Cascade leading to the emergence of small structures in vortex ring collisions

When vortex rings collide head-on at high enough Reynolds numbers, they ultimately annihilate through a violent interaction which breaks down their cores into a turbulent cloud. We experimentally show that this very strong interaction, which leads to the production of fluid motion at very fine scales, uncovers direct evidence of an iterative cascade of instabilities in a bulk fluid. When the coherent vortex cores approach each other, they deform into tentlike structures and the mutual strain causes them to locally flatten into extremely thin vortex sheets. These sheets then break down into smaller secondary vortex filaments, which themselves rapidly flatten and break down into even smaller tertiary filaments. By performing numerical simulations of the full Navier-Stokes equations, we also resolve one iteration of this instability and highlight the subtle role that viscosity must play in the rupturing of a vortex sheet. The concurrence of this observed iterative cascade of instabilities over various scales with those of recent theoretical predictions could provide a mechanistic framework in which the evolution of turbulent flows can be examined in real time as a series of discrete dynamic instabilities.

Figure 1

Experimental system. (a) Montage of images showing the head-on collision of two vortex rings formed at $\text{Re}=8000$ and $\text{SR}=2.5$, where both rings are dyed with food coloring. Upon first colliding, the vortex rings stretch and grow radially before rapidly breaking down to fine-scale smoke. (b) Image and (c) schematic of the two vortex cannons and high-speed scanning laser sheet fluorescence microscopy setup, which enables each scan of the flow to be reconstructed into a 3D volume.

Figure 2

Two vortex rings colliding. Four consecutive snapshots simultaneously taken from the front (top) and side (bottom) showing the progressive stages of the head-on collision between two identical vortex rings with $\text{Re}=4000$ and $\text{SR}=2.5$. Both cores are injected with a small volume of fluorescent dye (rhodamine-B) and illuminated from the side by a scanning green laser sheet, highlighting the core in bright yellow. (a) As the two cores approach, they stretch radially but remain circular. (b) With further stretching, the vortex rings develop long-wavelength perturbations that form tentlike structures. The inset shows a zoomed-in view of a developing tentlike structure. (c) Tips of the tentlike structures flatten due to the intense strains exerted by the circulating cores. The inset shows that the flattened core stretches into a vortex sheet and splits into two secondary vortex filaments. (d) The initial breakdown of the vortex cores at these local tentlike structures propagate along the vortex cores, leading to the annihilation of the vortex rings and the formation of a turbulent cloud.

Figure 3

The 3D reconstruction of the breakdown of two dyed vortex cores. Three snapshots show the late-stage dynamics of two colliding vortex rings where $\text{Re}=4000$ and $\text{SR}=2.5$. (a) The cores deflect toward one another, developing a tentlike structure. (b) The tentlike structure grows in amplitude, and the lower core flattens into a vortex sheet at the region where the two cores are closest. (c) The cores contact and break down. In all panels, the cyan arrows indicate length scales (a) $1.7\mathrm{mm}$, (b) $1.4\mathrm{mm}$, and (c) $1.4\mathrm{mm}$.

Figure 4

Extreme stretching of a colliding vortex core. (a) Cross-sectional slice of a deforming vortex core extracted from 3D flow visualization during the collision of a dyed vortex ring with an invisible undyed vortex ring. The dyed vortex core, rotating in the clockwise direction, is shown in blue and is stretched by the undyed vortex core, which rotates in the counterclockwise direction. The centerline of the vortex core is indicated by the black dashed line and the plane perpendicular to the midpoint of the centerline is shown by the red lines. The grid spacing is $1\times 1$ ${\mathrm{mm}}^2$. (b) Centerline length vs time for the dyed core. (c) Core thickness along the midpoint of the centerline vs time. The green dashed lines correspond to the times indicated in (a), the data in both plots are averaged over ten adjoining slices, and for this collision $\text{Re}=4000$ and $\text{SR}=2.5$.

Figure 5

An iterative cascade of instabilities leads to the emergence of small-scale flow structures. The 3D reconstruction of the breakdown dynamics follows the collision of a dyed vortex ring (top) with an invisible undyed one (bottom). For this typical example, $\text{Re}=4000$ and $\text{SR}=2.5$. (a) The upper dyed vortex develops a tentlike perturbation that deflects toward the undyed lower vortex. (b) The tentlike structure flattens into a very thin vortex sheet. Most of the dye collects at the edges of the sheet. (c) The thin vortex sheet ruptures and a hole is formed, which extends down the rest of the core. (d) The vortex sheet splits into two smaller secondary filaments, which unravel the vortex core at both ends. (e) A secondary vortex splits into smaller tertiary vortex filaments. (f) A complex structure of interacting secondary and tertiary vortex filaments emerges. The inset shows a zoomed-in and contrast-enhanced view showing a network of interacting coherent secondary and tertiary vortex filaments. The magnified view is a subvolume of the main panel indicated by the dashed box and is viewed through the dashed magenta window. (g) and (h) Full view of the emerging tertiary filaments. The contrast is enhanced to highlight the faint tertiary filaments. (i) The essence of the dynamics is captured entirely in one snapshot, simultaneously showing three generations of the iterative cascade. In every panel, the size of a distinct feature is indicated by a cyan scale bar: (a) $1.3\mathrm{mm}$, (b) $1.2\mathrm{mm}$, (c) $1.2\mathrm{mm}$, (d) $1.7\mathrm{mm}$, (e) $2.9\mathrm{mm}$, (f) $2.1\mathrm{mm}$, (inset) $1.1\mathrm{mm}$, (g) $1.1\mathrm{mm}$, (h) $0.6\mathrm{mm}$, and (i) $2.8\mathrm{mm}$.

Figure 6

Numerical simulation of colliding vortex rings. (a) Volumetric plot of dye concentration in a simulation of two colliding vortex rings whose cores are initially seeded with red and blue dye, respectively. As the vortex cores approach, the blue vortex flattens into thin vortex sheets, which break down into secondary filaments. (b) Zoomed-in cross-sectional view of the developing secondary filaments indicated by the white box in (a). The top panels plot the dye concentration and the bottom panels plot the vorticity modulus. The formation and rupture of sheets into filaments is clearly shown by the top (blue) vortex, while the lower vortex shows that if no rupture occurs, the sheet eventually curls back into a vortex when the strain is released.

Figure 7

Vortex cannon assembly: (a) image and (b) cross-sectional schematic of the vortex cannon assembly. The piston shaft is loaded with permanent magnets and is impulsively driven by the magnetic fields generated by a series of current-carrying coils in the linear motor. The linear motor is servo controlled and the feedback position of the piston is measured by a noncontact linear encoder. The core of the vortex ring is dyed by injecting fluorescent dye at the outer edge of the nozzle.

Figure 8

Vortex core tracking in two dimensions with PIV and dye. (a) Vorticity distribution and core trajectory of two colliding vortex rings at several time steps obtained via PIV through a 2D cross section perpendicular to the collision plane. The plotted vorticity distribution is the result of fitting the raw vorticity data to a 2D Gaussian function for each core. As the vortex rings collide, they grow radially and the cores contract, amplifying their vorticity before breaking down. The time steps correspond to $t=0$, 0.225, 0.5, 0.675, 0.8, 0.9, 1, and 1.25 s, and for all data in this figure $\text{Re}=4000$ and $\text{SR}=2.5$. (b) Vortex ring radius vs time for ten collisions where the vortex core is tracked from the vorticity data via PIV (black) and from dyeing the core with fluorescent dye (color). (c) Vortex core radius vs vortex ring radius for ten collisions; the core is tracked by fitting to the vorticity data (black) and by injecting the vortex ring with various amounts of fluorescent dye: 0.08 mL (red), 0.1 mL (orange), 0.1 mL (green), and 0.12 mL (blue).

Figure 9

Vorticity evolution in the collision simulations. (a) Evolution of the maximum (orange) and mean (blue) vorticity modulus with time, relative to the initial values. (b) Weight of the mean azimuthal component of vorticity relative to the mean vorticity modulus as a function of time. A value of $1/\sqrt{3}\approx 0.577$ would correspond to a vorticity distribution with no preferential direction.