Heavy Inertial Particles in Superfluid Turbulence: Coflow and Counterflow

We use pseudospectral direct numerical simulations of the 3D Hall-Vinen-Bekharevich-Khalatnikov model of superfluids to obtain the statistical properties of heavy inertial particles, in both coflow and counterflow superfluid turbulence (ST); particle motion is governed by generalized Maxey-Riley-Gatignol equations. We characterize the anisotropy of counterflow ST and calculate particle statistics for different values of (a) the temperature-dependent mutual-friction coefficients and the normal-fluid density, (b) the counterflow velocity $\bf U_{ns}$, and (c) the Stokes numbers of the particles. We calculate various probability distribution functions (PDFs) and cumulative PDFs (CPDFs) to characterize particle statistics: (1) The mean angle $\theta(\tau)$, between particle positions, separated by the time $\tau$ , exhibits different scaling regions in dissipation and inertial ranges; and $\theta(\tau)$, at large $\tau$ , depends on the magnitude of $\bf U_{ns}$ . (2) The CPDFs of the angle $\phi_n$ , between the normal-fluid and the particle velocities, and the angle $\phi_s$ , between the superfluid and particle velocities, exhibit power-law regimes, the extents of which depend upon the Stokes numbers. (3) CPDFs of the curvature $\kappa$ and the magnitude $\theta$ of the torsion of particle trajectories have power-law tails with universal exponents. (4) The CPDFs of persistence times, which we condition on the invariants of the velocity-gradient tensor, have exponentially decaying tails, whose decay times indicate how long particles spend in vortical regions. (5) We characterize the irreversibility of this turbulence via the statistics of particle-energy increments; we compare our results with experiments. (6) We define PDFs of particle first-passage times and obtain their dependence on the integral scale of this turbulence. (7) We compute the multiscaling of velocity structure functions.