Friction scaling laws for transport in active turbulence

Understanding the role of frictional drag in diffusive transport is an important problem in the field of active turbulence. Using a generic continuum model that applies well to living systems, we investigate the role of Ekman friction on the passive transport of Lagrangian tracers that go with the local flow. We find that the crossover from ballistic to diffusive regime happens at a timescale ${\tau }_c$ that attains a minimum at zero friction, meaning that both injection and dissipation of energy delay the relaxation of tracers. We explain this by proposing that ${\tau }_c\sim {\ell }^{*}/u_{\text{rms}}$, where ${\ell }^{*}$ is a dominant length scale extracted from energy spectrum peak, and $u_{\text{rms}}$ is a velocity scale that sets the kinetic energy at steady state; both scales monotonically decrease with friction. Finally, we predict friction scaling laws for ${\ell }^{*},u_{\text{rms}}$, and the diffusion coefficient $\mathcal{D}\sim {\ell }^{*}{u}_{\text{rms}}/2$, that are valid over a wide range of fluid friction. The findings of our report should apply to transport phenomena in generic active systems such as dense bacterial suspensions, microtubule networks, or even artificial swimmers, to name a few.

Figure 1

Color map of the 2D vorticity field along with the distribution of passive tracers, which are indicated as black dots, at some steady state. Grid resolution is ${512}^2$ and Ekman friction $\alpha =-1$.

Figure 2

Mean-square displacement $\langle \mathrm{\Delta }{\mathit{x}}^2\rangle$ vs time $t$ for a typical Ekman friction $\alpha$. Inset: Crossover time ${\tau }_c$ develops a minimum at zero friction (see text for details). Error bars indicate one standard deviation.

Figure 3

Energy spectrum $E_k$ for different values of Ekman friction $\alpha$. The peak shows the location of dominant length scale $k^{*}=1/{\ell }^{*}$. Inset: The exponent $\delta$ at low wave numbers is seen to be linear in $\alpha$.

Figure 4

Crossover time ${\tau }_c$ and the scale combination ${\ell }^{*}/u_{\text{rms}}$. The prefactor turns out to be 2.3 for the best fit.

Figure 5

Velocity autocorrelation of tracers for different values of Ekman friction $\alpha$. It is evident that the relaxation (e-folding) time is fastest for the case $\alpha =0$ and becomes progressively higher for both self-propulsion $\left(\alpha <0\right)$ and damping $\left(\alpha >0\right)$.

Figure 6

Characteristic length $\left({\ell }^{*}\right)$ and velocity $\left(u_{\text{rms}}\right)$ scales along with their predicted scalings shown as dashed lines.

Figure 7

Diffusion coefficient $\mathcal{D}$ vs Ekman friction $\alpha$ for different values of the free parameter $\beta$. Both the estimates of $\mathcal{D}$, namely, from the simulations $\langle \mathrm{\Delta }{\mathit{x}}^2\rangle /4t$, and from characteristic scales $u_{\text{rms}}{\ell }^{*}/2$, agree very well with our predicted scaling law [Eq. (20), dashed line]. The figure shows the generality of our scaling law as it holds over variation in the free parameters of our governing model, in this case $\beta$.