Use of explicit filtering, second-order scheme and SGS models in LES of turbulent flow

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TU Delft, The Netherlands, 2006

USE OF EXPLICIT FILTERING, SECOND-ORDER

SCHEME AND SGS MODELS IN LES OF TURBULENT

CHANNEL FLOW

Tellervo T. Brandt∗

∗_{Helsinki University of Technology, Laboratory of Aerodynamics}

P.O. BOX 4400, FIN-02015 TKK, Finland e-mail: tellervo.brandt@tkk.fi web page: http://www.aero.hut.fi/

Key words: LES, explicit filtering, SGS modeling, SFS modeling, channel flow

Abstract. In large eddy simulations (LES) using low-order finite-difference-type methods, the numerical error can be large in comparison to the effect of the sub-grid scale (SGS) model. In this paper, we study two approaches to explicit filtering which can reduce the numerical error involved. In the first approach, the non-linear convection term of the Navier–Stokes equations is filtered explicitly, and in the second one, the small-scale shear stress is devided into sub-filter (SFS) and sub-grid (SGS) components, and filtering is provided via the model. The aims are to clarify the effect of SGS modelling in the first approach and to compare the two approaches. For SGS and SFS modelling, the standard and dynamic Smagorinsky models and the scale-similarity model are applied.

When the convection term is filtered explicitly, the effect of modelling on the simulation results reduces and differences between the models are rather small. Filtering with a smooth three-dimensional filter has a strong effect on the flow statistics, and none of the models is able to compensate for this. Although the high frequencies are damped efficiently, the total simulation error does not decrease. When filtering is provided only via modelling, improved results are obtained if both SGS and SFS modelling are applied. However in this approach, all terms in the equations do not have the same frequency content, and the high frequencies that are badly described by the discrete gird are not damped as efficiently as when the convection term is explicitly filtered.

1 INTRODUCTION

and SGS scales, the numerical error becomes a problem: it has been shown in a priori tests that the error can be larger than the effect of the SGS model.1, 2 _{However, explicit}

filtering of the resolved flow field can improve the situation.1, 2

Explicit filtering of the whole velocity field in the end of each time step has been the traditional approach, but if a non-sharp filter is applied, it leads to multiple filtering of the velocity field from the previous time levels,3 _{and in a priori tests, it has led to unphysical}

behaviour of the SGS term.4 _{Explicit filtering of the non-linear convection term of the}

Navier–Stokes equations has been suggested as an alternative approach.3 _{The approach}

has been successfully applied in actual simulations using a fourth-order discretization method,5 _{and it has given promising results also for a second-order scheme in a priori}

tests.4 _{However, it has been noticed that in simulations with a second-order scheme and}

the standard Smagorinsky model, which are still widely applied methods in simulations involving complex geometries, it is difficult to reduce the total simulation error by explicit filtering.6 _{In this previous study, filtering with a smooth filter reduced the effect of the}

SGS model and most of the changes in the flow statistics were due to the explicit filtering operation. To clarify the effect of SGS modelling on this behaviour, these simulations are repeated here using also the dynamic version of the Smagorinsky model and the scale-similarity model. These models are selected because they are still rather simple to implement and the extra computation time is not too large. Since explicit filtering itself increases the computation time,7 _{it is not realistic to assume that it could be applied in}

complex applications together with a complex SGS model.

Filtering is actually present already in the derivation of the LES equations, and dis-cretization is performed after this. However in practice, these two are often mixed up together. Approaches to explicit filtering where filtering and discretization are treated as two different operations have been suggested.8 _{In these approaches, the sub-filter scale}

(SFS) stresses that can be described by the discrete grid are separated from the SGS stresses which are beyond grid resolution. In the application of the approach, explicit filtering is present only in the SFS and SGS models.9 _{It has been noticed that with the}

second-order method, rather sophisticated SFS modelling is required to obtain improved results.9 _{In this paper, one aim is to compare the two approaches to explicit filtering, and}

thus the same models are applied as in the cases with explicit filtering of the convection term.

2 APPLIED MODELS AND NUMERICAL METHODS

In the present study, a fully developed turbulent channel flow between two infinite parallel walls was applied as the test case. The Reynolds number was Reτ = 395 based

on the channel half-height and the friction velocity uτ = q

τwall/ρ, where τwall is the wall

shear stress and ρ the density.

The filtered Navier–Stokes equations which are being solved in LES are written here in the non-dimensional form as

∂ ˜ui ∂t = − ∂P ∂xi − ∂ ˜p ∂xi + ∂ ∂xj  _{− ˜uiu˜j− τij+} 1 Reτ ∂ ˜ui ∂xj +∂ ˜uj ∂xi ! _{, (1)}

where (x1, x2, x3) = (x, y, z) refer to non-dimensional streamwise, wall-normal and

span-wise spatial coordinates, respectively, t to time, (˜u1, ˜u2, ˜u3) = (u, v, w) to resolved velocity

vector, P to mean pressure, ˜p to fluctuating resolved pressure, and τij is a model for the

SGS stress tensor ugiuj − ˜uiu˜j. Here, the equations are scaled by the channel half-height,

0.5h, and friction velocity, uτ.

In this paper, the standard10 _{and dynamic versions}11 _{of the Smagorinsky model, and}

a scale-similarity model12 _{were applied to model the SGS shear stress. In Smagorinsky}

models, τij is written as −τij = µT2Sij = (CS∆S)2 q 2SijSij | {z } =µT ∂ ˜ui ∂xj +∂ ˜uj ∂xi ! , (2)

where CS is the model parameter and ∆S the model length scale. In this study, ∆S was

set proportional to grid spacing as

∆S = (∆x ∆y ∆z)1/3. (3)

or, in cases where explicit filtering was applied, to the explicit filter width. When the standard version of the model was applied, the value 0.085 was used for CS, and the value

was reduced near the solid walls using the van Driest damping. In the dynamic version of the model, CS was evaluated using the procedure proposed by Lilly.13 In the dynamic

model, a test filter is required in the evaluation of CS. When no explicit filtering was

applied, a three-dimensional fourth-order commutative14 _{test filter with the width of two}

grid spacings was used, and when explicit filtering was applied, a test filter with the width of four grid spacings, i.e. twice the explicit filter width, was applied.

In the scale-similarity model (SSM), the SGS shear stress is modelled as

τij = ˜uiu˜j − ˜uiu˜j, (4)

a mixed model,15 _{where SSM models the SFS stress and a Smagorinsky model the SGS}

stress, was applied.

In the present simulations, the second-order central-difference scheme was applied on a staggered grid system16 _{for spatial discretization, and for time integration, a third-order,}

three-stage Runge–Kutta method17 _{was applied. The non-dimensional mean-pressure}

gradient was fixed, and the fluctuating pressure was solved from a Poisson equation. The applied grid resolution and the dimensions of the computational domain are given in Table 1.

Table 1: Applied grid resolution and dimensions of the computational domain.

x z y

extent of the domain (scaled by h) 6.0 3.2 1.0 number of grid points 108 108 90

size of grid cells in wall units (∆+_{) 44 23 min 1.0, max 20}

wall units: x+= Reτx, where x is scaled by the channel half-height.

The same three-dimensional and fourth-order commutative filter that was applied in the SGS models was used for explicit filtering. With this filter function, the error due to changing the order of filtering and differentiation is of fourth order and the shape of the filter transfer function is closer to the spectral cutoff filter than that of the Trapezoidal or Simpson filters which are the traditional smooth filters.

3 APPROACHES TO EXPLICIT FILTERING

In this paper, two approaches to filtering in LES are applied. In the first approach, filtering is applied explicitly to the non-linear convection term of the Navier–Stokes equa-tions and to the SGS model.3 _{This approach guarantees that all the terms in the equations}

have the same frequency content. The filtered Navier–Stokes equations are then written as ∂ ˜ui ∂t = − ∂P ∂xi − ∂ ˜p ∂xi + ∂ ∂xj  _{− ˜uiu˜j− τij+} 1 Reτ ∂ ˜ui ∂xj +∂ ˜uj ∂xi ! _{, (5)}

where the overline refers to the explicit filtering operation, and the SGS stress which is modelled by τij is redefined as ˜uiu˜j −ugiuj. Equations (5) are written from the point of

view of the implementation of filtering. Because of the filtering, the resolved flowfield ˜

ui in these equations does not have same frequency content as the field obtained from

Equations (1).

are treated separately.8 _{Here, the filtered Navier–Stokes equations are written as} ∂ ˜ui ∂t = − ∂P ∂xi − ∂p˜ ∂xi + ∂ ∂xj  _−u˜g_iu˜j−Te_ij+ 1 Reτ ∂u˜i ∂xj + ∂u˜j ∂xi ! _{, (6)}

where tilde refers to reduction to the discrete grid and the notation ˜ui refers to the

resolved velocity field that contains all the scales larger than the chosen filter width. The chosen notation differs from the one applied in Equations (5) where overline meant that filtering is performed explicitly. In contrast here, filtering is explicitly present only in the models.8, 9 _{In Equation (6), the SGS stress, T}

ij, is written as −Tij = uiuj − ˜uiu˜j =  uiuj− ˜uiu˜j  +u˜iu˜j− ˜uiu˜j  = = −τij+  ˜ uiu˜j − ˜uiu˜j  , (7)

where the first term is the actual SGS stress and the second one the sub-filter scale (SFS) stress. If spectral methods were applied, it would be possible to recover the SFS stresses from the resolved velocity field, but with finite-difference-type methods, this is not possible due to the numerical error involved with these schemes.9

4 EXPLICIT FILTERING OF THE CONVECTION TERM

In this section, we discuss results obtained with the different models with and without explicit filtering of the non-linear convection term. The mean-velocity profiles are given in Figure 1. In the upper part of the figure, no explicit filtering is applied, and in the lower part, the non-linear convection term and the SGS model are filtered explicitly using the filter width of two grid spacings. When no explicit filtering is applied, there are clear differences between the mean bulk velocities obtained using different models, and the dynamic Smagorinsky model (DSM) produces the value closest to the reference data which are from the direct numerical simulation (DNS) of Moser et. al .18 _{With both}

Smagorinsky models, the viscous sublayer is slightly too thick. When compared to the case with no model, SSM has almost no effect on the mean-velocity profile.

In the lower part of Figure 1, we notice that explicit filtering reduces the differences between the SGS models and the curves are closer to the case with no modelling, meaning that the effect of modelling is decreased. As noticed in the previous study with the standard Smagorinsky model, filtering changes the slope of the profile which becomes too low.6 _{Here, we notice that despite of the use of somewhat better models, the slope still}

thickness of the viscous sublayer. With the two other models, the thickness of the viscous sublayer slightly improves when compared to the case with no filtering.

The resolved anisotropic streamwise Reynolds stress obtained using no explicit filtering is plotted in the upper part of Figure 2. The one obtained with SSM is underpredicted, whereas the ones produced by the Smagorinsky models are overpredicted. Here, the effect of modelling is the largest with SSM. The corresponding results for cases where explicit filtering was applied to the non-linear convection term and to the SGS stresses, are shown in the lower part of Figure 2. Here, we first notice that filtering increases the overprediction of the Reynolds stress with all models. It was previously noticed that this is due to the filtering itself,6 _{and here the trend continues also with DSM and SSM.}

With DSM and SSM, filtering slightly decreases the effect of modelling, whereas with the standard Smagorinsky model, the effect of modelling is increased because it is directly controlled via the model length scale.

The SGS shear stresses, τ12, obtained with and without explicit filtering are plotted

in the upper and lower parts of Figure 3, respectively. We notice that with DSM and SSM, τ12clearly decreases when filtering is applied, which is in agreement with the results

obtained for the mean-velocity profile and for the Reynolds stress. With the standard Smagorinsky model, τ12 increases because of the increased model length scale.

Based on the results of this subsection, it seems that although DSM and SSM are better models than the standard Smagorinsky model, the same deficiency as noticed with the standard Smagorinsky model is seen with these models when the non-linear convection term is filtered. In SSM, there is no direct interaction between the model and explicit filtering. In DSM, modelling and explicit filtering are coupled via the increased test-filter width. It has been noticed that as long as the width of the explicit filter is correctly treated, the results are not sensitive to the choice of the test filter.3 _{Since, in addition,}

explicit filtering reduces the effect of the model, it is understandable that the effect of the model did not increase. With the standard Smagorinsky model, the coupling between the model and filtering is easily set via the model length scale, but the modelling error limits the accuracy of the results.

The one-dimensional streamwise energy spectra from the near-wall region are given in Figure 4. In the upper part of the figure where no explicit filtering is applied, the differences between the models are rather small, whereas in the lower part, the standard Smagorinsky model damps down the spectrum most. It was noticed in the previous study with this model that damping of the low frequencies is mainly due to the implicit filtering provided by the model.6 _{Here, we see that with SSM, the low frequencies are least affected.}

5 FILTERING VIA THE SGS MODEL

In this section, we apply the second approach to explicit filtering discussed in Section 3. The small-scale stress is divided into SFS and actual SGS components, filtering is present only in the models, and the convection term is not filtered explicitly. First, modelling is only applied to the SGS component of the shear stress using the Smagorinsky models, then only the SFS component is modelled using SSM, and finally, both SFS and SGS stresses are modelled using SSM and a Smagorinsky model as a mixed model.

In the upper part of Figure 5, mean-velocity profiles from the cases using a Smagorinsky model for the SGS stress are depicted. First, no extra filtering was provided via the model. The model length scale in the standard Smagorinsky model, ∆S, was proportional to grid

spacing and the the test filter in DSM, ∆test, had the width of two grid spacings. Then,

the model length scale in the standard Smagorinsky model was increased to two grid spacings and in DSM, the width of the test filter was increased to four grid spacings. This corresponds to a situation where the explicit filter has the width of two grid spacings. In the standard Smagorinsky model, filtering provided by the model is actually implicit filtering since no filtering operation is made explicitly.

In Figure 5, the larger model length scale in the standard Smagorinsky model increases the thickness of the viscous sublayer and makes the mean bulk velocity overpredicted. This is due to the modelling error.19 _{In the dynamic Smagorinsky model, increasing the test}

filter width has only a small effect on the velocity profile. However, the thickness of the viscous sublayer is better predicted in the case with the wider test filter.

In the lower part of Figure 5, we have the velocity profiles from cases where SSM was used alone as an SFS model and as a mixed model together with a Smagorinsky model to model both SGS and SFS stresses. The filter in SSM had the width of two grid spacings, the model parameter in the standard Smagorinsky model was proportional to grid spacing, and the test-filter width in DSM was two grid spacings. This corresponds to a situation where the explicit filter has the width of two grid spacings, and the Smagorinsky models are applied to model the SGS effects and SSM the SFS effects. As seen in Figure 5, the use of the mixed model improves the prediction of the viscous sublayer which is too thin when only SSM is applied. DSM together with the SSM produces the best profile. When compared to the cases with only a SGS model in the upper part of the figure, here the mean bulk velocity is a bit low but the thickness of the logarithmic layer is better predicted.

In Figure 5, the filtering provided by modelling does not change the slope of the velocity profile in the way that explicit filtering of the convection term did in Figure 1. In addition, the behaviour in the viscous sublayer improves which does not happen in Figure 1.

increased test filter width has only a small effect on the Reynolds stress, and it becomes slightly more overpredicted. In the lower part of the figure, the use of SSM together with the standard Smagorinsky model produces the best Reynolds stress. However, the differences between the cases are small. By comparison of the upper and lower figures, we see that the use of SSM as SFS model clearly improved the prediction of the Reynolds stress. The results are similar for the other diagonal stress components. When compared to the case with explicit filtering of the non-linear convection term in Figure 2, the clear overprediction produced by the filtering is not visible in any of the cases in Figure 6.

In Figure 7, we have the SGS and SFS stresses from the different cases. As the model length scale is increased, the SGS shear stress of the standard Smagorinsky model is increased strongly. When the larger test filter is applied in DSM, the SGS shear stress inconsistently decreases. Thus, the improved results obtained using this test filter were probably due to the decreased effect of the model. In the lower part of figure, the SFS stress produced by SSM is much larger than the SGS stress produced by the Smagorinsky models. Thus, using the SFS model increases the effect of modelling.

The one-dimensional streamwise energy spectra are depicted in Figure 8. As can be expected, increasing the length scale of the Smagorinsky model damps down the whole spectrum. In the lower part of the figure, the mixed models affect less the low frequencies and damp down the high frequencies better than the Smagorinsky models alone in the upper figure. However, the differences are rather small, and when compared to the case with explicit filtering of the convection term in Figure 4, the damping of high frequencies is clearly not as efficient.

6 CONCLUSIONS

In this paper, two approaches to explicit filtering were applied. In the first one, the non-linear convection term of the Navier–Stokes equations and the SGS model were filtered explicitly and in the second one, filtering was provided only via the applied small-scale models. The first aim of the paper was to clarify the effect of SGS modelling on the behaviour of explicit filtering of the convection term. Then, the roles of different mod-els in the second approach were studied and comparisons were made between the two approaches.

It was noticed previously with the standard Smagorinsky model that explicit filtering of the convection term changes the slope of the mean-velocity profile, overpredicts the Reynolds stresses and decreases the effect of SGS modelling.6 _{In the present study, it was}

error due to the small and badly described high-frequency components.

When explicit filtering was applied only via the SGS or SFS modelling, the flow statis-tics were better predicted than in the case with explicit filtering of the convection term, and the best results were obtained with a mixed model where there is a model for both SGS and SFS stresses. The changes in the mean-velocity profile and the overprediction of the Reynolds stresses noticed with the explicit filtering of the convection term were not present. The small scales were not damped as efficiently as when the convection term was filtered, and thus the filtering provided by the applied models is not the same as explicit limiting of the high frequencies. However, the use of mixed models improved the predic-tion of the spectra when compared to using Smagorinsky models or SSM alone. If one considers the total simulation error and the computing time, this latter approach seems to be better than filtering of the non-linear convection term. However, the original reason for using explicit filtering was the large numerical error, and in the latter approach, it is not clear that this error was decreased.

ACKNOWLEDGEMENTS

This work has been funded by the Finnish national Graduate School in Computational Fluid Dynamics. The computer capacity was provided by CSC, Scientific Computing Ltd. The used channel-flow code is based on a code written by Dr. Boersma from TU Delft. REFERENCES

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