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Self-sustainment of coherent structures in counter-rotating Taylor–Couette flow

Published online by Cambridge University Press:  07 November 2022

B. Wang Open the ORCID record for B. Wang [Opens in a new window] ,
R. Ayats Open the ORCID record for R. Ayats [Opens in a new window] ,
K. Deguchi Open the ORCID record for K. Deguchi [Opens in a new window] ,
F. Mellibovsky Open the ORCID record for F. Mellibovsky [Opens in a new window]  and
A. Meseguer Open the ORCID record for A. Meseguer [Opens in a new window]
B. Wang
Affiliation:
Departament de Física, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
R. Ayats
Affiliation:
Institute of Science and Technology Austria (ISTA), 3400 Klosterneuburg, Austria
K. Deguchi
Affiliation:
School of Mathematics, Monash University, VIC 3800, Australia
F. Mellibovsky*
Affiliation:
Departament de Física, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
A. Meseguer
Affiliation:
Departament de Física, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
*
†Email address for correspondence: fernando.mellibovsky@upc.edu
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Abstract

We investigate the local self-sustained process underlying spiral turbulence in counter-rotating Taylor–Couette flow using a periodic annular domain, shaped as a parallelogram, two of whose sides are aligned with the cylindrical helix described by the spiral pattern. The primary focus of the study is placed on the emergence of drifting–rotating waves (DRW) that capture, in a relatively small domain, the main features of coherent structures typically observed in developed turbulence. The transitional dynamics of the subcritical region, far below the first instability of the laminar circular Couette flow, is determined by the upper and lower branches of DRW solutions originated at saddle-node bifurcations. The mechanism whereby these solutions self-sustain, and the chaotic dynamics they induce, are conspicuously reminiscent of other subcritical shear flows. Remarkably, the flow properties of DRW persist even as the Reynolds number is increased beyond the linear stability threshold of the base flow. Simulations in a narrow parallelogram domain stretched in the azimuthal direction to revolve around the apparatus a full turn confirm that self-sustained vortices eventually concentrate into a localised pattern. The resulting statistical steady state satisfactorily reproduces qualitatively, and to a certain degree also quantitatively, the topology and properties of spiral turbulence as calculated in a large periodic domain of sufficient aspect ratio that is representative of the real system.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Turbulent Flows: Intermittency
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 951 , 25 November 2022 , A21
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© The Author(s), 2022. Published by Cambridge University Press

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References

Andereck, C.D., Liu, S.S. & Swinney, H.L. 1986 Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech. 164, 155–183.CrossRefGoogle Scholar
Avila, M., Mellibovsky, F., Roland, N. & Hof, B. 2013 Streamwise-localized solutions at the onset of turbulence in pipe flow. Phys. Rev. Lett. 110 (22), 224502.CrossRefGoogle ScholarPubMed
Ayats, R., Deguchi, K., Mellibovsky, F. & Meseguer, A. 2020 a Fully nonlinear mode competition in magnetised Taylor–Couette flow. J. Fluid Mech. 897, A14.CrossRefGoogle Scholar
Ayats, R., Meseguer, A. & Mellibovsky, F. 2020 b Symmetry-breaking waves and space–time modulation mechanisms in two-dimensional plane Poiseuille flow. Phys. Rev. Fluids 5 (9), 094401.CrossRefGoogle Scholar
Barkley, D. & Tuckerman, L.S. 2005 Computational study of turbulent laminar patterns in Couette flow. Phys. Rev. Lett. 94 (1), 014502.CrossRefGoogle ScholarPubMed
Barnett, J., Gurevich, D.R. & Grigoriev, R.O. 2017 Streamwise localization of traveling wave solutions in channel flow. Phys. Rev. E 95 (3), 033124.CrossRefGoogle ScholarPubMed
Bender, C.M. & Orszag, S.A. 1999 Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, 1st edn. Springer.CrossRefGoogle Scholar
Berghout, P., Dingemans, R.J., Zhu, X., Verzicco, R., Stevens, R.J.A.M., van Saarloos, W. & Lohse, D. 2020 Direct numerical simulations of spiral Taylor–Couette turbulence. J. Fluid Mech. 887, A18.CrossRefGoogle Scholar
Boberg, L. & Brosa, U. 1988 Onset of turbulence in a pipe. Z. Naturforsch. 43a, 697–726.CrossRefGoogle Scholar
Brand, E. & Gibson, J.F. 2014 A doubly localized equilibrium solution of plane Couette flow. J. Fluid Mech. 750, R3.CrossRefGoogle Scholar
Canuto, C.G., Hussaini, M.Y., Quarteroni, A. & Zang, T.A. 2007 Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer.CrossRefGoogle Scholar
Canuto, C.G., Hussaini, M.Y., Quarteroni, A. & Zang, T.A. 2010 Spectral Methods: Fundamentals in Single Domains, 2nd edn. Springer.Google Scholar
Chantry, M., Tuckerman, L.S. & Barkley, D. 2017 Universal continuous transition to turbulence in a planar shear flow. J. Fluid Mech. 824, R1.CrossRefGoogle Scholar
Chantry, M., Willis, A.P. & Kerswell, R.R. 2014 Genesis of streamwise-localized solutions from globally periodic traveling waves in pipe flow. Phys. Rev. Lett. 112 (16), 164501.CrossRefGoogle ScholarPubMed
Chossat, P. & Iooss, G. 1994 The Couette–Taylor Problem. Springer.CrossRefGoogle Scholar
Clever, R.M. & Busse, F.H. 1992 Three-dimensional convection in a horizontal fluid layer subjected to a constant shear. J. Fluid Mech. 234, 511–527.CrossRefGoogle Scholar
Clever, R.M. & Busse, F.H. 1997 Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137–153.CrossRefGoogle Scholar
Coles, D. & Van Atta, C.W. 1967 Digital experiment in spiral turbulence. Phys. Fluids Suppl. 10 (9), S120–S121.CrossRefGoogle Scholar
Coughlin, K. & Marcus, P.S. 1996 Turbulent bursts in Couette–Taylor flow. Phys. Rev. Lett. 77 (11), 2214–2217.CrossRefGoogle ScholarPubMed
Crowley, C.J., Krygier, M.C., Borrero-Echeverry, D., Grigoriev, R.O. & Schatz, M.F. 2020 A novel subcritical transition to turbulence in Taylor–Couette flow with counter-rotating cylinders. J. Fluid Mech. 892, A12.CrossRefGoogle Scholar
Deguchi, K. 2016 The rapid-rotation limit of the neutral curve for Taylor–Couette flow. J. Fluid Mech. 808, R2.CrossRefGoogle Scholar
Deguchi, K. & Altmeyer, S. 2013 Fully nonlinear mode competitions of nearly bicritical spiral or Taylor vortices in Taylor–Couette flow. Phys. Rev. E 87 (4), 043017.CrossRefGoogle ScholarPubMed
Deguchi, K. & Hall, P. 2014 The high-Reynolds-number asymptotic development of nonlinear equilibrium states in plane Couette flow. J. Fluid Mech. 750, 99–112.CrossRefGoogle Scholar
Deguchi, K. & Hall, P. 2015 Asymptotic descriptions of oblique coherent structures in shear flows. J. Fluid Mech. 782, 356–367.CrossRefGoogle Scholar
Deguchi, K., Meseguer, A. & Mellibovsky, F. 2014 Subcritical equilibria in Taylor–Couette flow. Phys. Rev. Lett. 112 (18), 184502.CrossRefGoogle ScholarPubMed
Deguchi, K. & Nagata, M. 2011 Bifurcations and instabilities in sliding Couette flow. J. Fluid Mech. 678, 156–178.CrossRefGoogle Scholar
Dessup, T., Tuckerman, L.S., Wesfreid, J.E., Barkley, D. & Willis, A.P. 2018 Self-sustaining process in Taylor–Couette flow. Phys. Rev. Fluids 3 (12), 123902.CrossRefGoogle Scholar
Dickey, D.A. & Fuller, W.A. 1979 Distribution of the estimators for autoregressive time-series with a unit root. J. Am. Stat. Assoc. 74 (366), 427–431.Google Scholar
Dickey, D.A. & Fuller, W.A. 1981 Likelihood ratio statistics for autoregressive time-series with a unit-root. Econometrica 49 (4), 1057–1072.CrossRefGoogle Scholar
Dong, S. 2009 Evidence for internal structures of spiral turbulence. Phys. Rev. E 80 (6), 067301.CrossRefGoogle ScholarPubMed
Dong, S. & Zheng, X. 2011 Direct numerical simulation of spiral turbulence. J. Fluid Mech. 668, 150–173.CrossRefGoogle Scholar
Drazin, P.G. & Reid, W.H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Duguet, Y., Schlatter, P. & Henningson, D.S. 2010 Formation of turbulent patterns near the onset of transition in plane Couette flow. J. Fluid Mech. 650, 119–129.CrossRefGoogle Scholar
Eckhardt, B., Schneider, T.M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.CrossRefGoogle Scholar
Esser, A. & Grossmann, S. 1996 Analytic expression for Taylor–Couette stability boundary. Phys. Fluids 8 (7), 1814–1819.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91 (22), 224502.CrossRefGoogle Scholar
Feynman, R.P. 1964 Lecture Notes in Physics, vol. 2. Addison-Wesley.Google Scholar
Graham, M.D. & Floryan, D. 2021 Exact coherent states and the nonlinear dynamics of wall-bounded turbulent flows. Annu. Rev. Fluid Mech. 53, 227–253.CrossRefGoogle Scholar
Grossmann, S. 2000 The onset of shear flow turbulence. Rev. Mod. Phys. 72 (2), 603–618.CrossRefGoogle Scholar
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.CrossRefGoogle Scholar
Hamilton, J.M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence. J. Fluid Mech. 287, 317–348.CrossRefGoogle Scholar
Iooss, G. & Pérouème, M.C. 1993 Perturbed homoclinic solutions in reversible $1:1$ resonance vector fields. J. Differ. Equ. 102 (1), 62–88.CrossRefGoogle Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70 (3), 703–716.CrossRefGoogle Scholar
Jones, C.A. 1985 The transition to wavy Taylor vortices. J. Fluid Mech. 157, 135–162.CrossRefGoogle Scholar
Kawahara, G., Uhlmann, M. & Van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203–225.CrossRefGoogle Scholar
Kelley, C.T. 2003 Solving Nonlinear Equations With Newton's Method. SIAM.CrossRefGoogle Scholar
Kerswell, R.R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18 (6), R17–R44.CrossRefGoogle Scholar
Kirchgässner, K. 1982 Wave-solutions of reversible systems and applications. J. Differ. Equ. 45 (1), 113–127.CrossRefGoogle Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22 (4), 047505.CrossRefGoogle ScholarPubMed
Krygier, M.C., Pughe-Sanford, J.L. & Grigoriev, R.O. 2021 Exact coherent structures and shadowing in turbulent Taylor–Couette flow. J. Fluid Mech. 923, A7.CrossRefGoogle Scholar
Kuznetsov, Y.A. 2004 Elements of Applied Bifurcation Theory, 3rd edn. Springer.CrossRefGoogle Scholar
Lemoult, G., Shi, L., Avila, K., Jalikop, S.V., Avila, M. & Hof, B. 2016 Directed percolation phase transition to sustained turbulence in Couette flow. Nat. Phys. 12 (3), 254–258.CrossRefGoogle Scholar
Lin, C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.Google Scholar
Lustro, J.R.T., Kawahara, G., van Veen, L., Shimizu, M. & Kokubu, H. 2019 The onset of transient turbulence in minimal plane Couette flow. J. Fluid Mech. 862, R2.CrossRefGoogle Scholar
Martinand, D., Serre, E. & Lueptow, R.M. 2014 Mechanisms for the transition to waviness for Taylor vortices. Phys. Fluids 26 (9), 094102.CrossRefGoogle Scholar
Mellibovsky, F. & Eckhardt, B. 2011 Takens–Bogdanov bifurcation of travelling-wave solutions in pipe flow. J. Fluid Mech. 670, 96–129.CrossRefGoogle Scholar
Mellibovsky, F. & Eckhardt, B. 2012 From travelling waves to mild chaos: a supercritical bifurcation cascade in pipe flow. J. Fluid Mech. 709, 149–190.CrossRefGoogle Scholar
Mellibovsky, F. & Meseguer, A. 2015 A mechanism for streamwise localisation of nonlinear waves in shear flows. J. Fluid Mech. 779, R1.CrossRefGoogle Scholar
Meseguer, A. 2020 Fundamentals of Numerical Mathematics for Physicists and Engineers. John Wiley & Sons.CrossRefGoogle Scholar
Meseguer, A., Avila, M., Mellibovsky, F. & Marques, F. 2007 Solenoidal spectral formulations for the computation of secondary flows in cylindrical and annular geometries. Eur. Phys. J. Spec. Top. 146, 249–259.CrossRefGoogle Scholar
Meseguer, A., Mellibovsky, F., Avila, M. & Marques, F. 2009 a Families of subcritical spirals in highly counter-rotating Taylor–Couette flow. Phys. Rev. E 79 (3), 036309.CrossRefGoogle ScholarPubMed
Meseguer, A., Mellibovsky, F., Avila, M. & Marques, F. 2009 b Instability mechanisms and transition scenarios of spiral turbulence in Taylor–Couette flow. Phys. Rev. E 80 (4), 046315.CrossRefGoogle ScholarPubMed
Moser, R.D., Moin, P. & Leonard, A. 1983 A spectral numerical method for the Navier–Stokes equations with applications to Taylor–Couette flow. J. Comput. Phys. 52 (3), 524–544.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.CrossRefGoogle Scholar
Paranjape, C.S., Duguet, Y. & Hof, B. 2020 Oblique stripe solutions of channel flow. J. Fluid Mech. 897, A7.CrossRefGoogle Scholar
Patton, A., Politis, D.N. & White, H. 2009 Correction to “Automatic block-length selection for the dependent bootstrap” by D. Politis and H. White. Econom. Rev. 28 (4), 372–375.CrossRefGoogle Scholar
Politis, D.N. & Romano, J.P. 1994 The stationary bootstrap. J. Am. Stat. Assoc. 89 (428), 1303–1313.CrossRefGoogle Scholar
Politis, D.N. & White, H. 2004 Automatic block-length selection for the dependent bootstrap. Econom. Rev. 23 (1), 53–70.CrossRefGoogle Scholar
Prigent, A., Grégoire, G., Chaté, H., Dauchot, O. & van Saarloos, W. 2002 Finite-wavelength modulation within turbulent shear flows. Phys. Rev. Lett. 89 (1), 014501.CrossRefGoogle ScholarPubMed
Reetz, F., Kreilos, T. & Schneider, T.M. 2019 Exact invariant solution reveals the origin of self-organized oblique turbulent-laminar stripes. Nat. Commun. 10, 2277.CrossRefGoogle ScholarPubMed
Ritter, P., Zammert, S., Song, B., Eckhardt, B. & Avila, M. 2018 Analysis and modeling of localized invariant solutions in pipe flow. Phys. Rev. Fluids 3 (1), 013901.CrossRefGoogle Scholar
Sano, M. & Tamai, K. 2016 A universal transition to turbulence in channel flow. Nat. Phys. 12 (3), 249–253.CrossRefGoogle Scholar
Schneider, T.M., Gibson, J.F., Lagha, M., De Lillo, F. & Eckhardt, B. 2008 Laminar-turbulent boundary in plane Couette flow. Phys. Rev. E 78 (3), 037301.CrossRefGoogle ScholarPubMed
Shi, L., Avila, M. & Hof, B. 2013 Scale invariance at the onset of turbulence in Couette flow. Phys. Rev. Lett. 110 (20), 204502.CrossRefGoogle ScholarPubMed
Skufca, J.D., Yorke, J.A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96 (17), 174101.CrossRefGoogle Scholar
Trefethen, L.N. & Bau, D. 1997 Numerical Linear Algebra. SIAM.CrossRefGoogle Scholar
Tuckerman, L.S., Chantry, M. & Barkley, D. 2020 Patterns in wall-bounded shear flows. Annu. Rev. Fluid Mech. 52, 343–367.CrossRefGoogle Scholar
Tuckerman, L.S., Kreilos, T., Schrobsdorff, H., Schneider, T.M. & Gibson, J.F. 2014 Turbulent-laminar patterns in plane Poiseuille flow. Phys. Fluids 26 (11), 114103.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883–900.CrossRefGoogle Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15 (6), 1517–1534.CrossRefGoogle Scholar
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 204501.CrossRefGoogle ScholarPubMed
Wedin, H. & Kerswell, R.R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar
Zammert, S. & Eckhardt, B. 2014 Streamwise and doubly-localised periodic orbits in plane Poiseuille flow. J. Fluid Mech. 761, 348–359.CrossRefGoogle Scholar

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