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

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.

Axisymmetric and three-dimensional instabilities in an Ekman boundary layer flow

“…The values of R m 5 and 10 are chosen in order to keep curvature eects reasonably low. Indeed, for more signi®cant curvature eects R m 1:22, Ro 0:36, we have shown (Crespo del Arco et al, 1996) that the transition originates from an inertial instability in the entrance region that excited the Ekman layers. It was shown that this external perturbation provoked sub-critical modes of instability in the layers.…”

“…Computations were performed with spatial resolutions of N Â M Â K 48 Â 48 Â 48 to 48 Â 48 Â 128, where N and M are the number of Chebyshev polynomials in the axial and radial directions and K the number of the Fourier modes used in the azimuthal expansion. The former axisymmetric study in a rotating annular cavity of Crespo del Arco et al (1996) indicates that a spatial resolution of 48 Â 48 in r; z for the considered geometrical (R m ; L) and physical (E; C w parameters, constitutes a good compromise between accuracy and computational cost. The time step used in the computations is governed by the spatial resolution in r; z, Dt 4:0 Â 10 À3 .…”

“…In the limit Ro 3 0, the wavelength of the vortices is independent of the radius and the wave front is a straight line normal to the ¯ow. In Crespo del Arco et al (1996), a pseudospectral numerical method was used to obtain axisymmetric unsteady solutions that arise in an annular con®guration chosen to mimic the experiments of Owen et al (1985), which models situations of practical interest. Thus, the curvature eects were included and oscillatory ¯ow regimes were computed.…”

“…In the numerical simulations, at short distances from the axis, the inner velocity pro®le was chosen as a parabolic Poiseuille type pro®le in which case an entrance region develops which limits the radial displacement of the vortices. In such situations, Crespo del Arco et al (1996) reported a transition to an oscillatory and a nonperiodic regime via a complex bifurcation scenario. In the unstable ¯ow, six pairs of counter-rotating circular vortices, propagating radially outward, are found in the Ekman boundary layers.…”

“…In Crespo del Arco et al (1996), we have con®rmed and analyzed the axisymmetric instability that prevails at R m 5 (R m is a characteristic curvature parameter, R m a b=DR) and the supercritical conditions as derived from Lilly's theoretical analysis valid in the limit of Ro 3 0. The purpose of the present study is to extend the analysis of the oscillatory transition to the regime of three-dimensional spiral patterns.…”