Stochastic suspensions of heavy particles
Introduction
The current understanding of passive turbulent transport profited significantly from studies of the advection by random fields. In particular, flows belonging to the so-called Kraichnan ensemble–i. e. spatially self-similar Gaussian velocity fields with no time correlation–which was first introduced in the late 1960s by Kraichnan [1], led in the mid-1990s to a first analytical description of anomalous scaling in turbulence (see [2] for a review). More recently, much work is devoted to a generalization of this passive advection to heavy particles that, conversely to tracers, do not follow the flow exactly but lag behind it due to their inertia. The particle dynamics is thus dissipative even if the carrier flow is incompressible. This paper provides an overview of several recent results on the dynamics of very heavy particles suspended in random flows belonging to the Kraichnan ensemble.
The recent shift of focus to the transport of heavy particles is motivated by the fact that in many natural and industrial flows finite-size and mass effects of the suspended particles cannot be neglected. Important applications encompass rain formation [3], [4], [5] and suspensions of biological organisms in the ocean [6], [7], [8]. For practical purposes, the formation of particle clusters due to inertia is of central importance as the presence of such inhomogeneities significantly enhances interactions between the suspended particles. However, detailed and reliable predictions on collision or reaction rates, which are crucial to many applications, are still missing.
Two mechanisms compete in the formation of clusters. First, particles much denser than the fluid are ejected from the eddies of the carrier flow and concentrate in the strain-dominated regions [9]. Second, the dissipative dynamics leads the particle trajectories to converge onto a fractal, dynamically evolving attractor [10], [11]. In many studies, a carrier velocity field with no time correlation–and thus no persistent structures–is used to isolate the latter effect. As interactions between three or more particles are usually subdominant, most of the interesting features of monodisperse suspensions can be captured by focusing on the relative motion of two particles separated by : where dots denote time derivatives and the particle response time. The fluid velocity difference is a Gaussian vector field with correlation In order to model turbulent flows, the tensorial structure of the spatial correlation is chosen to ensure incompressibility, isotropy and scale invariance, namely where relates to the Hölder exponent of the fluid velocity field and measures the intensity of its fluctuations. In particular, corresponds to a spatially differentiable velocity field, mimicking the dissipative range of a turbulent flow, while models rough flows as in the inertial range of turbulence. In this paper we mostly focus on space dimensions and ; extensions to higher dimensions are just sketched.
The above depicted model flow has the advantage that the particle dynamics is a Markov process. In particular, Gaussianity and -correlation in time of the fluid velocity field imply that the probability density of finding the particles at separation and with relative velocity at time , when and is a solution of the Fokker–Planck equation with the initial condition . To maintain a statistical steady state, the Fokker–Planck equation (4) as well as the stochastic differential equation (1) should be supplemented by boundary conditions, here chosen to be reflective at a given distance .
For smooth flows (), the intensity of inertia is generally measured by the Stokes number , defined as the ratio between the particle response time and the fluid characteristic time scale. For , particles recover the incompressible dynamics of tracers. In the opposite limit where is very large, inertia effects dominate and the dynamics approaches that of free particles. In the above depicted model, the Stokes number is defined by nondimensionalizing by the typical fluid velocity gradient, i.e. . Note that by rescaling the physical time by , it is straightforward to recognize that the dynamics depends solely on .
Similarly it can be checked that in rough flows ()–with an additional rescaling of the distances by a factor –the dynamics of a particle pair at a distance only depends on the local Stokes number . This dimensionless quantity, first introduced in [12] and later used in [13], is a generalization of the Stokes number to cases in which the fluid turnover times depend on the observation scale. At large scales, and inertia becomes negligible. Particle dynamics thus approaches that of tracers. At small scales, and the particle and fluid motions decorrelate, so that the inertial particles move ballistically. In both the large and small Stokes number asymptotics, particles distribute uniformly in space, while inhomogeneities are expected at intermediate values of .
The paper is organized as follows. In Section 2, an approach originally proposed in [14] is used to reduce the dynamics of the particle separation to a system of three stochastic equations with additive noises. This formulation is useful for both numerical and analytical purposes, particularly when studying the statistical properties of particle pairs. In Section 3, we introduce the correlation dimension to quantify clustering as well as the approaching rate which measures collisions. Numerical results for these quantities are reported. In Section 4 we introduce the notion of time-dependent Stokes number which makes particularly transparent the interpretation of the behaviour of the long-time separation between particles. We show how Richardson dispersion, as for tracers, is recovered in the long-time asymptotics. Section 5 briefly summarizes some exact results that can be obtained for the one-dimensional case. Sections 6 Small Stokes number asymptotics, 7 Large Stokes number asymptotics are dedicated to the small and large Stokes number asymptotics, respectively. In particular, the former presents an original perturbative approach which turned out to predict, in agreement with numerical computations, the behaviour of the correlation dimension that characterizes particle clusters. Finally, Section 8 encompasses conclusions, open questions and discusses the relevance of the considered model for real suspensions in turbulent flows.
Section snippets
Reduced dynamics for the two-point motion
In this Section we focus on planar suspensions (). Following the approach proposed in [14] and with the notation , the change of variables is introduced to reduce the original system of stochastic equations to the following one of only three equations: where and denote two independent white noises. Reflective boundary conditions at in
Correlation dimension and approaching rate
Particle clustering is often quantified by the radial distribution function , which is defined as the ratio between the number of particles inside a thin shell of radius centred on a given particle and the number which would be in this shell if the particles were uniformly distributed. This quantity enters models for the collision kernel [17]. Following [10], [13], [16], [18], we consider a different, but related way to characterize particle clustering. Instead of the radial distribution
Stretching rate and relative dispersion
This section is devoted to the study of the behaviour of the distance between two particles at intermediate times such that . For convenience, we drop the reflective boundary condition at and consider particles evolving in an unbounded domain.
We first consider a differentiable fluid velocity field (). In this case, the time evolution of the distance is given by (11), so that and the particle separation can be measured by the stretching
Exact results in one dimension
A number of analytical results were derived for one-dimensional flows [29], [30], [31]. Although such flows are always compressible, their study helps improving the intuition for the dynamics of inertial particles in higher-dimensional random flows. In particular, several results on caustic formation hold also in two-dimensional (incompressible) flows because the typical velocity fluctuations, which lead to caustic formation, are effectively one-dimensional.
Here, we focus on one-dimensional
Small Stokes number asymptotics
This section reports some asymptotic results related to the limit of small particle inertia. The first part summarizes the approach developed by Mehlig, Wilkinson, and collaborators for differentiable flows (). In analogy to the WKB approximation in quantum mechanics (see, e.g. [32]), the authors construct perturbatively the steady solution to the Fokker–Planck equation associated to the reduced system (8), (9). In the second part of this section original results are reported where the
Large Stokes number asymptotics
Particles with huge inertia () take an infinite time to relax to the velocity of the carrier fluid. They become therefore uncorrelated with the underlying flow and evolve with ballistic dynamics, moving freely and maintaining, almost unchanged, their initial velocities. This limit is particularly appealing for deriving asymptotic theories [16]. In this section, we focus on two aspects, namely the problem of the recovery of homogeneous/uniform distribution for and the problem of the
Remarks and conclusions
Before concluding this paper the results discussed so far are commented in the light of what is known about real turbulent suspensions, which are relevant to most applications. Let us start by recalling the main features of turbulent flows. Turbulence is a multiscale phenomenon [46] which spans length scales ranging from a large (energy injection) scale to the very small (dissipative) scale , often called the Kolmogorov scale. This hierarchy of length scales is associated with a hierarchy of
Acknowledgments
We acknowledge useful discussions with S. Musacchio and M. Wilkinson. Part of this work was done while K.T. was visiting Lab. Cassiopée in the framework of the ENS-Landau exchange programme. J.B. was partially supported by ANR “DSPET” BLAN07-1_192604.
References (52)
- et al.
Turbulence effects on droplet growth and size distribution in clouds— a review
J. Aerosol. Sci.
(1997) - et al.
Preferential concentration of particles by turbulence
Int. J. Multiphase Flow
(1994) - et al.
Heavy particles in incompressible flows: The large Stokes number asymptotics
Physica D
(2007) - et al.
Anomalous scaling laws in multifractal objects
Phys. Rep.
(1987) Aggregation-disorder transition induced by random forces
J. Phys. A
(1985)Small-scale structure of a scalar field convected by turbulence
Phys. Fluids
(1968)- et al.
Particles and fields in fluid turbulence
Rev. Modern Phys.
(2001) - et al.
Microphysics of Clouds and Precipitation
(1996) - et al.
Acceleration of rain initiation by cloud turbulence
Nature
(2002) - et al.
Small-scale turbulence and plankton contact rates
J. Plankton Res.
(1988)
Feeding conditions of Arcto-Norwegian cod larvae compared with the Rothschild–Osborn theory on small-scale turbulence and plankton contact rates
J. Plankton Res.
Predator–prey encounters in turbulent waters
Phys. Rev. E
Stokes and Reynolds number dependence of preferential particle concentration in simulated three-dimensional turbulence
Phys. Fluids
Fractal clustering of inertial particles in random flows
Phys. Fluids
Clustering of heavy particles in random self-similar flow
Phys. Rev. E
The top Lyapunov exponent for stochastic flow modeling the upper ocean turbulence
SIAM J. Appl. Math.
Clustering by mixing flows
Phys. Rev. Lett.
Effect of preferential concentration on turbulent collision rates
Phys. Fluids
Clustering and collisions of heavy particles in random smooth flows
Phys. Fluids
Ergodic theory of chaos and strange attractors
Rev. Modern Phys.
Are the dimensions of a set and its image equal under typical smooth functions?
Ergodic Theory Dynam. Syst.
On the collision rate of small particles in isotropic turbulence. Part I. Zero-inertia case
Phys. Fluids
On the collision rate of small particles in isotropic turbulence. Part II. Finite-inertia case
Phys. Fluids
Coagulation by random velocity fields as a Kramers problem
Phys. Rev. Lett.
Aggregation of inertial particles in random flows
Phys. Rev. E
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