On the Heat Transfer in Rayleigh-Benard systems

TL;DR

Problem: determine Nu–Ra scaling in Rayleigh-Bénard convection amid competing theories. Approach: combine direct numerical simulations with a Bolgiano-based ansatz by introducing a local Bolgiano length and a boundary-layer–buoyancy coupling. Findings: DNS show $Nu \sim Ra^{2/7}$ independent of dimensionality; the thermal boundary-layer thickness $\lambda$ aligns with the Bolgiano dissipation scale $r_B$, and a z-dependent Bolgiano length explains near-boundary dynamics, predicting a Kraichnan asymptote at $Ra \sim 10^{11}$. Significance: provides a unified dynamical framework that reconciles prior models and clarifies when Bolgiano scaling governs heat transport in high-$Ra$ convection.

Abstract

In this paper we discuss some theoretical aspects concerning the scaling laws of the Nusselt number versus the Rayleigh number in a Rayleigh-Benard cell. We present a new set of numerical simulations and compare our findings against the predictions of existing models. We then propose a new theory which relies on the hypothesis of Bolgiano scaling. Our approach generalizes the one proposed by Kadanoff, Libchaber and coworkers and solves some of the inconsistencies raised in the recent literature.