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The influence of swirl and confinement on the stability of counterflowing streams

Published online by Cambridge University Press:  26 April 2006

J. D. Goddard ,
A. K. Didwania  and
C.-Y. Wu
J. D. Goddard
Affiliation:
Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla CA 92093–0310, USA
A. K. Didwania
Affiliation:
Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla CA 92093–0310, USA
C.-Y. Wu
Affiliation:
Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla CA 92093–0310, USA
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Abstract

A linear stability analysis of laterally confined swirling flow is given, of the type described by Long's equation in the inviscid limit or by the von Kármán similarity equations in the absence of lateral confinement. The flow of interest involves identical counterflowing fluid streams injected with equal velocity W0 through opposing porous disks, rotating with angular velocities Ω and ±Ω, respectively, about a common normal axis. By means of mass transfer experiments on an aqueous system of this type we have detected an apparent hydrodynamic instability having the appearance of an inviscid supercritical bifurcation at a certain |Ω| > 0. As an attempt to elucidate this phenomenon, linear stability analyses are performed on several idealized flows, by means of a numerical Galerkin technique. An analysis of high-Reynolds-number similarity flow predicts oscillatory instability for all non-zero Ω. The spatial structure of the most unstable modes suggests that finite container geometry, as represented by the confining cylindrical sidewalls, may have a strong influence on flow stability. This is borne out by an inviscid stability analysis of a confined flow described by Long's equation. This analysis suggests a novel bifurcation of the inviscid variety, which serves qualitatively to explain the results of our mass transfer experiments.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 251 , June 1993 , pp. 149 - 172
Copyright
© 1993 Cambridge University Press

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References

Acrivos, A. & Goddard, J. D. 1966 Asymptotic expansions for laminar forced-convection heat and mass transfer, Part 1. Boundary layer flows. J. Fluid Mech. 24, 339.Google Scholar
Batchelor, G. K. 1951 Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow. Q. J. Mech. Appl. Maths 4, 29.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bodonyi, R. J. & Ng, B. S. 1983 On the stability of the similarity solutions for swirling flow above an infinite rotating disk. J. Fluid Mech. 144, 311.Google Scholar
Brady, J. F. & Durlofsky, L. 1987 On rotating disk flow. J. Fluid Mech. 175, 363.Google Scholar
Brattkus, K. & Davis, S. H. 1990 The linear stability of plane stagnation-point flow against general disturbances. Q. J. Mech. Appl. Maths 43, 471.Google Scholar
Chen, K. K. & Libby, P. A. 1968 Boundary layers with small departures from Falkner-Skan profile. J. Fluid Mech. 33, 273.Google Scholar
Chen, Z. H., Liu, G. E. & Sohrab, S. H. 1987 Premixed flames in counterflowing jets under rigid-body rotation. Combust. Sci. Tech. 51, 39.Google Scholar
Drazin, P. G. & Reid, W. H. 1989 Hydrodynamic Stability. Cambridge University Press.
Erdélyi, A. (ed.) 1954 Tables of Integral Transforms. McGraw-Hill.
Faller, A. J. 1991 Instability and transition of disturbed flow over a rotating disk. J. Fluid Mech. 230, 245.Google Scholar
Fletcher, A. J. 1984 Computational Galerkin Method. Springer.
Fraenkel, L. E. 1953 On the flow of rotating fluid past bodies in a pipe. Proc. R. Soc. A 233, 506.Google Scholar
Goddard, J. D., Melville, J. B. & Zhang, K. 1987 Similarity solution for stratified rotating-disk flow. J. Fluid Mech. 182, 427.Google Scholar
Goldshtik, M. A. & Javorsky, N. J. 1989 On the flow between a porous rotating disk and a plane. J. Fluid Mech. 207, 1.Google Scholar
Golubitsky, M. & Schaeffer, D. G. 1985 Singularities and Groups in Bifurcation Theory. Springer.
Gordon, L. S., Newman, J. S. & Tobias, C. W. 1966 The role of ionic migration in electrolytic mass transport; diffusivities of [Fe(CN)6]3- and [Fe(CN)6]4- in KOH and NaOH solutions. Ber. Bunsengellschaft 4, 414–420.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Gregory, N., Stuart, J. T. & Walker, W. S. 1955 On the stability of three-dimensional boundary layers with application to a rotating disk. Phil. Trans. R. Soc. Lond. A 248, 155.Google Scholar
Kim, J. S., Libby, P. A. & Williams, F. A. 1992 Influence of swirl on the structure and extinction of strained premixed flames. Phys. Fluids A 4, 391.Google Scholar
Malik, M. R. 1986 The neutral curve for stationary disturbance in rotating-disk flow. J. Fluid Mech. 164, 275.Google Scholar
Parter, S. V. & Rajagopal, K. R. 1984 Swirling flow between rotating plates. Arch. Rat. Mech. Anal. 86, 305.Google Scholar
Sivashinsky, G. I. & Sohrab, S. H. 1987 The influence of rotation on premixed flames in stagnation point flow. Combust. Sci. Tech. 53, 67.Google Scholar
Szeri, A. Z., Giron, A. & Schneider, S. J. 1983 Flow between rotating disks. Part 2. Stability. J. Fluid Mech. 134, 133.Google Scholar
Wu, C.-Y. 1991 An electrochemical study of mass transfer between rotating counterflowing fluid streams. 1 thesis, University of Southern California.
Yih, C.-S. 1980 Stratified Flows. Academic.
Zandbergen, P. J. & Dijkstra, D. 1987 Von Kármán swirling flows. Ann. Rev. Fluid Mech. 19, 465.Google Scholar
Zhang, K. & Goddard, J. D. 1989 Viscous interlayer structure and transport properties in von Kármán swirling flows. Phys. Fluids A 1, 132.Google Scholar