Key Takeaway: The Peclet number (Pe) in Rayleigh-Bénard convection is a function of the relative strengths of various terms in the momentum equation, with buoyancy and viscous terms playing smaller roles in turbulent and viscous regimes, respectively.

Abstract

We derive a formula for the Peclet number (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 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 ReRa−0.14, where Re and Ra are Reynolds and Rayleigh numbers, respectively, and that the viscous dissipation rate ϵu = (U3/d)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.