Stability of lid-driven shallow cavity heated from below

Abstract

Hydrodynamic and thermal stability of combined thermal buoyancy and lid-driven shear flow in a shallow cavity is analyzed by means of linearized perturbation theory. The analysis considers a cavity heated from below and cooled at the upper moving lid. A numerical procedure, which has generality with respect to boundary conditions, Reynolds, and Prandtl numbers, is described for solution of the linearized model equations. A direct numerical integration (Runge-Kutta with Newton-Raphson) method is used to solve the differential conservation equations. This method gives an exact result for the classical Benard problem where the flow becomes unstable at a critical Rayleigh number, Ra c = 1707.76. The numerical results show the existence of two critical wave numbers depending on whether the dominant force driving the flow is due to buoyancy or shear. For Pr ⩽ 0.1 the instability is due to the buoyancy force for constant heat flux boundary conditions, while for Pr = 1 the instability is due to the shear force. Increasing the Reynolds number stabilizes the flow, and reducing the Prandtl number makes the flow more unstable.