Abstract
This paper focuses on the notion of suitable weak solutions for the three-dimensional incompressible Navier–Stokes equations and discusses the relevance of this notion to Computational Fluid Dynamics. The purpose of the paper is twofold (i) to recall basic mathematical properties of the three-dimensional incompressible Navier-Stokes equations and to show how they relate to LES (ii) to introduce an entropy viscosity technique based on the notion of suitable weak solution and to illustrate numerically this concept.
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References
C. Bardos, A.Y. le Roux, and J.-C. Nédélec. First order quasilinear equations with boundary conditions. Comm. Partial Differential Equations, 4(9):1017–1034, 1979.
S. Bertoluzza. The discrete commutator property of approximation spaces. C. R. Acad. Sci. Paris, Sér. I, 329(12):1097–1102, 1999.
L. Caffarelli, R. Kohn, and L. Nirenberg. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math., 35(6):771–831, 1982.
A.F. Emery. An evaluation of several differencing methods for inviscid fluid flow problems. J. Computational Phys., 2:306–331, 1968.
S. Gottlieb, C.-W. Shu, and E. Tadmor. Strong stability-preserving high-order time discretization methods. SIAM Rev., 43(1):89–112 (electronic), 2001.
J.-L. Guermond, and S. Prudhomme. On the construction of suitable solutions to the navier–stokes equations and questions regarding the definition of large eddy simulation. Physica D, 207:64–78, 2005.
J.-L. Guermond. Finite-element-based Faedo-Galerkin weak solutions to the Navier-Stokes equations in the three-dimensional torus are suitable. J. Math. Pures Appl. (9), 85(3):451–464, 2006.
J.-L. Guermond. Faedo-Galerkin weak solutions of the Navier–Stokes equations with Dirichlet boundary conditions are suitable. J. Math. Pures Appl., 88:87–106, 2007.
J.-L. Guermond. On the use of the notion of suitable weak solutions in CFD. Int. J. Numer. Methods Fluids, 57:1153–1170, 2008.
J.-L. Guermond and R. Pasquetti. Entropy-based nonlinear viscosity for fourier approximations of conservation laws. C. R. Math. Acad. Sci. Paris, 346:801–806, 2008.
J.-L. Guermond and R. Pasquetti. Entropy viscosity method for high-order approximations of conservation laws. In E. Ronquist, editor, ICOSAHOM09, Lecture Notes in computational Science and Engineering, Berlin, 2010. Springer-Verlag.
J. Hoffman and C. Johnson. Computational turbulent incompressible flow, volume 4 of Applied Mathematics: Body and Soul. Springer, Berlin, 2007.
E. Hopf. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr., 4:213–231, 1951.
C. Johnson and A. Szepessy. On the convergence of a finite element method for a nonlinear hyperbolic conservation law. Math. Comp., 49(180):427–444, 1987.
C. Johnson, A. Szepessy, and P. Hansbo. On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws. Math. Comp., 54(189):107–129, 1990.
S. N. Kružkov. First order quasilinear equations with several independent variables. Mat. Sb. (N.S.), 81 (123):228–255, 1970.
A. Kurganov, G. Petrova, and B. Popov. Adaptive semidiscrete central-upwind schemes for nonconvex hyperbolic conservation laws. SIAM J. Sci. Comput., 29(6):2381–2401 (electronic), 2007.
J. Leray. Essai sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math., 63:193–248, 1934.
R. J. LeVeque. Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002.
F. Lin. A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math., 51(3):241–257, 1998.
R. Liska and B. Wendroff. Comparison of several difference schemes on 1D and 2D test problems for the Euler equations. SIAM J. Sci. Comput., 25(3):995–1017 (electronic), 2003.
V. Scheffer. Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys., 55(2):97–112, 1977.
P. Woodward and P. Colella. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys., 54(1):115–173, 1984.
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Guermond, JL., Pasquetti, R., Popov, B. (2011). From suitable weak solutions to entropy viscosity. In: Salvetti, M., Geurts, B., Meyers, J., Sagaut, P. (eds) Quality and Reliability of Large-Eddy Simulations II. ERCOFTAC Series, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0231-8_34
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DOI: https://doi.org/10.1007/978-94-007-0231-8_34
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-0230-1
Online ISBN: 978-94-007-0231-8
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