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New formulations for the mean wall-normal velocity and Reynolds shear stress in a turbulent boundary layer under zero pressure gradient

Published online by Cambridge University Press:  10 August 2023

Tie Wei Open the ORCID record for Tie Wei [Opens in a new window] ,
Zhaorui Li Open the ORCID record for Zhaorui Li [Opens in a new window]  and
Yanxing Wang Open the ORCID record for Yanxing Wang [Opens in a new window]
Tie Wei*
Affiliation:
Department of Mechanical Engineering, New Mexico Tech, Socorro, NM 87801, USA
Zhaorui Li
Affiliation:
Department Of Engineering, Texas A&M University-Corpus Christi, 6300 Ocean Drive, Corpus Christi, TX 78412, USA
Yanxing Wang
Affiliation:
Department of Mechanical Engineering, New Mexico State University, 1780 E University Ave, Las Cruces, NM 88003, USA
*
†Email address for correspondence: tie.wei@nmt.edu
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Abstract

In this study, a novel analytical solution is derived rigorously for the mean wall-normal velocity in a turbulent boundary layer (TBL) under zero pressure gradient (ZPG), without relying on any a priori assumptions regarding the mean streamwise velocity. By neglecting the higher-order terms in the exact solution, an approximate formulation for the mean wall-normal velocity is obtained. The accuracy of this approximation is then validated through a comparison with data from direct numerical simulations (DNS). Drawing upon the distinct behaviours of force balance in the inner and outer regions of the ZPG TBL, simplified approximate mean momentum equations are derived by decomposing the Reynolds shear stress into inner and outer parts. The inner and outer Reynolds shear stresses are then obtained through integration of the corresponding approximate mean momentum equations. The outer Reynolds shear stress is revealed to exhibit a strong connection to the mean wall-normal velocity. Specifically, it is observed that $R_{uv-{out}}/u^2_\tau \approx 1 - UV/(U_e V_e)$, where $u_\tau$ represents the friction velocity, and $U_e$ and $V_e$ denote the mean streamwise and wall-normal velocities at the boundary layer edge, respectively. Finally, an approximate formulation for the Reynolds shear stress across the entire TBL is developed by synthesizing the inner and outer parts. Extensive validation against both DNS and experimental data, spanning a wide range of Reynolds numbers, demonstrates the excellent agreement achieved.

JFM classification

Boundary Layers: Boundary layer structure
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 969 , 25 August 2023 , A3
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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