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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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References

REFERENCES

Afzal, N. 1982 Fully developed turbulent flow in a pipe: an intermediate layer. Arch. Appl. Mech. 52 (6), 355–377.Google Scholar
Bobke, A., Vinuesa, R., Örlü, R. & Schlatter, P. 2017 History effects and near equilibrium in adverse-pressure-gradient turbulent boundary layers. J. Fluid Mech. 820, 667–692.CrossRefGoogle Scholar
Castillo, L. & George, W.K. 2001 Similarity analysis for turbulent boundary layer with pressure gradient: outer flow. AIAA J. 39 (1), 41–47.CrossRefGoogle Scholar
Clauser, F.H. 1954 Turbulent boundary layers in adverse pressure gradients. J. Aeronaut. Sci. 21, 91–108.CrossRefGoogle Scholar
Coles, D.E. & Hirst, E.A. 1969 Computation of Turbulent Boundary Layers – 1968 AFOSR-IFP-Stanford Conference. Thermosciences Division, Department of Mechanical Engineering, Stanford University.Google Scholar
Corrsin, S. 1943 Investigation of flow in an axially symmetric heated jet of air. NACA Tech. Rep. WR W-94. California Institute of Technology.Google Scholar
Corrsin, S. & Kistler, A.L. 1955 Free-stream boundaries of turbulent flows. NACA Tech. Rep. 1244. California Institute of Technology.Google Scholar
Cuvier, C., et al. 2017 Extensive characterisation of a high Reynolds number decelerating boundary layer using advanced optical metrology. J. Turbul. 18, 929–972.CrossRefGoogle Scholar
Eisfeld, B. 2021 Characteristics of incompressible free shear flows and implications for turbulence modeling. AIAA J. 59 (1), 180–195.Google Scholar
Elsberry, K., Loeffler, J., Zhou, M.D. & Wygnanski, I. 2000 An experimental study of a boundary layer that is maintained on the verge of separation. J. Fluid Mech. 423, 227–261.CrossRefGoogle Scholar
Fife, P. 2006 Scaling approaches to steady wall-induced turbulence. arXiv:2301.08740.Google Scholar
Fife, P., Klewicki, J., McMurtry, P. & Wei, T. 2005 Multiscaling in the presence of indeterminacy: wall-induced turbulence. Multiscale Model. Simul. 4 (3), 936–959.CrossRefGoogle Scholar
Fife, P., Klewicki, J. & Wei, T. 2009 Time averaging in turbulence settings may reveal an infinite hierarchy of length scales. Discret. Contin. Dyn. A 24 (3), 781.CrossRefGoogle Scholar
George, W.K., Stanislas, M. & Laval, J.P. 2012 New insights into adverse pressure gradient boundary layers. In Progress in Turbulence and Wind Energy IV (ed. M. Oberlack, J. Penke, A. Talamelli, L. Castillo & M. Hölling), Springer Proceedings in Physics, vol. 141. Springer.CrossRefGoogle Scholar
Görtler, H. 1942 Berechnung von Aufgaben der freien Turbulenz auf Grund eines neuen Näherungsansatzes. Z. Angew. Math. Mech. 22 (5), 244–254.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
von Kármán, T. 1930 Mechanische Ähnlichkeit und Turbulenz. In Proceedings of the 3rd International Congress on Applied Mechanics, Stockholm, Sweden, pp. 85–93. Weidmannsche Buchh.Google Scholar
Kitsios, V., Atkinson, C., Sillero, J.A., Borell, G., Gungor, A.G., Jimenez, J. & Soria, J. 2016 Direct numerical simulation of a self-similar adverse pressure gradient turbulent boundary layer. Intl J. Heat Fluid Flow 61, 129–136.Google Scholar
Kitsios, V., Sekimoto, A., Atkinson, C., Sillero, J.A., Borrell, G., Gungor, A.G., Jiménez, J. & Soria, J. 2017 Direct numerical simulation of a self-similar adverse pressure gradient turbulent boundary layer at the verge of separation. J. Fluid Mech. 829, 392–419.Google Scholar
Klebanoff, P. 1955 Characteristics of turbulence in a boundary layer with zero pressure gradient. NACA Tech. Rep. 1247. National Bureau of Standards.Google Scholar
Krug, D., Philip, J. & Marusic, I. 2017 Revisiting the law of the wake in wall turbulence. J. Fluid Mech. 811, 421–435.CrossRefGoogle Scholar
Liu, X., Thomas, F.O. & Nelson, R.C. 2002 An experimental investigation of the planar turbulent wake in constant pressure gradient. Phys. Fluids 14 (8), 2817–2838.CrossRefGoogle Scholar
Long, R.R. & Chen, T.-C. 1981 Experimental evidence for the existence of the ‘mesolayer’ in turbulent systems. J. Fluid Mech. 105, 19–59.Google Scholar
Maciel, Y., Rossignol, K.-S. & Lemay, J. 2006 A study of a turbulent boundary layer in stalled-airfoil-type flow conditions. Exp. Fluids 41 (4), 573–590.Google 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
Marušić, I. & Perry, A.E. 1995 A wall-wake model for the turbulence structure of boundary layers. Part 2. Further experimental support. J. Fluid Mech. 298, 389–407.CrossRefGoogle Scholar
Mehdi, F., Klewicki, J.C. & White, C.M. 2010 Mean momentum balance analysis of rough-wall turbulent boundary layers. Physica D 239 (14), 1329–1337.CrossRefGoogle Scholar
Mellor, G.L. & Gibson, D.M. 1966 Equilibrium turbulent boundary layers. J. Fluid Mech. 24 (2), 225–253.CrossRefGoogle Scholar
Millikan, C.B. 1938 A critical discussion of turbulent flows in channels and circular tubes. In Proceedings of the 5th International Congress for Applied Mechanics (ed. J.P. Den Hartog & H. Peters), pp. 386–392. Wiley.Google Scholar
Monkewitz, P.A., Chauhan, K.A. & Nagib, H.M. 2007 Comparison of mean flow similarity laws in zero pressure gradient turbulent boundary layers. Phys. Fluids 20, 105102.CrossRefGoogle Scholar
Nagano, Y., Tagawa, M. & Tsuji, T. 1993 Effects of adverse pressure gradients on mean flows and turbulence statistics in a boundary layer. In Turbulent Shear Flows (ed. F. Durst, R. Friedrich, B.E. Launder, F.W. Schmidt, U. Schumann & J.H. Whitelaw), vol. 8, pp. 7–21. Springer.Google Scholar
Nickels, T.B. 2004 Inner scaling for wall-bounded flows subject to large pressure gradients. J. Fluid Mech. 521, 217–239.CrossRefGoogle Scholar
Perry, A.E., Marusic, I. & Li, J.D. 1994 Wall turbulence closure based on classical similarity laws and the attached eddy hypothesis. Phys. Fluids 6, 1024–1035.CrossRefGoogle Scholar
Perry, A.E. & Schofield, W.H. 1973 Mean velocity and shear stress distributions in turbulent boundary layers. Phys. Fluids 16, 2068–2074.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Romero, S.K., Zimmerman, S.J., Philip, J. & Klewicki, J.C. 2021 Stress equation based scaling framework for adverse pressure gradient turbulent boundary layers. Intl J. Heat Fluid Flow 93, 108885.CrossRefGoogle Scholar
Rotta, J. 1950 Über die Theorie turbulenter Grenzschichten. Tech. Rep. Mitteilungen aus dem Max-Planck-Institut für Strömungsforschung Nr. 1. (Translated as: On the theory of turbulent boundary layers. NACA Technical Memorandum No. 1344, 1953.).Google Scholar
Schatzman, D.M. & Thomas, F.O. 2017 An experimental investigation of an unsteady adverse pressure gradient turbulent boundary layer: embedded shear layer scaling. J. Fluid Mech. 815, 592–642.CrossRefGoogle Scholar
Schlichting, H. 1979 Boundary-Layer Theory. McGraw-Hill, Inc.Google Scholar
Schofield, W.H. 1981 Equilibrium boundary layers in moderate to strong adverse pressure gradient. J. Fluid Mech. 113, 91–122.CrossRefGoogle Scholar
Schofield, W.H. & Perry, A.E. 1972 The turbulent boundary layer as a wall confined wake. Tech. Rep. 134. Australian Department of Supply.Google Scholar
Sekimoto, A., Kitsios, V., Atkinson, C., Sillero, J.A., Borell, G., Gungor, A.G., Jimenez, J. & Soria, J. 2019 Outer scaling of self-similar adverse-pressure-gradient turbulent boundary layers. arXiv:1912.05143.Google Scholar
Skåre, P.E. & Krogstad, P.A. 1994 A turbulent equilibrium boundary layer near separation. J. Fluid Mech. 272, 319–348.CrossRefGoogle Scholar
Slotnick, J.P. & Heller, G. 2019 Emerging opportunities for predictive CFD for off-design commercial airplane flight characteristics. In 54th 3AF International Conference on Applied Aerodynamics, Paris, pp. 25–27.Google 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:physics/9708016.Google Scholar
Tennekes, H. & Lumley, J.L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Townsend, A.A. 1956 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Wei, T. 2019 Multiscaling analysis of buoyancy-driven turbulence in a differentially heated vertical channel. Phys. Rev. Fluids 4, 073502.Google Scholar
Wei, T. 2020 Analyses of buoyancy-driven convection. Adv. Heat Transfer 52, 1–93.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.Google Scholar
Wei, T., Li, Z. & Livescu, D. 2022 a Scaling patch analysis of planar turbulent mixing layers. Phys. Fluids 34, 115120.CrossRefGoogle Scholar
Wei, T. & Livescu, D. 2021 a Scaling of the mean transverse flow and Reynolds shear stress in turbulent plane jet. Phys. Fluids 33 (3), 035142.CrossRefGoogle Scholar
Wei, T. & Livescu, D. 2021 b Scaling patch analysis of turbulent planar plume. Phys. Fluids 33 (5), 055101.CrossRefGoogle Scholar
Wei, T., Livescu, D. & Liu, X. 2022 b Scaling patch analysis of planar turbulent wake. Phys. Fluids 34, 065116.Google Scholar
Zagarola, M.V. & Smits, A.J. 1998 a Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 33–79.CrossRefGoogle Scholar
Zagarola, M.V. & Smits, A.J. 1998 b A new mean velocity scaling for turbulent boundary layers. ASME Paper No. FEDSM98-4950.Google Scholar
Zhou, A., Pirozzoli, S. & Klewicki, J. 2017 Mean equation based scaling analysis of fully-developed turbulent channel flow with uniform heat generation. Intl J. Heat Mass Transfer 115, 50–61.CrossRefGoogle Scholar