Rayleigh and Prandtl number scaling in the bulk of Rayleigh–Bénard turbulence

Abstract

The Ra and Pr number scaling of the Nusselt number Nu, the Reynolds number Re, the temperature fluctuations, and the kinetic and thermal dissipation rates is studied for (numerical) homogeneous Rayleigh–Bénard turbulence, i.e., Rayleigh–Bénard turbulence with periodic boundary conditions in all directions and a volume forcing of the temperature field by a mean gradient. This system serves as model system for the bulk of Rayleigh–Bénard flow and therefore as model for the so-called “ultimate regime of thermal convection.” With respect to the Ra dependence of Nu and Re we confirm our earlier results [D. Lohse and F. Toschi, “The ultimate state of thermal convection,” Phys. Rev. Lett. 90, 034502 (2003)] which are consistent with the Kraichnan theory [R. H. Kraichnan, “Turbulent thermal convection at arbitrary Prandtl number,” Phys. Fluids 5, 1374 (1962)] and the Grossmann–Lohse (GL) theory [S. Grossmann and D. Lohse, “Scaling in thermal convection: A unifying view,” J. Fluid Mech. 407, 27 (2000); “Thermal convection for large Prandtl number,” Phys. Rev. Lett. 86, 3316 (2001); “Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection,” Phys. Rev. E 66, 016305 (2002); “Fluctuations in turbulent Rayleigh–Bénard convection: The role of plumes,” Phys. Fluids 16, 4462 (2004)], which both predict $\mathrm{Nu}\sim {\mathrm{Ra}}^{1∕2}$ and $\mathrm{Re}\sim {\mathrm{Ra}}^{1∕2}$ . However the Pr dependence within these two theories is different. Here we show that the numerical data are consistent with the GL theory $\mathrm{Nu}\sim {\mathrm{Pr}}^{1∕2}$ , $\mathrm{Re}\sim {\mathrm{Pr}}^{-1∕2}$ . For the thermal and kinetic dissipation rates we find ${ϵ}_{\theta }∕\left(\kappa {\Delta }^2{L}^{-2}\right)\sim {\left(\mathrm{Re}\mathrm{Pr}\right)}^{0.87}$ and ${ϵ}_u∕\left({\nu }^3{L}^{-4}\right)\sim {\mathrm{Re}}^{2.77}$ , both near (but not fully consistent) the bulk dominated behavior, whereas the temperature fluctuations do not depend on Ra and Pr. Finally, the dynamics of the heat transport is studied and put into the context of a recent theoretical finding by Doering et al. [“Comment on ultimate state of thermal convection” (private communication)].