Optimal turbulent transport in microswimmer suspensions

Microswimmer suspensions self-organize into complex spatiotemporal flow patterns, including vortex lattices and mesoscale turbulence. Here we explore the consequences for the motion of passive tracers, based on a continuum model for the microswimmer velocity field. We observe two qualitatively different regimes distinguished via the dimensionless Kubo number $K$. At advection strengths right above the transition to turbulence, the flow field evolves very slowly ($K\gg 1$) and the spatial vortex structures lead to dominant trapping effects. In contrast, deep in the turbulent state, much faster dynamics ($K\ll 1$) consistent with the so-called sweeping hypothesis leads to transport properties completely determined by the temporal correlations. In between ($K\approx 1$), we observe a regime of optimal transport, signaled by a maximum of the diffusion coefficient.

Figure 1

Transition to turbulence. (a) Numerically obtained threshold value of the nonlinear advection strength ${\lambda }^{★}$ as a function of $a$ ($b=1.6$). Below ${\lambda }^{★}$, the system settles into a stationary vortex lattice, whereas above ${\lambda }^{★}$, a mesoscale-turbulent state emerges. Snapshots of the vorticity field $\omega ={(\mathbf{\nabla }\times \mathbf{v})}_z$ at $a=0.5$ in the lattice state at (b) $\lambda =1.7$ and in the mesoscale-turbulent state at (c) $\lambda =4$. Tracers move along instantaneous streamlines (solid lines) of the flow, which continuously break up and reconnect in the turbulent state. (d) MSD as a function of time lag $\tau$ for different $\lambda$ and $a=0.5$. (e) Sample trajectories for $\lambda =1.7$ (red), $\lambda =2$ (violet), $\lambda =4$ (blue), and $\lambda =10$ (green) and elapsed time $\mathrm{\Delta }t=250$. The same scale is used for (b), (c), and (e), where the gray bar indicates a length of $2\pi$

Figure 2

(a) Diffusion coefficient $D$ as a function of $\lambda$ for different values of $a$ ($b=1.6$). (b) Kubo number $K$ as a function of $\lambda$. Inset: Inverse Kubo number $K^{-1}$ as a function of $\lambda$. The dashed line gives the result obtained via Kraichnan's random sweeping hypothesis with $c=0.33$, see Eq. (9). (c) Dimensionless diffusion coefficient $D/\left(\ell v\right)$ as a function of $K$. The scaling in the different regimes is given as solid and dashed line, respectively. Error bars represent the standard error.

Figure 3

Eulerian correlations for $a=0.5$ and $b=1.6$. (a) Normalized Eulerian temporal correlation function $C_E\left(\tau \right)$ for different values of $\lambda$. Below the onset of turbulence at ${\lambda }^{★}$, the system is in a stationary state, i.e., temporal correlations do not decay. (b) Longitudinal correlation function $f\left(r\right)$ for different values of $\lambda$. Due to the finite-wavelength instability, $f\left(r\right)$ shows oscillatory behavior, especially for smaller values of $\lambda$. The longitudinal correlation function below the onset of turbulence at ${\lambda }^{★}, f\left(r\right)$ can be calculated analytically for a stationary square vortex lattice, see Appendix pp2. The result is included as black dashed line.

Figure 4

Quantities calculated in the Eulerian frame of reference for $b=1.6$. (a) Velocity scale $v$ as a function of $\lambda$ for different values of $a$. Above the transition to turbulence, which occurs at ${\lambda }^{★}\approx 1.7\cdots 2.8$ (dependent on $a$), $v$ decreases with increasing $\lambda$. (b) The integral length scale $\ell$ as a function of $\lambda$ for different values of $a$. Below ${\lambda }^{★}$, we observe $\ell =2$, consistent with a square vortex lattice, see Appendix pp2. In the turbulent state, energy is transferred to larger structures, resulting in $\ell$ increasing with $\lambda$. (c) Eulerian correlation time ${\tau }_E$ as a function of $\lambda$ for different values of $a$. Below the onset of turbulence at ${\lambda }^{★}$, the system is stationary, which means ${\tau }_E$ must diverge. Above ${\lambda }^{★}$, the dynamics gets faster with increasing $\lambda$, resulting in decreasing ${\tau }_E$.

Figure 5

(a) Normalized Lagrangian correlation function $C_L$ as a function of time lag $\tau$ for different values of $\lambda$. The other parameters are $a=0.5$ and $b=1.6$. (b) Lagrangian correlation time $\tau$ as a function of $\lambda$ for different values of $a$ and $b=1.6$.

Figure 6

Energy spectrum $\tilde{E}\left(k\right)$ as a function of the wave number for $\lambda =10, a=0.5$, and $b=1.6$. The spectrum increases as $\propto k^3$, shows a clear maximum and then decreases exponentially for larger $k$.

Figure 7

Correlation function $C_E\left(\tau \right)$ for $\lambda =5, \lambda =7$, and $\lambda =10$. The blue curve shows the correlation function calculated directly from the numerical results. The red curve shows the correlation function calculated via Eq. (E6) with $c=0.33$. The energy spectrum $\tilde{E}\left(k\right)$ was calculated from the spatial correlation function $f\left(r\right)$ determined from the numerical simulations. The other parameters are $a=0.5$ and $b=1.6$.

Figure 8

(a) Diffusion coefficient $D$ as a function of nonlinear advection strength $\lambda$ for different values of $b$. (b) Kubo number $K$ as a function of $\lambda$ for different values of $b$. $K$ is finite in the mesoscale-turbulent state for $\lambda >{\lambda }^{★}$ and diverges when approaching the stationary vortex lattice state for $\lambda <{\lambda }^{★}$. Inset: Inverse Kubo number $K^{-1}$ as a function of $\lambda$, which shows linear behavior for larger values of $\lambda$. The dashed line gives the result obtained via Kraichnan's random sweeping hypothesis with $c=0.33$. (c) Dimensionless diffusion coefficient $D/\left(\ell v\right)$ as a function of Kubo number $K$. The scaling in the different regimes is given as solid and dashed line, respectively. The remaining parameter is $a=0.5$.