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Instability and transition of disturbed flow over a rotating disk

Published online by Cambridge University Press:  26 April 2006

Alan J. Faller
Alan J. Faller
Affiliation:
Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA
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Abstract

Disturbed flow over a rotating disk can lead to transition of the von Kármán boundary layer at a much lower Reynolds number, Re, (i.e. smaller radius) than that due to the well-known Type 1 stationary mode of instability. This early transition is due to the excitation of the Type 2 instability, similar to that found in the Ekman layer. Detailed numerical values of the growth rates, phase speeds, group velocities, neutral curves, and other characteristics of these two instabilities have been calculated over a wide range of parameters. Neutral curves for the Ekman and Bödewadt boundary layers also are presented. The minimum critical Reynolds numbers for the von Kármán, Ekman and Bödewadt layers are Rec(2) = 69.4, 54.3, and 15.1 with wavelengths L = 22.5, 20.1, and 16.6 and at angles ε = −19.0°, −23.1°, and −33.2°, respectively. These minimum critical values frequently do not well describe laboratory observations, however, because at larger Re other modes grow more rapidly and dominate the flow.

The computed results are in excellent agreement with laboratory observations wherever comparison is possible. The growth of representative Type 1 instabilities with radius is shown to lead to N-factors greater than 9 at Re = 520 as appears to be necessary for transition to turbulence by the interaction of Type 1 with the basic flow. The growth of Type 2 instabilities with radius can lead to three additional mechanisms of transition. The necessary levels of excitation of Type 2 for these different mechanisms are estimated.

A sequence of photographs from a ciné film illustrate one of the transition mechanisms discussed: the interaction of Type 2 instabilities and a secondary instability that is nearly perpendicular to the Type 2 vortices.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 230 , September 1991 , pp. 245 - 269
Copyright
© 1991 Cambridge University Press

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References

BuUhler, K. & Zierep, J. 1983 Transition to turbulence in a spherical gap. 4th Intl Symp. on Turbulent Shear Flows, Sept. 1983, Karlsruhe.
Bushnell, D. M., Malik, M. R. & Harvey, W. D. 1988 Transition prediction in external flows via linear stability theory. IUTAM Symp. Transanicum III, GoUttingen.
Cebeci, T. & Stewartson, K. 1980 On stability and transition in three-dimensional flows. AIAA J. 18, 398–405.Google Scholar
Faller, A. J. 1963 An experimental study of the instability of the laminar Ekman boundary layer. J. Fluid Mech. 15, 560–576.Google Scholar
Faller, A. J. 1990 Instability and transition of disturbed flow on a rotating disk. Tech. Note.BN-1113 Inst. Phys. Sci. & Tech., University of Maryland.Google Scholar
Faller, A. J. & Kaylor, R. E. 1966a A Investigations of stability and transition in rotating boundary layers. In Dynamics of Fluid and Plasmas. (ed. S. I. Pai). Academic.
Faller, A. J. & Kaylor, R. E. 1966b A numerical study of the instability of the laminar Ekman boundary layer. J. Atmos. Sci. 23, 466–480.Google Scholar
Faller, A. J., Yang, S. T. & Piomelli, U. 1989 Instability of the KEB boundary layers. Tech. Note. BN-1102. Inst. Phys. Sci. & Tech., University of Maryland.Google Scholar
Gregory, N., Stuart, J. T. & Walker, W. S. 1955 On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk. Phil. Trans. R. Soc. Lond. A 248, 155–199 (referred to herein as GSW).Google Scholar
Iooss, G., Nielsen, H. G. B. & True, H. 1978 Bifurcation of the stationary Ekman layer flow into a stable periodic flow. Arch. Rat. Mech. Anal.68, 227–256.Google Scholar
Lilly, D. K. 1966 On the instability of Ekman boundary flow. J. Atmos. Sci.23, 481–194.Google Scholar
Mack, L. M. 1985 The wave pattern produced by point source on a rotating disk. AIAA Aerospace Sciences Meeting, Jan. 14–17, Reno, Nevada, AIAA 85–0490.
Malik, M. R. 1986 The neutral curve for stationary disturbances in rotating-disk flow. J. Fluid Mech 164, 275–287.Google Scholar
Malik, M. R., Wilkinson, S. P. & Orszag, S. A. 1981 Instability and transition in rotating disk flow. AIAA J. 19, 1131–1138.Google Scholar
Smith, N. H. 1947 Exploratory investigation of boundary layer oscillations on a rotating disk. NACA Tech. Note 1227.Google Scholar
Spooner, G. F. & Criminale, W. 0. 1982 The evolution of disturbances in the Ekman boundary layer. J. Fluid Mech.115, 327–346.Google Scholar
Wilkinson, S. P. & Malik, M. R. 1985 Stability experiments in the flow over a rotating disk. AIAA J.23, 588–595.Google Scholar