**1. Introduction**

24 Numerical Simulations

26 Computational Simulations and Applications

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Most of nature and industry flows are turbulence. There are three kinds of numerical simulation methods for turbulent flows (Lesieur 1990; Pope 2000; Sagaut 2000, 2006): direct numerical simulation (DNS), Reynolds-averaged Navier-Stokes equations (RANS) and large eddy simulation (LES). DNS is a straightforward way to simulate turbulent flows. Full Navier-Stokes equations are discretized and solved numerically without any model, empirical parameter or approximation. Theoretically speaking, results of DNS exactly reflect the real flow and the whole range of turbulence scales are computed. With DNS, people can compute and visualize any quantity of interest, including some that are too difficult or impossible to be measured by experiments. But as we all know the computation cost is very high. For high Reynolds number flow, even modern computer technology can not satisfy the computation requirement.

In RANS, the flow quantities are decomposed into two parts: the average or mean term and the fluctuating term by applying Reynolds averaging. The effect of the fluctuating quantities on the mean flow quantities is described by the so called Reynolds stress tensor, which is must be modelled in terms of the mean velocities. Typical models can be grouped loosely into three categories: algebraic models, one-equation models and two-equation models. RANS is simple and robust. It is widely used in engineering problem. The general limitation of RANS is the fact that the model must represent a very wide range of scales. While the small scales tend to be universal, and depend on viscosity, the larger scales depend largely on flow condition and boundaries. So there is no one universal model for all flows. For different flows, the model must be modified to obtain good results. Another issue is that usually a time averaging is adopted in RANS. So RANS has difficult to handle unsteady flows.

In LES, a filter is applied to separate the large scales from small scales. Then only the large, energy carrying scales (or called resolved scales) of turbulence are computed exactly by solving the governing equations. While the small, fluctuating scales are modelled, which is also called subgrid scales (SGS). Compared to RANS, LES has several advantages: 1) LES can capture the large scales directly which are the main energy container of turbulence and response for the momentum and energy transfer. 2) The dissipation of turbulence energy is believed to be done by small scales. Since small scales are thought to be homogenous,

Study of Some Key Issues for

Applying LES to Real Engineering Problems 29

theoretical analysis of numerical errors in LES has been proposed by Ghosal (Ghosal 1996) and later Chow and Moin (Chow and Moin 2003). They believed that 2nd order discretization scheme is not suitable for LES because it introduces errors larger than the SGS term. High order schemes are necessary. By applying the eddy-damped quasi-normal Markovian (EDQNM) theory to LES, a so called dynamic error analysis has been performed by Park and Mahesh (Park and Mahesh 2007) Their results show that low order scheme is acceptable for LES. The study of Yang and Fu (Yang and Fu 2008) show that there are complicated interactions between SGS model and numerical errors. A good SGS model can not only represent the effect of small scales to large scales, but also can dump the unphysical energy introduced by numerical scheme. So by careful designed SGS model, low order discretization scheme can also obtain reasonable result. Fauconnier *et al* (Fauconnier *et al* 2009) also point out that low-order methods may have advantages over high order scheme because the dissipation error of SGS model can cancel part of the numerical errors resulting in a reduction of the total errors on some quantities. Of course the disadvantage is that the accuracy of small scales is not

In Large Eddy Simulation (LES) a filtering operation is applied to separate the large scales

where *G* is the filter kernel and *D* is the filtering domain. The filter is characterized by a filter

For our study, the fluid is assumed to be incompressible; the viscosity is constant; there are no body forces; and the flow is initially homogenous, isotropic, i.e. there are no mean velocity gradients. So the incompressible Navier-Stokes equations after applying a low-pass

*<sup>i</sup> j j* 1 *ij i i*

*t x xx x x x*

 

*u u u u P u*

Above equations are also called the incompressible LES equations. The *u*, *P*,

velocity, pressure, density and kinematic viscosity, respectively. *ij*

*j i j j i j*

*ij uu u u <sup>i</sup> <sup>j</sup> <sup>i</sup> <sup>j</sup>*

It represents the effect of the unresolved (small) scales. It is the only unclosed term in the above LES equations (2.2) and should be parameterized in terms of the resolved (large)

In order to isolate other effects, the simplest homogenous, isotropic turbulence is chose as our simulation case. The advantage is that we can obtain the statistical quantities of this turbulence easily in spectral space, such s energy spectrum, total kinetic energy etc. So in

( ) ( ') ( , '; ) ' *<sup>D</sup> <sup>f</sup> <sup>x</sup> <sup>f</sup> x G x x dx* (2.1)

*i i u x* 

. (2.3)

. (2.2)

 , are the

is the subgrid stress

is called as the filter cut-off wave

from the small scales (Leonard 1974). In general, a filtered variable can be written as

controlled. So the best is high order scheme plus high accurate SGS model.

**2. Governing equations and numerical methods** 

width . The corresponding wave number *<sup>c</sup> k*

number.

(SGS) tensor

scales.

filter can be written as

universal, and less affected by flow and boundary conditions, the SGS model can be simple and requires fewer *ad hoc* parameters when it is applied to different flows. This is the big advantage of LES over RANS. That also is the reason why simple Smagrinsky model can obtain reasonable results in different flows. 3) LES can solve the unsteady flow directly. In additional, LES requires much less computation resource when compared to DNS because only large scales are computed.

Although LES has some advantages, for a long time RANS methods were used almost exclusively for the analysis of turbulent flows for practical engineering problems. LES has largely been used to study simple turbulent flows(Mahesh et al 2004; Georgiadis 2008; Bouffanais 2010). The primary reason is the computational cost. Until recently, the field of LES is attracting more and more people's attention. Not only its own scientific researcher who is applying LES to study the turbulence, but more industrial partners and engineers have started implementing LES to study real complex flows. There are two main reasons: 1) the urgent requirements from industry. the characteristics of lots nature or real engineering flows are determined by unsteady large scale motion, such as the external flow around ground vehicle, high attack angle airfoil flow etc. RANS models usually have difficult to handle such flows. But in order to improve the performance of airplane, to reduce the drag and noise around vehicle, we have to investigate such flow in depth. (2) rapid increases in computing power, memory, and storage, plus high efficient and high order computation algorithm. Indeed in the past few years applying LES to real engineering flows has becomes a research *hot spot*, such as LES of airfoil (Mary&Sagaut 2001; Dahlstrom&Davidson 2003; Mellen et al 2003), ground vehicle (McCallen *et al.* 2006; Kitoh *et al.* 2009; Krajnovic&Davidson 2005; Rodi 2006; Tsubokura *et al.* 2009; Minguez *et al.* 2008), combustion and reacting flows (Moin 2002), weather forecasting etc. But the application of LES is still limited. There are some key issues needed to solved before LES can be successfully applied to real engineering turbulence(Georgiadis 2008; Bouffanais 2010), such as the suitable SGS model, the choice of filter, the wall model, the transition model, the effect of numerical errors and the interactions between these issues. However as Bouffanais (Bouffanais 2010) pointed out that despite the numerous challenges still facing LES, one can fairly admit that LES has become one of the most promising and successful methodology available to simulate industrial turbulent flows.

In this chapter, three key issues of LES are discussed briefly: the SGS model, the filter and the numerical errors. First, the SGS model is the most important item in LES and has been extensive studied. There are thousand of different models which have been proposed during the past. But most of them are limited to simple geometry and have difficult to be applied to engineering problems. Right now the most widely used SGS models in complicated turbulence are still the simple Smagrinsky model (Smagorinsky 1963) and the so called the monotone integrated LES (MILES) model. So a simple, robust, efficient and can handling complicated geometry SGS model is what we need. The second problem is the choice of filter. In simple geometry, usually a smooth filter is adopted which is defined continuous in the whole domain. But in complicated geometry, only local discrete filter can be used. Obvious the order of filtering will be decreased. Its effect on SGS model and final simulation result need to be investigated. The third is the numerical errors of different discretization schemes. The effect of numerical errors on LES is a delicate issue and has been ignored for a long time because in simple geometry very high order can be achieved by pseudo-spectral mothod or other algorithm. But for complex problem, usually only second order can be achieved. The interaction between numerical scheme and SGS model is complicated. A first extensive

universal, and less affected by flow and boundary conditions, the SGS model can be simple and requires fewer *ad hoc* parameters when it is applied to different flows. This is the big advantage of LES over RANS. That also is the reason why simple Smagrinsky model can obtain reasonable results in different flows. 3) LES can solve the unsteady flow directly. In additional, LES requires much less computation resource when compared to DNS because

Although LES has some advantages, for a long time RANS methods were used almost exclusively for the analysis of turbulent flows for practical engineering problems. LES has largely been used to study simple turbulent flows(Mahesh et al 2004; Georgiadis 2008; Bouffanais 2010). The primary reason is the computational cost. Until recently, the field of LES is attracting more and more people's attention. Not only its own scientific researcher who is applying LES to study the turbulence, but more industrial partners and engineers have started implementing LES to study real complex flows. There are two main reasons: 1) the urgent requirements from industry. the characteristics of lots nature or real engineering flows are determined by unsteady large scale motion, such as the external flow around ground vehicle, high attack angle airfoil flow etc. RANS models usually have difficult to handle such flows. But in order to improve the performance of airplane, to reduce the drag and noise around vehicle, we have to investigate such flow in depth. (2) rapid increases in computing power, memory, and storage, plus high efficient and high order computation algorithm. Indeed in the past few years applying LES to real engineering flows has becomes a research *hot spot*, such as LES of airfoil (Mary&Sagaut 2001; Dahlstrom&Davidson 2003; Mellen et al 2003), ground vehicle (McCallen *et al.* 2006; Kitoh *et al.* 2009; Krajnovic&Davidson 2005; Rodi 2006; Tsubokura *et al.* 2009; Minguez *et al.* 2008), combustion and reacting flows (Moin 2002), weather forecasting etc. But the application of LES is still limited. There are some key issues needed to solved before LES can be successfully applied to real engineering turbulence(Georgiadis 2008; Bouffanais 2010), such as the suitable SGS model, the choice of filter, the wall model, the transition model, the effect of numerical errors and the interactions between these issues. However as Bouffanais (Bouffanais 2010) pointed out that despite the numerous challenges still facing LES, one can fairly admit that LES has become one of the most promising and successful methodology

In this chapter, three key issues of LES are discussed briefly: the SGS model, the filter and the numerical errors. First, the SGS model is the most important item in LES and has been extensive studied. There are thousand of different models which have been proposed during the past. But most of them are limited to simple geometry and have difficult to be applied to engineering problems. Right now the most widely used SGS models in complicated turbulence are still the simple Smagrinsky model (Smagorinsky 1963) and the so called the monotone integrated LES (MILES) model. So a simple, robust, efficient and can handling complicated geometry SGS model is what we need. The second problem is the choice of filter. In simple geometry, usually a smooth filter is adopted which is defined continuous in the whole domain. But in complicated geometry, only local discrete filter can be used. Obvious the order of filtering will be decreased. Its effect on SGS model and final simulation result need to be investigated. The third is the numerical errors of different discretization schemes. The effect of numerical errors on LES is a delicate issue and has been ignored for a long time because in simple geometry very high order can be achieved by pseudo-spectral mothod or other algorithm. But for complex problem, usually only second order can be achieved. The interaction between numerical scheme and SGS model is complicated. A first extensive

only large scales are computed.

available to simulate industrial turbulent flows.

theoretical analysis of numerical errors in LES has been proposed by Ghosal (Ghosal 1996) and later Chow and Moin (Chow and Moin 2003). They believed that 2nd order discretization scheme is not suitable for LES because it introduces errors larger than the SGS term. High order schemes are necessary. By applying the eddy-damped quasi-normal Markovian (EDQNM) theory to LES, a so called dynamic error analysis has been performed by Park and Mahesh (Park and Mahesh 2007) Their results show that low order scheme is acceptable for LES. The study of Yang and Fu (Yang and Fu 2008) show that there are complicated interactions between SGS model and numerical errors. A good SGS model can not only represent the effect of small scales to large scales, but also can dump the unphysical energy introduced by numerical scheme. So by careful designed SGS model, low order discretization scheme can also obtain reasonable result. Fauconnier *et al* (Fauconnier *et al* 2009) also point out that low-order methods may have advantages over high order scheme because the dissipation error of SGS model can cancel part of the numerical errors resulting in a reduction of the total errors on some quantities. Of course the disadvantage is that the accuracy of small scales is not controlled. So the best is high order scheme plus high accurate SGS model.
