Stochastic suspensions of heavy particles

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 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.

Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles

“…Let us list some of them: What about the expressions for the other 2d − 1 Lyapunov exponents (the 2d exponents have to sum to −d/τ [10])? Can one establish the existence of the large deviation regime for the finite-time Lyapunov exponents (the corresponding rate function for the top exponent was numerically studied in d = 2 model in [2,4]; it gives access to more subtle information about the clustering of inertial particles than the top Lyapunov exponent itself)? Is the SDE modeling the inertial particle dispersion in fully developed turbulence, that was introduced and studied numerically in [3,4], amenable to rigorous analysis?…”

“…Moreover, ϕ 4 → ∞ as r → ∞ in X 4 since then x must approach ∞ as r → ∞, and y is bounded above. Dropping insignificant terms in the expression for M ϕ 4 , we see that in X 4 : …”

Intermittency in the velocity distribution of heavy particles in turbulence

“…Indeed as r increases the associated eddy turnover time grows as r 2/3 (where we used the Kolmogorov 1941 scaling (K41)) so that the effective strength of inertia reduces. Similarly to random self-similar carrier flows (see Bec et al 2008), this effect can be defined as the ratio between the particle response time and the turnover time associated to the scale r, where ε denotes the mean dissipation rate of kinetic energy. We check whether the local scaling exponent…”

Inertial-particle dynamics in turbulent flows: caustics, concentration fluctuations and random uncorrelated motion

“…Spatial correlation dimension d 2 (•) as a function of 2 for the model described in section 2.1. The dashed red line shows the small-theory discussed inBec et al (2008) andWilkinson et al (2010).…”