Heat Transport by Coherent Rayleigh-B\'enard Convection

TL;DR

This paper investigates steady solutions of 2D Rayleigh-Bénard convection, revealing how heat flux scales with Rayleigh number and identifying a coherent solution that may underpin turbulent convection.

Contribution

It provides new steady solutions for high Rayleigh numbers and links these solutions to turbulent convection characteristics.

Findings

Heat flux scales as Nu ~ Ra^{0.28} and Nu ~ Ra^{0.31} in different regimes.

A coherent solution with specific wavenumber scaling is identified as relevant to turbulence.

The optimal solution is unstable but may influence turbulent plume formation.

Abstract

Steady but generally unstable solutions of the 2D Boussinesq equations are obtained for no-slip boundary conditions and Prandtl number 7. The primary solution that bifurcates from the conduction state at Rayleigh number $Ra \approx 1708$ has been calculated up to $Ra\approx 5. 10^6$ and shows heat flux $Nu \sim 0.143\, Ra^{0.28}$ with a delicate spiral structure in the temperature field. Another solution that maximizes $Nu$ over the horizontal wavenumber has been calculated up to $Ra=10^9$ and its heat flux scales as $Nu \sim 0.115\, Ra^{0.31}$ for $10^7 < Ra \le 10^9$, quite similar to 3D turbulent data. The latter is a simple yet multi-scale coherent solution whose horizontal wavenumber scales as $0.133 \, Ra^{0.217}$ in that range. That optimum solution is unstable to larger scale perturbations and in particular to mean shear flows, yet it appears to be relevant as a backbone for turbulent solutions, possibly setting the scale, strength and spacing of elemental plumes.