Unveiling the competitive role of global modes in the pattern formation of rotating sphere flows

Abstract

The wake flow past a streamwise rotating sphere is a canonical model of numerous applications, such as particle-driven flows, sport aerodynamics and freely rising or falling bodies, where the changes in particles’ paths are related to the destabilization of complex flow regimes and associated force distributions. Herein, we examine the spatio-temporal pattern formation, previously investigated by Lorite-Díez & Jiménez-González (J. Fluid Mech., vol. 896, 2020, A18) and Pier (J. Fluids Struct., vol. 41, 2013, pp. 43–50), from a dynamical system perspective. A systematic study of the mode competition between rotating waves, which arise from the linearly unstable modes of the steady-state, exhibits their connection to previously observed helical patterns present within the wake. The organizing centre of the dynamics turns out to be a triple Hopf bifurcation associated with three non-axisymmetric, oscillating modes with respective azimuthal wavenumbers $m=-1,-1$ and $-2$ . The unfolding of the normal form unveils the nonlinear interaction between the rotating waves to engender more complex states. It reveals that for low values of the rotation rate, the flow field exhibits a similar transition to the flow past the static sphere, but accompanied by a rapid variation of the frequencies of the flow with respect to the rotation. The transition from the single helix pattern to the double helix structure within the wake displays several regions with hysteric behaviour. Eventually, the interaction between single and double helix structures within the wake lead towards temporal chaos, which here is attributed to the Ruelle–Takens–Newhouse route. The onset of chaos is detected by the identification of an invariant state of the normal form constituted by three incommensurate frequencies. The evolution of the chaotic attractor is determined using of time-stepping simulations, which were also performed to confirm the existence of bi-stability and to assess the fidelity of the computations performed with the normal form.

Mode selection in concentric jets: the steady–steady 1 : 2 resonant mode interaction with O(2) symmetry

“…Similar observations were drawn in the case of the wake behind rotating spheres (Sierra-Ausín et al. 2022), where the core of the instability was also found near the downstream part of the recirculating flow region. Therein, it was concluded that the instability is supported by the recirculating flow region.…”

“…Other configurations where the primary or secondary instability involves modes with azimuthal component are swirling jets (Meliga, Gallaire & Chomaz 2012) and the wake flow past a rotating sphere (Sierra-Ausín et al. 2022). The unsteady modes and display a similar structure to the unsteady modes discussed in § 4.1.…”

Asymmetric cellular bi-stability in the gap between tandem cylinders

“…Bi-stability has also been observed in the case of a streamwise rotating sphere, for certain Reynolds number and rotation rates (Lorite-Díez & Jiménez-Gonzáles 2020; Sierra-Ausín et al. 2022). Lorite-Díez & Jiménez-Gonzáles (2020) found a switch between a quasi-periodic low-drag state and a more irregular high-drag state.…”

Permeability sets the linear path instability of buoyancy-driven disks

“…When two or more modes present similar thresholds, the dynamics results from a competition of the different modes (Fabre, Auguste & Magnaudet 2008), which cannot be unveiled only through linear stability analysis. In this case, weakly nonlinear analyses can shed light on the emerging trajectories or wake patterns, as performed by Meliga, Chomaz & Sipp (2009a), Citro et al (2016) and Sierra-Ausín et al (2022) for the wake of fixed and rotating spheres.…”