Unsteady convectively driven flow along a vertical plate immersed in a stably stratified fluid

This paper revisits the classical problem of convectively driven one-dimensional (parallel) flow along an infinite vertical plate. We consider flows induced by an impulsive (step) change in plate temperature, a sudden application of a plate heat flux, and arbitrary temporal variations in plate temperature or plate heat flux. Provision is made for pressure work and vertical temperature advection in the thermodynamic energy equation, processes that are generally neglected in previous one-dimensional studies of this problem. The pressure work term by itself provides a relatively minor refinement of the Boussinesq model, but can be conveniently combined with the vertical temperature advection term to form a single term for potential temperature advection. Vertical motion of air in a statically stable environment (stable potential temperature stratification) is associated with a simple negative feedback mechanism: warm air rises, expands and cools relative to the environment, whereas cool air subsides, compresses and warms relative to the environment. Exact solutions of the viscous equations of motion are obtained by the method of Laplace transforms for the case where the Prandtl number is unity. Pressure work and vertical temperature advection are found to have a significant impact on the structure of the solutions at later times.