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Axially homogeneous Rayleigh–Bénard convection in a cylindrical cell

  • Laura E. Schmidt (a1), Enrico Calzavarini (a2), Detlef Lohse (a1), Federico Toschi (a3) (a4) and Roberto Verzicco (a1) (a5)...
Abstract
Abstract

Previous numerical studies have shown that the ‘ultimate regime of thermal convection’ can be attained in a Rayleigh–Bénard cell when the kinetic and thermal boundary layers are eliminated by replacing both lateral and horizontal walls with periodic boundary conditions (homogeneous Rayleigh–Bénard convection). Then, the heat transfer scales like and turbulence intensity as , where the Rayleigh number indicates the strength of the driving force (for fixed values of , which is the ratio between kinematic viscosity and thermal diffusivity). However, experiments never operate in unbounded domains and it is important to understand how confinement might alter the approach to this ultimate regime. Here we consider homogeneous Rayleigh–Bénard convection in a laterally confined geometry – a small-aspect-ratio vertical cylindrical cell – and show evidence of the ultimate regime as is increased: in spite of the lateral confinement and the resulting kinetic boundary layers, we still find at . Further, it is shown that the system supports solutions composed of modes of exponentially growing vertical velocity and temperature fields, with as the critical parameter determining the properties of these modes. Counter-intuitively, in the low- regime, or for very narrow cylinders, the numerical simulations are susceptible to these solutions, which can dominate the dynamics and lead to very high and unsteady heat transfer. As is increased, interaction between modes stabilizes the system, evidenced by the increasing homogeneity and reduced fluctuations in the root-mean-square velocity and temperature fields. We also test that physical results become independent of the periodicity length of the cylinder, a purely numerical parameter, as the aspect ratio is increased.

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Corresponding author
Email address for correspondence: enrico.calzavarini@polytech-lille.fr
References
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1. Ahlers G., Grossmann S. & Lohse D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503.
2. Arakeri J. H., Avila F. E., Dada J. M. & Tovar R. O. 2000 Convection in a long vertical tube due to unstable stratification – a new type of turbulent flow? Curr. Sci. 79 (6), 859866.
3. Batchelor G. K. & Nitsche J. M. 1991 Instability of stationary unbounded stratified fluid. J. Fluid Mech. 227, 357391.
4. Batchelor G. K. & Nitsche J. M. 1993 Instability of stratified fluid in a vertical cylinder. J. Fluid Mech. 252, 419448.
5. Biferale L., Calzavarini E., Toschi F. & Tripiccione R. 2003 Universality of anisotropic fluctuations from numerical simulations of turbulent flows. Europhys. Lett. 64 (4), 461467.
6. Calzavarini E., Doering C. R., Gibbon J. D., Lohse D., Tanabe A. & Toschi F. 2006 Exponentially growing solutions of homogeneous Rayleigh–Bénard flow. Phys. Rev. E 73, R035301.
7. Calzavarini E., Lohse D., Toschi F. & Tripiccione R. 2005 Rayleigh and Prandtl number scaling in the bulk of Rayleigh–Bénard turbulence. Phys. Fluids 17, 055107.
8. Celani A., Mazzino A., Seminara A. & Tizzi M. 2007 Droplet condensation in two-dimensional Bolgiano turbulence. J. Turbul. 8, 19.
9. Chavanne X., Chilla F., Castaing B., Hebral B., Chabaud B. & Chaussy J. 1997 Observation of the ultimate regime in Rayleigh–Bénard convection. Phys. Rev. Lett. 79, 36483651.
10. Chavanne X., Chilla F., Chabaud B., Castaing B. & Hebral B. 2001 Turbulent Rayleigh–Bénard convection in gaseous and liquid He. Phys. Fluids 13, 13001320.
11. Cholemari M. & Arakeri J. 2005 Experiments and a model of turbulent exchange flow in a vertical pipe. Intl J. Heat Mass Transfer 48 (21–22), 44674473.
12. Cholemari M. & Arakeri J. 2009 Axially homogeneous, zero mean flow buoyancy-driven turbulence in a vertical pipe. J. Fluid Mech. 621, 69102.
13. Dubrulle B. 2001 Momentum transport and torque scaling in Taylor–Couette flow from an analogy with turbulent convection. Eur. Phys. J. B 21, 295.
14. Garaud P., Ogilvie G., Miller N. & Stellmach S. 2010 A model of the entropy flux and Reynolds stress in turbulent convection. Mon. Not. R. Astron. Soc. 407, 24512467.
15. Gibert M., Pabiou H., Chilla F. & Castaing B. 2006 High-Rayleigh-number convection in a vertical channel. Phys. Rev. Lett. 96, 084501.
16. Gibert M., Pabiou H., Tisserand J.-C., Gertjerenken B., Castaing B. & Chillà F. 2009 Heat convection in a vertical channel: plumes versus turbulence diffusion. Phys. Fluids 21, 035109.
17. van Gils D., Huisman S. G., Bruggert G. W., Sun C. & Lohse D. 2011 Torque scaling in turbulent Taylor–Couette flow with co- and counter-rotating cylinders. Phys. Rev. Lett. 106, 024502.
18. Grossmann S. & Lohse D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 2756.
19. Grossmann S. & Lohse D. 2001 Thermal convection for large Prandtl number. Phys. Rev. Lett. 86, 33163319.
20. Grossmann S. & Lohse D. 2002 Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66, 016305.
21. Grossmann S. & Lohse D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23, 045108.
22. Halesa L. 1937 Convection currents in geysers. Mon. Not. R. Astron. Soc. Geophys. Suppl. 4, 122.
23. Jones C. A. & Moore D. R. 1979 The stability of axisymmetric convection. Geophys. Astrophys. Fluid Dyn. 11, 245270.
24. Kadanoff L. P. 2001 Turbulent heat flow: structures and scaling. Phys. Today 54 (8), 3439.
25. Kim J. & Moin P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308323.
26. Kraichnan R. H. 1962 Turbulent thermal convection at arbritrary Prandtl number. Phys. Fluids 5, 13741389.
27. Landau L. D. & Lifshitz E. M. 1987 Fluid Mechanics. Pergamon.
28. Lohse D. & Toschi F. 2003 The ultimate state of thermal convection. Phys. Rev. Lett. 90, 034502.
29. Lohse D. & Xia K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335364.
30. Perrier F., Morat P. & LeMouel J. L. 2002 Dynamics of air avalanches in the access pit of an underground quarry. Phys. Rev. Lett. 89, 134501.
31. Siggia E. D. 1994 High Rayleigh number convection. Annu. Rev. Fluid Mech. 26, 137168.
32. Simitev R. D. & Busse F. H. 2010 Problems of astrophysical turbulent convection: thermal convection in a layer without boundaries. In Center for Turbulence Research, Proceedings of the Summer Program, 2010, Stanford University, CA (ed. Parviz Moin, Johan Larsson & Nagi Mansour), website where the proccedings can be found:http://www.stanford.edu/group/ctr/Summer/SP10/.
33. Spiegel E. A. 1971 Convection in stars. Annu. Rev. Astron. Astrophys. 9, 323352.
34. Taylor G. I. 1954 Diffusion and mass transport in tubes. Proc. Phys. Soc. B 67, 857869.
35. Tisserand J.-C., Creyssels M., Gibert M., Castaing D. & Chillà F. 2010 Convection in a vertical channel. New J. Phys. 12, 075024.
36. Verzicco R. & Camussi R. 2003 Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell. J. Fluid Mech. 477, 1949.
37. Verzicco R. & Orlandi P. 1996 A finite-difference scheme for three-dimensional incompressible flow in cylindrical coordinates. J. Comput. Phys. 123, 402413.
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