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The elephant mode between two rotating disks

Published online by Cambridge University Press:  25 February 2008

BERTRAND VIAUD ,
ERIC SERRE  and
JEAN-MARC CHOMAZ
BERTRAND VIAUD
Affiliation:
Centre de Recherche de l'Armée de l'Air, CReA BA701 13661 Salon de Provence, France
ERIC SERRE
Affiliation:
Laboratoire MSNM-GP, CNRS Universités Aix Marseille, IMT Chateau-Gombert 13451 Marseille, France
JEAN-MARC CHOMAZ
Affiliation:
Laboratoire d'Hydrodynamique-LadHyX, CNRS-Ecole Polytechnique, F-91128 Palaiseau, France
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Abstract

Spectral direct numerical simulations (DNS) are carried out for a source–sink flow in an annular cavity between two co-rotating disks. When the Reynolds number based on the forced inflow is increased, a self-sustained crossflow instability of finite amplitude is observed. We show that this nonlinear global mode is made up of a front located at the upstream boundary of the absolutely unstable domain, followed by a saturated spiral mode, and that its properties are in good agreement with results of the local stability theory. This structure is characteristic of the so-called elephant mode of Pier & Huerre (J. Fluid Mech. vol. 435, 2001, p. 145). The global bifurcation is subcritical since only large-amplitude initial perturbations are found to trigger the elephant mode. Small-amplitude perturbations induce a long-lasting transient growth but lead eventually to a damped linear global mode, showing that non-parallel effects counteract the absolute instability and restabilize the flow. A similar linear global stabilization due to non-parallel effects has been found in the case of the flow above a single rotating disk. For the single-disk geometry, the existence of an elephant mode would imply, together with results of Davies & Carpenter (2003) a subcritical global instability, which has not yet been demonstrated. Although the present geometry differs from the single-disk case, the existence of a subcritical global bifurcation is now established, allowing a precise analysis of the transition scenarios.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 598 , 10 March 2008 , pp. 451 - 464
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Andersen, A., Lautrup, B. & Bohr, T. 2003 Averaging method for the nonlinear laminar Ekman layers. J.Fluid Mech 487, 81–90.CrossRefGoogle Scholar
Bassom, A. P., Kuzanyan, K. M. & Soward, A. M. 1999 A nonlinear dynamo wave riding on a spatially varying background. Proc. R. Soc. Lond. A 455, 1443–1481.CrossRefGoogle Scholar
Brancher, P. & Chomaz, J. M. 1997 Absolute and convective secondary instabilities in spatially periodic shear flows. Phys. Rev. Lett 78, 658–661.CrossRefGoogle Scholar
Chomaz, J. M. & Couairon, A. 1999 Against the wind. Phys. Fluids 11, 2977–2983.CrossRefGoogle Scholar
Chomaz, J. M., Couairon, A. & Julien, S. 1999 Absolute and convective nature of the Eckhaus and zigzag instability with throughflow. Phys. Fluids 11, 3369–3373.CrossRefGoogle Scholar
Couairon, A. & Chomaz, J. M. 1997 Pattern selection in the presence of a crossflow. Phys. Rev. Lett 79, 2666–2669.CrossRefGoogle Scholar
Couairon, A. & Chomaz, J. M. 1999 a Fully nonlinear global modes in slowly varying flows. Phys. Fluids 12, 3688–3703.CrossRefGoogle Scholar
Couairon, A. & Chomaz, J. M. 1999 b Primary and secondary nonlinear global instability. Physica D 132, 428–456.Google Scholar
Couairon, A. & Chomaz, J. M. 2001 Pushed global modes in weakly inhomogeneous subcritical flows. Physica D 158, 129–150.Google Scholar
Crespo del Arco, E., Maubert, P., Randriamampianina, A. & Bontoux, P. 1996 Spatio-temporal behaviour in a rotating annulus with a source-sink flow. J. Fluid Mech 32, 1–27.Google Scholar
Crespo del Arco, E., Serre, E., Bontoux, P. & Launder, B.E. 2005 Stability, transition and turbulence in rotating cavities. In Advances in Fluid Mechanics (ed. Rahman, M.), vol. 41. WIT press.Google Scholar
Davies, C. & Carpenter, P. W. 2003 Global behaviour corresponding to the absolute instability of the rotating-disk boundary layer. J. Fluid Mech 486, 287–329.CrossRefGoogle Scholar
Davies, C., Thomas, C. & Carpenter, P. W. 2007 Global stability of the rotating-disk boundary layer. J. Engng Maths 57, 219–236.CrossRefGoogle Scholar
Delbende, Y., 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, 229–254.CrossRefGoogle Scholar
Healey, J. J. 2007 Enhancing the absolute instability of a boundary layer by adding a far-away plate. J. Fluid Mech 579, 29–61.CrossRefGoogle Scholar
Hide, R. 1968 On source-sink flows stratified in a rotating annulus. J. Fluid Mech 32, 737–764.CrossRefGoogle Scholar
Huerre, P. 1988 On the absolute/convective nature of primary and secondary instabilities. In Propagation in Systems Far From Equilibrium (ed. Wesfreid, J. E., Brand, H. R., Manneville, P., Albinet, G. & Boccara, N.). Springer.Google Scholar
Juniper, M. P. 2006 The effects of confinement on the stability of two-dimensional shear flows. J. Fluid Mech 565, 171–195.CrossRefGoogle Scholar
von Kármán, Th. 1921 Über laminare und turbulente Reibung. Z. Angew. Math. Mech 1, 233–252.CrossRefGoogle Scholar
Koch, W. 2002 On the spatio-temporal stability of primary and secondary crossflow vortices in a three-dimensional boundary layer. J. Fluid Mech 456, 85–111.CrossRefGoogle Scholar
Lingwood, R. J. 1995 Absolute instability of the boundary layer on a rotating disk. J. Fluid Mech 299, 17–33.CrossRefGoogle Scholar
Lingwood, R. J. 1996 An experimental study of the absolute instability of the rotating-disk boundary-layer flow. J. Fluid Mech 314, 373–405.CrossRefGoogle Scholar
Lingwood, R. J. 1997 Absolute instability of the Ekman layer and related rotating flows. J. Fluid Mech 331, 405–428.CrossRefGoogle Scholar
Mack, L.M. 1985 The wave pattern produced by a point source on a rotating-disk flow. AIAA Paper 85-0490.CrossRefGoogle Scholar
Meunier, N., Proctor, M. R. E., Sokoloff, D. D., Soward, A. M. & Tobias, S. M. 1997 Asymptotic properties of a nonlinear alpha omega-dynamo wave: Period, amplitude and latitude dependence. Geophys. Astrophys. Fluid Dyn 86, 249–285.CrossRefGoogle Scholar
Othman, H. & Corke, T. C. 2006 Experimental investigation of absolute instability of a rotating-disk boundary layer. J. Fluid Mech 565, 63–94.CrossRefGoogle Scholar
Owen, J. M. & Rogers, R. H. 1995 In Heat Transfer in Rotating-Disk System (ed. Morris, W. D.), vol. 2. Wiley.Google Scholar
Pier, B. 2003 Finite amplitude crossflow vortices, secondary instability and transition in the rotating-disk boundary layer. J. Fluid Mech 487, 315–343.CrossRefGoogle Scholar
Pier, B. & Huerre, P. 2001 a Nonlinear self-sustained structures and fronts in spatially developping wake flows. J. Fluid Mech 435, 145–174.CrossRefGoogle Scholar
Pier, B. & Huerre, P. 2001 b Nonlinear synchronization in open flows. J. Fluids Struct 15, 471–480.CrossRefGoogle Scholar
Pier, B., Huerre, P., Chomaz, J. M. & Couairon, A. 1998 Steep nonlinear global modes in spatially developping media. Phys. Fluids 10, 2433–2435.CrossRefGoogle Scholar
Raspo, I., Hugues, S., Serre, E., Randriamampianina, A. & Bontoux, P. 2002 A spectral projection method for the simulation of complex three-dimensional rotating flows. Comput. Fluids 31, 745–767.CrossRefGoogle Scholar
Ruith, M.R., Chen, E. & Meiburg, E. 2004 Developement of boundary conditions for dns of three-dimensional vortex breakdown phenomena in semi-infinite domains. Comput. Fluids 33, 1225–1250.CrossRefGoogle Scholar
van Saarloos, W. 2003 Front propagation into unstable states. Phys. Rep. Rev 386, 29–222.CrossRefGoogle Scholar
Serre, E., Hugues, S., Crespodel Arco, E., Randriamampianina, A. & Bontoux, P. 2001 Axisymmetric and three-dimensional instabilities in an Ekman boundary-layer flow. Intl J. Heat Fluid Flow 22, 82–93.CrossRefGoogle Scholar
Serre, E., Tuliszka-Sznitko, E. & Bontoux, P. 2004 Coupled numerical and theoretical study of the flow transition between a rotating and a stationary disk. Phys. Fluids 16, 688–706.CrossRefGoogle Scholar
Tobias, S. M., Proctor, M. R. E. & Knobloch, E. 1997 The role of absolute instability in the solar dynamo. Astron. Astrophys 318, L55–L58.Google Scholar
Tobias, S. M., Proctor, M. R. E. & Knobloch, E. 1998 Convective and absolute instabilities of fluid flows in finite geometry. Physica D 113, 43–72.Google Scholar
Turkyilmazoglu, M. & Gajjar, J. S. B. 2000 Direct spatial resonnance in the laminar boundary layer due to a rotating disk. Sadhana – Acad. Proc 25, 601–617.CrossRefGoogle Scholar