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Symmetry breaking in vortex-source and Jeffery—Hamel flows

Published online by Cambridge University Press:  26 April 2006

M. Goldshtik ,
F. Hussain  and
V. Shtern
M. Goldshtik
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston, TX 77204–7492, USA Permanent address: Institute of Themophysics, Novosibirsk 630090, USSR.
F. Hussain
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston, TX 77204–7492, USA
V. Shtern
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston, TX 77204–7492, USA Permanent address: Institute of Themophysics, Novosibirsk 630090, USSR.
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Abstract

The stability and bifurcations associated with the loss of azimuthal symmetry of planar flows of a viscous incompressible fluid, such as vortex-source and Jeffery–Hamel flows, are studied by employing linear, weakly nonlinear and fully nonlinear analyses, and features of new solutions are explained. We address here steady self-similar solutions of the Navier–Stokes equations and their stability to spatially developing disturbances. By considering bifurcations of a potential vortex-source flow, we find secondary solutions. They include asymmetric vortices which are generalizations of the classical point vortex to vortical flows with non-axisymmetric vorticity distributions. Another class of solutions we report relates to transition trajectories that connect new bifurcation-produced solutions with the primary ones. Such solutions provide far-field asymptotes for a number of jet-like flows. In particular, we consider a flow which is a combination of a jet and a sink, a tripolar jet, a jet emerging from a slit in a plane wall, a jet emerging from a plane channel and the reattachment phenomenon in the Jeffery–Hamel flow in divergent channels.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 232 , November 1991 , pp. 521 - 566
Copyright
© 1991 Cambridge University Press

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References

Abramowitz, M. & Stegun, A. (ed.) 1964 Handbook of Mathematical Functions. National Bureau of Standards. 1064 pp.
Akatnov, N. I. 1953 Propagation of plane laminar jet near rigid wall. Trans. Leningrad Politechnical Inst. Mashgiz. pp. 24–31.Google Scholar
Aref, H. 1983 Integrable, chaotic and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345–389.Google Scholar
Banks, W. H. H., DRAZIN, P. G. & ZATURSKA, M. B. 1988 On perturbations of Jeffery-Hamel flow. J. Fluid Mech. 186, 559–581.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press. 615 pp.
Batchelor, G. K. & Gill, A. E. 1962 Analysis of stability of axisymmetric jets. J. Fluid Mech. 14, 529–551.Google Scholar
Dean, W. R. 1934 Note on the divergent flow of fluid. Phil. Mag. (7), 18, 759–777.Google Scholar
Dean, W. R. & Montagnon, P. E. 1949 On the steady motion of a viscous fluid in a corner. Proc. Camb. Phil. Soc. 45, 389–394.Google Scholar
Deem, G. S. & Zabusky, N. J. 1978 Vortex waves: stationary ‘V-states’, interactions, recurrence and breaking. Phys. Rev. Lett. 13, 859–862.Google Scholar
Fraenkel, L. E. 1962 Laminar flow in symmetrical channels with slightly curved walls. I. On the Jeffery-Hamel solutions for the flow between plane walls. Proc. R. Soc. Lond. A 267, 119–138.Google Scholar
Fraenkel, L. E. 1963 Laminar flow in symmetrical channels with slightly curved walls. II An asymptotic series for the stream function. Proc. R. Soc. Lond. A 272, 406–428.Google Scholar
Georgiou, G. A. & Eagles, P. M. 1985 The stability of flows in channels with small wall curvature. J. Fluid Mech. 159, 259–287.Google Scholar
Glauert, M. B. 1956 The wall jet. J. Fluid Mech. 1, 626–643.Google Scholar
Goldshtik, M. A. 1990 Viscous flow paradoxes. Ann. Rev. Fluid Mech. 22, 441–472.Google Scholar
Goldshtik, M. A. & Shtern, V. N. 1977 Hydrodynamic Stability and Turbulence. Novosibirsk. Nauka. 366 pp. (In Russian.)Google Scholar
Goldshtik, M. A. & Shtern, V. N. 1989 Loss of symmetry in viscous flow from a linear source. Fluid Dyn. 24, 151–199.Google Scholar
Goldshtik, M. A. & Shtern, V. N. 1990 Collapse in conical viscous flows. J. Fluid Mech. 218, 483–508.Google Scholar
Hamel, G. 1916 Spiralförmige Bewegungen zäher Flüssigkeiten. Jahresbericht Deutsch. Math. Verein. 25, 34–60.Google Scholar
Hooper, A. P., Duffy, B. R. & Moffatt, H. K. 1982 Flow of fluid of nonuniform viscosity in converging and diverging channels. J. Fluid Mech. 117, 283–304.Google Scholar
Hussain, F. 1986 Coherent structures and turbulence. J. Fluid Mech. 173, 303–356.Google Scholar
Hussain, F. & Husain, H. S. 1989 Elliptic jets. Part 1. Characteristics of unexited and exited jets. J. Fluid Mech. 208, 257–320.Google Scholar
Jeffery, G. B. 1915 The two-dimensional steady motion of a viscous fluid. Phil. Mag. (6) 29, 455–465.Google Scholar
Joseph, D. D. 1976 Stability of Fluid Motions. Part 1. Springer. 638 pp.
Landau, L. D. 1944 On turbulence problem. Sov. Phys. Dokl. 44, 339–342.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon.
Loitsyansky, L. G. 1966 Mechanics of Liquids and Gas. Pergamon.
Millsaps, K. & Pohlhausen, K. 1953 Thermal distributions in Jeffery-Hamel flows between nonparallel plane walls. J. Aeronaut. Sci. 20, 187–196.Google Scholar
Moffatt, H. K. & Duffy, B. R. 1980 Local similarity solutions and their limitations. J. Fluid Mech. 96, 299–313.Google Scholar
Oseen, C. W. 1927 Exakte Lösungen der hydrodynamischen Differential-gleichungen. Arkiv fur Matematik, Astr. Fysik I (14), 1–24; II (22), 1–9.Google Scholar
Pshenichnikov, A. F. & Yatsenko, S. S. 1974 Convective diffusion from compact source of surface-active matter. Hydrodynamics (Scientific papers of Perm University), No. 5, pp. 175–181. (In Russian).Google Scholar
Rosenhead, L. 1940 The steady two-dimensional radial flow of viscous fluid between two inclined plane walls. Proc. R. Land. A 175, 436–467.Google Scholar
Schlichting, H. 1979 Boundary-layer Theory. McGraw-Hill.
Schneider, W., Zauner, E. & Bohm, H. 1987 The recirculatory flow induced by a laminar axisymmetric jet issuing from a wall. Trans. ASME I: J. Fluid Engng. 109, 237–241.Google Scholar
Sobey, I. J. & Drazin, P. G. 1986 Bifurcations of two-dimensional channel flows. J. Fluid Mech. 171, 263–287.Google Scholar
Uribe, F. J., Diaz, E., Bravo, A., Peralta Fabi, R. & Soto, R. 1989 Bifurcation diagram for the Jeffery—Hamel flow. Winter Meeting on Statistical Physical. World Scientific.
Widnall, S. E. & Sullivan, J. P. 1972 On the stability of vortex rings. Proc. R. Soc. Lond. A 332, 335–353.Google Scholar
Yudovich, V. I. 1965 Stability of steady flows of viscous incompressible fluid. Sov. Phys. Dokl. 10(4), 293–295.Google Scholar