High-order large eddy simulation of incompressible turbulent flow by variational multiscale method

doi:10.1007/s10483-026-3426-8

Abstract

Abstract:

This paper introduces an $h ⁢ p$ -refinement high-order finite element discretization based on a modified hierarchical Jacobian polynomial basis into a residual-based large eddy simulation (LES) by variational multiscale method (VMM). A high-order finite element basis is achieved by means of p-refinement by introducing a modified hierarchical Jacobian polynomial basis. The benchmark turbulent channel flow is computed using in-house finite element codes with three different orders of the Jacobian polynomial basis. The near-wall flow field and turbulent statistical results obtained using the same degrees of freedom (DOFs) are compared. The results show that the accuracy is significantly improved with the increasing order. The transportation equation of the turbulent kinetic energy is then derived with the VMM, and statistical dissipations are presented to clarify the improvement in the numerical accuracy by high-order simulations. In addition, the matrix structures of linear systems built using different orders of polynomial basis are presented, and the variation in the computational cost with the order of the finite element basis is explained. Furthermore, comparison of the computational costs for cases with equivalently accurate results is made to assess the computational efficiency of the high-order numerical method. Consequently, the results provide compelling evidence for the power of the high-order LES technique with the $h ⁢ p$ -refinement discretization, which not only reduces the DOFs to be solved, but also achieves the higher computational efficiency than low-order methods.

Key words: large eddy simulation (LES), variational multiscale method (VMM), hp-refinement discretization, high-order finite element basis, computational efficiency

2010 MSC Number:

O357.5

Linfeng CHEN, Xianglu JIA, Peng XU, Yuhong DONG, Jiafeng WU, Chen NIU, Jie ZHU, Fuchang ZHOU. High-order large eddy simulation of incompressible turbulent flow by variational multiscale method. Applied Mathematics and Mechanics (English Edition), 2026, 47(9): 2109-2122.

TrendMD

Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks

URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3426-8

https://www.amm.shu.edu.cn/EN/Y2026/V47/I9/2109

Figures/Tables 11

Fig. 1

Fig. 2

Table 1

Computational parameters"

$N x × N y × N z$	P	DOFs	$α s$	$Δ ⁢ x min +$	$Δ ⁢ z min +$	$Δ ⁢ y min +$
$24 × 24 × 24$	1	57 600	1.5	47.12	31.42	5.035
$12 × 12 × 12$	2	57 600	1.5	94.25	62.83	11.308
$8 × 8 × 8$	3	57 600	1.5	141.37	94.25	19.060

Table 1

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 2

Computational costs"

$N x × N y × N z$	P	Quadrature point per element	Core	Hour	Core hour per 100 time units
$24 × 24 × 24$	1	8	4	56.5	226
$12 × 12 × 12$	2	27	4	108.9	435.6
$8 × 8 × 8$	3	64	4	387.8	1 551.2

Table 2

Fig. 7

Table 3

Computational parameters and time costs"

$N x × N y × N z$	P	DOF	Quadrature point per element	Core	Hour	Core hour/100 time units
$64 × 64 × 64$	1	1 064 960	8	16	535	8 560
$18 × 18 × 18$	2	191 808	27	8	425	3 400
$8 × 8 × 8$	3	57 600	64	4	387.8	1 551.2

Table 3

Fig. 8

[1]	HUGHES, T. J. R., FEIJOO, G., MAZZEI, L., and QUINCY, J. The variational multiscale method - a paradigm for computational mechanics. Computer Methods in Applied Mechanics and Engineering, 166(1-2), 3–24 (1998)
[2]	CALO, V. Residual-Based Multiscale Turbulence Modeling: Finite Volume Simulations of Bypass Transition, Ph.D. dissertation, Stanford, CA (2005)
[3]	HUGHES, T. J., MAZZEI, L., and JANSEN, K. E. Large eddy simulation and the variational multiscale method. Computing and Visualization in Science, 3, 47–59 (2000)
[4]	CHEN, L., HULSHOFF, S., and DONG, Y. Turbulent kinetic dissipation analysis for residual-based large eddy simulation of incompressible turbulent flow by variational multiscale method. Computer Methods in Applied Mechanics and Engineering, 388, 114280 (2022)
[5]	COLOMES, O., BADIA, S., CODINA, R., and PRINCIPE, J. Assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows. Computer Methods in Applied Mechanics and Engineering, 285, 32–63 (2015)
[6]	SAGAUT, P. Large Eddy Simulation for Incompressible Flows, Springer Berlin, Heidelberg (2006)
[7]	WANG, Z. and OBERAI, A. A. Spectral analysis of the dissipation of the residual-based variational multiscale method. Computer Methods in Applied Mechanics and Engineering, 199(13-16), 810–818 (2010)
[8]	PRINCIPE, J., CODINA, R., and HENKE, F. The dissipative structure of variational multiscale methods for incompressible flows. Computer Methods in Applied Mechanics and Engineering, 199(13-16), 791–801 (2010)
[9]	CODINA, R., PRINCIPEA, J., GUASCHB, O., and BADIAA, S. Time dependent subscales in the stabilized finite element approximation of incompressible flow problems. Computer Methods in Applied Mechanics and Engineering, 196(21), 2413–2430 (2007)
[10]	BAZILEVS, Y., MICHLER, C., CALO, V. M., and HUGHES, T. J. Weak Dirichlet boundary conditions for wall-bounded turbulent flows. Computer Methods in Applied Mechanics and Engineering, 196(49-52), 4853–4862 (2007)
[11]	MOTLAGH, Y. G., AHN, H. T., HUGHES, T. J. R., and CALO, V. M. Simulation of laminar and turbulent concentric pipe flows with the isogeometric variational multiscale method. Computers & Fluids, 71(2), 146–155 (2013)
[12]	BAUER, G., GAMNITZER, P., GRAVEMEIER, V., and WALL, W. A. An isogeometric variational multiscale method for large-eddy simulation of coupled multi-ion transport in turbulent flow. Journal of Computational Physics, 251(4), 194–208 (2013)
[13]	TRELLES, J. P. and MODIRKHAZENI, S. M. Variational multiscale method for nonequilibrium plasma flows. Computer Methods in Applied Mechanics and Engineering, 282, 87–131 (2014)
[14]	RIBER, S., VALETTE, R., MESRI, Y., and HACHEM, E. Adaptive variational multiscale method for Bingham flows. Computers & Fluids, 138, 51–60 (2016)
[15]	CHEN, L., HULSHOFF, S., and WANG, Y. 2D residual-based LES of flow around a pipeline close to a flat seabed. Computer Methods in Applied Mechanics and Engineering, 363, 112788 (2020)
[16]	CHEN, L., WANG, Y., SUN, S., and WANG, S. The effect of boundary shear flow on hydrodynamic forces of a pipeline over a fully scoured seabed. Ocean Engineering, 206, 107326 (2020)
[17]	GRAVEMEIER, V. and WALL, W. A. Variational multiscale methods for premixed combustion based on a progress-variable approach. Combustion and Flame, 158(6), 1160–1170 (2011)
[18]	RASTHOFER, U., WALL, W. A., and GRAVEMEIER, V. An extended algebraic variational multiscale-multigrid-multifractal method (XAVM⁴) for large-eddy simulation of turbulent two-phase flow. Journal of Computational Physics, 359, 1–19 (2018)
[19]	YAN, J., LIN, S., BAZILEVS, Y., and WAGNER, G. J. Isogeometric analysis of multi-phase flows with surface tension and with application to dynamics of rising bubbles. Computers & Fluids, 179, 777–789 (2019)
[20]	AKKERMAN, I., DUNAWAY, J., KVANDAL, J., SPINKS, J., and BAZILEVS, Y. Toward free-surface modeling of planing vessels: simulation of the fridsma hull using ALE-VMS. Computational Mechanics, 50(6), 719–727 (2012)
[21]	BAZILEVS, Y., TAKIZAWA, K., TEZDUYAR, T. E., HSU, M. C., KOSTOV, N., and MCINTYRE, S. Aerodynamic and FSI analysis of wind turbines with the ALE-VMS and ST-VMS methods. Archives of Computational Methods in Engineering, 21(4), 359–398 (2014)
[22]	YAN, J., KOROBENKO, A., DENG, X., and BAZILEVS, Y. Computational free-surface fluid-structure interaction with application to floating offshore wind turbines. Computers & Fluids, 141, 155–174 (2016)
[23]	ZHU, L., GORAYA, S. A., and MASUD, A. Interface-capturing method for free-surface plunging and breaking waves. Journal of Engineering Mechanics, 145(11), 04019088 (2019)
[24]	YAN, J., KOROBENKO, A., TEJADA-MARTÍNEZ, A. E., GOLSHAN, R., and BAZILEVS, Y. A new variational multiscale formulation for stratified incompressible turbulent flows. Computers & Fluids, 158, 150–156 (2016)
[25]	BAZILEVS, Y., MICHLER, C., CALO, V. M., and HUGHES, T. J. R. Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes. Computer Methods in Applied Mechanics and Engineering, 199(13-16), 780–790 (2010)
[26]	BAZILEVS, Y. and AKKERMAN, I. Large eddy simulation of turbulent Taylor-Couette flow using isogeometric analysis and the residual-based variational multiscale method. Journal of Computational Physics, 229(9), 3402–3414 (2010)
[27]	AKKERMAN, I., BAZILEVS, Y., KEES, C. E., and FARTHING, M. W. Isogeometric analysis of free-surface flow. Journal of Computational Physics, 230(11), 4137–4152 (2011)
[28]	MOXEY, D., CANTWELL, C. O., BAO, Y., CASSINELLI, A., CASTIGLIONI, G., CHUN, S., JUDA, E., KAZEMI, E., LACKHOVE, K., MARCON, J., MENGALDO, G., SERSON, D., TURNER, M., XU, H., PEIRÓ, J., KIRBY, R. M., and SHERWIN, S. J. Nektar++: enhancing the capability and application of high-fidelity spectral/h⁢p element methods. Computer Physics Communications, 249(4), 107110 (2020)
[29]	XU, H., CANTWELL, C., MONTESERIN, C., ESKILSSON, C., ENGSIG-KARUP, A., and SHERWIN, S. Spectral/h⁢p element methods: recent developments, applications, and perspectives. Journal of Hydrodynamics, 30(2), 1–22 (2018)
[30]	AKKERMAN, I., BAZILEVS, Y., CALO, V. M., HUGHES, T. J. R., and HULSHOFF, S. The role of continuity in residual-based variational multiscale modeling of turbulence. Computational Mechanics, 41(3), 371–378 (2008)
[31]	BEHR, M. A., FRANCA, L. P., and TEZDUYAR, T. E. Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows. Computer Methods in Applied Mechanics and Engineering, 104(1), 31–48 (1993)
[32]	FRANCA, L. P. and FREY, S. L. Stabilized finite element methods: II. the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 99(2-3), 209–233 (1992)
[33]	SHAKIB, F. Finite Element Analysis of the Compressible Euler and Navier-Stokes Equations, Ph.D. dissertation, Stanford University (1989)
[34]	TAYLOR, C. A., HUGHES, T. J. R., and ZARINS, C. K. Finite element modeling of blood flow in arteries. Computer Methods in Applied Mechanics and Engineering, 158, 155–196 (1998)
[35]	JANSEN, K. E., WHITING, C. H., and HULBERT, G. M. A generalized-α method for integrating the filtered Navier-Stokes equations with a stabilized finite element method. Computer Methods in Applied Mechanics and Engineering, 190(3-4), 305–319 (2000)
[36]	KIM, J., MOIN, P., and MOSER, R. Turbulence statistics in fully developed channel flow at low Reynolds number. Journal of Fluid Mechanics, 177, 133–166 (1987)