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

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) $P(\varphi )$, of the angle $\varphi$ between the Eulerian velocity $\mathbf{u}$ and the particle velocity ${\bf v}$,atthispointandtime,showsapower-lawregioninwhich$P(\phi) \sim \phi^{-\gamma}$,withanewuniversalexponent$\gamma \simeq 4$. 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 $\kappa \to \mathrm{\infty }$, and $P(\theta) \sim \theta^{-h_\theta}$,as$\theta \to \infty$,withexponents$h_\kappa \simeq 2.5$and$h_\theta \simeq 3$ that are universal to the extent that they do not depend on the Stokes number $\mathrm{S}\mathrm{t}$ (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 $N_{\mathrm{I}}(t,\mathrm{S}\mathrm{t})$ 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.