Scaling of large-scale quantities in Rayleigh-B\'enard convection

TL;DR

This paper derives a formula for the Péclet number in Rayleigh-Bénard convection, validated with simulations, revealing how different forces dominate in turbulent and viscous regimes and quantifying the nonlinearity reduction due to wall effects.

Contribution

The paper introduces a new formula for the Péclet number based on force balance analysis, validated with numerical data, and quantifies the scaling of nonlinearity and viscous dissipation in convection.

Findings

Pressure gradient dominates acceleration in turbulence

Viscous and buoyancy forces balance in viscous regime

Nonlinearity decreases with wall effects, scaling as Re Ra^{-0.14}

Abstract

We derive a formula for the P\'eclet number ($\mathrm{Pe}$) by estimating the relative strengths of various terms of the momentum equation. Using direct numerical simulations in three dimensions we show that in the turbulent regime, the fluid acceleration is dominated by the pressure gradient, with relatively small contributions arising from the buoyancy and the viscous term, in the viscous regime, acceleration is very small due to a balance between the buoyancy and the viscous term. Our formula for $\mathrm{Pe}$ describes the past experiments and numerical data quite well. We also show that the ratio of the nonlinear term and the viscous term is $\mathrm{Re} \mathrm{Ra}^{-0.14}$, where $\mathrm{Re}$ and $\mathrm{Ra}$ are Reynolds and Rayleigh numbers respectively, and that the viscous dissipation rate $\epsilon_u = (U^3/d) \mathrm{Ra}^{-0.21}$, where $U$ is the root mean square velocity and $d$ is the distance between the two horizontal plates. The aforementioned decrease in nonlinearity compared to free turbulence arises due to the wall effects.