Instabilities of interacting vortex rings generated by an oscillating disk

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 $K_C=2\pi A/c$ and the Stokes number $\beta =f_0{c}^2/\nu$, where $\nu$ is the kinematic viscosity of the fluid. We observe two distinct flow regimes. First, for sufficiently small $\beta$, 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 $\beta$, 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 ($K_C,\beta$) 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.

Figure 1

Geometric definition of the disk (actually an oblate spheroid in rigorous geometric terms): (a) side view, where $z$ is the symmetry axis along which the disk oscillates, $b$ is the minor diameter, and the hollow arrow denotes the direction of oscillation; (b) top view, where $c$ is the major diameter of the spheroid.

Figure 2

Floquet instabilities to the two-dimensional vortex ring flow. Neutral stability points are marked with open circles, and instabilities are represented by filled circles with various colors to distinguish their wave numbers. The solid line, ${\text{Re}}_L=\beta K_C^{3/2}=180$ as defined in (8) bounds the symmetry instability region ($m=0$) to the lower left. The dashed line, ${\text{Re}}_H=\beta K_C^2=467$ as defined in (9) bounds the region to the upper-right of $m=1$ asymmetry from higher wave number $m>1$ modes in the high-$\beta$ regime. The instabilities between these two lines can be divided (by the empirical dot-dashed line, $\beta K_C^{7/4}=312$) into two groups associated with the in-phase branch ‘I’ and the out-of-phase branch “O” of instability as shown in Fig. 3, which plots the stability properties at $\beta =500$ (marked with shading) for various $K_C$. Also shown are two three-dimensional structures of the vortex rings generated by an oscillating disk in the low-$\beta$ regime, visualized using $Q=20$ [24], obtained by solving the full nonlinear three-dimensional Navier-Stokes equations, colored by the pressure distribution: for a typical axisymmetric flow at $K_C=1.88,\beta =50$ (marked by $+$); and for a typical asymmetric flow with azimuthal wave number $m=1$ at $K_C=3.14,\beta =100$ (marked by $☆$).

Figure 3

Variation of Floquet multiplier magnitude $|{\mu }_m|$ with azimuthal wave number $m$ for different $K_C$ at $\beta =500$, as marked with shading on Fig. 2. Note the two distinct branches of high wave number $m>1$ instability (subdivided by a dot-dashed line in Fig. 2): the in-phase branch “I” at lower $m$ and the out-of-phase branch “O” at higher $m$.

Figure 4

Evolution of the vortical field for the axisymmetric base flow (top row) and the $\theta$ component of the perturbation velocity for $m=4$ (bottom row) during a cycle for $\beta =500$ and $K_C=0.628$. From left to right, the disk starts from its lowest position then moves upwards, and the images are evenly spaced by $T_0/8$, where $T_0$ is the period of oscillation. Blue (dark gray) and red (light gray) denote negative values and positive values, respectively.

Figure 5

Evolution of the vortical field for the axisymmetric base flow (top row) and the $\theta$ component of the perturbation velocity for $m=7$ (bottom row) during a cycle for $\beta =500$ and $K_C=0.816$. From left to right, the disk starts from its lowest position then moves upwards, and the images correspond to eight evenly spaced phases in a period of the disk oscillation. Blue (dark gray) and red (light gray) denote negative values and positive values, respectively.

Figure 6

Three-dimensional structures of the vortex rings generated by an oscillating disk, visualized using $Q=20$ [24], obtained by solving the fully three-dimensional Navier-Stokes equations, colored by pressure, with $\beta$ fixed at $\beta =500$ while $K_C$ varies. (a) $K_C=0.628$, (b) $K_C=0.735$, (c) $K_C=0.754$, (d) $K_C=0.816$, (e) $K_C=0.879$, (f) $K_C=0.942$. Note that the vortical structures above and below the disk are in phase for in panels (a)–(c) and out of phase in panels (d)–(f) corresponding to branch “I” and branch “O” instabilities, respectively.