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Numerical investigation of supercritical Taylor-vortex flow for a wide gap

Published online by Cambridge University Press:  20 April 2006

H. Fasel  and
O. Booz
H. Fasel
Affiliation:
Institut A für Mechanik, Universität Stuttgart, Stuttgart, Germany Present address: University of Arizona, Aerospace and Mechanical Engineering Department, Tucson, AZ 85721.
O. Booz
Affiliation:
Institut A für Mechanik, Universität Stuttgart, Stuttgart, Germany
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Abstract

For a wide gap (R1/R2 = 0.5) and large aspect ratios L/d, axisymmetric Taylor-vortex flow has been observed in experiments up to very high supercritical Taylor (or Reynolds) numbers. This axisymmetric Taylor-vortex flow was investigated numerically by solving the Navier–Stokes equations using a very accurate (fourth-order in space) implicit finite-difference method. The high-order accuracy of the numerical method, in combination with large numbers of grid points used in the calculations, yielded accurate and reliable results for large supercritical Taylor numbers of up to 100Tac (or 10Rec). Prior to this study numerical solutions were reported up to only 16Tac. The emphasis of the present paper is placed upon displaying and elaborating the details of the flow field for large supercritical Taylor numbers. The flow field undergoes drastic changes as the Taylor number is increased from just supercritical to 100Tac. Spectral analysis (with respect to z) of the flow variables indicates that the number of harmonics contributing substantially to the total solution increases sharply when the Taylor number is raised. The number of relevant harmonics is already unexpectedly high at moderate supercritical Ta. For larger Taylor numbers, the evolution of a jetlike or shocklike flow structure can be observed. In the axial plane, boundary layers develop along the inner and outer cylinder walls while the flow in the core region of the Taylor cells behaves in an increasingly inviscid manner.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 138 , January 1984 , pp. 21 - 52
Copyright
© 1984 Cambridge University Press

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References

Batchelor, G. K. 1956 On steady laminar flow with closed streamlines at large Reynolds number J. Fluid Mech. 1, 177–190.Google Scholar
Booz, O. 1980 Numerische Lösung der Navier–Stokes Gleichungen für Taylor-Wirbelströmungen im weiten Spalt. Dissertation, Universität Stuttgart.
Burkhalter, J. D. & Koschmieder, E. L. 1973 Steady supercritical Taylor vortex flow J. Fluid Mech. 58, 547–560.Google Scholar
Burkhalter, J. D. & Koschmieder, E. L. 1974 Steady supercritical Taylor vortices after sudden starts Phys. Fluids 17, 1929–1935.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Cole, J. A. 1976 Taylor-vortex instability and annulus-length effects J. Fluid Mech. 75, 1–15.Google Scholar
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, 336–368.Google Scholar
Davey, A., Diprima, R. C. & Stuart, J. T. 1968 On the instability of Taylor vortices J. Fluid Mech. 31, 17–52.Google Scholar
Diprima, R. C. & Rogers, E. H. 1969 Computing problems in nonlinear hydrodynamic stability. In High Speed Computing in Fluid Dynamics: Phys. Fluids Suppl. II, 155–165.
Donnelly, R. J. 1958 Experiments on the stability of viscous flow between rotating cylinders, I. Torque measurements.Proc. R. Soc. Lond A 246, 312–325.
Donnelly, R. J. & Simon, N. J. 1960 An empirical torque relation for supercritical flow between rotating cylinders J. Fluid Mech. 7, 401–418.Google Scholar
Eagles, P. M. 1974 On the torque of wavy vortices J. Fluid Mech. 62, 1–9.Google Scholar
Fasel, H. 1974 Untersuchungen zum Problem des Grenzschichtumschlages durch numerische Integration der Navier–Stokes-Gleichungen. Dissertation, Universität Stuttgart.
Fasel, H. 1976 Investigation of the stability of boundary layers by a finite difference model of the Navier–Stokes equations J. Fluid Mech. 78, 355–383.Google Scholar
Jones, C. A. 1981 Nonlinear Taylor vortices and their stability J. Fluid Mech. 102, 249–261.Google Scholar
Kirchgässner, K. 1961 Die Instabilität der Strömung zwischen zwei rotierenden Zylindern gegenüber Taylor-Wirbeln für beliebige Spaltbreiten Z. angew. Math. Phys. 12, 14–29.Google Scholar
Kirchgässner, K. 1966 Verzweigungslösungen eines stationären hydrodynamischen Randwert-problems. Habilitationsschrift, Universität Freiburg.
Kirchgässner, K. & Sorger, P. 1969 Branching analysis for the Taylor problem Q. J. Mech. Appl. Maths 22, 183–209.Google Scholar
Kogelman, S. & Diprima, R. C. 1970 Stability of spatially periodic supercritical flows in hydrodynamics Phys. Fluids 13, 1–11.Google Scholar
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection J. Fluid Mech. 4, 225–260.Google Scholar
Meyer, K. A. 1967 Time-dependent numerical study of Taylor vortex flow Phys. Fluids 10, 1874–1879.Google Scholar
Meyer, K. A. 1969 Three-dimensional study of flow between concentric rotating cylinders. In High Speed Computing in Fluid Dynamics: Phys. Fluids Suppl. II, 165–170.
Meyer-Spasche, R. & Keller, H. B. 1980 Computations of the axisymmetric flow between rotating cylinders J. Comp. Phys. 35, 100–109.Google Scholar
Nakaya, C. 1974 Domain of stable periodic vortex flows in a viscous fluid between concentric circular cylinders J. Phys. Soc. Japan 36, 1164–1173.Google Scholar
Park, K. & Donnelly, R. J. 1980 Work in progress reported at the Division of Fluid Mechanics – American Physical Society Meeting, Cornell University.
Reynolds, W. C. & Potter, M. C. 1967 A finite-amplitude state-selection theory for Taylor-vortex flow. Stanford University Paper.Google Scholar
Roache, P. J. 1976 Computational Fluid Dynamics. Hermosa.
Rogers, E. H. & Beard, D. W. 1969 A numerical study of wide-gap Taylor vortices J. Comp. Phys. 4, 1–18.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, 281–289.Google Scholar
Snyder, H. A. 1969 Wave number selection at finite amplitude in rotating Couette flow J. Fluid Mech. 35, 273–298.Google Scholar
Snyder, H. A. & Lambert, R. B. 1966 Harmonic generation in Taylor vortices between rotating cylinders J. Fluid Mech. 26, 545–562.Google Scholar
Strawbridge, D. R. & Hooper, G. T. J. 1968 Numerical solutions of the Navier–Stokes equations for axisymmetric flows J. Mech. Engng. Sci. 10, 389–401.Google Scholar
Stuart, J. T. 1958 On the nonlinear mechanics of hydrodynamic stability J. Fluid Mech. 4, 1–21.Google Scholar
Stuart, J. T. 1960 On the nonlinear mechanisms of wave disturbances in stable and unstable parallel flows J. Fluid Mech. 9, 353–370.Google Scholar
Stuart, J. T. 1971 Nonlinear stability theory Ann. Rev. Fluid Mech. 3, 347–370.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders.Phil. Trans. R. Soc. Lond A 223, 289–343.
Velte, W. 1966 Stabilität und Verzweigung stationärer Lösungen der Navier–Stokesschen Gleichungen beim Taylor-Problem Arch. Rat. Mech. Anal. 22, 1–14.Google Scholar