Coherent structures in interacting vortex rings

We investigate experimentally the nonlinear structures that develop from interacting vortex rings induced by a sinusoidally oscillating ellipsoidal disk in fluid at rest. We vary the scaled amplitude or Keulegan-Carpenter number $0.3<N_{\text{KC}}=2\pi A/c<1.5$, where $A$ is the oscillation amplitude and $c$ is the major diameter of the disk, and the scaled frequency or Stokes number $100<\beta =f{c}^2/\nu <1200$, where $f$ is the frequency of oscillation and $\nu$ is the kinematic viscosity. Broadly consistent with global linear stability analyses, highly organized nonlinear structures with clear azimuthal wave number emerge as sequential vortex rings are shed from the disk. These organized structures exhibit wave numbers ranging from $m=2$ to $m=9$ and can be further divided into two distinct classes, distinguished by the phase and symmetry properties above and below the disk. We find some discrepancies between experiments and linear stability analysis, due to the inherent nonlinear mechanisms in the experiments, particulary on the boundary between the two branches, presenting unevenly distributed flow structures along the azimuthal direction.

Figure 1

Vortical structures obtained by three-dimensional simulations for (a) $(N_{\text{KC}},\beta )=(0.735,500)$ and (b) $(N_{\text{KC}},\beta )=(0.879,500)$ [26]. Note that the vortical structures shed above and below the disk are in phase or symmetric in (a) and out of phase or asymmetric in (b). $N_{\text{KC}}$ and $\beta$ are defined in Eq. (1).

Figure 2

Regime diagram for axisymmetry-breaking instabilities in the $\beta \text{−}{N}_{\text{KC}}$ plane. The dashed line, showing ${\text{Re}}_H=\beta N_{\text{KC}}^2=467$, bounds, to the lower left, the regime with $m=1$ instability. The solid line, showing ${\text{Re}}_L=\beta N_{\text{KC}}^{3/2}=180$, bounds, to the upper right, the symmetry regime $\left(m=0\right)$. Between these two lines, the circles filled with different colors mark different wave numbers visualized by experiments and the blocks with different colors represent the regimes with different wave numbers determined by linear stability analyses. The region between the two lines can be further divided into an in-phase branch and an out-of-phase branch by the dash-dotted line. The insets show the experimental visualizations of a typical symmetric flow at $(N_{\text{KC}},\beta )=(0.565,405)$ (marked by +) and an asymmetric flow at $(N_{\text{KC}},\beta )=(0.942,816)$ (marked by $*)$.

Figure 3

Top views of the experimental observation of unevenly azimuthally distributed multimode structures in unstable vortex rings, corresponding to the black closed circles in Fig. 2: (a) $(N_{\text{KC}},\beta )=(0.691,550)$ and (b) $(N_{\text{KC}},\beta )=(0.791,450)$.

Figure 4

Top views of structures in the in-phase branch (between solid and dash-dotted lines in Fig. 2) with (a) $(N_{\text{KC}},\beta )=(0.352,926)$, (b) $(N_{\text{KC}},\beta )=(0.352,1013)$, (c) $(N_{\text{KC}},\beta )=(0.455,947)$, (d) $(N_{\text{KC}},\beta )=(0.502,947)$, (e) $(N_{\text{KC}},\beta )=(0.565,752)$, and (f) $(N_{\text{KC}},\beta )=(0.942,289)$.

Figure 5

Top views of structures in the out-of-phase branch as shown in Fig. 2, with (a) $(N_{\text{KC}},\beta )=(0.691,658)$, (b) $(N_{\text{KC}},\beta )=(0.791,534)$, (c) $(N_{\text{KC}},\beta )=(0.791,591)$, and (d) $(N_{\text{KC}},\beta )=(0.942,422)$.

Figure 6

Side views of the flow structures for (a) $(N_{\text{KC}},\beta )=(0.455,947)$ and (b) $(N_{\text{KC}},\beta )=(0.791,591)$. Equivalent top-view images are shown in Figs. 4 and 5 respectively.