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On wavy instabilities of the Taylor-vortex flow between corotating cylinders

Published online by Cambridge University Press:  21 April 2006

Masato Nagata
Masato Nagata
Affiliation:
Department of Mathematical Sciences, University of St. Andrews, North Haugh, St Andrews KY16 9SS, Fife, Scotland
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Abstract

At least four wavy instabilities are found numerically by analysing the linear stability of Taylor-vortex flow (TVF) in the limit of a small gap between two concentric cylinders which rotate differentially in the same direction. Two of the wavy instabilities, including the one leading to conventional wavy vortex (WVF), have the same axial wavelength as TVF at the onset of instability, while the other two are characterized by subharmonic modes with axial wavelengths twice as long as those of TVF. The two subharmonic instabilities appear to correspond to the wavy-inflow-boundary flow (WIB) and the wavy-outflow-boundary flow (WOB) observed in the experiment of Andereck, Liu & Swinney (1986). The phase velocities, measured in the rotating frame of reference, of all the wavy instabilities are non-zero at the onset except that the phase velocity of WVF vanishes in the region where the average rotation rate Ω of the cylinders is small. By using this simple bifurcation property of WVF for small Ω, time-independent finite-amplitude non-axisymmetric solution branches bifurcating from TVF are followed numerically. The most interesting findings are that some of the solution branches cross the line Ω = 0, producing three-dimensional nonlinear solutions in plane Couette flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 188 , March 1988 , pp. 585 - 598
Copyright
© 1988 Cambridge University Press

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References

Andereck, C. D., Dickman, R. & Swinney, H. L. 1983 New flows in a circular Couette system with co-rotating cylinders. Phys. Fluids 26, 1395–1401.Google Scholar
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
Busse, F. H. 1978 Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 1927–1967.Google Scholar
Cowley, S. J. & Smith, F. T. 1985 On the stability of Poiseuille—Couette flow: a bifurcation from infinity. J. Fluid Mech. 156, 83–100.Google Scholar
Craik, A. D. D. 1971 Non-linear resonant instability in boundary layers. J. Fluid Mech. 50, 393–413.Google Scholar
Iooss, G. 1986 Secondary bifurcations of the Taylor vortices into wavy inflow and outflow boundaries. J. Fluid Mech. 173, 273–288.Google Scholar
Jones, C. A. 1985 The transition to wavy Taylor vortices. J. Fluid Mech. 157, 135–162.Google Scholar
Nagata, M. 1986 Bifurcations in Couette flow between almost corotating cylinders. J. Fluid Mech. 169, 229–250.Google Scholar
Nagata, M. 1987 Three-dimensional finite amplitude solutions in plane Couette flow. Phys. Rev. (submitted).Google Scholar
Orszag, S. A. & Kells, L. C. 1980 Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech. 96, 159–206.Google Scholar
Rosenblat, S. & Davis, S. H. 1979 SIAM J. Appl. Maths 37, 1–19.