**3. SGS model**

30 Computational Simulations and Applications

such case it is convenience to write the governing equations in spectral space. Note the continuity equation can be combined with the pressure term through the projecting operation (Lesieur 1990). So the governing equation in spectral space can be simplified as

projection operator, which ensures the continuity equation automatically satisfied. And the

For spatial discretization, a computation method similar to Rogallo's (Rogallo 1981) is adapted here. For the viscous term in the left hand side of equation (2.4), Rogallo proposed

> 

11 22

*ii ii*

*a b* Order

. (2.6)

(2.7)

*f f ff*

So the only term needed to be discretized is the nonlinear term in the right hand side, () () *m jj <sup>m</sup> N ik u u* **k k** , which usually is solved by high order spectral scheme. But for engineering problem, spectral method is not available at most cases. Finite difference scheme or finite volume scheme is used instead. Among them, Padé compact scheme is widely adapted due to its flexibility in handling complex geometry and to obtaining high order. For one dimensional derivative, the Padé scheme proposed by Lele (Lele 1992) can be

1 1 2 4

Padé2 0 1 0 2 Padé4 1/4 3/2 0 4 Padé6 1/3 14/9 1/9 6

For the temporal advancement of the nonlinear term, an explicit second-order Runge-Kutta scheme, also known as predictor-corrector scheme, is used. It's simple and efficient. Briefly, equation (2.5) with only nonlinear term on the right hand side can be seen as the following

> *u N t*

where *N* represents the nonlinear term. By applying second-order R-K scheme to above

 *fffa b* 

Different coefficient defines different order of compact scheme. The highest order is 6th for 3

2 2 <sup>ˆ</sup> ( ,) ( ) ( ) *k t k t eu t eP N i im <sup>m</sup> <sup>t</sup>*

 

'''

*iii*

 

point stencil. The parameters of Lele (Lele 1992) are shown in Table 2.1.

Table 2.1. Parameters for Padé compact scheme.

*t* 

an integrated factor method which can solve it analytically.

*k* **k** .

expressed as

Scheme

equations, we get

form

where ' ' means the Fourier transform, the tensor <sup>2</sup> ( ) *P k im im i m* **k**

<sup>2</sup> <sup>ˆ</sup> ( ,) ( ) ( ) *i im j <sup>j</sup> <sup>m</sup> k u t P ik u u*

**k kk** . (2.4)

**k kk** . (2.5)

*k k* is called the

In LES, only the large, energy carrying scales of turbulence are computed exactly. Specify in LES equation (2.2), the large scales are the filtered velocities, *ui* , which are also called the resolved scales. The small ones, *ui* , (unresolved, or subgrid scales) have been removed from the equation and needed to be modelled, i.e. *ij* in equation (2.2).

The SGS model is the key issue in LES. Since only large scales are resolved in LES, the energy transfer from large scales to small scales is cut off. The energy will accumulate at the cut-off wave number and lead to the unphysical solution. So the main role of SGS model is to provide necessary small scales dissipation and thus remove the accumulated energy. There are many different approaches for the modelling of the SGS stress tensor. Traditionally they are divided into three main categories: eddy viscosity models, similarity models and mixed models. Discussion of standard LES models can be found in some review paper, such as Piomelli (Piomelli 1999), Mathew (Mathew 2010) etc. Below we only discuss the eddy viscosity model briefly.

The eddy viscosity models assume:

$$
\sigma\_{i\rangle} = \overline{u\_i u\_j} - \overline{u}\_i \overline{u}\_j = -2\nu\_t \overline{S}\_{i\rangle} + \mathbf{1}/3 \,\mathcal{S}\_{i\rangle} \mathbf{r}\_{kk} \tag{3.1}
$$

which relate the SGS stresses to the large scale strain-rate tensor *S*ij , where *S*ij is

$$\mathbf{s}\_{ij} = \frac{\mathbf{1}}{\mathbf{2}} \left( \frac{\partial \mathbf{u}\_i}{\partial \mathbf{x}\_j} + \frac{\partial \mathbf{u}\_j}{\partial \mathbf{x}\_i} \right) \tag{3.2}$$

and *ν*t is the eddy viscosity. Like RANS, equation (3.1) was developed by analogizing to the molecular viscosity. So different eddy viscosity models are actually different methods to calculate the *ν*t.

The Smagorinsky model (Smagorinsky 1963) is perhaps the most successful SGS eddy viscosity models, which takes eddy viscosity proportional to the product of *Δ*2 and *s* ,

$$\nu\_t = \left(\mathbb{C}\_s \boldsymbol{\Delta}\right)^2 \left|\overline{\boldsymbol{s}}\right|\tag{3.3}$$

where *C*s is called the Smagorinsky constant, *Δ* is the grid size and 1/2 2 *ij ij s ss* is the magnitude of the strain-rate tensor. By choosing different *C*s for various flows, Smagorinsky model has been used with considerable success. For isotropic decaying turbulence, the value of the Smagorinsky constant is taken to be around 0.18∼0.23 (Lilly (Lilly 1996)), but in shear flow or near boundaries, *C*s must be decreased and values 0.06∼0.1 are preferred (Piomelli *et al.* (Piomelli *et al.*1988)).

Study of Some Key Issues for

*u* from the filtered velocity *u*

them is

Applying LES to Real Engineering Problems 33

Domaradzki *et al* (Domaradzki & Saiki 1997; Domaradzki et al 2002) proposed a method to estimate the small scales. This is the basic idea of the velocity estimation model. In order to describe the different velocities, we use *u* to present the full, unfiltered turbulence velocity; *u* means the filtered velocity; and *u*' is the small scale velocity. And the relation among

VEM contains two steps: First is the defiltering operation, i.e. try to recover the full velocity

which is inverse operation of (2.1). Bertero and Boccacci (Bertero & Boccacci 1998) give a detailed discussion about it. Since any filtering will loss part of original information, the defiltering can not recover the full velocity according to Riemann–Lebesgue theory. Only some very special filtering function and variables can return to its original state. Most results of defiltering can only be approximate. The velocity u in equation (3.5) actually only contains large scales information, so we denote it by <sup>0</sup> *u* . If we using tophat filter and the

filtering size is twice as the grid size, then the filtering operation can be expressed as

0 00

By defiltering, the large scales <sup>0</sup> *u* is closed to the original *u*, but there is no small scales. If we approximate the u using <sup>0</sup> *u* , i.e. let <sup>0</sup> *u u* . And then calculate the SGS tensor *τ* directly

number is not too high, the result is good enough. But if the Re number is high, the error is somehow too large. The information of the small scales is needed. The second step of VEM is to estimate the small scales. For full developed turbulence, the small scales are thought to

where *N*' is the growth rate of subgrid scales due to the nonlinear interactions among resolved scales and *θ* is the time scale related to the eddy turn-over time. The detailed description of the full process can be found in paper (Domaradzki & Saiki 1997; Domaradzki

be homogeneous, so a simplified way to estimate small scales can be described as

The energy spectral of full (DNS), filtered, defiltered and estimated velocity are shown in Fig. 3.1

and *a*, *b*, *c* are constants. Then the defiltering operation is

'' *u N*

et al 2002). Thus the final velocity can be expressed as

Correspondingly, the SGS stress tensor is

from the definition 00 0 0 

*uuu* ' (3.4)

<sup>1</sup> *u Gu* , (3.5)

*u au bu cu i ii* 1 1 , (3.6)

*i ii* 1 1 *au bu cu u* . (3.7)

. (3.8)

<sup>0</sup> *uuu u* ' . (3.9)

*uu u u* . (3.10)

*uu u u* . Through practice, it is found that if the Reynolds

Smagorinsky model can properly account for the global energy transfer. It is simple and robust, which make it the most widely used SGS model. But the modeled SGS quantities correlate poorly with the actual SGS quantities obtained from DNS. Moreover it is an absolutely dissipative model and tends to overestimate the SGS dissipation. It only allows one way energy flux, i.e. from large scales to small ones, and it fails to predict the inverse energy transfer from the subgrid scales to the resolved scales (backscatter) which is found in most flows. Many *ad hoc* corrections and variation of eddy viscosity models are proposed to solve the difficulties mentioned above. Among them the dynamic model of Germano *et al.* (Germano *et al.*1991) and its variations are the most attractive ones. The dynamic model calculates the eddy viscosity dynamically and obtains good results in different turbulence. But it still has some problems when applied to complex engineering flows.

#### **3.1 Velocity Estimation Model (VEM)**

To construct a reasonable and reliable SGS model, to properly predict the interactions between large scales and small scales is the key, which means we need to know more detailed information about the nonlinear interactions between large and small scales. Fortunately during the last several years there are many investigations in a variety of turbulent flows, including isotropic and channel flow, at low Reynolds numbers using direct numerical simulation databases and experimental measurements (Zhou 1993; Hartel *et al.* 1994; Domaradzki & Rogallo 1990). Their studies show that the large scales contain enough information. Many of the observed features of the exact SGS interactions can be inferred from the dynamics of the resolved scales alone. Thus it implies a possible way to improving SGS model, i.e. to estimate the small scales from large scales by using the observed properties of the nonlinear interactions. Based on that concept, Domaradzki *et al* (Domaradzki & Saiki 1997; Domaradzki et al 2002) develop the velocity estimation model in both spectral and physical space. Stolz and Adam also proposed similar model called deconvolution model (Stolz 1999).

The velocity estimation model is based on two observations: first, the dynamics of small scales are strongly determined by the large, energy carrying eddies; second, the contribution of small scales to large scales are mostly contained within wavenumbers that are twice that of the cutoff wavenumber, *k*c. These two observations rely on the properties of nonlinear interactions in turbulent flows and have been elucidated by a large number of theoretical, numerical and experimental investigations (Zhou 1993; Domaradzki & Rogallo 1990; Domaradzki & Saiki 1997; Domaradzki et al 2002). Basically these studies showed that most of the subgrid scale transfer happens in the range of 0.5*k*c ~*k*c and is determined by scales in the range of *k*c ~ 2*k*c. This implies that only a limited range of wavenumbers needs to be considered. Especially in VEM the modes beyond 2*k*c are ignored. With a proper estimation of the velocity field the subgrid scale stress tensor could be determined directly from the resolved scales and provides enough dissipation for LES.

The eddy viscosity models basically try to solve the imaginary *<sup>t</sup>* by related to the large scale strain-rate tensor. If we know the full velocity field of the turbulence flow, the *ij* can

be calculated directly from the definition equation (2.3) and do not need any assumption. Since the velocity in LES is the filtered velocity, a simple way to recover the full velocity is defiltering, i.e., an inversion of the filtering operation (2.1). Such a procedure is also called deconvolution. But the defiltered velocity does not contain any small scales information.

Smagorinsky model can properly account for the global energy transfer. It is simple and robust, which make it the most widely used SGS model. But the modeled SGS quantities correlate poorly with the actual SGS quantities obtained from DNS. Moreover it is an absolutely dissipative model and tends to overestimate the SGS dissipation. It only allows one way energy flux, i.e. from large scales to small ones, and it fails to predict the inverse energy transfer from the subgrid scales to the resolved scales (backscatter) which is found in most flows. Many *ad hoc* corrections and variation of eddy viscosity models are proposed to solve the difficulties mentioned above. Among them the dynamic model of Germano *et al.* (Germano *et al.*1991) and its variations are the most attractive ones. The dynamic model calculates the eddy viscosity dynamically and obtains good results in different turbulence.

To construct a reasonable and reliable SGS model, to properly predict the interactions between large scales and small scales is the key, which means we need to know more detailed information about the nonlinear interactions between large and small scales. Fortunately during the last several years there are many investigations in a variety of turbulent flows, including isotropic and channel flow, at low Reynolds numbers using direct numerical simulation databases and experimental measurements (Zhou 1993; Hartel *et al.* 1994; Domaradzki & Rogallo 1990). Their studies show that the large scales contain enough information. Many of the observed features of the exact SGS interactions can be inferred from the dynamics of the resolved scales alone. Thus it implies a possible way to improving SGS model, i.e. to estimate the small scales from large scales by using the observed properties of the nonlinear interactions. Based on that concept, Domaradzki *et al* (Domaradzki & Saiki 1997; Domaradzki et al 2002) develop the velocity estimation model in both spectral and physical space. Stolz and Adam also proposed similar model called

The velocity estimation model is based on two observations: first, the dynamics of small scales are strongly determined by the large, energy carrying eddies; second, the contribution of small scales to large scales are mostly contained within wavenumbers that are twice that of the cutoff wavenumber, *k*c. These two observations rely on the properties of nonlinear interactions in turbulent flows and have been elucidated by a large number of theoretical, numerical and experimental investigations (Zhou 1993; Domaradzki & Rogallo 1990; Domaradzki & Saiki 1997; Domaradzki et al 2002). Basically these studies showed that most of the subgrid scale transfer happens in the range of 0.5*k*c ~*k*c and is determined by scales in the range of *k*c ~ 2*k*c. This implies that only a limited range of wavenumbers needs to be considered. Especially in VEM the modes beyond 2*k*c are ignored. With a proper estimation of the velocity field the subgrid scale stress tensor could be determined directly from the

scale strain-rate tensor. If we know the full velocity field of the turbulence flow, the *ij*

be calculated directly from the definition equation (2.3) and do not need any assumption. Since the velocity in LES is the filtered velocity, a simple way to recover the full velocity is defiltering, i.e., an inversion of the filtering operation (2.1). Such a procedure is also called deconvolution. But the defiltered velocity does not contain any small scales information.

*<sup>t</sup>* by related to the large

can

But it still has some problems when applied to complex engineering flows.

**3.1 Velocity Estimation Model (VEM)** 

deconvolution model (Stolz 1999).

resolved scales and provides enough dissipation for LES.

The eddy viscosity models basically try to solve the imaginary

Domaradzki *et al* (Domaradzki & Saiki 1997; Domaradzki et al 2002) proposed a method to estimate the small scales. This is the basic idea of the velocity estimation model. In order to describe the different velocities, we use *u* to present the full, unfiltered turbulence velocity; *u* means the filtered velocity; and *u*' is the small scale velocity. And the relation among them is

$$
\mu = \overline{\mu} + \mu^\* \tag{3.4}
$$

VEM contains two steps: First is the defiltering operation, i.e. try to recover the full velocity *u* from the filtered velocity *u*

$$
\mu \approx \mathcal{G}^{-1} \overline{\mathcal{u}} \tag{3.5}
$$

which is inverse operation of (2.1). Bertero and Boccacci (Bertero & Boccacci 1998) give a detailed discussion about it. Since any filtering will loss part of original information, the defiltering can not recover the full velocity according to Riemann–Lebesgue theory. Only some very special filtering function and variables can return to its original state. Most results of defiltering can only be approximate. The velocity u in equation (3.5) actually only contains large scales information, so we denote it by <sup>0</sup> *u* . If we using tophat filter and the filtering size is twice as the grid size, then the filtering operation can be expressed as

$$
\overline{u} = au\_{i-1} + bu\_i + cu\_{i+1} \,\prime \tag{3.6}
$$

and *a*, *b*, *c* are constants. Then the defiltering operation is

$$a\widetilde{u}\_{i-1}^0 + b\widetilde{u}\_i^0 + c\widetilde{u}\_{i+1}^0 = \overline{u} \ . \tag{3.7}$$

By defiltering, the large scales <sup>0</sup> *u* is closed to the original *u*, but there is no small scales. If we approximate the u using <sup>0</sup> *u* , i.e. let <sup>0</sup> *u u* . And then calculate the SGS tensor *τ* directly from the definition 00 0 0 *uu u u* . Through practice, it is found that if the Reynolds number is not too high, the result is good enough. But if the Re number is high, the error is somehow too large. The information of the small scales is needed. The second step of VEM is to estimate the small scales. For full developed turbulence, the small scales are thought to be homogeneous, so a simplified way to estimate small scales can be described as

$$
\mu' = \theta \mathcal{N}'.\tag{3.8}
$$

where *N*' is the growth rate of subgrid scales due to the nonlinear interactions among resolved scales and *θ* is the time scale related to the eddy turn-over time. The detailed description of the full process can be found in paper (Domaradzki & Saiki 1997; Domaradzki et al 2002). Thus the final velocity can be expressed as

$$
\mu \approx \tilde{\mu} = \tilde{\mu}^0 + \mu^\prime. \tag{3.9}
$$

Correspondingly, the SGS stress tensor is

$$
\pi = \overline{\tilde{\iota}\iota\tilde{\iota}} - \overline{\tilde{\iota}} \cdot \overline{\tilde{\iota}}\ . \tag{3.10}
$$

The energy spectral of full (DNS), filtered, defiltered and estimated velocity are shown in Fig. 3.1

Study of Some Key Issues for

condition.

Applying LES to Real Engineering Problems 35

the high wavenumber modes *k*c<*k*<2*k*c provides a natural dissipation mechanism for the large scales, which also automatically includes the effect of reversing energy (backscatter). The energy accumulated at the subgrid scales *k*c<*k*<2*k*c is removed by truncation (filtering) at prescribed time intervals. In the physical space, the explanation for TNS is also

of solving the LES equations on the coarse mesh, full Navier-Stokes equations are solved on the fine mesh with a corresponding filtering operation in physical space. It should be noticed that the filtering time interval plays a critical role in TNS. In order to avoid underdissipating or over-dissipating, appropriate interval must be carefully chosen (Domaradzki & Yang 2004). The suitable interval depends on the filter type, grid resolution and flow

Compared to other LES models, TNS does not have the closure problem because it has no SGS term in the equation. It satisfies the Galilean transformation properties of the Navier-Stokes equations. It is easy to implement with fewer empirical parameters and can be easily extended to other types of turbulence without too much modification. When the explicit filtering is used, the TNS model also shows its advantages over the other models. For instance, as mentioned by Lund (Lund 1997), adaptation of the second explicit filtering leads

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

This is not Galilean invariant in most cases. In TNS, this problem is naturally avoided since

TNS was tested in several different turbulent flows. Here only the results of the simplest homogeneous, isotropic decaying turbulence are discussed. For this simple flow there are lot of DNS and experiment data which can be used to test LES model. Here the DNS data of Horiuti (Horiuti 1999) is used, which have a resolution of 2563. The initial condition for LES is obtained from DNS by truncating the full 2563 DNS field to 323 in spectral space, see Fig.

. (3.11)

*k* , denoted as a coarse

*k* . Instead

straightforward. In the traditional LES, the mesh size is *LES c*

Fig. 3.2. The sketch of TNS in spectral space.

the SGS term actually to be

no such term exists.

mesh; while the TNS operates on a fine mesh with the size of 2 2 *TNS c LES*

VEM was implemented in both spectral and physical space. It was applied to different flows, such as homogenous tubulence, incompressible channel flow, Rayleigh-Bénard convection flow, and obtained quite good results. But the disadvantage of VEM is that the procedure it uses is quite complicated and need much more CPU time than Smagorinsky model.

Fig. 3.1. Sketch of energy spectral for full (DNS), filtered, defiltered and estimated velocity.

#### **3.2 Truncated Navier–Stokes (TNS) equations approach**

As we can see, the traditional eddy viscosity models use the filtered velocity to calculate *τ*ij, while the VEM tries to recover the full velocity from the filtered velocity and then use it to calculate *τ*ij. So one may think: if we can get the full velocity from the experiment data or DNS directly, can we just skip the filtering and defiltering steps? Based on that concept, Domaradzki *et al* (Domaradzki et al 2002; Domaradzki & Yang 2004) developed a new TNS approach from VEM model. TNS uses the full velocity. It actually solves the N-S equations directly instead of LES equations. So it does not have the SGS term. Due to limitation of grid, it is an under-resolved DNS run. According to the energy transfer theory, the energy will accumulate at high modes. A mechanism is needed to provide necessary dissipation to remove the accumulated energy, such as filtering /truncation. A similar model in engine application is the MILES model, which depends on numerical scheme to provide implicit dissipation.

TNS model is still based on the same two observations of energy transfer as VEM. The large energy carrying eddies can determine the dynamics of the small scales; in return, the contribution of the small scales to the large scales are mostly contained within wavenumber range between the cutoff wavenumber, *k*c, and 2*k*c. Correspondingly, a scale decomposition is performed in TNS as shown in Fig. 3.2: a range of physical (large) scales up to the traditional LES wave number cutoff *k*c, and a range of modeled (SGS or estimated) scales between *k*c and 2*k*c. The nonlinear interaction between the low wavenumber modes *k*<*k*c and

VEM was implemented in both spectral and physical space. It was applied to different flows, such as homogenous tubulence, incompressible channel flow, Rayleigh-Bénard convection flow, and obtained quite good results. But the disadvantage of VEM is that the procedure it uses is quite complicated and need much more CPU time than Smagorinsky

> **DNS filtered velocity VEM**

**estimated small scales**

**k**

Fig. 3.1. Sketch of energy spectral for full (DNS), filtered, defiltered and estimated velocity.

As we can see, the traditional eddy viscosity models use the filtered velocity to calculate *τ*ij, while the VEM tries to recover the full velocity from the filtered velocity and then use it to calculate *τ*ij. So one may think: if we can get the full velocity from the experiment data or DNS directly, can we just skip the filtering and defiltering steps? Based on that concept, Domaradzki *et al* (Domaradzki et al 2002; Domaradzki & Yang 2004) developed a new TNS approach from VEM model. TNS uses the full velocity. It actually solves the N-S equations directly instead of LES equations. So it does not have the SGS term. Due to limitation of grid, it is an under-resolved DNS run. According to the energy transfer theory, the energy will accumulate at high modes. A mechanism is needed to provide necessary dissipation to remove the accumulated energy, such as filtering /truncation. A similar model in engine application is the MILES model, which depends on numerical scheme to provide implicit

TNS model is still based on the same two observations of energy transfer as VEM. The large energy carrying eddies can determine the dynamics of the small scales; in return, the contribution of the small scales to the large scales are mostly contained within wavenumber range between the cutoff wavenumber, *k*c, and 2*k*c. Correspondingly, a scale decomposition is performed in TNS as shown in Fig. 3.2: a range of physical (large) scales up to the traditional LES wave number cutoff *k*c, and a range of modeled (SGS or estimated) scales between *k*c and 2*k*c. The nonlinear interaction between the low wavenumber modes *k*<*k*c and

5 10 15 20 25 30 <sup>10</sup>-4

**kc**

model.

dissipation.

**E(k)**

10-3

**3.2 Truncated Navier–Stokes (TNS) equations approach** 

10-2

10-1

10<sup>0</sup>

the high wavenumber modes *k*c<*k*<2*k*c provides a natural dissipation mechanism for the large scales, which also automatically includes the effect of reversing energy (backscatter). The energy accumulated at the subgrid scales *k*c<*k*<2*k*c is removed by truncation (filtering) at prescribed time intervals. In the physical space, the explanation for TNS is also straightforward. In the traditional LES, the mesh size is *LES c k* , denoted as a coarse mesh; while the TNS operates on a fine mesh with the size of 2 2 *TNS c LES k* . Instead of solving the LES equations on the coarse mesh, full Navier-Stokes equations are solved on the fine mesh with a corresponding filtering operation in physical space. It should be noticed that the filtering time interval plays a critical role in TNS. In order to avoid underdissipating or over-dissipating, appropriate interval must be carefully chosen (Domaradzki & Yang 2004). The suitable interval depends on the filter type, grid resolution and flow condition.

Fig. 3.2. The sketch of TNS in spectral space.

Compared to other LES models, TNS does not have the closure problem because it has no SGS term in the equation. It satisfies the Galilean transformation properties of the Navier-Stokes equations. It is easy to implement with fewer empirical parameters and can be easily extended to other types of turbulence without too much modification. When the explicit filtering is used, the TNS model also shows its advantages over the other models. For instance, as mentioned by Lund (Lund 1997), adaptation of the second explicit filtering leads the SGS term actually to be

$$
\pi\_{i\bar{j}} = \overline{u\_i u\_j} - \overline{\overline{u}\_i \overline{u}\_j} \ . \tag{3.11}
$$

This is not Galilean invariant in most cases. In TNS, this problem is naturally avoided since no such term exists.

TNS was tested in several different turbulent flows. Here only the results of the simplest homogeneous, isotropic decaying turbulence are discussed. For this simple flow there are lot of DNS and experiment data which can be used to test LES model. Here the DNS data of Horiuti (Horiuti 1999) is used, which have a resolution of 2563. The initial condition for LES is obtained from DNS by truncating the full 2563 DNS field to 323 in spectral space, see Fig.

Study of Some Key Issues for

Smagrinsky model shows too much dissipation.

**E(t)/E(0)**

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

study is focus on the importance of the filter shape.

effect of the filter shape on LES is rarely discussed in the literature.

Fig. 3.4. The history of total energy decaying.

**4. The choice of filters** 

model to study the filter effect.

Applying LES to Real Engineering Problems 37

normalized energy decay for all models. Note here we simply divide the energy *E*(t) by the initial total energy *E*(0) to get normalized energy. Again TNS gets the best results while the

> **DNS VEM C-L Smag**

**time**

0 0.511.5 0.2

The filter shape and filtering width are the two free parameters in LES. Each affects the LES results greatly. Designing suitable filter type and filtering width is important to get reasonable results. In dynamic Smagrinsky model and similarity model, the effect of the filtering width has been studied by Lund (Lund 1997), De Stefano and Vasilyev (De Stefano and Vasilyev 2002) etc. In order to separate these two effects from one another, the present

Theoretically, the filtering operation should be repeated every time step because the nonlinear term continuously generates high frequency modes that need to be dissipated (Lund 1997). Depending on the type of filter, the SGS model should be adjusted in order to represent the dynamics of the unresolved scales correctly. Consequently the nature of the LES solution strongly depends on the filter shape. But for the traditional LES, especially for the eddy viscosity model, there is no explicit filtering process during the calculation in spite of the reasons mentioned above, i.e., the simulation result is independent of the filter. In the conventional practice, the filter has been only used as a concept (Fröhlich & Rodi 2001). The

On the other hand, a suitable LES model is needed to test different filters. As just mentioned, most traditional eddy viscosity models do not have explicit filtering in the solution procedure. In the similarity models and the dynamical Smagorinsky model the filtering width of the test (second) filter plays a key role besides the filter shape. A more appropriate LES model, which can directly validate different filters, is therefore required. From section 3.2 we found that the filtering plays a key role in TNS. The dynamics of the large scales and the energy budget strongly depend on the filter shape. It is a very good

There have been many filters proposed in the literatures that can be categorized into two groups: *smooth filters* and *discrete filters*. At the early stage of development, LES was mostly

5.1. Notice the energy at the cutoff *k*c=15 may not be small enough compared to the energy peak. Usually for LES models in order to get good results, the energy at cutoff should be two orders of magnitude less than the energy peak. The initial parameters are summarized in Table 3.1


Table 3.1. Initial parameters.

Fig. 3.3 shows the initial and final energy spectrum for TNS and DNS results. Note that in order to compare the results of other LES models are also presented, including Smagorinsky model and Chollet-Lesieur (C-L) eddy viscosity model (Chollet & Lesieur 1981). The C-L model in spectral space can be expressed as

$$
\mu\_t(k) = \nu^\*(k \;/\ k\_c) \left[ \mathcal{E}(k\_c) / k\_c \right]^{1/2}. \tag{3.12}
$$

*ν*\* is the normalized eddy viscosity, which is defined as

$$\boldsymbol{\dot{\nu}}^{\*}\left(\boldsymbol{k}\;/\;\boldsymbol{k}\_{c}\right) = \boldsymbol{\text{K}}\boldsymbol{\alpha}^{-3/2}\left[0.441 + 15.2 \exp(-3.03 \boldsymbol{k}\_{c}\;/\;\boldsymbol{k})\right].\tag{3.13}$$

where *K*o is the Kolmogoroff constant, and is usually set to 1.4.

Fig. 3.3. Initial and final energy spectrum for DNS, LES and TNS.

s

As we can see from Fig. 3.3, all models obtain reasonable results compared to DNS, especially at low modes they match each other quite well. However at high modes TNS spectrum matches the result of DNS best, and the *k*−5/3 spectral form is preserved. For Smagrinsky model, as indicated by many studies, the dissipation is overestimated and biggest. C-L shows good results but not as good as TNS. Fig. 3.4 shows the history of

5.1. Notice the energy at the cutoff *k*c=15 may not be small enough compared to the energy peak. Usually for LES models in order to get good results, the energy at cutoff should be two orders of magnitude less than the energy peak. The initial parameters are summarized

1/720 0.686 0.152 0.51 0.24 245 118 0.68

Fig. 3.3 shows the initial and final energy spectrum for TNS and DNS results. Note that in order to compare the results of other LES models are also presented, including Smagorinsky model and Chollet-Lesieur (C-L) eddy viscosity model (Chollet & Lesieur 1981). The C-L

*t cc* ( ) ( / ) ( )/ *<sup>c</sup>*

\* 3/2 ( / ) 0.441 15.2exp( 3.03 / ) *c c*

**k**

5 10 15 20 25 30 <sup>10</sup>-4

As we can see from Fig. 3.3, all models obtain reasonable results compared to DNS, especially at low modes they match each other quite well. However at high modes TNS spectrum matches the result of DNS best, and the *k*−5/3 spectral form is preserved. For Smagrinsky model, as indicated by many studies, the dissipation is overestimated and biggest. C-L shows good results but not as good as TNS. Fig. 3.4 shows the history of

**Initial Condition DNS VEM C-L Smag -5/3**

Fig. 3.3. Initial and final energy spectrum for DNS, LES and TNS.

\* 1/2

Re*<sup>L</sup>* Re

*k k k Ek k* . (3.12)

*k k Ko k k* . (3.13)

*ett t*

where *K*o is the Kolmogoroff constant, and is usually set to 1.4.

*ν*\* is the normalized eddy viscosity, which is defined as

 

*L*

in Table 3.1

Table 3.1. Initial parameters.

*E*<sup>0</sup>

model in spectral space can be expressed as

s

**E(k)**

10-3

10-2

10-1

normalized energy decay for all models. Note here we simply divide the energy *E*(t) by the initial total energy *E*(0) to get normalized energy. Again TNS gets the best results while the Smagrinsky model shows too much dissipation.

Fig. 3.4. The history of total energy decaying.
