Small particles in suspension in a turbulent fluid have trajectories that do not follow the pathlines of the flow exactly. We investigate the statistics of the angle of deviation φ between the particle and fluid velocities. We show that, when the effects of particle inertia are small, the probability distribution function (PDF) Pφ of this deviation angle shows a power-law region in which Pφ∼φ-4. We also find that the PDFs of the trajectory curvature κ and modulus θ of the torsion have power-law tails that scale, respectively, as Pκ∼κ-5/2, as κ→∞, and Pθ∼θ-3, as θ→∞: These exponents are in agreement with those previously observed for fluid pathlines. We propose a way to measure the complexity of heavy-particle trajectories by the number NI(t,St) of points (up until time t) at which the torsion changes sign. We present numerical evidence that nI(St)≡limt→∞NI(t,St)t∼St-Δ for large St, with Δ≃0.5.

We present an extensive numerical study of the time irreversibility of the dynamics of heavy inertial particles in three-dimensional, statistically homogeneous, and isotropic turbulent flows. We show that the probability density function (PDF) of the increment, W(tau), of a particle's energy over a time scale tau is non-Gaussian, and skewed toward negative values. This implies that, on average, particles gain energy over a period of time that is longer than the duration over which they lose energy. We call this slow gain and fast loss. We find that the third moment of W(tau) scales as tau(3) for small values of tau. We show that the PDF of power-input p is negatively skewed too; we use this skewness Ir as a measure of the time irreversibility and we demonstrate that it increases sharply with the Stokes number St for small St; this increase slows down at St similar or equal to 1. Furthermore, we obtain the PDFs of t(+) and t(-), the times over which p has, respectively, positive or negative signs, i.e., the particle gains or loses energy. We obtain from these PDFs a direct and natural quantification of the slow gain and fast loss of the energy of the particles, because these PDFs possess exponential tails from which we infer the characteristic loss and gain times t(loss) and t(gain), respectively, and we obtain t(loss) < t(gain) for all the cases we have considered. Finally, we show that the fast loss of energy occurs with greater probability in the strain-dominated region than in the vortical one; in contrast, the slow gain in the energy of the particles is equally likely in vortical or strain-dominated regions of the flow.

We obtain the probability distribution functions (PDFs) of the time that a Lagrangian tracer or a heavy inertial particle spends in vortical or strain-dominated regions of a turbulent flow, by carrying out direct numerical simulations of such particles advected by statistically steady, homogeneous, and isotropic turbulence in the forced, three-dimensional, incompressible Navier-Stokes equation. We use the two invariants, Q and R, of the velocity-gradient tensor to distinguish between vortical and strain-dominated regions of the flow and partition the Q-R plane into four different regions depending on the topology of the flow; out of these four regions two correspond to vorticity-dominated regions of the flow and two correspond to strain-dominated ones. We obtain Q and R along the trajectories of tracers and heavy inertial particles and find out the time t(pers) for which they remain in one of the four regions of the Q-R plane. We find that the PDFs of tpers display exponentially decaying tails for all four regions for tracers and heavy inertial particles. From these PDFs we extract characteristic time scales, which help us to quantify the time that such particles spend in vortical or strain-dominated regions of the flow.

We perform direct numerical simulations of a bidisperse suspension of heavy spherical particles in forced, homogeneous, and isotropic three-dimensional turbulence. We compute the joint distribution of relative particle distances and longitudinal relative velocities between particles of different inertia. For a pair of particles with small difference in their inertias we compare our results with recent theoretical predictions [Meibohm et al., Phys. Rev. E 96, 061102 (2017)] for the shape of this distribution. We also compute the moments of relative velocities as a function of particle separation and compare with the theoretical predictions. We observe good agreement. For a pair of particles that are very different from each other-one is heavy and the other one has negligible inertia-we give a theory to calculate their root-mean-square relative velocity. This theory also agrees well with the results of our simulations.

We use direct numerical simulations to calculate the joint probability density function of the relative distance R and relative radial velocity component V-R for a pair of heavy inertial particles suspended in homogeneous and isotropic turbulent flows. At small scales the distribution is scale invariant, with a scaling exponent that is related to the particle-particle correlation dimension in phase space, D-2. It was argued [K. Gustavsson and B. Mehlig, Phys. Rev. E 84, 045304 (2011); J. Turbul. 15, 34 (2014)] that the scale invariant part of the distribution has two asymptotic regimes: (1) vertical bar V-R vertical bar << R, where the distribution depends solely on R, and (2) vertical bar V-R vertical bar >> R, where the distribution is a function of vertical bar V-R vertical bar alone. The probability distributions in these two regimes are matched along a straight line: vertical bar V-R vertical bar = z*R. Our simulations confirm that this is indeed correct. We further obtain D-2 and z* as a function of the Stokes number, St. The former depends nonmonotonically on St with aminimum at about St approximate to 0.7 and the latter has only a weak dependence on St.

We study small-scale and high-frequency turbulent fluctuations in three-dimensional flows under Fourier-mode reduction. The Navier-Stokes equations are evolved on a restricted set of modes, obtained as a projection on a fractal or homogeneous Fourier set. We find a strong sensitivity (reduction) of the high-frequency variability of the Lagrangian velocity fluctuations on the degree of mode decimation, similarly to what is already reported for Eulerian statistics. This is quantified by a tendency towards a quasi-Gaussian statistics, i.e., to a reduction of intermittency, at all scales and frequencies. This can be attributed to a strong depletion of vortex filaments and of the vortex stretching mechanism. Nevertheless, we found that Eulerian and Lagrangian ensembles are still connected by a dimensional bridge-relation which is independent of the degree of Fourier-mode decimation.

We present an overview of the statistical properties of turbulence in two-dimensional (2D) fluids. After a brief recapitulation of well-known results for statistically homogeneous and isotropic 2D fluid turbulence, we give an overview of recent progress in this field for such 2D turbulence in conducting fluids, fluids with polymer additives, binary-fluid mixtures, and superfluids; we also discuss the statistical properties of particles advected by 2D turbulent fluids.