From steady to unsteady horizontal gradient-driven convection at high Prandtl number

A 2-D rectangular cavity subject to a horizontal temperature gradient imposed on its upper boundary induces a naturally convective flow, leading to two surprising results. First, at large Rayleigh numbers (Ra) two boundary layers appear driving the flow in such a way that heat is removed from the cold part of the fluid and released into the hot region of the fluid, generating a dynamical “heat pump”. Second, a narrow region of pulsating oscillatory flow appears in the vicinity of the boundary layer which is at the vertical wall adjoining the cold edge of the upper boundary. These pulsations occur when Ra exceeds a critical value, Rac, and are associated with large downward and upward vertical flows that appear simultaneously near this vertical wall. The rest of the cavity is principally a “dead” cold region with weak flows. This region is the heat source of the heat pump. It occupies the main part of the cavity which progressively gets colder while the horizontal thermal boundary layer progressively gets hotter with increasing Ra. We explain the physics of these phenomena via numerical calculations that employ a spectral Chebyshev method. It is shown, for the first time, that the curve of Rac versus the aspect ratio is non-monotonic. It is also shown that the fundamental frequency of the oscillations, ω, increases monotonically with the aspect ratio of the cavity that we have considered. Physical reasons are advanced to explain our observations.