title: Instabilities of interacting vortex rings generated by an oscillating disk.
abstract: We propose a natural model to probe in a controlled fashion the instability of interacting vortex rings shed from the edge of an oblate spheroid disk of major diameter c, undergoing oscillations of frequency f_{0} and amplitude A. We perform a Floquet stability analysis to determine the characteristics of the instability modes, which depend strongly on the azimuthal (integer) wave number m. We vary two key control parameters, the Keulegan-Carpenter number K_{C}=2πA/c and the Stokes number β=f_{0}c^{2}/ν, where ν is the kinematic viscosity of the fluid. We observe two distinct flow regimes. First, for sufficiently small β, and hence low frequency of oscillation corresponding to relatively weak interaction between sequentially shedding vortex rings, symmetry breaking occurs directly to a single unstable mode with m=1. Second, for sufficiently large yet fixed values of β, corresponding to a higher oscillation frequency and hence stronger ring-ring interaction, the onset of asymmetry is predicted to occur due to two branches of high m instabilities as the amplitude is increased, with m=1 structures being dominant only for sufficiently large values of K_{C}. These two branches can be distinguished by the phase properties of the vortical structures above and below the disk. The region in (K_{C},β) parameter space where these two high m instability branches arise can be described accurately in terms of naturally defined Reynolds numbers, using appropriately chosen characteristic length scales. We subsequently carry out direct numerical simulations of the fully three-dimensional flow to verify the principal characteristics of the Floquet analysis, in particular demonstrating that high wave-number symmetry-breaking generically occurs when vortex rings sequentially interact sufficiently strongly.
authors.affiliation: State Key Laboratory of Fluid Power & Mechatronic Systems, Zhejiang University, Hangzhou 310027, People's Republic of China; Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou 310027, People's Republic of China; and Department of Mechanics, Zhejiang University, Hangzhou 310027, People's Republic of China.
authors.forename: Jian
authors.initials: J
authors.lastname: Deng
authors.affiliation: State Key Laboratory of Fluid Power & Mechatronic Systems, Zhejiang University, Hangzhou 310027, People's Republic of China; Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou 310027, People's Republic of China; and Department of Mechanics, Zhejiang University, Hangzhou 310027, People's Republic of China.
authors.forename: Lubao
authors.initials: L
authors.lastname: Teng
authors.affiliation: BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, United Kingdom and Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom.
authors.forename: C P
authors.initials: CP
authors.lastname: Caulfield
authors.affiliation: School of Engineering and Computer Science, Durham University, Durham DH1 3LE, United Kingdom.
authors.forename: Xuerui
authors.initials: X
authors.lastname: Mao