Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows

Jean-Luc Guermond; Serge Prudhomme

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

  • Volume: 37, Issue: 6, page 893-908
  • ISSN: 0764-583X

Abstract

top
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 perturbed Navier–Stokes equations and we show that, as the cutoff wavenumber goes to infinity, the solution of the model converges (up to subsequences) to a weak solution which is dissipative in the sense defined by Duchon and Robert (2000).

How to cite

top

Guermond, Jean-Luc, and Prudhomme, Serge. "Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.6 (2003): 893-908. <http://eudml.org/doc/245067>.

@article{Guermond2003,
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 perturbed Navier–Stokes equations and we show that, as the cutoff wavenumber goes to infinity, the solution of the model converges (up to subsequences) to a weak solution which is dissipative in the sense defined by Duchon and Robert (2000).},
author = {Guermond, Jean-Luc, Prudhomme, Serge},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Navier–Stokes equations; turbulence; large Eddy simulation; convergence; Fourier-Galerkin approximation; weak solution},
language = {eng},
number = {6},
pages = {893-908},
publisher = {EDP-Sciences},
title = {Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows},
url = {http://eudml.org/doc/245067},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Guermond, Jean-Luc
AU - Prudhomme, Serge
TI - Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 6
SP - 893
EP - 908
AB - 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 perturbed Navier–Stokes equations and we show that, as the cutoff wavenumber goes to infinity, the solution of the model converges (up to subsequences) to a weak solution which is dissipative in the sense defined by Duchon and Robert (2000).
LA - eng
KW - Navier–Stokes equations; turbulence; large Eddy simulation; convergence; Fourier-Galerkin approximation; weak solution
UR - http://eudml.org/doc/245067
ER -

References

top
  1. [1] N.A. Adams and S. Stolz, A subgrid-scale deconvolution approach for shock capturing. J. Comput. Phys. 178 (2002) 391–426. Zbl1139.76319
  2. [2] C. Basdevant, B. Legras, R. Sadourny and M. Béland, A study of barotropic model flows: intermittency, waves and predictability. J. Atmospheric Sci. 38 (1981) 2305–2326. 
  3. [3] H. Brezis, Analyse Fonctionnelle, Théorie et Applications. Masson, Paris (1983). Zbl0511.46001MR697382
  4. [4] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982) 771–831. Zbl0509.35067
  5. [5] G.-Q. Chen, Q. Du and E. Tadmor, Spectral viscosity approximations to multidimensionnal scalar conservation laws. Math. Comp. 61 (1993) 629–643. Zbl0799.35148
  6. [6] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, A connection between the Camassa-Holm equation and turbulent flows in channels and pipes. Phys. Fluids 11 (1999) 2343–2353. Zbl1147.76357
  7. [7] J.P. Chollet and M. Lesieur, Parametrization of small scales of three-dimensional isotropic turbulence utilizing spectral closures. J. Atmospheric Sci. 38 (1981) 2747–2757. 
  8. [8] G.-H. Cottet, D. Jiroveanu and B. Michaux, Vorticity dynamics and turbulence models for Large-Eddy Simulations. ESAIM: M2AN 37 (2003) 187–207. Zbl1044.35051
  9. [9] C.R. Doering and J.D. Gibbon, Applied analysis of the Navier–Stokes equations. Cambridge texts in applied mathematics, Cambridge University Press (1995). Zbl0838.76016
  10. [10] J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity 13 (2000) 249–255. Zbl1009.35062
  11. [11] 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. Zbl1094.76030MR2053583
  12. [12] S. Kaniel, On the initial value problem for an incompressible fluid with nonlinear viscosity. J. Math. Mech. 19 (1970) 681–706. Zbl0195.10801
  13. [13] G.-S. Karamanos and G.E. Karniadakis, A spectral vanishing viscosity method for large-eddy simulations. J. Comput. Phys. 163 (2000) 22–50. Zbl0984.76036
  14. [14] N.K.-R. Kevlahan and M. Farge, Vorticity filaments in two-dimensional turbulence: creation, stability and effect. J. Fluid Mech. 346 (1997) 49–76. Zbl0910.76026
  15. [15] R.H. Kraichnan, Eddy viscosity in two and three dimensions. J. Atmospheric Sci. 33 (1976) 1521–1536. 
  16. [16] 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. Zbl0202.37301
  17. [17] 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. Zbl0202.37802MR226907
  18. [18] E. Lamballais, O. Métais and M. Lesieur, Spectral-dynamic model for large-eddy simulations of turbulent rotating channel flow. Theoret. Comput. Fluid Dynamics 12 (1998) 149–177. Zbl0941.76044
  19. [19] A. Leonard, Energy cascade in Large-Eddy simulations of turbulent fluid flows. Adv. Geophys. 18 (1974) 237–248. 
  20. [20] J. Leray, Essai sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math. 63 (1934) 193–248. JFM60.0726.05
  21. [21] M. Lesieur and R. Roggalo, Large-eddy simulations of passive scalar diffusion in isotropic turbulence. Phys. Fluids A 1 (1989) 718–722. 
  22. [22] J.-L. Lions, Quelques résultats d’existence dans des équations aux dérivées partielles non linéaires. Bull. Soc. Math. France 87 (1959) 245–273. Zbl0147.07902
  23. [23] J.-L. Lions, Sur certaines équations paraboliques non linéaires. Bull. Soc. Math. France 93 (1965) 155–175. Zbl0132.10601
  24. [24] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Vol 1. Dunod, Paris (1969). Zbl0189.40603MR259693
  25. [25] Y. Maday, M. Ould Kaber and E. Tadmor, Legendre pseudospectral viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 30 (1993) 321–342. Zbl0774.65072
  26. [26] 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. 
  27. [27] V. Scheffer, Hausdorff measure and the Navier–Stokes equations. Comm. Math. Phys. 55 (1977) 97–112. Zbl0357.35071
  28. [28] V. Scheffer, Nearly one-dimensional singularities of solutions to the Navier-Stokes inequality. Comm. Math. Phys. 110 (1987) 525–551. Zbl0653.35072
  29. [29] J. Smagorinsky, General circulation experiments with the primitive equations, part i: the basic experiment. Monthly Wea. Rev. 91 (1963) 99–152. 
  30. [30] E. Tadmor, Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal. 26 (1989) 30–44. Zbl0667.65079

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.