Heat Transport by Coherent Rayleigh-Bénard Convection

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 $Nu\sim 0.143R{a}^{0.28}$ 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 $Nu\sim 0.115R{a}^{0.31}$ 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 $0.133R{a}^{0.217}$ 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.

arXiv comment7 pages, one column, 5 figures, Revtex 4-1. Added discussion and references for `ultimate regime' and Grossmann-Lohse scaling theory. Updated best upper bound results

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• Scaling and spatial intermittency of thermal dissipation in turbulent convection
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• Optimal heat transfer and optimal exit times
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• Energy dissipation rate limits for flow through rough channels and tidal flow across topography- Corrigendum
• Marginally-Stable Thermal Equilibria of Rayleigh-Bénard Convection
• On high Taylor number Taylor vortices
• Plane-layer Rayleigh-Bénard convection up to $Ra=10^{11}$: Near-wall fluctuations and role of initial conditions
• Bifurcation analysis of two-dimensional Rayleigh--Bénard convection using deflation