Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T20:45:21.247Z Has data issue: false hasContentIssue false
  • English
  • Français

Mathematical analysis of a spectral hyperviscosityLES model for the simulation of turbulent flows

Published online by Cambridge University Press:  15 November 2003

Jean-Luc Guermond  and
Serge Prudhomme
Jean-Luc Guermond
Affiliation:
LIMSI (CNRS-UPR 3152), BP 133, 91403, Orsay, France. [email protected]. ICES, formerly TICAM, The University of Texas at Austin, TX 78712, USA
Serge Prudhomme
Affiliation:
ICES, formerly TICAM, The University of Texas at Austin, TX 78712, USA On leave at Universidad de los Andes, Bogotá, Colombia. [email protected].
Get access

Abstract

This paper presents a model based on spectral hyperviscosity for the simulation of 3D turbulent incompressible flows. One particularity of this model is that the hyperviscosity is active only at the short velocity scales, a feature which is reminiscent of Large Eddy Simulation models. We propose a Fourier–Galerkin approximation of the perturbedNavier–Stokes equations and we show that, as the cutoff wavenumbergoes to infinity, the solution of the modelconverges (up to subsequences) to a weak solution which is dissipativein the sense defined by Duchon and Robert (2000).

Keywords

Navier–Stokes equationsturbulencelarge Eddy simulation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 6 , November 2003 , pp. 893 - 908
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adams, N.A. and Stolz, S., A subgrid-scale deconvolution approach for shock capturing. J. Comput. Phys. 178 (2002) 391–426 . CrossRef
Basdevant, C., Legras, B., Sadourny, R. and Béland, M., A study of barotropic model flows: intermittency, waves and predictability. J. Atmospheric Sci. 38 (1981) 2305–2326 . 2.0.CO;2>CrossRef
H. Brezis, Analyse Fonctionnelle, Théorie et Applications. Masson, Paris (1983).
Caffarelli, L., Kohn, R. and Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982) 771–831 . CrossRef
Chen, G.-Q., Du, Q. and Tadmor, E., Spectral viscosity approximations to multidimensionnal scalar conservation laws. Math. Comp. 61 (1993) 629–643 . CrossRef
Chen, S., Foias, C., Holm, D.D., Olson, E., Titi, E.S. and Wynne, S., A connection between the Camassa-Holm equation and turbulent flows in channels and pipes. Phys. Fluids 11 (1999) 2343–2353 . CrossRef
Chollet, J.P. and Lesieur, M., Parametrization of small scales of three-dimensional isotropic turbulence utilizing spectral closures. J. Atmospheric Sci. 38 (1981) 2747–2757 . 2.0.CO;2>CrossRef
Cottet, G.-H., Jiroveanu, D. and Michaux, B., Vorticity dynamics and turbulence models for Large-Eddy Simulations. ESAIM: M2AN 37 (2003) 187–207 . CrossRef
C.R. Doering and J.D. Gibbon, Applied analysis of the Navier–Stokes equations. Cambridge texts in applied mathematics, Cambridge University Press (1995).
Duchon, J. and Robert, R., Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity 13 (2000) 249–255 . CrossRef
J.-L. Guermond, J.T. Oden and S. Prudhomme, Mathematical perspectives on the Large Eddy Simulation models for turbulent flows. J. Math. Fluid Mech. (2003). In press.
Kaniel, S., On the initial value problem for an incompressible fluid with nonlinear viscosity. J. Math. Mech. 19 (1970) 681–706 .
Karamanos, G.-S. and Karniadakis, G.E., A spectral vanishing viscosity method for large-eddy simulations. J. Comput. Phys. 163 (2000) 22–50 . CrossRef
Kevlahan, N.K.-R. and Farge, M., Vorticity filaments in two-dimensional turbulence: creation, stability and effect. J. Fluid Mech. 346 (1997) 49–76 . CrossRef
Kraichnan, R.H., Eddy viscosity in two and three dimensions. J. Atmospheric Sci. 33 (1976) 1521–1536 . 2.0.CO;2>CrossRef
O.A. Ladyženskaja, Modification of the Navier–Stokes equations for large velocity gradients, in Seminars in Mathematics V.A. Stheklov Mathematical Institute, Vol. 7, Boundary value problems of mathematical physics and related aspects of function theory, Part II, O.A. Ladyženskaja Ed., New York, London (1970). Consultant Bureau.
O.A. Ladyženskaja, New equations for the description of motion of viscous incompressible fluds and solvability in the large of boundary value problems for them, in Proc. of the Stheklov Institute of Mathematics, number 102 (1967), Boundary value problems of mathematical physics, O.A. Ladyženskaja Ed., V, Providence, Rhode Island (1970). AMS.
Lamballais, E., Métais, O. and Lesieur, M., Spectral-dynamic model for large-eddy simulations of turbulent rotating channel flow. Theoret. Comput. Fluid Dynamics 12 (1998) 149–177 . CrossRef
Leonard, A., Energy cascade in Large-Eddy simulations of turbulent fluid flows. Adv. Geophys. 18 (1974) 237–248 . CrossRef
Leray, J., Essai sur le mouvement d'un fluide visqueux emplissant l'espace. Acta Math. 63 (1934) 193–248 . CrossRef
Lesieur, M. and Roggalo, R., Large-eddy simulations of passive scalar diffusion in isotropic turbulence. Phys. Fluids A 1 (1989) 718–722 . CrossRef
Lions, J.-L., Quelques résultats d'existence dans des équations aux dérivées partielles non linéaires. Bull. Soc. Math. France 87 (1959) 245–273 . CrossRef
Lions, J.-L., Sur certaines équations paraboliques non linéaires. Bull. Soc. Math. France 93 (1965) 155–175 . CrossRef
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Vol 1. Dunod, Paris (1969).
Maday, Y., Ould Kaber, M. and Tadmor, E., Legendre pseudospectral viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 30 (1993) 321–342 . CrossRef
D. McComb and A. Young, Explicit-scales projections of the partitioned non-linear term in direct numerical simulation of the Navier–Stokes equation, in Second Monte Verita Colloquium on Fundamental Problematic Issues in Fluid Turbulence, Ascona, March 23–27 (1998). Available on the Internet at http://xxx.soton.ac.uk/abs/physics/9806029.
Scheffer, V., Hausdorff measure and the Navier–Stokes equations. Comm. Math. Phys. 55 (1977) 97–112 . CrossRef
Scheffer, V., Nearly one-dimensional singularities of solutions to the Navier-Stokes inequality. Comm. Math. Phys. 110 (1987) 525–551 . CrossRef
Smagorinsky, J., General circulation experiments with the primitive equations, part i: the basic experiment. Monthly Wea. Rev. 91 (1963) 99–152 . 2.3.CO;2>CrossRef
Tadmor, E., Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal. 26 (1989) 30–44 . CrossRef