Statistical Properties of the Intrinsic Geometry of Heavy-particle Trajectories in Two-dimensional, Homogeneous, Isotropic Turbulence

Abstract

We obtain, by extensive direct numerical simulations, trajectories of heavy inertial particles in two-dimensional, statistically steady, homogeneous, and isotropic turbulent flows, with friction. We show that the probability distribution function $\mathcal{P}(\kappa )$, of the trajectory curvature $\kappa$, is such that, as $\kappa \to \infty$, $\mathcal{P}(\kappa )\sim {\kappa }^{-h_{\mathrm{r}}}$, with $h_{\mathrm{r}}=2.07\pm 0.09$. The exponent $h_{\mathrm{r}}$ is universal, insofar as it is independent of the Stokes number ($\mathrm{St}$) and the energy-injection wave number. We show that this exponent lies within error bars of their counterparts for trajectories of Lagrangian tracers. We demonstrate that the complexity of heavy-particle trajectories can be characterized by the number $N_{\mathrm{I}}(t,\mathrm{St})$ of inflection points (up until time $t$) in the trajectory and $n_{\mathrm{I}}(\mathrm{St})\equiv {\lim }_{t\to \infty }\frac{N_{\mathrm{I}}(t,\mathrm{St})}{t}\sim {\mathrm{St}}^{-\Delta }$, where the exponent $\Delta =0.33\pm 0.02$ is also universal.

arXiv commentsubmitted to Phys. Rev. Lett. The supplementary information is added as an appendix