Scaling in thermal convection: A unifying theory

Siegfried Grossmann; Detlef Lohse

doi:10.1017/S0022112099007545

paper

Scaling in thermal convection: A unifying theory

arXiv:chao-dyn/9909032 · doi:10.1017/S0022112099007545

Abstract

A systematic theory for the scaling of the Nusselt number $N u$ and of the Reynolds number $R e$ in strong Rayleigh-Benard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large scale convection roll (``wind of turbulence'') and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number versus Prandtl number phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra smaller than 10^{11}) the leading terms are $N u \sim R a^{1/4} P r^{1/8}$ , $R e \sim R a^{1/2} P r^{- 3/4}$ for $P r < 1$ and $N u \sim R a^{1/4} P r^{- 1/12}$ , $R e \sim R a^{1/2} P r^{- 5/6}$ for $P r > 1$ . In most measurements these laws are modified by additive corrections from the neighboring regimes so that the impression of a slightly larger (effective) Nu vs Ra scaling exponent can arise. -- The presented theory is best summarized in the phase diagram figure 1.

30 pages, latex, 7 figures, under review at Journal of Fluid Mech

Scaling in thermal convection: A unifying theory

Abstract

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