22 research outputs found
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 .Comment: 21 pages, 13 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
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
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 . 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
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
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