Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-21T10:22:47.179Z Has data issue: false hasContentIssue false
  • English
  • Français

Spatio—temporal behaviour in a rotating annulus with a source—sink flow

Published online by Cambridge University Press:  26 April 2006

E. Crespo Del Arco ,
P. Maubert ,
A. Randriamampianina  and
P. Bontoux
E. Crespo Del Arco
Affiliation:
Departamento de Física Fundamental, UNED, Apdo. 60.141, 28080 Madrid, Spain
P. Maubert
Affiliation:
Institut de Recherche sur les Phénoménes, Hors Equilibre, UMR 138 CNRS, and Institut de Mécanique des Fluides, 1, Rue Honnorat, 13003 Marseille, France
A. Randriamampianina
Affiliation:
Institut de Recherche sur les Phénoménes, Hors Equilibre, UMR 138 CNRS, and Institut de Mécanique des Fluides, 1, Rue Honnorat, 13003 Marseille, France
P. Bontoux
Affiliation:
Institut de Recherche sur les Phénoménes, Hors Equilibre, UMR 138 CNRS, and Institut de Mécanique des Fluides, 1, Rue Honnorat, 13003 Marseille, France
Get access Rights & Permissions [Opens in a new window]

Abstract

The axisymmetric flows arising in a rotating annulus with a superimposed forced flow are investigated with a pseudo-spectral numerical method. The flow enters the annulus at the inner radius with a radial velocity, then develops into a geostrophic flow azimuthally directed and flanked by two Ekman (nonlinear) boundary layers, and finally exits the outer radius, with a radially directed velocity. In this study the rotation rate of the cavity is fixed and very high. When the forced flow is weak, the flow is steady. On increasing the mass flow rate, the flow evolves to a chaotic temporal behaviour through several bifurcations, which perturbs the basic spatial configuration of the flow. The first bifurcation drives the steady state into an oscillatory regime, associated with a break of symmetry with respect to the midheight of the annulus. The entry flow travels radially through the cavity as in the steady flow, but it wavers and then is alternately sucked towards each Ekman layer. The frequency of this oscillation is close to the rotation rate frequency of the cavity, which is characteristic of inertial waves in rotating flows. A second transition to a quasi-periodic regime is characterized by the appearance of a second frequency. Further increases in the flow rate lead to a period-five state, via a locking of both frequencies, and then to a chaotic motion. This second frequency is of the order of the inverse of the Ekman spin-up characteristic time, suggesting that this instability is originated by the relaxation of the perturbations in the flow field. These perturbations of the unsteady flow field are corotating vortices along the rigid boundary walls. They are excited by the entry flow and their strength diminishes with increasing radius due to the low value of the Reynolds number. The parameters characterizing the unstable flows are also consistent with this explanation. The conclusion is that in this configuration, the origin of the described dynamical behaviour is not the instability of the Ekman boundary layers, as could be expected, but the instability of the entry flow. The reason is the importance of the nonlinear inertial terms in cavities with small radius of curvature.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 328 , 10 December 1996 , pp. 271 - 296
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

Bennetts, D. A. & Jackson, W. D. N. 1974 Source—sink flows in a rotating annulus: a combined laboratory and numerical study. J. Fluid Mech. 66, 689–705.Google Scholar
Caldwell, D. R & Van Atta, C. W. 1970 Characteristics of Ekman boundary layer instabilities. J. Fluid Mech. 44, 79–95.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.
Cerasoli, A. P. 1975 Free shear layer instability due to probes in rotating source—sink flow. J. Fluid Mech. 72, 559–586.Google Scholar
Chaouche, A. M. 1990 Une méthode de collocation-Chebyshev pour la simulation des écoulements axisymétriques dans un domaine annulaire en rotation. La Recherche Aèrospatiale 5, 1–13.Google Scholar
Chaouche, A., Randriamampianina, A. & Bontoux, P. 1990 A collocation method based on an influence matrix technique for axisymmetric flows in an annulus. Comput. Meth. Appl. Mech. Engng 80, 237–244.Google Scholar
Cherdron, W., Durst, F. & Whitelaw, J. H. 1978 Asymmetric flows and instabilities in symmetric ducts with sudden expansions. J. Fluid Mech. 84, 13–31.Google Scholar
Chew, J. W., Owen, J. M. & Pincombe, J. R. 1984 Numerical predictions for laminar source—sink flow in a rotating cylindrical cavity. J. Fluid Mech. 143, 451–466.Google Scholar
Ehrenstein, U. & Peyret, R. 1989 A Chebyshev-collocation method for the Navier—Stokes equations with applications to double-diffusive convection. Intl J. Numer. Meth. Fluids 9, 427–452.Google Scholar
Faller, A. J. 1963 An experimental study of the instability of the laminar Ekman boundary layer. J. Fluid Mech. 15, 560–576.Google Scholar
Faller, A. J. 1991 Instability and transition of disturbed flow over a rotating disk. J. Fluid Mech. 230, 245–269.Google Scholar
Faller, A. J. & Kaylor, R. E. 1966 A numerical study of the instability of the laminar Ekman boundary layer. J. Atmos. Sci. 23, 466–480.Google Scholar
Fearn, R. M., Mullin, T. & Cliffe, K. A. 1990 Nonlinear flow phenomena in a symmetric sudden expansion. J. Fluid Mech. 211, 595–608.Google Scholar
Gottlieb, D. & Orszag, S. A. 1977 Numerical Analysis of Spectral Methods: Theory and Applications. CBMS-SIAM Publications.
Greenspan, H. P. 1969 The Theory of Rotating Fluids. Cambridge University Press.
Haidvogel, D. B. & Zang, T. A. 1979 The accurate solution of Poisson equation by expansion in Chebyshev polynomials. J. Comput. Phys. 30, 137–180.Google Scholar
Hide, R. 1968 On source-sink flows in a rotating fluid. J. Fluid Mech. 32, 737–764.Google Scholar
Hopfinger, E. J. & Linden, P. F. 1990 The effect of background rotation on fluid motions: a report on Euromech 245. J. Fluid Mech. 211, 417–435.Google Scholar
Hyun, J. M. 1984 Source—sink flows of a stratified fluid in a rotating annulus. J. Fluid Mech. 145, 111–125.Google Scholar
Iooss, G., Nielsen, H. B. & True, H. 1978 Bifurcation of the stationary Ekman flow into a stable periodic flow. Arch. Rat. Mech. Anal. 68, 227–256.Google Scholar
Kleiser, L. & Schumann, U. 1980 Treatment of incompressibility and boundary conditions in 3D numerical spectral simulations of plane channel flows. In Numerical Methods in Fluid Mechanics (ed. E H. Hirschel), pp. 165–173. Vieweg.
Le Quéré, P. & Pécheux, J. 1989 Numerical simulations of multiple flow transitions in axisymmetric annulus convection. J. Fluid Mech. 206, 517–544.Google Scholar
Lilly, D. K. 1966 On the instability of Ekman boundary flow. J. Atmos. Sci. 23, 481–494.Google Scholar
Maubert, P. 1993 Etude de régimes d’écoulements et de transferts thermiques dans les systèmes en rotation soumis à un flux forcé. Doctorate thesis, Université d'Aix-Marseille II, France.
Maubert, P., Chaouche, A. M., Crespo del Arco, E. & Bontoux, P. 1992 Régimes d’écoulements instationnaires dans une cavité annulaire tournante soumise à un flux radial forcé. C. R. Acad. Sci. Paris 315, S.II, 1593–1600.Google Scholar
Orszag, S. A. & Kells, L. C. 1980 Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech. 96, 159–205.Google Scholar
Ouazzani, J. 1984 Méthode pseudo-spectrale pour la résolution des équations d'un mélange de gaz binaire. Doctorate thesis, Université de Nice, France.
Owen, J. M. & Pincombe, J. R. 1979 Vortex breakdown in a rotating cylindrical cavity. J. Fluid Mech. 90, 109–127.Google Scholar
Owen, J. M. & Pincombe, J. R. 1980 Velocity measurements inside a rotating cavity with a radial outflow of fluid. J. Fluid Mech. 99, 111–127.Google Scholar
Owen, J. M., Pincombe, J. R. & Rogers, R. H. 1985 Source—sink flow inside a rotating cylindrical cavity. J. Fluid Mech. 155, 233–265.Google Scholar
Pulicani, J. P., Crespo del Arco, E., Randriamampianina, A., Bontoux, P. & Peyret, R. 1990 Spectral simulations of oscillatory convection at low Prandtl number. Intl J. Numer. Meth. Fluids 10, 481–517.Google Scholar
Savas, O. 1987 Stability of Bödewadt flow. J. Fluid Mech. 183, 77–94.Google Scholar
Schlichting, H. 1960 Boundary Layer Theory. McGraw Hill.
Tatro, P. R. & Mollö-Christensen, E. L. 1967 Experiments on Ekman layer instability. J. Fluid Mech. 28, 531–543.Google Scholar
Thomas, P. J. 1994 Pattern formation of granules on the bottom of a differentially rotating tank. J. Fluid Mech. 274, 23–41.Google Scholar
Vanel, J. M., Peyret, R. & Bontoux, P. 1986 A pseudo-spectral solution of vorticity streamfunction equations using the influence matrix technique. In Numerical Methods for Fluids Dynamics II (ed. K. W Morton & M. J Baines), pp 463–475. Clarendon Press.
Weidman, P. D. 1976 On the spin-up and spin-down of a rotating fluid. Part 2. Measurements and stability. J. Fluid Mech. 77, 709–735.Google Scholar