Modulated waves in a periodically driven annular cavity

Abstract

Time-periodic flows with spatio-temporal symmetry Z 2 × O(2) -invariance in the spanwise direction generating the O(2) symmetry group and a half-period-reflection symmetry in the streamwise direction generating a spatio-temporal Z 2 symmetry group -are of interest largely because this is the symmetry group of periodic laminar two-dimensional wakes of symmetric bodies. Such flows are the base states for various three-dimensional instabilities; the periodically shedding two-dimensional circular cylinder wake with three-dimensional modes A and B being the generic example. However, it is not easy to physically realize the ideal flows owing to the presence of end effects and finite spanwise geometries. Flows past rings are sometimes advanced as providing a relevant idealization, but in fact these have symmetry group O(2) and only approach Z 2 × O(2) symmetry in the infinite aspect ratio limit. The present work examines physically realizable periodically driven annular cavity flows that possess Z 2 × O(2) spatio-temporal symmetry. The flows have three distinct codimension-1 instabilities: two synchronous modes (A and B), and two manifestations of a quasi-periodic (QP) mode, either as modulated standing waves or modulated travelling waves. It is found that the curvature of the system can determine which of these modes is the first to become unstable with increasing Reynolds number, and that even in the nonlinear regime near onset of three-dimensional instabilities the dynamics are dominated by mixed modes with complicated spatio-temporal structure. Supplementary movies illustrating the spatio-temporal dynamics are available at journals.cambridge.org/flm.

Mode competition in cylindrical flows driven by sidewall oscillations

“…The inset in Fig. 3 resulting in flows with complex dynamics have been reported in an annulus of radius ratio close to one also driven by axial oscillations of the sidewall [8]. In this problem, the azimuthal wave numbers of the bifurcated solutions are at least one order of magnitude larger than in our problem, so there is a strong competition between the bifurcating modes.…”

“…Experiments and numerical simulations have been conducted in these flows, looking for the three-dimensional dynamics after breaking the O(2) symmetry [3,4,7,8,15]. In these different settings, the analysis of the three-dimensional dynamics has faced different setbacks.…”

“…In the wake flows, the lack of enough control parameters does not allow the separated analysis of the different instability modes, which can be only observed at secondary bifurcations that become chaotic rapidly when increasing the control parameter. In other flows [8,15], the dynamics are dominated by a variety of mixed modes with complicated spatiotemporal structures, even very near the onset of three-dimensional instabilities. The fluid problem analyzed in this study, a Newtonian fluid enclosed in a cylindrical cavity whose sidewall oscillates harmonically in the axial direction, solves the aforementioned problems.…”

“…Furthermore, these flows have an additional space-time symmetry: a reflection about the wake centerline followed by a half-period temporal evolution. In several wake flows, two distinct synchronous modes that break into the spanwise direction (with real Floquet exponent) have been observed experimentally [1][2][3][4], computed as direct instabilities from the flow [2,[5][6][7][8], and studied theoretically [9][10][11]. These modes are associated with breaking or preserving the spatiotemporal symmetry.…”

“…A periodically driven annular cavity solves this issue and has been considered in Ref. [8]. This case is a continuous deformation of the planar driven cavity flow in the sense that the infinite spanwise direction becomes a periodic angular direction.…”