A universal velocity transformation for attached turbulent boundary layers subjected to favorable or adverse pressure gradients

A velocity transformation is derived that transforms the mean flow in a boundary layer with streamwise pressure gradients to the equilibrium log law. The derivation assumes incompressible flow, two-dimensional mean flow, attached thin boundary layer, and arbitrary streamwise pressure gradient. The derived transformation accounts for the effects of flow history, turbulent mixing, and viscosity. The scaling is applied to spatially and temporally developing boundary layer flows subjected to favorable and adverse pressure gradients. The log law fails in these flows when scaled with viscous wall scaling, but the transformed velocity profiles collapse and follow the equilibrium log law. This is the first time the idea of velocity transformation is applied in the domain of incompressible flows and to non-equilibrium flows. The previous applications of velocity transformations, i.e., the van Driest transformation, are limited to equilibrium, high-Mach-number flows. Also, unlike prior studies on the topic that focused on adjusting the log law coefficients or tuning empirical damping or wake functions, this work corrects deviations of the untransformed velocity profiles by accounting for history effects.