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Geometry, Mechanics, and Dynamics pp 223–239Cite as

The Variational Multiscale Formulation of LES with Application to Turbulent Channel Flows

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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.

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References

  • Abraham, R. and J. E. Marsden [1966], Foundations of Mechanics. W. A. Benjamin and Co., Reading, Massachusetts.

    Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Dubois, T., F. Jauberteau, and R. Temam [1999], Dynamic Multilevel Methods and the Numerical Simulation of Turbulence. Cambridge University Press, Cambridge, U.K.

    Google Scholar 

  • Germano, M., U. Piomelli, P. Moin, and W. Cabot [1991], A dynamic subgridscale eddy viscosity model, Physics of Fluids 3(7), 1760–1765.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • Hughes, T. J. R. [2000], The Finite Element Method—Linear Static and Dynamic Finite Element Analysis. Dover Publications, Mineola, New York.

    Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Hughes, T. J. R. and J. E. Marsden [1978], Classical elastodynamics as a symmetric hyperbolic system, Journal of Elasticity 8, 97–110.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Lilly, D. [1992], A proposed modification of the Germano subgrid-scale closure method, Physics of Fluids A 4, 633–635.

    Article  Google Scholar 

  • Lopez, V. and R. Moser [1999]. Private communication, 1999.

    Google Scholar 

  • 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.

    Google Scholar 

  • Marsden, J. E., G. W. Patrick, and S. Shkoller [1998], Multisymplectic geometry, variational integrators, and nonlinear PDEs, Communications in Mathematical Physics 199, 351–395.

    Article  MathSciNet  Google Scholar 

  • Moin, P. and J. Kim [1982], Numerical investigation of turbulent channel flow, Journal of Fluid Mechanics 118, 341–377.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Smagorinsky, J. [1963], General circulation experiments with the primitive equations, I: The basic experiment, Monthly Weather Review 91, 99–164.

    Google Scholar 

  • Van Driest, E. [1956], On turbulent flow near a wall, Journal of the Aerospace Sciences 23, 1007–1011.

    MATH  Google Scholar 

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Authors
  1. Thomas J. R. Hughes
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  2. Assad A. Oberai
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Editor information

Editors and Affiliations

  1. Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA, 90089-1191, USA

    Paul Newton

  2. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089-1191, USA

    Paul Newton

  3. Department of Applied and Computational Mathematics Engineering Quadrangle, Princeton University, Princeton, NJ, 08544-1000, USA

    Philip Holmes

  4. Department of Mathematics, University of California, Berkeley, Berkeley, CA, 94720, USA

    Alan Weinstein

Additional information

To Jerry Marsden on the occasion of his 60th birthday

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© 2002 Springer-Verlag New York, Inc.

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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

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  • 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

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