Hostname: page-component-8678677fb7-9rf5j Total loading time: 0 Render date: 2026-10-16T19:04:18.119Z Has data issue: false hasContentIssue false
  • English
  • Français

Secondary instability and tertiary states in rotating plane Couette flow

Published online by Cambridge University Press:  14 November 2014

C. A. Daly ,
Tobias M. Schneider ,
Philipp Schlatter  and
N. Peake
C. A. Daly*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Tobias M. Schneider
Affiliation:
Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany ECPS, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Philipp Schlatter
Affiliation:
Linné FLOW Centre, KTH Mechanics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
N. Peake
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: cdaly87@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent experimental studies have shown rich transition behaviour in rotating plane Couette flow (RPCF). In this paper we study the transition in supercritical RPCF theoretically by determination of equilibrium and periodic orbit tertiary states via Floquet analysis on secondary Taylor vortex solutions. Two new tertiary states are discovered which we name oscillatory wavy vortex flow (oWVF) and skewed vortex flow (SVF). We present the bifurcation routes and stability properties of these new tertiary states and, in addition, we describe a bifurcation procedure whereby a set of defected wavy twist vortices is approached. Further to this, transition scenarios at flow parameters relevant to experimental works are investigated by computation of the set of stable attractors which exist on a large domain. The physically observed flow states are shown to share features with states in our set of attractors.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Instability: Nonlinear instability Instability: Transition to turbulence

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 761 , 25 December 2014 , pp. 27 - 61
Check for updates
Copyright
© 2014 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.)

Article purchase

Temporarily unavailable

References

Abcha, N., Crumeyrolle, O., Ezersky, A. B. & Mutabazi, I. 2013 Velocity field of the spiral vortex flow in the Couette–Taylor system. Eur. Phys. J. E 36 (3), 20.Google Scholar
Abshagen, J., Lopez, J. M., Marques, F. & Pfister, G. 2005 Symmetry breaking via global bifurcations of modulated rotating waves in hydrodynamics. Phys. Rev. Lett. 94 (7), 074501.CrossRefGoogle ScholarPubMed
Alfredsson, P. H. & Tillmark, N. 2005 Instability, transition and turbulence in plane Couette flow with system rotation. In IUTAM Symposium on Laminar Turbulent Transition and Finite Amplitude Solutions (ed. Mullin, T. & Kerswell, R. R.), Fluid Mechanics and its Applications, vol. 77. Springer.Google Scholar
Altmeyer, S., Hoffmann, Ch., Heise, M., Abshagen, J., Pinter, A., Lücke, M. & Pfister, G. 2010 End wall effects on the transition between Taylor vortices and spiral vortices. Phys. Rev. E 81 (6), 066313.CrossRefGoogle ScholarPubMed
Andereck, C. D., Dickman, R. & Swinney, H. L. 1983 New flows in a circular Couette system with co-rotating cylinders. Phys. Fluids 26 (6), 1395–1401.Google Scholar
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.Google Scholar
Antonijoan, J. & Sánchez, J. 2000 Transitions from Taylor vortex flow in a co-rotating Taylor–Couette system. Phys. Fluids 12 (12), 3147–3159.Google Scholar
Benjamin, T. B. & Mullin, T. 1982 Notes on the multiplicity of flows in the Taylor experiment. J. Fluid Mech. 121, 219–230.Google Scholar
Bot, P. & Mutabazi, I. 2000 Dynamics of spatio-temporal defects in the Taylor–Dean system. Eur. Phys. J. B 13, 141–155.Google Scholar
Clever, R. M. & Busse, F. H. 1997 Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137–153.Google Scholar
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385–425.Google Scholar
Czarny, O., Serre, E., Bontoux, P. & Lueptow, R. M. 2003 Interaction between Ekman pumping and the centrifugal instability in Taylor–Couette flow. Phys. Fluids 15 (2), 467–477.Google Scholar
Davey, A., DiPrima, R. C. & Stuart, J. T. 1968 On the instability of Taylor vortices. J. Fluid Mech. 31, 17–52.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.Google Scholar
Duguet, Y., Pringle, C. T. & Kerswell, R. R. 2008 Relative periodic orbits in transitional pipe flow. Phys. Fluids 20 (11), 114102.Google Scholar
Eagles, P. M. 1971 On stability of Taylor vortices by fifth-order amplitude expansions. J. Fluid Mech. 49, 529–550.Google Scholar
Eckhaus, W. 1965 Studies in Nonlinear Stability Theory. Springer.Google Scholar
Ezersky, A. B., Abcha, N. & Mutabazi, I. 2010 The structure of spatio-temporal defects in a spiral pattern in the Couette–Taylor flow. Phys. Lett. A 37 (33), 3297–3303.Google Scholar
Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91 (22), 224502.Google Scholar
Gibson, J. F.2008 Channelflow: a spectral Navier–Stokes simulator in C++. Tech. Rep. Georgia Institute of Technology.Google Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state-space in plane Couette flow. J. Fluid Mech. 611, 107–130.Google Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 683, 243–266.Google Scholar
Hegseth, J. J., Baxter, G. W. & Andereck, C. D. 1996 Bifurcations from Taylor vortices between corotating concentric cylinders. Phys. Rev. E 53, 507–521.CrossRefGoogle ScholarPubMed
Heise, M., Hoffmann, Ch., Will, Ch., Altmeyer, S., Abshagen, J. & Pfister, G. 2013 Co-rotating Taylor–Couette flow enclosed by stationary disks. J. Fluid Mech. 716, 507–521.Google Scholar
Hiwatashi, K., Alfredsson, P. H., Tillmark, N. & Nagata, M. 2007 Experimental observations of instabilities in rotating plane couette flow. Phys. Fluids 19, 48–103.Google Scholar
Hollerbach, R. & Fournier, A. 2004 End-effects in a rapidly rotating cylindrical Taylor–Couette flow. In MHD Couette Flows: Experiments and Models (ed. Rosner, R., Rüdiger, G. & Bonanno, A.), vol. 733, pp. 114–121. AIP Conf. Proc.Google Scholar
Itano, T. & Generalis, S. C. 2009 Hairpin vortex solution in planar Couette flow: a tapestry of knotted vortices. Phys. Rev. Lett. 102, 114–501.Google Scholar
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300.Google Scholar
Koschmieder, E. L. 1993 Bénard Cells and Taylor Vortices. Cambridge University Press.Google Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near the onset of chaos in plane Couette flow. Chaos 22 (4), 047505.Google Scholar
Lezius, D. K. & Johnston, J. P. 1976 Roll-cell instabilities in rotating laminar and turbulent channel flows. J. Fluid Mech. 77, 153–175.Google Scholar
Lopez, J. M. & Marques, F. 2005 Finite aspect ratio Taylor–Couette flow: Shil’nikov dynamics of 2-tori. Physica D 211, 168–191.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.Google Scholar
Mullin, T. 2010 The rich structure of transition in a shear flow. J. Fluid Mech. 648, 1–4.Google Scholar
Nagata, M. 1986 Bifurcations in Couette flow between almost corotating cylinders. J. Fluid Mech. 169, 229–250.Google Scholar
Nagata, M. 1988 On wavy instabilities of the Taylor-vortex flow between corotating cylinders. J. Fluid Mech. 188, 585–598.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.Google Scholar
Nagata, M. 1997 Three-dimensional travelling-wave solutions in plane Couette flow. Phys. Rev. E 55, 2023–2025.CrossRefGoogle Scholar
Nagata, M. 1998 Tertiary solutions and their stability in rotating plane Couette flow. J. Fluid Mech. 358, 357–378.Google Scholar
Nagata, M. 2013 A note on the mirror-symmetric coherent structure in plane Couette flow. J. Fluid Mech. 727, R1.Google Scholar
Nagata, M. & Kawahara, G. 2004 Three-dimensional periodic solutions in rotating/non-rotating plane Couette flow. In Advances in Turbulence X, Proceedings of the 10th European Turbulence Conference (ed. Andersson, H. I. & Krogstad, P. Å.), vol. 77. CIMNE.Google Scholar
Nana, L., Ezersky, A. B. & Mutabazi, I. 2009 Secondary structures in a one-dimensional complex Ginzburg–Landau equation with homogeneous boundary conditions. Proc. R. Soc. Lond. A 465, 2251–2265.Google Scholar
Pringle, C. T. & Kerswell, R. R. 2007 Asymmetric, helical, and mirror-symmetric travelling waves in pipe flow. Phys. Rev. Lett. 99 (7), 074502.CrossRefGoogle ScholarPubMed
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Schneider, T. M., Gibson, J. F., Lahga, M., De Lillo, F. & Eckhardt, B. 2008 Laminar turbulent boundary in plane Couette flow. Phys. Rev. E 78.Google Scholar
Shilnikov, L. P. 1965 A case of the existence of a denumerable set of periodic motions. Sov. Math. Dokl. 6, 163–166.Google Scholar
Suryadi, A., Segalini, A. & Alfredsson, P. H. 2014 Zero absolute vorticity: insight from experiments in rotating laminar plane Couette flow. Phys. Rev. E 89 (3), 033003.CrossRefGoogle ScholarPubMed
Suryadi, A., Tillmark, N. & Alfredsson, P. H. 2013 Velocity measurements of streamwise roll cells in rotating plane Couette flow. Exp. Fluids 15 (11), 1617.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. 223, 289–343.Google Scholar
Tsukahara, T., Tillmark, N. & Alfredsson, P. H. 2010 Flow regimes in plane Couette flow with system rotation. J. Fluid Mech. 648, 5–33.Google Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.Google Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140–4143.Google Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.Google Scholar
Wang, J., Gibson, J. F. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 204501.Google Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.Google Scholar
Weisshaar, E., Busse, F. H. & Nagata, M. 1991 Twist vortices and their instabilities in the Taylor–Couette system. J. Fluid Mech. 226, 549–564.Google Scholar
Wiggins, S. 2003 Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer.Google Scholar