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Analysis of Nonlinear Spectral Eddy-Viscosity Models of Turbulence

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Abstract

Fluid turbulence is commonly modeled by the Navier-Stokes equations with a large Reynolds number. However, direct numerical simulations are not possible in practice, so that turbulence modeling is introduced. We study artificial spectral viscosity models that render the simulation of turbulence tractable. We show that the models are well posed and have solutions that converge, in certain parameter limits, to solutions of the Navier-Stokes equations. We also show, using the mathematical analyses, how effective choices for the parameters appearing in the models can be made. Finally, we consider temporal discretizations of the models and investigate their stability.

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References

  1. Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. A Series of Comprehensive Studies in Mathematics, vol. 223. Springer, Berlin (1976)

    MATH  Google Scholar 

  2. Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  3. Constantin, P., Foias, C.: Navier-Stokes Equations. University of Chicago Press, Chicago (1988)

    MATH  Google Scholar 

  4. Chen, G.-Q., Du, Q., Tadmor, E.: Spectral viscosity approximations to multidimensional scalar conservation laws. Math. Comput. 61(204), 629–643 (1993)

    MATH  MathSciNet  Google Scholar 

  5. DiBenedetto, E.: Degenerate Parabolic Equations. Universitext. Springer, New York (1993)

    MATH  Google Scholar 

  6. Diening, L., Prohl, A., Ružička, M.: Semi-implicit Euler scheme for generalized Newtonian fluids. SIAM J. Numer. Anal. 44(3), 1172–1190 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Evans, L.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. Am. Math. Soc., Providence (1998)

    MATH  Google Scholar 

  8. Fefferman, C.: The multiplier problem for the ball. Ann. Math. (2) 94, 330–336 (1971)

    Article  MathSciNet  Google Scholar 

  9. Guermond, J., Prudhomme, S.: Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulence flows. M2AN Math. Model. Numer. Anal. 37(6), 893–908 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Jansen, K., Tejada-Martinez, A.: An evaluation of the variational multiscale model for large-eddy simulation while using a hierarchical basis. In: AIAA Annual Meeting and Exhibit, Paper No. 2002-0283 (2002)

  11. Karamanos, G., Karniadakis, G.: A spectral vanishing viscosity method for large-eddy simulations. J. Comput. Phys. 163, 22–50 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kraichnan, R.: Eddy-viscosity concept in spectral space. J. Atmos. Sci. 33, 1521–1536 (1976)

    Article  Google Scholar 

  13. Krantz, S.: A Panorama of Harmonic Analysis. Carus Mathematical Monographs, vol. 27. Math. Assoc. Am., Washington (1999)

    MATH  Google Scholar 

  14. Ladyzhenskaya, O.: Modifications of the Navier-Stokes equations for large velocity gradients. Boundary Value Problems of Mathematical Physics and Related Aspects of Function Theory. Consultants Bureau, New York (1970)

    Google Scholar 

  15. Layton, W.: A mathematical introduction to large eddy simulation. Technical report, University of Pittsburgh, TR-MATH 03-03 (2003)

  16. Leray, J.: Essay sur les mouvements plans d’une liquide visqueux que limitent des parois. J. Math. Pur. Appl., Paris Ser. IX(13), 331–418 (1934)

    Google Scholar 

  17. Lieb, E., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14. Am. Math. Soc., Providence (2001)

    MATH  Google Scholar 

  18. Lions, J.-L.: Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires. Dunod, Paris (1968)

    Google Scholar 

  19. Machihara, S., Ozawa, T.: Interpolation inequalities in Besov spaces. Proc. Am. Math. Soc. 131(5), 1553–1556 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Malek, J., Necas, J., Rokyta, M., Ruzicka, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Chapman & Hall, London (1996)

    MATH  Google Scholar 

  21. Prodi, G.: Un teorema di unicitá per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48(4), 173–182 (1959)

    MATH  MathSciNet  Google Scholar 

  22. Serrin, J.: The initial value problem for the Navier-Stokes equations. In: Nonlinear Problems Proc. Sympos., Madison, Wis, pp. 69–98. University of Wisconsin Press, Madison (1963)

    Google Scholar 

  23. Smagorinsky, J.: General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weather Rev. 91, 99–152 (1963)

    Article  Google Scholar 

  24. Tao, T.: Nonlinear Dispersive Equations: Local and Global Analysis. CBMS Regional Conference Series in Mathematics, vol. 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC (2006)

  25. Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1979)

    MATH  Google Scholar 

  26. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68. Springer, New York (1997)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Scientific Computing, Florida State University, Tallahassee, FL, 32306-4120, USA

    Max Gunzburger

  2. Department of Computational Science and Engineering, Yonsei University, Seoul, Korea

    Eunjung Lee

  3. Department of Mathematics, Florida State University, Tallahassee, FL, 32306-4510, USA

    Yuki Saka & Xiaoming Wang

  4. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Catalin Trenchea

Authors
  1. Max Gunzburger
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  2. Eunjung Lee
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  3. Yuki Saka
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  4. Catalin Trenchea
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  5. Xiaoming Wang
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Correspondence to Max Gunzburger.

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Dedicated to the memory of David Gottlieb.

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Gunzburger, M., Lee, E., Saka, Y. et al. Analysis of Nonlinear Spectral Eddy-Viscosity Models of Turbulence. J Sci Comput 45, 294–332 (2010). https://doi.org/10.1007/s10915-009-9335-8

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  • DOI: https://doi.org/10.1007/s10915-009-9335-8

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