Scaling of unsteady natural convection boundary layers with a non-instantaneous initiation

A scaling analysis is presented for the transient boundary layer established on a vertical wall following non-instantaneous heating in the form of an imposed wall temperature which increases linearly up to a prescribed steady value over a prescribed time. The scaling analysis is verified by comparison with numerical solutions of the full equations of motion and energy. The analysis reveals that, if the period of temperature growth on the wall is sufficiently long, the boundary layer reaches a quasi-steady state before the temperature growth is completed. In this mode the thermal boundary layer at first grows in thickness and then contracts with increasing time, and the fluid acceleration also changes character. If the wall temperature growth period is sufficiently short, the boundary layer commences differently, but after the wall temperature growth is completed, the boundary layer develops as though the start up had been instantaneous. In both cases, the ultimate steady state is the same as if the start up had been instantaneous; however the different transient nature may have implications for the stability of the boundary layer or of the subsequent development of the flow in a cavity of which the wall is one boundary.