High-Reynolds-number turbulent boundary layers under adverse pressure gradients. Part 2. A composite mean velocity profile

Abstract

A robust composite mean velocity profile is developed for turbulent boundary layers (TBLs) subjected to adverse pressure gradients (APGs), extending the composite formulation for generic pressure-gradient TBLs proposed by \citeauthor{nickels} (\textit{J.\ Fluid Mech.}, vol.\ 521, 2004). Several modifications are introduced to capture key features of APG flows. A new parameter accounts for pressure-gradient history effects in the wake region, a velocity-overshoot function is incorporated in the inner region, and the wake function is reformulated using an independent, physically motivated definition of boundary-layer thickness. A compilation of APG TBL datasets from the literature, including the new dataset presented in Part~1, is used to assess and refine the formulation. The resulting composite profile contains three physically meaningful parameters that capture pressure-gradient effects on the mean velocity profile, determined through nonlinear curve fitting. These parameters provide a framework for identifying `well-behaved' APG TBLs and quantifying the strength of pressure-gradient history effects. The profile also enables reliable estimation of the friction velocity and boundary-layer thickness in well-behaved APG TBLs, providing a practical tool for scaling analyses when these quantities are not directly measurable. Its analytical form yields improved estimates of mean velocity gradients, facilitating evaluation of the indicator function and identification of inflection points. Finally, the formulation predicts both the coefficients and spatial extent of the logarithmic region of the mean streamwise velocity profile, enabling assessment of its universality in high-Reynolds-number APG TBLs. This shows that the von K'arm'an coefficient approaches an invariant value of $κ\approx 0.39$ at sufficiently high Reynolds numbers, independent of pressure-gradient effects.