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On the instability of Taylor vortices

Published online by Cambridge University Press:  28 March 2006

A. Davey ,
R. C. Di Prima  and
J. T. Stuart
A. Davey
Affiliation:
Aerodynamics Division, National Physical Laboratory, Teddington, Middlesex
R. C. Di Prima
Affiliation:
Department of Mathematics, Rensselaer Polytechnic Institute, Troy, New York, 12181
J. T. Stuart
Affiliation:
Department of Mathematics, Imperial College, London, S. W. 7
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Abstract

It is known experimentally that laminar circular Couette flow between two concentric circular cylinders, the outer of which is fixed, becomes unstable when the speed of the inner cylinder is high enough. The flow is then replaced by a new circumferential flow with superimposed toroidal (or Taylor) vortices spaced periodically along the axis. At a higher speed still the new flow develops another instability, and is replaced by a flow in which the axially periodic vortices are simultaneously periodic travelling waves in the azimuth.

In the present paper an attack is made on the problem of instability of the Taylor-vortex flow against perturbations which are periodic both in the axial and azimuthal co-ordinates and, moreover, travel with some phase velocity in the latter. Subject to a number of assumptions and approximations, which are detailed in the paper, it is found that the Taylor-vortex flow is stable against perturbations with the same axial wavelength and phase, but unstable against perturbations differing in phase by ½π. After instability the new flow no longer has planes separating neighbouring vortices, but has wavy surfaces travelling in the azimuth. This feature is in accord with much (though not all) of the experimental evidence.

The critical Taylor number (proportional to the square of the speed) at which the Taylor vortices become unstable is found theoretically to be about 8% above the value for which Taylor vortices first appear. This must be compared with a value in the range 5-20% for the experiments which our work models most closely. The azimuthal wave-number given a slight preference by theory is 1, in agreement with those experiments.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 31 , Issue 1 , 8 January 1968 , pp. 17 - 52
Copyright
© 1968 Cambridge University Press

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References

Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon J. Fluid Mech. 14, 593–629.Google Scholar
Bisshopp, F. 1963 Flow between concentric rotating cylinders. A note on the narrow gap approximation. Technical Report 51, ONR contract Nonr 562(07), Brown University.Google Scholar
Capriz, G., Ghelardoni, G. & Lombardi, G. 1964 Numerical solution of a stability problem in hydrodynamics Estratto da Calcolo, 1, 19–48.Google Scholar
Capriz, G., Ghelardoni, G. & Lombardi, G. 1966 Numerical study of the stability problem for Couette flow Phys. Fluids, 9, 1934–6.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Coles, D. 1960 Paper presented at the Xth International Congress of Applied Mechanics, Stresa (p. 163). Elsevier.
Coles, D. 1965 Transition in circular Couette flow J. Fluid Mech. 21, 385–425.Google Scholar
Davey, A. 1962 The growth of Taylor vortices in flow between rotating cylinders J. Fluid Mech. 14, 336–68.Google Scholar
Di Prima, R. C. 1963 Stability of curved flows J. Appl. Mech. 30, 486–92.Google Scholar
Donnelly, R. J. 1958 Experiments on the stability of viscous flow between rotating cylinders I. Torque measurements. Proc. Roy. Soc. A 246, 312–25.Google Scholar
Donnelly, R. J. 1963 Experimental confirmation of the Landau law in Couette flow. Phys. Rev. Lett. 10, 7, 282–4.Google Scholar
Donnelly, R. J. & Schwarz, K. W. 1965 Experiments on the stability of viscous flow between rotating cylinders VI. Finite-amplitude experiments. Proc. Roy. Soc. A 283, 531–49.Google Scholar
Donnelly, R. J. & Simon, N. J. 1960 An empirical torque relation for supercritical flow between rotating cylinders J. Fluid Mech. 7, 401–18.Google Scholar
Gross, A. G. 1964 Numerical investigation of the stability of Couette flow. Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, New York.
Ince, E. L. 1956 Ordinary Differential Equations. Dover: New York.
Krueger, E. R. 1962 The stability of Couette flow and spiral flow. Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, New York.
Krueger, E. R., Gross, A. G. & di Prima, R. C. 1966 On the relative importance of Taylor-vortex and non-axisymmetric modes in flow between rotating cylinders J. Fluid Mech. 24, 521–38.Google Scholar
Landau, L. D. 1944 On the problem of turbulence C. R. Acad. Sci. U.R.S.S. 44, 311. See also Fluid Mechanics, by L. D. Landau and E. M. Lifsbitz. Oxford: Pergamon Press, 1959.Google Scholar
Meyer, K. 1966 A two-dimensional time-dependent numerical study of rotational Couette flow. Los Alamos Report LA-3497, Los Alamos, New Mexico.Google Scholar
Minorsky, N. 1962 Nonlinear Oscillations. Princeton, New Jersey: Van Nostrand.
Nissan, A. H., Nardacci, J. L. & Ho, C. Y. 1963 The onset of different modes of instability for flow between rotating cylinders. A.I.Ch.E. J. 9, 5, 620–4.Google Scholar
Reynolds, W. C. & Potter, M. C. 1967 To be published.
Roberts, P. H. 1965 Appendix. The solution of the characteristic-value problems. Proc. Roy. Soc. A 283, 550–6.Google Scholar
Schultz-Grunow, F. & Hein, H. 1956 Beitrag zur Couetteströmung Z. Flugwiss. 4, 28–30.Google Scholar
Schwarz, K. W., Springett, B. E. & Donnelly, R. J. 1964 Modes of instability in spiral flow between rotating cylinders J. Fluid Mech. 20, 281–9.Google Scholar
Segel, L. A. 1965 The non-linear interaction of a finite number of disturbances to a layer of fluid heated from below J. Fluid Mech. 21, 359–84.Google Scholar
Snyder, H. A. & Lambert, R. B. 1966 Harmonic generation in Taylor vortices between rotating cylinders J. Fluid Mech. 26, 545–62.Google Scholar
Snyder, H. A. & Karlsson, S. K. 1965 Nonaxisymmetric modes of secondary flow Bull. Am. Phys. Soc. 10, 24.Google Scholar
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability J. Fluid Mech. 4, 1–21.Google Scholar
Stuart, J. T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows, Part I J. Fluid Mech. 9, 353–70.Google Scholar
Stuart, J. T. 1961 On three-dimensional non-linear effects in the stability of parallel flows. Advances in Aeronautical Sciences, vols. 3–4, pp. 121–42. Oxford: Pergamon Press.
Stuart, J. T. 1964 Contribution to Stability of Systems, Discussion at I. Mech. Eng. May 21/22. Published in 1965, I. Mech. E. Proc. 178, 3M, 54–5.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Roy. Soc. A 223, 289–343.Google Scholar
Watson, J. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows, Part II J. Fluid Mech. 9, 371–89.Google Scholar
Whitham, G. B. 1963 The Navier-Stokes equations of motion. In Laminar Boundary Layers (ed. Rosenhead, L.). Oxford: Clarendon Press.