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 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.

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 $R$:

$$\ddot{R}=-\frac{1}{\tau }[\dot{R}-\delta u(R,t)]\text{,}$$

where dots denote time derivatives and $\tau$ the particle response time. The fluid velocity difference $\delta u$ is a Gaussian vector field with correlation

$$〈\delta u^i(r,t)\delta u^j(r^{\prime },t^{\prime })〉=2b^{ij}(r-r^{\prime })\delta (t-t^{\prime })\text{.}$$

In order to model turbulent flows, the tensorial structure of the spatial correlation $b^{ij}\left(r\right)$ is chosen to ensure incompressibility, isotropy and scale invariance, namely

$$b^{ij}\left(r\right)=D_1{r}^{2h}[(d-1+2h){\delta }^{ij}-2h{r}^i{r}^j/r^2]\text{,}$$

where $h$ relates to the Hölder exponent of the fluid velocity field and $D_1$ 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.

The above depicted model flow has the advantage that the particle dynamics is a Markov process. In particular, Gaussianity and $\delta$-correlation in time of the fluid velocity field imply that the probability density $p(\mathrm{r,v},t|r_0,v_0,t_0)$ of finding the particles at separation $R\left(t\right)=r$ and with relative velocity $\dot{R}\left(t\right)=v$ at time $t$, when $R\left(t_0\right)=r_0$ and $\dot{R}\left(t_0\right)=v_0$ is a solution of the Fokker–Planck equation

$${\partial }_{t}p+\sum _i(\underset{r}{\overset{i}{\partial }}-\frac{1}{\tau }{\partial }_v^i)\left(v^{i}p\right)-\sum _{i,j}\frac{b^{ij}\left(r\right)}{{\tau }^2}\underset{v}{\overset{i}{\partial }}\underset{v}{\overset{j}{\partial }}p=0\text{,}$$

with the initial condition $p(r,v,t_0)=\delta (r-r_0)\delta (v-v_0)$. 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 $L$.

For smooth flows ($h=1$), the intensity of inertia is generally measured by the Stokes number $\mathrm{St}$, defined as the ratio between the particle response time $\tau$ and the fluid characteristic time scale. For $\mathrm{St}\to 0$, particles recover the incompressible dynamics of tracers. In the opposite limit where $\mathrm{St}$ 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 $\tau$ by the typical fluid velocity gradient, i.e. $\mathrm{St}=D_1\tau$. Note that by rescaling the physical time by $\tau$, it is straightforward to recognize that the dynamics depends solely on $\mathrm{St}$.

Similarly it can be checked that in rough flows ($h<1$)–with an additional rescaling of the distances by a factor ${\left(D_1\tau \right)}^{1/(2-2h)}$–the dynamics of a particle pair at a distance $r$ only depends on the local Stokes number $\mathrm{St}\left(r\right)=D_1\tau /r^{2(1-h)}$. 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, $\mathrm{St}\left(r\right)\to 0$ and inertia becomes negligible. Particle dynamics thus approaches that of tracers. At small scales, $\mathrm{St}\left(r\right)\to \infty$ 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 $\mathrm{St}\left(r\right)$.

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.