On unsteady free convection in vertical slots due to prescribed fluxes of heat or mass at the vertical walls

Abstract

Unsteady convection of an initially homogeneous fluid in a vertical slot is investigated theoretically in the limit of large Rayleigh and Prandtl/Schmidt numbers. The motion is driven by prescribed fluxes of heat or mass at the vertical walls of the slot. The ‘heat-up’ problem is considered, i.e. the fluxes are specified to change instantaneously from zero to finite constant values. Perturbation methods are used to compute approximate solutions for the initial period and for the slow approach to the asymptotic state. Numerical solutions of the full problem are also given. It is shown that a significant stratification is set up after short time and that the system thereafter evolves as a strongly stratified fluid on a timescale that is proportional to $Ra^{\frac{2}{9}}$ . During the latter part of the process, linear buoyancy layers of thickness $\sim Ra^{\frac{2}{9}}$ appear on the vertical walls. On the horizontal walls, there are nonlinear boundary layers of thickness $\sim Ra^{-\frac{1}{9}}$ , whose structure is akin to that of a Stewartson E ¼ layer. The theoretical predictions are found to be in good agreement with experimental results.