A lattice method for the numerical modelling of inertial particles 

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RELATIVE DISPERSION RATES OF LAGRANGIAN TRAJECTORIES 

(2014) successfully proposed to interpret the explosive separation as an iterative ballistic process.
This Richardson dispersion regime is also associated with another issue about the re-versibility of pair separations: in three dimensions, the Richardson constant gR is not the same when considering the forward in time evolution of pairs and its backward in time equivalent. Heuristically, this can be understood by the fact that the odd number of di-mension allows a fluid ellipsoid to be elongated along more dimensions than those along which it can be squeezed. This would not be the case in two dimensions.

Introduction

Turbulent mixing is of particular concern in situations such as the formation of clouds through condensation of small water droplets (Grabowski & Wang, 2013), gas accretion in planet formation (Johansen et al., 2007) or phytoplankton and nutrients distribution in the oceans (Mann & Lazier, 2013). Mixing refers to the evolution of an initial distribution of a scalar field (temperature, salinity, or the concentration of any substance…) by the fluid.
The mechanical stress induced by the stirred flow tends to deform the initial distribution of this field, and the multi-scale nature of turbulent flows gives rise to very complex shapes and patterns of the concentration field (Celani et al., 2001). For example, while the scalar is also submitted to molecular diffusion, which tends to smooth out concentration gradi-ents, mixing by stretching and compression in directions orthogonal to each other cause to reinforce these gradients by creating elongated concentration filaments.
Because incompressible flows preserve volumes, an homogeneous initial scalar concen-tration remains uniform at any later time. However, non-uniform initial patches of concen-trations will be deformed by the swirling eddies and create locally high gradients. Figure 3.1 displays an instantaneous field of scalar concentration advected by a turbulent flow: large fronts and cliffs are seen along with rather uniform regions. These gradients form because turbulence brings close together trajectories of fluid elements carrying different scalar tra-jectories and history. Scalar differences over small scales grow in intensity while the front boundaries become thinner, until they are eventually dissipated by molecular viscosity. These large differences are responsible for strongly intermittent statistics in the scalar dis-tribution (Sreenivasan & Antonia, 1997), i.e. the probability density function (pdf) of scalar value shows a departure from a Gaussian behaviour, and displays exponential tails (Pumir et al., 1991) that result from rare, extreme events. They prove that a restrictive vision considering a large number of small-scales, uncorrelated stretching events for the scalar distribution, which would yield Gaussian pdf through the central-limit theorem, is not correct.
Interestingly, the passive scalar is strongly intermittent both in 2D and 3D even in the absence of intermittency in the velocity field itself in 2D, and also in simple random Gaussian velocity fields (Shraiman & Siggia, 2000). Similarly to the velocity increments in 3D or the vorticity in the 2D direct cascade, the scaling exponents of the scalar structure function S✓n(r) = h(✓(x) − ✓(x + r))ni / r⇣(n) are not linear: ⇣(n) 6= n/3 (see section 2.1.1). Scale invariance is thus broken and Sn reads (Celani & Vergassola, 2001):
The difference ⇣dim(n) − ⇣(n) is the correction for the anomalous scaling, and the broken scale invariance manifests in the presence of L although r ⌧ L.
Some analytical models for the velocity correlations, like the Kraichnan model (Kraich-nan, 1994), allow to recover predictions about the behaviour of limn!1 ⇣(n). In the Kraich-nan ensemble, the two-points, two-times velocity correlation are:
for r smaller than the integral scale. ⇠ denotes the degree of roughness of the flow: the velocity field smoothness increases with ⇠ and is differentiable for ⇠ = 2.
In this model, and under the additional assumption of high dimensionality, d � ⇣(2), it was analytically shown in Balkovsky & Lebedev (1998) that there exists a critical order nc such that 8n > nc, ⇣(n) is independent of n This asymptotic behaviour seems also to be observed with direct numerical simulations of two dimensional inverse cascade, where it was estimated in Celani et al. (2000).
As vorticity and passive scalar share the same transport equation, it is tempting to com-pare their scaling laws to see if the exhibit similarities. However, the direct link between ! and u make the equation for ! non-linear, which can lead to discrepancies for small scale quantities. For example, Dubos & Babiano (2003) have shown using numerical simulations that this difference is responsible for faster temporal fluctuations of the vorticity gradients. In Boffetta et al. (2002), a correspondence is made between the intermittency of vorticity and that of a passive scalar transported by the flow, showing that ⇣p! = ⇣p✓ 8p. This corre-spondence may be explained using the following ad-hoc argument (Tsang et al., 2005) based on the Lyapunov exponent λ (see section 2.3). Since λ ⇠ Dkruk2E 1/2 and assuming, then λ ⇠ kf−⇠/2. Thus λ (and ru) characterising small separations stretch-ing, are determined by large scale structures, and the small scale vorticity components behave like scalar advected by the large scale flow.
Structure functions of order n are linked to the equal time n-point correlation function of the scalar field. For example, consider the following equality for n = 2:
The last equality results from homogeneity and isotropy and C2(r, t) = h✓(x + r, t)✓(x, t)i.
Averages are taken over the positions x.
The generalisation of this quantity to n-points displays the link with the n-point joint transition probability for the Lagrangian motion. For a set of n particles initially at position x0, . . . , xn at instant t0:
where p(. . . ) expresses the joint probability that the n trajectories initially at positions x01 . . . , x0n are transported at x1, . . . , xn at time t.
The quantity Cn can be related to the joint motion of n particles. It has a geometrical interpretation in terms of Lagrangian trajectories. For example, in Celani & Vergassola (2001), the intermittency of the passive scalar advection is attributed to long lasting clus-tering of n-tuple of particles. In Bianchi et al. (2016), the shape of spherical puffs of particles emitted is monitored as a function of time, showing that although the puff is initially spherical, the quick and strong distortions prevent the cloud to return back to a spherical shape at later times. It is also shown not to affect much large scale transport statistics, like the pdf of durations of hits and between hits of a downstream target.
In particular, the two-point scalar correlation allows one to express pair dispersion statistics. This correspondence was for example used in Boffetta & Celani (2000) to link frequent pairs encounter and scalar fronts formation. This object, C2(r) may be analytically derived only under drastic constrain on the flow, like for example the Kraichnan ensemble, In such a flow, Celani et al. (2007) have studied scaling properties of a scalar continuously emitted from a point source and derived an exact relation for the two-points equal-time scalar correlation function C(x1, x2, t) = h✓(x1, t)✓(x2, t)i, demonstrating the persistence of inhomogeneities at small scales.
Figure 3.1: Illustration of a scalar field mixed by turbulent flow, representing a 2D slice from a 3D DNS simulation at Nx3 = 40963 with a mean gradient scalar source. When initial inhomogeneities or inhomogeneous scalar sources are present, mixing by eddies create fronts where the scalar variations over very small scales are of the same order than the rms value itself.

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Concentration and mass fluctuations of particles emitted by a continuous point source

The large scalar temporal correlations play a role in what is called vortex trapping. Indeed, particles trapped inside a long-lived coherent vortex allows for interactions which take place at a distance of the size of the eddy to last longer and affect both the suspended solid phase as well as the carrier flow. This phenomenon is for example currently believed to play a key role in planetesimal accretions. Indeed, in planet formation, one challenging step is the understanding of the formation of planetesimals of the kilometre size, and the existence of long-living vortices in protoplanetary disks capable to concentrate large dust concentrations constitute a promising theory (Meheut et al., 2012).
The challenge of the system that is studied in this chapter, i.e. the continuous emission from a source, lies in the fact that both spatial and temporal correlations play a role in the n-point concentrations. The addition of the time difference in the correlations adds a non trivial complexity. In this chapter, the problem of continuous mass release is addressed.
Having introduced to the reader the dispersion dynamics of both Lagrangian tracers and scalar, the chosen framework to study the dispersion from a continuous point source in two-dimensional turbulence is now described.

Fluid phase integration

The flow regime we considered is the inverse turbulent cascade. Direct numerical sim-ulations have been performed using pseudo-spectral (Fourier) scheme in a d-dimensional periodic domain. The flow is forced at high wave-numbers. To maximize the inertial range and to minimise the range of scales affected by viscosity, we chose to implement hyper-viscosity, which translates into a higher power of the Laplacian p > 2 in the Navier-Stokes Figure 3.2: Illustration of recurrence phenomenon. In a few integral times TL, particle distribution becomes nearly uniform in the domain due to large amount of returns near the source. The cut-off distance is Rmax = 2L and width of the window is L.

Table of contents :

1 Introduction and context 
2 Definitions and concepts 
2.1 Navier–Stokes equations
2.1.1 Structure functions and intermittency
2.2 Turbulence in two dimensions
2.2.1 Two dimensional Navier–Stokes equations
2.2.2 The double cascade framework
2.2.3 Energy and enstrophy budgets
2.3 Relative dispersion rates of Lagrangian trajectories
2.3.1 Lyapunov exponents
2.3.2 Separation rates
I Turbulent dispersion and mixing 
3 Tracers dispersion in two dimensional turbulence 
3.1 Introduction
3.1.1 Diffusion at long times
3.1.2 Continuous source
3.2 middling version
3.2.1 Fluid phase integration
3.2.2 Injection mechanism
3.2.3 Removal mechanism
3.3 Results
3.3.1 One point dispersion
3.3.2 Two-point correlation
3.3.3 Phenomenological description
3.4 Brief conclusion
II Inertial particle-laden flows 
4 A lattice method for the numerical modelling of inertial particles 
4.1 Inertial particles dynamics
4.1.1 Individual particles
4.2 The modelling of dispersed multiphase flows
4.2.1 From microscopic description to macroscopic quantities
4.3 Description of the method
4.4 Application to a one-dimensional random flow
4.4.1 Particle dynamics for d = 1
4.4.2 Lattice-particle simulations
4.5 Application to incompressible two-dimensional flows
4.5.1 Cellular flow
4.5.2 Heavy particles in 2D turbulence
4.6 Conclusions
5 Turbulence modulation by small heavy particles 
6 Conclusions and perspectives
6.1 Turbulent transport of particles emitted from a point source
6.2 Modelisation of small inertial particles
6.3 Turbulence modulation by small heavy particles

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