Abstract
We begin by recalling old times when the senior author and Jerry Marsden collaborated on research in mechanics in the 1970’s. We note that both our recent interests have been focused on turbulence, although with different approaches. The common theme is reliance on variational structure. We then get down to business and describe our approach—the variational multiscale formulation of LES. Application is made to turbulent two-dimensional equilibrium and three-dimensional non-equilibrium channel flows. Simple, constant-coefficient Smagorinsky-type eddy viscosities, without wall damping functions, are used to model the transfer of energy from small resolved scales to unresolved scales, an approach which is not viable within the traditional LES framework. Nevertheless, very good results are obtained.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abraham, R. and J. E. Marsden [1966], Foundations of Mechanics. W. A. Benjamin and Co., Reading, Massachusetts.
Chorin, A. J., T. J. R. Hughes, J. E. Marsden, and M. F. McCracken [1978], Product formulas and numerical algorithms, Communications on Pure and Applied Mathematics 31, 205–256.
Coleman, G., J. Kim, and A.-T. Le [1996], A Numerical Study of Threedimensional wall-Bounded flows, International Journal of Heat and Fluid Flow 17, 333–342.
Dubois, T., F. Jauberteau, and R. Temam [1993], Solution of the incompressible Navier-Stokes equations by the nonlinear Galerkin method, Journal of Scientific Computing 8, 167–194.
Dubois, T., F. Jauberteau, and R. Temam [1998], Incremental unknowns, multilevel methods and the numerical simulations of turbulence, Computer Methods in Applied Mechanics and Engineering 159, 123–189.
Dubois, T., F. Jauberteau, and R. Temam [1999], Dynamic Multilevel Methods and the Numerical Simulation of Turbulence. Cambridge University Press, Cambridge, U.K.
Germano, M., U. Piomelli, P. Moin, and W. Cabot [1991], A dynamic subgridscale eddy viscosity model, Physics of Fluids 3(7), 1760–1765.
Ghosal, S., T. Lund, P. Moin, and K. Akselvoll [1995], A dynamic localization model for large-Eddy simulation of turbulent flows, Journal of Fluid Mechanics 286, 229–255.
Hughes, T. J. R. [1995], Multiscale Phenomena: Green’s functions, The Dirichletto-Neumann formulation, subgrid scale models, bubbles, and the origins of stabilized methods, Computer Methods in Applied Mechanics and Engineering 127, 387–401.
Hughes, T. J. R. [2000], The Finite Element Method—Linear Static and Dynamic Finite Element Analysis. Dover Publications, Mineola, New York.
Hughes, T. J. R., G. R. Feijóo, L. Mazzei, and J.-B. Quincy [1998], The variational multiscale method—a paradigm for computational mechanics, Computer Methods in Applied Mechanics and Engineering 166(1–2), 3–24.
Hughes, T. J. R., T. Kato, and J. E. Marsden [1977], Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63, 273–294.
Hughes, T. J. R. and J. E. Marsden [1978], Classical elastodynamics as a symmetric hyperbolic system, Journal of Elasticity 8, 97–110.
Hughes, T. J. R., L. Mazzei, A. A. Oberai, and A. A. Wray [2001a], The multiscale formulation of large eddy simulation: decay of homogeneous isotropic turbulence, Physics of Fluids 13(2), 505–512.
Hughes, T. J. R., A. A. Oberai, and L. Mazzei [2001b], Large eddy simulation of turbulent channel flows by the variational multiscale method, Physics of Fluids 13(6), 1784–1799.
Hughes, T. J. R., G. Engel, L. Mazzei, and M. Larson [2000a], The Continuous Galerkin method is locally conservative, Journal of Computational Physics 163, 467–488.
Hughes, T. J. R., L. Mazzei, and K. Jansen [2000b], Large eddy simulation and the variational multiscale method, Computing and Visualization in Science 3, 47–59.
Kane, C., J. E. Marsden, M. Ortiz, and M. West [2000], Variational integrators and the newmark algorithm for conservative and dissipative mechanical systems, International Journal for Numerical Methods in Engineering 49, 1295–1325.
Kim, J., P. Moin, and R. Moser [1987], Turbulence statistics in fully developed channel flow at low Reynolds number, Journal of Fluid Mechanics 177, 133–166.
Lilly, D. [1992], A proposed modification of the Germano subgrid-scale closure method, Physics of Fluids A 4, 633–635.
Lopez, V. and R. Moser [1999]. Private communication, 1999.
Marsden, J. E. and T. J. R. Hughes [1983, 1994], Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs, New Jersey. Reprinted by Dover Publications, Mineola, New York, 1994.
Marsden, J. E., G. W. Patrick, and S. Shkoller [1998], Multisymplectic geometry, variational integrators, and nonlinear PDEs, Communications in Mathematical Physics 199, 351–395.
Moin, P. and J. Kim [1982], Numerical investigation of turbulent channel flow, Journal of Fluid Mechanics 118, 341–377.
Moser, R., J. Kim, and N. Mansour [1999], Direct numerical simulation of turbulent channel flows up to Reτ=590, Physics of Fluids 11, 943–945.
Moser, R., P. Moin, and A. Leonard [1983], A spectral numerical method for the Navier-Stokes equations with applications to Taylor-Couette flow, Journal of Computational Physics 52, 524–544.
Smagorinsky, J. [1963], General circulation experiments with the primitive equations, I: The basic experiment, Monthly Weather Review 91, 99–164.
Van Driest, E. [1956], On turbulent flow near a wall, Journal of the Aerospace Sciences 23, 1007–1011.
Editor information
Editors and Affiliations
Additional information
To Jerry Marsden on the occasion of his 60th birthday
Rights and permissions
Copyright information
© 2002 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Hughes, T.J.R., Oberai, A.A. (2002). The Variational Multiscale Formulation of LES with Application to Turbulent Channel Flows. In: Newton, P., Holmes, P., Weinstein, A. (eds) Geometry, Mechanics, and Dynamics. Springer, New York, NY. https://doi.org/10.1007/0-387-21791-6_7
Download citation
DOI: https://doi.org/10.1007/0-387-21791-6_7
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-95518-6
Online ISBN: 978-0-387-21791-8
eBook Packages: Springer Book Archive