Spatial distributions of heavy particles suspended in an incompressible isotropic and homogeneous turbulent flow are investigated by means of high resolution direct numerical simulations. In the dissipative range, it is shown that particles form fractal clusters with properties independent of the Reynolds number. Clustering is there optimal when the particle response time is of the order of the Kolmogorov time scale ${\tau }_{\eta }$. In the inertial range, the particle distribution is no longer scale invariant. It is, however, shown that deviations from uniformity depend on a rescaled contraction rate, which is different from the local Stokes number given by dimensional analysis. Particle distribution is characterized by voids spanning all scales of the turbulent flow; their signature in the coarse-grained mass probability distribution is an algebraic behavior at small densities.

We present the results of direct numerical simulations of heavy particle transport in homogeneous, isotropic, fully developed turbulence, up to resolution $512^3$ ($R_\lambda\approx 185$). Following the trajectories of up to 120 million particles with Stokes numbers, St, in the range from 0.16 to 3.5 we are able to characterize in full detail the statistics of particle acceleration. We show that: (i) the root-mean-squared acceleration $a_{\rm rms}$ sharply falls off from the fluid tracer value at quite small Stokes numbers; (ii) at a given St the normalized acceleration $a_{\rm rms}/(\epsilon^3/\nu)^{1/4}$ increases with $R_\lambda$ consistently with the trend observed for fluid tracers; (iii) the tails of the probability density function of the normalized acceleration $a/a_{\rm rms}$ decrease with St. Two concurrent mechanisms lead to the above results: preferential concentration of particles, very effective at small St, and filtering induced by the particle response time, that takes over at larger St.

The statistical properties of velocity and acceleration fields along the trajectories of fluid particles transported by a fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. We present results for Lagrangian velocity structure functions, the acceleration probability density function, and the acceleration variance conditioned on the instantaneous velocity. These are compared with predictions of the multifractal formalism, and its merits and limitations are discussed.

We present a collection of eight data sets from state-of-the-art experiments and numerical simulations on turbulent velocity statistics along particle trajectories obtained in different flows with Reynolds numbers in the range $R_{\lambda }\in [120:740]$. Lagrangian structure functions from all data sets are found to collapse onto each other on a wide range of time lags, pointing towards the existence of a universal behavior, within present statistical convergence, and calling for a unified theoretical description. Parisi-Frisch multifractal theory, suitably extended to the dissipative scales and to the Lagrangian domain, is found to capture the intermittency of velocity statistics over the whole three decades of temporal scales investigated here.

The statistical properties of fluid particles transported by a three-dimensional fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. Single trajectory statistics are investigated in a time range spanning more than three decades, from less than a tenth of the Kolmogorov time scale, ${\tau }_{\eta }$ , up to one large-eddy turnover time. Our analysis reveals the existence of relatively rare trapping events in vortex filaments which give rise to enhanced intermittency on Lagrangian observables up to $10{\tau }_{\eta }$ . Lagrangian velocity structure function attain scaling properties in agreement with the multifractal prediction only for time lags larger than those affected by trapping.

The statistics of velocity differences between pairs of heavy inertial point particles suspended in an incompressible turbulent flow is studied and found to be extremely intermittent. The problem is particularly relevant to the estimation of the efficiency of collisions among heavy particles in turbulence. We found that when particles are separated by distances within the dissipative subrange, the competition between regions with quiet regular velocity distributions and regions where very close particles have very different velocities (caustics) leads to a quasi bi-fractal behaviour of the particle velocity structure functions. Contrastingly, we show that for particles separated by inertial-range distances, the velocity-difference statistics can be characterized in terms of a local roughness exponent, which is a function of the scale-dependent particle Stokes number only. Results are obtained from high-resolution direct numerical simulations up to 2048³ collocation points and with millions of particles for each Stokes number.