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) $P(\phi)$, of the angle $\phi$ between the Eulerian velocity ${ \bf u}$ and the particle velocity ${ \bf v}$, at this point and time, shows a power-law region in which $P(\phi) \sim \phi^{ -\gamma}$, with a new universal exponent $\gamma \simeq 4$. Furthermore, the PDFs of the trajectory curvature $\kappa$ and modulus $\theta$ of the torsion $\vartheta$ have power-law tails that 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 are universal to the extent that they do not depend on the Stokes number ${ \rm St}$ (given our error bars). We also show that $\gamma$, $h_\kappa$ and $h_\theta$ 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 $n_{ \rm I}({ \rm St}) \equiv \lim_{ t\to\infty} \frac{ N_{ \rm I}(t,{ \rm St})}{ t} \sim { \rm St}^{ -\Delta}$, with $\Delta \simeq 0.4$ a universal exponent.
Categories: physics.flu-dyn nlin.CD