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Outer scaling of the mean momentum equation for turbulent boundary layers under adverse pressure gradient

Published online by Cambridge University Press:  28 February 2023

Tie Wei Open the ORCID record for Tie Wei [Opens in a new window]  and
Tobias Knopp Open the ORCID record for Tobias Knopp [Opens in a new window]
Tie Wei*
Affiliation:
Department of Mechanical Engineering, New Mexico Tech, Socorro, NM 87801, USA
Tobias Knopp
Affiliation:
Institute of Aerodynamics and Flow Technology, DLR (German Aerospace Center), Bunsenstr. 10, 37073 Gottingen, Germany
*
†Email address for correspondence: tie.wei@nmt.edu
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Abstract

A new scaling of the mean momentum equation is developed for the outer region of turbulent boundary layers (TBLs) under adverse pressure gradient (APG). The maximum Reynolds shear stress location, denoted as $y_{m}$, is employed to determine the proper scales for the outer region of an APG TBL. An outer length scale is proposed as $\delta _e - y_{m}$, where $\delta _e$ is the boundary layer thickness. An outer velocity scale for the mean streamwise velocity deficit is proposed as $U_e - U_{m}$, where $U_e$ and $U_m$ are the mean streamwise velocities at the boundary layer edge and $y_{m}$, respectively. An outer velocity scale for the mean wall-normal velocity deficit is proposed as $V_e - V_{m}$, where $V_e$ and $V_{m}$ are the wall-normal velocities at $\delta _e$ and $y_{m}$, respectively. The maximum Reynolds shear stress is found to scale as $(\delta _e - y_{m}) U_e \,{\rm d}U_e/{{\rm d}x}$. The new outer scaling collapses well the experimental and numerical data on APG TBLs over a wide range of Reynolds numbers and strengths of pressure gradient. Approximations of the new scaling are developed for TBLs under strong APG and at high Reynolds numbers. The relationships between the new scales and previously proposed scales are discussed.

JFM classification

Boundary Layers: Free shear layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 958 , 10 March 2023 , A9
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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