Scaling in thermal convection: a unifying theory

Abstract

A systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh–Bénard 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 Ra versus Prandtl number Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively, and by whether the thermal or the kinetic boundary layer is thicker. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra [lsim ] 10¹¹) the leading terms are Nu ∼ Ra^1/4Pr^1/8, Re ∼ Ra^1/2Pr^−3/4 for Pr [lsim ] 1 and Nu ∼ Ra^1/4Pr^−1/12, Re ∼ Ra^1/2Pr^−5/6 for Pr [gsim ] 1. In most measurements these laws are modified by additive corrections from the neighbouring regimes so that the impression of a slightly larger (effective) Nu vs. Ra scaling exponent can arise. The most important of the neighbouring regimes towards large Ra are a regime with scaling Nu ∼ Ra^1/2Pr^1/2, Re ∼ Ra^1/2Pr^−1/2 for medium Pr (‘Kraichnan regime’), a regime with scaling Nu ∼ Ra^1/5Pr^1/5, Re ∼ Ra^2/5Pr^−3/5 for small Pr, a regime with Nu ∼ Ra^1/3, Re ∼ Ra^4/9Pr^−2/3 for larger Pr, and a regime with scaling Nu ∼ Ra^3/7Pr^−1/7, Re ∼ Ra^4/7Pr^−6/7 for even larger Pr. In particular, a linear combination of the ¼ and the 1/3 power laws for Nu with Ra, Nu = 0.27Ra^1/4 + 0.038Ra^1/3 (the prefactors follow from experiment), mimics a 2/7 power-law exponent in a regime as large as ten decades. For very large Ra the laminar shear boundary layer is speculated to break down through the non-normal-nonlinear transition to turbulence and another regime emerges. The theory presented is best summarized in the phase diagram figure 2 and in table 2.