The three effects of the pressure force on turbulent boundary layers

This study aims to isolate the three effects of the pressure force on the inner and outer layers: the local direct impact (characterized by the pressure gradient parameter, β), the local disequilibrating effect (represented here by the normalized streamwise derivative dβ/dX), and the upstream cumulative effect, while also accounting for the inevitable Reynolds number influence. To achieve this objective, we draw on several non-equilibrium and near-equilibrium databases from the literature, and employ a methodology based on the selection of pressure-gradient parameters — distinct for the inner and outer layers — that capture the local direct impact and the local disequilibration effect of the pressure gradient. The pressure force impact on the inner and outer regions is represented by two parameters: the friction–viscous pressure-gradient parameter, β_i, and the pressure-gradient parameter based on Zagarola–Smits velocity, β_ZS, respectively. In the non-equilibrium flow cases, both β_i and β_ZS exhibit similar distributions, initially increasing and then decreasing. However, the rate of change of these parameters along the streamwise direction varies among the flows, indicating differing levels of pressure force disequilibration. In addition, we employ two near-equilibrium cases with minimal variations of the pressure gradient parameters for comparisons. In the outer layer, it is found that both the local and cumulative disequilibrating effects modify the mean velocity and Reynolds stress profiles at identical β_ZS values. The faster the variations in pressure force impact, the more delayed the response of both mean flow and turbulence. Cumulative effects prove to be significant. In the inner layer, which responds much faster to changes in pressure force, the local disequilibrating effect still modifies the mean velocity profile in the viscous sublayer. Notably, when the mean velocity defect is significant, the behavior of u-structures in the inner layer appears to be governed by how outer turbulence responds to pressure force effects. In contrast, the size of uv-structures in the inner layer scales with the mixed pressure-friction length (ν/u_τ+ν/u^pi). Unlike inner u-structures, they are independent of large-scale outer structures and flow history.