Turbulent Boundary Layer With Negligible Wall Stress

The turbulent boundary layer subjected to strong adverse pressure gradient near the separation region has been analyzed at large Reynolds numbers by the method of matched asymptotic expansions. The two regions consisting of outer nonlinear wake layer and inner wall layer are analyzed in terms of pressure scaling velocities $Up=(νp′∕ρ)1∕3$ in the wall region and $Uδ=(δp′∕ρ)1∕2$ in the outer wake region, where $p′$ is the streamwise pressure gradient and $ρ$ is the fluid density. In this work, the variables $δ$⁠, the outer boundary layer thickness, and $Uδ$⁠, the outer velocity scale, are independent of $ν$⁠, the molecular kinematic viscosity, which is a better model of fully developed mean turbulent flow. The asymptotic expansions have been matched by Izakson–Millikan–Kolmogorov hypothesis leading to open functional equations. The solution for the velocity distribution gives new composite log-half-power laws, based on the pressure scales, providing a better model of the flow, where the outer composite log-half-power law does not depend on the molecular kinematic viscosity. These new composite laws are better and one may be benefited from their limiting relations that for weak pressure gradient yield the traditional logarithmic laws and for strong adverse pressure gradient yield the half-power laws. During matching of the nonlinear outer layer two cases arise: One where $Uδ∕Ue$ is small and second where $Uδ∕Ue$ of order unity (where $Ue$ is the velocity at the edge of the boundary layer). In the first case, the lowest order nonlinear outer flow under certain conditions shows equilibrium. The outer flow subjected to the constant eddy viscosity closure model is governed by the Falkner–Skan equation subjected to the matching condition of finite slip velocity on the surface. The jet- and wakelike solutions are presented, where the zero velocity slip implying the point of separation, which compares well with Coles traditional wake function. In the second case, higher order terms in the asymptotic solutions for nearly separating flow have been estimated. The proposed composite log-half-power law solution and the limiting half-power law have been well supported by extensive experimental and direct numerical simulation data. For moderate values of the pressure gradient the data show that the proposed composite log-half-power laws are a better model of the flow.