Numerical studies of finite element variational multiscale methods for turbulent flow simulations. (English) Zbl 1406.76029

Summary: Different realizations of variational multiscale (VMS) methods within the framework of finite element methods are studied in turbulent channel flow simulations. One class of VMS methods uses bubble functions to model resolved small scales whereas the other class contains the definition of the resolved small scales by an explicit projection in its set of equations. All methods are employed with eddy viscosity models of Smagorinsky type. The simulations are performed on grids for which a Direct Numerical Simulation blows up in finite time.


76F65 Direct numerical and large eddy simulation of turbulence
76M10 Finite element methods applied to problems in fluid mechanics
76M30 Variational methods applied to problems in fluid mechanics


Full Text: DOI


[1] Mohammadi, B.; Pironneau, O., Analysis of the K-epsilon turbulence model, (1994), John Wiley & Sons
[2] Sagaut, P., Large eddy simulation for incompressible flows, (2006), Springer-Verlag Berlin, Heidelberg, New York
[3] Hughes, T., Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid-scale models, bubbles and the origin of stabilized methods, Comput. methods appl. mech. engrg., 127, 387-401, (1995) · Zbl 0866.76044
[4] J.-L. Guermond, Stabilization of Galerkin approximations of transport equations by subgrid modeling, M2AN 33 (1999) 1293-1316.
[5] Hughes, T.; Mazzei, L.; Jansen, K., Large eddy simulation and the variational multiscale method, Comput. visual. sci., 3, 47-59, (2000) · Zbl 0998.76040
[6] Hughes, T.; Mazzei, L.; Oberai, A.; Wray, A., The multiscale formulation of large eddy simulation: decay of homogeneous isotropic turbulence, Phys. fluids, 13, 505-512, (2001) · Zbl 1184.76236
[7] Hughes, T.; Oberai, A.; Mazzei, L., Large eddy simulation of turbulent channel flows by the variational multiscale method, Phys. fluids, 13, 1784-1799, (2001) · Zbl 1184.76237
[8] V. John, Large Eddy simulation of turbulent incompressible flows. Analytical and Numerical Results for a Class of LES Models, Lecture Notes in Computational Science and Engineering, vol. 34, Springer-Verlag, Berlin, Heidelberg, New York, 2004. · Zbl 1035.76001
[9] Dunca, A.; John, V.; Layton, W., The commutation error of the space averaged navier – stokes equations on a bounded domain, (), 53-78 · Zbl 1096.35101
[10] Berselli, L.; John, V., Asymptotic behavior of commutation errors and the divergence of the Reynolds stress tensor near the wall in the turbulent channel flow, Math. methods appl. sci., 29, 1709-1719, (2006) · Zbl 1370.76065
[11] Berselli, L.; Grisanti, C.; John, V., Analysis of commutation errors for functions with low regularity, J. comput. appl. math., 206, 1027-1045, (2007) · Zbl 1125.76032
[12] John, V.; Kaya, S., A finite element variational multiscale method for the navier – stokes equations, SIAM J. sci. comput., 26, 1485-1503, (2005) · Zbl 1073.76054
[13] Gravemeier, V., Scale-separating operators for variational multiscale large eddy simulation of turbulent flows, J. comput. phys., 212, 400-435, (2006) · Zbl 1161.76494
[14] Holmen, J.; Hughes, T.; Oberai, A.; Wells, G., Sensitivity of the scale partition for variational multiscale large-eddy simulation of channel flow, Phys. fluids, 16, 824-827, (2004) · Zbl 1186.76234
[15] Farhat, C.; Rajasekharan, A.; Koobus, B., A dynamic variational multiscale method for large eddy simulations on unstructured meshes, Comput. methods appl. mech. engrg., 195, 1667-1691, (2006) · Zbl 1116.76046
[16] Gravemeier, V., The variational multiscale method for laminar and turbulent flow, Arch. comput. methods engrg., 13, 249-324, (2006) · Zbl 1177.76341
[17] V. Gravemeier, The Variational Multiscale Method for Laminar and Turbulent Incompressible Flow. Ph.D. Thesis, Institute of Structural Mechanics, University of Stuttgart, 2003. · Zbl 1177.76341
[18] Gravemeier, V.; Wall, W.; Ramm, E., A three-level finite element method for the instationary incompressible navier – stokes equation, Comput. methods appl. mech. engrg., 193, 1323-1366, (2004) · Zbl 1085.76038
[19] Gravemeier, V.; Wall, W.; Ramm, E., Large eddy simulation of turbulent incompressible flows by a three-level finite element method, Int. J. numer. methods fluids, 48, 1067-1099, (2005) · Zbl 1070.76034
[20] Franca, L.; Oliveira, S., Pressure bubbles stabilization features in the Stokes problem, Comput. methods appl. mech. engrg., 192, 1929-1937, (2003) · Zbl 1029.76032
[21] Franca, L.; Frey, S., Stabilized finite element methods: II. the incompressible navier – stokes equations, Comput. methods appl. mech. engrg., 99, 209-233, (1992) · Zbl 0765.76048
[22] Braack, M.; Burman, E.; John, V.; Lube, G., Stabilized finite element methods for the generalized Oseen problem, Comput. methods appl. mech. engrg., 196, 853-866, (2007) · Zbl 1120.76322
[23] Collis, S., Monitoring unresolved scales in multiscale turbulence modeling, Phys. fluids, 13, 1800-1806, (2001) · Zbl 1184.76110
[24] K. Jansen, A. Tejada-Martinez, An Evaluation of the Variational Multiscale Model for Large-Eddy Simulation While Using a Hierarchical Basis, AIAA Paper 2002-0283.
[25] S. Ramakrishnan, S. Collis, Multiscale Modeling for Turbulence Simulation in Complex Geometries, AIAA Paper, 2004.
[26] John, V.; Roland, M., Simulations of the turbulent channel flow at \(\mathit{Re}_\tau = 180\) with projection-based finite element variational multiscale methods, Int. J. numer. methods fluids, 55, 407-429, (2007) · Zbl 1127.76026
[27] John, V.; Kindl, A., Variants of projection-based finite element variational multiscale methods for the simulation of turbulent flows, Int. J. numer. methods fluids, 56, 1321-1328, (2008) · Zbl 1161.76028
[28] Germano, M.; Piomelli, U.; Moin, P.; Cabot, W., A dynamic subgrid-scale eddy viscosity model, Phys. fluids A, 3, 1760-1765, (1991) · Zbl 0825.76334
[29] Lilly, D., A proposed modification of the Germano subgrid-scale closure method, Phys. fluids A, 4, 633-635, (1992)
[30] Bazilevs, Y.; Calo, V.; Cottrell, J.; Hughes, T.; Reali, A.; Scovazzi, G., Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. methods appl. mech. engrg., 197, 173-201, (2007) · Zbl 1169.76352
[31] Hughes, T.; Sangalli, G., Variational multiscale analysis: the fine-scale green’s function, projection, optimization, localization, and stabilized methods, SIAM J. numer. anal., 45, 539-557, (2007) · Zbl 1152.65111
[32] Senoner, J.-M.; Garcia, M.; Mendez, S.; Staffelbach, G.; Vermorel, O.; Poinsot, T., The growth of rounding errors and the repetitivity of large eddy simulation, Aiaa j., 46, 1773-1781, (2008)
[33] Adams, R., Sobolev spaces, (1975), Academic Press New York · Zbl 0314.46030
[34] John, V., Reference values for drag and lift of a two-dimensional time dependent flow around a cylinder, Int. J. numer. methods fluids, 44, 777-788, (2004) · Zbl 1085.76510
[35] John, V.; Matthies, G.; Rang, J., A comparison of time-discretization/linearization approaches for the time-dependent incompressible navier – stokes equations, Comput. methods appl. mech. engrg., 195, 5995-6010, (2006) · Zbl 1124.76041
[36] John, V., Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D navier – stokes equations, Int. J. num. methods fluids, 40, 775-798, (2002) · Zbl 1076.76544
[37] John, V., On the efficiency of linearization schemes and coupled multigrid methods in the simulation of a 3D flow around a cylinder, Int. J. numer. methods fluids, 50, 845-862, (2006) · Zbl 1086.76039
[38] Brezzi, F.; Franca, L.; Hughes, T.; Russo, A., \(b = \int g\), Comput. methods appl. mech. engrg., 145, 329-339, (1997) · Zbl 0904.76041
[39] John, V., On large eddy simulation and variational multiscale methods in the numerical simulation of turbulent incompressible flows, Appl. math., 51, 321-353, (2006) · Zbl 1164.76348
[40] Smagorinsky, J., General circulation experiments with the primitive equations, Mon. weather rev., 91, 99-164, (1963)
[41] Russo, A., Bubble stabilization of finite element methods for the linearized incompressible navier – stokes equations, Comput. methods appl. mech. engrg., 132, 335-343, (1996) · Zbl 0887.76038
[42] Codina, R.; Principe, J.; Guasch, O.; Badia, S., Time dependent subscales in the stabilized finite element approximation of incompressible flow problems, Comput. methods appl. mech. engrg., 196, 2413-2430, (2007) · Zbl 1173.76335
[43] Franca, L.; Madureira, A.; Valentin, F., Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions, Comput. methods appl. mech. engrg., 194, 3006-3021, (2005) · Zbl 1091.76034
[44] Araya, R.; Barrenechea, G.; Valentin, F., Stabilized finite element methods based on multiscale enrichment for the Stokes problem, SIAM J. numer. anal., 44, 322-348, (2006) · Zbl 1121.65116
[45] Moser, D.; Kim, J.; Mansour, N., Direct numerical simulation of turbulent channel flow up to \(\mathit{Re}_\tau = 590\), Phys. fluids, 11, 943-945, (1999) · Zbl 1147.76463
[46] Choi, H.; Moin, P., Effects of the computational time step on numerical solutions of turbulent flow, J. comput. phys., 113, 1-4, (1994) · Zbl 0807.76051
[47] Gresho, P.; Sani, R., Incompressible flow and the finite element method, (2000), Wiley Chichester · Zbl 0988.76005
[48] Codina, R., Stabilized finite element approximation of transient incompressible flows using orthogonal subscales, Comput. methods appl. mech. engrg., 191, 4295-4321, (2002) · Zbl 1015.76045
[49] Gravemeier, V., Variational multiscale large eddy simulation of turbulent flow in a diffuser, Comput. mech., 39, 477-495, (2007) · Zbl 1160.76022
[50] John, V.; Matthies, G., Moonmd – a program package based on mapped finite element methods, Comput. visual sci., 6, 163-170, (2004) · Zbl 1061.65124
[51] Gravemeier, V., A consistent dynamic localization model for large eddy simulation of turbulent flows based on a variational formulation, J. comput. phys., 218, 677-701, (2006) · Zbl 1161.76495
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