Volume 237, Issues 14–17, 15 August 2008, Pages 2037–2050

Euler Equations: 250 Years On — Proceedings of an international conference

Edited By Gregory Eyink, Uriel Frisch, René Moreau and Andreı˘ Sobolevskiı˘

Stochastic suspensions of heavy particles

  • a Laboratoire Cassiopée, Observatoire de la Côte d’Azur, CNRS, Université de Nice Sophia-Antipolis, Bd. de l’Observatoire, 06300 Nice, France
  • b SMC INFM-CNR c/o Dip. di Fisica Università di Roma “La Sapienza”, Piazzale A. Moro 2, 00185 Roma, Italy
  • c CNR, Istituto dei Sistemi Complessi, Via dei Taurini 19, 00185 Roma, Italy
  • d The Future of Humanity Institute, University of Oxford, Suite 8, Littlegate House 16/17, St Ebbe’s Street, Oxford, OX1 1PT, United Kingdom
  • e James Franck Institute, University of Chicago, Chicago, IL 60637, USA
  • f Landau Institute for Theoretical Physics, Moscow, Kosygina 2, 119334, Russia

Abstract

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 dynamic 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.

PACS

  • 47.27.-i;
  • 47.51.+a;
  • 47.55.-t

Keywords

  • Stochastic flows;
  • Inertial particles;
  • Kraichnan model;
  • Lyapunov exponent

1. 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] and [5] and suspensions of biological organisms in the ocean [6], [7] and [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] and [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 View the MathML source:

equation1
View the MathML source
where dots denote time derivatives and τ the particle response time. The fluid velocity difference View the MathML source is a Gaussian vector field with correlation
equation2
View the MathML source
In order to model turbulent flows, the tensorial structure of the spatial correlation View the MathML source is chosen to ensure incompressibility, isotropy and scale invariance, namely
equation3
View the MathML source
where h relates to the Hölder exponent of the fluid velocity field and D1 measures the intensity of its fluctuations. In particular, h=1 corresponds to a spatially differentiable velocity field, mimicking the dissipative range of a turbulent flow, while h<1 models rough flows as in the inertial range of turbulence. In this paper we mostly focus on space dimensions d=1 and d=2; extensions to higher dimensions are just sketched.