The flow near the end of a shallow laterally heated cavity enters a nonlinear convective regime when the Rayleigh number R, based on cavity height, is of the same order of magnitude as the aspect ratio L (length/height). In the case of thermally insulated horizontal boundaries the end-region solution determines a correction to the flow and temperature fields throughout the cavity. Numerical solutions are obtained for the end-region flow for several different Prandtl numbers and for a range of values of the scaled Rayleigh number RIL using a Dufort-Frankel multigrid method. The results are compared with asymptotic predictions of the motion in the conductive limit RIL-O and the boundary-layer limit RIL o.

No abstract

The temporal evolution of thermally driven flow in a shallow laterally heated cavity is studied for the nonlinear regime where the Rayleigh number R based on cavity height is of the same order of magnitude as the aspect ratio L (length/height). The horizontal surfaces of the cavity are assumed to be thermally insulating. For a certain class of initial conditions the evolution is found to occur over two non-dimensional timescales, of order one and of order L2. Analytical solutions for the motion throughout most of the cavity are found for each of these timescales and numerical solutions are obtained for the nonlinear time-dependent motion in end regions near each lateral wall. This provides a complete picture of the evolution of the steady-state flow in the cavity for cases where instability in the form of multicellular convection does not occur. The final steady state evolves on a dimensional timescale proportional to l2/κ, where l is the length of the cavity, κ is the thermal diffusivity of the fluid and the constant of proportionality depends on the ratio R/L.