Heat transport by coherent Rayleigh-Benard convection

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 approximate to 1708 has been calculated up to Ra approximate to 5.10(6) and its Nusselt number is Nu similar to 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 scales as Nu similar to 0.115 Ra-0.31 for 10(7) < Ra <= 10(9), quite similar to 3D turbulent data that show Nu similar to 0.105 Ra-0.31 in that range. The optimum solution is a simple yet multi-scale coherent solution whose horizontal wavenumber scales as 0.133 Ra-0.217. 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. (C) 2015 AIP Publishing LLC.