Hostname: page-component-5c6d5d7d68-thh2z Total loading time: 0 Render date: 2024-08-16T07:49:51.888Z Has data issue: false hasContentIssue false
  • English
  • Français

Curvature effects on the structure of near-wall turbulence

Published online by Cambridge University Press:  06 October 2023

Davide Selvatici Open the ORCID record for Davide Selvatici [Opens in a new window] ,
Maurizio Quadrio Open the ORCID record for Maurizio Quadrio [Opens in a new window]  and
Alessandro Chiarini Open the ORCID record for Alessandro Chiarini [Opens in a new window]
Davide Selvatici*
Affiliation:
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy
Maurizio Quadrio
Affiliation:
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy
Alessandro Chiarini*
Affiliation:
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy
*
‡Present address: Physics of Fluids Group, Max Planck Center Twente for Complex Fluid Dynamics, J. M. Burgers Center for Fluid Dynamics, University of Twente, P. O. Box 217, Enschede 7500 AE, The Netherlands.
†Present address: Complex Fluids and Flows Unit, Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan. Email address for correspondence: alessandro.chiarini@polimi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

The interaction between near-wall turbulence and wall curvature is described for the incompressible flow in a plane channel with a small concave–convex–concave bump on the bottom wall, with height comparable to the wall-normal location of the main turbulent structures. The analysis starts from a database generated by a direct numerical simulation and hinges upon the anisotropic generalised Kolmogorov equations, i.e. the exact budget equations for the second-order structure function tensor. The influence of the bump on the wall cycle and on the energy production, redistribution and transfers is described in the physical and scale spaces. Over the upstream side of the bump, the energy drained from the mean flow to sustain the streamwise fluctuations decreases, and the streaks of high and low streamwise velocity weaken and are stretched spanwise. After the bump tip, instead, the production of streamwise fluctuations grows and the streaks intensify, progressively recovering their characteristic spanwise scale. The wall-normal fluctuations, and thus the quasi-streamwise vortices, are sustained by the mean flow over the upstream side of the bump, while energy flows from the vertical fluctuations to the mean field over the downstream side. On the concave portion of the upstream side, the near-wall fluctuations form structures of spanwise velocity which are consistent with Taylor–Görtler vortices at an early stage of development. Their evolution is described by analysing the scale-space pressure–strain term. A schematic description of the bump flow is presented, in which various regions are identified according to the signs of curvature and streamwise pressure gradient.

JFM classification

Waves/Free-surface Flows: Channel flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 972 , 10 October 2023 , A39
Check for updates
Copyright
© The Author(s), 2023. 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.)

References

Banchetti, J., Luchini, P. & Quadrio, M. 2020 Turbulent drag reduction over curved walls. J. Fluid Mech. 896, 1–23.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
Benmalek, A. & Saric, W.S. 1994 Effects of curvature variations on the nonlinear evolution of Görtler vortices. Phys. Fluids 6 (10), 3353–3367.CrossRefGoogle Scholar
Berger, S.A. & Jou, L.-D. 2000 Flows in stenotic vessels. Annu. Rev. Fluid Mech. 32 (1), 347–382.CrossRefGoogle Scholar
Blackwelder, R.F. & Kaplan, R.E. 1976 On the wall structure of the turbulent boundary layer. J. Fluid Mech. 76, 89–112.CrossRefGoogle Scholar
Bradshaw, P. 1973 Effects of streamline curvature on turbulent flow. AGARD 169.Google Scholar
Breuer, M., Peller, N., Rapp, C. & Manhart, M. 2009 Flow over periodic hills: a numerical and experimental study in a wide range of Reynolds number. Comput. Fluids 38 (2), 433–457.CrossRefGoogle Scholar
Chiarini, A., Gatti, D., Cimarelli, A. & Quadrio, M. 2022 a Structure of turbulence in the flow around a rectangular cylinder. J. Fluid Mech. 946, A35.CrossRefGoogle Scholar
Chiarini, A., Mauriello, M., Gatti, D. & Quadrio, M. 2022 b Ascending-descending and direct-inverse cascades of Reynolds stresses in turbulent Couette flow. J. Fluid Mech. 930, A9.CrossRefGoogle Scholar
Cimarelli, A., De Angelis, E. & Casciola, C.M. 2013 Paths of energy in turbulent channel flows. J. Fluid Mech. 715, 436–451.CrossRefGoogle Scholar
Cimarelli, A., De Angelis, E., Jimenez, J. & Casciola, C.M. 2016 Cascades and wall-normal fluxes in turbulent channel flows. J. Fluid Mech. 796, 417–436.CrossRefGoogle Scholar
Cimarelli, A., Leonforte, A., De Angelis, E., Crivellini, A. & Angeli, D. 2019 On negative turbulence production phenomena in the shear layer of separating and reattaching flows. Phys. Lett. A 383 (10), 1019–1026.CrossRefGoogle Scholar
Dagaut, J., Negretti, M.E., Balarac, G. & Brun, C. 2021 Linear to turbulent Görtler instability transition. Phys. Fluids 33 (1), 014102.CrossRefGoogle Scholar
Danaila, L., Anselmet, F., Zhou, T. & Antonia, R.A. 2001 Turbulent energy scale budget equations in a fully developed channel flow. J. Fluid Mech. 430, 87–109.CrossRefGoogle Scholar
Floryan, J.M. 1991 On the Görtler instability of boundary layers. Prog. Aerosp. Sci. 28 (3), 235–271.CrossRefGoogle Scholar
Floryan, J.M. & Saric, W.S. 1982 Stability of Görtler vortices in boundary layers. AIAA J. 20 (3), 316–324.CrossRefGoogle Scholar
Fröhlich, J., Mellen, C.P., Rodi, W., Temmerman, L. & Leschziner, M.A. 2005 Highly resolved large-eddy simulation of separated flow in a channel with streamwise periodic constrictions. J. Fluid Mech. 526, 19–66.CrossRefGoogle Scholar
Gallaire, F., Marquillie, M. & Ehrenstein, U. 2007 Three-dimensional transverse instabilities in detached boundary layers. J. Fluid Mech. 571, 221–233.CrossRefGoogle Scholar
Gatti, D., Chiarini, A., Cimarelli, A. & Quadrio, M. 2020 Structure function tensor equations in inhomogeneous turbulence. J. Fluid Mech. 898, A5.CrossRefGoogle Scholar
Gatti, D., Remigi, A., Chiarini, A., Cimarelli, A. & Quadrio, M. 2019 An efficient numerical method for the generalized Kolmogorov equation. J. Turbul. 20 (8), 457–480.CrossRefGoogle Scholar
Görtler, H. 1941 Instabilität laminarer Grenzschichten an konkaven Wänden gegenüber gewissen dreidimensionalen Störungen. Z. Angew. Math. Mech. 21 (4), 250–252.CrossRefGoogle Scholar
Hall, P. 1985 The Görtler vortex instability mechanism in three-dimensional boundary layers. Proc. R. Soc. Lond. A 399 (1816), 135–152.Google Scholar
Hamilton, J., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.CrossRefGoogle Scholar
Hill, R.J. 2001 Equations relating structure functions of all orders. J. Fluid Mech. 434, 379–388.CrossRefGoogle Scholar
Jeong, J., Hussain, F., Schoppa, W. & Kim, J. 1997 Coherent structures near the wall in a turbulent channel flow. J. Fluid Mech. 332, 185–214.CrossRefGoogle Scholar
Jiménez, J. 2016 Optimal fluxes and Reynolds stresses. J. Fluid Mech. 809, 585–600.CrossRefGoogle Scholar
Jiménez, J. 2018 Coherent structures in wall-bounded turbulence. J. Fluid Mech. 842 (P1), 1–99.CrossRefGoogle Scholar
Jiménez, J. 2022 The streaks of wall-bounded turbulence need not be long. J. Fluid Mech. 945, R3.CrossRefGoogle Scholar
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359.CrossRefGoogle Scholar
Kähler, C.J., Scharnowski, S. & Cierpka, C. 2016 Highly resolved experimental results of the separated flow in a channel with streamwise periodic constrictions. J. Fluid Mech. 796, 257–284.CrossRefGoogle Scholar
Kawata, T. & Alfredsson, P.H. 2018 Inverse interscale transport of the Reynolds shear stress in plane Couette turbulence. Phys. Rev. Lett. 120 (24), 244501.CrossRefGoogle ScholarPubMed
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
Kuban, L., Laval, J.P., Elsner, W., Tyliszczak, A. & Marquillie, M. 2012 LES modeling of converging-diverging turbulent channel flow. J. Turbul. 13 (11), 1–19.CrossRefGoogle Scholar
Lee, M. & Moser, R.D. 2019 Spectral analysis of the budget equation in turbulent channel flows at high $Re$. J. Fluid Mech. 860, 886–938.CrossRefGoogle Scholar
Luchini, P. 2013 Linearized no-slip boundary conditions at a rough surface. J. Fluid Mech. 737, 349–367.CrossRefGoogle Scholar
Luchini, P. 2016 Immersed-boundary simulation of turbulent flow past a sinusoidally undulated river bottom. Eur. J. Mech. B/Fluids 55, 340–347.CrossRefGoogle Scholar
Luchini, P. 2020 CPL. Available at https://CPLcode.net.Google Scholar
Luchini, P. 2021 Introducing CPL. arXiv:2012.12143.Google Scholar
Luchini, P. & Bottaro, A. 1998 Görtler vortices: a backward-in-time approach to the receptivity problem. J. Fluid Mech. 363, 1–23.CrossRefGoogle Scholar
Lui, M., Martino, S., Salerno, M. & Quadrio, M. 2019 On the turbulence modeling of blood flow in a stenotic vessel. J. Biomech. Engng 142, 011009.CrossRefGoogle Scholar
Mansour, N., Kim, J. & Moin, P. 1988 Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech. 194, 15–44.CrossRefGoogle Scholar
Marati, N., Casciola, C.M. & Piva, R. 2004 Energy cascade and spatial fluxes in wall turbulence. J. Fluid Mech. 521, 191–215.CrossRefGoogle Scholar
Marquillie, M., Ehrenstein, U. & Laval, J.P. 2011 Instability of streaks in wall turbulence with adverse pressure gradient. J. Fluid Mech. 681, 205–240.CrossRefGoogle Scholar
Marquillie, M., Laval, J.-P. & Dolganov, R. 2008 Direct numerical simulation of a separatedchannel flow with a smooth profile. J. Turbul. 9, 1–23.CrossRefGoogle Scholar
Mollicone, J.-P., Battista, F., Gualtieri, P. & Casciola, C.M. 2017 Effect of geometry and Reynolds number on the turbulent separated flow behind a bulge in a channel. J. Fluid Mech. 823, 100–133.CrossRefGoogle Scholar
Mollicone, J.-P., Battista, F., Gualtieri, P. & Casciola, C.M. 2018 Turbulence dynamics in separated flows: the generalised Kolmogorov equation for inhomogeneous anisotropic conditions. J. Fluid Mech. 841, 1012–1039.CrossRefGoogle Scholar
Moser, R.D. & Moin, P. 1987 The effects of curvature in wall-bounded turbulent flows. J. Fluid Mech. 175, 479–510.CrossRefGoogle Scholar
Panton, R. 1997 Self-sustaining mechanisms of wall turbulence. In Advances in Fluid Mechanics, vol. 15. Computational Mechanics.Google Scholar
Peerhossaini, H. & Wesfreid, J.E. 1988 On the inner structure of streamwise Görtler rolls. Intl J. Heat Fluid Flow 9 (1), 12–18.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Robinson, S.K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639.CrossRefGoogle Scholar
Saric, W.S. 1994 Görtler vortices. Annu. Rev. Fluid Mech. 26 (1), 379–409.CrossRefGoogle Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.CrossRefGoogle Scholar
Smits, A.J. & Wood, D.H. 1985 The response of turbulent boundary layers to sudden perturbations. Annu. Rev. Fluid Mech., 17, 321–358.CrossRefGoogle Scholar
Webster, D.R., Degraaff, D.B. & Eaton, J.K. 1996 a Turbulence characteristics of a boundary layer over a swept bump. J. Fluid Mech. 323, 1–22.CrossRefGoogle Scholar
Webster, D.R., Degraaff, D.B. & Eaton, J.K. 1996 b Turbulence characteristics of a boundary layer over a two-dimensional bump. J. Fluid Mech. 320, 53–69.CrossRefGoogle Scholar
Wu, X. & Squires, K.D. 1998 Numerical investigation of the turbulent boundary layer over a bump. J. Fluid Mech. 362, 229–271.CrossRefGoogle Scholar
Xu, D., Liu, J. & Wu, X. 2020 Görtler vortices and streaks in boundary layer subject to pressure gradient: excitation by free stream vortical disturbances, nonlinear evolution and secondary instability. J. Fluid Mech. 900, A15.CrossRefGoogle Scholar
Xu, D., Zhang, Y. & Wu, X. 2017 Nonlinear evolution and secondary instability of steady and unsteady Görtler vortices induced by free-stream vortical disturbances. J. Fluid Mech. 829, 681–730.CrossRefGoogle Scholar