zbMATH — the first resource for mathematics

Quadrupolar flows around spots in internal shear flows. (English) Zbl 07190261
Summary: Turbulent spots occur in shear flows confined between two walls and are surrounded by robust quadrupolar flows. Although the far-field decay of such large-scale flows has been reported to be exponential, we predict a different algebraic decay for the case of plane Couette flow. We address this problem theoretically, by modelling an isolated spot as an obstacle in a linear plane shear flow with free-slip boundary conditions at the walls. By seeking invariant solutions in a co-moving Lagrangian frame and using geometric scale separation, a set of differential equations governing large-scale flows is derived from the Navier-Stokes equations and solved analytically. The wall-normal velocity turns out to be exponentially localised in the plane, while the quadrupolar in-plane velocity field, after wall-normal averaging, features a superposition of algebraic and exponential decays. The algebraic decay exponent is \(-3\). The quadrupolar angular dependence stems from (i) the shearing of the streamwise velocity and (ii) the breaking of the spanwise homogeneity. Near the spot, exponentially decaying solutions can generate reversed quadrupolar flows. Eventually, by noting that the algebraically decaying in-plane flow is two-dimensional and harmonic, we suggest a topological origin to the quadrupolar large-scale flow.

76F06 Transition to turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI
[1] Abramowitz, M. & Stegun, I. A.1965Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Courier Corporation. · Zbl 0171.38503
[2] Alavyoon, F., Henningson, D. S. & Alfredsson, P. H.1986Turbulent spots in plane Poiseuille flow-flow visualization. Phys. Fluids29 (4), 1328-1331.
[3] Barkley, D. & Tuckerman, L. S.2007Mean flow of turbulent-laminar patterns in plane Couette flow. J. Fluid Mech.576, 109-137. · Zbl 1124.76018
[4] Blake, J. R.1971A note on the image system for a Stokeslet in a no-slip boundary. Math. Proc. Camb. Phil. Soc.70 (2), 303-310. · Zbl 0244.76016
[5] Bottin, S., Dauchot, O. & Daviaud, F.1997Intermittency in a locally forced plane Couette flow. Phys. Rev. Lett.79 (22), 4377-4380.
[6] Bottin, S., Dauchot, O. & Daviaud, F.1998Experimental evidence of streamwise vortices as finite amplitude solutions in transitional plane Couette flow. Phys. Fluids10 (10), 2597-2607.
[7] Brand, E. & Gibson, J. F.2014A doubly localized equilibrium solution of plane Couette flow. J. Fluid Mech.750, R3.
[8] Carlson, D. R., Widnall, S. E. & Peeters, M. F.1982A flow-visualization study of transition in plane Poiseuille flow. J. Fluid Mech.121, 487-505.
[9] Chaikin, P. M. & Lubensky, T. C.1995Principles of Condensed Matter Physics. Cambridge University Press.
[10] Chantry, M., Tuckerman, L. S. & Barkley, D.2016Turbulent-laminar patterns in shear flows without walls. J. Fluid Mech.791, R8. · Zbl 1382.76106
[11] Chantry, M., Tuckerman, L. S. & Barkley, D.2017Universal continuous transition to turbulence in a planar shear flow. J. Fluid Mech.824, R1.
[12] Coles, D.1965Transition in circular Couette flow. J. Fluid Mech.21, 385-425. · Zbl 0134.21705
[13] Couliou, M. & Monchaux, R.2015Large-scale flows in transitional plane Couette flow: a key ingredient of the spot growth mechanism. Phys. Fluids27 (3), 034101. · Zbl 1383.76188
[14] Couliou, M. & Monchaux, R.2016Spreading of turbulence in plane Couette flow. Phys. Rev. E93, 013108. · Zbl 1383.76188
[15] Couliou, M. & Monchaux, R.2017Growth dynamics of turbulent spots in plane Couette flow. J. Fluid Mech.819, 1-20. · Zbl 1383.76188
[16] Couliou, M. & Monchaux, R.2018Childhood of turbulent spots in a shear flow. Phys. Rev. F3 (12), 123901. · Zbl 1383.76188
[17] Dauchot, O. & Daviaud, F.1995aFinite amplitude perturbation and spots growth mechanism in plane Couette flow. Phys. Fluids7, 335-343.
[18] Dauchot, O. & Daviaud, F.1995bStreamwise vortices in plane Couette flow. Phys. Fluids7, 901-903.
[19] Daviaud, F., Hegseth, J. & Bergé, P.1992Subcritical transition to turbulence in plane Couette flow. Phys. Rev. Lett.69, 2511-2514.
[20] Duguet, Y., Maitre, O. L. & Schlatter, P.2011Stochastic and deterministic motion of a laminar-turbulent front in a spanwisely extended Couette flow. Phys. Rev. E84 (6), 066315.
[21] Duguet, Y. & Schlatter, P.2013Oblique laminar-turbulent interfaces in plane shear flows. Phys. Rev. Lett.110, 034502.
[22] Duguet, Y., Schlatter, P. & Henningson, D. S.2010Formation of turbulent patterns near the onset of transition in plane Couette flow. J. Fluid Mech.650, 119-129. · Zbl 1189.76254
[23] Eckhardt, B. & Pandit, R.2003Noise correlations in shear flows. Eur. Phys. J. B33 (3), 373-378.
[24] Emmons, H. W.1951The laminar-turbulent transition in a boundary layer-part I. J. Aeronaut. Sci.18 (7), 490-498. · Zbl 0043.19109
[25] Gradshteyn, I. S. & Ryžhik, I. M.2014Table of Integrals, Series, and Products. Academic Press. · Zbl 0918.65002
[26] Grenier, E. & Nguyen, T. T.2019Green function of Orr-Sommerfeld equations away from critical layers. SIAM J. Math. Anal.51 (2), 1279-1296. · Zbl 1414.35149
[27] Guckenheimer, J. & Holmes, P.2013Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer. · Zbl 0515.34001
[28] Henningson, D. S. & Kim, J.1991On turbulent spots in plane Poiseuille flow. J. Fluid Mech.228, 183-205. · Zbl 0723.76048
[29] Howison, S.2005Practical Applied Mathematics: Modelling, Analysis, Approximation. Cambridge University Press. · Zbl 1226.00032
[30] Ishida, T., Duguet, Y. & Tsukahara, T.2016Transitional structures in annular Poiseuille flow depending on radius ratio. J. Fluid Mech.794, R2.
[31] Jiménez, J.2018Coherent structures in wall-bounded turbulence. J. Fluid Mech.842, P1. · Zbl 1419.76316
[32] Jost, J.1995Riemannian Geometry and Geometric Analysis. Springer. · Zbl 0828.53002
[33] Kelvin, Lord1887On ship waves. Proc. Inst. Mech. Engrs38, 641-649.
[34] Klotz, L., Lemoult, G., Frontczak, I., Tuckerman, L. S. & Wesfreid, J. E.2017Couette-Poiseuille flow experiment with zero mean advection velocity: subcritical transition to turbulence. Phys. Rev. F2, 043904.
[35] Lagha, M. & Manneville, P.2007Modeling of plane Couette flow. I. Large scale flow around turbulent spots. Phys. Fluids19, 094105. · Zbl 1182.76422
[36] Lefschetz, S.1949Introduction to Topology. Princeton University Press.
[37] Lemoult, G., Aider, J. L. & Wesfreid, J. E.2013Turbulent spots in a channel: large-scale flow and self-sustainability. J. Fluid Mech.731, R1. · Zbl 1294.76165
[38] Li, F. & Widnall, S. E.1989Wave patterns in plane Poiseuille flow created by concentrated disturbances. J. Fluid Mech.208, 639-656. · Zbl 0681.76055
[39] Liron, N. & Mochon, S.1976Stokes flow for a Stokeslet between two parallel flat plates. J. Engng Maths10 (4), 287-303. · Zbl 0377.76030
[40] Lundbladh, A. & Johansson, A. V.1991Direct simulation of turbulent spots in plane Couette flow. J. Fluid Mech.229, 499-516. · Zbl 0850.76256
[41] Marqués, F.1990On boundary conditions for velocity potentials in confined flows: application to Couette flow. Phys. Fluids2 (3), 729-737. · Zbl 0703.76028
[42] Prigent, A., Grégoire, G., Chaté, H. & Dauchot, O.2003Long-wavelength modulation of turbulent shear flows. Physica D174 (1-4), 100-113. · Zbl 1036.76023
[43] Prigent, A., Grégoire, G., Chaté, H., Dauchot, O. & Van Saarloos, W.2002Large-scale finite-wavelength modulation within turbulent shear flows. Phys. Rev. Lett.89, 014501. · Zbl 1036.76023
[44] Reynolds, O.1883An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. R. Soc. Lond. A174, 935-982. · JFM 16.0845.02
[45] Ritter, P., Zammert, S., Song, B., Eckhardt, B. & Avila, M.2018Analysis and modeling of localized invariant solutions in pipe flow. Phys. Rev. F3, 013901.
[46] Romanov, V. A.1973Stability of plane-parallel Couette flow. Funct. Anal. Applics7, 137-146. · Zbl 0287.76037
[47] Samanta, D., Lozar, A. D. & Hof, B.2011Experimental investigation of laminar turbulent intermittency in pipe flow. J. Fluid Mech.681, 193-204. · Zbl 1241.76047
[48] Schumacher, J. & Eckhardt, B.2001Evolution of turbulent spots in a parallel shear flow. Phys. Rev. E63, 046307.
[49] Tardu, S.2012Forcing a low Reynolds number channel flow to generate synthetic turbulent-like structures. Comput. Fluids55, 101-108. · Zbl 1291.76189
[50] Taylor, G. I.1938The spectrum of turbulence. Proc. R. Soc. Lond.164 (919), 476-490. · JFM 64.1454.02
[51] Tillmark, N. & Alfredsson, P. H.1992Experiments on transition in plane Couette flow. J. Fluid Mech.235, 89-102.
[52] Wang, Z.2019 Localised and bifurcating structures in planar shear flows. PhD thesis, Nanyang Technological University.
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.