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Energy thickness in turbulent boundary layer flows

Published online by Cambridge University Press:  13 January 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, 37073 Gottingen, Germany
Zhaorui Li
Affiliation:
Department of Engineering, Texas A&M University-Corpus Christi, 6300 Ocean Drive, Corpus Christi, TX 78412, USA
*
†Email address for correspondence: tie.wei@nmt.edu
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Abstract

In this study, we investigate the properties of energy thickness $\delta _3$ in turbulent boundary layer (TBL) flows, a parameter derived solely from the mean streamwise velocity ($U$) profile. Through an analysis of the energy integral equation for zero pressure gradient TBLs, we establish a close relationship between turbulent kinetic energy (TKE) production and $\delta _3$, offering a practical method to estimate TKE production, which is particularly useful in physical experiments where direct measurements are challenging. The significance of $\delta _3$ becomes even more pronounced in TBLs under pressure gradient. Through extensive analysis of numerical and experimental data, we show that the ratio between $\delta _3$ and the momentum thickness $\delta _2$ is a promising criterion for predicting flow separation. Moreover, we derive a new energy integral equation for TBLs under arbitrary pressure gradients, and provide approximations for TKE productions terms by $R_{uv}\,\partial U/\partial y$ and $R_{uu}\,\partial U/\partial x$, and dissipation term by the mean shear. Here, $x, y$ represent the streamwise and wall-normal directions, respectively, and $R_{uu}$ and $R_{uv}$ are the Reynolds normal and shear stresses. The accuracy and robustness of the new energy integral equation and the approximation equations are validated using direct numerical simulations data. Our results show that the TKE production by $R_{uv}\,\partial U/\partial y$ and the overall productions consistently remain positive, reflecting a continuous conversion of mean kinetic energy into TKE across all TBLs. However, under strong favourable pressure gradients, TKE production by $R_{uu}\,\partial U/\partial x$ becomes negative, indicating a reverse energy transfer from TKE to mean kinetic energy.

JFM classification

Boundary Layers: Boundary layer separation Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1003 , 25 January 2025 , A7
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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