Clustering and collisions of heavy particles in random smooth flows

Finite-size impurities suspended in incompressible flows distribute inhomogeneously, leading to a drastic enhancement of collisions. A description of the dynamics in the full position-velocity phase space is essential to understand the underlying mechanisms, especially for polydisperse suspensions. These issues are here studied for particles much heavier than the fluid by means of a Lagrangian approach. It is shown that inertia enhances collision rates through two effects: correlation among particle positions induced by the carrier flow and uncorrelation between velocities due to their finite size. A phenomenological model yields an estimate of collision rates for particle pairs with different sizes. This approach is supported by numerical simulations in random flows.Comment: 12 pages, 9 Figures (revTeX 4) final published versio

Nonlinear diffusion model for Rayleigh-Taylor mixing

The complex evolution of turbulent mixing in Rayleigh-Taylor convection is studied in terms of eddy diffusiviy models for the mean temperature profile. It is found that a non-linear model, derived within the general framework of Prandtl mixing theory, reproduces accurately the evolution of turbulent profiles obtained from numerical simulations. Our model allows to give very precise predictions for the turbulent heat flux and for the Nusselt number in the ultimate state regime of thermal convection.Comment: 4 pages, 4 figure, PRL in pres

Inertial particles driven by a telegraph noise

We present a model for the Lagrangian dynamics of inertial particles in a compressible flow, where fluid velocity gradients are modelled by a telegraph noise. The model allows for an analytic investigation of the role of time correlation of the flow in the aggregation-disorder transition of inertial particle. The dependence on Stokes and Kubo numbers of the Lyapunov exponent of particle trajectories reveals the presence of a region in parameter space (St, Ku) where the leading Lyapunov exponent changes sign, thus signaling the transition. The asymptotics of short and long-correlated flows are discussed, as well as the fluid-tracer limit.Comment: 8 pages, 6 figure

Lyapunov exponents of heavy particles in turbulence

Lyapunov exponents of heavy particles and tracers advected by homogeneous and isotropic turbulent flows are investigated by means of direct numerical simulations. For large values of the Stokes number, the main effect of inertia is to reduce the chaoticity with respect to fluid tracers. Conversely, for small inertia, a counter-intuitive increase of the first Lyapunov exponent is observed. The flow intermittency is found to induce a Reynolds number dependency for the statistics of the finite time Lyapunov exponents of tracers. Such intermittency effects are found to persist at increasing inertia.Comment: 4 pages, 4 figure

Power-Law Distributions in a Two-sided Market and Net Neutrality

"Net neutrality" often refers to the policy dictating that an Internet service provider (ISP) cannot charge content providers (CPs) for delivering their content to consumers. Many past quantitative models designed to determine whether net neutrality is a good idea have been rather equivocal in their conclusions. Here we propose a very simple two-sided market model, in which the types of the consumers and the CPs are {\em power-law distributed} --- a kind of distribution known to often arise precisely in connection with Internet-related phenomena. We derive mostly analytical, closed-form results for several regimes: (a) Net neutrality, (b) social optimum, (c) maximum revenue by the ISP, or (d) maximum ISP revenue under quality differentiation. One unexpected conclusion is that (a) and (b) will differ significantly, unless average CP productivity is very high

Acceleration statistics of heavy particles in turbulence

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: ({\it i}) The root-mean-squared acceleration $a_{\rm rms}$ sharply falls off from the fluid tracer value already at quite small Stokes numbers; ({\it ii}) At a given $St$ the normalised acceleration $a_{\rm rms}/(\epsilon^3/\nu)^{1/4}$ increases with $R_\lambda$ consistently with the trend observed for fluid tracers; ({\it iii}) The tails of the probability density function of the normalised 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 $St$.Comment: 10 pages, 3 figs, 2 tables. A section with new results has been added. Revised version accepted for pubblication on Journal of Fluid Mechanic

Heavy particle concentration in turbulence at dissipative and inertial scales

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.Comment: 4 RevTeX pgs + 4 color Figures included, 1 figure eliminated second part of the paper completely revise

Small scale statistics of viscoelastic turbulence

The small scale statistics of homogeneous isotropic turbulence of dilute polymer solutions is investigated by means of direct numerical simulations of a simplified viscoelastic fluid model. It is found that polymers only partially suppress the turbulent cascade below the Lumley scale, leaving a remnant energy flux even for large elasticity. As a consequence, fluid acceleration in viscoelastic flows is reduced with respect to Newtonian turbulence, whereas its rescaled probability density is left unchanged. At large scales the velocity field is found to be unaffected by the presence of polymers.Comment: 7 pages, 4 figure

Intermittency in two-dimensional Ekman-Navier-Stokes turbulence

We study the statistics of the vorticity field in two-dimensional Navier-Stokes turbulence with a linear Ekman friction. We show that the small-scale vorticity fluctuations are intermittent, as conjectured by Nam et al. [Phys. Rev. Lett. vol.84 (2000) 5134]. The small-scale statistics of vorticity fluctuations coincides with the one of a passive scalar with finite lifetime transported by the velocity field itself.Comment: 4 pages, 7 figure