Unsteady natural convection boundary-layer flow of a linearly-stratified fluid with image on an evenly heated semi-infinite vertical plate

The transient natural convection boundary-layer flow adjacent to a vertical plate heated with a uniform flux in an initially linearly-stratified ambient fluid with Prandtl number (Pr) smaller than one is investigated by scaling analysis and direct numerical simulation. The dominant parameters characterizing the flow behavior are the plate temperature, maximum vertical velocity, thermal boundary-layer thickness, whole and inner velocity boundary-layer thicknesses, and the corresponding time scales featuring these stages. Scaling laws relating these parameters to the flow governing parameters, that is Prandtl number and the dimensionless temperature stratification parameter have been obtained and validated against an exact solution and against a series of direct numerical simulations. It is shown that the scaling laws provide a good description of the flow behavior for start-up, transition and fully developed steady state. Both the scaling and numerical simulations show that the boundary layer is one dimensional away from the plate origin during start-up and at steady state, and two dimensional near the plate origin.