Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-18T19:20:13.322Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcation phenomena in Taylor–Couette flow with buoyancy effects

Published online by Cambridge University Press:  21 April 2006

K. S. Ball  and
B. Farouk
K. S. Ball
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
B. Farouk
Affiliation:
Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A numerical study has been conducted to determine the various modes of Taylor–Couette flow that exist between concentric vertical cylinders, as the aspect ratio Γ (height to gap width, H/d) and the Reynolds number Re (based on the inner cylinder speed) are varied. Furthermore, the effects of the introduction of buoyancy on the development of the flow are examined. This is accomplished by considering both cylinders to be isothermal, with the rotating inner cylinder at a higher temperature than the stationary outer cylinder. Results are presented for a wide range of the Grashof number Gr (based on the temperature difference ΔT across the annular gap). The structure of the Taylor vortices is observed to be distorted considerably with the buoyant flows, and the nature of the onset and subsequent development of the vortices is altered. The hysteresis between the different modes of cellular flow, characteristic of the bifurcation phenomena, is also substantially modified.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 197 , December 1988 , pp. 479 - 501
Copyright
© 1988 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.)

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.Google Scholar
Astill, K. N.1964 Studies of the developing flow between concentric cylinders with the inner cylinder rotating. Trans. ASME C: J. Heat Transfer 86, 383–392.Google Scholar
Ball, K. S.1987 Mixed convection heat transfer in rotating systems Ph.D. thesis, Drexel University.
Ball, K. S. & Farouk, B.1986 Numerical studies of mixed convection flows in the annulus between vertical concentric cylinders with rotating inner cylinder. In Proc. Eighth Intl Heat Trans. Conf. San Francisco (ed. C. L. Tien, V. P. Carey & J. K. Ferrell), pp. 435–440. Hemisphere.
Ball, K. S. & Farouk, B.1987 On the development of Taylor vortices in a vertical annulus with a heated rotating inner cylinder. Intl J. Num. Meth. Fluids 7, 857–867.Google Scholar
Becker, K. M. & Kaye, J.1962a Measurements of diabatic flow in an annulus with an inner rotating cylinder. Trans. ASME C: J. Heat Transfer 84, 97–105.Google Scholar
Becker, K. M. & Kaye, J.1962b The influence of a radial temperature gradient on the instability of fluid flow in an annulus with an inner rotating cylinder. Trans. ASME C: J. Heat Transfer 84, 106–110.Google Scholar
Benjamin, T. B.1978a Bifurcation phenomena in steady flows of a viscous fluid. I. Theory. Proc. R. Soc. Lond. A 359, 1–26.Google Scholar
Benjamin, T. B. 1978b Bifurcation phenomena in steady flows of a viscous fluid. II. Experiments. Proc. R. Soc. Lond. A 359, 27–43.Google Scholar
Benjamin, T. B. & Mullin, T.1981 Anomalous modes in the Taylor experiment. Proc. R. Soc. Lond. A 377, 221–249.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
Bettes, T.1982 Chemical vapor deposition, trends and equipment. Semiconductor Intl, Denver. March, 59–72.
Bjorklund, I. S. & Kays, W. M.1959 Heat transfer between concentric rotating cylinders. Trans. ASME C: J. Heat Transfer 81, 175–186.Google Scholar
Bollen, L. J. M.1978 Epitaxial silicon, state-of-the-art. Acta Electronica 21, 185–199.Google Scholar
Chossat, P., Demay, Y. & Iooss, G.1985 Recent results about secondary bifurcations in the Couette—Taylor problem. Fourth Taylor Vortex Flow Working Party, Karlsruhe.Google Scholar
Chossat, P. & Iooss, G.1985 Primary and secondary bifurcation in the Couette—Taylor problem. Japan J. Appl. Maths 2, 37–68.Google Scholar
Cliffe, K. A.1983 Numerical calculations of two-cell and simple-cell Taylor flows. J. Fluid Mech. 135, 219–233.Google Scholar
Cliffe, K. A.1988 Numerical calculations of the primary flow exchange process in the Taylor problem. J. Fluid Mech. (In press.)Google Scholar
Cliffe, K. A. & Mullin, T.1985 A numerical and experimental study of anomalous modes in the Taylor experiment. J. Fluid Mech. 153, 243–258.Google Scholar
Demay, Y. & Iooss, G.1984 Calcul des solutions bifurquées pour le probléme de Couette—Taylor avec les deux cylindres en rotation. J. Méc. Théor. Appl. Numéro Spécial, pp. 193–216.Google Scholar
De Vahl Davis, G., Leonardi, E. & Reizes, J. A. 1984 Convection in a rotating annular cavity. In Heat and Mass Transfer in Rotating Machinery (ed. D. E. Metzger & N. H. Afgan), pp. 131–142. Hemisphere.
Diprima, R. C. & Swinney, H. L.1985 Instabilities and transition in flow between concentric rotating cylinders. In Hydrodynamic Instabilities and the Transition to Turbulence, 2nd edn (ed. H. L. Swinney & J. P. Gollub), pp. 139–180. Springer.
Drazin, P. G. & Reid, W. H.1984 Hydrodynamic Stability. Cambridge University Press.
Fung, Y. T. & Kurzweg, U. M.1975 Stability of swirling flows with radius dependent density. J. Fluid Mech. 72, 243–255.Google Scholar
Gardner, S. R. M. & Sabersky, R. H.1978 Heat transfer in an annular gap. Intl J. Heat Mass Transfer 21, 1459–1466.Google Scholar
Gazley, C.1958 Heat transfer characteristics of the rotational and axial flow between concentric cylinders. Trans. ASME C: J. Heat Transfer 80, 79–90.Google Scholar
Golubitsky, M. & Schaeffer, D. G.1985 Singularities and Groups in Bifurcation Theory, vol. 1. Springer.
Gosman, A. D., Pun, W. M., Runchal, A. K., Spalding, D. B. & Wolfstein, M. W. 1969 Heat and Mass Transfer in Recirculating Flows. Cambridge University Press.
Greenspan, H. P.1968 The Theory of Rotating Fluids. Cambridge University Press.
Hall, P.1980 Centrifugal instabilities in finite containers: a periodic model. J. Fluid Mech. 99, 575–596.Google Scholar
Hall, P.1982 Centrifugal instabilities of circumferential flows in finite cylinders: the wide gap problem. Proc. R. Soc. Lond. A 384, 359–379.Google Scholar
Holman, J. P.1981 Heat Transfer. McGraw-Hill Press.
Hughes, C. T., Leonardi, E., de Vahl Davis, G. & Reizes, J. A. 1985 A numerical study of the multiplicity of flows in the Taylor experiment. Phys. Chem. Hydrodyn. 6, 637–645.Google Scholar
Jones, I. P. & Cliffe, K. A.1983 Numerical studies for the flow due to rotating cylinders and disks. In Proceedings of the Fifth Workshop on Gases in Strong Rotation, Charlottesville, Virginia (ed. H. G. Wood), pp. 223–246.
Karlsson, S. K. F. & Snyder, H. A.1965 Observations on a thermally induced instability between rotating cylinders. Ann. Phys. 31, 314–324.Google Scholar
Kaye, J. & Elgar, E. C.1958 Modes of adiabatic and diabatic fluid flow in an annulus with an inner rotating cylinder. Trans. ASME C: J. Heat Transfer 80, 753–765.Google Scholar
Kreith, F.1968 Convection heat transfer in rotating systems. In Advances in Heat Transfer, vol. 5 (ed. T. F. Irvine & J. P. Hartnett), pp. 129–250. Academic.
Leonardi, E., Reizes, J. A. & de Vahl Davis, G. 1982 Heat transfer in a vertical rotating annulus – a numerical study. In Proc. 7th Intl Heat Transfer Conf. Paper no. FC12.
Lorenzen, A. & Mullin, T.1985 Anomalous modes and finite-length effects in Taylor—Couette flow. Phys. Rev. A 31, 3463–3465.Google Scholar
Mullin, T.1982 Mutations of steady cellular flows in the Taylor experiment. J. Fluid Mech. 121, 207–218.Google Scholar
Mullin, T., Pfister, G. & Lorenzen, A.1982 New observations on hysteresis effects in Taylor—Couette flow. Phys. Fluids 25, 1134–1136.Google Scholar
Patankar, S. V.1980 Numerical Heat Transfer and Fluid Flow. Hemisphere Press.
Schaeffer, D. G.1980 Qualitative analysis of a model for boundary effects in the Taylor problem. Math. Proc. Camb. Phil. Soc. 87, 307–337.Google Scholar
Singer, P. H.1984 Techniques of low pressure chemical vapor deposition. Semiconductor Intl, Denver. May, 72–77.
Snyder, H. A.1965 Experiments on the stability of two types of spiral flow. Ann. Phys. 31, 292–313.Google Scholar
Snyder, H. A. & Karlsson, S. K. F.1964 Experiments on the stability of Couette motion with a radial thermal gradient. Phys. Fluids 7, 1696–1706.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
Streett, C. L. & Hussaini, M. Y.1987 A numerical simulation of finite-length Taylor—Couette flow. AIAA Paper 87–1444.Google Scholar
Walowit, J., Tsao, S. & DiPrima, R. C.1964 Stability of flow between arbitrarily spaced concentric cylindrical surfaces, including the effect of a radial temperature gradient. Trans. ASME E: J. Appl. Mech. 31, 585–593.Google Scholar
Wan, C. C. & Coney, J. E. R.1982 An experimental study of diabatic spiral vortex flow. Intl J. Heat Fluid Flow 3, 31–38.Google Scholar
Withjack, E. M. & Chen, C. F.1974 An experimental study of Couette instability of stratified fluids. J. Fluid Mech. 66, 725–737.Google Scholar