Scaling analysis of buoyant boundary layer flow induced by flux heating with a spatial gradient at Pr ≪ 1, Pr ∼ 1 and Pr ≫ 1

Abstract

Buoyancy induced convective boundary layer flow is investigated with a scale law analysis in this study. Thermal forcing at the heated wall adopts the second-type boundary condition, and a spatial gradient is specified for the heat flux profile. A full range of Prandtl number is considered and separately analyzed, i.e. Pr << 1, Pr ∼ 1 and Pr >> 1, since the convective flow fundamentally differentiates for the three Pr ranges. Important scale laws quantifying different states of the transient flow are determined by the scaling analysis and they are validated against numerical calculations. The comparison suggests that the proposed scale laws could satisfactorily quantify the boundary layer flow and the corresponding regression constants R2 are around 0.999. The results indicate that scale laws for the three Pr ranges differ only in the Pr term and the Rayleigh number, streamwise location and heating parameter dependencies are identical. It is also demonstrated that, in the start-up state, the boundary layer growth is always two-dimensional at non-zero spatial gradient conditions and the heated wall temperature increases temporally according to τ1/2, which are profoundly different from the extensively investigated isothermal problems. The present work also identifies a special flow condition. That is, when the heat flux gradient equals 2, the heated wall temperature and characteristic velocity of the flow are reduced to vary linearly with streamwise location and the thermal boundary layer thickness becomes independent of streamwise location.