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) $P(\varphi )$, of the angle $\varphi$ between the Eulerian velocity $\mathbf{u}$ and the particle velocity $\mathbf{v}$, at this point and time, shows a power-law region in which $P(\varphi )\sim {\varphi }^{-\gamma }$, 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 $\mathrm{St}$ (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 $N_{\mathrm{I}}(t,\mathrm{St})$ of points (up until time $t$) at which $\vartheta$ changes sign. We show that $n_{\mathrm{I}}(\mathrm{St})\equiv {\lim }_{t\to \infty }\frac{N_{\mathrm{I}}(t,\mathrm{St})}{t}\sim {\mathrm{St}}^{-\Delta }$, with $\Delta \simeq 0.4$ a universal exponent.