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

Published online by Cambridge University Press:  15 October 2025

Tie Wei Open the ORCID record for Tie Wei [Opens in a new window] ,
Tobias Knopp Open the ORCID record for Tobias Knopp [Opens in a new window]  and
Zhaorui Li Open the ORCID record for Zhaorui Li [Opens in a new window]
Tie Wei*
Affiliation:
Department of Mechanical Engineering, New Mexico Institute of Mining and Technology, 801 Leroy PL, Socorro, NM 87801, USA
Tobias Knopp
Affiliation:
Institute of Aerodynamics and Flow Technology, DLR (German Aerospace Center), Bunsenstr. 10, Gottingen 37073, Germany
Zhaorui Li
Affiliation:
Department of Engineering, Texas A&M University-Corpus Christi, 6300 Ocean Drive, Corpus Christi, TX 78412, USA
*
Corresponding author: Tie Wei, tie.wei@nmt.edu
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Abstract

This study applies the scaling patch approach to investigate the influence of pressure gradients on the mean-momentum balance in turbulent boundary layers (TBLs). Under strong pressure gradients, the force balance in the outer region is dominated by advective and pressure forces, with gradients of Reynolds stresses playing a minimal role. To retain the relevance of Reynolds stress gradients within the scaling patch framework, we propose a redistribution of the component $U_e \textrm {d}U_e/\textrm {d}x$ from the advective term to the pressure-gradient term. Here, $U_e$ is the mean streamwise velocity at the boundary layer edge. This reformulation enhances the outer-scaling framework of Wei & Knopp (2023 J. Fluid Mech. 958, 1–21), ensuring consistency across a wide range of pressure gradients, including those involving flow separation. Remarkably, the new outer-scaled gradient of Reynolds shear stress in TBLs under a pressure gradient closely resembles that observed in zero-pressure-gradient TBLs. In the inner region, the impact of pressure gradient is well captured by the Stratford–Mellor parameter $\beta _{\textit{in}}$. For weak pressure gradients ($|\beta _{\textit{in}}| \ll 0.07$), traditional inner scaling remains valid. However, for stronger pressure gradients $|\beta _{\textit{in}}| \gtrsim 0.07$, the near-wall dynamics is governed by a balance between pressure gradient and viscous force, as described by Stratford (1959 J. Fluid Mech. 5, 1–16) and Mellor (1966 J. Fluid Mech. 24, 255–274). In this sub-layer, viscosity and the imposed wall pressure gradient dictate the relevant velocity and length scales. Moreover, when $|\beta _{\textit{in}}| \gtrsim 0.7$ and the wall pressure $P_{w\textit{all}}$ gradient $\textrm { d}P_{w\textit{all}}/\textrm {d}x \gt 0$, a distinct sub-layer emerges outside the pressure–viscous balance region, characterised 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 and establish a refined framework for modelling their dynamics.

JFM classification

Boundary Layers: Boundary Layers Boundary Layers: Boundary layer structure

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JFM Papers
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Journal of Fluid Mechanics , Volume 1021 , 25 October 2025 , A21
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© The Author(s), 2025. Published by Cambridge University Press

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