## Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows.(English)Zbl 1169.76352

Summary: We present an LES-type variational multiscale theory of turbulence. Our approach derives completely from the incompressible Navier-Stokes equations and does not employ any ad hoc devices, such as eddy viscosities. We tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows. In the calculations, we employed linear, quadratic and cubic NURBS. A dispersion analysis of simple model problems revealed NURBS elements to be superior to classical finite elements in approximating advective and diffusive processes, which play a significant role in turbulence computations. The numerical results are very good and confirm the viability of the theoretical framework.

### MSC:

 76F65 Direct numerical and large eddy simulation of turbulence 76F05 Isotropic turbulence; homogeneous turbulence 76M30 Variational methods applied to problems in fluid mechanics 76M10 Finite element methods applied to problems in fluid mechanics 76M25 Other numerical methods (fluid mechanics) (MSC2010)
Full Text:

### References:

 [1] I. Akkerman, Y. Bazilevs, V.M. Calo, T.J.R. Hughes, S. Hulshoff, The role of continuity in residual-based variational multiscale modeling of turbulence. Comput. Mech., 2007, doi: doi:10.1007/s00466-007-0193-7. · Zbl 1162.76355 [2] Barenblatt, G.I., Similarity, self-similarity, and intermediate asymptotics, (1996), Cambridge University Press · Zbl 0907.76002 [3] Y. Bazilevs. Isogeometric Analysis of Turbulence and Fluid-Structure Interaction, PhD Thesis, ICES, UT Austin, 2006. [4] Bazilevs, Y.; Calo, V.M.; Zhang, Y.; Hughes, T.J.R., Isogeometric fluid – structure interaction analysis with applications to arterial blood flow, Comput. mech., 38, 310-322, (2006) · Zbl 1161.74020 [5] Bazilevs, Y.; Beirao da Veiga, L.; Cottrell, J.A.; Hughes, T.J.R.; Sangalli, G., Isogeometric analysis: approximation, stability and error estimates for h-refined meshes, Math. mod. methods appl. sci., 16, 1031-1090, (2006) · Zbl 1103.65113 [6] Bazilevs, Y.; Hughes, T.J.R., Weak imposition of Dirichlet boundary conditions in fluid mechanics, Comp. fluids, 36, 12-26, (2007) · Zbl 1115.76040 [7] Y. Bazilevs, C. Michler, V.M. Calo, T.J.R. Hughes, Weak Dirichlet boundary conditions for wall-bounded turbulent flows, Comp. Methods Appl. Mech. Engrg., in press, 2007, doi:10.1016/j.cma.2007.06.026. · Zbl 1173.76397 [8] Behr, M.; Hastreiter, D.; Mittal, S.; Tezduyar, T.E., Incompressible flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries, Comp. methods appl. mech. engrg., 123, 309-316, (1996) [9] Bochev, P.B.; Hughes, T.J.R.; Scovazzi, G., A multiscale discontinuous Galerkin method, Springer lecture notes in computer science, vol. 3743, (2006), Springer · Zbl 1142.65442 [10] Brezzi, F.; Franca, L.P.; Hughes, T.J.R.; Russo, A., $$b = \int g$$, Comp. methods appl. mech. engrg., 145, 329-339, (1997) · Zbl 0904.76041 [11] Brillouin, L., Wave propagation in periodic structures, (1953), Dover Publications Inc. · Zbl 0050.45002 [12] Brooks, A.N.; Hughes, T.J.R., Streamline upwind/petrov – galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier – stokes equations, Comp. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041 [13] Buffa, A.; Hughes, T.J.R.; Sangalli, G., Analysis of the multiscale discontinuous Galerkin method for convection – diffusion problems, SIAM J. numer. anal., 44, 4, 1420-1440, (2006) · Zbl 1153.76038 [14] V.M. Calo, Residual-based Multiscale Turbulence Modeling: Finite Volume Simulation of Bypass Transition, PhD Thesis, Department of Civil and Environmental Engineering, Stanford University, 2004. [15] Chung, J.; Hulbert, G.M., A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method, J. appl. mech., 60, 371-375, (1993) · Zbl 0775.73337 [16] Codina, R.; Principe, J.; Guasch, O.; Badia, S., Time dependent subscales in the stabilized finite element approximation of incompressible flow problems, Comp. methods appl. mech. engrg., 196, 2413-2430, (2007) · Zbl 1173.76335 [17] Cohen, E.; Riesenfeld, R.; Elber, G., Geometric modeling with splines. an introduction, (2001), A.K. Peters Ltd. Wellesley, MA · Zbl 0980.65016 [18] S.S. Collis, Multiscale Methods for Turbulence Simulation and Control, Technical Report 034, MEMS, Rice University, 2002. [19] Cottrell, J.A.; Reali, A.; Bazilevs, Y.; Hughes, T.J.R., Isogeometric analysis of structural vibrations, Comp. methods appl. mech. engrg., 195, 5257-5297, (2006) · Zbl 1119.74024 [20] C. Farhat, B. Koobus. Finite volume discretization on unstructured meshes of the multiscale formulation of large eddy simulations, in: F.G. Rammerstorfer, H.A. Mang, J. Eberhardsteiner (Eds.), Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V), Vienna University of Technology, Austria, July 7-12, 2002. [21] G.E. Farin, NURBS Curves and Surfaces: From Projective Geometry to Practical Use. A.K. Peters, Ltd., Natick, MA, 1995. · Zbl 0848.68112 [22] Gravemeier, V., The variational multiscale method for laminar and turbulent flow, Arch. comput. methods engrg. - state of the art rev., 13, 249-324, (2006) · Zbl 1177.76341 [23] Gresho, P.M.; Sani, R.L., Incompressible flow and the finite element method, (1998), Wiley New York, NY · Zbl 0941.76002 [24] Hauke, G.; Doweidar, M.H.; Miana, M., The multiscale approach to error estimation and adaptivity, Comp. methods appl. mech. engrg., 195, 1573-1593, (2006) · Zbl 1122.76057 [25] Hauke, G.; Doweidar, M.H.; Miana, M., Proper intrinsic scales for a-posteriori multiscale error estimation, Comp. methods appl. mech. engrg., 195, 3983-4001, (2006) · Zbl 1134.76028 [26] Hoffman, J.; Johnson, C., Stability of the dual navier – stokes equations and efficient computation of Mean output in turbulent flow using adaptive DNS/LES, Comp. methods appl. mech. engrg., 195, 1709-1721, (2006) · Zbl 1115.76037 [27] Holmen, J.; Hughes, T.J.R.; Oberai, A.A.; Wells, G.N., Sensitivity of the scale partition for variational multiscale LES of channel flow, Phys. fluids, 16, 3, 824-827, (2004) · Zbl 1186.76234 [28] Hughes, T.J.R., Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comp. methods appl. mech. engrg., 127, 387-401, (1995) · Zbl 0866.76044 [29] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (2000), Dover Publications Mineola, NY [30] Hughes, T.J.R.; Cottrell, J.A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Comp. methods appl. mech. engrg., 194, 4135-4195, (2005) · Zbl 1151.74419 [31] Hughes, T.J.R.; Feijóo, G.; Mazzei, L.; Quincy, J.B., The variational multiscale method – aparadigm for computational mechanics, Comp. methods appl. mech. engrg., 166, 3-24, (1998) · Zbl 1017.65525 [32] Hughes, T.J.R.; Mallet, M., A new finite element formulation for fluid dynamics: III. the generalized streamline operator for multidimensional advective – diffusive systems, Comp. methods appl. mech. engrg., 58, 305-328, (1986) · Zbl 0622.76075 [33] Hughes, T.J.R.; Mazzei, L.; Jansen, K.E., Large-eddy simulation and the variational multiscale method, Comp. vis. sci., 3, 47-59, (2000) · Zbl 0998.76040 [34] Hughes, T.J.R.; Mazzei, L.; Oberai, A.A.; Wray, A.A., The multiscale formulation of large eddy simulation: decay of homogenous isotropic turbulence, Phys. fluids, 13, 2, 505-512, (2001) · Zbl 1184.76236 [35] Hughes, T.J.R.; Oberai, A.A., Calculation of shear stresses in fourier – galerkin formulations of turbulent channel flows: projection, the Dirichlet filter and conservation, J. comput. phys., 188, 1, 281-295, (2003) · Zbl 1020.76025 [36] Hughes, T.J.R.; Oberai, A.A.; Mazzei, L., Large-eddy simulation of turbulent channel flows by the variational multiscale method, Phys. fluids, 13, 6, 1784-1799, (2001) · Zbl 1184.76237 [37] Hughes, T.J.R.; 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 [38] Hughes, T.J.R.; Scovazzi, G.; Bochev, P.B.; Buffa, A., A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method, Comp. methods appl. mech. engrg., 195, 2761-2787, (2006) · Zbl 1124.76027 [39] Hughes, T.J.R.; Scovazzi, G.; Franca, L.P., Multiscale and stabilized methods, (), (Chapter 4) [40] Hughes, T.J.R.; Stewart, J., A space-time formulation for multiscale phenomena, J. comput. appl. math., 74, 217-229, (1996) · Zbl 0869.65061 [41] Hughes, T.J.R.; Wells, G.N.; Wray, A.A., Energy transfers and spectral eddy viscosity of homogeneous isotropic turbulence: comparison of dynamic smagorinsky and multiscale models over a range of discretizations, Phys. fluids, 16, 4044-4052, (2004) · Zbl 1187.76226 [42] Jansen, K.E.; Collis, S.S.; Whiting, C.H.; Shakib, F., A better consistency for low-order stabilized finite element methods, Comp. methods appl. mech. engrg., 174, 153-170, (1999) · Zbl 0956.76044 [43] Jansen, K.E.; Whiting, C.H.; Hulbert, G.M., A generalized-α method for integrating the filtered navier – stokes equations with a stabilized finite element method, Comp. methods appl. mech. engrg., 190, 305-319, (1999) · Zbl 0973.76048 [44] H. Jeanmart, G.S. Winckelmans. Comparison of recent dynamic subgrid-scale models in the case of the turbulent channel flow, in: Proceedings of the Summer Program, Center for Turbulence Research, Stanford University and NASA Ames, 2002. · Zbl 1080.76531 [45] Johnson, C., Numerical solution of partial differential equations by the finite element method, (1987), Cambridge University Press Sweden [46] G.P. Johnson, V.M. Calo, K. Gaither. Interactive visualization and analysis of transitional flow, IEEE Trans. Vis. Comp. Graphics, submitted for publication. [47] G.P. Johnson, K. Gaither, V.M. Calo. Visualizing turbulent flow, in: 15th IEEE Visualization Conference, Austin, TX, October 2004, p. 598.22. [48] Koobus, B.; Farhat, C., A variational multiscale method for the large eddy simulation of compressible turbulent flows on unstructured meshes – application to vortex shedding, Comp. methods appl. mech. engrg., 193, 1367-1383, (2004) · Zbl 1079.76567 [49] Kravchenko, A.G.; Moin, P.; Moser, R., Zonal embedded grids for numerical simulation of wall-bounded turbulent flows, J. comput. phys., 127, 412-423, (1996) · Zbl 0862.76062 [50] Kravchenko, A.G.; Moin, P.; Shariff, K., B-spline method and zonal grids for simulation of complex turbulent flows, J. comput. phys., 151, 757-789, (1999) · Zbl 0942.76058 [51] Kwok, W.Y.; Moser, R.D.; Jiménez, J., A critical evaluation of the resolution properties of B-spline and compact finite difference methods, J. comput. phys., 174, 510-551, (2001) · Zbl 0995.65089 [52] Langford, J.A.; Moser, R.D., Optimal LES formulations for isotropic turbulence, J. fluid mech., 398, 321-346, (1999) · Zbl 0983.76043 [53] Lele, S.K., Compact finite difference schemes with spectral-like resolution, J. comput. phys., 103, 16-42, (1992) · Zbl 0759.65006 [54] Lesieur, M.; Métais, O.; Comte, P., Large eddy simulation of turbulence, (2005), Cambridge University Press Cambridge, England [55] Moser, R.; Kim, J.; Mansour, R., DNS of turbulent channel flow up to re=590, Phys. fluids, 11, 943-945, (1999) · Zbl 1147.76463 [56] A.A. Oberai, T.J.R. Hughes, The variational multiscale formulation of LES: channel flow at Reτ=590, in: 40th AIAA Ann. Mtg., Reno, NV, 2002, AIAA 2002-1056. [57] Piegl, L.; Tiller, W., The NURBS book (monographs in visual communication), (1997), Springer-Verlag New York [58] Pope, S.B., Turbulent flows, (2000), Cambridge University Press Cambridge · Zbl 0802.76033 [59] S. Ramakrishnan, S.S. Collis, Variational multiscale modeling for turbulence control, in: AIAA 1st Flow Control Conference, St. Louis, MO, June 2002, AIAA 2002-3280. [60] S. Ramakrishnan and S.S. Collis. Multiscale modeling for turbulence simulation in complex geometries, in: 40th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, Jan. 2004, AIAA 2004-0241. [61] Ramakrishnan, S.; Collis, S.S., Turbulence control simulation using the variational multiscale method, Aiaa j., 42, 4, 745-753, (2004) [62] Ramakrishnan, S.; Collis, S.S., Partition selection in multiscale turbulence modeling, Phys. fluids, 18, 7, (2006) · Zbl 1185.76728 [63] Rogers, D.F., An introduction to NURBS with historical perspective, (2001), Academic Press San Diego, CA [64] Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. stat. comput., 7, 856-869, (1986) · Zbl 0599.65018 [65] Sagaut, P.; Deck, S.; Terracol, M., Multiscale and multiresolution approaches in turbulence, (2006), Imperial College Press · Zbl 1107.76003 [66] G. Scovazzi. Multiscale Methods in Science and Engineering, PhD Thesis, Department of Mechanical Engineering, Stanford University. http://www.cs.sandia.gov/ gscovaz, 2004. [67] Shakib, F.; Hughes, T.J.R.; Johan, Z., A new finite element formulation for computational fluid dynamics: X. the compressible Euler and navier – stokes equations, Comp. methods appl. mech. engrg., 89, 141-219, (1991) · Zbl 0838.76040 [68] Shariff, K.; Moser, R.D., Two-dimensional mesh embedding for B-spline methods, J. comput. phys., 145, 471-488, (1998) · Zbl 0910.65083 [69] Sheffer, V., An inviscid flow with compact support in space-time, J. geom. anal., 3, 343-401, (1993) · Zbl 0836.76017 [70] Simo, J.C.; Hughes, T.J.R., Computational inelasticity, (1998), Springer-Verlag New York · Zbl 0934.74003 [71] Texas Advanced Computing Center (TACC). http://www.tacc.utexas.edu. [72] Tezduyar, T.E., Computation of moving boundaries and interfaces and stabilization parameters, Int. J. numer. methods fluids, 43, 555-575, (2003) · Zbl 1032.76605 [73] Vichnevetsky, R.; Bowles, J.B., Fourier analysis of numerical approximations of hyperbolic equations, (1982), Society for Industrial and Applied Mathematics Philadelphia · Zbl 0495.65041 [74] Whitham, G.B., Linear and nonlinear waves, (1974), John Wiley and Sons New York · Zbl 0373.76001 [75] C.H. Whiting, Stabilized Finite Element Methods for Fluid Dynamics using a Hierarchical Basis, PhD Thesis, Rensselaer Polytechnic Institute, September 1999. [76] Whiting, C.H.; Jansen, K.E., A stabilized finite element method for the incompressible navier – stokes equations using a hierarchical basis, Int. J. numer. methods fluids, 35, 93-116, (2001) · Zbl 0990.76048 [77] Zhang, Y.; Bazilevs, Y.; Goswami, S.; Bajaj, C.; Hughes, T.J.R., Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow, Comp. methods appl. mech. engrg., 196, 2943-2959, (2007) · Zbl 1121.76076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.