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

Toward a phenomenological approach to the clustering of heavy particles in turbulent flows

A simple model accounting for the ejection of heavy particles from the vortical structures of a turbulent flow is introduced. This model involves a space and time discretization of the dynamics and depends on only two parameters: the fraction of space-time occupied by rotating structures of the carrier flow and the rate at which particles are ejected from them. The latter can be heuristically related to the response time of the particles and hence measure their inertia. It is shown that such a model reproduces qualitatively most aspects of the spatial distribution of heavy particles transported by realistic flows. In particular the probability density function of the mass $m$ in a cell displays an power-law behavior at small values and decreases faster than exponentially at large values. The dependence of the exponent of the first tail upon the parameters of the dynamics is explicitly derived for the model. The right tail is shown to decrease as $\exp (-C m \log m)$. Finally, the distribution of mass averaged over several cells is shown to obey rescaling properties as a function of the coarse-grain size and of the ejection rate of the particles. Contrarily to what has been observed in direct numerical simulations of turbulent flows (Bec et al., http://arxiv.org/nlin.CD/0608045), such rescaling properties are only due in the model to the mass dynamics of the particles and do not involve any scaling properties in the spatial structure of the carrier flow.Comment: 16 pages, 8 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

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

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

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

Term Structure and Cyclicity of Value-at-Risk: Consequences for the Solvency Capital Requirement

This paper explores empirically the link between French equities returns Value-at-Risk (VaR) and the state of financial markets cycle. The econometric analysis is based on a simple vector autoregression setup. Using quarterly data from 1970Q4 to 2008Q3, it turns out that the k-year VaR of French equities is strongly dependent on the cycle phase: the expected losses as measured by the VaR are twice smaller in recession times than expansion periods. These results strongly suggest that the European rules regarding the solvency capital requirements for insurance companies should adapt to the state of the financial market’s cycle. To this end, we propose a cycle-dependent measure of the Solvency Capital Requirement.expected equities returns, Value at Risk, investment horizon, vector auto-regression