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A log-layer analogy for fluid acceleration in the inner layer of wall-bounded turbulence with pressure gradients

Published online by Cambridge University Press:  28 April 2025

Peng E.S. Chen ,
Wen Zhang ,
Minping Wan Open the ORCID record for Minping Wan [Opens in a new window] ,
Xiang I.A. Yang Open the ORCID record for Xiang I.A. Yang [Opens in a new window] ,
Yipeng Shi Open the ORCID record for Yipeng Shi [Opens in a new window]  and
Shiyi Chen
Peng E.S. Chen
Affiliation:
Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, PR China
Wen Zhang*
Affiliation:
Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, PR China
Minping Wan*
Affiliation:
Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, PR China
Xiang I.A. Yang*
Affiliation:
Mechanical Engineering, Pennsylvania State University, State College, PA 16802, USA
Yipeng Shi
Affiliation:
State Key Laboratory of Turbulence and Complex Systems, Beijing 100871, PR China
Shiyi Chen
Affiliation:
Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, PR China Eastern Institute for Advanced Study, Eastern Institute of Technology, Ningbo 315200, PR China
*
Corresponding authors: Minping Wan, [email protected]; Wen Zhang, [email protected]; Xiang I.A. Yang, [email protected]
Corresponding authors: Minping Wan, [email protected]; Wen Zhang, [email protected]; Xiang I.A. Yang, [email protected]
Corresponding authors: Minping Wan, [email protected]; Wen Zhang, [email protected]; Xiang I.A. Yang, [email protected]
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Abstract

History effects play a significant role in determining the velocity in boundary layers with pressure gradients, complicating the identification of a velocity scaling. This work pivots away from traditional velocity analysis to focus on fluid acceleration in boundary layers with strong adverse pressure gradients. We draw parallels between the transport equation of the velocity in an equilibrium spatially evolving boundary layer and the transport equation of the fluid acceleration in temporally evolving boundary layers with pressure gradients, establishing an analogy between the two. To validate our analogy, we show that the laminar Stokes solution, which describes the flow immediately after the application of a pressure gradient force, is consistent with the present analogy. Furthermore, fluid acceleration exhibits a linear scaling in the wall layer and transitions to logarithmic scaling away from the wall after the initial period, mirroring the velocity in an equilibrium boundary layer, lending further support to the analogy. Finally, by integrating fluid acceleration, a velocity scaling is derived, which compares favourably with data as well.

JFM classification

Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1010 , 10 May 2025 , A13
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

Alving, A.E. & Fernholz, H.H. 1995 Mean-velocity scaling in and around a mild, turbulent separation bubble. Phys. Fluids 7 (8), 1956–1969.CrossRefGoogle Scholar
Bailey, S.C.C. et al. 2013 Obtaining accurate mean velocity measurements in high Reynolds number turbulent boundary layers using Pitot tubes. J. Fluid Mech. 715, 642–670.CrossRefGoogle Scholar
Baskaran, V., Smits, A.J. & Joubert, P.N. 1987 A turbulent flow over a curved hill. Part 1. Growth of an internal boundary layer. J. Fluid Mech. 182, 47–83.CrossRefGoogle 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
Bose, S.T. & Park, G.I. 2018 Wall-modeled large-eddy simulation for complex turbulent flows. Ann. Rev. Fluid Mech. 50 (1), 535–561.CrossRefGoogle ScholarPubMed
Bradshaw, P. 1974 Possible origin of Prandtl’s mixing-length theory. Nature 249 (5453), 135–136.CrossRefGoogle Scholar
Brucker, K.A. & Sarkar, S. 2010 A comparative study of self-propelled and towed wakes in a stratified fluid. J. Fluid Mech. 652, 373–404.CrossRefGoogle Scholar
de Bruyn Kops, S.M. & Riley, J.J. 2019 The effects of stable stratification on the decay of initially isotropic homogeneous turbulence. J. Fluid Mech. 860, 787–821.CrossRefGoogle Scholar
Chen, P.E.S., Wu, W., Griffin, K.P., Shi, Y. & Yang, X.I.A. 2023 a A universal velocity transformation for boundary layers with pressure gradients. J. Fluid Mech. 970, A3.CrossRefGoogle Scholar
Chen, P.E.S., Zhu, X., Shi, Y. & Yang, X.I.A. 2023 b Quantifying uncertainties in direct-numerical-simulation statistics due to wall-normal numerics and grids. Phys. Rev. Fluids 8 (7), 074602.CrossRefGoogle Scholar
Clauser, F.H. 1954 Turbulent boundary layers in adverse pressure gradients. J. Atmos. Sci. 21 (2), 91–108.Google Scholar
Clauser, F.H. 1956 The turbulent boundary layer. Adv. Appl. Mech. 4, 1–51.CrossRefGoogle Scholar
Coleman, G.N., Kim, J. & Spalart, P.R. 2000 A numerical study of strained three-dimensional wall-bounded turbulence. J. Fluid Mech. 416, 75–116.CrossRefGoogle Scholar
Coleman, G.N., Kim, J. & Spalart, P.R. 2003 Direct numerical simulation of a decelerated wall-bounded turbulent shear flow. J. Fluid Mech. 495, 1–18.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
Deshpande, R., van den Bogaard, A., Vinuesa, R., Lindić, L. & Marusic, I. 2023 Reynolds-number effects on the outer region of adverse-pressure-gradient turbulent boundary layers. Phys. Rev. Fluids 8 (12), 124604.CrossRefGoogle Scholar
Dróżdż, A., Elsner, W. & Drobniak, S. 2015 Scaling of streamwise Reynolds stress for turbulent boundary layers with pressure gradient. Eur. J. Mech. B/Fluids 49, 137–145.CrossRefGoogle Scholar
El Telbany, M.M.M. & Reynolds, A.J. 1980 Velocity distributions in plane turbulent channel flows. J. Fluid Mech. 100 (1), 1–29.CrossRefGoogle Scholar
Galbraith, R.A.M.D., Sjolander, S. & Head, M.R. 1977 Mixing length in the wall region of turbulent boundary layers. Aeronaut. Q. 28 (2), 97–110.CrossRefGoogle Scholar
Garratt, J.R. 1990 The internal boundary layer – a review. Boundary-Layer Meteorol. 50 (1–4), 171–203.CrossRefGoogle Scholar
George, W.K. & Castillo, L. 1997 Zero-pressure-gradient turbulent boundary layer. Appl. Mech. Rev. 50 (12), 689–729.CrossRefGoogle Scholar
Graham, J. et al. 2016 A web services accessible database of turbulent channel flow and its use for testing a new integral wall model for LES. J. Turbul. 17 (2), 181–215.CrossRefGoogle Scholar
Guerrero, B., Lambert, M.F. & Chin, R.C. 2023 Transient behaviour of decelerating turbulent pipe flows. J. Fluid Mech. 962, A44.CrossRefGoogle Scholar
He, K., Seddighi, M. & He, S. 2016 DNS study of a pipe flow following a step increase in flow rate. Intl J. Heat Fluid Flow 57, 130–141.CrossRefGoogle Scholar
He, S. & Seddighi, M. 2013 Turbulence in transient channel flow. J. Fluid Mech. 715, 60–102.CrossRefGoogle Scholar
He, S. & Seddighi, M. 2015 Transition of transient channel flow after a change in Reynolds number. J. Fluid Mech. 764, 395–427.CrossRefGoogle Scholar
Johnstone, R., Coleman, G.N. & Spalart, P.R. 2010 The resilience of the logarithmic law to pressure gradients: evidence from direct numerical simulation. J. Fluid Mech. 643, 163–175.CrossRefGoogle Scholar
Kader, B.A. & Yaglom, A.M. 1978 Similarity treatment of moving-equilibrium turbulent boundary layers in adverse pressure gradients. J. Fluid Mech. 89 (2), 305–342.CrossRefGoogle Scholar
Kays, W.M., Crawford, M.E. & Weigand, B. 1980 Convective Heat and Mass Transfer. McGraw-Hill.Google Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.CrossRefGoogle Scholar
Knopp, T. 2022 An empirical wall law for the mean velocity in an adverse pressure gradient for RANS turbulence modelling. Flow Turbul. Combust. 109 (3), 571–601.CrossRefGoogle Scholar
Knopp, T., Reuther, N., Novara, M., Schanz, D., Schülein, E., Schröder, A. & Kähler, C.J. 2021 Experimental analysis of the log law at adverse pressure gradient. J. Fluid Mech. 918, A17.CrossRefGoogle Scholar
Kozul, M., Chung, D. & Monty, J.P. 2016 Direct numerical simulation of the incompressible temporally developing turbulent boundary layer. J. Fluid Mech. 796, 437–472.CrossRefGoogle Scholar
Lee, J.H. 2017 Large-scale motions in turbulent boundary layers subjected to adverse pressure gradients. J. Fluid Mech. 810, 323–361.CrossRefGoogle Scholar
Lee, J.-H. & Sung, H.J. 2009 Structures in turbulent boundary layers subjected to adverse pressure gradients. J. Fluid Mech. 639, 101–131.CrossRefGoogle Scholar
Lee, M. & Moser, R.D. 2015 Direct numerical simulation of turbulent channel flow up to Re τ ≈ 5200. J. Fluid Mech. 774, 395–415.CrossRefGoogle Scholar
Li, J.J.L., Yang, X.I.A. & Kunz, R.F. 2024 Direct numerical simulation of temporally evolving stratified wakes with ensemble average. J. Fluid Mech. 980, A3.CrossRefGoogle Scholar
Li, M., de Silva, C.M., Chung, D., Pullin, D.I., Marusic, I. & Hutchins, N. 2022 Modelling the downstream development of a turbulent boundary layer following a step change of roughness. J. Fluid Mech. 949, A7.CrossRefGoogle Scholar
Lozano-Durán, A., Giometto, M.G., Park, G.I. & Moin, P. 2020 Non-equilibrium three-dimensional boundary layers at moderate Reynolds numbers. J. Fluid Mech. 883, A20.CrossRefGoogle Scholar
Lozano-Durán, A. & Jiménez, J. 2014 Effect of the computational domain on direct simulations of turbulent channels up to Re τ = 4200. Phys. Fluids 26 (1), 011702.CrossRefGoogle Scholar
Luchini, P. 2017 Universality of the turbulent velocity profile. Phys. Rev. Lett. 118 (22), 224501.CrossRefGoogle ScholarPubMed
Lv, Y., Huang, X.L.D., Yang, X & Yang, X.I.A. 2021 Wall-model integrated computational framework for large-eddy simulations of wall-bounded flows. Phys. Fluids 33 (12), 125120.CrossRefGoogle Scholar
Marusic, I., Monty, J.P., Hultmark, M. & Smits, A.J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.CrossRefGoogle Scholar
Materny, M., Dróżdż, A., Drobniak, S. & Elsner, W. 2008 Experimental analysis of turbulent boundary layer under the influence of adverse pressure gradient. Arch. Mech. 60 (6), 449–466.Google Scholar
Mathur, A., Gorji, S., He, S., Seddighi, M., Vardy, A.E., O’Donoghue, T. & Pokrajac, D. 2018 Temporal acceleration of a turbulent channel flow. J. Fluid Mech. 835, 471–490.CrossRefGoogle Scholar
Mellor, G.L. & Gibson, D.M. 1966 Equilibrium turbulent boundary layers. J. Fluid Mech. 24 (2), 225–253.CrossRefGoogle Scholar
Monkewitz, P.A. & Nagib, H.M. 2023 The hunt for the Kármán ‘constant’ revisited. J. Fluid Mech. 967, A15.CrossRefGoogle Scholar
Monty, J.P., Harun, Z. & Marusic, I. 2011 A parametric study of adverse pressure gradient turbulent boundary layers. Intl J. Heat Fluid Flow 32 (3), 575–585.CrossRefGoogle Scholar
Morrill-Winter, C., Philip, J. & Klewicki, J. 2017 An invariant representation of mean inertia: theoretical basis for a log law in turbulent boundary layers. J. Fluid Mech. 813, 594–617.CrossRefGoogle Scholar
Parthasarathy, A. & Saxton-Fox, T. 2023 A family of adverse pressure gradient turbulent boundary layers with upstream favourable pressure gradients. J. Fluid Mech. 966, A11.CrossRefGoogle Scholar
Perry, A.E., Bell, J.B. & Joubert, P.N. 1966 Velocity and temperature profiles in adverse pressure gradient turbulent boundary layers. J. Fluid Mech. 25 (2), 299–320.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Pozuelo, R., Li, Q., Schlatter, P. & Vinuesa, R. 2022 An adverse-pressure-gradient turbulent boundary layer with nearly constant β ≈ 1.4 up to Re θ ≈ 8700. J. Fluid Mech. 939, A34.CrossRefGoogle Scholar
Prandtl, L. 1925 A report on testing for built-up turbulence. Z. Angew. Math. Mech. 5 (2), 136–139.CrossRefGoogle Scholar
Raupach, M.R., Finnigan, J.J. & Brunet, Y. 1996 Coherent eddies and turbulence in vegetation canopies: the mixing-layer analogy. Boundary-Layer Meteorol. 78 (3–4), 351–382.CrossRefGoogle Scholar
Romero, S., Zimmerman, S., Philip, J., White, C. & Klewicki, J. 2022 a Properties of the inertial sublayer in adverse pressure-gradient turbulent boundary layers. J. Fluid Mech. 937, A30.CrossRefGoogle Scholar
Romero, S.K., Zimmerman, S.J., Philip, J. & Klewicki, J.C. 2022 b Stress equation based scaling framework for adverse pressure gradient turbulent boundary layers. Intl J Heat Fluid Flow 93, 108885.CrossRefGoogle Scholar
Sanmiguel Vila, C., Örlü, R., Vinuesa, R., Schlatter, P., Ianiro, A. & Discetti, S. 2017 Adverse-pressure-gradient effects on turbulent boundary layers: statistics and flow-field organization. Flow Turbul. Combust. 99 (3–4), 589–612.CrossRefGoogle ScholarPubMed
Schlatter, P., Örlü, R., Li, Q., Brethouwer, G., Fransson, J.H.M., Johansson, A.V., Alfredsson, P.H. & Henningson, D.S. 2009 Turbulent boundary layers up to Re θ = 2500 studied through simulation and experiment. Phys. Fluids 21 (5), 51702.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2016 Boundary-Layer Theory. Springer.Google Scholar
Simpson, R.L. 1989 Turbulent boundary-layer separation. Annu. Rev. Fluid Mech. 21 (1), 205–232.CrossRefGoogle Scholar
Skote, M., Henningson, D.S. & Henkes, R.A.W.M. 1998 Direct numerical simulation of self-similar turbulent boundary layers in adverse pressure gradients. Flow Turbul. Combust. 60 (1), 47–85.CrossRefGoogle Scholar
Smits, A.J., McKeon, B.J. & Marusic, I. 2011 High-Reynolds number wall turbulence. Ann. Rev. Fluid Mech. 43 (1), 353–375.CrossRefGoogle Scholar
Smits, A.J. & Wood, D.H. 1985 The response of turbulent boundary layers to sudden perturbations. Ann. Rev. Fluid Mech. 17 (1), 321–358.CrossRefGoogle Scholar
So, R.M.C. & Speziale, C.G. 1999 A review of turbulent heat transfer modeling. Annu. Rev. Heat Transfer 10 (10), 177–220.CrossRefGoogle Scholar
Spalart, P.R. & Watmuff, J.H. 1993 Experimental and numerical study of a turbulent boundary layer with pressure gradients. J. Fluid Mech. 249, 337–371.CrossRefGoogle Scholar
de Stadler, M.B., Sarkar, S. & Brucker, K.A. 2010 Effect of the Prandtl number on a stratified turbulent wake. Phys. Fluids 22 (9), 095102.CrossRefGoogle Scholar
Stratford, B.S. 1959 The prediction of separation of the turbulent boundary layer. J. Fluid Mech. 5 (1), 1–16.CrossRefGoogle Scholar
Subrahmanyam, M.A., Cantwell, B.J. & Alonso, J.J. 2022 A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers. J. Fluid Mech. 933, A16.CrossRefGoogle Scholar
Tanarro, Á., Vinuesa, R. & Schlatter, P. 2020 Effect of adverse pressure gradients on turbulent wing boundary layers. J. Fluid Mech. 883, A8.CrossRefGoogle Scholar
Taylor, P.S. & Seddighi, M. 2024 Turbulent–turbulent transient concept in pulsating flows. J. Fluid Mech. 982, A20.CrossRefGoogle Scholar
Townsend, A.A.R. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Vinuesa, R., Örlü, R., Sanmiguel Vila, C., Ianiro, A., Discetti, S. & Schlatter, P. 2017 Revisiting history effects in adverse-pressure-gradient turbulent boundary layers. Flow Turbul. Combust. 99 (3–4), 565–587.CrossRefGoogle ScholarPubMed
Vinuesa, R., Rozier, P.H., Schlatter, P. & Nagib, H.M. 2014 Experiments and computations of localized pressure gradients with different history effects. AIAA J. 52 (2), 368–384.CrossRefGoogle Scholar
Vishwanathan, V., Fritsch, D.J., Lowe, K.T. & Devenport, W.J. 2023 History effects and wall-similarity of non-equilibrium turbulent boundary layers in varying pressure gradient over rough and smooth surfaces. Intl J. Heat Fluid Flow 102, 109145.CrossRefGoogle Scholar
Volino, R.J. 2020 Non-equilibrium development in turbulent boundary layers with changing pressure gradients. J. Fluid Mech. 897, A2.CrossRefGoogle Scholar
Wei, T., Li, Z., Knopp, T. & Vinuesa, R. 2023 The mean wall-normal velocity in turbulent boundary layer flows under pressure gradient. J. Fluid Mech. 975, A27.CrossRefGoogle Scholar
Yang, X.I.A., Sadique, J., Mittal, R. & Meneveau, C. 2015 Integral wall model for large eddy simulations of wall-bounded turbulent flows. Phys. Fluids 27 (2), 025112.CrossRefGoogle Scholar
Zhang, W., Zhu, X., Yang, X.I.A. & Wan, M. 2022 Evidence for Raupach et al.’s mixing-layer analogy in deep homogeneous urban-canopy flows. J. Fluid Mech. 944, A46.CrossRefGoogle Scholar