This Rapid Communication investigates boundary layer flows with a pressure gradient using a 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 ${\beta }_{\text{RC}}$, 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 ${\beta }_{\text{RC}}$ (an important effect of an adverse pressure gradient), the friction coefficient is closely related to ${\beta }_{\text{RC}}$ as $C_f\propto 1/{\beta }_{\text{RC}}$, 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 $U_e{V}_e/u_{\tau }^2=H+(1+\delta /{\delta }_1+H){\beta }_{\text{RC}}$, where $U_e$ is the streamwise velocity at the edge of the boundary layer. At zero pressure gradient, this relation reduces to $U_{\infty }{V}_{\infty }/u_{\tau }^2=H$, as recently derived by Wei and Klewicki [Phys. Rev. Fluids 1, 082401 (2016)].

• Received 28 April 2017

DOI:https://doi.org/10.1103/PhysRevFluids.2.092601