Preferential transport of swimmers in heterogeneous two-dimensional turbulent flow

We investigate the performance of active swimmers in a strongly heterogeneous two-dimensional weakly turbulent flow. The flow is heterogeneous in Reynolds number along one direction. Using a hybrid experimental-numerical model, we demonstrate that there are three regimes of preferential transport for rodlike swimmers as the swimmers' intrinsic speed increases. Using Lagrangian statistics along swimmers' trajectories, we reveal that the three regimes are due to the relative strengths of three different effects: the intrinsic speed of the swimmers, the reorientation ability of the shear layer at the interface of two flow regions, and the attracting Lagrangian coherent structures of the flow field. Our results elucidate the mechanism of preferential transport for swimmers in heterogeneous flow. We hope to raise researchers' attention to the dynamics of swimmers in strongly heterogeneous flow environments.

Figure 1

Schematic for the experimental setup. The pair of electrodes conduct dc current horizontally through the saltwater. The vertical magnetic field from the permanent magnets and the horizontal dc current together generate the Lorentz force on the fluid that is nearly entirely in the plane. The fluid layer has a very high aspect ratio that can be considered as quasi-two-dimensional. A piece of acrylic board is put above the left half of the setup to increase bottom friction. The quantity of magnets in the left half is only 1/5 of that in the right half, so that the energy injection in the left half is reduced. The reduced energy injection as well as the increased friction damping make root mean square velocity of the left hand side flow much smaller than that of the right hand side flow.

Figure 2

(a) Snapshot of ${\mathrm{FTLT}}^{-}$ field, where the color bar shows local ${\mathrm{FTLT}}^{-}$ values (negative). Larger absolute FTLT values indicate stronger dynamics. The solid line is the middle interface. Dash lines roughly mark the boundaries of the shear layer. FTLE has a unit of $s^{-1}$. (b) The angle $\beta$ between the middle interface ($y$ axis) and the eigenvectors corresponding to the larger eigenvalues of local strain rate tensors, averaged over time and $y$ direction.

Figure 3

Ratio between the time-averaged swimmer density in the faster side ($\langle C_f\rangle$) and the time-averaged swimmer density in the slower side ($\langle C_s\rangle$). From blue to red, the $\alpha$ of swimmers increases from 0 to 1.

Figure 4

Schematic for the effect of LCSs and the shear layer on rodlike swimmers at different regimes. The intermediate swimmer (light green) is at regime I and it is affected by both LCSs and the shear layer. Once it walks on an LCS to reach the interface, it is reoriented by the shear layer and encounters another nearby LCS where local velocity points toward the faster side. Thus recycling is completed. The fast swimmer (dark green) is in regime II and it is only affected by the LCSs. In this case, LCSs on the faster side facilitate the preferential transport toward the slower side. Once the swimmers' intrinsic velocity increase and reach regime III, the flow will have a negligible effect on the swimmers and preferential transport will disappear.

Figure 5

(a) Time-averaged local ${\mathrm{FTLE}}^{-}$ values for swimmers stay in the band within 3 cm distance from the middle interface, normalized by the initial averaged ${\mathrm{FTLE}}^{-}$. (b) The forward time Lagrangian statistics of the angle $\theta$ between the middle interface ($y$ axis) and the orientation of swimmers $\mathbf{p}$ for swimmers that are in the right 3 cm band and have the $x$ component of $\mathbf{p}$ pointing toward the left at $t=0$. If the $x$ component of $\mathbf{p}$ is pointing toward the right (faster side), $\theta$ is considered as negative; if the $x$ component of $\mathbf{p}$ is pointing toward the left (slower side), $\theta$ is considered as positive. The subfigure shows the schematic of calculating $\theta$. (c) The forward time Lagrangian statistics of $\theta$ for swimmers that cross the interface at $t=0$ from the faster side to the slower side. Panels (b) and (c) show that the shear layer is effective in reorienting swimmers with $v_s=0.7U_f$ (in regime I) but not in reorienting swimmers with $v_s=1.4U_f$ (in regime II). (d) The ratio of swimmers that return to their original side when they cross the middle interface at $t=0$. The direction of $\mathbf{p}$ in the legend indicates the direction of $v_s$ when swimmers cross the interface at $t=0$.