Periodic Homogenization for Inertial Particles

We study the problem of homogenization for inertial particles moving in a periodic velocity field, and subject to molecular diffusion. We show that, under appropriate assumptions on the velocity field, the large scale, long time behavior of the inertial particles is governed by an effective diffusion equation for the position variable alone. To achieve this we use a formal multiple scale expansion in the scale parameter. This expansion relies on the hypo-ellipticity of the underlying diffusion. An expression for the diffusivity tensor is found and various of its properties studied. In particular, an expansion in terms of the non-dimensional particle relaxation time $\tau$ (the Stokes number) is shown to co-incide with the known result for passive (non-inertial) tracers in the singular limit $\tau \to 0$. This requires the solution of a singular perturbation problem, achieved by means of a formal multiple scales expansion in $\tau.$ Incompressible and potential fields are studied, as well as fields which are neither, and theoretical findings are supported by numerical simulations.Comment: 31 pages, 7 figures, accepted for publication in Physica D. Typos corrected. One reference adde

Multiple-scale analysis and renormalization for pre-asymptotic scalar transport

Pre-asymptotic transport of a scalar quantity passively advected by a velocity field formed by a large-scale component superimposed to a small-scale fluctuation is investigated both analytically and by means of numerical simulations. Exploiting the multiple-scale expansion one arrives at a Fokker--Planck equation which describes the pre-asymptotic scalar dynamics. Such equation is associated to a Langevin equation involving a multiplicative noise and an effective (compressible) drift. For the general case, no explicit expression for both the effective drift and the effective diffusivity (actually a tensorial field) can be obtained. We discuss an approximation under which an explicit expression for the diffusivity (and thus for the drift) can be obtained. Its expression permits to highlight the important fact that the diffusivity explicitly depends on the large-scale advecting velocity. Finally, the robustness of the aforementioned approximation is checked numerically by means of direct numerical simulations.Comment: revtex4, 12 twocolumn pages, 3 eps figure

Interference phenomena in scalar transport induced by a noise finite correlation time

The role played on the scalar transport by a finite, not small, correlation time, $\tau_u$, for the noise velocity is investigated, both analytically and numerically. For small $\tau_u$'s a mechanism leading to enhancement of transport has recently been identified and shown to be dominating for any type of flow. For finite non-vanishing $\tau_u$'s we recognize the existence of a further mechanism associated with regions of anticorrelation of the Lagrangian advecting velocity. Depending on the extension of the anticorrelated regions, either an enhancement (corresponding to constructive interference) or a depletion (corresponding to destructive interference) in the turbulent transport now takes place.Comment: 8 pages, 3 figure

Modeling the Pollution of Pristine Gas in the Early Universe

We conduct a comprehensive theoretical and numerical investigation of the pollution of pristine gas in turbulent flows, designed to provide new tools for modeling the evolution of the first generation of stars. The properties of such Population III (Pop III) stars are thought to be very different than later generations, because cooling is dramatically different in gas with a metallicity below a critical value Z_c, which lies between ~10^-6 and 10^-3 solar value. Z_c is much smaller than the typical average metallicity, , and thus the mixing efficiency of the pristine gas in the interstellar medium plays a crucial role in the transition from Pop III to normal star formation. The small critical value, Z_c, corresponds to the far left tail of the probability distribution function (PDF) of the metallicity. Based on closure models for the PDF formulation of turbulent mixing, we derive equations for the fraction of gas, P, lying below Z_c, in compressible turbulence. Our simulation data shows that the evolution of the fraction P can be well approximated by a generalized self-convolution model, which predicts dP/dt = -n/tau_con P (1-P^(1/n)), where n is a measure of the locality of the PDF convolution and the timescale tau_con is determined by the rate at which turbulence stretches the pollutants. Using a suite of simulations with Mach numbers ranging from M = 0.9 to 6.2, we provide accurate fits to n and tau_con as a function of M, Z_c/, and the scale, L_p, at which pollutants are added to the flow. For P>0.9, mixing occurs only in the regions surrounding the pollutants, such that n=1. For smaller P, n is larger as mixing becomes more global. We show how the results can be used to construct one-zone models for the evolution of Pop III stars in a single high-redshift galaxy, as well as subgrid models for tracking the evolution of the first stars in large cosmological simulations.Comment: 37 pages, accepted by Ap

Particles and fields in fluid turbulence

The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in non-equilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.Comment: 165 pages, review article for Rev. Mod. Phy