Abstract
There is a strong analogy between Rayleigh-Bénard convection and the Taylor-Couette system. This analogy is well known when dealing with the primary instability, and is based on the existence of an unstable stratification in both systems. We show that the analogy can be extended beyond the primary instability modes, to the weakly non-linear regime and even further to the fully turbulent one.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Couette, “Sur un nouvel appareil pour l’étude du frottement des fluides”, Ann. Chim. Phys. (6) 21, 433–510, 70 (1890).
A. Mallock, “Determination of the viscosity of water”, Phil. Trans. Roy. Soc. A 187, 41–56 (1896).
L. Rayleigh, “On the dynamics of revolving fluids”, Proc. R. Soc. London, Ser. A 93, 148–54 (1916).
L. Rayleigh, “On convective currents in a horizontal layer of fluid, when the higher temperature is on the under side”, Phil. Mag (6) 32, 529–46 (1916).
G.I. Taylor, “Stability of a viscous liquid contained between two rotating cylinders”, Phil. Trans. Roy. Soc. London, Ser. A 223, 289–343 (1923).
H. Jeffreys, “Some cases of instability in fluid motion”, Proc. R. Soc. London, Ser. A 118, 195–208 (1928).
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford (1961).
F.H. Busse, “Bounds for turbulent shear flow”, J. Fluid Mech. 41, 219 (1970).
E. Guyon, J.P. Hulin, and L. Petit, Hydrodynamique Physique, Savoirs actuels, Paris (2002).
E.L. Koscmieder, Bénard Cells and Taylor Vortices, Cambridge University Press, Cambridge (1993).
B.J. Bayly, “Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows”, Phys. Fluids A 31(1), 56–64 (1988).
I. Mutabazi, C. Normand, and J.E. Wesfreid, “Gap size effects on centrifugally and rotaionally driven instabilities”, Phys. Fluids A 4, 1199–1205 (1992).
P. Manneville, Structures Dissipatives, Chaos et Turbulence, Aléa Saclay, Saclay (1991).
A. Esser and S. Grossmann, “Analytic expression for Taylor-Couette stability boundary”, Phys. Fluids 8, 1814 (1996).
B. Dubrulle, O. Dauchot, F. Daviaud, F. Hersant, P.-Y. Longaretti, D. Richard, and J-P. Zahn,“Stability and turbulent transport in rotating shear flows: prescription from analysis of cylindrical and plane Couette flows”, submitted to Phys. Fluids (2004).
D.K. Lezius and J.P. Johnston, “Roll-cell instabilities in rotating laminar and turbulent channel flows”, J. Fluid Mech. 77, 573 (1976).
O.J.E. Matsson and P.H. Alfredsson, “Curvature-and rotation-induced instabilities in channel flow”, J. Fluid Mech. 210, 537–563 (1990).
J.E. Wesfreid, Y. Pomeau, M. Dubois, C. Normand, and P. Bergé, “Critical effects in Rayleigh-Bénard convection”, J. Physique (Paris) 39, 725 (1978).
H. Yahata, “Slowly-varying amplitude of the Taylor vortices near the instability point”, Prog. Theor. Phys. 57, 347 (1977).
P. Tabeling, “Dynamics of the phase variable in the Taylor vortex system”, J. Phys. Lett. 44, 16 (1983).
P. Hall, “Evolution equations for Taylor vortices in the small-gap limit”, Phys. Rev. A 29, 2921 (1984).
Y. Demay and G. Iooss, “Calcul des solutions bifurquées pour le problème de Couette-Taylor avec les 2 cylindres en rotation”, J. Méca. Théor. Appl., numéro spécial, 193 (1984).
Y. Demay, G. Iooss, and P. Laure, “Wave patterns in the small gap Couette-Taylor problem”, Eur. J. Mech. B 11, 621 (1992).
M.A. Dominguez-Lerma, G. Ahlers, and D. S. Cannell, “Marginal stability curve and linear growth rate for rotating Couette-Taylor flow and Rayleigh-Bénard convection”, Phys. Fluids 27, 856 (1984).
G. Ahlers, “Experiments on bifurcation and one-dimensional patterns in nonlinear systems far from equilibrium”, in Lectures in the Sciences of Complexity, ed. by D.L. Stein, Addison-Wesley, Redwood City, CA (1989).
Y. Pomeau and P. Manneville, “Stability and fluctuations of a spatially periodic convective flow”, J. Phys. Lett. 40, 610 (1979).
J.E. Wesfreid and V. Croquette, “Forced Phase Diffusion in Rayleigh-Bénard Convection”, Phys. Rev. Lett. 45, 6340 (1980).
H. Paap and H. Riecke, “Drifting vortices in ramped Taylor vortex flow. Quantitative results from phase equation”, Phys. Fluids A 3, 1519 (1991).
M. Wu and C.D. Andereck, “Phase modulation of Taylor vortex flow”, Phys. Rev. A 43, 2074 (1991).
M. Wu and C.D. Andereck, “Phase dynamics of wavy vortex flow”, Phys. Rev. Lett. 67, 1258 (1991).
M. Wu and C.D. Andereck, “Phase dynamics in the Taylor-Couette system”, Phys. Fluids A 4, 2432 (1992).
D. Coles, “Transition in circular Couette flow”, J. Fluid Mech. 21, 385–425 (1965).
C.D. Andereck, S.S. Liu, and H.L. Swinney, “Flow regimes in a circular Couette system with independently rotating cylinders”, J. Fluid Mech. 164, 155–183 (1986).
R. Tagg, W.S. Edwards, and H.L. Swinney, “Nonlinear standing waves in Couette-Taylor flow”, Phys. Rev. A 39, 3734 (1989).
R. Tagg, “The Couette-Taylor problem”, Nonlinear Science Today 4, 1 (1994).
F.H. Busse, “Transition to turbulence in Rayleigh-Bénard convection”, in Hydrodynamic Instabilities and the Transition to Turbulence eds. H.L. Swinney and J.P. Gollub, Springer, New York (1981).
B. Dubrulle and F. Hersant, “Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection”, Eur. Phys. J. B 26, 379 (2002).
A. Schlüter, D. Lortz, and F. Busse, “Transition to turbulence in Rayleigh-Bénard convection”, J. Fluid Mech. 23, 129 (1965).
J.K. Platten and J.C. Legros, Convection in liquids, Springer, New York (1984).
D.P. Lathrop, J. Fineberg, and H.L. Swinney, “Transition to shear-driven turbulence in Couette-Taylor flow”, Phys. Rev A 46, 6390 (1992).
G.P. King, Y. Li, W. Lee, H.L. Swinney, and P.S. Marcus, “Wave speeds in wavy Taylor vortex flow”, J. Fluid Mech. 41, 365 (1984).
A. Barcilon and J. Brindley, “Organized structures in turbulent Taylor-Couette flows at a very high Taylor number”, J. Fluid Mech. 143, 429 (1984).
F. Wendt, “Turbulente Stromungen zwischen zwei rotierenden konaxialen Zylindern”, Ingenieur-Archiv. 4, 577 (1933).
G.I. Taylor,“Fluid friction between rotating cylinders I. — Torque measurements”, Proc. R. Soc. London A 157, 546 (1936).
P. Tong, W.I. Goldburg, J.S. Huang, and T.A. Witten, “Anisotropy in turbulent drag reduction”, Phys. Rev. Lett. 65, 2780 (1990).
G.S. Lewis and H.L. Swinney, “Velocity structure functions, scaling, and transitions in high-Reynolds-number Couette-Taylor flow”, Phys Rev. E 59, 5457 (1999).
F. Heslot, B. Castaing, and A. Libchaber, “Transitions to turbulence in helium gas”, Phys. Rev. A 36, 5870 (1987).
B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X-Z. Wu, S. Zaleski, and G. Zanetti, “Scaling of hard thermal turbulence in Rayleigh-Bénard convection”, J. Fluid Mech. 204, 1 (1989).
R. Kraichnan, “Mixing-length analysis of turbulent thermal convection at arbitrary Prandtl number”, Phys. Fluids 5, 1374 (1962).
X. Chavanne, F. Chilla, B. Castaing, B. Hébral, B. Chabaud, and J. Chaussy, “Observation of the ultimate regime in Rayleigh-Bénard convection”, Phys. Rev. Lett. 79, 3648 (1997).
B. Dubrulle, “Scaling laws predictions from a solvable model of turbulent thermal convection”, Europhys. Letters 51, 513 (2000).
B. Dubrulle, “Logarithmic corrections to scaling in turbulent thermal convection”, Euro. Phys. J. B 21, 295 (2001).
F.H. Busse, “Transition to turbulence in Rayleigh-Bénard convection”, in Nonlinear Physics of Complex Systems ed. G. Parisi, S.C. Muller, and W. Zimmermann, Lecture Notes in Physics 476, 1, Springer, New York (1996).
F.H. Busse, “Transition to turbulence in Rayleigh-Bénard convection”, Arch. Rat. Mech. Anals 47, 28 (1972).
J.J. Niemela, L. Skrbek, K.R. Sreenivasan, and R.J. Donnelly, “Turbulent convection at very high Rayleigh number”, Nature 404, 837 (2000).
J.W. Deardoff and G.E Willis, “Investigation of turbulent thermal convection between horizontal plates”, J. Fluid Mech. 28, 675 (1967).
E. Guyon and P. Pieranski, “Convective instabilities in nematic liquid crystals”, Physica 73, 184 (1974).
I.H. Herron, “Stability criteria for flow along a convex wall”, Phys. Fluids A 3, 1825 (1991).
D.J. Tritton and P.A. Davies, in Hydrodynamic Instabilities and the Transition to Turbulence, H.L. Swinney and J.P. Gollub (eds.), Springer, New York (1981).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
Prigent, A., Dubrulle, B., Dauchot, O., Mutabazi, I. (2006). The Taylor-Couette Flow: The Hydrodynamic Twin of Rayleigh-Bénard Convection. In: Mutabazi, I., Wesfreid, J.E., Guyon, E. (eds) Dynamics of Spatio-Temporal Cellular Structures. Springer Tracts in Modern Physics, vol 207. Springer, New York, NY. https://doi.org/10.1007/978-0-387-25111-0_13
Download citation
DOI: https://doi.org/10.1007/978-0-387-25111-0_13
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-40098-3
Online ISBN: 978-0-387-25111-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)