Log-linear law of the mean streamwise velocity in turbulent boundary layers with moderate adverse pressure gradients

An essential feature of canonical zero-pressure-gradient (ZPG) turbulent boundary layers (TBLs) is that the mean streamwise velocity exhibits a logarithmic dependence on the wall-normal distance, known as the log law. In this study, we demonstrate that this conventional log law is not suitable for turbulent boundary layers subjected to pressure gradients (PGs). Instead, a log–linear law is theoretically derived for TBLs under moderate adverse pressure gradients (APGs), based on the total shear-stress balance and a rescaled eddy-viscosity model. The validity of the proposed log–linear law is assessed using available datasets of incompressible APG TBLs with the Clauser pressure-gradient parameter $\beta$ ranging from 0.73 to 9.0. Compared with the conventional log law, the present log–linear formulation shows significantly improved agreement with the measured mean velocity profiles. In the limiting case of $\beta \to 0$, the proposed law naturally recovers the classical log law.