Lagrangian description of the unsteady flow induced by a single pulse of a jellyfish

Lagrangian statistics and pair dispersion induced by an isolated pulse of a small jellyfish, Aurelia aurita, were quantified and characterized using 3D particle tracking velocimetry (3D-PTV). Probability density functions (PDF) of the Lagrangian velocity components indicated more intense mixing in the radial direction and revealed three stages dominated by flow acceleration, mixing, and dissipation. Time evolution of the Lagrangian acceleration variance further illustrates each phase. During the mixing phase, the flow shares characteristics of homogeneous isotropic turbulence. In addition, we show that a single pulse may induce rich wake dynamics characterized by pair dispersion with a super-diffusive $t^3$ regime due to large-scale flow inhomogeneity, followed by a coherent $t^2$-Batchelor scaling and then $t^1$-Brownian motions. The first trend occurred in the accelerated flow, whereas the second dynamic was observed in the mixed wake and depended on the initial separation. The Brownian motion was present in the late stage dominated by flow dissipation. Kolmogorov microscales during the fully mixed phase were obtained with three distinct approaches, namely, Heisenberg-Yaglom relation of the Lagrangian acceleration variance, the fluctuating rate of the strain tensor in the Eulerian frame of reference as well as the Batchelor scaling in pair dispersion, which showed good agreement.

Figure 1

Jellyfish wake in Lagrangian and Eulerian frames of reference. (a) Basic setup. (b) four-image stereo view. (c) Sample flow trajectories; colors denote axial velocity $u$ normalized by the maximum body velocity $U=28$ mm ${\mathrm{s}}^{-1}$. (d) Single-particle trajectory in the vicinity of the bell margin; the color represents acceleration magnitude $\parallel a\parallel /g$, every 20 instants. (e) Velocity and acceleration of the trajectory from Fig. 1 in the axial and radial directions. (f) 3D Eulerian instantaneous velocity field with in-plane vorticity field, indicating the presence of a vortex ring. (g) Axial velocity field along $x$ for various radial locations as a function of time; the solid and dotted lines indicate instants at maximum and minimum bell diameters.

Figure 2

Lagrangian statistics. The bivariate PDFs of Lagrangian velocity and temporal evolution of the jellyfish propulsion: (a) axial and (b) radial components. (c) Axial velocity PDFs at instants of acceleration ($t/T_p=0.4$), fully mixed ($t/T_p=1.8$), and dissipation ($t/T_p=3.2$) stages. (d) Temporal evolution of the peak of velocity PDFs in the axial and radial directions.

Figure 3

Lagrangian acceleration statistics. PDF of axial Lagrangian acceleration $a_x$ at instants of acceleration ($t/T_p=0.4$) and dissipation ($t/T_p=3.2$) stages. (Inset at right shows the second-order PDF)

Figure 4

Variance of the axial (solid curve) and radial (dotted curve) compensated accelerations ${[\langle a^2\rangle v^{1/2}/a_0]}^{2/3}$ based on the Heisenberg-Yaglom relation [41, 42]. $\mathrm{\Delta }{t}_{\text{A}}$, $\mathrm{\Delta }{t}_{\text{FM}}$, and $\mathrm{\Delta }{t}_{\text{D}}$ indicate the intervals of the acceleration, fully mixed, and dissipation phases. The vertical solid and dotted lines indicate instants at maximum and minimum bell diameters. Inset shows the ratio of axial and radial acceleration variances.

Figure 5

Pair dispersion. The wake induced by the jellyfish showed rich Lagrangian dynamics exhibiting (a) $t^3$-Richardson scaling at $t/T_p=0.4$, (b) coherent $t^2$-Batchelor scaling as a function of $\parallel \mathbf{r}\left(0\right)\parallel$ at $t/T_p=1.8$, and (c) $t^1$-Brownian motion.

Figure 6

Normalized pair dispersion $R^2$ in the mixed phase $\mathrm{\Delta }{t}_{\text{FM}}$. It includes nine initial separations ranging from $\parallel \mathbf{r}\parallel /d=0.1$ to 0.5 every $\parallel \mathrm{\Delta }\mathbf{r}\parallel /d=0.05$. The data and Batchelor's prediction [7] collapse onto a universal power law. The top inset shows the same curves scaled by the Kolmogorov length $\eta$. The bottom inset shows the energy dissipation rate computed using Eulerian-based $\langle {\epsilon }_e\rangle$ and Lagrangian-based $\langle \epsilon \rangle$. The notable match suggests the accurate estimate of the Kolmogorov microscales.