Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-26T00:48:51.662Z Has data issue: false hasContentIssue false
  • English
  • Français

On the stability of circular Couette flow with radial heating

Published online by Cambridge University Press:  26 April 2006

Mohamed Ali  and
P. D. Weidman
Mohamed Ali
Affiliation:
Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309, USA Permanent address: Helwan University, Faculty of Engineering Technology, El-Mattaria, Cairo, Egypt.
P. D. Weidman
Affiliation:
Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of circular Couette flow with radial heating across a vertically oriented annulus with inner cylinder rotating and outer cylinder stationary is investigated using linear stability theory. Infinite aspect ratio and constant fluid properties are assumed and critical stability boundaries are calculated for a conduction-regime base flow. Buoyancy is included through the Boussinesq approximation and stability is tested with respect to both toroidal and helical disturbances of uniform wavenumber. Symmetries of the linearized disturbance equations based on the sense of radial heating and the sense of cylinder rotation and their effect on the kinematics and morphology of instability waveforms are presented. The numerical investigation is primarily restricted to radius ratios 0.6 and 0.959 at Prandtl numbers 4.35, 15 and 100. The results follow the development of critical stability from Taylor cells at zero heating through a number of asymmetric modes to axisymmetric cellular convection at zero rotation. Increasing the Prandtl number profoundly destabilizes the flow in both wide and narrow gaps and the number of contending critical modes increases with increasing radius ratio. Specific calculations made to compare with the stability measurements of Snyder & Karlsson (1964) and Sorour & Coney (1979) exhibit good agreement considering the idealizations built into the linear stability analysis.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 220 , November 1990 , pp. 53 - 84
Copyright
© 1990 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

Ali, M. E.: 1988 The stability of Taylor–Couette flow with radial heating. Ph.D. thesis, University of Colorado, Boulder, CO.
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
Bahl, S. K.: 1972 The effect of radial temperature gradient on the stability of a viscous flow between two rotating coaxial cylinders. Trans. ASME E: J. Appl. Mech. 39, 593–595.Google Scholar
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. Eight 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
Ball, S. K. & Farouk, B., 1988 Bifurcation and phenomena in Taylor–Couette flow with buoyancy effects. J. Fluid Mech. 197, 479–501.Google Scholar
Ball, K. S. & Farouk, B., 1989 A flow visualization study of the effects of buoyancy on Taylor vortices. Phys. Fluids A 1, 1502–1507.Google Scholar
Barratt, P. J. & Zuniga, I., 1984 A theoretical investigation of Bénard–Couette instabilities in nematic liquid crystals. J. Phys. D: Appl. Phys. 17.Google Scholar
Batchelor, G. K.: 1954 Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Qt. Appl. Maths 7, 209–233.Google Scholar
Becker, K. M. & Kaye, J., 1962 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 80, 106–110.Google Scholar
Chandrasekhar, S.: 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Choi, I. G. & Korpela, S. A., 1980 Stability of the conduction regime of natural convection in a tall vertical annulus. J. Fluid Mech. 99, 725–738.Google Scholar
Cognet, G.: 1984 Les esapes ver la turbulence dans l'ecoulement de Couette–Taylor entre cylindres coaxiaux. J. Méc. Theor. Appl. Numero special, pp. 7–44.Google Scholar
Di Prima, R. C. & Swinney, H. L. 1981 Instabilities and transition in flow between concentric rotating cylinders. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub). Topics in Applied Physics, vol. 45, pp. 139–180. Springer.
Eckert, E. R. G. & Carlson, W. O. 1961 Natural convection in an air layer enclosed between two vertical plates with different temperatures. Intl J. Heat Mass Transfer 2, 106–120.Google Scholar
Garg, V. K.: 1981 Stability of developing flow in a pipe: non-axisymmetric disturbances. J. Fluid Mech. 110, 209–216.Google Scholar
Garg, V. K. & Rouleau, W. T., 1972 Linear spatial stability of pipe Poiseuille flow. J. Fluid Mech. 54, 113–127.Google Scholar
Hart, J. E.: 1971 Stability of the flow in a differentially heated inclined box. J. Fluid Mech. 47, 547–576.Google Scholar
Keller, H. B.: 1976 Numerical solution of two point boundary value problems. SIAM Regional Conference Series in Applied Mathematics, vol. 24.Google Scholar
Kreith, F.: 1968 Convection heat transfer in rotating systems. Adv. Heat Transfer 5, 129–251.Google Scholar
Lee, Y., Korpela, S. A. & Horn, R. N., 1982 Structure of multicellular natural convection in a tall vertical annulus. In Proc. 7th Intl Heat Transfer Conf. Munich, vol. 2, pp. 221–226.
Lee, Y. N. & Minkowycz, W. J., 1989 Heat transfer characteristics of the annulus of two-coaxial cylinders with one cylinder rotating. Intl J. Heat Mass Transfer 32, 711–722.Google Scholar
Mcfadden, G. B., Coriell, S. R., Boisvert, R. F. & Glicksman, M. E., 1984a Asymmetric instabilities in a buoyancy-driven flow in a tall vertical annulus. Phys. Fluids 27, 1359–1361.Google Scholar
Mcfadden, G. B., Corell, S. R. & Boisvert, R. F., Glicksman, M. E., Fang, Q. T.: 1984b Morphological stability in the presence of fluid flow in the melt. Metall. Trans. 15A, 2117–2124.Google Scholar
Powell, M. J. D.: 1970 Numerical Methods for Nonlinear Algebraic Equations. Gordon and Breach.
Roberts, P. H.: 1965 The solution of the characteristic value problem. Appendix to: Donnelly, R. J. & Schwarz, K. W. Experiments on the stability of viscous flow between rotating cylinders. VI. Finite-amplitude experiments. Proc. R. Soc. Lond. A 283, 531–556.Google Scholar
Roesner, K. G.: 1978 Hydrodynamic stability of cylindrical Couette flow. Arch. Mech. 30, 619–627.Google Scholar
Scott, M. R. & Watts, H. A., 1977 Computational solution of linear two point boundary value problems via orthonormalization. SIAM J. Numer. Anal. 14, 40–70.Google Scholar
Singer, P. H.: 1984 Techniques of low pressure chemical vapor deposition. Semiconductor Intl Denver, May, 72–77.Google Scholar
SLATEC COMMON MATH LIBRARY, National Energy Software Center, Argonne National Laboratory, Argonne, IL, written by K. L. Hiebert, based on Powell, 1970.
Snyder, H. A.: 1962 Experiments on the stability of spiral flow at low axial Reynolds numbers. Proc. R. Soc. Lond. A 265, 198–214.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
Sorour, M. M.: 1977 Hydrodynamic instability, with special reference to the effect of heat transfer, in a concentric annulus having an inner rotating wall. Ph.D. thesis, University of Leeds.
Sorour, M. M. & Coney, J. E. R. 1979 The effect of temperature gradient on the stability of flow between vertical concentric rotating cylinders. J. Mech. Engng Sci. 21, 403–409.Google Scholar
Soundalgekar, V. M., Takhar, H. S. & Smith, J. T., 1981 Effects of radial temperature gradient on the stability of viscous flow in an annulus with a rotating inner cylinder. Warme Stoff. 15, 233–238.Google Scholar
Sparrow, E. M., Munro, W. D. & Jonsson, V. K., 1964 Instability of the flow between rotating cylinders: the wide gap problem. J. Fluid Mech. 20, 35–46.Google Scholar
Stuart, J. T.: 1986 Taylor vortex flow: a dynamical system. SIAM Rev. 28, 315–342.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.Google Scholar
Thomas, R. W. & De Vahl Davis, G. 1970 Natural convection in annular and rectangular cavities: a numerical study. Proc. 4th Intl Heat Transfer Conf., Paris, paper NC 2.4.Google Scholar
De Vahl, Davis G. & Thomas, R. W., 1969 Natural convection between concentric vertical cylinders. Phys. Fluids Suppl. II, 198–207.Google Scholar
Vives, C.: 1988 Effects of a forced Couette flow during the controlled solidification of a pure metal. Intl J. Heat Mass Transfer 31, 2047–2061.Google Scholar
Walowit, J., Tsao, S. & Di Prima, R. C. 1964 Stability of flow between arbitrarily spaced concentric cylinder surfaces, including the effect of a radial temperature gradient. Trans. ASME E: J. Appl. Mech. 31, 585–593.Google Scholar
Weidman, P. D.: 1989 Experimental techniques in laboratory rotating flows. In Frontiers in Fluid Mechanics (ed. M. Gad-El-Hak). Lecture Notes in Engineering, vol. 45. Springer.
Weidman, P. D. & Ali, M. E., 1989 The stability of Taylor–Couette flow with radial heating. In Proc. 2nd Workshop on Instabilities and Nonequilibrium Structures, Universidad de Santa Maria, Valpariso, Chile, Dec. 17–22, 1987. Reidel.
Weidman, P. D. & Mehrdadtehranfar, G., 1985 Instability of natural convection in a tall vertical annulus. Phys. Fluids 28, 776–787.Google Scholar
Yih, C.-S.: 1961 Dual role of viscosity in the stability of revolving fluids of variable density. Phys. Fluids 4, 806–811.Google Scholar