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

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 snumericald 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 fD. Lohse and F. Toschi, “The ultimate state of thermal convection,” Phys. Rev. Lett. 90, 034502 s2003dg which are consistent with the Kraichnan theory fR. H. Kraichnan, “Turbulent thermal convection at arbitrary Prandtl number,” Phys. Fluids 5, 1374 s1962dg and the Grossmann–Lohse sGLd theory fS. Grossmann and D. Lohse, “Scaling in thermal convection: A unifying view,” J. Fluid Mech. 407, 27 s2000d; “Thermal convection for large Prandtl number,” Phys. Rev. Lett. 86, 3316 s2001d; “Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection,” Phys. Rev. E 66, 016305 s2002d; “Fluctuations in turbulent Rayleigh–Bénard convection: The role of plumes,” Phys. Fluids 16, 4462 s2004dg, which both predict Nu,Ra1/2 and Re,Ra1/2. However the Pr dependence within these two theories is different. Here we show that the numerical data are consistent with the GL theory Nu,Pr1/2, Re,Pr−1/2. For the thermal and kinetic dissipation rates we find eu / skD2L−2d,sRe Prd0.87 and eu / sn3L−4d,Re2.77, both near sbut not fully consistentd 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. f“Comment on ultimate state of thermal convection” sprivate communicationdg. © 2005 American Institute of Physics. fDOI: 10.1063/1.1884165g