Anomalous Diffusion and Lévy Walks Distinguish Active from Inertial Turbulence

Lagrangian Manifestation of Anomalies in Active Turbulence

“…The most striking of these being that unlike high Reynolds number flows, experiments suggest non-universal signatures of Lévy walks and anomalous diffusion in measurements of mean-square-displacements (MSDs) in active suspensions [18][19][20][21]. These observations were substantiated in a recent theoretical work [22] which showed not only the robustness of the anomalous diffusion and its coincidence with the emergence of novel, streaky structures in the flow for optimal activity, but also why in earlier theoretical studies [23] such a scaling regime remained masked.…”

“…1(b), which contribute most to the anomalous diffusive behaviour in highly active suspensions. This is because a simple decomposition of the flow field in terms of vortical and straining regions fails to capture a recently discovered [22] additional emergent geometrical feature (with no counterpart in inertial turbulence): Streaks which are striped structures with alternate signs of ω as seen in Fig. 2(a).…”

“…The simulations are performed for 5 × 10 5 iterations with time-steps δt = 0.001 and, in some cases, δt = 0.0002 for higher temporal resolution. We choose Γ 0 = 0.045, Γ 2 = Γ 3 0 , β = 0.5, and λ = 3.5, consistent with earlier studies [16,22,[31][32][33]. The flow, after a spinup time of 2 × 10 4 iterations, is seeded with 1 × 10 5 randomly distributed tracers which evolve as dx/dt = u(x(t)), with x(t) being the tracer location at time t. We use a fourth-order Runge-Kutta scheme, along with a bilinear interpolation scheme to obtain the fluid velocity at the particle positions u(x(t)), to evolve the tracers.…”

“…Walking-segments and turns can be identified using a threshold θ(t) > θ c , which in turn gives the step-lengths d and waiting-times τ of the segments between successive turns. We choose θ c = 30 • for this study [22] and have checked that our results remain unchanged for 25 • ≤ θ c ≤ 45 • .…”

“…1(a) and 1(b). Identifying the "turns" [21,22] is then crucial to segment trajectories into their constituent step-lengths (and waiting-times) for the analysis that follows. This is done by first calculating a turning angle θ at each point along the trajectory at time intervals ∆t as cos (θ(t)) = ∆r(t)•∆r (t+∆t) |∆r(t)||∆r(t+∆t)| , where ∆r(t) = r(t) − r(t − ∆t); our results are insensitive for a wide range of ∆t.…”