**4. The choice of filters**

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 study is focus on the importance of the filter shape.

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

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 model to study the filter effect.

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

Study of Some Key Issues for

or close to it at the same time.

filter as the physical filter

Applying LES to Real Engineering Problems 39

because for most flows transformation to the spectral space is difficult. People are trying to find a filter that is defined in the physical space while has the property of the spectral filter

Actually the filtering operation (2.1)(Leonard 1974) is a linear spatial averaging operation,

<sup>1</sup> <sup>1</sup> <sup>2</sup> *L I IL I IL IL G G GG*

where *I* is the unity operator. The product of *L***G** and the first few terms of the above expansion (4.2) actually defines a suitable new filter (Domaradzki et al 2002) (the product of *L***G** and the full equation (4.2) is equal to *I*, of course). If only first few terms are selected, the new filter is close to the original filter *L***G** and the extra computation cost is small. When more terms are selected, the new filter is closer to unity. It has less effect on the large scales but needs much more computation time. Domaradzki *et. al*. (Domaradzki et al 2002) found that the combination of the first three terms in Eq. 4.2 is the best choice and denoted this

where '^' is the original Tophat filter or Gaussian filter. One thing need to be pointed out is that the results of the Tophat, Gaussian and Physical filters strongly depend on the filter width . However the objective of present work is to highlight the importance of the filter shape as mentioned above. In the following analysis, the filter width is fixed for these

**k**

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

A formal inverse of it in a power series expansion can be expressed as

smooth filters, which is equal to 2 times of grid size .

**E(k)**

10-4

Fig. 4.1. Effect of the smooth filters on the *k*-5/3 spectrum.

10-3

10-2

10-1

10<sup>0</sup>

( ) ( ) ( , '; ) ( ') ' *<sup>G</sup> <sup>D</sup> <sup>f</sup> x L <sup>f</sup> x Gxx <sup>f</sup> x dx* (4.1)

(4.2)

<sup>ˆ</sup> ˆ ˆ 3 3 ˆ ˆˆ *u u uu i ii* (4.3)

**k-5/3 Spectral TopHat Gaussian Physical**

performed in spectral space in which the filters were defined as continuous in the whole domain. These filters are referred to as smooth filters. The most commonly used ones are the Tophat (box), the Gaussian, and the sharp spectral (Fourier cutoff) filters. Recently, the application of LES to solve real engineering problems in the complex flow has become realistic and, in fact, popular because of the urgent need from the industry. Finite difference scheme instead of pseudo-spectral method is now widely adopted in the numerical approach due to its flexibility in handling complex geometry and obtaining high order schemes (Lele 1992, Visbal & Gaitonde 2002). The finite difference discretization scheme, together with the limited grid resolution, can be seen as an implicit filtering as mentioned by many researchers (De Stefano and Vasilyev2002; Fröhlich & Rodi 2001; Lund 2003; Vasilyev 1998). However this kind of implicit filtering has some problems because of the interactions among the modified terms in the governing equations, the numerical error, and the order of the filter, etc.(Lund 2003; Vasilyev 1998). In order to avoid some of these problems, researchers tend to employ the explicit filtering to exert direct influence on the simulation result. These filters are usually defined on several adjacent points and, hence, denoted as discrete filters thereafter. There are several advantages using explicit filter: First it is easier to control the truncation and aliasing errors by removing the high wave number modes which is beyond the bandwidth allowed by the mesh. Second it can dump the oscillation at high frequency which comes from the numerical discretization scheme, boundary condition, etc. The amplitude of these oscillations usually is comparable to or even larger than that of the small scales after sufficiently long computation time, which tends to contaminate the final result of the simulation. By using the same explicit filter, it also makes the comparison with experiment or DNS data more direct.

#### **4.1 Smooth filters**

The smooth filters include the spectral filter, the Gaussian filter, the Tophat filter and those are defined continuously in the whole computation domain. The definitions for the first three can be easily found in some books (Pope 2000). Table 4.1 shows these filter functions in physical and spectral space respectively.


Table 4.1. Smooth filters in physical and spectral space / *<sup>c</sup> k* .

The main problem for the Tophat and Gaussian filters is that they remove too much energy of the large scales (Domaradzki et al 2002; Yang & Domaradzki 2004). The spectral filter is thought as the best among these three for LES because it keeps all the large scales while removes all the small scales. However filters defined in the physical space are much flexible

performed in spectral space in which the filters were defined as continuous in the whole domain. These filters are referred to as smooth filters. The most commonly used ones are the Tophat (box), the Gaussian, and the sharp spectral (Fourier cutoff) filters. Recently, the application of LES to solve real engineering problems in the complex flow has become realistic and, in fact, popular because of the urgent need from the industry. Finite difference scheme instead of pseudo-spectral method is now widely adopted in the numerical approach due to its flexibility in handling complex geometry and obtaining high order schemes (Lele 1992, Visbal & Gaitonde 2002). The finite difference discretization scheme, together with the limited grid resolution, can be seen as an implicit filtering as mentioned by many researchers (De Stefano and Vasilyev2002; Fröhlich & Rodi 2001; Lund 2003; Vasilyev 1998). However this kind of implicit filtering has some problems because of the interactions among the modified terms in the governing equations, the numerical error, and the order of the filter, etc.(Lund 2003; Vasilyev 1998). In order to avoid some of these problems, researchers tend to employ the explicit filtering to exert direct influence on the simulation result. These filters are usually defined on several adjacent points and, hence, denoted as discrete filters thereafter. There are several advantages using explicit filter: First it is easier to control the truncation and aliasing errors by removing the high wave number modes which is beyond the bandwidth allowed by the mesh. Second it can dump the oscillation at high frequency which comes from the numerical discretization scheme, boundary condition, etc. The amplitude of these oscillations usually is comparable to or even larger than that of the small scales after sufficiently long computation time, which tends to contaminate the final result of the simulation. By using the same explicit filter, it also makes the comparison

The smooth filters include the spectral filter, the Gaussian filter, the Tophat filter and those are defined continuously in the whole computation domain. The definitions for the first three can be easily found in some books (Pope 2000). Table 4.1 shows these filter functions in

> *c c*

*k k*

*k k* 

*k*

2 2

The main problem for the Tophat and Gaussian filters is that they remove too much energy of the large scales (Domaradzki et al 2002; Yang & Domaradzki 2004). The spectral filter is thought as the best among these three for LES because it keeps all the large scales while removes all the small scales. However filters defined in the physical space are much flexible

*k* 

ˆ( ) exp <sup>24</sup> *<sup>k</sup> G k*

Spectral space Physical space

*k* . sin ( ) *x*

<sup>1</sup> , 0.5 ; ( )

0, *<sup>x</sup> G x*

6 6 ( ) exp *<sup>x</sup> G x* 

 

*x* 

*otherwise*

2 2

2

*G x*

with experiment or DNS data more direct.

physical and spectral space respectively.

Spectral filter 1, ; ˆ( ) 0,

Tophat filter sin 0.5 ˆ( ) 0.5

*G k*

*G k*

Table 4.1. Smooth filters in physical and spectral space / *<sup>c</sup>*

**4.1 Smooth filters** 

Gaussian filter

because for most flows transformation to the spectral space is difficult. People are trying to find a filter that is defined in the physical space while has the property of the spectral filter or close to it at the same time.

Actually the filtering operation (2.1)(Leonard 1974) is a linear spatial averaging operation,

$$
\overline{f}\left(\mathbf{x}\right) = L\_G\left(f\left(\mathbf{x}\right)\right) = \int\_D G\left(\mathbf{x}, \mathbf{x}'; \overline{\Delta}\right) f\left(\mathbf{x}'\right) d\mathbf{x}' \tag{4.1}
$$

A formal inverse of it in a power series expansion can be expressed as

$$L\_{\rm G}^{-1} = \left(I - \left(I - L\_{\rm G}\right)\right)^{-1} = I + \left(I - L\_{\rm G}\right) + \left(I - L\_{\rm G}\right)^2 + \cdots \tag{4.2}$$

where *I* is the unity operator. The product of *L***G** and the first few terms of the above expansion (4.2) actually defines a suitable new filter (Domaradzki et al 2002) (the product of *L***G** and the full equation (4.2) is equal to *I*, of course). If only first few terms are selected, the new filter is close to the original filter *L***G** and the extra computation cost is small. When more terms are selected, the new filter is closer to unity. It has less effect on the large scales but needs much more computation time. Domaradzki *et. al*. (Domaradzki et al 2002) found that the combination of the first three terms in Eq. 4.2 is the best choice and denoted this filter as the physical filter

$$
\overline{\mu} = \Im \hat{u}\_i - \Im \hat{\hat{u}}\_i + \hat{\hat{\hat{u}}}\_i \tag{4.3}
$$

where '^' is the original Tophat filter or Gaussian filter. One thing need to be pointed out is that the results of the Tophat, Gaussian and Physical filters strongly depend on the filter width . However the objective of present work is to highlight the importance of the filter shape as mentioned above. In the following analysis, the filter width is fixed for these smooth filters, which is equal to 2 times of grid size .

Fig. 4.1. Effect of the smooth filters on the *k*-5/3 spectrum.

Study of Some Key Issues for

**E(k)**

downwards periodically.

**K(t)**

0.2

Fig. 4.4. The decay of the total kinetic energy for the smooth filters.

0.3

0.4

0.5

0.6

0.7

10-3

Fig. 4.3. Energy spectrum at final time for LES case 2.

10-2

10-1

Applying LES to Real Engineering Problems 41

**k**

**DNS Spectral Tophat Physical**

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

The history of total kinetic energy decay for the smooth filters is plotted in Fig. 4.4. Corresponding to Fig. 4.3, the physical filter obtains better result compared to DNS. Since the spectral filter does not provide sufficient dissipation as shown in Fig. 4.3, its total energy is the biggest among all the results. The Tophat filter removes a part of large-scale energy each time when the filtering operation is applied. That is why the total energy jumps

**t**

0 0.511.5 0.1

**Initial Condition**

**DNS Spectral Tophat Physical**

The effects of above filters on the *k***-5/3** spectrum are shown in Fig.4.1. As can be seen, the Tophat and Gaussian filters remove too much energy of the low modes. The spectral filter only keeps the large scales. The physical filter strongly damps the small scales while affecting the large scales very little, which make it a good filter for LES.

Beside *a prior* test, the effects of these filters on a real three dimensional LES are also examined. Again the simplest homogeneous, isotropic decaying turbulence is utilized as the test case with two different initial conditions. The first has the initial condition of spectrum 4 22 *E k Ak k k* ( ) exp 2 / *<sup>p</sup>* , where *k***p** is the peak mode and equals to 4. The grid resolution is 643 (In the following, the mesh size is 643 for all LES unless further specified). For comparison, the 2563 DNS result is also included. The final energy spectrum is plotted in Fig. 4.2. It also shows that the Tophat filter removes too much energy (Since the Gaussian filter performs very similarly to the Tophat filter, we did not include it in the figure). The spectral and physical filters show very good agreement with the DNS data.

Fig. 4.2. Energy spectrum at final time for LES case 1.

The second case has a more critical initial condition as shown in Fig. 4.3. The initial condition is obtained from the 2563 DNS data of Horiuti (Horiuti 1999) same as section 3. It is a challenging case for LES because the energy at cut-off mode *k*c is not in the inertial range. The Tophat filter shows too much dissipation same as above. However the spectral filter delivers some undesirable behaviors this time. By removing all the small scales, it also shut down the energy transfer from the large to small scales completely. It will take some time for LES to rebuild the nonlinear interactions between the large and small scales, which leads to insufficient dissipation. Thus, the energy accumulates near the cutoff wavenumber as shown in Fig. 4.3. The physical filter provides the best result compared to the DNS data as also observed in paper ((Domaradzki et al 2002; Yang & Domaradzki 2004). The main reason is that the physical filter keeps a small part of the small scales which facilitates the energy transfer. Since this initial condition is a better case to test filters, we only run LES with case two in the following discussion.

The effects of above filters on the *k***-5/3** spectrum are shown in Fig.4.1. As can be seen, the Tophat and Gaussian filters remove too much energy of the low modes. The spectral filter only keeps the large scales. The physical filter strongly damps the small scales while

Beside *a prior* test, the effects of these filters on a real three dimensional LES are also examined. Again the simplest homogeneous, isotropic decaying turbulence is utilized as the test case with two different initial conditions. The first has the initial condition of spectrum 4 22 *E k Ak k k* ( ) exp 2 / *<sup>p</sup>* , where *k***p** is the peak mode and equals to 4. The grid resolution is 643 (In the following, the mesh size is 643 for all LES unless further specified). For comparison, the 2563 DNS result is also included. The final energy spectrum is plotted in Fig. 4.2. It also shows that the Tophat filter removes too much energy (Since the Gaussian filter performs very similarly to the Tophat filter, we did not include it in the figure). The

**k**

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

The second case has a more critical initial condition as shown in Fig. 4.3. The initial condition is obtained from the 2563 DNS data of Horiuti (Horiuti 1999) same as section 3. It is a challenging case for LES because the energy at cut-off mode *k*c is not in the inertial range. The Tophat filter shows too much dissipation same as above. However the spectral filter delivers some undesirable behaviors this time. By removing all the small scales, it also shut down the energy transfer from the large to small scales completely. It will take some time for LES to rebuild the nonlinear interactions between the large and small scales, which leads to insufficient dissipation. Thus, the energy accumulates near the cutoff wavenumber as shown in Fig. 4.3. The physical filter provides the best result compared to the DNS data as also observed in paper ((Domaradzki et al 2002; Yang & Domaradzki 2004). The main reason is that the physical filter keeps a small part of the small scales which facilitates the energy transfer. Since this initial condition is a better case to test filters, we only run LES

**Initial Condition**

**DNS Spectral Tophat Physical**

affecting the large scales very little, which make it a good filter for LES.

spectral and physical filters show very good agreement with the DNS data.

**E(k)**

10-3

with case two in the following discussion.

Fig. 4.2. Energy spectrum at final time for LES case 1.

10-2

10-1

Fig. 4.3. Energy spectrum at final time for LES case 2.

The history of total kinetic energy decay for the smooth filters is plotted in Fig. 4.4. Corresponding to Fig. 4.3, the physical filter obtains better result compared to DNS. Since the spectral filter does not provide sufficient dissipation as shown in Fig. 4.3, its total energy is the biggest among all the results. The Tophat filter removes a part of large-scale energy each time when the filtering operation is applied. That is why the total energy jumps downwards periodically.

Fig. 4.4. The decay of the total kinetic energy for the smooth filters.

Study of Some Key Issues for

utilized rather than the spectral smooth filters

1 1/4 1/2 1/4

Table 4.2. The weight parameters of the V-filters.

3 1/8 5/8 3/8 -1/8 6 -1/16 1/4 5/8 1/4 -1/16

9 -1/32 5/32 11/16 5/16 -5/32 1/32 10 1/64 -3/32 15/64 11/16 15/64 -3/32 1/64

order are 3 and 5 respectively) are tested as well as the symmetric ones.

**4.2 Discrete filters** 

(Mathew *et al.* 2003).

Applying LES to Real Engineering Problems 43

In order to handle complex geometry, finite difference scheme is widely used instead of the spectral method. The solution is available only on a set of discrete grid points. At most time, the filter for the whole domain does not exist due to the inhomogeneous and boundary condition. The discrete filters, including the discrete Tophat filter, the Padé filter (Visbal & Gaitonde 2002) and the filter series proposed by Vasilyev *et al.* (Vasilyev *et al.* 1998) are

The main problem of the discrete filter is the commutation error between differentiation and filtering operation. Fortunately Vasilyev *et al.* (Vasilyev *et al.* 1998) gave out a solution which can control the commutation error to any specified order. Another problem is that if the order of the filter is too low, the error introduced by filtering may become larger than the magnitude of SGS term. Hence for traditional LES, the filter order is usually required to be higher than that of SGS term. But the filtering operation in TNS only acts as a dissipation source. The numerical error can be included into it as part of the dissipation. Low order filters can also obtain good results. It is similar to the strategy used by Mathew *et al.*

A one-dimensional filter given by Vasilyev *et al.* (Vasilyev *et al.* 1998) is defined as:

*L j l j l l K f w f* 

In order to control the commutation error to a specified order, the filter is required to have a different number of vanishing moments. Correspondingly, the weight factors *w***i** should satisfy a set of constrains. These filters are referred as V-filters in the following analysis.

Case *w*<sup>3</sup> *w*<sup>2</sup> *w*<sup>1</sup> *w*<sup>0</sup> *w*<sup>1</sup> *w*<sup>2</sup> *w*<sup>3</sup> *w*<sup>4</sup> *w*<sup>5</sup>

7 31/32 5/32 -5/16 5/16 -5/32 1/32

Several sets of weights for the V-filters are given in Table 4.2 which is similar to the Table 1 in Vasilyev *et al*'s paper (Vasilyev *et al.* 1998). The equation (4.4) defines a symmetric (center) scheme if K equals to L. The case 1, 6 and 10 are symmetric and have a commutation error of order 2, 4, 6 respectively. In order to handle boundary points, Vasilyev *et al* also proposed several asymmetric filters, i.e., K and L is different. For high asymmetric V-filters, such as the one side filter - case 7, it is found that too much unphysical energy is introduced to the high modes as also mentioned by Vasilyev *et al* (Vasilyev *et al.* 1998). This property is not desirable for TNS because it will lead to unphysical solution. Thus high asymmetric filters (case 2, 4 5, 7, 8) are not included in the following analysis. Only case 3 and 9 (whose

Fig. 4.7 presents the filtering results of different V-filters applied to the *k***-5/3** spectrum. For comparison, the result of the smooth physical filter is also included. Case 1 in fact is a discrete version of the Tophat filter using trapezoidal rule. Similar to the smooth one, it

(4.4)

The effect of different grid resolutions is also investigated. The result of LES with grid 323 is shown in Fig4.5 and the result for 1283 is plotted in Fig. 4.6. The behaviors of these smooth filters in coarse mesh (323) are almost the same as those in grid 643. Physical filter still gets the best results. For the fine mesh (1283), all filters obtain good result except that the Tophat filter still dissipates a little more. The effect of the SGS model becomes small when the grid resolution increases, which is well known.

Fig. 4.5. Energy spectrum at final time for the smooth filters with grid 323.

Fig. 4.6. Energy spectrum at final time for the smooth filters with grid 1283.

### **4.2 Discrete filters**

42 Computational Simulations and Applications

The effect of different grid resolutions is also investigated. The result of LES with grid 323 is shown in Fig4.5 and the result for 1283 is plotted in Fig. 4.6. The behaviors of these smooth filters in coarse mesh (323) are almost the same as those in grid 643. Physical filter still gets the best results. For the fine mesh (1283), all filters obtain good result except that the Tophat filter still dissipates a little more. The effect of the SGS model becomes small when the grid

**k**

**k**

10 20 30 40 5060 <sup>10</sup>-4

**Initial Spectrum**

**DNS Spectral Tophat Physical**

Fig. 4.6. Energy spectrum at final time for the smooth filters with grid 1283.

5 10 15 10-4

**Initial Spectrum**

**DNS Spectral Tophat Physical**

Fig. 4.5. Energy spectrum at final time for the smooth filters with grid 323.

resolution increases, which is well known.

10-1

**E(k)**

10-3

**E(k)**

10-3

10-2

10-1

10-2

In order to handle complex geometry, finite difference scheme is widely used instead of the spectral method. The solution is available only on a set of discrete grid points. At most time, the filter for the whole domain does not exist due to the inhomogeneous and boundary condition. The discrete filters, including the discrete Tophat filter, the Padé filter (Visbal & Gaitonde 2002) and the filter series proposed by Vasilyev *et al.* (Vasilyev *et al.* 1998) are utilized rather than the spectral smooth filters

The main problem of the discrete filter is the commutation error between differentiation and filtering operation. Fortunately Vasilyev *et al.* (Vasilyev *et al.* 1998) gave out a solution which can control the commutation error to any specified order. Another problem is that if the order of the filter is too low, the error introduced by filtering may become larger than the magnitude of SGS term. Hence for traditional LES, the filter order is usually required to be higher than that of SGS term. But the filtering operation in TNS only acts as a dissipation source. The numerical error can be included into it as part of the dissipation. Low order filters can also obtain good results. It is similar to the strategy used by Mathew *et al.* (Mathew *et al.* 2003).

A one-dimensional filter given by Vasilyev *et al.* (Vasilyev *et al.* 1998) is defined as:

$$\overline{f}\_{j} = \sum\_{l=-K}^{L} w\_{l} f\_{j+l} \tag{4.4}$$

In order to control the commutation error to a specified order, the filter is required to have a different number of vanishing moments. Correspondingly, the weight factors *w***i** should satisfy a set of constrains. These filters are referred as V-filters in the following analysis.


Table 4.2. The weight parameters of the V-filters.

Several sets of weights for the V-filters are given in Table 4.2 which is similar to the Table 1 in Vasilyev *et al*'s paper (Vasilyev *et al.* 1998). The equation (4.4) defines a symmetric (center) scheme if K equals to L. The case 1, 6 and 10 are symmetric and have a commutation error of order 2, 4, 6 respectively. In order to handle boundary points, Vasilyev *et al* also proposed several asymmetric filters, i.e., K and L is different. For high asymmetric V-filters, such as the one side filter - case 7, it is found that too much unphysical energy is introduced to the high modes as also mentioned by Vasilyev *et al* (Vasilyev *et al.* 1998). This property is not desirable for TNS because it will lead to unphysical solution. Thus high asymmetric filters (case 2, 4 5, 7, 8) are not included in the following analysis. Only case 3 and 9 (whose order are 3 and 5 respectively) are tested as well as the symmetric ones.

Fig. 4.7 presents the filtering results of different V-filters applied to the *k***-5/3** spectrum. For comparison, the result of the smooth physical filter is also included. Case 1 in fact is a discrete version of the Tophat filter using trapezoidal rule. Similar to the smooth one, it

Study of Some Key Issues for

**K(t)**

0.2

Fig. 4.9. The decay of the total kinetic energy for the V-filters.

For a variable *f*, the filtered value can be expressed as:

coefficients *na* are listed in Table 4.3.

where  0.3

0.4

0.5

0.6

0.7

Applying LES to Real Engineering Problems 45

Fig.4.8 shows the final energy spectrum of the V-filters in the same homogeneous run as Fig.4.3. As expected, case 1 dissipates too much energy. High order filters obtains better results. Corresponding to Fig.4.7, the result of the low order asymmetric filter case 3 (3rd order) is a little better than that of the high order symmetric filter case 6 (4th order). And the behaviors of case9 and case 10 are very similar. The reason is still attributed to the fact that a small amount of energy is introduced at high modes for the asymmetric filters. The decay of the total energy is plotted in Fig.4.9. Except case 1, all other cases show good agreement with the filtered DNS data. But there are small jumps for low order filters (case 3 and case 6) because of the undesirable effect to the large scales as shown in Fig.4.7. The effects of grid

resolutions on V-filters are similar to the smooth ones, which are not included here.

**t**

1 1 <sup>0</sup> 2

*a*

(4.5)

*<sup>f</sup>* means a less

*<sup>N</sup> <sup>n</sup> <sup>f</sup> i i <sup>f</sup> i i <sup>n</sup> <sup>i</sup> <sup>n</sup> n*

 *ff f ff* 

dissipative filter. N is the order of filter scheme, 2N+1 points give a 2N order filter. The

The filtering results of the *k*-5/3 spectrum using different Padé filters are shown in Fig. 4.10. The smooth physical filter is also included as a benchmark. The 2nd order filter removes a small amount of the energy of the low modes, which may have undesirable effect on LES because of the unnecessary dissipation. The 4th order and above have little effect on the

 

*<sup>f</sup>* is an adjustable parameters between (-0.5, 0.5) and high value of

**DNS case1 case3 case6 case9 case10**

0 0.511.5 0.1

Another series of discrete filters is the Padé filters. The Padé compact difference scheme can be regarded as an implicit filter (Visbal & Gaitonde 2002; Vasilyev et al 1998). Based on that, a set of Padé explicit filters is proposed by Visbal and Gaitonde (Visbal & Gaitonde 2002).

removes too much energy of the large scales. It is interesting that the asymmetric filters like case 3 and 9 keep more energy than the symmetric ones (case 6 and 10). Note case 3 and case 6 still remove a small part of the large-scale energy. While case 7 introduces too much energy at high wavenumber modes which will lead to an unphysical solution in a real LES run.

Fig. 4.7. Effect of the V-filters on the *k*-5/3 spectrum.

Fig. 4.8. Energy spectrum at final time for the V-filters.

removes too much energy of the large scales. It is interesting that the asymmetric filters like case 3 and 9 keep more energy than the symmetric ones (case 6 and 10). Note case 3 and case 6 still remove a small part of the large-scale energy. While case 7 introduces too much energy at high wavenumber modes which will lead to an unphysical solution in a real LES

**k**

**k**

5 10 15 20 25 30 10-4

**Initial Spectrum**

**DNS case 1 case 3 case 6 case 9 case 10**

5 10 15 20 25 30

**k-5/3 Physical case 1 case 3 case 6 case 7 case 9 case 10**

run.

**E(k)**

10-4

**E(k)**

10-3

Fig. 4.8. Energy spectrum at final time for the V-filters.

10-2

10-1

Fig. 4.7. Effect of the V-filters on the *k*-5/3 spectrum.

10-3

10-2

10-1

10<sup>0</sup>

Fig.4.8 shows the final energy spectrum of the V-filters in the same homogeneous run as Fig.4.3. As expected, case 1 dissipates too much energy. High order filters obtains better results. Corresponding to Fig.4.7, the result of the low order asymmetric filter case 3 (3rd order) is a little better than that of the high order symmetric filter case 6 (4th order). And the behaviors of case9 and case 10 are very similar. The reason is still attributed to the fact that a small amount of energy is introduced at high modes for the asymmetric filters. The decay of the total energy is plotted in Fig.4.9. Except case 1, all other cases show good agreement with the filtered DNS data. But there are small jumps for low order filters (case 3 and case 6) because of the undesirable effect to the large scales as shown in Fig.4.7. The effects of grid resolutions on V-filters are similar to the smooth ones, which are not included here.

Fig. 4.9. The decay of the total kinetic energy for the V-filters.

Another series of discrete filters is the Padé filters. The Padé compact difference scheme can be regarded as an implicit filter (Visbal & Gaitonde 2002; Vasilyev et al 1998). Based on that, a set of Padé explicit filters is proposed by Visbal and Gaitonde (Visbal & Gaitonde 2002). For a variable *f*, the filtered value can be expressed as:

$$a\sigma\_f \overline{f}\_{i-1} + \overline{f}\_i + a\sigma\_f \overline{f}\_{i+1} = \sum\_{n=0}^{N} \frac{a\_n}{2} (f\_{i+n} + f\_{i-n}) \tag{4.5}$$

where *<sup>f</sup>* is an adjustable parameters between (-0.5, 0.5) and high value of *<sup>f</sup>* means a less dissipative filter. N is the order of filter scheme, 2N+1 points give a 2N order filter. The coefficients *na* are listed in Table 4.3.

The filtering results of the *k*-5/3 spectrum using different Padé filters are shown in Fig. 4.10. The smooth physical filter is also included as a benchmark. The 2nd order filter removes a small amount of the energy of the low modes, which may have undesirable effect on LES because of the unnecessary dissipation. The 4th order and above have little effect on the

Study of Some Key Issues for

**E(k)**

**K(t)**

0.2

Fig. 4.12. The decay of the total kinetic energy for the Padé filters.

0.3

0.4

0.5

0.6

0.7

10-3

Fig. 4.11. Energy spectrum at final time for the Padé filters.

10-2

10-1

PV-filter.

Applying LES to Real Engineering Problems 47

infeasible for the inhomogeneous case. In the Section 4.1 the physical filter shows very good property but it is a smooth filter. So we modified it into a discrete version using the V-filters, i.e. in equation 4.3 we use the V-filters instead of the Tophat filter. Hereafter we denote it as

**k**

**DNS Pade 2nd Pade 4th Pade 6th Pade 8th Pade 10th**

**t**

0 0.511.5 0.1

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

**Initial Spectrum**

**DNS Pade2 Pade4 Pade6 pade8 Pade10**

Fig. 4.10. Effect of the V-filters on the *k*-5/3 spectrum.


Table 4.3. The Coefficients of the Padé filters.

large scales. But higher the order is, the Padé filter tends to keep more small scales compared to the physical filter. In turn it may not provide enough dissipation for TNS.

The final energy spectra of the LES run with the Padé filters are plotted in Fig. 4.11 and the time evolutions of the total energy are shown in Fig. 4.12. Corresponding to Fig.4.10 the 2nd order filter overestimates the dissipation and subsequently provides the worst results among these runs. The 4th order and above show very good results as compared to the DNS data. However the 6th order and above filters keep more small-scale energy than DNS which may imply them do not provide enough dissipation.

From above results, it is found that the Padé filters show better results than the V-filters. It could be attributed to the fact that the Padé filters consider the effects of adjacent points. On the other hand, the calculation of the V-filters is much simple and straightforward. For the Padé filters we need to solve a tri-diagonal system. It is time consuming and may be

**k-5/3 Physical Pade 2nd Pade 4th Pade 6th Pade 8th Pade 10th**

**E(k)**

10-3

Fig. 4.10. Effect of the V-filters on the *k*-5/3 spectrum.

1

1 <sup>2</sup> *<sup>f</sup> <sup>a</sup>*

15 17 32 16 *<sup>f</sup> <sup>a</sup>*

16

256

which may imply them do not provide enough dissipation.

*<sup>f</sup> a* 7 18

*<sup>f</sup> a* 105 302

Table 4.3. The Coefficients of the Padé filters.

F2 <sup>1</sup>

F4 5 3

F6 11 5

F8

F10

<sup>2</sup> *<sup>f</sup> <sup>a</sup>*

8 4 *<sup>f</sup> <sup>a</sup>*

16 8 *<sup>f</sup> <sup>a</sup>*

93 70 128

193 126 256

10-2

10-1

10<sup>0</sup>

**k**

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

Scheme 0*a* <sup>1</sup>*a* <sup>2</sup> *a* <sup>3</sup> *a* <sup>4</sup> *a* <sup>5</sup> *a*

1 1

3 3 16 8 *<sup>f</sup> <sup>a</sup>*

32

64

large scales. But higher the order is, the Padé filter tends to keep more small scales compared to the physical filter. In turn it may not provide enough dissipation for TNS. The final energy spectra of the LES run with the Padé filters are plotted in Fig. 4.11 and the time evolutions of the total energy are shown in Fig. 4.12. Corresponding to Fig.4.10 the 2nd order filter overestimates the dissipation and subsequently provides the worst results among these runs. The 4th order and above show very good results as compared to the DNS data. However the 6th order and above filters keep more small-scale energy than DNS

From above results, it is found that the Padé filters show better results than the V-filters. It could be attributed to the fact that the Padé filters consider the effects of adjacent points. On the other hand, the calculation of the V-filters is much simple and straightforward. For the Padé filters we need to solve a tri-diagonal system. It is time consuming and may be

8 4 *<sup>f</sup> <sup>a</sup>*

*<sup>f</sup> a* 1

*<sup>f</sup> a* 45(1 2 )

1 32 16

16 8

512

*<sup>f</sup> <sup>a</sup>*

*<sup>f</sup> <sup>a</sup>* <sup>1</sup>

*<sup>f</sup> a* 5( 1 2 )

128 64 *<sup>f</sup> a* 

256

*<sup>f</sup> a* 1 2

512 *<sup>f</sup> a*

<sup>2</sup> *<sup>f</sup> <sup>a</sup>*

*<sup>f</sup> a* 7 14

*<sup>f</sup> a* 15( 1 2 )

infeasible for the inhomogeneous case. In the Section 4.1 the physical filter shows very good property but it is a smooth filter. So we modified it into a discrete version using the V-filters, i.e. in equation 4.3 we use the V-filters instead of the Tophat filter. Hereafter we denote it as PV-filter.

Fig. 4.11. Energy spectrum at final time for the Padé filters.

Fig. 4.12. The decay of the total kinetic energy for the Padé filters.

Study of Some Key Issues for

**E(k)**

**K(t)**

0.2

0.3

0.4

0.5

0.6

0.7

10-3

Fig. 4.14. Energy spectrum at final time for the PV-filters.

10-2

10-1

Applying LES to Real Engineering Problems 49

**k**

**t**

<sup>0</sup> 0.5 <sup>1</sup> 1.5 0.1

For DNS, the numerical errors mainly are aliasing and truncation errors (Chow 2003). As for LES, the small scales must be modelled because of the limited grid resolution which can not

**DNS PV-case1 PV-case3 PV-case6 PV-case9 PV-case10**

Fig. 4.15. The decay of the total kinetic energy for the PV-filters.

**5. The effects of numerical errors on TNS** 

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

**Initial Spectrum**

**DNS PV-case1 PV-case6 PV-case9 PV-case10**

The results of a priori test and the LES run of the PV-filters are shown in Fig. 4.13, Fig. 4.14 and Fig. 4.15 respectively. For comparison, we also include the result of the original case 1 of the V-filter (V-case1) in Fig.4.13. It shows that the result of the 2nd order PVcase1 is improved significantly as compared to V-case1. As mentioned before V-case1 is actually a discrete version of the Tophat filter. Therefore the PV-case1 is a discrete version of the smooth physical filter. Since the result of smooth physical filter is much better than that of the Tophat filter as shown in Fig.4.1, it is no wonder that the PV-case1 is better than the V-case1. However, by comparing Fig.4.14 and Fig.4.3, it was found the results of this discrete version (PV-case1) are not as good as those of the smooth physical filter for a real LES run. Similar to Fig.4.7, the asymmetric filters (PV-case3 and PV-case9) show good behavior in Fig.4.13, keeping more energy than the symmetric ones (PV-case6 and PVcase10). But it was also found that small amount of nonphysical energy is introduced near the cutoff wavenumber. Correspondingly the behaviors of these asymmetric filters in a real LES run are not good as shown in Fig.4.14 and Fig.4.15 (PV-case3 is not shown in Fig.4.14 because the energy spectrum becomes so large that it is out of the scope range). The results of high order symmetric PV-filters are also improved compared to the original V-filters, but not too much.

Fig. 4.13. Effect of the PV-filters on the *k*-5/3 spectrum.

The results of a priori test and the LES run of the PV-filters are shown in Fig. 4.13, Fig. 4.14 and Fig. 4.15 respectively. For comparison, we also include the result of the original case 1 of the V-filter (V-case1) in Fig.4.13. It shows that the result of the 2nd order PVcase1 is improved significantly as compared to V-case1. As mentioned before V-case1 is actually a discrete version of the Tophat filter. Therefore the PV-case1 is a discrete version of the smooth physical filter. Since the result of smooth physical filter is much better than that of the Tophat filter as shown in Fig.4.1, it is no wonder that the PV-case1 is better than the V-case1. However, by comparing Fig.4.14 and Fig.4.3, it was found the results of this discrete version (PV-case1) are not as good as those of the smooth physical filter for a real LES run. Similar to Fig.4.7, the asymmetric filters (PV-case3 and PV-case9) show good behavior in Fig.4.13, keeping more energy than the symmetric ones (PV-case6 and PVcase10). But it was also found that small amount of nonphysical energy is introduced near the cutoff wavenumber. Correspondingly the behaviors of these asymmetric filters in a real LES run are not good as shown in Fig.4.14 and Fig.4.15 (PV-case3 is not shown in Fig.4.14 because the energy spectrum becomes so large that it is out of the scope range). The results of high order symmetric PV-filters are also improved compared to the original

**k**

5 10 15 20 25 30

**k-5/3 Physical PV-case1 PV-case3 PV-case6 PV-case9 PV-case10 original case1**

V-filters, but not too much.

**E(k)**

10-4

Fig. 4.13. Effect of the PV-filters on the *k*-5/3 spectrum.

10-3

10-2

10-1

Fig. 4.14. Energy spectrum at final time for the PV-filters.

Fig. 4.15. The decay of the total kinetic energy for the PV-filters.
