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 ≈ 1708 has been calculated up to Ra ≈ 5.106 and its Nusselt number is Nu ~ 0.143 Ra0.28 with a delicate spiral structure in the temperature field. Another solution that maximizes Nu over the horizontal wavenumber has been calculated up to Ra = 109 and scales as Nu ~ 0.115 Ra0.31 for 107 < Ra ≤ 109, quite similar to 3D turbulent data that show Nu ~ 0.105 Ra0.31 in that range. The optimum solution is a simple yet multi-scale coherent solution whose horizontal wavenumber scales as 0.133 Ra0.217. That 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.