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

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

Study of Some Key Issues for

**E(k)**

10-4

10-3

10-2

10-1

of aliasing error, which is calculated as

**k**

Fig. 5.2. The aliasing errors for DNS using spectral method.

with time evolution, as shown in Fig.5.1, but it is still very small.

nonlinear term is a key factor which affects the stability of computation.

smaller and the result is more close to spectral method.

(a) (b)

10 20 30 40 5060 <sup>10</sup>-5

**Dealiased Aliased**

Applying LES to Real Engineering Problems 51

aliasing errors destroy the energy conservation and cause the solution to departure from physical solution. Usually a random shift technology is used to eliminate aliasing error. In order to identify the importance of aliasing error, we run two DNS: one uses anti-aliasing technology (the result marked as Dealiased) and the other one does not (the result denoted by Aliased). The simulation results are shown in Fig. 5.1. In Fig. 5.1a the energy spectrum at t=1 (the upper line) and t=3 (the lower line) are shown. And Fig. 5.1b gives the relative value

> **Alias Error**

> > -0.1

As we can see, the aliasing error is relatively small for DNS with spectral method. The largest error happens at high modes, which is consistent with the analysis of Park and Mahesh (Park & Mahesh 2007). The aliasing error can contaminate the low modes gradually

For complex engineering problem, the spectral method is no longer suitable. Finite difference scheme or finite volume scheme is used instead. Same as section 4, the Padé compact scheme is used here. In order to examine the truncation error of different order of Padé scheme, one common used technology is to analysis the modified wavenumber in spectral space (Lele 1992). Similar to aliasing error, Padé scheme has little effect on low modes. The error is highest near the cut-off wavenumber. With higher order, the error is

In order to isolate the discretization error, here we only apply the Padé scheme to the nonlinear term, which is similar to the way used by Kravchenko and Moin (Kravchenko & Moin 1997). The reason is that nonlinear term has important impact on SGS term. Also

In Fig. 5.3 we compare the energy spectrum at different time for spectral method and different order Padé schemes. While the time evolution of total kinetic energy is shown in Fig. 5.4. As we can see, high order scheme, 6th order Padé scheme compares well with spectral method at all time. But for 4th order Padé scheme, at the initial stage (t=1), it

0

0.1

0.2

0.3

() () () () *k Ek Ek Ek dealias alias dealias* (5.1)

**t = 1. t = 3.**

**<sup>k</sup>**

10 20 30 40 5060 -0.2

resolve all the scales. Therefore LES has an additional source of error comes from the SGS models. In general, it is required that all the source of errors can not overwhelm the contribution of SGS model in LES (Ghosal 1996; Chow 2003). In addition, unlike DNS the cut-off mode in LES is still energetic. As a result, LES is more sensitive to numerical errors. The numerical errors must be well controlled.

Note the numerical error in this section refers to aliasing and truncation errors introduced by spatial numerical scheme. Usually time discretization also introduces some computational errors (He et al 2004). Guo-Wei He etc studied the time correlation of several SGS model on LES (He et al 2004). Here we only focus on the spatial discretization error. In addition, the effect of the floating error of the computer is assumed to be small and ignored thereafter.

#### **5.1 DNS results**

We conduct DNS run at first in order to avoid the effect of SGS model. Also DNS data can provide benchmark for LES. The initial condition is the homogenous isotropic turbulence same as section 3 and 4. The grid resolution is 128×128×128 which is found fine enough to resolve the large scales for this simple flow. At Fig.1 the DNS results of 1283 and 2563 grid are compared. It is clear shows that they have a very good match for low wavenumber modes. In order to save computation time, only 1283 runs are carried out thereafter.

Fig. 5.1. The result of 1283 DNS compared with 2563.

As mentioned above, DNS only suffers from aliasing and discretization errors that depend on the numerical scheme used. When spectral method is used, the discretization errors is very small. So the numerical error is dominated by aliasing error. The main reason for aliasing error arising is that the gird resolution is limited. The modes beyond the grid cut-off wavenumber are incorrectly 'aliased' to wavenumbers that are resolved. Usually the contribution of aliasing errors is largest at the highest wavenumbers where any energy above the wavenumber cutoff incorrectly adds on the resolved spectrum. Without control,

resolve all the scales. Therefore LES has an additional source of error comes from the SGS models. In general, it is required that all the source of errors can not overwhelm the contribution of SGS model in LES (Ghosal 1996; Chow 2003). In addition, unlike DNS the cut-off mode in LES is still energetic. As a result, LES is more sensitive to numerical errors.

Note the numerical error in this section refers to aliasing and truncation errors introduced by spatial numerical scheme. Usually time discretization also introduces some computational errors (He et al 2004). Guo-Wei He etc studied the time correlation of several SGS model on LES (He et al 2004). Here we only focus on the spatial discretization error. In addition, the effect of the floating error of the computer is assumed to be small and ignored

We conduct DNS run at first in order to avoid the effect of SGS model. Also DNS data can provide benchmark for LES. The initial condition is the homogenous isotropic turbulence same as section 3 and 4. The grid resolution is 128×128×128 which is found fine enough to resolve the large scales for this simple flow. At Fig.1 the DNS results of 1283 and 2563 grid are compared. It is clear shows that they have a very good match for low wavenumber

**k**

<sup>10</sup><sup>0</sup> <sup>10</sup><sup>1</sup> <sup>10</sup> <sup>10</sup> <sup>2</sup>

As mentioned above, DNS only suffers from aliasing and discretization errors that depend on the numerical scheme used. When spectral method is used, the discretization errors is very small. So the numerical error is dominated by aliasing error. The main reason for aliasing error arising is that the gird resolution is limited. The modes beyond the grid cut-off wavenumber are incorrectly 'aliased' to wavenumbers that are resolved. Usually the contribution of aliasing errors is largest at the highest wavenumbers where any energy above the wavenumber cutoff incorrectly adds on the resolved spectrum. Without control,

**128 DNS 256 DNS**

modes. In order to save computation time, only 1283 runs are carried out thereafter.

The numerical errors must be well controlled.

**E(k)**


Fig. 5.1. The result of 1283 DNS compared with 2563.

10-4

10-3

10-2

10-1

thereafter.

**5.1 DNS results** 

aliasing errors destroy the energy conservation and cause the solution to departure from physical solution. Usually a random shift technology is used to eliminate aliasing error. In order to identify the importance of aliasing error, we run two DNS: one uses anti-aliasing technology (the result marked as Dealiased) and the other one does not (the result denoted by Aliased). The simulation results are shown in Fig. 5.1. In Fig. 5.1a the energy spectrum at t=1 (the upper line) and t=3 (the lower line) are shown. And Fig. 5.1b gives the relative value of aliasing error, which is calculated as

$$
\sigma(k) = \left( E(k)\_{\text{dealias}} - E(k)\_{\text{alias}} \right) / E(k)\_{\text{dealias}} \tag{5.1}
$$

Fig. 5.2. The aliasing errors for DNS using spectral method.

As we can see, the aliasing error is relatively small for DNS with spectral method. The largest error happens at high modes, which is consistent with the analysis of Park and Mahesh (Park & Mahesh 2007). The aliasing error can contaminate the low modes gradually with time evolution, as shown in Fig.5.1, but it is still very small.

For complex engineering problem, the spectral method is no longer suitable. Finite difference scheme or finite volume scheme is used instead. Same as section 4, the Padé compact scheme is used here. In order to examine the truncation error of different order of Padé scheme, one common used technology is to analysis the modified wavenumber in spectral space (Lele 1992). Similar to aliasing error, Padé scheme has little effect on low modes. The error is highest near the cut-off wavenumber. With higher order, the error is smaller and the result is more close to spectral method.

In order to isolate the discretization error, here we only apply the Padé scheme to the nonlinear term, which is similar to the way used by Kravchenko and Moin (Kravchenko & Moin 1997). The reason is that nonlinear term has important impact on SGS term. Also nonlinear term is a key factor which affects the stability of computation.

In Fig. 5.3 we compare the energy spectrum at different time for spectral method and different order Padé schemes. While the time evolution of total kinetic energy is shown in Fig. 5.4. As we can see, high order scheme, 6th order Padé scheme compares well with spectral method at all time. But for 4th order Padé scheme, at the initial stage (t=1), it

Study of Some Key Issues for

reader should refer to their paper.

**E(k)**

importance of aliasing error.

cases.

**5.2 LES results** 

10-3

10-2

10-1

Applying LES to Real Engineering Problems 53

computation diverging. We observe the same problem here. Usually an artificial viscosity is needed to dump the oscillation, which is beyond the research scope of this article. In Fig.5.5 the energy spectrum before computation fail is shown. It is clear seen that the nonphysical energy accumulates quickly at high modes which lead to the computation divergence. Park and Mahesh (Park & Mahesh 2007) gave a detail description about this problem. Interested

**k**

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

The results for Padé schemes shown in Fig. 5.3 actually contain all the numerical errors, i.e. discretization and aliasing errors. Unlike spectral method, many researches thought that finite difference scheme can automatically decrease the aliasing error and therefore no antialiasing method is needed. Since the result of 6th order Padé scheme is the closest one to the spectral method, here we also apply the same anti-aliasing technology as used in spectral method to 6th order Padé scheme (denoted by Dealiased). The new result is then compared with the original one as shown in Fig. 5.6. Compared to spectral method, the magnitude of aliasing error does decrease. Note that actually it is very hard to isolate aliasing error from discretization error. The aliasing error shown above is not the actual aliasing error for 6th order Padé scheme. But anyway we can use that technology to determine the relative

In general, the aliasing error in DNS can be well controlled. The main error is the truncation error of discretization scheme. Low order scheme is not suitable for simulation in some

For the same simulations above, we also run the LES. The mesh size is 64×64×64. Spatial discretization adapts the same spectral and Padé compact schemes as in DNS. Note that LES has additional numerical error source which comes from the SGS model besides the aliasing error and discretization error for DNS. Again TNS approach is used to modelling the small scales. TNS model depends on periodic filtering to remove high modes energy. There are

Fig. 5.5. The time evolution of energy spectrum for DNS using 2nd order Padé scheme.

**t = 0.2 t = 0.5**

compares well with spectral method at low modes, but shows some difference at high modes. The nonphysical energy introduced by discretization error accumulates at high modes and affects low modes gradually with time developing (t=3)

Fig. 5.3. The energy spectrum for DNS using different discretization schemes at t=1 and t=3 (from above to low).

Fig. 5.4. The time evolution of total kinetic energy for DNS using different discretization schemes.

Note that 2nd order Padé scheme actually is the 2nd order center difference scheme. It is well known that it has the problem of even-odd oscillation which will leads to the

compares well with spectral method at low modes, but shows some difference at high modes. The nonphysical energy introduced by discretization error accumulates at high

**k**

**Spectral Pade6 Pade4**

**t**

Fig. 5.4. The time evolution of total kinetic energy for DNS using different discretization

Note that 2nd order Padé scheme actually is the 2nd order center difference scheme. It is well known that it has the problem of even-odd oscillation which will leads to the

0123

10 20 30 40 5060 <sup>10</sup>-5

Fig. 5.3. The energy spectrum for DNS using different discretization schemes at t=1 and t=3

**Spectral Pade6 Pade4**

modes and affects low modes gradually with time developing (t=3)

**E(k)**

**K(t)**

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(from above to low).

schemes.

10-4

10-3

10-2

10-1

computation diverging. We observe the same problem here. Usually an artificial viscosity is needed to dump the oscillation, which is beyond the research scope of this article. In Fig.5.5 the energy spectrum before computation fail is shown. It is clear seen that the nonphysical energy accumulates quickly at high modes which lead to the computation divergence. Park and Mahesh (Park & Mahesh 2007) gave a detail description about this problem. Interested reader should refer to their paper.

Fig. 5.5. The time evolution of energy spectrum for DNS using 2nd order Padé scheme.

The results for Padé schemes shown in Fig. 5.3 actually contain all the numerical errors, i.e. discretization and aliasing errors. Unlike spectral method, many researches thought that finite difference scheme can automatically decrease the aliasing error and therefore no antialiasing method is needed. Since the result of 6th order Padé scheme is the closest one to the spectral method, here we also apply the same anti-aliasing technology as used in spectral method to 6th order Padé scheme (denoted by Dealiased). The new result is then compared with the original one as shown in Fig. 5.6. Compared to spectral method, the magnitude of aliasing error does decrease. Note that actually it is very hard to isolate aliasing error from discretization error. The aliasing error shown above is not the actual aliasing error for 6th order Padé scheme. But anyway we can use that technology to determine the relative importance of aliasing error.

In general, the aliasing error in DNS can be well controlled. The main error is the truncation error of discretization scheme. Low order scheme is not suitable for simulation in some cases.

### **5.2 LES results**

For the same simulations above, we also run the LES. The mesh size is 64×64×64. Spatial discretization adapts the same spectral and Padé compact schemes as in DNS. Note that LES has additional numerical error source which comes from the SGS model besides the aliasing error and discretization error for DNS. Again TNS approach is used to modelling the small scales. TNS model depends on periodic filtering to remove high modes energy. There are

Study of Some Key Issues for

**E(k)**

**E(k)**

model.

10-3

schemes and TNS models.

10-2

10-3

10-2

**k**

**k**

**E(k)**

10-3

10-2

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

**DNS d4f10 d4f8 d4f6 d4f4 d4f2**

(a) spectral method (b) 2nd order Padé scheme

(c) 4th order Padé scheme (d) 6th order Padé scheme

Fig. 5.7. The final energy spectrum for LES using spectral method, different order Padé

**DNS Spectral Pade6 Pade4 Pade2**

**k**

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

Fig. 5.8. The final energy spectrum for different discretization scheme with the same TNS

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

**DNS f10 f8 f6 f4 f2**

Applying LES to Real Engineering Problems 55

**E(k)**

**E(k)**

10-3

10-2

10-3

10-2

**k**

**k**

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

**DNS d6f10 d6f8 d6f6 d6f4 d6f2**

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

**DNS d2f10 d2f8 d2f6 d2f4 d2f2**

Fig. 5.6. The aliasing error for DNS using 6th order Padé scheme.

many filters available as mentioned in section 4. Because we pay more attention to the possibility of applying LES to real engineering problem, only discrete filters are adapted here. Since Padé discrete filters show good results in section 4, we only focus on them to simply the research. Note that different order of Padé filter means the error introduced by SGS model is different.

In Fig. 5.7a, b, c, d it shows the final energy spectrum for spectral method and different Padé compact schemes (marked with 'd') combined with TNS model using different order Padé discrete filters (denoted as 'f'), i.e. 'd6f4' means 6th order Padé compact scheme and 4th order filter. As we can see, TNS model plays a key role in the simulations. Low order filter provides too much dissipation and performs poorly for all discretization schemes. High order filter (4th order and above) obtains good results compared with DNS. However higher the filter order is, it keeps more high modes energy and thus dissipates less. In return it decreases the performance of LES. TNS model with 4th order filter provides the best results. Compared to DNS, LES is not so sensitive to the order of the discretization scheme. Even

2nd order scheme can obtain reasonable results. The result is not as good as of high order scheme though, but much better than of DNS which diverges when using 2nd order scheme. The reason is that the discretization and aliasing errors cause the energy to accumulate at high modes; while TNS model removes high modes energy through filtering. The dissipation provided by SGS model dumps the unphysical energy increment. This result may help explain why low order scheme can obtain good results when applied to engineering problem.

In order to further comparing the effect of different discretization scheme on LES, here we run a series cases with the same TNS model (all using 6th order Padé filter) while the spatial discretization uses spectral method and different order of Padé schemes. The results are shown in Fig. 5.8 and Fig. 5.9. Similar to DNS, low order Padé discretization scheme leads energy to accumulate at high modes. Higher the order is, the result is more close to DNS. The spectral method is the best one. But different form DNS, low order (2nd order) scheme obtains reasonable result and does not cause divergence like DNS case in Fig. 5.5. It is because the TNS model removes the unphysical energy accumulated at high modes and thus dumps the discretization error.

**Alias Error**


many filters available as mentioned in section 4. Because we pay more attention to the possibility of applying LES to real engineering problem, only discrete filters are adapted here. Since Padé discrete filters show good results in section 4, we only focus on them to simply the research. Note that different order of Padé filter means the error introduced by

In Fig. 5.7a, b, c, d it shows the final energy spectrum for spectral method and different Padé compact schemes (marked with 'd') combined with TNS model using different order Padé discrete filters (denoted as 'f'), i.e. 'd6f4' means 6th order Padé compact scheme and 4th order filter. As we can see, TNS model plays a key role in the simulations. Low order filter provides too much dissipation and performs poorly for all discretization schemes. High order filter (4th order and above) obtains good results compared with DNS. However higher the filter order is, it keeps more high modes energy and thus dissipates less. In return it decreases the performance of LES. TNS model with 4th order filter provides the best results. Compared to DNS, LES is not so sensitive to the order of the discretization scheme. Even 2nd order scheme can obtain reasonable results. The result is not as good as of high order scheme though, but much better than of DNS which diverges when using 2nd order scheme. The reason is that the discretization and aliasing errors cause the energy to accumulate at high modes; while TNS model removes high modes energy through filtering. The dissipation provided by SGS model dumps the unphysical energy increment. This result may help explain why low order scheme can obtain good results when applied to

In order to further comparing the effect of different discretization scheme on LES, here we run a series cases with the same TNS model (all using 6th order Padé filter) while the spatial discretization uses spectral method and different order of Padé schemes. The results are shown in Fig. 5.8 and Fig. 5.9. Similar to DNS, low order Padé discretization scheme leads energy to accumulate at high modes. Higher the order is, the result is more close to DNS. The spectral method is the best one. But different form DNS, low order (2nd order) scheme obtains reasonable result and does not cause divergence like DNS case in Fig. 5.5. It is because the TNS model removes the unphysical energy accumulated at high modes and

0

0.1

0.2

0.3

**k**

10 20 30 40 5060 -0.2

**t = 1. t = 3.**

**k**

(a) (b)

Fig. 5.6. The aliasing error for DNS using 6th order Padé scheme.

10 20 30 40 5060 <sup>10</sup>-5

**Dealiased Original**

SGS model is different.

engineering problem.

thus dumps the discretization error.

**E(k)**

10-4

10-3

10-2

10-1

Fig. 5.7. The final energy spectrum for LES using spectral method, different order Padé schemes and TNS models.

Fig. 5.8. The final energy spectrum for different discretization scheme with the same TNS model.

Study of Some Key Issues for

it does not have any empirical parameter.

other SGS models, the effect of filters may be different.

investigated briefly.

Applying LES to Real Engineering Problems 57

LES, namely the SGS models, the choice of filters and the effects of numerical errors, are

The TNS approach is a simple and promising model. It actually is a DNS run with periodic processing of high modes. The basic assumption in TNS is that dynamics of small scales are strongly determined by the large eddies and the contribution of small scales to large scales are mostly contained within wavenumbers that are twice that of the cut-off wavenumber. By periodic removal of high mode energy using a filter, TNS provides necessary dissipation for small scales and thus avoids energy accumulation at high wavenumbers. It provides natural dissipation mechanism for low modes by nonlinear interactions between low modes and high modes at the cost of doubling the mesh. It can be expended to other flow easily because

The filtering operation plays a key role in TNS. It is responsible for removing the accumulated energy of the small scales at prescribed intervals. The filter type has direct effect on the final simulation results. Here a set of smooth filters and discrete filters are tested using TNS model. For the smooth filters, the physical filter is implemented in the physical space while it has similar property to the spectral filter at the same time. Also it keeps a part of the small-scale energy, which benefits the energy transfer. The Tophat and Gaussian filters are easy to implement but have serious undesirable effects on the large-scale energy. The discrete Padé filters exhibit advantages over the V-filters because the effect of adjacent points can be taken into account. The Padé filters can keep most of large-scale energy while maintaining sufficient amount of the small-scale energy at the same time. They show very good performance in all the runs, even better than the smooth physical filter. However the second order Padé filter still has effect on the large scales and the high order ones tend to keep too much small-scale energy, both of them may decrease the performance of TNS. The V-filters can be easily implemented to any specified order. However, high asymmetric filters should be avoided because they introduce too much nonphysical energy at high modes. The second order symmetric V-filter is actually a discrete version of the Tophat filter. It has the same problem as the smooth one, i.e., removing too much large-scale energy. When the smooth physical filter idea is applied to the V-filters, the results for the low order V-filters are improved significantly, but it is not necessary for high order V-filters. Note that the filter type should be adjusted according to the SGS model. So for LES with

The effects of numerical errors on LES are investigated briefly here, including discretization error, alias errors as well as SGS model error. As for DNS, no matter using spectral method or Padé compact scheme, the aliasing error can be well controlled. The final result is more affected by the truncation error of discretization scheme. High order scheme is preferred. Low order scheme is not suitable in some case. For example in our simulation, 2nd order discretization scheme leads energy to accumulate at high modes quickly and causes the solution unstable. As for LES, the interaction between numerical error and SGS model is complicated. Different model, such as TNS with different order filtering, can have different results. Low order filter brings in too much dissipation which is not a suitable property for LES; while the dissipation provided by high order filter is too small which also decreases the performance of TNS model. High order discretization schemes (4th order Padé scheme and above) plus middle order filters (4th or 6th order) can obtain good results compared with DNS. A more advantage is that TNS model not only avoids the small scales energy accumulating, but also dumps the side effect of discretization and alias errors to the high modes because TNS model removes the high modes energy periodically by filtering. This

Fig. 5.9. The time evolution of total energy for different discretization scheme with the same TNS model.

Same to Fig. 5.6, we also try to investigate the effect of anti-aliasing method on the final results (only the case using 6th order Padé scheme, 6th TNS model is considered). The comparison is shown in Fig. 5.10. The aliasing error is still small. But for LES the energy at cut-off wavenumber is considerable large compared to DNS, the relative value of aliasing error increases.

Fig. 5.10. The aliasing error for LES using 6th Padé scheme and TNS model with 6th order filter.

#### **6. Conclusion**

In the past ten years significant progress has been made in the LES technology. LES has become one of the most promising and successful methodology available to solve the complex turbulent flows. However there are still some challenges before LES can be a mature tool to predict engineering problems, including reliable subgrid scale model, the choice of filter, near-wall treatment, numerical errors etc. And an accurate, fast and robust numerical algorithm for complex geometry is also needed. In this article, three key issues of

**DNS Spectral Pade6 Pade4 Pade2**

**K(t)**

0

**k**

(a) (b)

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

**Dealias Alias**

TNS model.

error increases.

**E(k)**

filter.

**6. Conclusion** 

10-3

10-2

10-1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

**t**

Fig. 5.9. The time evolution of total energy for different discretization scheme with the same

Same to Fig. 5.6, we also try to investigate the effect of anti-aliasing method on the final results (only the case using 6th order Padé scheme, 6th TNS model is considered). The comparison is shown in Fig. 5.10. The aliasing error is still small. But for LES the energy at cut-off wavenumber is considerable large compared to DNS, the relative value of aliasing

> **Alias Error**

Fig. 5.10. The aliasing error for LES using 6th Padé scheme and TNS model with 6th order

In the past ten years significant progress has been made in the LES technology. LES has become one of the most promising and successful methodology available to solve the complex turbulent flows. However there are still some challenges before LES can be a mature tool to predict engineering problems, including reliable subgrid scale model, the choice of filter, near-wall treatment, numerical errors etc. And an accurate, fast and robust numerical algorithm for complex geometry is also needed. In this article, three key issues of

0

0.2

0.4

0.6

0.8

**<sup>k</sup>**

5 10 15 20 25 30 -0.2

**t = 1. t = 3.**

0 0.5 1 1.5 2 2.5 3

LES, namely the SGS models, the choice of filters and the effects of numerical errors, are investigated briefly.

The TNS approach is a simple and promising model. It actually is a DNS run with periodic processing of high modes. The basic assumption in TNS is that dynamics of small scales are strongly determined by the large eddies and the contribution of small scales to large scales are mostly contained within wavenumbers that are twice that of the cut-off wavenumber. By periodic removal of high mode energy using a filter, TNS provides necessary dissipation for small scales and thus avoids energy accumulation at high wavenumbers. It provides natural dissipation mechanism for low modes by nonlinear interactions between low modes and high modes at the cost of doubling the mesh. It can be expended to other flow easily because it does not have any empirical parameter.

The filtering operation plays a key role in TNS. It is responsible for removing the accumulated energy of the small scales at prescribed intervals. The filter type has direct effect on the final simulation results. Here a set of smooth filters and discrete filters are tested using TNS model. For the smooth filters, the physical filter is implemented in the physical space while it has similar property to the spectral filter at the same time. Also it keeps a part of the small-scale energy, which benefits the energy transfer. The Tophat and Gaussian filters are easy to implement but have serious undesirable effects on the large-scale energy. The discrete Padé filters exhibit advantages over the V-filters because the effect of adjacent points can be taken into account. The Padé filters can keep most of large-scale energy while maintaining sufficient amount of the small-scale energy at the same time. They show very good performance in all the runs, even better than the smooth physical filter. However the second order Padé filter still has effect on the large scales and the high order ones tend to keep too much small-scale energy, both of them may decrease the performance of TNS. The V-filters can be easily implemented to any specified order. However, high asymmetric filters should be avoided because they introduce too much nonphysical energy at high modes. The second order symmetric V-filter is actually a discrete version of the Tophat filter. It has the same problem as the smooth one, i.e., removing too much large-scale energy. When the smooth physical filter idea is applied to the V-filters, the results for the low order V-filters are improved significantly, but it is not necessary for high order V-filters. Note that the filter type should be adjusted according to the SGS model. So for LES with other SGS models, the effect of filters may be different.

The effects of numerical errors on LES are investigated briefly here, including discretization error, alias errors as well as SGS model error. As for DNS, no matter using spectral method or Padé compact scheme, the aliasing error can be well controlled. The final result is more affected by the truncation error of discretization scheme. High order scheme is preferred. Low order scheme is not suitable in some case. For example in our simulation, 2nd order discretization scheme leads energy to accumulate at high modes quickly and causes the solution unstable. As for LES, the interaction between numerical error and SGS model is complicated. Different model, such as TNS with different order filtering, can have different results. Low order filter brings in too much dissipation which is not a suitable property for LES; while the dissipation provided by high order filter is too small which also decreases the performance of TNS model. High order discretization schemes (4th order Padé scheme and above) plus middle order filters (4th or 6th order) can obtain good results compared with DNS. A more advantage is that TNS model not only avoids the small scales energy accumulating, but also dumps the side effect of discretization and alias errors to the high modes because TNS model removes the high modes energy periodically by filtering. This

Study of Some Key Issues for

Applying LES to Real Engineering Problems 59

Hartel, C. Kleiser, L. Unger F. Friedrich, R. (1994). Subgrid-scale energy transfer in the near-

He, G. Wang, M. Lele, S. (2004). On the computation of space-time correlations in decaying isotropic turbulence. *Physics of Fluids*, 2004, Vol.14, No.11, (2004), pp.3859-3867 Horiuti, K. (1999). Transformation properties of subgrid-scale models in a frame of reference

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interaction between TNS model and numerical error benefits LES. So low order discretization scheme plus a more dissipative TNS model can get good enough result. In addition, the energy at cut-off wavenumber for LES is still relative large, so the effect of aliasing error increases. We have to admit that due to the time and our knowledge limitation, the study on the numerical error is not full enough. There are some excellent works by other researches available.
