Scaling of the translational velocity of vortex rings behind conical objects

Ring vortices are efficient at transporting fluid across long distances. They can be found in nature in various ways: they propel squids, inject blood in the heart, and entertain dolphins. These vortices are generally produced by ejecting a volume of fluid through a circular orifice. The impulse given to the vortex rings moving away results in a propulsive force on the vortex generator. Propulsive vortex rings have been widely studied and characterized. After four convective times, the vortex moves faster than the shear layer it originates from and separates from it. When the vortex separates, the circulation of the vortex reaches a maximum value, and the nondimensional energy attains a minimum. The simultaneity of these three events obfuscates the causality between them. To analyze the temporal evolution of the nondimensional energy of ring vortices independent of their separation, we analyze the spatiotemporal development of vortices generated in the wake of cones. Cones with different apertures and diameters were accelerated from rest to produce a wide variety of vortex rings. The energy, circulation, and velocity of these vortices were extracted based on time-resolved velocity field measurements. The vortex rings that form behind the cones have a self-induced velocity that causes them to follow the cone, and they continue to grow as the cone travels well beyond the limiting vortex formation timescales observed for propulsive vortices. The nondimensional circulation, based on the vortex diameter, and the nondimensional energy of the drag vortex rings converge after three convective times to values comparable to their propulsive counterparts. This result proves that vortex pinch-off does not cause the nondimensional energy to reach a minimum value. The limiting values of the nondimensional circulation and energy are mostly independent of the cone geometry and translational velocity and fall within an interval of $10\%$ around the mean value. The velocity of the vortex shows only $6\%$ of variation and is the most unifying quantity that governs the formation of vortex rings behind cones.

Figure 1

(a) Sketch of the cone and the ring vortex generated in its wake indicating the relevant experimental parameters: cone diameter $D$, cone aperture $\alpha$, and constant final cone translation velocity $U$. The direction of $U$ indicates the upward motion of the cone in the quiescent volume of water. (b) Raw image of the seeding particles accumulating in the main vortex ring and highlighting the small-scale shear layer vortices. This raw image was taken after seeding particles had settled on the cone and was not used for processing.

Figure 2

Growth of a vortex ring in the wake of a translating cone of $D=6\mathrm{cm}$, $\alpha ={45}^{\circ }$, ${\text{Re}}_D=3\times {10}^4$. The green line delimits the contour of the vortex based on FTLE. The light gray area from (b) to (f) highlights the nonvortical volume of fluid in the vortex and outlines the spreading of vorticity over time.

Figure 3

Vorticity field and vortex boundary in the wake of a translating cone with $\alpha ={45}^{\circ }$, $D=6\mathrm{cm}$, and $U=0.5\mathrm{m}{\mathrm{s}}^{-1}$ at (a) $T^{*}=1$, (b) $T^{*}=2$, (c) $T^{*}=3$, and (d) $T^{*}=4$. (e) Temporal evolution of the nondimensional vortex circulation $\mathrm{\Gamma }/UD$ and the volume of nonvortical fluid $V_0$ relative to the total vortex volume $V$. The gray area indicates the acceleration phase of the cone.

Figure 4

(a) Trajectory of the vortex center relative to the cone for $\alpha ={45}^{\circ }$, $D=6\mathrm{cm}$, and $U=0.5\mathrm{m}{\mathrm{s}}^{-1}$. (b) Parametrization of the vortex center. (c) Circulation of the vortex, nondimensionalized using the cone diameter $D$ and the vortex diameter $D_0$. The gray area indicates the acceleration phase of the cone.

Figure 5

(a) Temporal evolution of the nondimensional energy and relative vorticity distribution. The gray area indicates the acceleration phase of the cone. (b) Theoretical and measured velocity of the vortex, relative to the maximum cone velocity. The evolution of the translation velocity of the cone is added for reference.

Figure 6

(a) Vortex center trajectories for four aperture angles, at $D=6\mathrm{cm}$ and $U=0.5\mathrm{m}{\mathrm{s}}^{-1}$. (b–d) Maximum circulation, minimum nondimensional energy, and maximum theoretical vortex velocity for all experiments. The gray lines indicate the average value across all parameter variation. The error bars reflect the uncertainty on the converged values quantified as explained in Appendix pp2. The shape of the markers indicates the final translation velocity of the cone: $▿$: $U=0.35\mathrm{m}{\mathrm{s}}^{-1}$, $\square$: $U=0.5\mathrm{m}{\mathrm{s}}^{-1}$, Δ: $U=0.65\mathrm{m}{\mathrm{s}}^{-1}$. The size of the markers reflects the diameter of the cone: $\circ$: $D=3\mathrm{cm}$, $◯$: $D=6\mathrm{cm}$, $◯$: $9\mathrm{cm}$.

Figure 7

(a) Vorticity field at $T^{*}=3$ and contours used to compute vortex characteristics including the FTLE ridge (—), isocontour of vorticity $|\omega D/U|=1$ (—), and a rectangular box (—). (b) Temporal evolution of the nondimensional circulation, (c) nondimensional energy, and (d) theoretical vortex velocity relative to the cone velocity. The inset in (b) zooms in on the convergence for the circulation calculated within the FTLE boundary. The black line indicates the average value between $T^{*}=3$ and $T^{*}=5$ and is considered as the converged circulation result. The amplitude of the variations are outlined by the gray rectangle, and this is considered as the uncertainty for the measured circulation values.