Deviation-angle and trajectory statistics for inertial particles in turbulence.

Small particles in suspension in a turbulent fluid have trajectories that do not follow the pathlines of the flow exactly. We investigate the statistics of the angle of deviation ϕ between the particle and fluid velocities. We show that, when the effects of particle inertia are small, the probability distribution function (PDF) P_{ϕ} of this deviation angle shows a power-law region in which P_{ϕ}∼ϕ^{-4}. We also find that the PDFs of the trajectory curvature κ and modulus θ of the torsion ϑ have power-law tails that scale, respectively, as P_{κ}∼κ^{-5/2}, as κ→∞, and P_{θ}∼θ^{-3}, as θ→∞: These exponents are in agreement with those previously observed for fluid pathlines. We propose a way to measure the complexity of heavy-particle trajectories by the number N_{I}(t,St) of points (up until time t) at which the torsion changes sign. We present numerical evidence that n_{I}(St)≡lim_{t→∞}N_{I}(t,St)/t∼St^{-Δ} for large St, with Δ≃0.5.