Existence for a one-equation turbulent model with strong nonlinearities

Abstract

The purpose of this article is to improve the existence theory for the steady problem of an one-equation turbulent model. For this study, we consider a very general model that encompasses distinct situations of turbulent flows described by the k-epsilon model. Although the boundary-value problem we consider here is motivated by the modelling of turbulent flows through porous media, the importance of our results goes beyond this application. In particular, our results are suited for any turbulent flows described by the k-epsilon model whose mean flow equation incorporates a feedback term, as the Coriolis force, the Lorentz force or the Darcy–Forchheimer’s drag force. The consideration of feedback forces in the mean flow equation will affect the equation for the turbulent kinetic energy (TKE) with a new term that is known as the production and represents the rate at which TKE is transferred from the mean flow to the turbulence. For the associated boundary-value problem, we prove the existence of weak solutions by assuming that the feedback force and the turbulent dissipation are strong nonlinearities, i.e. when no upper restrictions on the growth of these functions with respect to the mean velocity and to the turbulent kinetic energy, respectively, are required. This result improves, in particular, the existence theory for the classical turbulent k-epsilon model which corresponds to assume that both the feedback force and the production term are absent in our model.

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References

  1. 1.

    Antohe, B.V., Lage, J.L.: A general two-equation macroscopic turbulence model for incompressible flow in porous media. Int. J. Heat Mass Transfer 40, 3013–3024 (1997)

    Article  MATH  Google Scholar 

  2. 2.

    Antontsev, S.N., Díaz, J.I., de Oliveira, H.B.: Stopping a viscous fluid by a feedback dissipative field: I. The stationary Stokes problem. J. Math. Fluid Mech. 6, 439–461 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Antontsev, S.N., Diaz, J.I., de Oliveira, H.B.: Stopping a viscous fluid by a feedback dissipative field: II. The stationary Navier-Stokes problem. Rend. Mat. Acc. Lincei 15, 257–270 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Bernis, F.: Elliptic and parabolic semilinear problems without conditions at infinity. Arch. Ration. Mech. Anal. 106, 217–241 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Boccardo, L., Gallouët, T.: Strongly nonlinear elliptic equations having natural growth terms and \(L^1\) data. Nonlinear Anal. 19(6), 573–579 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Brezis, H., Browder, F.E.: Strongly non-linear elliptic boundary value problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5, 587–603 (1978)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Chacón-Rebollo, T., Lewandowski, R.: Mathematical and numerical foundations of turbulence models and applications. Springer, New York (2014)

    Google Scholar 

  8. 8.

    Ciarlet, P.: The finite element method for elliptic problems. Elsevier, Amsterdam (1978)

    Google Scholar 

  9. 9.

    Dreyfuss, P.: Results for a turbulent system with unbounded viscosities: weak formulations, existence of solutions, boundedness and smoothness. Nonlinear Anal. 68, 1462–1478 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Druet, P.-É., Naumann, J.: On the existence of weak solutions to a stationary one-equation RANS model with unbounded eddy viscosities. Ann. Univ. Ferrara 55, 67–87 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Gallouët, T., Lederer, J., Lewandowski, R., Murat, F., Tartar, L.: On a turbulent system with unbounded eddy viscosities. Nonlinear Anal. 52, 1051–1068 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Landes, R.: On the existence of weak solutions of perturbated systems with critical growth. J. Reine Angew. Math. 393, 21–38 (1989)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Lederer, J., Lewandowski, R.: A RANS 3D model with unbounded eddy viscosities. Ann. Inst. H. Poincaré Anal. Non Linéaire 24, 413–441 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    de Lemos, M.J.S.: Turbulence in Porous Media, 2nd edn. Elsevier, Waltham (2012)

    Google Scholar 

  15. 15.

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

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

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

    Google Scholar 

  17. 17.

    Nakayama, A., Kuwahara, F.: A macroscopic turbulence model for flow in a porous medium. J. Fluid Eng. 121, 427–433 (1999)

    Article  Google Scholar 

  18. 18.

    Naumann, J.: Existence of weak solutions to the equations of stationary motion of heat-conducting incompressible viscous fluids. Progr. Nonlinear Differ. Equ. Appl. 64, 373–390 (2005)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Naumann, J., Wolf, J.: On Prandtl’s turbulence model: existence of weak solutions to the equations of stationary turbulent pipe-flow. Discrete Contin. Dyn. Syst. Ser. S 6(5), 1371–1390 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    H.B. de Oliveira and A. Paiva. On a one equation turbulent model with feedbacks. In Differential and Difference Equations with Applications, S. Pinelas et al. (eds.), Springer Proc. Math. Stat. 164 (2016), 51–61

  21. 21.

    H.B. de Oliveira and A. Paiva. A stationary turbulent one-equation model with applications in porous media. J. Math. Fluid Mech. First Online: 12 May 2017

  22. 22.

    Pedras, M.H.J., de Lemos, M.J.S.: On the definition of turbulent kinetic energy for flow in porous media. Int. Commun. Heat Mass Transfer 27(2), 211–220 (2000)

    Article  Google Scholar 

  23. 23.

    Rakotoson, J.-M.: Quasilinear elliptic problems with measures as data. Differ. Integral Equ. 4(3), 449–457 (1991)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Temam, R.: Navier-Stokes equations. Elsevier, Amsterdam (1979)

    Google Scholar 

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Correspondence to H. B. de Oliveira.

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Oliveira, H.B.d., Paiva, A. Existence for a one-equation turbulent model with strong nonlinearities. J Elliptic Parabol Equ 3, 65–91 (2017). https://doi.org/10.1007/s41808-017-0005-y

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Keywords

  • Turbulence
  • k-epsilon modelling
  • Strong nonlinearities
  • Existence

Mathematics Subject Classification

  • 76F60
  • 76S05
  • 35J57
  • 35D30
  • 76D03