Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-20T01:35:35.212Z Has data issue: false hasContentIssue false
  • English
  • Français

Spatio-temporal mode dynamics and higher order transitions in high aspect ratio Newtonian Taylor–Couette flows

Published online by Cambridge University Press:  17 November 2009

CARI S. DUTCHER  and
SUSAN J. MULLER
CARI S. DUTCHER
Affiliation:
Department of Chemical Engineering, University of California at Berkeley, Berkeley, CA 94720, USA
SUSAN J. MULLER*
Affiliation:
Department of Chemical Engineering, University of California at Berkeley, Berkeley, CA 94720, USA
*
†Email address for correspondence: muller2@berkeley.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Spatial and temporal frequency dynamics were experimentally tracked via flow visualization for Newtonian fluids as a function of the inner cylinder Reynolds number (Rei) in the flow between concentric, independently rotating cylinders with a radius ratio of 0.912 and an aspect ratio of 60.7. Eight transitions from laminar to turbulent flow were characterized in detail for a stationary outer cylinder, producing highly resolved space–time and frequency–time plots for wavy, modulated and weakly turbulent states. A previously unreported early-modulated wavy vortex flow was found in our high aspect ratio geometry both with and without the presence of a dislocation. The envelope of stability for this flow state was shown to cross into the co-rotating regime, and is present up to Reo ~ 60, where Reo is the outer cylinder Reynolds number. This early modulation is independent of acceleration in the range 0.18 < dRei/dτ < 2.9, where τ is the time nondimensionalized with a viscous time scale. While many of the flow states have been previously observed in geometries with somewhat different radius ratios, we provide new characterization of transitional structures for Reo = 0 in the range 0 < Re* < 21.4, where Re* = Rei/Rec and Rec is the value of Rei at the primary instability. Special attention has been given to ramp rate. For quasi-static ramps, axisymmetric states are stable over the ranges of Re* = [(0–1.17), > 15.4], states characterized by a single distinct temporal frequency for Re* = [(1.17–1.41), (3.56–5.20), (7.85–15.4)], states with multiple temporal frequencies for Re* = [(1.41–3.56), (5.20–7.85)], and a transition from laminar to weakly turbulent vortices occurs at Re* = 5.49. All flow states are characterized by symmetry/symmetry-breaking features as well as azimuthal and axial wavenumbers.

JFM classification

Instability: Instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 641 , 25 December 2009 , pp. 85 - 113
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Abcha, N., Latrache, N., Dumouchel, F. & Mutabazi, I. 2008 Qualitative relation between reflected light intensity by Kalliroscope flakes and velocity field in the Couette–Taylor flow system. Exp. Fluids 45, 85–94.CrossRefGoogle Scholar
Abshagen, J., Lopez, J. M., Marques, F. & Pfister, G. 2005 Mode competition of rotating waves in reflection-symmetric Taylor–Couette flow. J. Fluid Mech. 540, 269–299.CrossRefGoogle Scholar
Abshagen, J., Lopez, J. M., Marques, F. & Pfister, G. 2008 Bursting dynamics due to a homoclinic cascade in Taylor–Couette flow. J. Fluid Mech. 613, 357–384.CrossRefGoogle Scholar
Abshagen, J., Pfister, G. & Mullin, T. 2001 Gluing bifurcations in a dynamically complicated extended flow. Phys. Rev. Lett. 87 (22), 224501.CrossRefGoogle Scholar
Abshagen, J., Schulz, A. & Pfister, G. 1996 The Couette–Taylor flow: a paradigmatic system for instabilities, pattern formation and routes to chaos. In Nonlinear Physics of Complex Systems, pp. 63–72. Lecture Notes in Physics 476. Springer.CrossRefGoogle Scholar
Ahlers, G., Cannell, D. S. & Dominguez-Lerma, M. A. 1983 Possible mechanism for transitions in wavy Taylor-vortex flow. Phys. Rev. A 27, 1225–1227.CrossRefGoogle Scholar
Andereck, C. D., Dickman, R. & Swinney, H. L. 1983 New flows in a circular Couette system with co-rotating cylinders. Phys. Fluids 26, 1395–1401.CrossRefGoogle 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.CrossRefGoogle Scholar
Avila, M., Marques, F., Lopez, J. M., & Meseguer, A. 2007 Stability control and catastrophic transition in a forced Taylor–Couette system. J. Fluid Mech. 590, 471–496.CrossRefGoogle Scholar
Baer, S. M. & Gaekel, E. M. 2008 Slow acceleration and deacceleration through a hopf bifurcation: power ramps, target nucleation, and elliptic bursting. Phys. Rev. E 78 (036205).CrossRefGoogle ScholarPubMed
Barcilon, A., Brindley, J., Lessen, M. & Mobbs, F. R. 1979 Marginal instability in Taylor–Couette flows at a very high Taylor number. J. Fluid Mech. 94, 453–463.CrossRefGoogle Scholar
Baxter, G. W. & Andereck, C. D. 1986 Formation of dynamical domains in a circular Couette System. Phys. Rev. Lett. 57, 3046–3049.CrossRefGoogle Scholar
Benjamin, T. B. & Mullin, T. 1982 Notes on the multiplicity of flows in the Taylor experiment. J. Fluid Mech. 121, 219–230.CrossRefGoogle Scholar
Brandstater, A. & Swinney, H. L. 1987 Strange attractors in weakly turbulent Couette–Taylor flow. Phys. Rev. A 35, 2207–2220.CrossRefGoogle ScholarPubMed
Cole, J. A. 1976 Taylor-vortex instability and annulus-length effects. J. Fluid Mech. 75, 1–15.CrossRefGoogle Scholar
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385–425.CrossRefGoogle Scholar
Coles, D. 1967 A note on Taylor instability in circular Couette flow. J. Appl. Mech. 34, 529–532.CrossRefGoogle Scholar
Coughlin, K. T., Marcus, P. S., Tagg, R. P. & Swinney, H. L. 1991 Distinct quasiperiodic modes with like symmetry in a rotating fluid. Phys. Rev. Lett. 66, 1161–1164.CrossRefGoogle Scholar
Czarny, O. & Lueptow, R. M. 2007 Time scales for transition in Taylor–Couette flow. Phys. Fluids 19, 054103.CrossRefGoogle Scholar
Czarny, O., Serre, E., Bontoux, P. & Lueptow, R. M 2004 Ekman vortices and the centrifugal instability in counter-rotating cylindrical Couette flow. Theor. Comput. Fluid Dyn. 18, 151–168.CrossRefGoogle Scholar
DiPrima, R. C., Eagles, P. M. & Ng, B. S. 1984 The effect of radius ratio on the stability of Couette flow and Taylor vortex flow. Phys. Fluids 27, 2403–2411.CrossRefGoogle Scholar
Donnelly, R. J. & Fultz, D. 1960 Experiments on the stability of viscous flow between rotating cylinders. II. Visual observations. Proc. R. Soc. Lond. A 258, 101–122.Google Scholar
Donnelly, R. J. & LaMar, M. M. 1988 Flow and stability of helium II between concentric cylinders. J. Fluid Mech. 186, 163–198.CrossRefGoogle Scholar
Donnelly, R. J., Park, K., Shaw, R. & Walden, R. W. 1980 Early nonperiodic transitions in Couette flow. Phys. Rev. Lett. 44, 987–989.CrossRefGoogle Scholar
Dutcher, C. S. & Muller, S. J. 2007 Explicit analytic formulas for Newtonian Taylor–Couette primary instabilities. Phys. Rev. E 75, 047301.CrossRefGoogle ScholarPubMed
Esser, A. & Grossmann, S. 1996 Analytic expression for Taylor–Couette stability boundary. Phys. Fluids. 8, 1814–1819.CrossRefGoogle Scholar
Fenstermacher, P. R., Swinney, H. L. & Gollub, J. P. 1979 Dynamical instabilities and the transition to chaotic Taylor vortex flow. J. Fluid Mech. 94, 103–128.CrossRefGoogle Scholar
Gollub, J. P. & Swinney, H. L. 1975 Onset of turbulence in a rotating fluid. Phys. Rev. Lett. 35, 927–930.CrossRefGoogle Scholar
Gorman, M. & Swinney, H. L. 1982 Spatial and temporal characteristics of modulated waves in the circular Couette system. J. Fluid Mech. 117, 123–142.CrossRefGoogle 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
King, G. P. & Swinney, H. L. 1983 Limits of stability and irregular flow patterns in wavy vortex flow. Phys. Rev. A 27, 1240–1243.CrossRefGoogle Scholar
Koschmieder, E. L. 1979 Turbulent Taylor vortex flow. J. Fluid Mech. 93, 515–527.CrossRefGoogle Scholar
Koschmieder, E. L. 1993 Bénard Cells and Taylor Vortices. Cambridge University Press.Google Scholar
Lathrop, D. P., Fineberg, J. & Swinney, H. L. 1992 Transition to shear-driven turbulence in Couette–Taylor flow. Phys. Rev. A 46, 6390–6405.CrossRefGoogle ScholarPubMed
Lewis, G. S. & Swinney, H. L. 1999 Velocity structure functions, scaling, and transitions in high-Reynolds-number Couette–Taylor flow. Phys. Rev. E 59, 5457–5467.CrossRefGoogle ScholarPubMed
Meincke, O. & Egbers, C. 1999 Route into chaos in small and wide gap Taylor–Couette flow. Phys. Chem. Earth B 25, 467–471.CrossRefGoogle Scholar
Mullin, T. 1985 Onset of time dependence in Taylor–Couette flow. Phys. Rev. A 31, 1216–1218.CrossRefGoogle ScholarPubMed
Park, K. & Crawford, G. L. 1983 Deterministic transition in Taylor wavy-vortex flow. Phys. Rev. Lett. 50, 343–346.CrossRefGoogle Scholar
Park, K., Crawford, G. L. & Donnelly, R. J. 1981 Determination of transition in Couette flow in finite geometries. Phys. Rev. Lett. 47, 1448–1450.CrossRefGoogle Scholar
Pfister, G. & Gerdts, U. 1981 Dynamics of Taylor wavy vortex flow. Phys. Lett. 83A (1), 23–25.CrossRefGoogle Scholar
Sinha, M., Kevrekidis, I. G. & Smits, A. J. 2006 Experimental study of a Neimark–Sacker bifurcation in axially forced Taylor–Couette flow. J. Fluid Mech. 558, 1–32.CrossRefGoogle Scholar
Smith, G. P. & Townsend, A. A. 1982 Turbulent Couette flow between concentric cylinders at large Taylor numbers. J. Fluid Mech. 123, 187–217.CrossRefGoogle Scholar
Snyder, H. A. 1968 Stability of rotating Couette flow. II. Comparison with numerical results. Phys. Fluids 11, 1599–1605.CrossRefGoogle Scholar
Snyder, H. A. 1969 Wave-number selection at finite amplitude in rotating Couette flow. J. Fluid Mech. 35, 273–298.CrossRefGoogle Scholar
Tagg, R. 1994 The Couette–Taylor problem. Nonlinear Sci. Today 4, 2–25.Google Scholar
Takeda, Y. 1999 Quasi-periodic state and transition to turbulence in a rotating Couette system. J. Fluid Mech. 389, 81–99.CrossRefGoogle Scholar
Takeda, Y., Fischer, W. E., Kobashi, K. & Takeda, T. 1992 Spatial characteristics of dynamic properties of modulated wavy vortex flow in a rotating Couette system. Exp. Fluids 13, 199–207.CrossRefGoogle Scholar
Takeda, Y., Fischer, W. E., Sakakibara, J. & Ohmura, K. 1993 Experimental observation of the quasiperiodic modes in a rotating Couette system. Phys. Rev. E 47, 4130–4134.CrossRefGoogle Scholar
Taylor, G. I. 1923 Flow regimes in a circular Couette system with independently rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289–343.Google Scholar
von Stamm, J., Gerdts, U., Buzug, Th. & Pfister, G. 1996 Symmetry breaking and period doubling on a torus in the VLF regime in Taylor–Couette flow. Phys. Rev. E 54, 4938–4957.CrossRefGoogle Scholar
Walden, R. W. & Donnelly, R. J. 1979 Reemergent order of chaotic circular Couette flow. Phys. Rev. Lett 42, 301–304.CrossRefGoogle Scholar
Wang, L., Olsen, M. G. & Vigil, R. D. 2005 Reappearance of azimuthal waves in turbulent Taylor–Couette flow at large aspect ratio. Chem. Engng Sci. 60, 5555–5568.CrossRefGoogle Scholar
White, J. 2002 Experimental investigation of instabilities in Newtonian and viscoelastic Taylor–Couette flows. PhD dissertation, University of California, Berkeley.Google Scholar
Xiao, Q., Lim, T. T. & Chew, Y. T. 2002 Effect of acceleration on the wavy Taylor vortex flow. Exp. Fluids 32, 639–644.CrossRefGoogle Scholar
Zhang, L. H. & Swinney, H. L. 1985 Nonpropagating oscillatory modes in Couette–Taylor flow. Phys. Rev. A 31, 1006–1009.CrossRefGoogle ScholarPubMed