The relative dispersion of pairs of inertial particles in incompressible, homogeneous, and isotropic turbulence is studied by means of direct numerical simulations at two values of the Taylor-scale Reynolds number $Re_{\lambda} \sim 200$ and 400. The evolution of both heavy and light particle pairs is analysed at varying the particle Stokes number and the fluid-to-particle density ratio. For heavy particles, it is found that turbulent dispersion is schematically governed by two temporal regimes. The first is dominated by the presence, at large Stokes numbers, of small-scale caustics in the particle velocity statistics, and it lasts until heavy particle velocities have relaxed towards the underlying flow velocities. At such large scales, a second regime starts where heavy particles separate as tracers particles would do. As a consequence, at increasing inertia, a larger transient stage is observed, and the Richardson diffusion of simple tracers is recovered only at large times and large scales. These features also arise from a statistical closure of the equation of motion for heavy particle separation that is proposed, and which is supported by the numerical results. In the case of light particles with high density ratios, strong small-scale clustering leads to a considerable fraction of pairs that do not separate at all, although the mean separation increases with time. This effect strongly alters the shape of the probability density function of light particle separations.
Deep Dive into Turbulent pair dispersion of inertial particles.
The relative dispersion of pairs of inertial particles in incompressible, homogeneous, and isotropic turbulence is studied by means of direct numerical simulations at two values of the Taylor-scale Reynolds number $Re_{\lambda} \sim 200$ and 400. The evolution of both heavy and light particle pairs is analysed at varying the particle Stokes number and the fluid-to-particle density ratio. For heavy particles, it is found that turbulent dispersion is schematically governed by two temporal regimes. The first is dominated by the presence, at large Stokes numbers, of small-scale caustics in the particle velocity statistics, and it lasts until heavy particle velocities have relaxed towards the underlying flow velocities. At such large scales, a second regime starts where heavy particles separate as tracers particles would do. As a consequence, at increasing inertia, a larger transient stage is observed, and the Richardson diffusion of simple tracers is recovered only at large times and large
Suspensions of dust, droplets, bubbles, and other finite-size particles advected by incompressible turbulent flows are commonly encountered in many natural phenomena (see, e.g., Csanady 1980, Eaton & Fessler 1994, Falkovich et al.. 2002, Post & Abraham 2002, Shaw 2003, Toschi & Bodenschatz 2009). Understanding their statistical properties is thus of primary importance. From a theoretical point of view, the problem is more complicated than in the case of fluid tracers, i.e. point-like particles with the same density as the carrier fluid. Indeed, when the suspended particles have a finite size and a density ratio different from that of the fluid, they have inertia and do not follow exactly the flow. As a consequence, correlations between particle positions and structures of the underlying flow appear. It is for instance well known that heavy particles are expelled from vortical structures, while light particles tend to concentrate in their cores. This results in the formation of strong inhomogeneities in the particle spatial distribution, an effect often refered to as preferential concentration (see Douady et al. 1991, Squires & Eaton 1991, Eaton & Fessler 1994). This phenomenon has gathered much attention, as it is revealed by the amount of recently published theoretical work (Balkovsky et al. 2001, Zaichik et al. 2003, Falkovich & Pumir 2004), and numerical studies (Collins & Keswani 2004, Chun et al. 2005, Bec et al. 2007, Goto & Vassilicos 2008). Progresses in the statistical characterization of particle aggregates have been achieved by studying particles evolving in stochastic flows by Sigurgeirsson & Stuart 2002, Mehlig & Wilkinson 2004, Bec et al. 2005, Olla 2002 and in two-dimensional turbulent flows by Boffetta et al. 2004. Also, single trajectory statistics have been addressed both numerically and experimentally for small heavy particles (see, e.g., Bec et al. 2006, Cencini et al. 2006, Gylfason et al. 2006, Gerashchenko et al. 2008, Zaichik & Alipchenkov 2008, Ayyalasomayajula et al. 2008, Volk et al. 2008), and for large particles (Qureshi et al. 2007, Xu & Bodenschatz 2008). The reader is refered to Toschi & Bodenschatz 2009 for a review.
In this paper we are concerned with particle pair dispersion, that is with the statistics, as a function of time, of the separation distance R(t) = X 1 (t)-X 2 (t) between two inertial particles, labelled by the subscripts 1 and 2 (see Bec et al. 2008, Fouxon & Horvai 2008, Derevich 2008 for recent studies on that problem). In homogeneous turbulence, it is sufficient to consider the statistics of the instantaneous separation of the positions of the two particles. These are organised in different families according to the values of their Stokes number St, and of their density mismatch with the fluid, β.
For our purposes, the motion of particle pairs, with given (St, β) values and with initial separations inside a given spherical shell, R = |X 1 (t 0 ) -X 2 (t 0 )| ∈ [R 0 , R 0 + dR 0 ] is followed until particle separation reaches the large scale of the flow. With respect to the case of simple tracers, the time evolution of the inertial particle pair separation R(t) becomes a function not only of the initial distance R 0 , and of the Reynolds number of the flow, but also of the inertia parameters (St, β).
A key question that naturally arises is how to choose the initial spatial and velocity distributions of inertial pairs. Indeed, it is known that heavy (resp. light) particles tend to concentrate preferentially in hyperbolic (resp. elliptic) regions of the advecting flow, with spatial correlation effects that may extend up to the inertial range of scales, as shown in Bec et al. 2007. Moreover, when inertia is high enough, the particle pair velocity difference, δ R V = |V 1 (X 1 (t), t) -V 2 (X 2 (t), t)|, may not go smoothly to zero when the particle separations decreases, a phenomenon connected to the formation of caustics, see Wilkinson & Mehlig 2005, Falkovich & Pumir 2007. In our numerical simulations, particles of different inertia are injected into the flow and let evolve until they reach a stationary statistics for both spatial and velocity distributions. Only after this transient time, pairs of particles with fixed intial separation are selected and then followed in the spatial domain to study relative dispersion.
By reason of the previous considerations, the main issue is to understand the role played by the spatial inhomogeneities of the inertial particle concentration field and by the presence of caustics on the pair separations, at changing the degree of inertia. We remark that these two effects can be treated as independent only in the limit of very small and very large inertia. In the former case, particles tend to behave like tracers and move with the underlying fluid velocity: preferential concentration may affect only their separation. In the opposite limit, particles distribute almost homogeneously in the flow: however, due to their ballisti
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