Multifractal concentrations of inertial particles in smooth random flows

Collisionless suspensions of inertial particles (finite-size impurities) are studied in 2D and 3D spatially smooth flows. Tools borrowed from the study of random dynamical systems are used to identify and to characterise in full generality the mechanisms leading to the formation of strong inhomogeneities in the particle concentration. Phenomenological arguments are used to show that in 2D, heavy particles form dynamical fractal clusters when their Stokes number (non-dimensional viscous friction time) is below some critical value. Numerical simulations provide strong evidence for this threshold in both 2D and 3D and for particles not only heavier but also lighter than the carrier fluid. In 2D, light particles are found to cluster at discrete (time-dependent) positions and velocities in some range of the dynamical parameters (the Stokes number and the mass density ratio between fluid and particles). This regime is absent in 3D, where evidence is that the Hausdorff dimension of clusters in phase space (position-velocity) remains always above two. After relaxation of transients, the phase-space density of particles becomes a singular random measure with non-trivial multiscaling properties. Theoretical results about the projection of fractal sets are used to relate the distribution in phase space to the distribution of the particle positions. Multifractality in phase space implies also multiscaling of the spatial distribution of the mass of particles. Two-dimensional simulations, using simple random flows and heavy particles, allow the accurate determination of the scaling exponents: anomalous deviations from self-similar scaling are already observed for Stokes numbers as small as $10^{-4}$.Comment: 21 pages, 13 figure

Fractal clustering of inertial particles in random flows

It is shown that preferential concentrations of inertial (finite-size) particle suspensions in turbulent flows follow from the dissipative nature of their dynamics. In phase space, particle trajectories converge toward a dynamical fractal attractor. Below a critical Stokes number (non-dimensional viscous friction time), the projection on position space is a dynamical fractal cluster; above this number, particles are space filling. Numerical simulations and semi-heuristic theory illustrating such effects are presented for a simple model of inertial particle dynamics.Comment: 4 pages, 4 figures, Physics of Fluids, in pres

Forced Burgers Equation in an Unbounded Domain

The inviscid Burgers equation with random and spatially smooth forcing is considered in the limit when the size of the system tends to infinity. For the one-dimensional problem, it is shown both theoretically and numerically that many of the features of the space-periodic case carry over to infinite domains as intermediate time asymptotics. In particular, for large time $T$ we introduce the concept of $T$-global shocks replacing the notion of main shock which was considered earlier in the periodic case (1997, E et al., Phys. Rev. Lett. 78, 1904). In the case of spatially extended systems these objects are no anymore global. They can be defined only for a given time scale and their spatial density behaves as $\rho(T) \sim T^{-2/3}$ for large $T$. The probability density function $p(A)$ of the age $A$ of shocks behaves asymptotically as $A^{-5/3}$. We also suggest a simple statistical model for the dynamics and interaction of shocks and discuss an analogy with the problem of distribution of instability islands for a simple first-order stochastic differential equation.Comment: 9 pages, 10 figures, revtex4, J. Stat. Phys, in pres

Burgers Turbulence

The last decades witnessed a renewal of interest in the Burgers equation. Much activities focused on extensions of the original one-dimensional pressureless model introduced in the thirties by the Dutch scientist J.M. Burgers, and more precisely on the problem of Burgers turbulence, that is the study of the solutions to the one- or multi-dimensional Burgers equation with random initial conditions or random forcing. Such work was frequently motivated by new emerging applications of Burgers model to statistical physics, cosmology, and fluid dynamics. Also Burgers turbulence appeared as one of the simplest instances of a nonlinear system out of equilibrium. The study of random Lagrangian systems, of stochastic partial differential equations and their invariant measures, the theory of dynamical systems, the applications of field theory to the understanding of dissipative anomalies and of multiscaling in hydrodynamic turbulence have benefited significantly from progress in Burgers turbulence. The aim of this review is to give a unified view of selected work stemming from these rather diverse disciplines.Comment: Review Article, 49 pages, 43 figure

Finite-size effects in the dynamics of neutrally buoyant particles in turbulent flow

The dynamics of neutrally buoyant particles transported by a turbulent flow is investigated for spherical particles with radii of the order of the Kolmogorov dissipative scale or larger. The pseudo-penalisation spectral method that has been proposed by Pasquetti et al. (2008) is adapted to integrate numerically the simultaneous dynamics of the particle and of the fluid. Such a method gives a unique handle on the limit of validity of point-particle approximations, which are generally used in applicative situations. Analytical predictions based on such models are compared to result of very well resolved direct numerical simulations. Evidence is obtained that Faxen corrections give dominant finite-size corrections to velocity and acceleration fluctuations for particle diameters up to four times the Kolmogorov scale. The dynamics of particles with larger diameters is dominated by inertial-range physics, and is consistent with predictions obtained from dimensional analysis.Comment: 10 pages, 5 figure

Stochastic suspensions of heavy particles

Turbulent suspensions of heavy particles in incompressible flows have gained much attention in recent years. A large amount of work focused on the impact that the inertia and the dissipative dynamics of the particles have on their dynamical and statistical properties. Substantial progress followed from the study of suspensions in model flows which, although much simpler, reproduce most of the important mechanisms observed in real turbulence. This paper presents recent developments made on the relative motion of a pair of particles suspended in time-uncorrelated and spatially self-similar Gaussian flows. This review is complemented by new results. By introducing a time-dependent Stokes number, it is demonstrated that inertial particle relative dispersion recovers asymptotically Richardson's diffusion associated to simple tracers. A perturbative (homogeneization) technique is used in the small-Stokes-number asymptotics and leads to interpreting first-order corrections to tracer dynamics in terms of an effective drift. This expansion implies that the correlation dimension deficit behaves linearly as a function of the Stokes number. The validity and the accuracy of this prediction is confirmed by numerical simulations.Comment: 15 pages, 12 figure

Assets Returns Volatility and Investment Horizon: The French Case

This paper explores French assets returns predictability within a VAR setup. Using quarterly data from 1970Q4 to 2006Q4, it turns out that bonds, equities and bills returns are actually predictable. This feature implies that the investment horizon does indeed matter in the asset allocation. The VAR parameters estimates are then used to compute real returns conditional volatility across investment horizons. The results reveal the same kind of horizon effect as the one found in recent empirical studies using quarterly U.S. data. More specifically, the excess annualized standard deviation of French stocks returns with respect to bills and bonds returns decreases as the investment horizon grows. They suggest that long-horizon investors overstate the share of bonds in their portfolio choice when neglecting the horizon effect on risk of asset returns predictability.asset return predictability, investment horizon, vector autoregression

Statistical steady state in turbulent droplet condensation

Motivated by systems in which droplets grow and shrink in a turbulence-driven supersaturation field, we investigate the problem of turbulent condensation in a general manner. Using direct numerical simulations we show that the turbulent fluctuations of the supersaturation field offer different conditions for the growth of droplets which evolve in time due to turbulent transport and mixing. Based on that, we propose a Lagrangian stochastic model for condensation and evaporation of small droplets in turbulent flows. It consists of a set of stochastic integro-differential equations for the joint evolution of the squared radius and the supersaturation along the droplet trajectories. The model has two parameters fixed by the total amount of water and the thermodynamic properties, as well as the Lagrangian integral timescale of the turbulent supersaturation. The model reproduces very well the droplet size distributions obtained from direct numerical simulations and their time evolution. A noticeable result is that, after a stage where the squared radius simply diffuses, the system converges exponentially fast to a statistical steady state independent of the initial conditions. The main mechanism involved in this convergence is a loss of memory induced by a significant number of droplets undergoing a complete evaporation before growing again. The statistical steady state is characterised by an exponential tail in the droplet mass distribution. These results reconcile those of earlier numerical studies, once these various regimes are considered.Comment: 24 pages, 12 figure