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A model for two coupled turbulent fluids Part III: Numerical approximation by finite elements

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This paper introduces a scheme for the numerical solution of a model for two turbulent flows with coupling at an interface. We consider a variational formulation of the coupled model, where the turbulent kinetic energy equation is formulated by transposition. We prove the convergence of the approximation to this formulation for 2D flows by piecewise affine triangular elements. Our main contribution is to prove that the standard Galerkin - finite element approximation of the Laplace equation approximates in L2 norm its solution by transposition, for data with low smoothness. We include some numerical tests for simple geometries that exhibit the behaviour predicted by our analysis.

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References

  1. Bernardi, C., Chacón Rebollo, T., Murat, F., Lewandowsi, R.: Existence d’une solution pour un modèle de deux fluides turbulents couplés. C.R. Acad. Sc. Paris, Série I 328, 993–998 (1999)

    Google Scholar 

  2. Bernardi, C., Chacón Rebollo, T., Murat, F., Lewandowski, R.: A model for two coupled turbulent fluids. Part I : analysis of the system. In: Studies in Mathematics and its Applications, Vol. 31, D. Cioranescu and J.L. Lions, (eds.), Elsevier Science BV, 2002, pp. 69–102

  3. Bernardi, C., Chacón Rebollo, T., Murat, F., Lewandowski, R.: A model for two coupled turbulent fluids. Part II: numerical analysis of a spectral discretization. SIAM J. Numer. Anal. Vol 4, no. 6, 2368–2394 (2003)

  4. Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)

    MathSciNet  Google Scholar 

  5. Bramble, J.: A second order finite-difference analog of the first biharmonic boundary value problem. Numer. Math. 9, 236–249 (1966)

    Google Scholar 

  6. Ciarlet, Ph.: The Finite Element Method for Elliptic Problems. North-Holland, 1979

  7. Chacón Rebollo, T.: A term by term stabilization algorithm for finite element solution of incompressible flow problems. Numer. Math. 79, 283–319 (1998)

    Article  Google Scholar 

  8. Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics 1341, Springer-Verlag, 1988

  9. Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comp. 34, 441–463 (1980)

    MathSciNet  Google Scholar 

  10. Gómez Mármol, M., Ortegón Gallego, F.: Existence of solution to nonlinear elliptic systems arising in turbulence modelling. Math. Models and Methods in Appl. Sci. 10, 247–260 (2000)

    Article  MathSciNet  Google Scholar 

  11. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, 1985

  12. Lewandowski, R.: Analyse mathématique et océanographie. Collection Recherches en Mathématiques Appliquées. Masson, 1997

  13. Lewandowski, R.: The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity. Nonlinear Analysis TMA 28, 393–417 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier Villars, 1969

  15. Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications. Vol. 1. Dunod, 1968

  16. Lions, P.-L., Murat, F.: Solutions renormalisées d’équations elliptiques. To appear

  17. Mohammadi, B., Pironneau, O.: Analysis of the K-Epsilon Turbulence Model. Wiley-Masson, 1993

  18. Stampacchia, G.: Équations elliptiques du second ordre à coefficients discontinus. Presses de l’Université de Montréal, 1965

  19. Wilcox, D.C.: Turbulence Modelling for CFD. DCW Industries, 1993

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Correspondence to T. Chacón Rebollo.

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Mathematics Subject Classification (2000): 65 N30, 76M10

Revised version received March 24, 2003

This research was partially supported by Spanish Government REN2000-1162-C02-01 and REN2000-1168-C02-01 grants

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Bernardi, C., Rebollo, T., Mármol, M. et al. A model for two coupled turbulent fluids Part III: Numerical approximation by finite elements. Numer. Math. 98, 33–66 (2004). https://doi.org/10.1007/s00211-003-0490-9

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