Mean flow scaling in stably stratified temporally developing turbulent boundary layers
Stably stratified wall-bounded turbulence governs the dynamics of many environmental and engineering flows. A key challenge is characterizing how stratification modifies mean and turbulent profiles. Monin--Obukhov similarity theory (MOST) is the dominant modelling framework, although it has rarely been rigorously validated against well-controlled direct numerical simulation (DNS) data over a wide range of stratification levels. In this study, we exploit the temporally developing turbulent boundary layer (TTBL) framework to investigate stratified turbulent boundary layers from the weakly stable to the very stable regime, spanning a range of Reynolds and Richardson numbers, and isolating the effects due to buoyancy from other mechanisms such as flow rotation. We demonstrate that the TTBL set-up faithfully reproduces classical similarity theory results and that surface-based scaling of the mean velocity gradient holds over a wider range of $z/L$ ($L$ being the Obukhov length) than previously reported. This result is attributed to the similar decay rate of turbulent shear stress and heat flux in this canonical flow. Next, we show that, as stratification intensifies, the intercept of the mean velocity profile increases, until the separation of scales required for a logarithmic region to exist can no longer be sustained. We propose an empirical closure for this intercept shift in terms of the Reynolds number based on the Obukhov length. Finally, a simple damping of the MOST contribution to the mean velocity profile is proposed and validated, enabling accurate prediction of the wall friction coefficient ($C_f$) across the investigated regimes.