Universal Statistical Properties of Inertial-particle Trajectories in Three-dimensional, Homogeneous, Isotropic, Fluid Turbulence

Abstract

We uncover universal statistical properties of the trajectories of heavy inertial particles in three-dimensional, statistically steady, homogeneous, and isotropic turbulent flows by extensive direct numerical simulations. We show that the probability distribution functions (PDFs) $P(ϕ)$, of the angle $ϕ$ between the Eulerian velocity ${\bf u}$ and the particle velocity ${\bf v}$, at this point and time, shows a power-law region in which $P(ϕ) \sim ϕ^{-γ}$, with a new universal exponent $γ\simeq 4$. Furthermore, the PDFs of the trajectory curvature $κ$ and modulus $θ$ of the torsion $\vartheta$ have power-law tails that scale, respectively, as $P(κ) \sim κ^{-h_κ}$, as $κ\to \infty$, and $P(θ) \sim θ^{-h_θ}$, as $θ\to \infty$, with exponents $h_κ\simeq 2.5$ and $h_θ\simeq 3$ that are universal to the extent that they do not depend on the Stokes number ${\rm St}$ (given our error bars). We also show that $γ$, $h_κ$ and $h_θ$ can be obtained by using simple stochastic models. We characterize the complexity of heavy-particle trajectories by the number $N_{\rm I}(t,{\rm St})$ of points (up until time $t$) at which $\vartheta$ changes sign. We show that $n_{\rm I}({\rm St}) \equiv \lim_{t\to\infty} \frac{N_{\rm I}(t,{\rm St})}{t} \sim {\rm St}^{-Δ}$, with $Δ\simeq 0.4$ a universal exponent.