Pattern-forming fronts in Rayleigh-Bénard convection

We numerically investigate pattern-forming fronts in Rayleigh-Bénard convection across a range of Rayleigh numbers for 2D and 3D geometries. By focusing on the propagation of a chain

of convection rolls in a shallow fluid layer, we quantify the selected front velocity and wavenumber as functions of the reduced Rayleigh number. We find that the front velocity increases with a square-root scaling of the reduced Rayleigh number, with modest deviations at higher values of the Rayleigh number. Near the onset of convection, the wavenumber selected by the front increases linearly with the reduced Rayleigh number and transitions to a one-fourth scaling for larger values. The pattern-forming fronts exhibit two time scales: a fast time scale of new roll formation and a slower time scale of front advancement. The fronts undergo slow asymptotic relaxation to their final states with time near the threshold. We show that the selected wavenumber is sensitive to the details of the boundary conditions and the geometry of the convection domain. We compare with available experimental measurements and theoretical predictions where possible.