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Analytical equations for mean wall-normal velocity and momentum integral in compressible boundary-layer flows

Published online by Cambridge University Press:  01 December 2025

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] ,
Alessandro Ceci Open the ORCID record for Alessandro Ceci [Opens in a new window]  and
Sergio Pirozzoli Open the ORCID record for Sergio Pirozzoli [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, Corpus Christi, TX 78412, USA
Alessandro Ceci
Affiliation:
Dipartimento di Ingegneria Meccanica e Aerospaziale, Universitá di Roma `La Sapienza’, Via Eudossiana 18, Roma 00184, Italy
Sergio Pirozzoli
Affiliation:
Dipartimento di Ingegneria Meccanica e Aerospaziale, Universitá di Roma `La Sapienza’, Via Eudossiana 18, Roma 00184, Italy
*
Corresponding author: Tie Wei, tie.wei@nmt.edu
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Abstract

Analytical expressions for the mean wall-normal velocity and wall shear stress in compressible boundary layers are derived by integrating the mean continuity and momentum equations. In the constant-density limit, the momentum integral formulation recovers the classical Kármán–Pohlhausen equation for incompressible boundary-layer flows. In compressible regimes, particularly under strong pressure gradients, streamwise density gradients are shown to play a crucial role in shaping boundary-layer dynamics. The derived analytical equations are validated against high-fidelity direct numerical simulation data, demonstrating both accuracy and robustness. Furthermore, the analytical equations offer insights into the physical mechanisms of compressible boundary layers, particularly the influence of density gradients. The effect of compressibility on the wall-normal velocity is explicitly demonstrated, highlighting the distinct behaviour of compressible boundary layers compared with incompressible flows. Finally, an analytical expression for the skin-friction coefficient is developed, revealing its close connection to the mean wall-normal velocity at the boundary-layer edge.

JFM classification

Compressible Flows: Compressible boundary layers Compressible Flows: Supersonic flow

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1024 , 10 December 2025 , A26
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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