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

etc15 Tracking Number 173

We obtain new universal statistical properties of heavy-particle trajectories in three-dimensional, statistically steady, homogeneous, and isotropic turbulent flows by direct numerical simulations. We show that the probability distribution functions (PDFs) $P(\phi)$, of the angle $\phi$ between the Eulerian velocity ${\bf u}$ and the particle velocity ${\bf v}$, at a point and time, scales as $P(\phi) \sim \phi^{-\gamma}$, with a new universal exponent $\gamma \simeq 4$. The PDFs of the trajectory curvature $\kappa$ and modulus $\theta$ of the torsion $\vartheta$ scale, respectively, as $P(\kappa) \sim \kappa^{-h_\kappa}$, as $\kappa \to \infty$, and $P(\theta) \sim \theta^{-h_\theta}$, as $\theta \to \infty$, with exponents $h_\kappa \simeq 2.5$ and $h_\theta \simeq 3$ that do not depend on the Stokes number $St$. We also show that $\gamma$, $h_\kappa$ and $h_\theta$ can be obtained by using simple stochastic models. We show that the number $N_I(t,St)$ of points (up until time $t$), at which $\vartheta$ changes sign, is such that $n_I(St) \equiv \lim_{t\to\infty} \frac{N_I(t,St)}{t} \sim St^{-\Delta}$, with $\Delta \simeq 0.4$ a universal exponent.