Scaling Laws in Rayleigh‐Bénard Convection

Abstract

The heat transfer scaling theories for Rayleigh‐Bénard convection (RBC) are reviewed and discussed for configurations with and without rotation and magnetic fields. Scaling laws are a useful tool in studying and characterizing geophysical flows as they provide a basis for extrapolation to extreme parameter regimes that remain unobtainable by current computational and experimental efforts. Specifically, power law scalings that relate the efficiency of the heat transport, as measured by the nondimensional Nusselt number Nu, to the thermal driving are pursued. Relations of the functional form Nu∝false(Rafalse/Racfalse)α are considered. Given the strongly stabilizing influences of rotation and magnetic fields, thermal driving is considered in the context of the supercriticality of the system given by the ratio of the Rayleigh number Ra, measuring the thermal forcing, to the critical Rac, above which convection occurs. Analytical predictions for the exponent α are presented for the regimes of convection, rotating convection, and magnetoconvection, and the scalings are benchmarked against available numerical and experimental results in the accessible regimes. The exponents indicate that the thermal bottleneck to heat transport occurs within the thermal boundary layers for nonrotating RBC and the turbulent interior for rotating RBC. For magnetoconvection, a single exponent of α=1 is obtained for all theories and no bottleneck is identified.

Connections between nonrotating, slowly rotating, and rapidly rotating turbulent convection transport scalings

“…Further, the RRL heat transport scaling Eq. (35c) is also consistent with asympotically reduced theory and diffusivity-free formulations [4,[89][90][91]. Recent studies, such as Plumley et al [92,93], suggest that it is possible to reach the RRL scalings Eq.…”

“…The Nu, Re , and Pe transport scalings will be formulated in terms of Eqs. (1)'s nondimensional control parameters, which are the Prandtl, Rayleigh, and Ekman numbers [4]. The Prandtl number describes the fluid's thermophysical properties,…”

“…Dimensional analysis can be used, independently, to solve for the exponents ζ and χ that yield diffusivity-free expressions for the characteristic transport parameters [4], yielding…”

“…Accurate parametrizations are ubiquituously sought for the turbulent transport properties of fluid dynamical systems. In buoyancy-driven convection systems, the heat and momentum transport properties are the main foci of such investigations [1][2][3][4]. These transport estimates are essential for understanding the possible behaviors of a given system and for extrapolating these behaviors to extreme industrial, geophysical and astrophysical settings that are difficult to simulate directly (e.g., Refs.…”

Characteristics of a precessing flow under the influence of a convecting temperature field in a spheroidal shell

“…A great advantage of such scalings is that they can be extrapolated to regimes that are (currently) unreachable by either laboratory or numerical experiments. We refer to the recent review by Plumley & Julien (2019) for an overview on this subject with a focus on plane layer convection.…”