Integral analysis of adverse pressure gradient turbulent boundary layers

This study suggests that partial changes in adverse pressure gradient (APG) turbulent boundary layers (TBLs) relative to zero pressure gradient (ZPG) conditions can be obtained quantitatively by the wall-normal integral, while clarifying the partial influence of non-equilibrium effects. Specifically, the term $u_{\tau }^{2}/ ( {U_{e}V_{e}} )$ , which is found to describe the degree of scale separation under non-equilibrium conditions, is decomposed into three terms. Here, $u_{\tau }$ is the frictional velocity, $U_{e}$ is the streamwise velocity at the boundary layer edge, and $V_{e}$ is the normal velocity at the boundary layer edge. This equation includes a ZPG term, a pressure gradient term and a streamwise variation term, indicating that the pressure gradient promotes scale separation. The equation can be applied to ZPG TBLs and equilibrium APG TBLs by separately ignoring the pressure gradient term and the streamwise variation term. By using this equation to simplify the integral of the inertia term of the mean momentum equation, an expression for the Reynolds shear stress in the outer region can be obtained, which indicates how APG affects the Reynolds shear stress through the mean velocity. The above quantitative results support further study of non-equilibrium APG TBLs.