Upstream history quantification and scale-decomposed energy analysis for weak-to-strong adverse-pressure-gradient turbulent boundary layers

The present study delineates the effects of pressure gradient history and local disequilibration on the small and large-scale energy in turbulent boundary layers (TBLs) imposed with a broad range of adverse-pressure-gradients (APG). This is made possible by analyzing four published high-fidelity APG TBL databases, which span weak to strong APGs and cover dynamic conditions ranging from near-equilibrium to strong disequilibrium. These databases enable the development of a methodology to understand the effects of PG history and local disequilibration, the latter defined here as the local streamwise rate of change of the pressure force contribution in the force balance. The influence of PG history on TBL statistics is quantified by the accumulated PG parameter ($\overline{\beta }$), proposed previously by Vinuesa et al. (2017) to study integral quantities, which is compared here between cases at matched local PG strength ($\beta$), Reynolds number ($Re$) and $d\beta /dRe$ at nominally similar orders of magnitude. Here, $\beta$ denotes a general umbrella term used for pressure gradient parameters that is estimated using different scaling parameters in this study. While the effects of local disequilibration ($d\beta /dRe$) are investigated by considering TBL cases at matched $\beta$, $Re$, and fairly matched $\overline{\beta }$. This enables analysis of accumulated PG history and local disequilibration effects separately, where applicable, to highlight qualitative differences in statistical trends. It is found that $\overline{\beta }$ cannot unambiguously capture history effects when $d\beta /dRe$ levels are significantly high, as it does not account for the delayed response of the mean flow and turbulence, nor the attenuation of the pressure gradient effect with distance. In two comparisons of APG TBLs under strong non-equilibrium, the values of $\overline{\beta }$ and $d\beta /dRe$ expressed using Zagarola–Smits scaling were found to be consistent with the trends in mean velocity defect and Reynolds stresses noted previously for weak APG TBLs. While an increase in $\overline{\beta }$ is associated with energization of both the small and large scales in the outer regions of APG TBLs, it affects only the large scales in the near-wall region. This confirms the ability of near-wall small scales to rapidly adjust to changes in PG strength. By attempting to provide a structured parametric methodology to isolate effects of PG history and local disequilibration, this study reports the influence of these effects on turbulent flow statistics across the widest APG strengths documented in the literature.