Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-19T00:38:30.001Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of confined swirling flow with and without vortex breakdown

Published online by Cambridge University Press:  26 April 2006

A. Yu. Gelfgat ,
P. Z. Bar-Yoseph  and
A. Solan
A. Yu. Gelfgat
Affiliation:
Computational Mechanics Laboratory, Faculty of Mechanical Engineering, Technion — Israel Institute of Technology, Haifa 32000, Israel; e-mailmerbygr@cmlp.technion.ac.il
P. Z. Bar-Yoseph
Affiliation:
Computational Mechanics Laboratory, Faculty of Mechanical Engineering, Technion — Israel Institute of Technology, Haifa 32000, Israel; e-mailmerbygr@cmlp.technion.ac.il
A. Solan
Affiliation:
Computational Mechanics Laboratory, Faculty of Mechanical Engineering, Technion — Israel Institute of Technology, Haifa 32000, Israel; e-mailmerbygr@cmlp.technion.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

A numerical investigation of steady states, their stability, onset of oscillatory instability, and slightly supercritical unsteady regimes of an axisymmetric swirling flow of a Newtonian incompressible fluid in a closed circular cylinder with a rotating lid is presented for aspect ratio (height/radius) 1 [les ] γ [les ] 3.5. Various criteria for the appearance of vortex breakdown are discussed. It is shown that vortex breakdown takes place in this system not as a result of instability but as a continuous evolution of the stationary meridional flow with increasing Reynolds number. The dependence of the critical Reynolds number Recr and frequency of oscillations ωcr on the aspect ratio of the cylinder γ is obtained. It is found that the neutral curve Recr(γ) and the curve ωcr(γ) consist of three successive continuous segments corresponding to different modes of the dominant perturbation. The calculated critical parameters are in good agreement with the available experimental and numerical data for γ < 3. It is shown that the onset of the oscillatory instability does not depend on the existence of a separation bubble in the subcritical steady state. By means of a weakly nonlinear analysis it is shown that the axisymmetric oscillatory instability sets in as a result of a supercritical Hopf bifurcation for each segment of the neutral curve. A weakly nonlinear asymptotic approximation of slightly supercritical flows is carried out. The results of the weakly nonlinear analysis are verified by direct numerical solution of the unsteady Navier-Stokes equation using the finite volume method. The analysis of the supercritical flow field for aspect ratio less than 1.75, for which no steady vortex breakdown is found, shows the existence of an oscillatory vortex breakdown which develops as a result of the oscillatory instability.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 311 , 25 March 1996 , pp. 1 - 36
Copyright
© 1996 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arkadyev, A. Bar-Yoseph, P., Solan, A. & Roesner, K. G. 1993 Thermal effects on axisymmetric vortex breakdown in a spherical gap.. Phys. Fluids A 5, 1211–1223.Google Scholar
Bar-Yoseph, P. 1994 On multiple flow patterns and vortex breakdown phenomena in confined rotating flows. Comput. Fluid Dyn. J. 3, 273–292.Google Scholar
Bar-Yoseph, P. & Kryzhanovski, Yu. 1995 On the validity of power law models in the prediction of vortex breakdown phenomena. 6th Intl Symp. on Computational Fluid Dynamics (ed. M. Hafez), Lake Tahoe, Nevada, September 4-8, pp. 29–34; Comput. Fluid Dyn. (to appear).
Bar-Yoseph, P., Roesner, K. G. & Solan, A. 1992 Vortex breakdown in the polar region between rotating spheres.. Phys. Fluids A 4, 1677–1686.Google Scholar
Brown, G. L. & Lopez, J. M. 1990 Axisymmetric vortex breakdown. Part 2. Physical mechanisms. J. Fluid Mech. 221, 533–576.Google Scholar
Christensen, E. A. Sorensen, J. N., Brons, M. & Christiansen P. L. 1993 Low-dimensional representation of early transition in rotating fluid flow. Theor. Comput. Fluid Dyn. 5, 259–267.Google Scholar
Daube, O. 1992 Resolution of the 2D Navier-Stokes equations in velocity-vorticity form by means of an influence matrix technique. J. Comput. Phys. 103, 402–414.Google Scholar
Daube, O. & Sorensen, J. N. 1989 Simulation numérique de l'écoulement périodique axisymmétrique dans une cavité cylindrique. C. R. Acad. Sci. Paris 308, 463–469.Google Scholar
Davidson, P. A. 1989 The interaction between swirling and recirculating velocity components in unsteady, inviscid flow. J. Fluid Mech. 209, 35–55.Google Scholar
Delery, J. M. 1994 Aspects of vortex breakdown. Prog. Aerosp. Sci. 30 1–59.Google Scholar
Escudier, M. P. 1984 Observation of the flow produced in a cylindrical container by a rotating endwall. Exps. Fluids 2, 189–196.Google Scholar
Escudier, M. P. 1988 Vortex breakdown: observations and explanations. Proc. Aerosp. Sci. 25, pp. 189–229.Google Scholar
Gelfgat, A., Bar-Yoseph, P. & Solan, A. 1995 Numerical investigation of a confined swirling flow in a cylinder with rotating top and bottom by the Galerkin spectral method. 6th Intl Symp. on Computational Fluid Dynamics (ed. M. Hafez), Lake Tahoe, Nevada, September 4-8, pp. 355–360.
Gelfgat, A. Yu. & Tanasawa, I. 1993 Systems of basis functions for calculation of three-dimensional fluid flows in cylindrical containers with the Galerkin spectral method. Proc. Inst. Indust. Sci., Univ. Tokyo 45 (8), 60–63.Google Scholar
Gelfgat, A. Yu. & Tanasawa, I. 1994 Numerical analysis of oscillatory instability of buoyancy convection with the Galerkin spectral method. Numer. Heat Transfer A: Applications 25, 627–648.Google Scholar
Hassard, B. D., Kazarinoff, N. D. & Wan, Y.-H. 1981 Theory and Applications of Hopf Bifurcation. London Math. Soc. Lecture Note Series, vol.41.
Hopf, E. 1942 Abzweigung einer Periodischen Losung von Einer Stationaren Losung eines Differentialsystems. Ber. Verh. Sachs. Acad. Wiss. Leipsig Math.-Nat. 94, 3–22.Google Scholar
Hopfinger, E. J. & Linden, P. F. 1990 The effect of background rotation on fluid motions. J. Fluid Mech. 211, 417–435.Google Scholar
Janssen, R. J. A. Henkes, R. A. W. M., Hoogendoorn, C. J. 1993 Transition to time-periodicity of a natural convection flow in a 3D differentially heated cavity. Intl J. Heat Mass Transfer 36, 2927–2940.Google Scholar
Lopez, J. M. 1990 Axisymmetric vortex breakdown. Part 1. Confined swirling flow. J. Fluid Mech. 221, 533–552.Google Scholar
Lopez, J. M. 1994 On the bifurcation structure of axisymmetric vortex breakdown in a constricted pipe. Phys. Fluids 6, 3683–3693.Google Scholar
Lopez, J. M. & Perry, A. D. 1992 Axisymmetric vortex breakdown. Part 3. Onset of periodic flow and chaotic advection. J. Fluid Mech. 234, 449–471.Google Scholar
Lugt, H. J. & Abboud, M. 1987 Axisymmetric vortex breakdown with and without temperature effects in a container with a rotating lid. J. Fluid Mech. 179, 179–200.Google Scholar
Lugt, H. J. & Haussling, H. J. 1973 Development of flow circulation in a rotating tank. Acta Mechanica 18, 255–272.Google Scholar
Neitzel, G. P. 1988 Streak-line motion during steady and unsteady axisymmetric vortex breakdown. Phys. Fluids 31, 958–960.Google Scholar
Patankar, S. V. & Spalding, D. B. 1972 A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Intl J. Heat Mass Transfer 15, 1787–1806.Google Scholar
Roesner, K. G. 1990 Recirculation zones in a cylinder with rotating lid. In Topological Fluid Mechanics (ed. H. K. Moffat & A. Tsinober), pp. 699–708. Cambridge University Press.
Sørensen, J. N. & Daube, O. 1989 Direct simulation of flow structures initiated by a rotating cover in a cylindrical vessel. In Advances in Turbulence (ed. H. H. Fernholz & H. E. Fielder), pp. 383–390. Springer.
Tsitverblit, N. 1993 Vortex breakdown in a cylindrical container in the light of continuation of a steady solution. Fluid Dyn. Res. 11, 19–35.Google Scholar
Tsitverblit, N. & Kit, E. 1994 On the onset of nonsteadiness in the flows with vortex breakdown in a cylindrical container. Proc. IUTAM Symp on Nonlinear Instability of Nonparallel Flows (ed. S. P. Lin, W. R. C. Philips & D. T. Valentine), pp. 310–319. Springer.
Vogel, H. U. 1975 Rückströmungsblasen in Drallströmungen. In Festschrift zum 50-jährigen Bestehen des MPI für Strömungsforschung, pp. 263–272. Hubert, Göttingen.