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Instability of bounded flows with elliptical streamlines

Published online by Cambridge University Press:  26 April 2006

E. B. Gledzer  and
V. M. Ponomarev
E. B. Gledzer
Affiliation:
Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, 109017, Russia
V. M. Ponomarev
Affiliation:
Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, 109017, Russia
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Abstract

In connection with the recent investigations of the instability of unbounded elliptical flows, some methods are discussed for the study of the instability of bounded flows. The stability of a ‘basic flow’ which is two-dimensional and rotating, with elliptical streamlines similar to the elliptical section of an experimentally studied cavity, is investigated in the framework of linear theory (for circular rotation, the flow discussed is stable). The regions of instability for three-dimensional disturbances are found in the plane of the parameters defining the geometry of the system (the height of the ellipsoidal cavity and the degree of ellipticity). It is shown that two types of instability exist, characterized by either monotone or oscillatory growth of the amplitudes of small disturbances.

The influence of the Coriolis force field on this instability mechanism is also studied. Rotation of the system as a whole changes the regions of instability in parameter space characterizing the geometry of the cavity and the wavenumbers of unstable disturbances. As a result, the Coriolis force may stabilize or destabilize the basic flow for a given geometry.

The instability of rotating density-stratified flow with elliptical streamlines is also considered.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 240 , July 1992 , pp. 1 - 30
Copyright
© 1992 Cambridge University Press

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References

Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2171.Google Scholar
Bayly, B. J., Orszag, S. A. & Herbert, T. 1988 Instability mechanisms in shear-flow transition. Ann. Rev. Fluid Mech. 20, 359–391.Google Scholar
Boubnov, B. M., 1978 Effect of Coriolis force field on the motion of a fluid inside an ellipsoidal cavity. Izv. Atmos. Ocean. Phys. 14, 501–504.Google Scholar
Chernous'Ko, Yu. L. 1978 An experimental study of secondary multi-eddy flows in elliptical cylinders. Izv. Atmos. Ocean. Phys. 14, 151–153.Google Scholar
Craik, A. D. D. 1988 A class of exact solutions in viscous incompressible magnetohydrodynamics.. Proc. R. Soc. Lond. A 417, 235–244.Google Scholar
Craik, A. D. D. 1989 The stability of unbounded two- and three-dimensional flows subject to body forces: some exact solutions. J. Fluid Mech. 198, 275–295.Google Scholar
Craik, A. D. D. & Criminale, W. O. 1986 Evolution of wavelike disturbances in shear flows: a class of exact solutions of the Navier—Stokes equations.. Proc. R. Soc. Lond. A 406, 13–26.Google Scholar
Gledzer, E. B., Dolzhansky, F. V. & Obukhov, A. M. 1981 Hydrodynamic Type Systems and their Applications. Moscow: Nauka.
Gledzer, E. B., Dolzhansky, F. V., Obukhov, A. M. & Ponomarev, V. M. 1975 An experimental and theoretical study of the stability of motion of a liquid in an elliptical cylinder. Izv. Atmos. Ocean. Phys. 11, 617–622.Google Scholar
Gledzer, E. B., Makarov, A. L. & Ponomarev, V. M. 1980 Stability of elliptical fluid rotation in a Coriolis force field. Isv. Atmos. Ocean. Phys. 16, 280–282.Google Scholar
Gledzer, E. B., Novikov, Yu. V., Obukhov, A. M. & Chusov, M. A. 1974 An investigation of the stability of liquid flows in a three-axis ellipsoid. Isv. Atmos. Ocean. Phys. 10, 69–71.Google Scholar
Gledzer, E. B. & Ponomarev, V. M. 1977a Finite-dimensional approximation of the motions of an incompressible fluid in an ellipsoid cavity. Isv. Atmos. Ocean. Phys. 13, 565–569.Google Scholar
Gledzer, E. B. & Ponomarev, V. M. 1977b On the forced motion of fluid within an ellipsoid. Isv. Atmos. Ocean. Phys. 13, 687–689.Google Scholar
Greenhill, A. G. 1879 On the rotation of a liquid ellipsoid about its mean axis. Proc. Camb. Phil. Soc. 3, 233–246.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Herbert, T. 1983 Secondary instability of plane channel flow to subharmonic tree-dimensional disturbances. Phys. Fluids 26, 871–874.Google Scholar
Hough, S. S. 1895 The oscillations of a rotating ellipsoidal shell containing fluid.. Phil. Trans. R. Soc. Lond. A 186, 469–506.Google Scholar
Landman, M. J. & Saffman, P. G. 1987 The three-dimensional instability of strained vortices in a viscous fluid. Phys. Fluids 30, 2339–2342.Google Scholar
Malkus, W. V. R. 1989 An experimental study of global instabilities due to the tidal (elliptical) distortion of a rotating elastic cylinder. Geophys. Astrophys. Fluid Dyn. 30, 123–134.Google Scholar
Malkus, W. V. R. & Waleffe, F. 1991 The transition from order to disorder in elliptical flow: a direct path to shear flow turbulence. In Advances in Turbulence 3 (ed. A. H. Johansson & P. V. Alfredson). Springer
Mcewan, A. D. 1970 Inertial oscillations in a rotating fluid cylinder. J. Fluid Mech. 40, 603–640.Google Scholar
Obukhov, A. M. & Dolzhansky, F. V. 1975 On simple models for simulation of nonlinear processes in convection and turbulence. Geophys. Fluid Dyn. 6, 195–209.Google Scholar
Obukhov, A. M., Glukhovsky, A. B. & Chernous'ko, Yu. L. 1976 Reversal phenomena in the simplest fluid-dynamic systems. Isv. Atmos. Ocean. Phys. 12, 693–693.Google Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Pierrehumbert, R. T. 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57, 2157–2159.Google Scholar
Pierrehumbert, R. T. & Widnall, S. E. 1982 The two- and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–82.Google Scholar
Poincaré, H. 1910 Sur la precession des corps deformables. Bull. Astr. 27, 321–356.Google Scholar
Robinson, A. C. & Saffman, P. G. 1984 Three-dimensional stability of an elliptical vortex in a straining field. J. Fluid Mech. 142, 451–466.Google Scholar
Roesner, K. G. & Schmieg, H. 1980 Instabilities of spin-up and spin-down flows inside of liquid-filled ellipsoids. Proc. Colloque Pierre Curie, 1-5 Sept., 1980, Paris.Google Scholar
Tsai, C.-Y. & Widnall, S. E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73, 721–733.Google Scholar
Vladimirov, V. A. & Ilyjn, K. I. 1988 Three-dimensional instability of an elliptical Kirchhoff vortex. Mech. Zhid. i Gaza No. 3, 40–45 (in Russian).Google Scholar
Vladimirov, V. A., Ribak, L. Ya & Tarasov, V. F. 1983a Experimental and theoretical investigation of the stability of a linear vortex with a deformed core. Prikl. Mech. Tekhn. Fis. No. 3, 61–69 (in Russian).Google Scholar
Vladimirov, V. A., Tarasov, V. P. & Ribak, L. Ya. 1983b On stability of elliptically deformed rotation of ideal incompressible fluid in the field of Coriolis forces. Isv. Atmos. Ocean. Phys. 19, 586–594 (in Russian).Google Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices.. Phys. Fluids A 2, 76–80.Google Scholar