Nonlinear high-wavenumber Bnard convection

Weakly nonlinear two-dimensional roll cells in Bnard convectionare examined in the limit as the wavenumber a of the roll cellsbecomes large. In this limit the second harmonic contributionsto the solution become negligible, and a flow develops wherethe fundamental vortex terms and the correction to the meanare determined simultaneously, rather than sequentially as inthe weakly nonlinear case. Extension of this structure to Rayleighnumbers O(a³) above the neutral curve is shown to be possible,with the resulting flow field having a form very similar tothat for strongly nonlinear vortices in a centripetally unstableflow. The flow in this strongly nonlinear regime consists ofa core region, and boundary layers of thickness O(a^–1)at the walls. The core region occupies most of the thicknessof the fluid layer and only mean terms and cos az terms playa role in determining the flow; in the boundary layer all harmonicsof the vortex motion are present. Numerical solutions of thewall layer equations are presented and it is also shown thatthe heat transfer across the layer is significantly greaterthan in the conduction state.