Scaling of convective boundary layer flow induced by linear thermal forcing at Pr1

The transient convective boundary layer flow induced by heating an initially isothermal and stationary fluid with a linearly heated semi-infinite vertical plate is comprehensively investigated by a scaling analysis in the present study. The local temperature at the leading edge of the specified linear thermal profile can be any value that is no lower than the ambient fluid, which significantly differentiates the present study from previous similar investigations. Additionally, the convective flow associated with both Pr<1 and Pr>1 (where Pr is the Prandtl number) fluids is analyzed and quantified. The derived scaling relations demonstrate that the dynamics of the Pr<1 and Pr>1 fluids are fundamentally different and the two flows are described by different sets of scaling laws. The study also shows that the convective boundary layer flow first experiences an initial growth state, and it eventually transits to a fully steady state after the leading edge effect has been completely convected away. It is further revealed that unlike the extensively studied homogenously heating problem which is featured by a one-dimensional initial growth state and a two-dimensional fully developed state, the present flow is consistently two-dimensional if a nonzero background temperature stratification presents. The derived scaling relations are validated against the direct numerical simulation results, and a good agreement is obtained.