# 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 (2010)

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

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topGuermond, 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 37.6 (2010): 893-908. <http://eudml.org/doc/194197>.

@article{Guermond2010,

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},

keywords = {Navier–Stokes equations; turbulence; large Eddy simulation.; convergence; Fourier-Galerkin approximation; weak solution},

language = {eng},

month = {3},

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/194197},

volume = {37},

year = {2010},

}

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

DA - 2010/3//

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/194197

ER -

## References

top- N.A. Adams and S. Stolz, A subgrid-scale deconvolution approach for shock capturing. J. Comput. Phys.178 (2002) 391–426 . Zbl1139.76319
- 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 .
- H. Brezis, Analyse Fonctionnelle, Théorie et Applications. Masson, Paris (1983).
- 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
- G.-Q. Chen, Q. Du and E. Tadmor, Spectral viscosity approximations to multidimensionnal scalar conservation laws. Math. Comp.61 (1993) 629–643 . Zbl0799.35148
- 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. Fluids11 (1999) 2343–2353 . Zbl1147.76357
- 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 .
- G.-H. Cottet, D. Jiroveanu and B. Michaux, Vorticity dynamics and turbulence models for Large-Eddy Simulations. ESAIM: M2AN37 (2003) 187–207 . Zbl1044.35051
- 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
- J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity13 (2000) 249–255 . Zbl1009.35062
- 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. Zbl1094.76030
- S. Kaniel, On the initial value problem for an incompressible fluid with nonlinear viscosity. J. Math. Mech.19 (1970) 681–706 .
- 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
- 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
- R.H. Kraichnan, Eddy viscosity in two and three dimensions. J. Atmospheric Sci.33 (1976) 1521–1536 .
- 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.
- E. Lamballais, O. Métais and M. Lesieur, Spectral-dynamic model for large-eddy simulations of turbulent rotating channel flow. Theoret. Comput. Fluid Dynamics12 (1998) 149–177 . Zbl0941.76044
- A. Leonard, Energy cascade in Large-Eddy simulations of turbulent fluid flows. Adv. Geophys.18 (1974) 237–248 .
- J. Leray, Essai sur le mouvement d'un fluide visqueux emplissant l'espace. Acta Math.63 (1934) 193–248 .
- M. Lesieur and R. Roggalo, Large-eddy simulations of passive scalar diffusion in isotropic turbulence. Phys. Fluids A1 (1989) 718–722 .
- J.-L. Lions, Quelques résultats d'existence dans des équations aux dérivées partielles non linéaires. Bull. Soc. Math. France87 (1959) 245–273 . Zbl0147.07902
- J.-L. Lions, Sur certaines équations paraboliques non linéaires. Bull. Soc. Math. France93 (1965) 155–175 . Zbl0132.10601
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Vol 1. Dunod, Paris (1969). Zbl0189.40603
- 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
- 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.
- V. Scheffer, Hausdorff measure and the Navier–Stokes equations. Comm. Math. Phys.55 (1977) 97–112 . Zbl0357.35071
- V. Scheffer, Nearly one-dimensional singularities of solutions to the Navier-Stokes inequality. Comm. Math. Phys.110 (1987) 525–551 . Zbl0653.35072
- J. Smagorinsky, General circulation experiments with the primitive equations, part i: the basic experiment. Monthly Wea. Rev.91 (1963) 99–152 .
- E. Tadmor, Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal.26 (1989) 30–44 . Zbl0667.65079

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