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Influence of subgrid scale models on the buffer sublayer in channel flow

Subgrid-scale (SGS) models are critical in large-eddy simulations (LES) of turbulent flows. In this paper we conduct a comparative study on different SGS models, including one-k-equation, wall-adapting local eddy-viscosity (WALE), Sigma and shear-constrained model. Wall-resolved LES simulations of channel flows are performed with a finite volume code at shear Reynolds number Re_\tau = 395. In the simulations, the buffer sublayer turns out to be the most sensitive to the SGS model. Through the analysis of the mean velocity, the second-order moments, the SGS viscosity and the fluctuating vorticities, it is shown that the WALE and Sigma model outperform others significantly in terms of fluctuations while the constrained model improves slightly the mean velocity. The results also indicate that the SGS dissipation influences strongly the velocity fluctuations but not the mean flow nor the log-layer mismatch.


I. INTRODUCTION
The governing equations for LES simulaton of incompressible turbulent flows are the filtered Navier-Stokes equations 1,2 , where p and u are the filtered pressure and velocity fields, and τ ij = u i u j − u i u j is the SGS stress tensor. The tilde denotes the filter operator which will be hereafter omitted for simplicity. Modeling the SGS flow motions must be based on a functional form of τ using the filtered velocity u. This seemingly uncomplicated problem has been scrutinized for almost half century yet remains a challenge.
The first SGS model was proposed in 1960s by Smagorinsky 3 and Lilly 4 , and was inspired by the eddy-viscosity concept τ ij = −2ν SGS S ij . Here ν SGS = l 2 S |S| is the SGS viscosity, S ij = (u i,j + u j,i )/2 is the rate-of-strain tensor of the filtered velocity and l S = C S ∆ denotes the Smagorinsky length scale. Even though it is affected by a consistent overprediction of velocity in the logarithmic region of boundary-layer flows (the log-layer mismatch problem), the Smagorinsky model is still widely used by CFD practitioners due to its simplicity and numerical stability. Many variants have been ever since developed to improve the Smagorinsky model. Schumann 5 proposed a kinetic energy model where the SGS kinetic energy is solved by a transport equation and its square root is used as a velocity scale in the model.
Another important development is the dynamic Smagorinsky model 6 , which is based on the Germano identity and the scale-invariance assumption of the model coefficient near the cut-off scale. To account for situations where the scale-invariance assumption does not hold, e.g., near-surface regions in atmospheric boundary layer flows, Porté-Agel and Meneveau 7 proposed a scale-dependent dynamic SGS model. However, the averaging procedure in dynamic models renders difficult their application in complex flows. Another group of models is based on the invariants of a symmetric tensor that depends on the gradient of the filtered velocity 8 . This group includes, among many others, the wall-adapting local eddy-viscosity model (WALE) 9 , the Vreman's model 10 , the Verstappen's model 11 and the Sigma model 12 .
Similar to the Smagorinsky model, these models are local and hence are suitable for complex flows and geometries.
Another interesting idea on SGS modeling, which was also introduced by Schumann 5 , is to decompose the SGS stress into a mean and a fluctuating part following the concept of Reynolds' decomposition. The inhomogeneous effects near the wall, which cause problems in Smagorinsky model, are thus taken into account in the mean part. Based on this idea, many SGS models have been proposed, including the two-part model by Sullivan et al. 13 , the shear-improved model by Lévêque et al. 14  Note that CSGS model relies heavily on the accuracy of the mean or second-moment source data, which becomes impractical for flows without reliable information on these data.
The purpose of this paper is to evaluate the performance of recently proposed SGS models and to gain insights into the SGS modeling problems by performing wall-resolved LES simulations of channel flows at a moderate Reynolds number. Four SGS models are chosen for this comparative study: one-k-equation model (kEqn), WALE, sigma, and CSGS. The details of these four SGS models are given in Section II. The simulation parameters and configurations are described in Section III. The results and their discussion are presented in Sections IV and V.

A. kEqn model
The kEqn model 16,17 consists of solving a transport equation of the SGS kinetic energy where C k , C ǫ are constants. Typically, C k = 0.094 and C ǫ = 1.048. The terms in the right hand side denote the viscous and turbulent diffusion, the gradient diffusion (the energy transfer between the filtered and sub-grid scales), and the dissipation, respectively. The square root of k SGS is taken as the velocity scale in the definition of the SGS viscosity, Compared to the Smagorinsky model 3 , the kEqn model takes into account history and spatial effects. However, in practice, both models yield very close results. In the equilibrium state they are statistically equivalent.

B. WALE model
The WALE model was developed by Nicoud and Ducros 9 in order to remedy the imperfection of the Smagorinsky model. In the Smagorinsky model, the SGS viscosity does not go to zero as the wall is approached. This causes an overestimation of the SGS dissipation and the log-layer mismatch problem. The WALE model is constructed from the invariants of the square of the velocity gradient tensor and achieves the correct scaling behavior near the wall.
The WALE model preserves the property of locality, meaning that only local quantities are required to evaluate the model at any point in space and time. The model reads where D w is a differential operator, S d ij is the traceless symmetric part of the square of the velocity gradient tensor, and C w ≃ 0.165.

C. Sigma model
where σ 1,2,3 denote the three singular values of g and satisfy σ 1 > σ 2 > σ 3 > 0. By definition, the sigular values of g are the square roots of the eigenvalues of g t g.
The model constant C σ is typically taken to be 1.35. Note that Sigma model is designed for boundary layer flows with smooth walls. With rough walls, the near wall scaling may be different from that of the smooth wall and approriate adjustment of the model is required to better capture the near-wall dynamics.

D. CSGS model
Originally used in optimizaton problems, constraints were recently introduced to model the SGS flow motions 18 . Given that constraints reflect basically the current available knowledge of a problem, the philosophy of CSGS is to use a priori known flow statistics to guide the behavior of a model. This methodology was proposed for LES simulations 15,18 and has already been applied in many situations 19,20 . Constraints can be applied to different physical quantities, such as Reynolds shear stress or SGS dissipation, depending on the currently available knowledge and the importance of the quantities. Following the proposal in 15 , in this paper a constraint is placed on the Reynolds shear stress, since its inaccurate prediction is thought to be responsible for the log-layer mismatch near the wall. The constraint can be obtained from high-fidelity DNS, experimental/theoretical results, or even a RANS model in certain circumstances.
The idea of the Reynolds-stress constrained SGS model can be expressed by the following decomposition of the SGS stresses, where R ij is the actual knowledge of the Reynolds stress of the physical velocity field, R LES ij is the Reynolds stress of the resolved velocity field in LES, and τ ij is the sub-grid scale stress.
Since R ij is known and R LES ij can be evaluated by u ′ i u ′ j ( · denotes the ensemble average operator), the only term to be modeled is the fluctuating part τ ′ ij . There are many different implementations of τ ′ ij . In the paper by Chen et al. 15 , τ ′ ij is evaluated by the dynamic Smagorinky procedure. In this paper, a slightly different approach is employed: choose a baseline model (e.g., kEqn), evaluate the SGS stress τ kEqn of the baseline model, and calculate an additional term ( The additional term is the difference of the two mean SGS stresses, one from the existing knowledge and another from the baseline model. Actually, τ ′ ij is modeled by the fluctuating part of the baseline SGS stress, τ ′ = τ kEqn − τ kEqn , which fullfils the necessary condition τ ′ = 0. The derivation is as following: . This implementation is non-intrusive, meaning that it does not require any modification of the pre-existing LES code. For modular programming language, only a subroutine or module is needed, which calculates the additional term.

III. SIMULATION DETAILS
The computational domain size of the channel flow is taken to be (2π × π × 2)H, where H is the half-channel height. The coordinates (X,Y,Z) denote the streamwise, the spanwise and the wall normal direction, respectively. The spatial resolution is (96 × 96 × 96). The cell size is uniform in the X and Y directions, and it is linearly decreasing in the Z direction towards the wall. The difference between the maximum and minimun cell size in the Z direction is ∆z max /∆z min ≃ 10. The wall-unit size z + = zu * /ν of the first point is about it are the logarithmic layer and the outer layer. The buffer layer is shaded, since, as will be shown in this section, the SGS models play a key role in this region.
The SGS viscosities and stresses are the direct results of the SGS models, which are hence firstly studied. Figure 1 shows the profiles of the SGS viscosity ν SGS and the major component τ 13 of the SGS stresses (other components are negligibly small). The ν SGS in the CSGS model denotes the ν SGS in the baseline kEqn model. In the viscous sublayer, the SGS viscosity scales in all models with the distance as ν SGS ∼ z 3 , which agrees with the theoretical expectation 24 . Nevertheless, the magnitudes differ by up to one order of magnitude, the smallest being from the WALE model and the biggest from the kEqn and CSGS model.
Given that all models considered here are based on the eddy-viscosity assumption, τ = −2ν SGS |S|, the difference in ν SGS is translated into the SGS stress, as clearly shown in Fig. 1 (right). Interestingly, in the buffer layer, the differences of the SGS viscosity and stress among models are significantly higher than in other regions. Moreover, since the baseline model of the CSGS model is kEqn and the CSGS can be viewed simply as a correction to the kEqn model, the closeness in ν SGS between these two models is expected and indicates that the constraint has a negligible effect on the SGS kinetic energy and viscosity. However, the constraint has a large impact on the SGS stress in the constrained region (z + < 40), as is shown in the profile of τ 13 . How the turbulence statistics are affected by the difference in the magnitude of ν SGS and the correction from the constraint will be explained in the remainder of this section. A more dynamically relevant quantity is the mean velocity gradient, which is plotted in Fig. 3 (left). Except for the kEqn model in the buffer layer and the CSGS model near where the mean SGS stress is defined as τ 13 = − ν SGS du/dz . Note that the averaging operator · is omitted in above equation. The derivation of this equation assumes tempo- ral and horizontal statistical homogeneity, which is satisfied in the statistically stationary channel flow. Note that this relation is an exact integral form of the momentum equations and shows that, indepentent from the SGS modeling, the total stress is a linear function of the distance to the wall. Any deviation from this behavior is not due to the SGS model but to other numerical issues.
The total shear stress, and the resolved and full Reynolds shear stresses, are plotted in Fig. 4. Here only the xz or 13 component is shown. The other two shear components are theoretically zero. All total shear profiles agree very well with theoretical results, except in the near-wall region z < 0.2H. In this region, the total shear profiles in the WALE and Sigma model deviate slightly further from the theoretical curve than in CSGS and kEqn, overall within 2% relative error. This deviation due to numerical errors is very small compared to that for other statistics, which hints a small numerical viscosity in the simulations.
In the uw profile, WALE and Sigma are much closer to the DNS curve, while in the uw + τ 13 profile all models perform very closely. From all above results, it seems that, among the variables in Eq. (7), uw correlate highly with ν SGS , while the velocity gradient dU/dz is a result of much more complicated mechanism, including the governing equations and the SGS models. This complexity makes it extremely difficult to solve the log-layer mismatch problem, and also indicates that ensuring correct SGS and Reynolds shear stresses is only a necessary but not a sufficient condition for accurately predicting the mean flow.
The turbulence intensities are characterized by the resolved velocity variances, which are plotted in Fig. 5. and resolve the variances vv and ww better than the other models. It is interesting that the CSGS model performs even worse in uu than its baseline model kEqn, although CSGS predicts a better mean velocity profile(see Fig. 2). Note that the inaccurate predictions of variances are not necessarily due to the incorrectly modeled SGS dissipation, which can also be induced by the nonnegligible numerical viscosity inherent in the finite volume method.
Overall, it is shown that different models result in significant differences in the buffer layer, including the location of the peaks.  The reduced content in the near-wall flow structures for kEqn and CSGS may be due to the excessive over-estimation of the SGS dissipation, as shown in Fig. 3 (right).
The fluctuations of vorticity are next analyzed. Vorticity fluctuations are thought to be more relevent to small-scale turbulent structures than velocity flucturations 27 . Therefore, the vorticity fluctuations are considered as a better measure for the accuracy of SGS models.
The profiles of the root-mean-square (RMS) of the vorticity fluctuations are shown in Fig. 8.
The values for the WALE and Sigma model are considerably closer to the DNS results than for other models, notably in the buffer layer where the turbulence is the most intense. The   locations. It is shown that as the constraint interface moves away from the wall, the difference between the log layer and the DNS results becomes smaller. For z i = 0.01H, the profile is almost the same as the one from the simulation without constraint. These observations are in agreement with Chen et al. 15 , regardless of the pseudo-spectral method used in the simulations in the latter. However, the velocity gradient ( Fig. 10 left), the second moments and the fluctuating vorticities display negligible difference for different interface locations.
The interface location, nevertheless, changes significantly the mean-flow SGS dissipation in the near-wall region, especially the buffer layer, as shown in Fig. 10 (right). Here for WALE, the constraint increases the SGS dissipation in the buffer layer, whereas, for kEqn, it decreases the SGS dissipation (Fig. 3 right). This provides strong evidence that in the buffer layer, the kEqn model overpredict the SGS dissipation while WALE results in an under-estimation.

V. DISCUSSION AND CONCLUSIONS
In this paper a comparative study of various SGS models was conducted by performing achieved with a spectral code in 15 , the deteriorated performance of CSGS here is probably due to the finite volume method used in the simulations, which is also consistent with the observations by Verma et al. 28 . This seems consistent with the arguments in 29 that ensuring correct Reynolds shear stresses is only a necessary but not a sufficient condition for yielding accurate statistics. Overall, these differences reveal the crucial role of the SGS models in the buffer layer in wall-resolved LES simulations and more generally in the near-wall regions.
The SGS models considered in the paper also share common features. First, in the log layer and outer layer, all models resulted in a mean velocity profile which agrees reasonably well with the DNS results. A slight log-layer mismatch is present in each model. Different from the wall-modelled simulations in previous studies 7,25,26 , the mismatch here was shown to have little correlation with the SGS viscosity and dissipation. The underlying mechanisms for the mismatch are much more complicated. Second, most turbulence statistics are underestimated compared to the DNS data. This systematic underestimation is likely a result of the lesser resolutions, but may also be attributed to a fundamental problem intrinsic in the eddy-viscosity family, i.e., the assumption that the SGS stress is linearly proportional to the rate-of-strain tensor is incorrect 2 . Nonlinear gradient models had been proprosed 30 but they were reported to achieve only a marginal improvement. Another important missing component in the current majority SGS models is stochasticity. Although governed by deterministic equations, turbulence is stochastic. Previous studies 31-33 already showed in certain cases that stochastic effects improve the fluctuation magnitudes and their anisotropy.
Hence, building appropriate practical stochastic SGS models is desired to adequately capture the random backscattering of the SGS flow motions.
A constantly ignored quantity in the study of SGS models is the pressure fluctuation, which is often considered as less relevant to the flow dynamics. As Pope indicated in 24 , "the primary effect of the fluctuating pressure is to redistribute the energy among the components -to extract energy from uu and transfer it to vv and ww ." However, the interaction between the SGS model and the resolved pressure fluctuation is unknown. More than 30 years ago, Moin and Kim 27 had already speculated that an appreciable portion of the pressure fluctuation may reside in SGS motions and that "the splatting effect is an important property of the flow in the vicinity of the walls and should be taken into account in the modeling of near-wall turbulence." Different from the "primary" effect on the bulk flow, the splatting effect denotes the transfer of energy from ww to the other two components due to the presence of the wall. These two types of effects are seldom considered in most SGS models. According to the results from WALE and Sigma models, the energy from uu seems to be incorrectly transferred to the other two components, resulting in an overprediction of uu . This transfer of energy is mainly affected through the pressure-rate-of-strain tensor pS ij . To the author's best knowledge, no DNS or experimental data are available for this tensor, making it difficult to check the above speculation. Nevertheless, the discovery of a relationship between the pressure-rate-of-strain tensor and the velocity-accelaration correlation 34 offers a feasible way for the experimentalists to gain insights into this quantity and to check whether the normal SGS stress should be directly modeled.