Model flocks in a steady vortical flow

We modify the standard Vicsek model to clearly distinguish between intrinsic noise due to imperfect alignment between organisms and extrinsic noise due to fluid motion. We then consider the effect of a steady vortical flow, the Taylor-Green vortex, on the dynamics of the flock, for various flow speeds, with a fixed intrinsic particle speed. We pay particular attention to the morphology of the flow, and quantify its filamentarity. Strikingly, above a critical flow speed there is a pronounced increase in the filamentarity of the flock, when compared to the zero-flow case. This is due to the fact that particles appear confined to areas of low vorticity; a familiar phenomena, commonly seen in the clustering of inertial particles in vortical flows. Hence, the cooperative motion of the particles gives them an effective inertia, which is seen to have a profound effect on the morphology of the flock, in the presence of external fluid motion. Finally, we investigate the angle between the flow and the particles direction of movement and find it follows a power-law distribution.

Figure 1

Snapshots of the Vicsek model in the statistically steady-state (left, $\eta =0.1,R=0.2$; right, $\eta =0.1,R=0.8$). Arrows indicate the particles direction of motion.

Figure 2

The time-averaged Besag's function $\langle \widehat{L}\left(r\right)\rangle$, Eq. (5) plotted as a function of spatial scale $r$ for simulations with $\eta =0.2$. $\widehat{L}\left(r\right)>0$ indicates clustering of particles at a given scale, $\widehat{L}\left(r\right)<0$ indicates segregation.

Figure 3

An example of a flock formed from the standard Vicsek model. Red (filled) points indicate where the separation between that point and its nearest neighbor is less than $\ell /2$, where $\ell =L/\sqrt{N}$ is the expected separation between random points. Blue (open) points are those which do not satisfy this criteria. The black structure shows the $\alpha$-shape [24] constructed from the red points, which is used to quantify the morphology of the flock.

Figure 4

The Taylor-Green vortex flow. The pseudocolor plot displays the flow's vorticity, Eq. (9); black arrows indicate the velocity field, Eq. (8).

Figure 5

Snapshots of the Vicsek flock with external fluid forcing, in the statistically steady state (left, $\eta =0.1,R=0.2$; right, $\eta =0.1,R=0.8$) with $V_f=0.5$. A pseudocolor plot (made slightly translucent to improve visualisation of the particles) displays the flow's vorticity $\omega$, Eq. (9). Arrows indicate the particles direction of motion. Note the concentration of particles in regions where $\omega \simeq 0$.

Figure 6

The time-averaged Besag's function $\langle \widehat{L}\left(r\right)\rangle$, Eq. (5), plotted as a function of spatial scale $r$ for simulations with $V_f=0.5$. $\widehat{L}\left(r\right)>0$ indicates clustering of particles at a given scale; $\widehat{L}\left(r\right)<0$ indicates segregation.

Figure 7

(Left) The time-averaged characteristic size of the flock $\langle D\rangle$ [Eq. (6)], plotted as a function of the radius of interaction $R$, for varying flow velocities $V_f$. (Right) The corresponding plot of the variation of the flock filamentarity $\langle F\rangle$ [Eq. (7)]. Note a profound change is observed in the flock morphology $\left(R\ge 0.6\right)$ when $V_f\ge 0.2$.

Figure 8

The probability density function (PDF) of $\varphi$, the angle between the flow direction, and the particles intrinsic velocity, for the simulation with $V_f=0.2,\eta =0.1,R=0.6$. The inset shows the PDF plotted with a linear scale. The main figure shows a logscale plot, with a power-law fit $\mathrm{PDF}\left(\varphi \right)\sim {\varphi }^{-2.2}$ displayed as a red (dashed) line.

Figure 9

The power-law scaling, $\alpha$, of the probability density function (PDF) of $\varphi [\mathrm{PDF}\left(\varphi \right)={\varphi }^{\alpha }]$, where $\varphi$ is the angle between the flow direction and the particles intrinsic velocity. (Left) $V_f=0.2$, (right) $V_f=0.5$, plotted for varying radii of interaction $R$. The slope is clearly dependent on both $R$ and $V_f$, but appears independent of the intrinsic noise $\eta$.

Figure 10

Snapshots of the Vicsek flock with external fluid forcing given by Eq. (13), with $\eta =0.1,R=0.8$, and $V_f=0.5$. A pseudocolor plot (made slightly translucent to improve visualization of the particles) displays the flow's vorticity $\omega$. Arrows indicate the particles direction of motion.