Distribution of velocity gradients and rate of caustic formation in turbulent aerosols at finite Kubo numbers

Rate of formation of caustics in heavy particles advected by turbulence

The rate of collision and the relative velocities of the colliding particles in turbulent flows are a crucial part of several natural phenomena, e.g. rain formation in warm clouds and planetesimal formation in protoplanetary discs. The particles are often modelled as passive, but heavy and inertial. Within this model, large relative velocities emerge due to formation of singularities (caustics) of the gradient matrix of the velocities of the particles. Using extensive direct numerical simulations of heavy particles in both two (direct and inverse cascade) and three-dimensional turbulent flows, we calculate the rate of formation of caustics, [Formula: see text] as a function of the Stokes number ([Formula: see text]). The best approximation to our data is [Formula: see text], in the limit [Formula: see text] where [Formula: see text] is a non-universal constant. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.

Heavy particles in a persistent random flow with traps

We study a one-dimensional model for heavy particles in a compressible fluid. The fluid-velocity field is modeled by a persistent Gaussian random function, and the particles are assumed to be weakly inertial. Since one-dimensional fluid-velocity fields are always compressible, the model exhibits spatial trapping regions where particles tend to accumulate. We determine the statistics of fluid-velocity gradients in the vicinity of these traps and show how this allows one to determine the spatial Lyapunov exponent and the rate of caustic formation. We compare our analytical results with numerical simulations of the model and explore the limits of validity of the theory. Finally, we discuss implications for higher-dimensional systems.

Relative velocities in bidisperse turbulent suspensions

We investigate the distribution of relative velocities between small heavy particles of different sizes in turbulence by analyzing a statistical model for bidisperse turbulent suspensions, containing particles with two different Stokes numbers. This number, St, is a measure of particle inertia which in turn depends on particle size. When the Stokes numbers are similar, the distribution exhibits power-law tails, just as in the case of equal St. The power-law exponent is a nonanalytic function of the mean Stokes number St[over ¯], so that the exponent cannot be calculated in perturbation theory around the advective limit. When the Stokes-number difference is larger, the power law disappears, but the tails of the distribution still dominate the relative-velocity moments, if St[over ¯] is large enough.

Statistical model for collisions and recollisions of inertial particles in mixing flows

Finding a quantitative description of the rate of collisions between small particles suspended in mixing flows is a long-standing problem. Here we investigate the validity of a parameterisation of the collision rate for identical particles subject to Stokes force, based on results for relative velocities of heavy particles that were recently obtained within a statistical model for the dynamics of turbulent aerosols. This model represents the turbulent velocity fluctuations by Gaussian random functions. We find that the parameterisation gives quantitatively good results in the limit where the "ghost-particle approximation" applies. The collision rate is a sum of two contributions due to "caustics" and to "clustering". Within the statistical model we compare the relative importance of these two collision mechanisms. The caustic formation rate is high when the particle inertia becomes large, and we find that caustics dominate the collision rate as soon as they form frequently. We compare the magnitude of the caustic contribution to the collision rate to the formation rate of caustics.

Abrupt growth of large aggregates by correlated coalescences in turbulent flow

Smoluchowski's coagulation kinetics is here shown to fail when the coalescing species are dilute and transported by a turbulent flow. The intermittent Lagrangian motion involves correlated violent events that lead to an unexpected rapid occurrence of the largest particles. This new phenomena is here quantified in terms of the anomalous scaling of turbulent three-point motion, leading to significant corrections in macroscopic processes that are critically sensitive to the early-stage emergence of large embryonic aggregates, as in planet formation or rain precipitation.

Preferential Sampling and Small-Scale Clustering of Gyrotactic Microswimmers in Turbulence

Recent studies show that spherical motile microorganisms in turbulence subject to gravitational torques gather in down-welling regions of the turbulent flow. By analyzing a statistical model we analytically compute how shape affects the dynamics, preferential sampling, and small-scale spatial clustering. We find that oblong organisms may spend more time in up-welling regions of the flow, and that all organisms are biased to regions of positive fluid-velocity gradients in the upward direction. We analyze small-scale spatial clustering and find that oblong particles may either cluster more or less than spherical ones, depending on the strength of the gravitational torques.

Tumbling of small axisymmetric particles in random and turbulent flows

We analyze the tumbling of small nonspherical, axisymmetric particles in random and turbulent flows. We compute the orientational dynamics in terms of a perturbation expansion in the Kubo number, and obtain the tumbling rate in terms of Lagrangian correlation functions. These capture preferential sampling of the fluid gradients, which in turn can give rise to differences in the tumbling rates of disks and rods. We show that this is a weak effect in Gaussian random flows. But in turbulent flows persistent regions of high vorticity cause disks to tumble much faster than rods, as observed in direct numerical simulations [S. Parsa, E. Calzavarini, F. Toschi, and G. A. Voth, Phys. Rev. Lett. 109, 134501 (2012)]. For larger particles (at finite Stokes numbers), rotational and translational inertia affects the tumbling rate and the angle at which particles collide, due to the formation of rotational caustics.