Effect of turbulence on collisions of dust particles with planetesimals in protoplanetary disks

Planetesimals in gaseous protoplanetary disks may grow by collecting dust particles. Hydrodynamical studies show that small particles generally avoid collisions with the planetesimals because they are entrained by the flow around them. This occurs when $St$, the Stokes number, defined as the ratio of the dust stopping time to the planetesimal crossing time, becomes much smaller than unity. However, these studies have been limited to the laminar case, whereas these disks are believed to be turbulent. We want to estimate the influence of gas turbulence on the dust-planetesimal collision rate and on the impact speeds. We used three-dimensional direct numerical simulations of a fixed sphere (planetesimal) facing a laminar and turbulent flow seeded with small inertial particles (dust) subject to a Stokes drag. A no-slip boundary condition on the planetesimal surface is modeled via a penalty method. We find that turbulence can significantly increase the collision rate of dust particles with planetesimals. For a high turbulence case (when the amplitude of turbulent fluctuations is similar to the headwind velocity), we find that the collision probability remains equal to the geometrical rate or even higher for $St\geq 0.1$, i.e., for dust sizes an order of magnitude smaller than in the laminar case. We derive expressions to calculate impact probabilities as a function of dust and planetesimal size and turbulent intensity

Universality of Velocity Gradients in Forced Burgers Turbulence

It is demonstrated that Burgers turbulence subject to large-scale white-noise-in-time random forcing has a universal power-law tail with exponent -7/2 in the probability density function of negative velocity gradients, as predicted by E, Khanin, Mazel and Sinai (1997, Phys. Rev. Lett. 78, 1904). A particle and shock tracking numerical method gives about five decades of scaling. Using a Lagrangian approach, the -7/2 law is related to the shape of the unstable manifold associated to the global minimizer.Comment: 4 pages, 2 figures, RevTex4, published versio

Ergodic properties of a model for turbulent dispersion of inertial particles

We study a simple stochastic differential equation that models the dispersion of close heavy particles moving in a turbulent flow. In one and two dimensions, the model is closely related to the one-dimensional stationary Schroedinger equation in a random delta-correlated potential. The ergodic properties of the dispersion process are investigated by proving that its generator is hypoelliptic and using control theory

Caustics in turbulent aerosols

Networks of caustics can occur in the distribution of particles suspended in a randomly moving gas. These can facilitate coagulation of particles by bringing them into close proximity, even in cases where the trajectories do not coalesce. We show that the long-time morphology of these caustic patterns is determined by the Lyapunov exponents lambda_1, lambda_2 of the suspended particles, as well as the rate J at which particles encounter caustics. We develop a theory determining the quantities J, lambda_1, lambda_2 from the statistical properties of the gas flow, in the limit of short correlation times.Comment: 4 pages, 3 figure

Dynamics and statistics of heavy particles in turbulent flows

We present the results of Direct Numerical Simulations (DNS) of turbulent flows seeded with millions of passive inertial particles. The maximum Taylor's Reynolds number is around 200. We consider particles much heavier than the carrier flow in the limit when the Stokes drag force dominates their dynamical evolution. We discuss both the transient and the stationary regimes. In the transient regime, we study the growt of inhomogeneities in the particle spatial distribution driven by the preferential concentration out of intense vortex filaments. In the stationary regime, we study the acceleration fluctuations as a function of the Stokes number in the range [0.16:3.3]. We also compare our results with those of pure fluid tracers (St=0) and we find a critical behavior of inertia for small Stokes values. Starting from the pure monodisperse statistics we also characterize polydisperse suspensions with a given mean Stokes.Comment: 13 pages, 10 figures, 2 table

Transition from ergodic to explosive behavior in a family of stochastic differential equations

We study a family of quadratic stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, we find a critical parameter value $\alpha_{1}=\alpha_{2}$ such that when $\alpha_{2}>\alpha_{1}$ the system is ergodic and when $\alpha_{2}<\alpha_{1}$ solutions are not defined for all times. H\"{o}rmander's hypoellipticity theorem and geometric control theory are also utilized.Comment: 35 pages, 6 figure

Anisotropic clustering of inertial particles in homogeneous shear flow

Recently, clustering of inertial particles in turbulence has been thoroughly analyzed for statistically homogeneous isotropic flows. Phenomenologically, spatial homogeneity of particles configurations is broken by the advection of a range of eddies determined by the Stokes relaxation time of the particles which results in a multi-scale distribution of local concentrations and voids. Much less is known concerning anisotropic flows. Here, by addressing direct numerical simulations (DNS) of a statistically steady particle-laden homogeneous shear flow, we provide evidence that the mean shear preferentially orients particle patterns. By imprinting anisotropy on large scales velocity fluctuations, the shear indirectly affects the geometry of the clusters. Quantitative evaluation is provided by a purposely designed tool, the angular distribution function of particle pairs (ADF), which allows to address the anisotropy content of particles aggregates on a scale by scale basis. The data provide evidence that, depending on the Stokes relaxation time of the particles, anisotropic clustering may occur even in the range of scales where the carrier phase velocity field is already recovering isotropy. The strength of the singularity in the anisotropic component of the ADF quantifies the level of fine scale anisotropy, which may even reach values of more than 30% direction-dependent variation in the probability to find two close-by particles at viscous scale separation.Comment: To appear in Journal Fluid Mechanics 200

Superdiffusion of massive particles induced by multi-scale velocity fields

We study drag-induced diffusion of massive particles in scale-free velocity fields, where superdiffusive behavior emerges due to the scale-free size distribution of the vortices of the underlying velocity field. The results show qualitative resemblance to what is observed in fluid systems, namely the diffusive exponent for the mean square separation of pairs of particles and the preferential concentration of the particles, both as a function of the response time.Comment: 5 pages, 5 figures. Accepted for publication in EP