Marginally-Stable Thermal Equilibria of Rayleigh-B\'enard Convection

TL;DR

This paper introduces marginally-stable thermal equilibria (MSTE) as exact solutions to the Rayleigh-Bénard convection problem, revealing new insights into the temperature profiles and Nusselt number scaling in convection systems.

Contribution

The study derives and solves a quasilinear form of Rayleigh-Bénard convection using marginally-stable eigenmodes, providing a novel analytical framework for understanding thermal equilibria.

Findings

Mean temperature profiles have thinner boundary layers and higher Nusselt numbers than nonlinear simulations.

Nusselt number scales as Nu ~ Ra^{1/3} in the MSTE.

Using MSTE as initial conditions accelerates thermal equilibrium in 2D simulations.

Abstract

Natural convection is ubiquitous throughout the physical sciences and engineering, yet many of its important properties remain elusive. To study convection in a novel context, we derive and solve a quasilinear form of the Rayleigh-B\'enard problem by representing the perturbations in terms of marginally-stable eigenmodes. The amplitude of each eigenmode is determined by requiring that the background state maintains marginal stability. The background temperature profile evolves due to the advective flux of every marginally-stable eigenmode, as well as diffusion. To ensure marginal stability and to obtain the eigenfunctions at every timestep, we perform a one-dimensional eigenvalue solve on each of the allowable wavenumbers. The background temperature field evolves to an equilibrium state, where the advective flux from the marginally-stable eigenmodes and the diffusive flux sum to a constant. These marginally-stable thermal equilibria (MSTE) are exact solutions of the quasilinear equations. The mean temperature profile has thinner boundary layers and larger Nusselt numbers than thermally-equilibrated 2D and 3D simulations of the full nonlinear equations. We find the Nusselt number scales like $\rm{Nu} \sim\rm{Ra}^{1/3}$. When an MSTE is used as initial conditions for a 2D simulation, we find that Nu quickly equilibrates without the burst of turbulence often induced by purely conductive initial conditions, but we also find that the kinetic energy is too large and viscously attenuates on a long viscous time scale. This is due to the thin temperature boundary layers which diffuse heat very effectively, thereby requiring high-velocity advective flows to reach an equilibrium.