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Viscous Stabilizations for High Order Approximations of Saint-Venant and Boussinesq Flows

  • Richard PasquettiEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)

Abstract

Two viscous stabilization methods, namely the spectral vanishing viscosity (SVV) technique and the entropy viscosity method (EVM), are applied to flows of interest in geophysics. First, following a study restricted to one space dimension, the spectral element approximation of the shallow water equations is stabilized using the EVM. Our recent advances are here carefully described. Second, the SVV technique is used for the large-eddy simulation of the spatial and temporal development of the turbulent wake of a sphere in a stratified fluid. We conclude with a parallel between these two stabilization techniques.

Notes

Acknowledgements

Part of this work was made at the Dpt of Mathematics of National Taiwan University in the frame of the Inria project AMOSS.

References

  1. 1.
    C. Berthon, F. Marche, A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes. SIAM J. Sci. Comput. 30, 2587–2612 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    A. Bonito, J.L. Guermond, B. Popov, Stability analysis of explicit entropy viscosity methods for non-linear scalar conservation equations. Math. Comput. 83, 1039–1062 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    J.P. Chollet, M. Lesieur, Parametrisation of small scales of three-dimensional isotropic turbulence utilizing spectral closures. J. Atmos. Sci. 38, 2747–2757 (1981)CrossRefGoogle Scholar
  4. 4.
    P.J. Diamessis, J.A. Domaradzki, J.S. Hesthaven, A spectral multidomain penalty method model for the simulation of high Reynolds number localized incompressible stratified turbulence. J. Comput. Phys. 202, 298–322 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    D.G. Dommermuth, J.W. Rottman, G.E. Innis, E.V. Novikov, Numerical simulation of the wake of a towed sphere in a weakly stratified fluid. J. Fluid Mech. 473, 83–101 (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    J.-L. Guermond, A. Larios, T. Thompson, Validation of an entropy-viscosity model for large eddy simulation. Direct and Large Eddy-Eddy Simulation IX, ECOFTAC Series 20, 43–48 (2015)Google Scholar
  7. 7.
    J.-L. Guermond, R. Pasquetti, Entropy-based nonlinear viscosity for Fourier approximations of conservation laws. C.R. Acad. Sci. Paris Ser. I 346, 801–806 (2008)Google Scholar
  8. 8.
    J.L. Guermond, B. Popov, Viscous regularization of the Euler equations and entropy principles. SIAM J. Appl. Math. 74(2), 284–305 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    J.L. Guermond, R. Pasquetti, B. Popov, Entropy viscosity method for non-linear conservation laws. J. Comput. Phys. 230(11), 4248–4267 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    G.S. Karamanos, G.E. Karniadakis, A spectral vanishing viscosity method for large-eddy simulation. J. Comput. Phys. 163, 22–50 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    R.M. Kirby, S.J. Sherwin, Stabilisation of spectral / hp element methods through spectral vanishing viscosity: application to fluid mechanics. Comput. Methods Appl. Mech. Eng. 195, 3128–3144 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    K. Koal, J. Stiller, H.M. Blackburn, Adapting the spectral vanishing viscosity method for large-eddy simulations in cylindrical configurations. J. Comput. Phys. 231, 3389–3405 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Y. Maday, S.M.O. Kaber, E. Tadmor, Legendre pseudo-spectral viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 30, 321–342 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    R.C. Moura, S.J. Sherwin, J. Peiró, Eigensolution analysis of spectral/hp continuous Galerkin approximations to advection-diffusion problems: insights into spectral vanishing viscosity. J. Comput. Phys. 307, 401–422 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    R. Pasquetti, Spectral vanishing viscosity method for large-eddy simulation of turbulent flows. J. Sci. Comput. 27, 365–375 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    R. Pasquetti, Temporal/spatial simulation of the stratified far wake of a sphere. Comput. Fluids 40, 179–187 (2010)CrossRefzbMATHGoogle Scholar
  17. 17.
    R. Pasquetti, E. Séverac, E. Serre, P. Bontoux, M. Schäfer, From stratified wakes to rotor-stator flows by an SVV-LES method, Theor. Comput. Fluid Dyn. 22, 261–273 (2008)CrossRefzbMATHGoogle Scholar
  18. 18.
    R. Pasquetti, R. Bwemba, L. Cousin, A pseudo-penalization method for high Reynolds number unsteady flows. Appl. Numer. Math. 58(7), 946–954 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    R. Pasquetti, J.L. Guermond, B. Popov, Stabilized spectral element approximation of the Saint-Venant system using the entropy viscosity technique, in Lecture Notes in computational Science and Engineering: Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2014, vol. 106 (Springer, Berlin, 2015), pp. 397–404zbMATHGoogle Scholar
  20. 20.
    P. Sagaut, Large Eddy Simulation for Incompressible Flows (Springer, Berlin, Heidelberg, 2006)zbMATHGoogle Scholar
  21. 21.
    G.R. Spedding, F.K. Browand, A.M. Fincham, The evolution of initially turbulent bluff-body wakes at high internal Froude number. J. Fluid Mech. 337, 283–301 (1997)CrossRefGoogle Scholar
  22. 22.
    E. Tadmor, Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal. 26, 30–44 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    W.C. Thacker, Some exact solutions to the nonlinear shallow-water wave equations, J. Fluid Mech. 107, 499–508 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Y. Xing, X. Zhang, Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water equations on unstructured triangular meshes. J. Sci. Comput. 57, 19–41 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    C.J. Xu, R. Pasquetti, Stabilized spectral element computations of high Reynolds number incompressible flows. J. Comput. Phys. 196, 680–704 (2004)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Université Côte d’AzurCNRS, Inria, LJADNiceFrance
  2. 2.Lab. J.A. Dieudonné (CASTOR project)Nice Cedex 2France

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