Key ingredients for wall-modeled LES with the Lattice Boltzmann method: Systematic comparison of collision schemes, SGS models, and wall functions on simulation accuracy and efficiency for turbulent channel flow

Gregorio Gerardo Spinelli; Jana Gericke; Kannan Masilamani; Harald Günther Klimach

doi:10.3934/dcdss.2023212

Article Contents

2024, Volume 17, Issue 11: 3224-3251. Doi: 10.3934/dcdss.2023212 open access

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Key ingredients for wall-modeled LES with the Lattice Boltzmann method: Systematic comparison of collision schemes, SGS models, and wall functions on simulation accuracy and efficiency for turbulent channel flow

German Aerospace Center (DLR), Institute of Software Methods for Product Virtualization, Dresden, Germany

^*Corresponding author: Gregorio Gerardo Spinelli
^*Corresponding author: Gregorio Gerardo Spinelli

Received: January 2023

Revised: November 2023

Early access: December 12, 2023

Published online: December 12, 2023

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Abstract

In this study, we consider different combinations of collision schemes, wall functions, and subgrid scale (SGS) models to simulate the bi-periodic turbulent channel flow at $ {\mathrm{Re}_\tau} = 1000 $. The study is carried out on a $ D3Q27 $ lattice stencil, where the considered collision schemes are the Multiple Relaxation Times (MRT), the Hybrid Recursive Regularized Bhatnagar-Gross-Krook (HRR), and the parameterized Cumulant scheme. The considered SGS models are the Smagorinsky, the Wall-Adapting Local Eddy-viscosity (WALE), and the Vreman model. The Cumulant scheme utilizes its intrinsic implicit SGS model. The turbulent velocity profile is modeled with the following wall functions: the Reichardt, the Musker, and a combination of the Werner and Wengle and the Schmitt (Power-law) function. To assure an impartial comparison, all these ingredients are implemented in the same infrastructure, the open-source software Musubi. The comparison of the considered wall functions shows that the Musker function offers a good compromise between accuracy and performance. When comparing the considered SGS models, although the WALE model delivers the most accurate results, the Vreman model offers the fastest computation. On average, the parallel performance improves by $ 10 \% $. Amongst the considered collision schemes, the Cumulant outperforms the competitors accuracy-wise. However, for the considered test case, the fastest collision scheme is the MRT. Our investigations show that for the considered test case, the best results in terms of accuracy and performance are delivered by the combination of the Cumulant scheme with its implicit SGS model and the Musker wall function.

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Mathematics Subject Classification: Primary: 76F65; Secondary: 76G25.

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Figure 1. Computational domain for the bi-periodic turbulent channel flow. Walls are indicated in grey

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Figure 2. Normalized velocity profile along the wall normal direction for the considered domain sizes

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Figure 3. Normalized Reynolds stress in stream-wise direction along the wall normal direction for the considered domain sizes

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Figure 4. Normalized Reynolds stress in lateral and span-wise directions along the wall normal direction for the considered domain sizes

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Figure 5. Normalized Reynolds shear stress along the wall normal direction for the considered domain sizes

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Figure 6. Normalized velocity profile along the wall normal direction for the considered wall functions

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Figure 7. Normalized Reynolds stress in stream-wise direction along the wall normal direction for the considered wall functions

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Figure 8. Normalized Reynolds stress in lateral and span-wise directions along the wall normal direction for the considered wall functions

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Figure 9. Normalized Reynolds shear stress along the wall normal direction for the considered wall functions

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Figure 10. Variation of the $ L2 $-norm of the normalized Reynolds shear stress with respect to the mesh resolution for the considered wall functions

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Figure 11. Normalized velocity profile along the wall normal direction for the considered SGS models

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Figure 12. Normalized Reynolds stress in stream-wise direction along the wall normal direction for the considered SGS models

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Figure 13. Normalized Reynolds stress in lateral and span-wise directions along the wall normal direction for the considered SGS models

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Figure 14. Normalized Reynolds shear stress along the wall normal direction for the considered SGS models

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Figure 15. Variation of the $ L2 $-norm of the normalized Reynolds shear stress with respect to the mesh resolution for the considered SGS models

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Figure 16. Normalized velocity profile along the wall normal direction for the considered collision schemes

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Figure 17. Normalized Reynolds stress in stream-wise direction along the wall normal direction for the considered collision schemes

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Figure 18. Normalized Reynolds stress in lateral and span-wise directions along the wall normal direction for the considered collision schemes

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Figure 19. Normalized Reynolds shear stress along the wall normal direction for the considered collision schemes

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Figure 20. Variation of the $ L2 $-norm of the normalized Reynolds shear stress with respect to the mesh resolution for the considered collision schemes

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Table 1. The lattice weights $ w_i $ and discrete velocities $ \mathit{\boldsymbol{c_i}} $ for the most popular lattice stencils for three dimensions

Lattice stencil	$ i $	$ \mathit{\boldsymbol{c_i}} $	$ w_i $
$ D3Q19 $	$ 0 $	$ (0,0,0) $	$ 1/3 $
	$ 1-6 $	$ (\pm1,0,0), (0,\pm1,0), (0,0,\pm1) $	$ 1/18 $
	$ 7-18 $	$ (\pm1,\pm1,0), (\pm1,0,\pm1), (0,\pm1,\pm1) $	$ 1/36 $
$ D3Q27 $	$ 0 $	$ (0,0,0) $	$ 8/27 $
	$ 1-6 $	$ (\pm1,0,0), (0,\pm1,0), (0,0,\pm1) $	$ 2/27 $
	$ 7-18 $	$ (\pm1,\pm1,0), (\pm1,0,\pm1), (0,\pm1,\pm1) $	$ 1/54 $
	$ 19-26 $	$ (\pm1,\pm1,\pm1) $	$ 1/216 $

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Table 2. Flow parameters and target values for the bi-periodic turbulent channel flow

Parameter	Value	Unit
${\mathrm{Re}_\tau}$	1000.0	-
${\mathrm{Re}_\mathrm{bulk}}$	39278.3	-
${\mathrm{Ma}}$	0.1	-
$ u_m $	1.0	$ m/s $
${u_{\tau}}$	0.0509	$ m/s $
$ \nu $	5.09e-5	$ m^{2}/s $

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Table 3. Settings for the considered domain sizes, along with turbulent quantities and performance measurements obtained from the simulation with MRT scheme, Vreman SGS model and Musker wall function

Domain size	${y^+}$	Nr. of cells	${u_{\tau,\mathrm{sim}}}$	${u_{\tau,\mathrm{err}}}$	${\mathrm{Re}_{\tau,\mathrm{sim}}}$	${\mathrm{Re}_{\tau,\mathrm{err}}}$	MLUPs/n
$ L = 2 \pi H $	50	76,880	0.0538	$ 5.66 \% $	1056.0	$ 5.60 \% $	37.06
$ W = 2 \pi H $	25	625,000	0.0512	$ 0.55 \% $	1007.0	$ 0.70 \% $	37.67
$ L = 3 \pi H $	50	176,720	0.0527	$ 3.50 \% $	1034.6	$ 3.46 \% $	33.98
$ W = 3 \pi H $	25	1,413,760	0.0514	$ 0.94 \% $	1009.2	$ 0.92 \% $	36.18
$ L = 8 \pi H $	50	471,880	0.0526	$ 3.30 \% $	1032.7	$ 3.27 \% $	34.89
$ W = 3 \pi H $	25	3,775,040	0.0516	$ 1.34 \% $	1013.6	$ 1.36 \% $	35.76

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Table 4. Relative $ L2 $-norms of the velocity and Reynolds stresses profiles for the considered wall functions with respect to the DNS data

Wall function	${y^+}$	${u^+}$	${\langle u^{\prime}u^{\prime} \rangle^+}$	${\langle v^{\prime}v^{\prime} \rangle^+}$	${\langle w^{\prime}w^{\prime} \rangle^+}$	${\langle u^{\prime}v^{\prime} \rangle^+}$
Reichardt	50	0.04129	0.5073	1.4139	0.3591	0.5353
	25	0.01400	0.3404	0.2681	0.3739	0.3592
	12.5	0.03120	0.3081	0.2148	0.3034	0.1726
Musker	50	0.03731	0.5105	1.4707	0.3654	0.5511
	25	0.01553	0.4199	0.2697	0.4160	0.3351
	12.5	0.02014	0.2935	0.2073	0.2872	0.1738
Power-law	50	0.04448	0.5585	1.3370	0.3525	0.5486
	25	0.03506	0.4516	0.3163	0.4660	0.3614
	12.5	0.02901	0.2518	0.2005	0.2789	0.1729

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Table 5. Performance and accuracy of the considered wall functions in conjunction with MRT scheme and Vreman model

Wall function	${y^+}$	${u_{\tau,\mathrm{sim}}}$	${u_{\tau,\mathrm{err}}}$	${\mathrm{Re}_{\tau,\mathrm{sim}}}$	${\mathrm{Re}_{\tau,\mathrm{err}}}$	MLUPs/node
Reichardt	50	0.0517	$ 1.53 \% $	1016.2	$ 1.62 \% $	40.73
	25	0.0482	$ 5.34 \% $	946.3	$ 5.37 \% $	39.21
	12.5	0.0413	$ 18.89 \% $	810.9	$ 18.91 \% $	35.06
Musker	50	0.0538	$ 5.66 \% $	1056.0	$ 5.60 \% $	37.67
	25	0.0512	$ 0.55 \% $	1007.0	$ 0.70 \% $	37.06
	12.5	0.0452	$ 11.23 \% $	888.2	$ 11.18 \% $	34.62
Power-law	50	0.0549	$ 7.82 \% $	1078.7	$ 7.87 \% $	41.71
	25	0.0538	$ 5.66 \% $	1056.4	$ 5.64 \% $	38.04
	12.5	0.0422	$ 17.12 \% $	829.1	$ 17.09 \% $	37.31

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Table 6. Relative $ L2 $-norms of the velocity and Reynolds stresses profiles for the considered SGS models with respect to the DNS data

SGS model	${y^+}$	${u^+}$	${\langle u^{\prime}u^{\prime} \rangle^+}$	${\langle v^{\prime}v^{\prime} \rangle^+}$	${\langle w^{\prime}w^{\prime} \rangle^+}$	${\langle u^{\prime}v^{\prime} \rangle^+}$
Smagorinsky	50	0.01994	0.4713	1.1474	0.4143	0.6974
	25	0.01542	0.4260	0.3312	0.4647	0.3432
	12.5	0.03260	0.3760	0.3793	0.4308	0.3150
Vreman	50	0.03731	0.5105	1.4707	0.3654	0.5511
	25	0.01553	0.4199	0.2697	0 4160	0.3351
	12.5	0.02014	0.2935	0.2073	0.2872	0.1738
WALE	50	0.02261	0.4143	1.2735	0.3766	0.6810
	25	0.02003	0.4446	0.2641	0.4171	0.3067
	12.5	0.01483	0.2725	0.1578	0.2492	0.1618

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Table 7. Performance and accuracy of the considered SGS models with Musker wall function and MRT collision scheme

SGS models	${y^+}$	${u_{\tau,\mathrm{sim}}}$	${u_{\tau,\mathrm{err}}}$	${\mathrm{Re}_{\tau,\mathrm{sim}}}$	${\mathrm{Re}_{\tau,\mathrm{err}}}$	MLUPs/node
Smagorinsky	50	0.0519	$ 1.92 \% $	1019.0	$ 1.90 \% $	34.6
	25	0.0501	$ 1.02 \% $	984.5	$ 1.55 \% $	33.81
	12.5	0.0423	$ 16.93 \% $	829.8	$ 17.02 \% $	31.42
Vreman	50	0.0538	$ 5.66 \% $	1056.0	$ 5.60 \% $	37.67
	25	0.0512	$ 0.55 \% $	1007.0	$ 0.70 \% $	37.06
	12.5	0.0452	$ 11.23 \% $	888.2	$ 11.18 \% $	34.62
WALE	50	0.0522	$ 2.51 \% $	1025.9	$ 2.59 \% $	30.70
	25	0.0519	$ 1.92 \% $	1019.1	$ 1.91 \% $	33.20
	12.5	0.0468	$ 8.09 \% $	918.9	$ 8.11 \% $	31.04

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Table 8. Relative $ L2 $-norms of the velocity and Reynolds stresses profiles for the considered collision schemes with respect to the DNS data

Collision scheme	${y^+}$	${u^+}$	${\langle u^{\prime}u^{\prime} \rangle^+}$	${\langle v^{\prime}v^{\prime} \rangle^+}$	${\langle w^{\prime}w^{\prime} \rangle^+}$	${\langle u^{\prime}v^{\prime} \rangle^+}$
MRT	50	0.03731	0.5105	1.4707	0.3654	0.5511
	25	0.01553	0.4199	0.2697	0.4160	0.3351
	12.5	0.02014	0.2935	0.2073	0.2872	0.1738
Cumulant	50	0.02743	0.5230	0.6573	0.4750	0.6717
	25	0.01796	0.4447	0.2947	0.2360	0.4161
	12.5	0.01869	0.2182	0.0971	0.1052	0.1585
HRR	50	0.04407	0.5443	0.9579	0.8772	0.9041
	25	0.04101	0.4781	0.8412	0.7622	0.7787
	12.5	0.05266	0.4931	0.7226	0.6914	0.6786

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Table 9. Performance and accuracy of the considered collision schemes in conjunction with Musker wall function, and Vreman SGS model solely for MRT and HRR

Collision scheme	${y^+}$	${u_{\tau,\mathrm{sim}}}$	${u_{\tau,\mathrm{err}}}$	${\mathrm{Re}_{\tau,\mathrm{sim}}}$	${\mathrm{Re}_{\tau,\mathrm{err}}}$	MLUPs/node
MRT	50	0.0538	$ 5.66 \% $	1056.0	$ 5.60 \% $	37.67
	25	0.0512	$ 0.55 \% $	1007.0	$ 0.70 \% $	37.06
	12.5	0.0452	$ 11.23 \% $	888.2	$ 11.18 \% $	34.62
Cumulant	50	0.0521	$ 2.32 \% $	1022.7	$ 2.27 \% $	30.33
	25	0.0517	$ 1.53 \% $	1015.1	$ 1.51 \% $	31.60
	12.5	0.0465	$ 8.68 \% $	913.3	$ 8.67 \% $	32.64
HRR	50	0.0467	$ 8.29 \% $	916.2	$ 8.38 \% $	25.94
	25	0.0455	$ 10.65 \% $	892.9	$ 10.71 \% $	27.00
	12.5	0.0397	$ 22.03 \% $	780.1	$ 21.99 \% $	27.68

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