Outer scaling of the mean momentum equation for turbulent boundary layers under adverse pressure gradient

Abstract

A new scaling of the mean momentum equation is developed for the outer region of turbulent boundary layers (TBLs) under adverse pressure gradient (APG). The maximum Reynolds shear stress location, denoted as ym, is employed to determine the proper scales for the outer region of an APG TBL. An outer length scale is proposed as delta_e-y_m, where delta_e is the boundary layer thickness. An outer velocity scale for the mean streamwise velocity deficit is proposed as Ue-Um, where Ue and Um are the mean streamwise velocities at the boundary layer edge and ym, respectively. An outer velocity scale for the mean wall-normal velocity deficit is proposed as Ve-Vm, where Ve and Vm are the wall-normal velocities at delta_e and ym, respectively. The maximum Reynolds shear stress is found to scale as (delta_e - ym) Ue dUe/dx. The new outer scaling collapses well the experimental and numerical data on APG TBLs over a wide range of Reynolds numbers and strengths of pressure gradient. Approximations of the new scaling are developed for TBLs under strong APG and at high Reynolds numbers. The relationships between the new scales and previously proposed scales are discussed.