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Inviscid axisymmetric absolute instability of swirling jets

Published online by Cambridge University Press:  01 October 2008

J. J. HEALEY*
Affiliation:
Department of Mathematics, Keele University, Keele, Staffs. ST5 5BG, UKj.j.healey@keele.ac.uk

Abstract

The propagation characteristics of inviscid axisymmetric linearized disturbances to swirling jets are investigated for two families of model velocity profiles. Briggs' method is applied to dispersion relations to determine when the basic swirling jets are absolutely or convectively unstable. The method is also applied to the neutral inertial waves used by Benjamin to characterize the subcritical or supercritical nature of the flow. Although these waves are neutral, Briggs' method nonetheless indicates a qualitative change in behaviour at Benjamin's criticality condition. The first model jet has uniform axial velocity, rigid-body rotation and issues into still fluid. A known difficulty in the application of Briggs' method to the analytical dispersion relation for inviscid waves in this flow is resolved. The difficulty is that the pinch point can cross into the left half of the complex-wavenumber plane, where waves grow exponentially with radius and fail to satisfy homogeneous boundary conditions. In this paper the physical consequences of this behaviour are explained. It is shown that if the still fluid is of infinite extent in the radial direction, then the jet is convectively unstable to axisymmetric waves, but if the jet is confined by an outer cylinder concentric with the jet axis, then it becomes absolutely unstable to axisymmetric waves provided that the swirl (ratio of azimuthal to axial velocity) is large enough. This destabilizing effect of confinement occurs however large the radius of the outer cylinder. A second family of model swirling jets with smooth profiles and a finite thickness shear layer at the jet edge is also studied. The inviscid stability equations are solved numerically in this case. The results from the analytical dispersion relations are reproduced as the shear layer thickness tends to zero. However, increasing this thickness acts to destabilize the absolute instability of axisymmetric waves, apparently due to the centrifugal instability present in the shear layer. It is suggested that the transition from convective to absolute instability could be associated with the onset of an unsteady vortex breakdown. The swirl required to produce this transition can be either greater, or less, than the swirl required to produce the transition from supercritical to subcritical flow, depending on the details of the basic velocity profiles. A codimension-two point in parameter space can exist where the unsteady bifurcation associated with the convective–absolute transition coincides with the steady bifurcation associated with the supercritical–subcritical transition. Such codimension-two points can control a rich variety of nonlinear dynamical behaviour.

Type
Papers
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Batchelor, G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645658.CrossRefGoogle Scholar
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14, 593629.CrossRefGoogle Scholar
Benjamin, T. B. 1967 Some developments in the theory of vortex breakdown. J. Fluid Mech. 28, 6584.Google Scholar
Benjamin, T. B. 1978 Bifurcation phenomena in steady flows of a viscous fluid I. Theory. Proc. R. Soc. Lond. A 359, 126.Google Scholar
Benjamin, T. B. & Mullin, T. 1981 Anomalous modes in the Taylor experiment. Proc. R. Soc. Lond. A 377, 221249.Google Scholar
Beran, P. S. & Culick, F. E. C. 1992 The role of non-uniqueness in the development of vortex breakdown in tubes. J. Fluid Mech. 242, 491527.CrossRefGoogle Scholar
Billant, P., Chomaz, J.-M. & Huerre, P. 1998 Experimental study of vortex breakdown in swirling jets. J. Fluid Mech. 376, 183219.Google Scholar
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas. MIT Press.CrossRefGoogle Scholar
Couairon, A. & Chomaz, J.-M. 1999 Fully nonlinear global modes in slowly varying flows. Phys. Fluids 11, 36883703.CrossRefGoogle Scholar
Delbende, I., Chomaz, J.-M. & Huerre, P. 1998 Absolute/convective instabilities in the Batchelor vortex: A numerical study of the linear impulse response. J. Fluid Mech. 355, 229254.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability Theory. Cambridge University Press.Google Scholar
Escudier, M. P. 1984 Observations of the flow produced in a cylindrical container by a rotating endwall. Exp. Fluids 2, 189196.CrossRefGoogle Scholar
Escudier, M. P., Bornstein, J. & Maxworthy, T. 1982 The dynamics of confined vortices. Proc. R. Soc. Lond. A 382, 335360.Google Scholar
Gallaire, F. & Chomaz, J.-M. 2003 a Instability mechanisms in swirling flows. Phys. Fluids 15, 26222639.CrossRefGoogle Scholar
Gallaire, F. & Chomaz, J.-M. 2003 b Mode selection in swirling jet experiments: A linear stability analysis. J. Fluid Mech. 494, 223253.CrossRefGoogle Scholar
Gallaire, F. & Chomaz, J.-M. 2004 The role of boundary conditions in a simple model of incipient vortex breakdown. Phys. Fluids 16, 274286.CrossRefGoogle Scholar
Gallaire, F., Ruith, M., Meiburg, E., Chomaz, J.-M. & Huerre, P. 2006 Spiral vortex breakdown as a global mode. J. Fluid Mech. 549, 7180.Google Scholar
Gaster, M. 1968 Growth of disturbances in both space and time. Phys. Fluids 11, 723727.Google Scholar
Grabowski, W. & Berger, S. 1976 Solutions of the Navier–Stokes equations for vortex breakdown. J. Fluid Mech. 75, 525544.CrossRefGoogle Scholar
Harvey, J. K. 1960 Analysis of the vortex breakdown phenomenon. Part II. Report no. 103, Aero. Dept., Imperial College.Google Scholar
Harvey, J. K. 1962 Some observations of the vortex breakdown phenomenon. J. Fluid Mech. 14, 585592.Google Scholar
Healey, J. J. 2005 Long-wave theory for a new convective instability with exponential growth normal to the wall. Phil. Trans. R. Soc. Lond. A 363, 11191130.Google ScholarPubMed
Healey, J. J. 2006 a Inviscid long-wave theory for the absolute instability of the rotating-disk boundary layer. Proc. R. Soc. Lond. A 462, 14671492.Google Scholar
Healey, J. J. 2006 b A new type of convective instability with exponential growth perpendicular to the basic flow. J. Fluid Mech. 560, 279310.CrossRefGoogle Scholar
Healey, J. J. 2007 Enhancing the absolute instability of a boundary layer by adding a far-away plate. J. Fluid Mech. 579, 2961.CrossRefGoogle Scholar
Healey, J. J., Broomhead, D. S., Cliffe, K. A., Jones, R. & Mullin, T. 1991 The origins of chaos in a modified van der Pol oscillator. Physica D 48, 322339.Google Scholar
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463476.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151168.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech. 22, 473537.CrossRefGoogle Scholar
Ivanic, T., Foucault, E. & Pecheux, J. 2003 Dynamics of swirling jet flows. Exps. Fluids 35, 317324.CrossRefGoogle Scholar
Juniper, M. P. 2006 The effect of confinement on the stability of two-dimensional shear flows. J. Fluid Mech. 565, 171195.CrossRefGoogle Scholar
Juniper, M. P. 2007 The full impulse response of two-dimensional jet/wake flows and implications for confinement. J. Fluid Mech. 590, 163185.Google Scholar
Juniper, M. P. & Candel, S. M. 2003 The stability of ducted compound flows and consequences for the geometry of coaxial injectors. J. Fluid Mech. 482, 257269.Google Scholar
Leibovich, S. 1983 Vortex stability and breakdown: Survey and extension. AIAA J. 22, 11921206.Google Scholar
Leibovich, S. & Kribus, A. 1990 Large-amplitude wavetrains and solitary waves in vortices. J. Fluid Mech. 216, 459504.CrossRefGoogle Scholar
Liang, H. & Maxworthy, T. 2005 An experimental investigation of swirling jets. J. Fluid Mech. 525, 115159.CrossRefGoogle Scholar
Lim, D. W. & Redekopp, L. G. 1998 Absolute instability conditions for variable density, swirling jet flows. Eur. J. Mech. b Fluids 17, 165185.Google Scholar
Loiseleux, T., Chomaz, J.-M. & Huerre, P. 1998 The effect of swirl on jets and wakes: Linear instability of the Rankine vortex with axial flow. Phys. Fluids 10, 11201134.Google Scholar
Loiseleux, T., Delbende, I. & Huerre, P. 2000 Absolute and convective instabilities of a swirling jet/wake shear layer. Phys. Fluids 12, 375380.Google Scholar
Lopez, J. M. 1994 On the bifurcation structure of axisymmetric vortex breakdown in a constricted pipe. Phys. Fluids 6, 36833693.CrossRefGoogle Scholar
Monkewitz, P. A. & Sohn, K. D. 1988 Absolute instability in hot jets. AIAA J. 26, 911916.CrossRefGoogle Scholar
Olendraru, C., Sellier, A., Rossi, M. & Huerre, P. 1999 Inviscid instability of the Batchelor vortex: Absolute–convective transition and spatial branches. Phys. Fluids 11, 18051820.CrossRefGoogle Scholar
Pier, B., Huerre, P. & Chomaz, J.-M. 2001 Bifurcation to fully nonlinear synchronized structures in slowly varying media. Physica D 148, 4996.Google Scholar
Ruith, M. R., Chen, P., Meiburg, E. & Maxworthy, T. 2003 Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical solution. J. Fluid Mech. 486, 331378.CrossRefGoogle Scholar
Sarpkaya, T. 1971 On stationary and travelling vortex breakdowns. J. Fluid Mech. 45, 545559.CrossRefGoogle Scholar
Spall, R. E., Gatski, T. B. & Grosch, C. E. 1987 A criterion for vortex breakdown. Phys. Fluids 30, 34343440.Google Scholar
Squire, H. B. 1956 Rotating fluids. In Surveys in Mechanics (ed. Batchelor, G. K. & Davies, R. M.), pp. 139161. Cambridge University Press.Google Scholar
Squire, H. B. 1960 Analysis of the vortex breakdown phenomenon. In Miszallaneen der Angewandten Mechanik, pp. 306312. Berlin: Akademie.Google Scholar
Tsai, C.-Y. & Widnall, S. E. 1980 Examination of a group-velocity criterion for breakdown of vortex flow in a divergent duct. Phys. Fluids 23, 864870.Google Scholar
Wang, S. & Rusak, Z. 1996 On the stability of an axisymmetric rotating flow in a pipe. Phys. Fluids 8, 10071016.CrossRefGoogle Scholar
Wang, S. & Rusak, Z. 1997 a The effect of slight viscosity on a near-critical swirling flow in a pipe. Phys. Fluids 9, 19141927.CrossRefGoogle Scholar
Wang, S. & Rusak, Z. 1997 b The dynamics of a swirling flow in a pipe and transition to axisymmetric vortex breakdown. J. Fluid Mech. 340, 177223.CrossRefGoogle Scholar
Yu, M.-H. & Monkewitz, P. A. 1990 The effect of nonuniform density on the absolute instability of two-dimensional inertial jets and wakes. Phys. Fluids A 2, 11751181.CrossRefGoogle Scholar