Scaling of natural convection of an inclined flat plate : sudden cooling condition

The natural convection boundary layer adjacent to an inclined plate subject to sudden cooling boundary condition has been studied. It is found that the cold boundary layer adjacent to the plate is potentially unstable to Rayleigh-Bénard instability if the Rayleigh number exceeds a certain critical value. A scaling relation for the onset of instability of the boundary layer is achieved. The scaling relations have been developed by equating important terms of the governing equations based on the development of the boundary layer with time. The flow adjacent to the plate can be classified broadly into a conductive, a stable convective or an unstable convective regime determined by the Rayleigh number. Proper scales have been established to quantify the flow properties in each of these flow regimes. An appropriate identification of the time when the instability may set in is discussed. A numerical verification of the time for the onset of instability is also presented in this study. Different flow regimes based on the stability of the boundary layer have been discussed with numerical results.