Hostname: page-component-77f85d65b8-mrggf Total loading time: 0 Render date: 2026-03-26T12:58:53.031Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1003 , 25 January 2025 , A7
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abe, H. & Antonia, R.A. 2016 Relationship between the energy dissipation function and the skin friction law in a turbulent channel flow. J. Fluid Mech. 798, 140–164.CrossRefGoogle Scholar
Afzal, N. 1982 Fully developed turbulent flow in a pipe: an intermediate layer. Arch. Appl. Mech. 52 (6), 355–377.Google Scholar
Baidya, R., Philip, J., Hutchins, N., Monty, J.P. & Marusic, I. 2017 Distance-from-the-wall scaling of turbulent motions in wall-bounded flows. Phys. Fluids 29 (2), 020712.CrossRefGoogle Scholar
Baidya, R., Philip, J., Hutchins, N., Monty, J.P. & Marusic, I. 2021 Spanwise velocity statistics in high-Reynolds-number turbulent boundary layers. J. Fluid Mech. 913, A35.CrossRefGoogle Scholar
Bradshaw, P. 1965 The response of a retarded equilibrium turbulent boundary layer to the sudden removal of pressure gradient. Tech. Rep. NPL Aero. Rep. 1145.Google Scholar
Bradshaw, P. 1966 The turbulence structure of equilibrium boundary layers. Tech. Rep. NPL Aero. Rep. 1184.Google Scholar
Castillo, L., Wang, X. & George, W.K. 2004 Separation criterion for turbulent boundary layers via similarity analysis. Trans. ASME J. Fluids Engng 126 (3), 297–304.CrossRefGoogle Scholar
Clauser, F.H. 1954 Turbulent boundary layers in adverse pressure gradients. J. Aeronaut. Sci. 21, 91–108.CrossRefGoogle Scholar
Clauser, F.H. 1956 The turbulent boundary layer. Adv. Appl. Mech. 4, 1–51.CrossRefGoogle Scholar
Coleman, G.N., Rumsey, C.L. & Spalart, P.R. 2018 Numerical study of turbulent separation bubbles with varying pressure gradient and Reynolds number. J. Fluid Mech. 847, 28–70.CrossRefGoogle ScholarPubMed
Coles, D.E. & Hirst, E.A. 1969 Computation of Turbulent Boundary Layers – 1968 AFOSR-IFP-Stanford Conference. Vol. 2: Compiled Data. Stanford University.Google Scholar
Griffin, K.P., Fu, L. & Moin, P. 2021 General method for determining the boundary layer thickness in nonequilibrium flows. Phys. Rev. Fluids 6 (2), 024608.CrossRefGoogle Scholar
Gungor, A.G., Maciel, Y., Simens, M.P. & Soria, J. 2016 Scaling and statistics of large-defect adverse pressure gradient turbulent boundary layers. Intl J. Heat Fluid Flow 59, 109–124.CrossRefGoogle Scholar
Gungor, T.R., Gungor, A.G. & Maciel, Y. 2024 Turbulent boundary layer response to uniform changes of the pressure force contribution. J. Fluid Mech. 997, A75.CrossRefGoogle Scholar
Gungor, T.R., Maciel, Y. & Gungor, A.G. 2022 Energy transfer mechanisms in adverse pressure gradient turbulent boundary layers: production and inter-component redistribution. J. Fluid Mech. 948, A5.CrossRefGoogle Scholar
von Kármán, T. 1921 Über laminare und turbulente reibung. Z. Angew. Math. Mech. 1 (4), 233–252.CrossRefGoogle Scholar
von Kármán, T. 1934 Turbulence and skin friction. J. Aeronaut. Sci. 1 (1), 1–20.CrossRefGoogle Scholar
Laadhari, F. 2007 Reynolds number effect on the dissipation function in wall-bounded flows. Phys. Fluids 19 (3), 038101.CrossRefGoogle Scholar
Ludwieg, H. & Tillmann, W. 1949 Untersuchungen über die Wandschubspannung in turbulenten Reibungsschichten. Ing.-Arch. 17, 288–299.CrossRefGoogle Scholar
Maciel, Y., Wei, T., Gungor, A.G. & Simens, M.P. 2018 Outer scales and parameters of adverse-pressure-gradient turbulent boundary layers. J. Fluid Mech. 844, 5–35.CrossRefGoogle Scholar
Manhart, M. & Friedrich, R. 2002 DNS of a turbulent boundary layer with separation. Intl J. Heat Fluid Flow 23 (5), 572–581.CrossRefGoogle Scholar
Monin, A.S. & Yaglom, A.M. 1965 Statistical Fluid Mechanics, Volume I: Mechanics of Turbulence. MIT Press.Google Scholar
Perry, A.E. & Schofield, W.H. 1973 Mean velocity and shear stress distributions in turbulent boundary layers. Phys. Fluids 16 (12), 2068–2074.CrossRefGoogle Scholar
Pohlhausen, K. 1921 Zur näherungsweisen integration der differentialgleichung der iaminaren grenzschicht. Z. Angew. Math. Mech. 1 (4), 252–290.CrossRefGoogle Scholar
Renard, N. & Deck, S. 2016 A theoretical decomposition of mean skin friction generation into physical phenomena across the boundary layer. J. Fluid Mech. 790, 339–367.CrossRefGoogle Scholar
Rotta, J. 1953 On the theory of the turbulent boundary layer. NACA Tech. Mem. 1344. NACA.Google Scholar
Rotta, J. 1962 Turbulent boundary layers in incompressible flow. Prog. Aerosp. Sci. 2 (1), 1–95.CrossRefGoogle Scholar
Schlatter, P. & Örlü, R. 2010 Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116–126.CrossRefGoogle Scholar
Schlichting, H. 1979 Boundary-Layer Theory. McGraw-Hill.Google Scholar
Schubauer, G.B. & Klebanoff, P.S. 1950 Investigation of separation of the turbulent boundary layer. NASA Tech. Rep. TN 2133. NASA.Google Scholar
Schubauer, G.B. & Klebanoff, P.S. 1951 Investigation of separation of the turbulent boundary layer. NACA Tech. Rep. TR-1030. NACA.Google Scholar
Schubauer, G.B. & Sprangenberg, W.G. 1960 Forced mixing in boundary layers. J. Fluid Mech. 8, 10–32.CrossRefGoogle Scholar
Senthil, S., Kitsios, V., Sekimoto, A., Atkinson, C. & Soria, J. 2020 Analysis of the factors contributing to the skin friction coefficient in adverse pressure gradient turbulent boundary layers and their variation with the pressure gradient. Intl J. Heat Fluid Flow 82, 108531.CrossRefGoogle Scholar
Simpson, R.L. 1989 Turbulent boundary-layer separation. Annu. Rev. Fluid Mech. 21 (1), 205–232.CrossRefGoogle Scholar
Skåre, P.E. & Krogstad, P.A. 1994 A turbulent equilibrium boundary layer near separation. J. Fluid Mech. 272, 319–348.CrossRefGoogle Scholar
Sreenivasan, K.R. & Sahay, A. 1997 The persistence of viscous effects in the overlap region, and the mean velocity in turbulent pipe and channel flows. arXiv:9708016.Google Scholar
Stratford, B.S. 1959 The prediction of separation of the turbulent boundary layer. J. Fluid Mech. 5 (1), 1–16.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J.L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Wei, T. 2018 Integral properties of turbulent-kinetic-energy production and dissipation in turbulent wall-bounded flows. J. Fluid Mech. 854, 449–473.CrossRefGoogle Scholar
Wei, T., Fife, P., Klewicki, J. & McMurtry, P. 2005 Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows. J. Fluid Mech. 522, 303–327.CrossRefGoogle Scholar
Wei, T. & Klewicki, J. 2016 Scaling properties of the mean wall-normal velocity in zero-pressure-gradient boundary layers. Phys. Rev. Fluids 1 (8), 082401.CrossRefGoogle Scholar
Wei, T. & Klewicki, J. 2023 Integral relation in zero-pressure-gradient boundary layer flows. Phys. Rev. Fluids 8, 124601.CrossRefGoogle Scholar
Wei, T. & Knopp, T. 2023 Outer scaling of the mean momentum equation for turbulent boundary layers under adverse pressure gradient. J. Fluid Mech. 958, A9.CrossRefGoogle Scholar
Wei, T., Li, Z., Knopp, T. & Vinuesa, R. 2023 a The mean wall-normal velocity in turbulent-boundary-layer flows under pressure gradient. J. Fluid Mech. 975, A27.CrossRefGoogle Scholar
Wei, T., Li, Z. & Wang, Y. 2023 b New formulations for the mean wall-normal velocity and Reynolds shear stress in turbulent boundary layer under zero pressure gradient. J. Fluid Mech. 969, A3.CrossRefGoogle Scholar
Wei, T., Li, Z. & Wang, Y. 2024 New momentum integral equation applicable to boundary layer flows under arbitrary pressure gradients. J. Fluid Mech. 984, A64.CrossRefGoogle Scholar
Wieghardt, K. 1945 Turbulente grenzschichten. Tech. Rep.Google Scholar
Wieghardt, K. 1948 Über einen energiesatz zur berechnung laminarer grenzschichten. Ing.-Arch. 16 (3), 231–242.CrossRefGoogle Scholar