^{1}, Donghyun You

^{1,a)}and Sanjeeb T. Bose

^{2}

### Abstract

Large-eddy simulations of turbulent channel flow at *Re* _{τ} = 395 are performed using explicit filtering. Two different subfilter-scale models, the dynamic Smagorinsky mixed model and the dynamic global-coefficient mixed model, are formulated in accordance with the explicitly filtered governing equations. The use of explicit filtering separates the filtering operation from discretization, thereby producing a grid-independent solution. In explicit-filter large-eddy simulations, both the dynamic Smagorinsky mixed and the dynamic global-coefficient mixed subfilter-scale models are found to produce solutions close to the non-filtered direct numerical simulation data when explicit-filter widths in the streamwise and spanwise directions and at the center of the channel in the wall normal direction are about four times the grid spacings for direct numerical simulation. Solutions obtained using explicit-filter large-eddy simulation are compared with solutions obtained using implicit-filter large-eddy simulation in a range of filter and grid resolution. Explicit-filter large-eddy simulations performed on gradually refined grids with a fixed set of explicit-filter widths are found to produce grid-converged solutions for both models. Similarly in implicit-filter large-eddy simulation, predictive capabilities of subfilter-scale models in explicit-filter large-eddy simulation are found to be interfered with truncation errors when the ratio of the explicit-filter width to the grid spacing is small.

The authors are thankful to A. Sifounakis for fruitful technical discussion. The corresponding author acknowledges the support of the (U.S.) Army Research Office (USARO) under Grant No. W911NF1010348, with Dr. Frederick Ferguson as the program manager. Computing time was provided by the National Science Foundation (NSF) TeraGrid under Grant No. ASC110028 through the Pittsburgh Supercomputing Center.

I. INTRODUCTION

II. EXPLICIT-FILTER LES

A. Numerical method

B. Subfilter-scale stress models

1. DSMM

2. DGMM

III. RESULTS AND DISCUSSION

A. Grid independence of explicit-filter LES

B. Effect of explicit-filter resolution on LES prediction

C. Comparison of explicit-filter LES and implicit-filter LES

IV. CONCLUSIONS

### Key Topics

- Large eddy simulations
- 111.0
- Turbulence simulations
- 32.0
- Turbulent flows
- 14.0
- Numerical modeling
- 12.0
- Numerical solutions
- 11.0

## Figures

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DSMM for channel flow in configurations of A_{ exp } (solid lines), B_{ exp } (dashed lines), and C_{ exp } (dotted lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DSMM for channel flow in configurations of A_{ exp } (solid lines), B_{ exp } (dashed lines), and C_{ exp } (dotted lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of A_{ exp } (solid lines), B_{ exp } (dashed lines), and C_{ exp } (dotted lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of A_{ exp } (solid lines), B_{ exp } (dashed lines), and C_{ exp } (dotted lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Ensemble-averaged modeled stress (), resolved stress, and total stress (resolved stress + modeled stress) and (b) modified Leonard stress (), eddy-viscosity stress (η_{12}), and total modeled stress () in configurations of A_{ exp } (solid lines), B_{ exp } (dashed lines), and C_{ exp } (dotted lines) obtained using DSMM. Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Ensemble-averaged modeled stress (), resolved stress, and total stress (resolved stress + modeled stress) and (b) modified Leonard stress (), eddy-viscosity stress (η_{12}), and total modeled stress () in configurations of A_{ exp } (solid lines), B_{ exp } (dashed lines), and C_{ exp } (dotted lines) obtained using DSMM. Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Ensemble-averaged modeled stress (), resolved stress, and total stress (resolved stress + modeled stress) and (b) modified Leonard stress (), eddy-viscosity stress (η_{12}), and total modeled stress () in configurations of A_{ exp } (solid lines), B_{ exp } (dashed lines), and C_{ exp } (dotted lines) obtained using DGMM. Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Ensemble-averaged modeled stress (), resolved stress, and total stress (resolved stress + modeled stress) and (b) modified Leonard stress (), eddy-viscosity stress (η_{12}), and total modeled stress () in configurations of A_{ exp } (solid lines), B_{ exp } (dashed lines), and C_{ exp } (dotted lines) obtained using DGMM. Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DSMM for channel flow in configurations of D_{ exp } (solid lines), E_{ exp } (dashed lines), and F_{ exp } (dotted lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DSMM for channel flow in configurations of D_{ exp } (solid lines), E_{ exp } (dashed lines), and F_{ exp } (dotted lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of D_{ exp } (solid lines), E_{ exp } (dashed lines), and F_{ exp } (dotted lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of D_{ exp } (solid lines), E_{ exp } (dashed lines), and F_{ exp } (dotted lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Ensemble-averaged modeled stress (), resolved stress, and total stress (resolved stress + modeled stress) and (b) modified Leonard stress (), eddy-viscosity stress (η_{12}), and total modeled stress () in configurations of D_{ exp } (solid lines), E_{ exp } (dashed lines), and F_{ exp } (dotted lines) obtained using DSMM. Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Ensemble-averaged modeled stress (), resolved stress, and total stress (resolved stress + modeled stress) and (b) modified Leonard stress (), eddy-viscosity stress (η_{12}), and total modeled stress () in configurations of D_{ exp } (solid lines), E_{ exp } (dashed lines), and F_{ exp } (dotted lines) obtained using DSMM. Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Ensemble-averaged modeled stress (), resolved stress, and total stress (resolved stress + modeled stress) and (b) modified Leonard stress (), eddy-viscosity stress (η_{12}), and total modeled stress () in configurations of D_{ exp } (solid lines), E_{ exp } (dashed lines), and F_{ exp } (dotted lines) obtained using DGMM. Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Ensemble-averaged modeled stress (), resolved stress, and total stress (resolved stress + modeled stress) and (b) modified Leonard stress (), eddy-viscosity stress (η_{12}), and total modeled stress () in configurations of D_{ exp } (solid lines), E_{ exp } (dashed lines), and F_{ exp } (dotted lines) obtained using DGMM. Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DSMM for channel flow in configurations of C_{ exp } (solid lines), E_{ exp } (dashed lines), and G_{ exp } (dotted lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DSMM for channel flow in configurations of C_{ exp } (solid lines), E_{ exp } (dashed lines), and G_{ exp } (dotted lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of C_{ exp } (solid lines), E_{ exp } (dashed lines), and G_{ exp } (dotted lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of C_{ exp } (solid lines), E_{ exp } (dashed lines), and G_{ exp } (dotted lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

Ensemble-averaged modeled stress and resolved stress in configurations of C_{ exp } (solid lines), E_{ exp } (dashed lines), and G_{ exp } (dotted lines) obtained using (a) DSMM and (b) DGMM. Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

Ensemble-averaged modeled stress and resolved stress in configurations of C_{ exp } (solid lines), E_{ exp } (dashed lines), and G_{ exp } (dotted lines) obtained using (a) DSMM and (b) DGMM. Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of D_{ exp } (solid lines) and A_{ imp } (dashed lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of D_{ exp } (solid lines) and A_{ imp } (dashed lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of G_{ exp } (solid lines) and B_{ imp } (dashed lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of G_{ exp } (solid lines) and B_{ imp } (dashed lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streatwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of D_{ exp } (solid lines) and B_{ imp } (dashed lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streatwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of D_{ exp } (solid lines) and B_{ imp } (dashed lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of G_{ exp } (solid lines) and C_{ imp } (dashed lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

(a) Mean streamwise velocity and (b) RMS velocity fluctuations obtained using DGMM for channel flow in configurations of G_{ exp } (solid lines) and C_{ imp } (dashed lines). Symbols denote the non-filtered DNS data of Moser *et al.* ^{23}

## Tables

Grid parameters for explicit-filter LES of turbulent channel flow. *N* _{ x(y, z)} and are the number of grid points and explicit-filter widths in wall units, respectively. Grid parameters used for DNS by Moser *et al.* ^{23} are also provided for reference.

Grid parameters for explicit-filter LES of turbulent channel flow. *N* _{ x(y, z)} and are the number of grid points and explicit-filter widths in wall units, respectively. Grid parameters used for DNS by Moser *et al.* ^{23} are also provided for reference.

Root-mean-squared differences between the present explicit-filter LES solutions and the non-filtered DNS data of Moser *et al.* ^{23} calculated using Eq. (27) for the first- and second-order statistics.

Root-mean-squared differences between the present explicit-filter LES solutions and the non-filtered DNS data of Moser *et al.* ^{23} calculated using Eq. (27) for the first- and second-order statistics.

Grid parameters for implicit-filter LES of turbulent channel flow. *N* _{ x(y, z)} and are the number of grid points and implicit-filter widths in wall units, respectively.

Grid parameters for implicit-filter LES of turbulent channel flow. *N* _{ x(y, z)} and are the number of grid points and implicit-filter widths in wall units, respectively.

Root-mean-squared differences between the present implicit-filter LES solutions and the non-filtered DNS data of Moser *et al.* ^{23} calculated using Eq. (27) for the first- and second-order statistics. Implicit-filter LES were performed with DGMM.

Root-mean-squared differences between the present implicit-filter LES solutions and the non-filtered DNS data of Moser *et al.* ^{23} calculated using Eq. (27) for the first- and second-order statistics. Implicit-filter LES were performed with DGMM.

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