Velocity fluctuations in a dilute suspension of viscous vortex rings

We explore the velocity fluctuations in a fluid due to a dilute suspension of randomly distributed vortex rings at moderate Reynolds number, for instance, those generated by a large colony of jellyfish. Unlike previous analysis of velocity fluctuations associated with gravitational sedimentation or suspensions of microswimmers, here the vortices have a finite lifetime and are constantly being produced. We find that the net velocity distribution is similar to that of a single vortex, except for the smallest velocities which involve contributions from many distant vortices; the result is a truncated $5/3$-stable distribution with variance (and mean energy) linear in the vortex volume fraction $\varphi$. The distribution has an inner core with a width scaling as ${\varphi }^{3/5}$, then long tails with power law ${\left|u\right|}^{-8/3}$, and finally a fixed cutoff (independent of $\varphi$) above which the probability density scales as ${\left|u\right|}^{-5}$, where $u$ is a component of the velocity. We argue that this distribution is robust in the sense that the distribution of any velocity fluctuations caused by random forces localized in space and time has the same properties, except possibly for a different scaling after the cutoff.

Figure 1

(Left) A “suspension” of spotted jellyfish (Mastigias papua) at the Vancouver Aquarium. (Center) Fast swimming Nemopsis bachei expels a single vortex ring with each rapid pulse (reproduced with permission from [41]). (Right) Schematic of the problem: we seek the distribution of the fluid velocity $\mathit{u}$ at ${\mathit{r}}_0$ due to a randomly distributed suspension of viscous vortex rings in three dimensions.

Figure 2

(Left) Diagram of an early-stage vortex ring. (Center) Contours of the streamfunction $\mathrm{\Psi }$ normalized by ${\mathrm{\Gamma }}_0{R}_0$ in the laboratory frame from Eq. (4b) at time $t=R_0^2/\nu$. (Right) The same normalized streamfunction in a frame moving with the vortex ring.

Figure 3

Plot of the terms involving $\xi$ inside the parentheses of Eq. (5), along with the small and large $\xi$ approximations used to derive Eq. (7). We see that the approximations are very good outside of a transition region with $0.4\lesssim \xi \lesssim 2.5$.

Figure 4

The energy integrated over all space $E_1\left(t\right)$ for a single vortex ring normalized by ${\mathrm{\Gamma }}_0^2{R}_0$ and compared with the small and large-time asymptotics in Eq. (9).

Figure 5

The numerically evaluated velocity probability density function for a single vortex ring (solid line) compared with the analytic approximation Eq. (25) (dashed line). The approximation is about 40% higher than the numerical values on the segment with $0.001{\mathrm{\Gamma }}_0/R_0\lesssim u\lesssim 0.04{\mathrm{\Gamma }}_0/R_0$.

Figure 6

The probability density function for the $x$-component of velocity (normalized by ${\varphi }^{3/5}{\mathrm{\Gamma }}_0/R_0$) for various $\varphi$. We see that the core scales with ${\varphi }^{3/5}$. The dashed curve is the analytical expression ${\mathrm{\Phi }}_{5/3}(u_x;a)$, which agrees closely with the numerics. The dotted curve is a Gaussian distribution with unit standard deviation, included for reference.

Figure 7

The same distributions as in Fig. 6, but on a log-log scale. The additional dashed line verifies the $-8/3$ power law for large (but not very large) velocities.

Figure 8

Plot of the (normalized) probability density function for the $x$ component of velocity divided by $\varphi$ compared with Eq. (39) (the dashed line), showing close agreement, except at small velocities. In particular, regardless of $\varphi$, the distributions transition away from the $-8/3$ power law at around $u_x\approx 0.4{\mathrm{\Gamma }}_0/R_0$, regardless of $\varphi$.