Natural convection in a stably stratified fluid along vertical plates and cylinders with temporally periodic surface temperature variations

This paper describes one-dimensional (parallel) laminar natural convection in a viscous stably stratified fluid owing to temporally periodic variations in the surface temperature of infinite vertical plates and circular cylinders. Analytical solutions of the one-dimensional (parallel) Boussinesq equations of motion and thermodynamic energy are obtained for the periodic regime for arbitrary values of ambient stratification, Prandtl number and forcing frequency. The solutions for plates and cylinders are qualitatively similar and show that (i) the flows are composed of two waves that decay exponentially with distance from the surface: a fast long wave and a slow short wave; (ii) for forcing frequencies greater than the natural frequency of the corresponding inviscid system, these two waves propagate away from the surface; and (iii) for forcing frequencies less than this natural frequency, the short wave propagates away from the surface while the long wave propagates toward the surface. This latter case provides an example of a flow for which the conventional radiation condition is not appropriate. The analytical results are complemented, for the plate problem, with three-dimensional numerical simulations of flows that start from rest and are suddenly subjected to a periodic thermal forcing at the plate. The numerical results depict the transient (start-up) stage of the flow and the approach to a periodic regime. These results confirm that the analytical solutions provide the appropriate description of the periodic regime.