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Boundary Layer Analysis for Navier-Slip Rayleigh–Bénard Convection: The Non-existence of an Ultimate State

  • G.-M. Gie
  • J. P. WhiteheadEmail author
Article
  • 43 Downloads

Abstract

We discuss the asymptotic behavior, at small viscosity and/or diffusivity, of the Rayleigh–Bénard convection problem governed by the Boussinesq equations. The velocity vector field and the temperature are supplemented respectively with the Navier friction boundary conditions and the fixed flux boundary condition in a 3D periodic channel domain. By explicitly constructing the boundary layer correctors, which approximate the difference between the viscous/diffusive solutions and the corresponding limit solution, we validate the asymptotic expansions, and prove the vanishing viscosity and diffusivity limit with the optimal rate of convergence. Correctors in this setting include higher order diffusive effects than considered previously and accurately account for the interplay between the viscous and thermal layers. The boundary layer correctors satisfy a linear evolution equation indicating that for these boundary conditions, there is no turbulence in the boundary layer. The impact of this fact on the existence of an ‘ultimate state’ of turbulent convection is discussed, particularly in light of recent upper bounds on the heat transport that indicate such a state may exist in this setting.

Keywords

Boundary layers Navier boundary conditions Rayleigh–Bénard convection 

Notes

Acknowledgements

We thank the anonymous referee for some insightful comments and corrections which improved the presentation of this result. G-MG is partially supported by the Research—RI Grant, Office of the Executive Vice President for Research and Innovation, University of Louisville, and the Victor A. Olorunsola Endowed Research Award for Young Scholars, College of Arts and Sciences, University of Louisville. This collaboration arose following participation in a Mathematics Research Communities workshop sponsored by the American Mathematical Society, and further extended through a visit of G-MG sponsored by the Department of Mathematics at Brigham Young University.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Ahlers, G., Grossmann, S., Lohse, D.: Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–537 (2009)ADSCrossRefGoogle Scholar
  2. 2.
    Veiga, H.B., Crispo, F.: Sharp inviscid limit results under Navier type boundary conditions. An \(L^p\) theory. J. Math. Fluid Mech. 12(3), 397–411 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Beirão da Veiga, H., Crispo, F.: A missed persistence property for the Euler equations, and its effect on inviscid limits. Nonlinearity 25(6), 1661–1669 (2011)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bénard, H.: Les Tourbillons cellulaires dans une nappe liquide. Revue génórale des Sciences pures et appliquées 11, 1261–1271 (1900)Google Scholar
  5. 5.
    Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)zbMATHGoogle Scholar
  6. 6.
    Cannon, J.R., Di Benedetto, E.: The initial value problem for the Boussinesq equations with data in \(L^{p}\). In: Approximation methods for Navier–Stokes problems (Proceedings of Symposia, University of Paderborn, Paderborn, 1979), vol. 771, Lecture Notes in Mathematics, pp. 129–144. Springer, Berlin (1980)Google Scholar
  7. 7.
    Doering, C.R., Constantin, P.: Variational bounds on energy dissipation in incompressible flows. III. Convection. Phys. Rev. E 53(6), 5957–5981 (1996)ADSCrossRefGoogle Scholar
  8. 8.
    Fantuzzi, G.: Bounds for Rayleigh–Bénard convection between free-slip boundaries with an imposed heat flux. J. Fluid Mech. 837, R5 (2018)ADSCrossRefGoogle Scholar
  9. 9.
    Gie, G.-M.: Asymptotic expansion of the Stokes solutions at small viscosity: the case of non-compatible initial data. Commun. Math. Sci. 12(2), 383–400 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gie, G.-M., Jung, C.-Y.: Vorticity layers of the 2D Navier–Stokes equations with a slip type boundary condition. Asymptot. Anal. 84(1–2), 17–33 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gie, G.-M., Kelliher, J.P.: Boundary layer analysis of the Navier–Stokes equations with generalized Navier boundary conditions. J. Differ. Equ. 253(6), 1862–1892 (2012)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Grossmann, S., Lohse, D.: Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56 (2000)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Grossmann, S., Lohse, D.: Thermal convection for large Prandtl number. Phys. Rev. Lett. 86, 3316–3319 (2001)ADSCrossRefGoogle Scholar
  14. 14.
    Grossmann, S., Lohse, D.: Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66, 016305 (2002)ADSCrossRefGoogle Scholar
  15. 15.
    Grossmann, S., Lohse, D.: Fluctuations in turbulent Rayleigh–Bénard convection: the role of plumes. Phys. Fluids 16(12), 4462–4472 (2004)ADSzbMATHCrossRefGoogle Scholar
  16. 16.
    Grossmann, S., Lohse, D.: Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23, 045108 (2011)ADSCrossRefGoogle Scholar
  17. 17.
    He, X., Funfschilling, D., Nobach, H., Bodenschatz, E., Ahlers, G.: Transition to the ultimate state of turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 108(2), 024502 (2012)ADSCrossRefGoogle Scholar
  18. 18.
    He, X., Funfschilling, D., Nobach, H., Bodenschatz, E., Ahlers, G.: Comment on “Effect of boundary layers asymmetry on heat transfer efficiency in turbulent Rayleigh–Bénard convection at very high Rayleigh numbers”. Phys. Rev. Lett. 110(199401) (2013)Google Scholar
  19. 19.
    He, X., Bodenschatz, E., Ahlers, G.: Azimuthal diffusion of the large-scale-circulation plane, and absence of significant non-boussinesq effects, in turbulent convection near the ultimate-state transition. J. Fluid Mech. 791, R3 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    He, X., van Gils, D.P.M., Bodenschatz, E., Ahlers, G., et al.: Logarithmic spatial variations and universal f\(^{-1}\) power spectra of temperature fluctuations in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 112(17), 174501 (2014)ADSGoogle Scholar
  21. 21.
    Hou, T.Y., Li, C.: Global well-posedness of the viscous Boussinesq equations. Discrete Contin. Dyn. Syst. 12(1), 1–12 (2005)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Weiwei, H., Kukavica, I., Ziane, M.: On the regularity for the Boussinesq equations in a bounded domain. J. Math. Phys. 54(8), 081507 (2013). 10ADSMathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Iftimie, D., Sueur, F.: Viscous boundary layers for the Navier–Stokes equations with the Navier slip conditions. Arch. Ration. Mech. Anal. 199(1), 145–175 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Iftimie, D., Planas, G.: Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions. Nonlinearity 19(4), 899–918 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Johnston, H., Doering, C.R.: Comparison of turbulent thermal convection between conditions of constant temperature and constant flux. Phys. Rev. Lett. 102(064501), 1–p4 (2009)Google Scholar
  26. 26.
    Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Seminar on Nonlinear Partial Differential Equations (Berkeley, California, 1983), vol. 2 of Mathematical Sciences Research Institute Publications, pp. 85–98. Springer, New York (1984)Google Scholar
  27. 27.
    Kelliher, J.P.: Navier–Stokes equations with Navier boundary conditions for a bounded domain in the plane. SIAM J. Math. Anal. 38(1), 210–232 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Kraichnan, R.H.: Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5, 1374–1389 (1962)ADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Lai, M.-J., Pan, R., Zhao, K.: Initial boundary value problem for two-dimensional viscous Boussinesq equations. Arch. Ration. Mech. Anal. 199(3), 739–760 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Lions, J.-L.: Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal. Lecture Notes in Mathematics, vol. 323. Springer, Berlin (1973)Google Scholar
  31. 31.
    Masmoudi, N., Rousset, F.: Uniform regularity for the Navier–Stokes equation with Navier boundary condition. Arch. Ration. Mech. Anal. 203(2), 529–575 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Otero, J., Wittenberg, R.W., Worthing, R.A., Doering, C.R.: Bounds on Rayleigh–Bénard convection with an imposed heat flux. J. Fluid Mech. 473, 191–199 (2002)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Petschel, K., Stellmach, S., Wilczek, M., Lülff, J., Hansen, U.: Dissipation layers in Rayleigh–Bénard convection: a unifying view. Phys. Rev. Lett. 110(11), 114502 (2013)ADSCrossRefGoogle Scholar
  34. 34.
    Rayleigh, L.: On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Philos. Mag. J. Sci. 32(192), 529–546 (1916)zbMATHCrossRefGoogle Scholar
  35. 35.
    Seis, C.: Laminar boundary layers in convective heat transport. Commun. Math. Phys. 324(3), 995–1031 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Skrbek, L., Urban, P.: Has the ultimate state of turbulent thermal convection been observed? J. Fluid Mech. 785, 270–282 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Spiegel, E.A.: Convection in stars. 1. Basic Boussinesq convection. Ann. Rev. Astron. Astrophys. 9, 323–352 (1971)ADSCrossRefGoogle Scholar
  38. 38.
    Stevens, R.J.A.M., van der Poel, E.P., Grossmann, S., Lohse, D.: The unifying theory of scaling in thermal convection: the updated prefactors. J. Fluid Mech. 730, 295–308 (2013)ADSzbMATHCrossRefGoogle Scholar
  39. 39.
    Urban, P., Hanzelka, P., Kralik, T., Musilova, V., Srnka, A., Skrbek, L.: Effect of boundary layers asymmetry on heat transfer efficiency in turbulent Rayleigh Bénard convection at very high Rayleigh numbers. Phys. Rev. Lett. 109, 154301 (2012)ADSCrossRefGoogle Scholar
  40. 40.
    Urban, P., Hanzelka, P., Kralik, T., Musilova, V., Srnka, A., Skrbek, L.: Urban et al reply. Phys. Rev. Lett. 110(199402), 1 (2013)Google Scholar
  41. 41.
    Urban, P., Musilová, V., Skrbek, L.: Efficiency of heat transfer in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 107, 014302 (2011)ADSCrossRefGoogle Scholar
  42. 42.
    van der Poel, E.P., Ostilla-Monico, R., Verzicco, R., Lohse, D.: Effect of velocity boundary conditions on the heat transfer and flow topology in two-dimensional Rayleigh–Bénard convection. Phys. Rev. E 90(013017) (2014)Google Scholar
  43. 43.
    Wang, L., Xin, Z., Zang, A.: Vanishing viscous limits for 3D Navier–Stokes equations with a Navier-slip boundary condition. J. Math. Fluid Mech. 14(4), 791–825 (2012)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Wang, X.: A note on long time behavior of solutions to the Boussinesq system at large Prandtl number. In: Nonlinear Partial Differential Equations and Related Analysis, Contemporary Mathematics, vol. 371, pp. 315–323. American Mathematical Society, Providence (2005)Google Scholar
  45. 45.
    Whitehead, J.P., Doering, C.R.: The ultimate regime of two-dimensional Rayleigh–Bénard convection with stress-free boundaries. Phys. Rev. Lett. 106, 244501 (2011)ADSCrossRefGoogle Scholar
  46. 46.
    Whitehead, J.P., Doering, C.R.: Rigid rigorous bounds on heat transport in a slippery container. J. Fluid Mech. 707, 241–259 (2012)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Whitehead, J.P., Wittenberg, R.: A rigorous bound on the vertical transport of heat in Rayleigh–Bénard convection at infinite Prandtl number with mixed thermal boundary conditions. J. Math. Phys. 55(093104) (2014)Google Scholar
  48. 48.
    Wittenberg, R.W.: Bounds on Rayleigh–Bénard convection with imperfectly conducting plates. J. Fluid Mech. 665, 158–198 (2010)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Xiao, Y., Xin, Z.: On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition. Commun. Pure Appl. Math. 60(7), 1027–1055 (2007)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LouisvilleLouisvilleUSA
  2. 2.275 TMCBBrigham Young UniversityProvoUSA

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