1016/0032-5910(86)80012-2

[29] Germano M, Piomelli U, Moin P, Cabot WH. A dynamic subgrid-scale eddy viscosity model. Physics of Fluids. 1991;A(3):1760-1765

An Eulerian-Lagrangian Coupled Model for Droplets Dispersion from Nozzle Spray DOI: http://dx.doi.org/10.5772/intechopen.81110

[30] Gicquel LYM, Givi P, Jaberi FA, Pope SB. Velocity filtered density function for large-eddy simulation of turbulent flow. Physics of Fluids. 2002; 14(3):1196-1213

[14] Aguirre C. Dispersión et Mélange Atmosphérique Euléro-Lagrangien de Particules Fluides Réactives. Application à des cas simples et complexes [thesis]. Lyon, France: Université Claude

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

Asociación Argentina de Mecánica Computacional AMCA - Mecánica Computacional. 2017;VXXIII:

[22] Ayres D, Caldas M, Semião V, da Graça Carvalho M. Prediction of the droplet size and velocity joint

distribution for sprays. Fuel. 2001;80:

[23] Baetens K. Development and application of drift prediction models in fields praying [thesis]. Leuven: Faculteit

Bio-ingenieurs wetenschappen, Katholieke Universiteit; 2009

Rammler and Gates-Gaudin-

10.1016/j.matchar.2004.04.007

[26] Holterman H. Kinetic and evaporation of waterdrops in air, Wageningen: IMAG. Report 2003-12; Wageningen UR, InstituutvoorvMilieu

en Agritechniek; 2003. 67 p

[27] Sedano C, Aguirre A, Brizuela B. Numerical simulation of spray ejection from nozzle for herbicide application: Comparison of drag coefficient

expressions. Computers and Electronics in Agriculture. Elsevier. Forthcoming

[28] Turton R, Levenspiel O. A short note on the drag correlation for sphere. Powder Technology. 1986;47:83-86. DOI: 10.1016/0032-5910(86)80012-2

[29] Germano M, Piomelli U, Moin P, Cabot WH. A dynamic subgrid-scale eddy viscosity model. Physics of Fluids.

1991;A(3):1760-1765

[24] Macías-García A, Cuerda-Correa E, Díaz-Díez M. Application of the Rossin-

Schuhmann models to the particle size analysis of agglomerated cork. Material Characterization. 2004;52:159-164. DOI:

[25] Michelot C. Développement d'un modèle stochastique lagrangien.

Application à la dispersion et à la chimie de l'atmosphère [thesis]. Lyon, France: Université Claude Bernard; 1996. 180 p

1049-1068

383-394

[15] Fackrell J, Robins A. Concentration fluctuation and fluxes in plumes from point sources in a turbulent boundary layers. Journal of Fluid Mechanics. 1982;

[16] Gong W. A wind tunnel study of turbulent dispersion over two- and three-dimensional gentle Hills from upwind point sources in neutral flow. Boundary Layers Meteorology. 1991;54:

[17] Munnannur A, Reitz R. A new predictive model for fragmenting and non-fragmenting binary droplet collisions. International Journal of Multiphase Flow. 2007;33(8):873-896. DOI: 10.1016/j.ijmultiphaseflow.

[18] Kim S, Lee D, Lee C. Modeling of binary droplet collisions for application

[19] Zhang H, Li Y, Li J, Liu Q. Study on

to inter-impingement sprays. International Journal of Multiphase Flow. 2009;35:533-549. DOI: 10.1016/j.

ijmultiphaseflow.2009.02.010

separation abilities of moisture separators based on droplet collision models. Nuclear Engineering and Design. 2017;325:135-148. DOI: 10.1016/

[20] Rabe C, Malet J, Feuillebois F. Experimental investigation of water droplet binary collisions and description of outcomes with a symmetric Weber number. Physics of Fluids. 2010;22: 047101. DOI: 10.1063/1.3392768

[21] Sedano C, Aguirre C, Brizuela A. Simulación de la Eyección de Spray Líquido desde un Pico de Pulverizadora

para Aplicación de Herbicidas.

64

j.nucengdes.2017.09.030

Bernard; 2005. 337 p

117:1-26

211-230

2007.03.003

[31] Deardorff JW. Stratocumuluscapped mixed layer derived from a three dimensional model. Boundary Layer Meteorology. 1980;18:495-527

[32] Pope SB. Lagrangian PDF methods for turbulent flows. Annual Review of Fluid Mechanics. 1994;26:23-63

[33] Brazier-Smith PR, Jennings SG, Latham J. The interaction of falling water drops: Coalescence. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 1972;326(1566):393-408. DOI: 10.1098/ rspa.1972.0016

[34] Ashrgiz N, Poo JY. Coalescence and separation in binary collisions of liquid drops. Journal of Fluid Mechanics. 1990; 221:183-204

[35] Estrade JP, Carentz H, Lavergne G, Biscos Y. Experimental investigation of dynamic binary collision of ethanol droplets—A model for droplet coalescence and bouncing. International Journal of Heat and Fluid Flow. 1999; 20(5):486-491. DOI: 10.1016/ S0142-727X(99)00036-3

[36] Brenn G, Valkovska D, Danov K. The formation of satellite droplets by unstable binary drop collisions. Physics of Fluids. 1994;13(9):2463, 2001-2477. DOI: 10.1063/1.1384892

[37] Kollár LE, Farzaneh M, Karev AR. Modeling droplet collision and coalescence in an icing wind tunnel and the influence of these processes on droplet size distribution. International Journal of Multiphase Flow. 2005;31(1): 69-92. DOI: 10.1016/j.ijmultiphaseflow. 2004.08.007. ISSN 0301-9322

[38] Ko GH, Ryou HS. Modeling of droplet collision-induced breakup process. International Journal of Multiphase Flow. 2005;31:723-738. DOI: 10.1016/j.ijmultiphaseflow.2005. 02.004. ISSN 0301-9322

[39] Hu C, Xia S, Li C, Wu G. Threedimensional numerical investigation and modeling of binary alumina droplet collisions. International Journal of Heat and Mass Transfer. 2017;113:569-588. DOI: 10.1016/j.ijheatmasstransfer. 2017.05.094

[40] O'Rourke P. Collective drop effects in vaporizing liquid sprays [PhD dissertation]. Princeton, NJ: Dept. Mech. Aerospace Engg., Princeton University; 1981

[41] Georjon TL, Reitz RD. A dropshattering collision model for multidimensional spray computations. Atomization and Sprays. 1999;9(3)

[42] Almohammed N. Modelling and simulation of particle agglomeration droplet coalescence and particle-wall adhesion in turbulent multiphase flow [thesis]. Hamburg, Germany: Helmet-Schmidt University; 2018; 402 p

Section 2

Computational Fluid

Dynamics Applications

67

Section 2
