On the use of the notion of suitable weak solutions in CFD

The notion of suitable weak solutions for the three-dimensional incompressible Navier-Stokes equations together with some standard regularization techniques for constructing these solutions is reviewed. The novel result presented in this paper is that Faedo-Galerkin weak solutions to the Navier-Stokes equations are suitable provided they are constructed using finite-dimensional spaces having a discrete commutator property and satisfying a proper inf-sup condition. Low-order mixed finite element spaces appear to be acceptable for this purpose. Connections between the notion of suitable solutions and LES modeling are investigated. A proposal for a large eddy scale model based on the notion of suitable solutions is made and numerically illustrated.

[1]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[2]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[3]  Jean-Luc Guermond,et al.  MATHEMATICAL ANALYSIS OF A SPECTRAL HYPERVISCOSITY LES MODEL FOR THE SIMULATION OF TURBULENT FLOWS , 2003 .

[4]  R. Temam.  Navier-Stokes Equations , 1977 .

[5]  Jean-Luc Guermond,et al.  Finite-element-based Faedo–Galerkin weak solutions to the Navier–Stokes equations in the three-dimensional torus are suitable , 2006 .

[6]  Darryl D. Holm,et al.  A connection between the Camassa–Holm equations and turbulent flows in channels and pipes , 1999, chao-dyn/9903033.

[7]  Hiroko Morimoto,et al.  On the Navier-Stokes initial value problem , 1974 .

[8]  Claes Johnson,et al.  Computational Turbulent Incompressible Flow: Applied Mathematics: Body and Soul 4 , 2007 .

[9]  Darryl D. Holm,et al.  On a Leray–α model of turbulence , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[10]  Jean-Luc Guermond,et al.  Faedo-Galerkin weak solutions of the Navier-Stokes equations with dirichlet boundary conditions are suitable , 2007 .

[11]  Claude Basdevant,et al.  A Study of Barotropic Model Flows: Intermittency, Waves and Predictability , 1981 .

[12]  Jean Leray,et al.  Sur le mouvement d'un liquide visqueux emplissant l'espace , 1934 .

[13]  Volker John,et al.  Analysis of Numerical Errors in Large Eddy Simulation , 2002, SIAM J. Numer. Anal..

[14]  B. Legras,et al.  A comparison of the contour surgery and pseudo-spectral methods , 1993 .

[15]  Darryl D. Holm,et al.  The Navier–Stokes-alpha model of fluid turbulence , 2001, nlin/0103037.

[16]  R. Temam.  Une méthode d'approximation de la solution des équations de Navier-Stokes , 1968 .

[17]  Winding, Fingering and Reconnection Mechanisms of Closely Interacting Vortex Tubes in Three Dimensions. , 1991 .

[18]  Silvia Bertoluzza,et al.  The discrete commutator property of approximation spaces , 1999 .

[19]  Johan Hoffman,et al.  Stability of the dual Navier-Stokes equations and efficient computation of mean output in turbulent flow using adaptive DNS/LES , 2006 .

[20]  J. Lions.  Sur certaines équations paraboliques non linéaires , 1965 .

[21]  Fanghua Lin,et al.  A new proof of the Caffarelli‐Kohn‐Nirenberg theorem , 1998 .

[22]  E. Hopf,et al.  Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Erhard Schmidt zu seinem 75. Geburtstag gewidmet , 1950 .

[23]  O. A. Ladyzhenskai︠a︡,et al.  Boundary value problems of mathematical physics and related aspects of function theory , 1970 .

[24]  Claes Johnson,et al.  On the convergence of a finite element method for a nonlinear hyperbolic conservation law , 1987 .

[25]  Vladimir Scheffer.  Hausdorff measure and the Navier-Stokes equations , 1977 .

[26]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[27]  Jean-Luc Guermond,et al.  Entropy-based nonlinear viscosity for Fourier approximations of conservation laws , 2008 .

[28]  J. Guermond,et al.  On the construction of suitable solutions to the Navier-Stokes equations and questions regarding the definition of large eddy simulation , 2005 .

[29]  Darryl D. Holm,et al.  The Three Dimensional Viscous Camassa–Holm Equations, and Their Relation to the Navier–Stokes Equations and Turbulence Theory , 2001, nlin/0103039.

[30]  Jean-Luc Guermond,et al.  Mathematical Perspectives on Large Eddy Simulation Models for Turbulent Flows , 2004 .

[31]  J. Smagorinsky,et al.  GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS , 1963 .

[32]  R. Kohn,et al.  Partial regularity of suitable weak solutions of the navier‐stokes equations , 1982 .

[33]  Juhani Pitkäranta,et al.  An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation , 1986 .

[34]  B. Geurts.  Elements of direct and large-eddy simulation , 2003 .

[35]  Raoul Robert,et al.  Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations , 2000 .

[36]  Darryl D. Holm,et al.  Regularization modeling for large-eddy simulation , 2002, nlin/0206026.

[37]  Peter Hansbo,et al.  On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws , 1990 .