Properties of the mean momentum balance in turbulent boundary layer flows subjected to pressure gradients

We present a new application of the scaling patch approach to turbulent boundary layers (TBLs), focusing on how pressure gradients modify the structure of the mean momentum balance. Under strong pressure gradients, the outer-region force balance is dominated by advective and pressure terms, while Reynolds stress gradients play a minimal role. To retain their relevance within the scaling patch framework, we propose redistributing the component $U_e {\rm d}U_e/{\rm d}x$ from the advective to the pressure gradient contribution. This reformulation enhances the outer scaling framework of Wei & Knopp (2023), ensuring consistency across a broad range of pressure gradients, including flows with separation. Remarkably, the new outer-scaled gradient of Reynolds shear stress closely resembles that in zero-pressure-gradient TBLs. In the inner region, the impact of pressure gradient is well captured by the Stratford-Mellor parameter $\beta_\mathrm{in}$. For weak pressure gradients ($|\beta_\mathrm{in}| \ll 0.07$), traditional inner scaling remains valid. However, for stronger pressure gradients $|\beta_\mathrm{in}| \gtrsim 0.07$, the near-wall dynamics is governed by a balance between pressure gradient and viscous force. In this sub-layer, viscosity and the imposed wall pressure gradient dictate the relevant velocity and length scales. Moreover, when $|\beta_\mathrm{in}| \gtrsim 0.7$ and ${\rm d}P_\mathrm{wall}/{\rm d}x > 0$, a distinct sub-layer emerges outside the pressure-viscous balance region, characterized by a dominant balance between the imposed pressure gradient and the gradient of the Reynolds shear stress. In this region, the Reynolds shear stress increases linearly with distance from the wall. These findings provide new insights into the structure of TBLs under pressure gradients.