Heat Transport by Coherent Rayleigh-Bénard Convection (1502.02292v2)
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 $Ra\approx 5. 106$ 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 $Ra=109$ and its heat flux scales as $Nu \sim 0.115\, Ra{0.31}$ for $107 < Ra \le 109$, 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.