zbMATH — the first resource for mathematics

Self-similar invariant solution in the near-wall region of a turbulent boundary layer at asymptotically high Reynolds numbers. (English) Zbl 07173505
Summary: At sufficiently high Reynolds numbers, shear-flow turbulence close to a wall acquires universal properties. When length and velocity are rescaled by appropriate characteristic scales of the turbulent flow and thereby measured in inner units, the statistical properties of the flow become independent of the Reynolds number. We demonstrate the existence of a wall-attached non-chaotic exact invariant solution of the fully nonlinear three-dimensional Navier-Stokes equations for a parallel boundary layer that captures the characteristic self-similar scaling of near-wall turbulent structures. The branch of travelling wave solutions can be followed up to \(Re=1\,000\,000\). Combined theoretical and numerical evidence suggests that the solution is asymptotically self-similar and exactly scales in inner units for Reynolds numbers tending to infinity. Demonstrating the existence of invariant solutions that capture the self-similar scaling properties of turbulence in the near-wall region is a step towards extending the dynamical systems approach to turbulence from the transitional regime to fully developed boundary layers.
76 Fluid mechanics
Full Text: DOI
[1] Avila, M., Mellibovsky, F., Roland, N. & Hof, B.2013Streamwise-localized solutions at the onset of turbulence in pipe flow. Phys. Rev. Lett.110 (22), 224502.
[2] Bobke, A., Örlü, R. & Schlatter, P.2016Simulations of turbulent asymptotic suction boundary layers. J. Turbul.17 (2), 157-180.
[3] Chandler, G. J. & Kerswell, R. R.2013Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow. J. Fluid Mech.722, 554-595. · Zbl 1287.76124
[4] Cvitanović, P., Artuso, R., Mainieri, G., Tanner, G. & Vattay, G.2016 Chaos: Classical and Quantum. Niels Bohr Institute. Available at: .
[5] Deguchi, K.2015Self-sustained states at Kolmogorov microscale. J. Fluid Mech.781, R6. · Zbl 1359.76011
[6] Deguchi, K. & Hall, P.2014aFree-stream coherent structures in parallel boundary-layer flows. J. Fluid Mech.752, 602-625.
[7] Deguchi, K. & Hall, P.2014bThe high-Reynolds-number asymptotic development of nonlinear equilibrium states in plane Couette flow. J. Fluid Mech.750, 99-112.
[8] Eckhardt, B. & Zammert, S.2018Small scale exact coherent structures at large Reynolds numbers in plane Couette flow. Nonlinearity31 (2), R66-R77. · Zbl 1388.37079
[9] Gibson, J. F., Halcrow, J. & Cvitanović, P.2008Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech.611, 107-130. · Zbl 1151.76453
[10] Gibson, J. F., Reetz, F., Azimi, S., Ferraro, A., Kreilos, T., Schrobsdorff, H., Farano, M., Yesil, A. F., Schütz, S. S., Culpo, M.et al. 2020 Channelflow 2.0. (in preparation). Available at: .
[11] Hocking, L. M.1975Non-linear instability of the asymptotic suction velocity profile. Q. J. Mech. Appl. Maths28 (3), 341-353. · Zbl 0321.76021
[12] Jiménez, J.2018Coherent structures in wall-bounded turbulence. J. Fluid Mech.842, P1. · Zbl 1419.76316
[13] Jiménez, J. & Simens, M.2001Low-dimensional dynamics of a turbulent wall flow. J. Fluid Mech.435, 81-91. · Zbl 1022.76022
[14] Kawahara, G., Uhlmann, M. & van Veen, L.2012The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech.44 (1), 203-225. · Zbl 1352.76031
[15] Khapko, T., Schlatter, P., Duguet, Y. & Henningson, D. S.2016Turbulence collapse in a suction boundary layer. J. Fluid Mech.795, 356-379. · Zbl 1359.76144
[16] Kim, J., Moin, P. & Moser, R.1987Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech.177, 133-166. · Zbl 0616.76071
[17] Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W.1967The structure of turbulent boundary layers. J. Fluid Mech.30 (04), 741-773.
[18] Nagata, M.1990Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech.217, 519-527.
[19] Neelavara, S. A., Duguet, Y. & Lusseyran, F.2017State space analysis of minimal channel flow. Fluid Dyn. Res.49 (3), 035511.
[20] Park, J. S. & Graham, M. D.2015Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech.782, 430-454. · Zbl 1381.76097
[21] Pope, S. B.2000Turbulent Flows. Cambridge University Press.
[22] Rawat, S., Cossu, C., Hwang, Y. & Rincon, F.2015On the self-sustained nature of large-scale motions in turbulent Couette flow. J. Fluid Mech.782, 515-540. · Zbl 1381.76098
[23] Rawat, S., Cossu, C. & Rincon, F.2016Travelling-wave solutions bifurcating from relative periodic orbits in plane Poiseuille flow. C. R. Méc344 (6), 448-455.
[24] Reetz, F., Kreilos, T. & Schneider, T. M.2019Exact invariant solution reveals the origin of self-organized oblique turbulent-laminar stripes. Nature Commun.10 (1), 2277.
[25] Schlatter, P. & Örlü, R.2011Turbulent asymptotic suction boundary layers studied by simulation. J. Phys.: Conf. Ser.318 (2), 022020. · Zbl 1205.76139
[26] Schlichting, H.2004Boundary-layer Theory. Springer. · Zbl 1059.18007
[27] Schneider, T. M., Eckhardt, B. & Yorke, J. A.2007Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett.99, 34502.
[28] Skufca, J. D., Yorke, J. A. & Eckhardt, B.2006Edge of chaos in a parallel shear flow. Phys. Rev. Lett.96, 174101.
[29] Suri, B., Tithof, J., Grigoriev, R. O. & Schatz, M. F.2017Forecasting fluid flows using the geometry of turbulence. Phys. Rev. Lett.118 (11), 114501.
[30] Wang, J., Gibson, J. F. & Waleffe, F.2007Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett.98 (20), 204501.
[31] Yang, Q., Willis, A. P. & Hwang, Y.2019Exact coherent states of attached eddies in channel flow. J. Fluid Mech.862, 1029-1059. · Zbl 1415.76373
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.