Integral Analysis of Boundary Layer Flows with Pressure Gradient

Boundary layer flows with pressure gradient is investigated using a novel similarity/integral analysis of the continuity equation and momentum equation in the streamwise direction. The analysis yields useful analytical relations for $V_e$, the mean wall-normal velocity at the edge of the boundary layer, and for the skin friction coefficient, $C_f$, in terms of the boundary layer parameters and in particular $\beta_{_{RC}}$, the Rotta-Clauser pressure gradient parameter. The analytical results are compared with experimental and numerical data and are found to be valid. One of the main findings is that for large positive $\beta_{_{RC}}$, the friction coefficient is closely related to $\beta_{_{RC}}$ as $C_f \propto 1/\beta_{_{RC}}$, because $\delta/\delta_1$, $\delta_1/\delta_2=H$ and $d\delta/dx$ become approximately constant. Here $\delta$ is the boundary layer thickness, $\delta_1$ is the displacement thickness, $\delta_2$ is the momentum thickness and $H$ is the shape factor. Another finding is that the mean wall-normal velocity at the edge of the boundary layer is related to other flow variables as $U_e V_e/u^2_\tau = H + (1+\delta/\delta_1+H)\beta_{_{RC}}$, where $U_e$ is the streamwise velocity at the edge of the boundary layer.