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# Statistical models for the dynamics of heavy particles in turbulence

J. Bec1,2 K. Gustavsson3 and B. Mehlig3 1MINES Paris, PSL Research University,
CNRS, Cemef, Sophia-Antipolis, France F-06900; email: jeremie.bec@mines-
paristech.fr  2Université Côte d’Azur, Inria, CNRS, Cemef, Sophia-Antipolis,
France, F-06900  3Department of Physics, University of Gothenburg, 41296
Gothenburg, Sweden; email<EMAIL_ADDRESS>
###### Abstract

When very small particles are suspended in a fluid in motion, they tend to
follow the flow. How such tracer particles are mixed, transported, and
dispersed by turbulent flow has been successfully described by statistical
models. Heavy particles, with mass densities larger than that of the carrying
fluid, can detach from the flow. This results in preferential sampling, small-
scale fractal clustering, and large collision velocities. To describe these
effects of particle inertia, it is necessary to consider both particle
positions and velocities in phase space. In recent years, statistical phase-
space models have significantly contributed to our understanding of inertial-
particle dynamics in turbulence. These models help to identify the key
mechanisms and non-dimensional parameters governing the particle dynamics, and
have made qualitative, and in some cases quantitative predictions. This
article reviews statistical phase-space models for the dynamics of small, yet
heavy, spherical particles in turbulence. We evaluate their effectiveness by
comparing their predictions with results from numerical simulations and
laboratory experiments, and summarise their successes and failures.

###### doi:

10.1146/((please add article doi))

###### keywords:

turbulent particle suspensions, multiphase flow, statistical models, particle
inertia, fractal phase-space attractor, preferential sampling, caustics,
relative velocities, collisions, interactions

††journal: Xxxx. Xxx. Xxx. Xxx.

##  1 INTRODUCTION

The development of new experimental particle-tracking techniques and efficient
numerical-simulation methods on parallel computers have led to a better
understanding of the dynamics, statistics, and geometry of particle transport
and mixing in turbulence (Toschi & Bodenschatz, 2009). When the particles are
so small that their inertia is negligible, they follow the fluid flow as
Lagrangian tracers. Their spatial number density $n(\mbox{\boldmath$x$},t)$
obeys the advection equation

$\partial_{t}n(\mbox{\boldmath$x$},t)+\mbox{\boldmath$\nabla$}\cdot\left[\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)\,n(\mbox{\boldmath$x$},t)\right]=0\,,$
(1)

where $\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)$ is the Eulerian fluid
velocity at position $x$ and time $t$. In statistical models of turbulent
transport, $\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)$ is approximated by a
suitably chosen random field (Kraichnan, 1968). The analysis of such models
with tools from statistical physics and dynamical-systems theory has provided
valuable insight into the anomalous scaling laws governing the transported
densities, their geometrical conservation laws (Falkovich et al., 2001), and
their significance for turbulent mixing (Warhaft, 2000, Dimotakis, 2005) and
dispersion (Sawford, 2001, Salazar & Collins, 2009).

Many natural problems involve heavier particles — with mass densities larger
than that of the carrying fluid. Examples are water droplets in clouds (Shaw,
2003, Bodenschatz et al., 2010), and dust grains in planet-forming
circumstellar disks (Birnstiel et al., 2016). Although small in size, such
inertial particles can detach from the streamlines of the flow. Therefore one
must follow the particle velocities $v$, in addition to their positions $x$.
Turbulence stretches and folds the patterns formed by particles in the phase
space spanned by $x$ and $v$, so that their velocities become multi-valued in
$x$-space, or configuration space. As a consequence, Equation 1 fails to
describe the evolution of particle concentrations, simply because there is no
particle-velocity field. Instead, the particles form a fractal phase-space
attractor, resulting in violent and intermittent fluctuations of particle
separations and relative velocities. In short, the theoretical analysis of
inertial particles in turbulence requires models that account for the full
phase-space dynamics.

Here we review the main insights gained by analysing statistical phase-space
models for dilute suspensions of small, yet heavy, spherical particles in
homogeneous isotropic turbulence. We discuss how spatial clustering results
from centrifugal ejection from vortices and convergence to a fractal
attractor. Details depend on the scale at which the patterns are observed, and
whether or not the particles settle. Folds of fractal phase-space patterns,
akin to caustics in geometrical optics, give rise to anomalous scaling of the
particle-velocity structure functions, describing large relative velocities
between nearby particles which in turn accelerate particle collisions.
Statistical models show that the formation of these caustics depends
sensitively on the Stokes number, the particle-inertia parameter.

The statistical models reviewed in the following are highly idealised. We
address their strengths and weaknesses by comparing their predictions with
results obtained by direct numerical simulation of Navier–Stokes turbulence.

For other aspects of the physics and dynamics of dispersed multi-phase flow,
we refer the reader to other reviews. The significance of turbulence-induced
droplet collisions for cloud micro-physics is reviewed by Shaw (2003) and
Grabowski & Wang (2013). Balachandar & Eaton (2010) discuss the problems
arising from two-way coupling between particles and turbulence. Soldati &
Marchioli (2009) review particles in wall-bounded turbulence, and recent
advances in this area are discussed by Brandt & Coletti (2022), with detailed
comparisons between simulation and experimental results, and between point-
particle and particle-resolving simulations. Fox (2012) reviews numerical
methods for multiphase flow, and Monchaux et al. (2012) describe experimental
advances. Voth & Soldati (2017) review non-spherical particles in turbulence.

##  2 PARTICLE DYNAMICS

An incompressible fluid-velocity field
$\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)$ solves the Navier–Stokes
equations

$\mbox{\boldmath$\nabla$}\cdot\mbox{\boldmath$u$}=0\,,\quad\tfrac{{\rm
D}}{{\rm
D}t}\mbox{\boldmath$u$}\equiv\partial_{t}\mbox{\boldmath$u$}+(\mbox{\boldmath$u$}\cdot\mbox{\boldmath$\nabla$})\,\mbox{\boldmath$u$}=\tfrac{1}{\varrho_{\rm
f}}\,\mbox{\boldmath$\nabla$}\cdot\bbsigma\,,$ (2)

with fluid density $\varrho_{\rm f}$, and stress tensor
$\bbsigma=-p\,\mathbb{1}+2\nu{\varrho_{\rm f}}\mathbb{S}$. Here $p$ is
pressure, $\nu$ is the kinematic viscosity, and
$\mathbb{S}=\tfrac{1}{2}(\mathbb{A}+\mathbb{A}^{\sf T})$ is the symmetric part
of the fluid-velocity gradient tensor $\mathbb{A}$ with elements
$A_{ij}=\partial u_{i}/\partial x_{j}$. Its antisymmetric part is denoted by
$\mathbb{O}=\tfrac{1}{2}(\mathbb{A}-\mathbb{A}^{\sf T})$. The particles impose
boundary conditions upon Equation (2). For solid particles, the fluid velocity
on any point of the particle surface must equal the particle velocity of this
point.

The hydrodynamic force $\mbox{\boldmath$F$}_{\rm h}$ on a particle is given by
the integral of the normal component of $\bbsigma$ over the particle surface.
This non-linear coupling between particle and fluid dynamics poses fundamental
difficulties for modeling. One way out is to rely on particle-resolving
numerical simulations (Tenneti & Subramaniam, 2014, Maxey, 2017). For
turbulent suspensions with many particles, this is still very challenging. An
alternative is to use empirical force models, obtained either by fitting
simulation results for a single particle to a model (Goossens, 2019), or by
solving the Navier–Stokes equations in perturbation theory. The advantage of
empirical parameterisations is that one can go beyond the perturbative limit.
Disadvantages are that results are uncertain outside the fitting range, and
that this procedure does not yield immediate insight into the mechanisms at
play.

A standard perturbative method is to neglect the convective term in Equation
2, starting from the time-dependent Stokes equation. Including the
gravitational acceleration $g$, one obtains for a small sphere of radius $a$
and velocity $v$ (Maxey & Riley, 1983, Landau & Lifshitz, 1987):

On the r.h.s. we have the pressure-gradient force, Stokes’ force, Archimedes’
force which combines with gravity $\mbox{\boldmath$F$}_{g}=m_{\rm
p}\mbox{\boldmath$g$}$ to the buoyancy force (with particle mass $m_{\rm p}$
and gravitational acceleration $g$), the added-mass force, and the history
force arising from $\partial_{t}\mbox{\boldmath$u$}$ in Equation 2. Equation
LABEL:eq:bbo is frequently used to model the dynamics of particles in
turbulence (Brandt & Coletti, 2022), sometimes emphasising the significance of
the history force (Daitche & Tél, 2011, Olivieri et al., 2014, Guseva et al.,
2016, Prasath et al., 2019). However, convective inertia – neglected in
Equation LABEL:eq:bbo – weakens history effects (Lovalenti & Brady, 1993).
Faxén corrections to Equation LABEL:eq:bbo have been considered (Maxey &
Riley, 1983), but for small particles they are of the same order as shear-
induced inertia corrections, neglected in Equation LABEL:eq:bbo. Moreover,
lift forces are neglected, and it is hard to justify the above form of the
added-mass force for turbulent flow (Candelier et al., 2023). For these
reasons, Equation LABEL:eq:bbo fails to describe the dynamics of particles in
turbulence in general.

For small spherical particles with mass density much larger than that of the
fluid, all terms except Stokes’ force are negligible, fluid inertia does not
matter, and the equation of motion simplifies to

$\dot{}\mbox{\boldmath$x$}=\mbox{\boldmath$v$}\,,\quad\dot{}\mbox{\boldmath$v$}=-\left[\mbox{\boldmath$v$}-\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)\right]/\tau_{\rm
p}+\mbox{\boldmath$g$}\,.$ (4)

Here we included gravity, dots denote time derivatives, and $\tau_{\rm
p}=m_{\rm p}/(6\pi\nu\varrho_{\rm f}a)$ is the Stokes time.[] Stokes
timedamping time for the dynamics of inertial particle in the Stokes
approximation. It is common to use this single-particle equation for many
particles, neglecting particle-particle interactions (e.g. electrostatic or
hydrodynamic). Nevertheless, two nearby particles exhibit correlated motions,
because they experience correlated fluid velocities.

[t]

##  3 LENGTH AND TIME SCALES OF HOMOGENEOUS ISOTROPIC TURBULENCE

Turbulent velocity fluctuations are characterised by the root-mean-square
velocity $u_{\rm rms}=\langle u_{1}^{2}\rangle^{1/2}$ and the average kinetic-
energy dissipation rate $\varepsilon=\nu\,\mathrm{tr}\,\langle\mathbb{A}^{\sf
T}\mathbb{A}\rangle$. Turbulent fluctuations span a wide range of length
scales, from the Kolmogorov scale $\eta=(\nu^{3}/\varepsilon)^{1/4}$ below
which viscous dissipation dominates, to the largest scale $L=u_{\rm
rms}^{3}/\varepsilon$ at which the system is driven. At the spatial scale
$\ell$ in the inertial range $\eta\ll\ell\ll L$, velocity differences grow as
$u_{\ell}\sim(\varepsilon\ell)^{1/3}$ with associated eddy-turnover times
$\tau_{\ell}=\ell/u_{\ell}\sim\varepsilon^{-1/3}\ell^{2/3}$ (Frisch, 1995).
Extending this scaling to the edges of the inertial range gives the Kolmogorov
timescale $\tau_{\eta}=(\nu/\varepsilon)^{1/2}$ and the large-eddy turnover
time $\tau_{L}=u_{\rm rms}^{2}/\varepsilon$. Turbulence intensity is measured
by the Taylor-scale Reynolds number ${\rm Re}_{\lambda}=u_{\rm
rms}\lambda/\nu$, where $\lambda=u_{\rm
rms}/\langle\partial_{1}u_{1}^{2}\rangle^{1/2}=(15\,\nu/\varepsilon)^{1/2}u_{\rm
rms}$ is the Taylor microscale.

The non-dimensional parameters of the problem are summarised in Table 1. As
stated above, Equation 4 requires that the particles are small and much
heavier than the fluid. The particle Reynolds number ${\rm Re_{\rm
p}}={v}_{\rm s}a/\nu$ must be small. Here ${v}_{\rm s}$ estimates the
magnitude of the slip velocity $\mbox{\boldmath$u$}-\mbox{\boldmath$v$}$.
Since the particles detach more easily when $\tau_{\rm p}$ is larger, ${\rm
Re_{\rm p}}$ grows as $\tau_{\rm p}$ increases. Also the shear Reynolds number
must be small. In turbulence, it can be estimated as ${\rm
Re}_{s}\approx(a/\eta)^{2}$. To neglect molecular diffusion, one must assume a
large Péclet number $\mathrm{Pe}=a^{2}/(\mathscr{D}\tau_{\eta})$ (with
diffusion constant $\mathscr{D}$; see Balkovsky et al., 2001). Low mass-
loading ($\Phi_{\rm m}=(\varrho_{\rm p}/\varrho_{\rm f})n_{0}a^{3}\ll 1$ with
spatial number density $n_{0}$) ensures that the particles do not modify the
turbulence (Balachandar & Eaton, 2010). The Stokes number ${\rm St}=\tau_{\rm
p}/\tau_{\eta}$ is a measure of particle inertia (Snyder & Lumley, 1971). In
the limit ${\rm St}\to 0$ at constant non-dimensional settling speed ${\rm
Sv}=g\tau_{\rm p}/{u}_{\eta}$, Equation 4 simplifies to
$\mbox{\boldmath$v$}=\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)+{\rm
Sv}\,u_{\eta}\hat{\mbox{\boldmath$g$}}$ ($\hat{\mbox{\boldmath$g$}}$ is the
direction of gravity). If one keeps the Froude number ${\rm Fr}={\rm St}/{\rm
Sv}$ constant instead, then ${\rm St}\to 0$ constrains the particles to follow
the flow, $\mbox{\boldmath$v$}=\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)$
(Equation 1). Finally, for particles with different sizes, the parameter
$\theta{=|{\rm St}_{1}-{\rm St}_{2}|/({\rm St}_{1}+{\rm St}_{2})}$ measures
Stokes-number differences.

Table 1:  Non-dimensional parameters. | $a/\eta\ll 1$ | small particle size
---|---|---
| $\varrho_{\rm p}/\varrho_{\rm f}\gg 1$ | large mass-density ratio
Reynolds number | ${\rm Re}_{\lambda}=\sqrt{15}\,\tau_{L}/\tau_{\eta}$ | turbulence intensity
Particle Reynolds number | ${\rm Re_{\rm p}}={v}_{\rm s}a/\nu\ll 1$ | $\Bigg{\\}}$ convective inertia
Shear Reynolds number | ${\rm Re}_{\rm s}=a^{2}/(\nu\tau_{\eta})=(a/\eta)^{2}\ll 1$ |
Mass fraction | $\Phi_{\rm m}=(\varrho_{\rm p}/\varrho_{\rm f})n_{0}a^{3}\ll 1$ | turbulence modification
Péclet number | ${\rm Pe}=a^{2}/(\mathscr{D}\tau_{\eta})\gg 1$ | effect of diffusion
Settling number | ${\rm Sv}=\tau_{\rm p}g/u_{\eta}$ | settling speed
Stokes number | ${\rm St}=\tau_{\rm p}/\tau_{\eta}$ | particle inertia
| $\theta=|{\rm St}_{1}-{\rm St}_{2}|/({\rm St}_{1}+{\rm St}_{2})$ | particle-size dispersion

Light particles pose additional challenges (Mathai et al., 2020). Pressure-
gradient, added-mass, and buoyancy forces matter. For bubbles, the boundary
conditions at the surface can be different (Legendre & Magnaudet, 1997). The
history force is negligible in the high-frequency limit, but lift forces must
be accounted for (Mazzitelli & Lohse, 2004), and the flow around larger
bubbles can become unsteady (Mathai et al., 2015, 2016).

##  4 DISSIPATIVE DYNAMICAL SYSTEMS

###  4.1 Multifractal phase-space attractor

Figure 1: Snapshots of the phase-space attractor for an idealised model. The
equation of motion is a one-dimensional version of Equation 4, $\dot{x}=v$,
$\dot{v}=-[v-u(x,t)]/\tau_{\rm p}$, with a random velocity field $u(x,t)$
(Section 6). Shown are the positions $x(t)$ and velocities $v(t)$ of $10^{6}$
particles at different times (red dots). The magnifications illustrate fractal
clustering.

Equation 4 defines a dynamical system in six-dimensional phase space. How
phase-space structures stretch and contract under the dynamics is measured by
the Lyapunov exponents $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{6}$.
They describe how nearby particle trajectories diverge or converge. The growth
rate $\lim_{t\to\infty}t^{-1}\log[\mathscr{V}_{n}(t)/\mathscr{V}_{n}(0)]$ of
the volume $\mathscr{V}_{n}(t)$ of an $n$-dimensional phase-space region is
$\sigma_{n}=\lambda_{1}+\cdots+\lambda_{n}$ ($\sigma_{1}$ for distances,
$\sigma_{2}$ for areas, …). The inertial dynamics of small particles in
incompressible turbulence is chaotic because $\lambda_{1}>0$. At the same
time, since
$\sigma_{6}={\nabla_{\\!\scriptsize\mbox{\boldmath$x$}}\cdot\dot{\mbox{\boldmath$x$}}+\nabla_{\\!\scriptsize\mbox{\boldmath$v$}}\cdot\dot{\mbox{\boldmath$v$}}}=-{3}/\tau_{\rm
p}<0$, phase-space volumes contract. In short, the dynamics of this
dissipative chaotic system stretches, folds, and contracts, causing particle
trajectories to converge to a phase-space attractor that evolves in time
(Figure 1). Dynamical-systems theory allows to analyse this attractor, and to
draw conclusions about the physical properties of the system (Sections 8–10).

Attractors of dissipative chaotic systems are fractal, not smooth. The
distribution of phase-space distances $\delta
w=\sqrt{|\delta\mbox{\boldmath$x$}_{t}|^{2}+\tau^{2}|\delta\mbox{\boldmath$v$}_{t}|^{2}}\big{/}\ell$
has power-law form, Prob$(\delta w<\delta)\sim\delta^{D_{2}}$ for $\delta\ll
1$, with time and length scales $\tau$ and $\ell$. The exponent $D_{2}$ is the
fractal correlation dimension of the phase-space attractor. More generally,
the attractor is characterised by a spectrum $D_{p}$ of fractal dimensions
(Grassberger, 1983, Hentschel & Procaccia, 1983), defined in terms of the
fraction of particles
$\mathscr{M}_{\delta}(\mbox{\boldmath$x$}_{t},\mbox{\boldmath$v$}_{t},t)$
contained in a small ball of radius $\delta$ around
$(\mbox{\boldmath$x$}_{t},\mbox{\boldmath$v$}_{t})$. For positive integer $p$,
the $p$-th moment of $\mathscr{M}_{\delta}$ is the probability of finding
$p+1$ particles within $\delta$ from each other. The average of
$\mathscr{M}^{p}_{\delta}(\mbox{\boldmath$x$}_{t},\mbox{\boldmath$v$}_{t},t)$
along trajectories has power-law form in the statistically steady state,
$\langle\mathscr{M}^{p}_{\delta}\rangle\sim\delta^{\xi_{p}}$ for $\delta\ll
1$. The exponents $\xi_{p}\equiv p\,D_{p+1}$ define the fractal dimensions. In
analogy with multifractal turbulence models, one expects that $\xi_{p}$ are
non-linear, concave functions of $p$ (Paladin & Vulpiani, 1987, Meibohm et
al., 2020). This implies that the phase-space distribution is intermittent. []
Intermittent mass distributionthe probability density of
$\mathscr{M}_{\delta}$ has non- Gaussian tails. This results in large
fluctuations of particle separations and relative velocities (Sections 9 and
10).

[t]

##  5 MEASURING FRACTAL DIMENSIONS OF SPATIAL PATTERNS

The radial distribution function $g(R)$ measures the probability of finding
two particles at distance $R$ in configuration space. In three spatial
dimensions, the function is defined as $g(R)={(4\pi R^{2}})^{-1}\tfrac{{\rm
d}}{{\rm d}R}\langle\hat{\mathscr{M}}_{R}\rangle$, and it is normalised such
that $g(R)=1$ for uniformly distributed particles. When the particles form
fractal clusters $g(R)$ grows as a power law $\propto R^{\hat{D}_{2}-{3}}$ at
small distances, with spatial correlation dimension $\hat{D}_{2}<3$. A fractal
dimension introduced by Kaplan & Yorke (1979), $\hat{D}_{KY}$, measures the
dimensionality of spatial regions that neither expand nor contract. This
dimension can be calculated from the spatial Lyapunov exponents. Under quite
general circumstances one can show that $\hat{D}_{KY}=\hat{D}_{1}$ (Ledrappier
& Young, 1988).

###  5.1 Multi-valued velocities and projection

To determine the spatial distribution of particles, one must project the
phase-space patterns to configuration space ($x$-space). A fold of a smooth
phase-space curve causes a divergence of the projected spatial density, a
catastrophe or caustic just as in geometrical optics (Berry & Upstill, 1980),
and in the mass distribution in the early universe (Zel’Dovich, 1970). The
phase-space patterns of heavy particles in turbulence are fractal.
Nevertheless their folds (fractal catastrophes, see Meibohm et al., 2020)
result in large spatial particle-number densities (Crisanti et al., 1992,
Martin & Meiburg, 1994, Wilkinson & Mehlig, 2005).

Between two caustics, the projection to configuration space is many-to-one
(Falkovich et al., 2002). []CausticsLocations where phase-space folds cause
divergences of the particle-number density in configuration space. If the
particles approach on different branches of the phase-space attractor (Figure
1), they can collide at high velocities (Section 10). In Figure 1, fold
caustics are points on the $x$-axis. In reality, fold caustics are surfaces in
configuration space around regions of multi-valued particle velocities.
Higher-order catastrophes can occur, such as cusp catastrophes, swallow-tails,
and umbilics (Berry & Upstill, 1980). As the attractor folds, caustics occur
at rate $\mathscr{J}$, increasing the number of phase-space branches with
different velocities. This is balanced by the contraction of the dynamics in
the velocity direction, so that a steady state emerges.

Spatial distributions of particles in turbulence inherit their fractal
properties from the fractal phase-space attractor. The fraction of particles
$\hat{\mathscr{M}}_{\delta}(\mbox{\boldmath$x$}_{t},t)$ in a sphere of non-
dimensional radius $\delta$ around $\mbox{\boldmath$x$}_{t}$ obeys
$\langle\hat{\mathscr{M}}_{\delta}^{p}\rangle\sim\delta^{{p}\hat{D}_{{p}+1}}$
for $\delta\ll 1$, with spatial fractal dimension $\hat{D}_{{p}+1}$. The first
moment defines the the spatial correlation dimension ${\hat{D}}_{2}$ (Section
9). For a typical projection of a generic attractor to a three-dimensional
subspace, the projected correlation dimension obeys
$\hat{D}_{2}=\mbox{min}\\{D_{2},{3}\\}$ (Hunt & Kaloshin, 1997, Bec et al.,
2008). Meibohm et al. (2020) showed that this saturation of $\hat{D}_{2}$ at
the spatial dimension is due to caustics, they result in a uniform particle
distribution at large ${\rm St}$. Projection formulae for other fractal
dimensions are not known. Settling changes fractal clustering in configuration
space (Section 9), the spatial patterns become anisotropic (Bec et al., 2014,
Gustavsson et al., 2014, Ireland et al., 2016b).

Particles with different Stokes numbers cluster on different fractal
attractors, shifted with respect to each other (Chun et al., 2005, Bec et al.,
2005, Section 9). The phase-space attractors are shifted too, affecting the
relative particle-velocity distribution (Section 10).

Spatial clustering has been analysed using Voronoi tessellations of thin
slices of particle positions (Monchaux et al., 2010). This provides a robust
measure of particle clustering. But it does not yield information about the
fractal dimensions.

##  6 STATISTICAL MODELS

Statistical models avoid some of the challenges posed by ab-initio simulations
of particles in turbulence. The idea is to study the solutions of Equation 4
with a prescribed fluid velocity $\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)$,
using mathematical analysis, statistical closure, or Monte-Carlo simulation.
The goal is twofold. First, there is no other way at the moment of studying
how the 108 water droplets in a cubic metre of a turbulent atmospheric cloud
determine its radiative properties (Schneider et al., 2017). Second, the
analysis of idealised statistical models can explain the key mechanisms and
non-dimensional parameters that determine the particle dynamics. [] Lagrangian
fluid elementsinfinitesimal fluid elements that follow the flow like tracers.

This approach has been successful for inertialess tracer particles. Their
turbulent mixing is understood using Lagrangian models, which approximate the
turbulent velocity fluctuations seen by Lagrangian fluid elements (Pope,
1994). A different point of view was taken by Kraichnan (1968). He analysed
how the spatial concentration $n(\mbox{\boldmath$x$},t)$ of tracer particles
develops according to Equation 1, using a statistical model for the Eulerian
field $\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)$. The theoretical analysis
of Kraichnan’s model allows to relate the scalar concentration fluctuations to
the relative motion of the tracers (Falkovich et al., 2001). [] Kraichnan
modelapproximates $\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)$ as a Gaussian
random field, which is white-noise in time with power-law correlations in
space.

Heavy particles that do not follow the flow call for different levels of
sophistication in the modelling. One can solve the Navier–Stokes equations
without the particles by direct numerical simulation and integrate Equation
(4) with the numerically computed $\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)$
(Riley & Patterson, 1974, Squires & Eaton, 1991, Sundaram & Collins, 1997).
This method is called one-way-coupled DNS (Elgobashi, 2019). Although not ab-
initio, it has significantly advanced our understanding of the problem (Brandt
& Coletti, 2022).

[b]

##  7 MATCHING STATISTICAL MODELS TO TURBULENCE

In KS-models, a discrete set of Fourier modes $k_{n}$ is chosen to approximate
the turbulent kinetic-energy spectrum $E(\mbox{\boldmath$k$})\simeq
1.5\,\varepsilon^{-2/3}|\mbox{\boldmath$k$}|^{-5/3}$ (upper and lower cutoffs
are matched to $\eta$, and to the large scale $L$). Time correlations are
imposed by the frequency spectrum
$\omega_{n}=\omega_{0}\sqrt{k_{n}^{3}E(k_{n})}$, where $\omega_{0}$ determines
the Kubo number (Voßkuhle et al., 2015). The Stokes number is not defined in
terms of $\tau_{\eta}$, but using a typical wave vector (Ijzermans et al.,
2010), introducing an unknown factor in the definition of ${\rm St}$. The
three parameters $u_{0}$, $\ell_{0}$, $\tau_{0}$ of the single-scale model are
matched as follows. The magnitude of actual turbulent fluid-velocity gradients
is $\sim\\!\tau_{\eta}^{-1}$, and Lagrangian fluid elements separate and
decorrelate on time scales $\sim\\!\tau_{\eta}$. The single-scale model
exhibits the same behaviour for $\tau_{0}\gg\tau_{\eta}$. In this large-${\rm
Ku}$ limit, the Eulerian time scale $\tau_{0}$ does not matter, and one
matches the model time-scale $\ell_{0}/u_{0}$ to the Kolmogorov time
$\tau_{\eta}$ of the turbulent flow. To model single-particle statistics
(Section 8), the remaining parameter is chosen as $u_{0}=\sqrt{3}u_{\rm rms}$,
or equivalently $\ell_{0}=\lambda$, the Taylor microscale. For the dynamics of
particle separations (Sections 9 and 10), one matches $\ell_{0}$ to the length
scale where the smooth dissipation range turns into the inertial range, of the
order $\ell_{0}\sim 10\eta$. For KE-models, one matches the eddy-turnover time
$\tau_{\ell}$ to turbulence, as well as the correlation times $\tau_{O}$ and
$\tau_{S}$ of vorticity and strain. The resulting model contains unknown
higher-order structure functions of fluid-velocity differences. They are
approximated in terms of two-point functions. The latter are taken from DNS
(Bragg & Collins, 2014a, b). An alternative to DNS is to develop and analyse
idealised models for the turbulent flucutations. Maxey & Corrsin (1986) used
an ensemble of stationary cellular flows. Saffman & Turner (1956) averaged
over Gaussian-distributed fluid-velocity gradients $\mathbb{A}$ of time-
independent linear flows
$\mbox{\boldmath$u$}(\mbox{\boldmath$x$})=\mathbb{A}\,\mbox{\boldmath$x$}$.
Falkovich et al. (2007) used a one-dimensional model where $A=\partial
u/\partial x$ follows a telegraph process. Another possibility is to represent
$\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)$ as a sequence of non-linear
velocity fields $\mbox{\boldmath$u$}_{t}(\mbox{\boldmath$x$})$ changing at
random times $t$ (Pergolizzi, 2012). The most common approach, however, is to
approximate the turbulent velocity by an incompressible random field. To
ensure incompressibility, it is convenient to represent the fluid velocity as
the curl of a random vector potential,
$\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)=\mbox{\boldmath$\nabla$}\wedge\mbox{\boldmath$\Psi$}(\mbox{\boldmath$x$},t)$.
The vector potential $\mbox{\boldmath$\Psi$}(\mbox{\boldmath$x$},t)$ is
constructed from random spatial Fourier modes. []KS modelapproximates
$\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)$ as a Gaussian random function
with power-law correlations in space and time. Kinematic-simulation (KS)
models (Fung et al., 1992, Ducasse & Pumir, 2009, Ijzermans et al., 2010)
approximate turbulent inertial-range scaling by imposing a power-law
dependence of the correlation function
$\langle\Psi_{i}(\mbox{\boldmath$x$},t)\Psi_{j}(\mbox{\boldmath$x$}^{\prime},t^{\prime})\rangle$
upon $\mbox{\boldmath$x$}-\mbox{\boldmath$x$}^{\prime}$, as in the Kraichnan
model. While Kraichnan’s model is white noise in time, KS models have a non-
zero decay time.

To describe the particle dynamics in the dissipative range of turbulence, one
can simplify the model further, keeping only a single spatial scale
$\ell_{0}$. Time correlations can be generated by Ornstein-Uhlenbeck
processes, resulting in components $\Psi_{j}$ of the vector field that are
Gaussian distributed with zero mean and correlations (Maxey, 1987, Pinsky &
Khain, 1995, Sigurgeirsson & Stuart, 2002, Bec et al., 2005, 2008, Gustavsson
& Mehlig, 2016), []Single-scale modelapproximates
$\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)$ as a Gaussian random function
with the single spatial scale $\ell_{0}$.

$\displaystyle\left\langle\Psi_{i}(\mbox{\boldmath$x$},t)\Psi_{j}(\mbox{\boldmath$x$}^{\prime},t^{\prime})\right\rangle=\mathscr{C}\,(u_{0}\,\ell_{0})^{2}\,\delta_{ij}\,\exp\big{[}-{|\mbox{\boldmath$x$}-\mbox{\boldmath$x$}^{\prime}|^{2}}/(2\ell_{0}^{2})-{|t-t^{\prime}|}/{\tau_{0}}\big{]}\,.$
(15)

The normalisation constant $\mathscr{C}$ is chosen so that
$\langle|\mbox{\boldmath$u$}|^{2}\rangle=u_{0}^{2}$. The model has two time
scales: the Eulerian correlation time $\tau_{0}$ and the Kolmogorov time
$\tau_{\eta}=({\rm Tr}\,\,\langle\mathbb{A}^{\sf T}\mathbb{A}\rangle)^{-1/2}$
which evaluates to $\tau_{\eta}=\ell_{0}/(u_{0}\sqrt{{5}})$. The ratio of
these time scales defines the Kubo number ${\rm
Ku}=\frac{u_{0}\tau_{0}}{\ell_{0}}\propto\frac{\tau_{0}}{\tau_{\eta}}$
(Wilkinson et al., 2007). In order to match the model to turbulence, one
considers the limit of large Kubo numbers, where the dynamics becomes
independent of $\tau_{0}$. The remaining parameters, $u_{0}$ and $\ell_{0}$,
can then be matched to the relevant scales of the turbulent flow. In this
case, any dependence on ${\rm Re}_{\lambda}$ is contained in the choice of
$u_{0}$ and $\ell_{0}$. The model allows mathematical analysis in the limit of
small but finite ${\rm Ku}$ using perturbation theory (Gustavsson & Mehlig,
2016). The white-noise limit (the limit ${\rm Ku}\to 0$ at constant ${\rm
Ku}\,{\rm St}$) can be analysed using diffusion approximations (Wilkinson et
al., 2007). The persistent limit of ${\rm Ku}\to\infty$ at constant ${\rm St}$
can in some cases be treated analytically (Derevyanko et al., 2007,
Pergolizzi, 2012, Meibohm et al., 2023a).

The single-scale model assumes that the particle dynamics is mainly determined
by the smooth part of the turbulent velocity correlation function. At large
Stokes numbers or settling velocities, the inertial range may influence the
particle dynamics, and this can lead to new parameter dependencies, for
example on ${\rm Re}_{\lambda}$. KS models do have a range of spatial scales,
but they do not account for the sweeping of small turbulent eddies by large
ones. Instead, large eddies drag Lagrangian fluid elements through smaller
eddies. As a consequence, fluid-velocity gradients decorrelate too quickly
compared with turbulence (Voßkuhle et al., 2015). Both models have Gaussian
fluid-velocity fluctuations. This means that they neglect violent fluctuations
of the fluid-velocity gradients corresponding to persistent regions of
anomalously high turbulent vorticity. Under which circumstances this matters
is discussed in the following Sections.

Other models for dispersed multiphase flow are related to Langevin models for
the Lagrangian dynamics of fluid elements (Pope, 1994). Approximate Langevin
equations for inertial particles (Minier, 2016) allow to compute the
distribution of $\mbox{\boldmath$x$}(t)$, $\mbox{\boldmath$v$}(t)$, and
$\mbox{\boldmath$u$}(\mbox{\boldmath$x$}(t),t)$ by Monte-Carlo simulation. The
models reproduce single-time, single-particle statistics, but they are not
designed to describe two-point statistics such as relative particle-
velocities. Multi-valued particle velocities can be represented in a heuristic
fashion, by decomposing the particle velocities into two contributions, a
smooth velocity field plus spatially uncorrelated fluctuations intended to
represent the effect of caustics (Fevrier et al., 2005). Alternatively, one
may consider models for the distribution of $\mbox{\boldmath$x$}(t)$ and
$\mbox{\boldmath$v$}(t)$ alone. One can derive a kinetic equation (KE) for
$P(\mbox{\boldmath$x$},\mbox{\boldmath$v$};t)$ if one assumes that the
turbulent velocity fluctuations are Gaussian. Keeping only the leading terms
in a moment expansion, one obtains:

$\left\\{\partial_{t}+v_{i}\partial_{x_{i}}-\tfrac{1}{\tau_{\rm
p}}\partial_{v_{i}}v_{i}-\tfrac{1}{\tau_{\rm
p}}\partial_{v_{i}}\left[\partial_{v_{j}}\mu_{ij}(\mbox{\boldmath$x$},t)+\partial_{x_{j}}\lambda_{ij}(\mbox{\boldmath$x$},t)-\kappa_{i}(\mbox{\boldmath$x$},t)\right]\right\\}P(\mbox{\boldmath$x$},\mbox{\boldmath$v$};t)=0\,.$
(16)

Here $\mu_{ij}$, $\lambda_{ij}$, and $\kappa_{i}$ are unknown coefficients
that require closure approximations, and summation over repeated indices is
implied. This approach yields accurate numerical descriptions of single-
particle properties, also in spatially inhomogeneous turbulent flow (Reeks,
2021). Zaichik & Alipchenkov (2003) derived analogous equations for two-
particle statistics. Such KE models can be closed using DNS results for the
unknown []Closureapproximate evaluation of unknown terms in a moment
expansion. [] KE modelkinetic equation for distribution of particle positions
and velocities, requiring closure. coefficients (Bragg & Collins, 2014a, b),
to obtain numerical results for the statistics of separations and relative
velocities between heavy particles in turbulence. In Sections 9 and 10, we
compare the corresponding results to those of statistical models based on
Equation 15. In the white-noise limit, the kinetic equation for particle
separations and relative velocities reduces to a Fokker-Planck equation,
similar to the one analysed by Gustavsson et al. (2008).

##  8 PREFERENTIAL SAMPLING

###  8.1 Maxey’s centrifuge

Heavy particles can detach from the flow, they are not Lagrangian fluid
elements. This leads to a bias in the statistical properties of the fluid-
velocity gradients evaluated along particle paths (Maxey, 1987), called
preferential sampling. Maxey obtained an approximate equation-of-motion by
expanding Equation 4 at small ${\rm St}$. When ${\rm Sv}=0$, this yields
$\tfrac{{\rm d}}{{\rm d}t}\mbox{\boldmath$x$}\approx\mbox{\boldmath$v$}_{\rm
p}(\mbox{\boldmath$x$},t)=\mbox{\boldmath$u$}-\tau_{\rm p}\tfrac{\rm D}{{\rm
D}t}\mbox{\boldmath$u$}$. Since the effective particle-velocity field
$\mbox{\boldmath$v$}_{\rm p}(\mbox{\boldmath$x$},t)$ is compressible, []
Maxey’s centrifugeparticle inertia expels heavy particles from vortices.
$\mbox{\boldmath$\nabla$}\cdot\mbox{\boldmath$v$}_{\rm p}=-\tau_{\rm p}{\rm
Tr}\,\mathbb{A}^{2}=-\tau_{\rm p}\big{(}{\rm Tr}\,\mathbb{S}^{2}-{\rm
Tr}\,\mathbb{O}^{\sf T}\mathbb{O}\big{)}\neq 0$, he concluded that small-${\rm
St}$ particles are more likely to explore the sinks of
$\mbox{\boldmath$v$}_{\rm p}(\mbox{\boldmath$x$},t)$, regions of large strain
and small vorticity. This centrifuge effect is illustrated in Figure 2a,
showing a snapshot of particle positions together with the magnitude of
vorticity
$\omega\equiv\mbox{\boldmath$|$}\nabla\wedge\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)|=\sqrt{2{\rm
Tr}\,\mathbb{O}^{\sf T}\mathbb{O}}$. Heavy particles tend to avoid connected
regions of high vorticity of linear sizes smaller than $\sim{10\eta}$ (Section
6).

Falkovich & Pumir (2004) predicted that $\langle{\rm
Tr}\,\mathbb{A}^{2}\rangle{\tau_{\eta}^{2}}\propto{\rm St}$ along particle
paths. This is consistent with a perturbative analysis of the single-scale
model neglecting multi-valued particle velocities, which gives $\langle{\rm
Tr}\,\mathbb{A}^{2}\rangle\tau_{\eta}^{2}{\propto}{\rm St}$ for small ${\rm
Ku}$ and ${\rm St}$ (Gustavsson & Mehlig, 2016). Figure 2b shows $\langle{\rm
Tr}\,\mathbb{A}^{2}\rangle$ from DNS and simulations of the single-scale
model. The results are qualitatively similar, but there is one important
difference: the model underestimates preferential sampling (Gustavsson &
Mehlig, 2016) because turbulence has more persistent regions of much larger
vorticity that act as efficient centrifuges (Figure 2a). Note that the DNS
results decrease more slowly than predicted by perturbation theory as ${\rm
St}\to 0$. The reason for this difference is not understood. Figure 2b also
shows that the centrifuge effect becomes weaker at larger ${\rm St}$. This is
expected, because the history of the fluid-velocity gradients seen by the
particle matters more than the gradients at the present position (Gustavsson
et al., 2008, Bragg et al., 2015b). Note also that the DNS results in Figure
2b show a weak dependence on ${\rm Re}_{\lambda}$. A possible explanation is
that at larger ${\rm Re}_{\lambda}$ particles are more likely to be expelled
from straining regions too (Ireland et al., 2016a).

\begin{overpic}[width=455.24408pt]{Fig2} \end{overpic} Figure 2: Preferential
sampling. a Heavy particles avoid vortices. Shown are DNS results for particle
positions in a thin slice of three-dimensional configuration space. Different
levels of gray reflect particle depth within the slice [data from Bec et al.
(2014) with ${\rm St}=1$ and ${\rm Re}_{\lambda}\approx 460$]. Also shown is
the magnitude $\omega$ of vorticity (see text). b Average of ${\rm
Tr}\,\mathbb{A}^{2}$ along inertial-particle paths versus ${\rm St}$. DNS
results from Ireland et al. (2016a) ($\scriptstyle\Box$, $\Diamond$), Bec et
al. (2014) ($\scriptstyle\bigtriangleup$, $\scriptstyle\bigtriangledown$ ),
data from simulations of the single-scale model (Equation 15) for ${\rm
Ku}=10$ ($\bullet$), and the prediction of perturbation theory
$\tau_{\eta}^{2}\langle{\rm Tr}\,\mathbb{A}^{2}\rangle\propto{\rm St}$ (solid
line). c Voids in the spatial distribution of cloud droplets. Data from
(Karpińska et al., 2019). Image obtained from E. Bodenschatz.

An alternative preferential-sampling mechanism was suggested by Coleman &
Vassilicos (2009): at small ${\rm St}$, heavy particles cluster near points of
zero fluid acceleration, where $\tfrac{{\rm D}}{{\rm
D}t}\mbox{\boldmath$u$}=0$. At small Stokes numbers, this is consistent with
Maxey’s small-${\rm St}$ expansion. At large ${\rm St}$, by contrast, this
expansion fails, and preferential sampling of low-acceleration regions becomes
negligible (Bragg et al., 2015a).

Quantifying preferential sampling in laboratory experiments is hard because
one must simultaneously track particles and fluid-velocity gradients. The
measurements of Gibert et al. (2012) and Petersen et al. (2019) show a certain
bias towards straining regions at intermediate Stokes numbers.

###  8.2 Settling

Preferential sampling has a significant effect on the gravitational settling
of particles in turbulence. Averaging Equation 4 leads to
$\langle\mbox{\boldmath$v$}\cdot\hat{\mbox{\boldmath$g$}}\rangle=\tau_{\rm
p}\,\mbox{\boldmath$g$}+\langle\mbox{\boldmath$u$}\cdot\hat{\mbox{\boldmath$g$}}\rangle$,
where $\tau_{\rm p}\,\mbox{\boldmath$g$}$ is the Stokes settling speed in a
still fluid. Maxey (1987) suggested that turbulence enhances settling, because
particles expelled from vortices preferentially sample down-welling regions
where $\mbox{\boldmath$u$}\cdot\hat{\mbox{\boldmath$g$}}>0$. This preferential
sweeping is confirmed by DNS (Wang & Maxey, 1993, Bec et al., 2014, Rosa et
al., 2016), as well as experiments (Aliseda et al., 2002, Good et al., 2014,
Petersen et al., 2019).[] Preferential sweepingparticles preferentially sample
down-welling regions of turbulent flow. The relative increase in settling
speed is largest for settling numbers ${\rm Sv}\approx 1$, when the settling
velocity is of the order of the smallest turbulent eddies. For large ${\rm
Sv}$, Good et al. (2014) observed that turbulence can have the opposite
effect, reducing the settling speed. What causes this reduction is not
understood (Rosa et al., 2016), but model calculations using cellular flow
fields confirm this behaviour (Lillo et al., 2008, Afonso, 2008).

###  8.3 Large-scale preferential sampling

The mechanisms reviewed in Section 8.1 explain preferential sampling in the
dissipative range of turbulence, leading to spatial patterns on scales up to
$\ell_{0}\sim 10\eta$. Figure 2a shows strong preferential sampling also at
much larger scales. Similar structures – termed cloud holes by Karpińska et
al. (2019) – are observed in the spatial patterns of cloud droplets (Figure
2c). It is plausible that these voids are caused by rare fluctuations of
turbulent vorticity, intense and persistent vortex tubes that manage to expel
particles with moderate inertia.

At large Stokes numbers, inertial-range statistics may become important. The
mean-squared particle velocity, for example, is of the order of $\langle
v_{1}^{2}\rangle\sim u_{\rm rms}^{2}/(1+\tau_{\rm p}/\tau_{u})$, where
$\tau_{u}\\!\sim\\!\tau_{L}\\!=\\!\tau_{\eta}{\rm Re}_{\lambda}/\sqrt{15}$ is
the Lagrangian correlation time of the large-scale fluid velocity (Abrahamson,
1975, Ireland et al., 2016a). In other words, the ${\rm
Re}_{\lambda}$-dependence $\langle v_{1}^{2}\rangle$ is due to large-scale
motion. This effect can be accounted for by setting $u_{0}=\sqrt{{3}}\,u_{\rm
rms}$ and ${\ell_{0}}\\!\sim\\!u_{\rm rms}\,\tau_{L}$ in the single-scale
model, Equation 15. A related choice of scales is discussed in Section 9,
intended to describe inertial-range clustering in terms of a scale-dependent
Stokes number ${\rm St}_{L}\sim\tau_{\rm p}/\tau_{L}$. The only ${\rm
Re}_{\lambda}$-dependence is contained in the choice of the time scale
$\tau_{L}$.

The settling velocity of heavy particles, on the other hand, exhibits an
intricate ${\rm Re}_{\lambda}$-dependence (Bec et al., 2014), that cannot be
described by simply changing a time scale. The reason is preferential
sampling: rapidly settling particles explore vertical fluid-velocity
correlations at spatial separations much larger than the dissipation scale.
The settling particles experience an effective horizontal flow that is
compressible, and collect in the sinks of this flow. A Kraichnan model for the
effect of large eddies on the settling speed predicts
$\langle\mbox{\boldmath$v$}\cdot\hat{\mbox{\boldmath$g$}}\rangle/v_{\rm
s}-1\propto{\rm Re}_{\lambda}^{3/4}{\rm Fr}^{5/2}{\rm St}^{-2}$, in good
agreement with DNS (Bec et al., 2014).

##  9 FRACTAL CLUSTERING

###  9.1 Multiplicative amplification

Tracer particles in a compressible flow form fractal spatial patterns. The sum
of the spatial Lyapunov exponents is negative, so that the Kaplan-Yorke
dimension $\hat{D}_{\rm KY}$ is smaller than the spatial dimension (Sommerer &
Ott, 1993, Falkovich et al., 2001, Toschi & Bodenschatz, 2009). In
incompressible flow, tracer particles cannot cluster because the spatial
Lyapunov exponents sum to zero. The infinitesimal volume
$\hat{\mathscr{V}}_{3}=\mbox{det}[\mbox{\boldmath$R$}_{1},{\mbox{\boldmath$R$}_{2}},\mbox{\boldmath$R$}_{3}]$
spanned by three separation vectors $\mbox{\boldmath$R$}_{\alpha}$ between
four nearby particles neither contracts nor expands.

Inertial particles cluster even in incompressible flow. To understand the
mechanism, consider how $\hat{\mathscr{V}}_{3}$ changes. If the particles
remain close to each other, one can show that $\hat{\mathscr{V}}_{3}(t+\delta
t)=\det[\mathbb{I}+\mathbb{Z}(t)]\hat{\mathscr{V}}_{3}(t)$. Here, $\mathbb{Z}$
is the matrix of particle-velocity gradients $Z_{ij}=\partial v_{i}/\partial
x_{j}$. Since the product of many random factors
$\det[\mathbb{I}+\mathbb{Z}(t)]$ tends to be smaller than unity, spatial
volumes contract. This amplifies local particle-number density fluctuations,
resulting in spatial clustering by multiplicative amplification (Gustavsson &
Mehlig, 2016).

The dynamical equation for $\mathbb{Z}$ follows from Equation 4 by
differentiation, $\tfrac{\rm d}{{\rm
d}t}\mathbb{Z}=(\mathbb{A}-\mathbb{Z})/\tau_{\rm p}-\mathbb{Z}^{2}$ (Falkovich
et al., 2002). It must be integrated alongside the particle motion, therefore
fractal clustering is determined by the history of the fluid-velocity
gradients experienced by the particles (Gustavsson et al., 2008, Bragg et al.,
2015b). In the white-noise limit (Section 6), fractal clustering is entirely
due to multiplicative amplification (Wilkinson et al., 2007).

###  9.2 Correlation dimension

The spatial correlation dimension $\hat{D}_{2}$ has a straightforward
interpretation in terms of the pair-correlation function, and quantifies the
effect of fractal clustering on the collision rate (Section 10).
Unfortunately, however, $\hat{D}_{2}$ is hard to calculate. Systematic
perturbation expansions for the spatial correlation dimension are known only
for special cases, such as the white-noise limit. To make matters worse, the
perturbation series diverge asymptotically. In addition, it appears that the
perturbation theory misses non-analytical contributions proportional to the
rate of caustic formation (Gustavsson et al., 2015).

\begin{overpic}[width=426.79134pt]{Fig3} \end{overpic} Figure 3:  Fractal
clustering. a Correlation dimension for identical particles for ${\rm Sv}=0$.
Spatial $\hat{D}_{2}$ (large symbols), phase-space $D_{2}$ (small symbols)
versus ${\rm St}$. DNS results: ${\rm Re}_{\lambda}=89$ (Bhatnagar et al.,
2018), ${\rm Re}_{\lambda}=88$ (Bragg & Collins, 2014a), ${\rm
Re}_{\lambda}=65,185$ (Bec et al., 2007), ${\rm Re}_{\lambda}=400$ (Bec et
al., 2010). Model predictions: KE-model, evaluated by Bragg & Collins (2014a)
using DNS for the unclosed terms (blue,$\blacklozenge$), simulations of the
single-scale model (SSM, Equation 15) for ${\rm Ku}=10$ (red, $\bullet$ and
$\bullet$), together with a fit to the small-${\rm St}$ asymptote (solid
line). Also shown are simulations of the KS-model (Ijzermans et al., 2010)
(green, $\blacksquare$), see text. Small-${\rm St}$ theory of Chun et al.
(2005) evaluated by Bragg & Collins (2014a) using DNS for the unclosed terms
(magenta, $\blacktriangle$). Experimental data extracted from panel b by
fitting the data in the range $1\leq R/\eta\leq 10$ ($\star$). b Pair
correlation functions $g(R)$ for polydisperse suspensions of droplets in air
turbulence. Laboratory experiments (Saw et al., 2012), symbols, compared with
predictions of single-scale model (SSM) ${\rm Ku}=10$ and $\ell_{0}/\eta=20$.
First: for particles of two different sizes (dashed lines) with
$\overline{{\rm St}}\sim 0.11,\theta\sim 0.91$; $\overline{{\rm St}}\sim
0.19,\theta=0.53$; $\overline{{\rm St}}\sim 0.4,\theta=0.37$; $\overline{{\rm
St}}\sim 0.7,\theta=0.43$, read off from Fig. 7 in Saw et al. (2012). The
average Stokes number is defined as $\overline{{\rm St}}=\overline{\tau}_{\rm
p}/\tau_{\eta}$, where $\overline{\tau}_{\rm p}$ is the harmonic mean of the
different Stokes times, see (Meibohm et al., 2017). Second: distribution of
particle sizes extracted from their Fig. 3 (solid lines). c Effect of
settling, $\hat{D}_{2}$ for different values of ${\rm Fr}$, DNS results from
Bec et al. (2014), single-scale model (SSM) for ${\rm Ku}=10$ and
$\ell_{0}/\eta=10$. d Inertial-range clustering, pair-correlation function
$g(R)$ versus $R$ for ${\rm St}=1$. DNS at ${\rm Re}_{\lambda}=597$ (Ireland
et al., 2016a), ${\rm Re}_{\lambda}=460$ (Bec et al., 2014), ${\rm
Re}_{\lambda}=328$ (Ariki et al., 2018), and ${\rm Re}_{\lambda}=227$ (Ray &
Collins, 2011). Single-scale model for ${\rm Ku}=10$ and $\ell_{0}/\eta=20$,
and small-$R$ asymptote of $g(R)$, solid line (Section 4).

Figure 3a shows DNS measurements of $\hat{D}_{2}$ for different ${\rm
Re}_{\lambda}$. Also shown are results of model calculations: numerical
evaluations for the KE-model carried out by Bragg & Collins (2014a) using DNS
for the unclosed terms, for the single-scale model (Equation 15), and for the
KS model (Ijzermans et al., 2010). As pointed out in Section 6, there is some
arbitrariness in the definition of the Stokes number for the KS-model. Here we
simply rescaled ${\rm St}$ for the KS-model data (by a factor of $1.4$) to fit
the location of the minimum in the DNS data in Figure 3a. Overall, we observe
good agreement between model and DNS results. In particular, $\hat{D}_{2}$
depends only weakly on ${\rm Re}_{\lambda}$. This is expected, because
$\hat{D}_{2}$ characterises the small-$R$ asymptote of $g(R)$, dominated by
particle pairs that have spent a long time together at small relative
velocities, but only the tails of the relative-velocity distribution are
sensitive to ${\rm Re}_{\lambda}$. Therefore the small-$R$ behaviour of $g(R)$
is insensitive to ${\rm Re}_{\lambda}$-corrections. Figure 3a demonstrates
that fractal clustering is strongest at ${\rm St}\approx 1$, as expected,
because turbulent fluctuations affect the particles most when $\tau_{\rm p}$
and $\tau_{\eta}$ are comparable.

The largest deviations between model and DNS results are observed at small
Stokes numbers. When ${\rm St}\ll 1$, one may replace the inertial-particle
dynamics by advection in an effective velocity field $\mbox{\boldmath$v$}_{\rm
p}(\mbox{\boldmath$x$},t)$ with ${\rm St}$-dependent compressibility (Section
8). This suggests that $3-\hat{D}_{2}\propto{\rm St}^{2}$ (Balkovsky et al.,
2001, Elperin et al., 2002). Chun et al. (2005) came to the same conclusion,
expanding the relative-particle dynamics for small ${\rm St}$. There is,
however, no agreement on the numerical prefactor. An improved version of the
small-${\rm St}$ theory by Chun et al. (2005) is also shown in Figure 3a,
evaluated by (Bragg & Collins, 2014a) using DNS for the unclosed terms. The
result is consistent with a ${\rm St}^{2}$-dependence, as predicted by
perturbative analysis of the single-scale model (Gustavsson & Mehlig, 2016).
But the DNS results are not. The reason for this discrepancy is unclear.

A large-deviation argument indicates that $3-\hat{D}_{2}=2(3-\hat{D}_{\rm
KY})$ at small ${\rm St}$ (Fouxon, 2011). The same result holds in the white-
noise limit (Gustavsson et al., 2015, Gustavsson & Mehlig, 2016). DNS results
are consistent with this prediction (Bec et al., 2006, 2007, Calzavarini et
al., 2008).

Regarding the phase-space correlation dimension $D_{2}$, we observe good
overall agreement between DNS and model simulations (Figure 3a), although
small differences at large ${\rm St}$ remain unexplained. We also see that
$\hat{D}_{2}=\min\\{D_{2},3\\}$, consistent with the projection formula
discussed in Section 4. [] Caustics and $\hat{D}_{2}$caustics explain that the
spatial correla- tion dimension $\hat{D}_{2}$ equals the spatial di- mension
for ${\rm St}\\!>\\!{\rm St}_{\rm c}$.

Figure 3b shows the experimental data of Saw et al. (2012) for $g(R)$,
averaged over particle sizes, and results of simulations of the single-scale
model, in good agreement with experiments. Averaging over different particle
sizes reduces the local slope ${\rm d}\log g(R)/{\rm d}\log R$, reducing
clustering. This could explain why experimental data (with size dispersion)
display a larger value of $\hat{D}_{2}$ than observed in DNS (Figure 3a).
Larsen et al. (2018) observed spatial clustering of cloud droplets in the
dissipation range. Their results for $g(R)$ are broadly consistent with the
model predictions: below $R\sim\ell_{0}$, $g(R)$ increases as $R$ decreases.
However, since the Stokes numbers are very small (${\rm St}\approx 10^{-2}$),
the clustering is so weak that its effect on the collision rate (Section 10)
is negligible. []Size dispersionbidisperse sus- pensions contain particles of
two different sizes (with different Stokes numbers, ${\rm St}_{1}$ and ${\rm
St}_{2}$). Polydisperse suspensions involve a distribution of many different
sizes.

Settling affects fractal clustering, because settling changes how particles
sample fluid-velocity gradients. Since gravity breaks the rotational symmetry,
clustering becomes anisotropic (Bec et al., 2014, Gustavsson et al., 2014), so
that the pair correlation depends on the angle $\varphi$ between $g$ and
$\mbox{\boldmath$\hat{R}$}=\mbox{\boldmath$R$}/R$. This affects only the
prefactor, not the exponent: $g(R)=C(\varphi)R^{\hat{D}_{2}-{3}}$ (Bhatnagar,
2020). Figure 3c shows $\hat{D}_{2}$ versus ${\rm St}$ for different values of
the settling number, for DNS and model simulations. Overall we observe good
agreement. At small ${\rm St}$, fractal clustering decreases slightly as the
effect of gravity increases (smaller ${\rm Fr}$). But at large ${\rm St}$,
settling increases clustering by multiplicative amplification – rapidly
settling particles experience the flow as white noise (Bec et al., 2014,
Gustavsson et al., 2014). Note that the correlation dimension depends non-
monotonically on ${\rm Fr}$ in this regime.

Lu et al. (2010) measured the effect of electrical charges on spatial
clustering of droplets in turbulence. They determined $g(R)$ for highly
charged droplets of equal parity, and found that the resulting electrostatic
interactions affect the pair correlation function $g(R)$ at length scales
below $R/\eta\sim 4$, for ${\rm St}=0.3$ and ${\rm Re}_{\lambda}=80$. At
present there is no systematic study of how the spatial clustering depends on
the amount of charge, on ${\rm St}$, and on ${\rm Re}_{\lambda}$.

###  9.3 Inertial-range clustering

The fractal dimensions $\hat{D}_{\rm KY}$ and $\hat{D}_{2}$ discussed above
measure small-scale fractal clustering in the dissipative range of turbulence.
In turbulent flow, spatial clustering is observed not only in the dissipative
range but also in the inertial range. However, as seen from Figure 3d, the
radial distribution function $g(R)$ no longer behaves as a power-law when
$R\gg\eta$. In the inertial range, the turnover time
$\tau_{{R}}=({R}^{2}/\varepsilon)^{1/3}$ depends on the spatial scale ${R}$.
Based on this observation, Falkovich et al. (2003) suggested that the radial
distribution function behaves as $\log g(R)\propto{\rm
St}_{R}^{2}\sim\tau_{\rm p}^{2}/R^{4/3}$ (Balkovsky et al., 2001). The scale-
dependent Stokes number arises naturally in white-noise models. Simulations of
the Kraichnan model yield $\log g(R)/\log R\simeq\hat{D}_{2}({\rm
St}_{R})-3\propto{\rm St}_{R}^{2}$ (Bec et al., 2008). To which extent time
correlations change this picture is unclear. KS models cannot answer this
question, because they do not account for sweeping. In particular, the degree
of large-scale clustering obtained in KS-model simulations depends on the
parameter $\omega_{0}$, that is on the Kubo number (Section 6), see Chen et
al. (2006). There is numerical evidence for the scaling $\log g(R)\propto
R^{-4/3}$ from DNS (Bragg et al., 2015a, Ariki et al., 2018). KE-models yield
the same inertia-range scaling, but but they do not capture the Stokes-number
dependence (Bragg et al., 2015a).

An alternative picture was proposed by Bec et al. (2007). They argued and
observed in DNS that $g(R){R^{2}}$ depends on a non-dimensional ejection rate
$\Gamma(R,\tau_{\rm p})\propto u_{\rm rms}\varepsilon^{1/3}\tau_{\rm
p}/R^{5/3}$. How to reconcile this behaviour with the above ${\rm
St}_{R}$-scaling remains an open question. The function $\Gamma(R,\tau_{\rm
p})$ prescribes the scale-dependence not only of $g(R){R^{2}}=\tfrac{\rm
d}{{\rm d}R}\langle\hat{\mathscr{M}}_{R}\rangle$, but of the whole probability
distribution $p(\hat{\mathscr{M}}_{R})$. In DNS, deviations from a uniform
distribution are significant in the two tails of this distribution. The form
of the tails is explained by the statistical ejection model of Bec & Chétrite
(2007). They found that
$p(\hat{\mathscr{M}}_{R})\propto\hat{\mathscr{M}}_{R}^{\alpha}$ at small
$\hat{\mathscr{M}}_{R}$, and
$\propto\exp(-c\,\hat{\mathscr{M}}_{R}\log\hat{\mathscr{M}}_{R})$ for large
$\hat{\mathscr{M}}_{R}$, where $\alpha$ and $c$ depend only on
$\Gamma(R,\tau_{\rm p})$.

A certain form of scale invariance is recovered in the distribution of large
void sizes. DNS of two-dimensional (Boffetta et al., 2004, Goto & Vassilicos,
2006) and three-dimensional turbulence (Yoshimoto & Goto, 2007), as well as
experimental data (Sumbekova et al., 2017) suggest that the probability
distribution of very large void sizes behaves as a power-law. However,
different authors obtain different power-law exponents.

##  10 CAUSTICS AND RELATIVE VELOCITIES

\begin{overpic}[width=426.79134pt]{Fig4} \end{overpic} Figure 4:  Relative
velocities. a Multi-valued particle velocities (red and blue arrows) from a
thin slice of a three-dimensional DNS (Bec et al., 2014). The two colours
distinguish two particle clouds with different histories. b Experimental
observation of two nearby particle pairs (red and blue) with anomalously large
relative velocity, adapted from Fig. 2 in Bewley et al. (2013) with
permission. c Particle-velocity structure function $S_{2}(R)$ versus $R$ for
${\rm Sv}=0$. DNS from Bec et al. (2010) [${\rm Re}_{\lambda}=400$, small
empty symbols], and Bragg & Collins (2014b) [${\rm Re}_{\lambda}=597$, large
empty symbols], single-scale model (SSM, Equation 15) with ${\rm Ku}=10$ and
$\ell_{0}/\eta=15$ (small filled symbols), ${\rm St}\\!=\\!0$-result from
Equation 15 (solid red line), small-$R$ scaling prediction 17 (lines), and KE-
model predictions evaluated by Bragg & Collins (2014b) (large filled symbols)
c Joint distribution of $R$ and $V_{R}$ from DNS, adapted from Fig. 3(b) in
(Bhatnagar et al., 2018). The matching line $V_{R}\sim z^{\ast}R$ (see text)
is shown as a thick solid line.

Particle inertia causes large fluctuations of relative particle velocities
$\mbox{\boldmath$V$}=\mbox{\boldmath$v$}_{2}-\mbox{\boldmath$v$}_{1}$ in
turbulence. The reason is that phase-space manifolds fold at caustic
singularities, resulting in multi-valued particle velocities (Section 4). This
is illustrated in Figure 4a, which shows inertial-particle velocities in a
thin slice of configuration space. The two colours distinguish two particle
clouds that move on different branches of the phase-space attractor. Figure 4b
shows measured droplet trajectories in turbulent air (Bewley et al., 2013). As
in panel a, the two colours distinguish particle pairs originating from
different regions in phase space, allowing the pairs to pass close to each
other at anomalously large relative velocities.

###  10.1 Rate of caustic formation

The rate of caustic formation $\\!\mathscr{J}\\!$ depends sensitively on ${\rm
St}$, because dimensional analysis shows that the required strain for a
caustic to form is proportional to ${\rm St}^{-1}$. At small ${\rm St}$, the
rate of caustic formation is therefore determined by the tails of the
turbulent velocity-gradient distribution (Meibohm et al., 2023b, Bätge et al.,
2023). In the white-noise limit, diffusion approximations for the Gaussian
single-scale model yield
$\scalebox{0.75}[1.0]{$-$}\log(\mathscr{J}\\!\tau_{\rm p})\propto({\rm
Ku}\,{\rm St})^{-1}$ for ${\rm Ku}\,{\rm St}\ll 1$ (Wilkinson & Mehlig, 2005).
In the persistent limit of the Gaussian model, the rate has a different form,
$\scalebox{0.75}[1.0]{$-$}\log(\mathscr{J}\\!\tau_{\rm p})\propto{\rm
St}^{-2}$ (Falkovich et al., 2002, Derevyanko et al., 2007). Kinematic
simulations by Ducasse & Pumir (2009) and DNS by Bhatnagar et al. (2022) yield
$\scalebox{0.75}[1.0]{$-$}\log(\mathscr{J}\\!\tau_{\rm p})\propto{\rm
St}^{-1}$. In general, one expects
$\scalebox{0.75}[1.0]{$-$}\log(\mathscr{J}\\!\tau_{\rm p})\propto{F}({\rm
St})$ where ${F}\to\infty$ as ${\rm St}\to 0$. The form of ${F}$ depends on
the tails of the velocity-gradient distribution. The key point is nevertheless
that caustics form at small ${\rm St}$, leading to multivalued particle
velocities. For very small ${\rm St}$, however, phase-space contraction
reduces the effect of caustics, when $\mathscr{J}\\!\tau_{\rm p}\ll 1$.
Voßkuhle et al. (2014) concluded that caustics have a significant effect on
the collision rate for ${\rm St}{\gtrsim}0.3$. Exceptions to this rule are
model flows with bounded fluid-velocity gradients, where $\\!\mathscr{J}\\!$
vanishes below a certain Stokes number (Ijzermans et al., 2010).

[] Caustic formationFold caustics form when the particle- velocity gradients
diverge, equivalently when the infini- tesimal volume $\hat{\mathscr{V}}_{3}$
collapses. Perrin & Jonker (2014) found that caustics tend to occur in regions
of large turbulent strain. Meibohm et al. (2023a) explained this by computing
the optimal path to caustic formation in the persistent limit, at weak
particle inertia, and ${\rm Sv}=0$.

###  10.2 Relative particle velocities

Völk et al. (1980) estimated the relative velocities between identical
particles as $\langle V_{R}^{2}\rangle\sim\varepsilon\tau_{\rm p}$. Related
expressions for particles with different sizes were suggested by Mizuno et al.
(1988). These estimates fail to describe the relative particle-velocities at
small spatial separations, because the derivation ignores the dissipation-
range dynamics (Pan & Padoan, 2010). DNS results indicate that the relative-
velocity moments $m_{n}(R)\equiv\int_{-\infty}^{\infty}{\rm
d}V_{R}|V_{R}|^{p}P(R,V_{R})$ depend sensitively on both $R$ and ${\rm St}$.
Figure 4c shows DNS results for the relative particle-velocity structure
function $S_{2}(R)\equiv m_{2}(R)/m_{0}(R)$. Also shown are results of model
calculations, for the KE model as evaluated by (Bragg & Collins, 2014b), and
for the single-scale model based on Equation 15. For advected particles (${\rm
St}=0$), $S_{2}(R)\propto R^{2}$ in the smooth dissipative range. Particle
inertia enhances the relative particle velocities, more so at smaller
separations. The single-scale model describes this behaviour well, although
the weak dependence on ${\rm Re}_{\lambda}$ seen for DNS is not captured by
the model. The KE-model works well at ${\rm St}=3$, but fails at smaller
values of the Stokes number. Bragg & Collins (2014b) suggest that this may be
due to Gaussian decoupling approximations used in evaluating the model
(Section 6).

The structure function exhibits anomalous scaling at small $R$, $S_{n}(R)\sim
R^{\zeta_{n}}$. Gustavsson & Mehlig (2014) derived an approximate theory for
the exponents $\zeta_{n}$ by matching the tails of the joint distribution
$P(R,V_{R})$ at $|V_{R}|=z^{\ast}R$, with matching scale
$z^{\ast}\propto\tau_{\rm p}^{-1}$ (Figure 4d). Recognising that $P(R,V_{R})$
must have algebraic large-$V_{R}$ tails at small $R$, determined by the phase-
space correlation dimension $D_{2}$ (Section 4), one finds

$\displaystyle m_{n}(R){\ \simeq\ }b_{n}({\rm
St})(R/\eta)^{p+D_{2}-1}+c_{n}({\rm St})(R/\eta)^{{2}}$ (17)

for small separations $R$. Equation 17 predicts that $m_{0}(R)$ is
proportional to $R^{\mbox{\scriptsize min}\\{D_{2},{3}\\}-1}$ for small enough
$R$. The scaling of the radial distribution function
$g(R)=m_{0}(R)/R^{2}{\propto}R^{\hat{D}_{2}-{3}}$ implies that
$\hat{D}_{2}=\mbox{min}\\{D_{2},{3}\\}$ (Sections 4 and 9). Using this scaling
of $m_{0}(R)$, Equation 17 yields a bifractal law for the scaling exponents of
the structure functions $S_{n}(R)$, namely
$\zeta_{n}=\mbox{min}\\{n,3-{{\hat{D}}}_{2}{({\rm St})}\\}$ for $n>0$,
consistent with the DNS results for $n=2$ shown in Figure 4c. Note that the
saturation of $\zeta_{n}$ at large $n$ is a consequence of caustic
singularities. Analogous behaviour is observed in Burgers turbulence, where
the singularities are shocks. Note also that Simonin et al. (2006) suggested
that $S_{2}(R)$ approaches a constant as $R\to 0$, reflecting random
uncorrelated motion (Section 6). For $n>0$, this is true only when ${\rm St}$
is so large that $D_{2}>3$. [] Bifractal scalingThe scaling exponents of
structure functions equal $\zeta_{n}\\!=\\!h_{1}n$ for $n\\!<\\!n_{\star}$,
and $\zeta_{n}\\!=\\!h_{2}n\\!+\\!(h_{1}\\!-\\!h_{2})n_{\star}$ for
$n\\!>\\!n_{\star}$. Scaling laws of this kind arise in the $\beta$-model for
turbulent intermittency and in Burgers turbulence (Frisch, 1995). For the
bifractal law corresponding to Equation 17, $h_{2}=0$.

The two terms on the r.h.s. of Equation 17 correspond to different histories
of the relative dynamics. The first term is a smooth contribution, it comes
from particles that stayed together in the past because they approached along
the same phase-space branch. The second is a singular contribution,
corresponding to particles that followed different phase-space branches
(Figure 1). Its $R$-dependence, $\propto R^{2}$, reflects the uniform
distribution of nearby particles that came from far apart with uncorrelated
initial conditions.

There is no consensus regarding the particle relative-velocity distribution.
It has been modeled as lognormal (Carballido et al., 2010), or with stretched
exponential tails (Pan et al., 2014b). The matching ansatz outlined above
yields $P(R,V_{R})R^{{2}-D_{2}}=f(V_{R}/R)$ for small distances, with
$f(y)\approx{\rm min}\\{y,y^{D_{2}-{4}}\\}$. DNS results (Perrin & Jonker,
2015, Bhatnagar et al., 2018) at small ${\rm Re}_{\lambda}$ support this
prediction. Saw et al. (2014) measured $P(R,V_{R})$ for water droplets in air
turbulence. Their results do not show algebraic tails, likely because $R/\eta$
is too large. Measurements of Hammond & Meng (2021) with hollow glass spheres
show much broader distributions than those of Saw et al. (2014), although
their experiments explored similar parameter ranges. When the Stokes number is
large enough, the analysis of Gustavsson et al. (2008) suggests that relative
velocities are determined by the history of inertial-range fluctuations,
resulting in $P(R,V_{R})\propto\exp\\{-[V_{R}^{2}/(\tau_{\rm
p}\varepsilon)]^{2/3}\\}$. This is consistent with the results of Pan & Padoan
(2010, 2013).

Bhatnagar (2020) concluded that settling does not change the relative dynamics
very much, for identical particles. In poly-disperse suspensions, by contrast,
settling is expected to have a stronger effect on the relative velocities.

For ${\rm Sv}=0$, the effect of size dispersion upon the distribution of
$V_{R}$, is discussed by Pan et al. (2014a). In essence, particles with
different Stokes numbers tend to have larger relative velocities. Meibohm et
al. (2017) show that $S_{n}(R)$ approaches a constant for
$R<R_{\theta}\propto\theta$ with $\theta{=|{\rm St}_{1}-{\rm St}_{2}|/({\rm
St}_{1}+{\rm St}_{2})}$, causing an enhancement of the collision rate (Section
10.3).

Electrostatic charges affect the relative particle velocities (Alipchenkov et
al., 2004, Lu & Shaw, 2015), but it is not yet clear how electrostatic
interactions modify the relative-velocity theory reviewed above.

###  10.3 Models for the collision rate

Saffman & Turner (1956) computed how turbulence accelerates collisions between
small cloud droplets. They assumed that the droplets follow the flow (${\rm
St}=0$) and that they are uniformly distributed. Linearising the fluid
velocity, assuming time-independent gradients, Saffman & Turner (1956)
averaged over the gradients to obtain $\mathscr{R}_{\rm
ST}=C_{1}n_{0}(a_{1}+a_{2})^{3}/\tau_{\eta}$. Comparing with the collision
rate for droplets settling in a quiescent fluid, $\mathscr{R}_{g}\simeq
n_{0}v_{s}(a_{1}+a_{2})^{2}$ with settling speed $v_{s}=\tfrac{2\varrho_{\rm
p}}{9\varrho_{\rm f}{\nu}}|a_{2}^{2}-a_{1}^{2}|g$, leads to two conclusions:
$\mathscr{R}_{\rm ST}$ dominates for similar-sized droplets, and it increases
proportional to the turbulent shear rate.

To determine the effect of particle inertia, Sundaram & Collins (1997)
integrated Equation 4 for fluid velocities obtained from turbulent DNS. They
found that the collision rate

$\displaystyle\mathscr{R}({{\rm St}})$ $\displaystyle=-n_{0}\lim_{R\to
a_{1}+a_{2}}\int_{-\infty}^{0}\\!{\rm d}V_{R}\,V_{R}\,P(R,V_{R})$ (18)

with number density $n_{0},$ depends sensitively on the Stokes number. For
identical particles this is explained by the fact that caustics allow for
large relative velocities, and that the rate of caustic formation depends
sensitively on ${\rm St}$. A model for the ${\rm St}$-dependence follows from
Equation 17, using $\mathscr{R}({{\rm St}})\approx\tfrac{1}{2}m_{1}(2a)$. This
gives $\mathscr{R}({{\rm St}})\approx\mathscr{R}_{{\rm
smooth}}+\mathscr{J}({\rm St})\mathscr{R}_{\rm kin}$ (Falkovich et al., 2002).
Equation 17 implies that $\mathscr{R}_{{\rm smooth}}\propto a^{\hat{D}_{2}}$,
and $\mathscr{R}_{\rm kin}\propto a^{2}$, the collision rate in the kinetic
limit (Abrahamson, 1975). The second term dominates for small particles and
large enough ${\rm St}$. This occurs already for ${\rm St}>0.3$ (Voßkuhle et
al., 2011, 2014).

Some authors write $\mathscr{R}({{\rm
St}})\\!=\\!\tfrac{1}{2}g(2a)\langle|V_{R}|\rangle_{R=2a}$ (Hammond & Meng,
2021), in terms of the average $\langle\cdot\rangle_{R=2a}$ conditioned on the
particle separation at contact. This does not mean that spatial clustering and
relative velocities contribute in a multiplicative fashion, because
$\langle|V_{R}|\rangle_{R=2a}$ equals
$m_{1}(2a)/m_{0}(2a)=m_{1}(2a)/[(2a)^{2}g(2a)]$, so the factor $g(2a)$ cancels
out, see Fig. 9(d)–(f) in (Nair et al., 2022). Equation 17 shows that the
collision rate for identical particles depends on these two effects
additively. Particle-size differences increase the collision rate between
small particles because they increase their relative velocity at separations
smaller than $R_{\theta}$, where $S_{2}(R)$ approaches a constant (Section
10.2). Pan & Padoan (2014) discuss in detail how size dispersion competes with
particle inertia in determining $\mathscr{R}$.

[] Hydrodynamic interactionsfor two nearby particles, the presence of the
second particle changes the fluid velocity seen by the first one, causing them
to interact. [] Collision efficiencyaccounts for the effect of hydrodynamic
interactions on the collision rate The models described above neglect
hydrodynamic interactions, which make it harder for the particles to approach
(and to separate). In order to determine whether the particles collide or not,
one must also account for the fact that hydrodynamics breaks down when the
distance between the particle surfaces is smaller than the mean-free path of
the fluid (Sundararajakumar & Koch, 1996). For water droplets in air, this
changes the interplay between $\mathscr{R}({{\rm St}})$ and $\mathscr{R}_{g}$,
and reduce the overall collision rate (Dhanasekaran et al., 2021).

Hydrodynamic interactions can be incorporated in independent-particle models
in a heuristic fashion, by writing the collision rate as $e_{\rm
c}\,\mathscr{R}({{\rm St}})$ (Pinsky et al., 1999, 2007, Devenish et al.,
2012, Pumir & Wilkinson, 2016), where $e_{\rm c}$ is a collision efficiency
intended to account for hydrodynamic interactions. Klett & Davis (1973)
realised that the collision efficiency of similar-sized settling droplets
depends strongly on ${\rm Re}_{p}$, simply because identical droplets do not
even approach each other for ${\rm Re_{\rm p}}=0$. Klett & Davis (1973) and
later studies (Pinsky et al., 2007, Wang et al., 2008) accounted for
hydrodynamic interactions by perturbation theory in $a/R$. This describes the
droplet dynamics at separations of the order of several droplet radii
(Magnusson et al., 2022), but it does not allow to compute the collision
efficiency reliably. Studies resolving close droplet approaches were performed
only for ${\rm St}=0$ (Dhanasekaran et al., 2021, Dubey et al., 2022). In
summary, it is not understood how the collision efficiency depends on ${\rm
Re}_{p}$, ${\rm St}$, and ${\rm Sv}$.

Recent experiments (Yavuz et al., 2018, Bragg et al., 2022) measuring spatial
clustering of small particles in turbulence show that the pair correlation
function $g(R)$ is large at very small separations, much larger than predicted
by the theories summarised in Section 9. The mechanisms for this strong
clustering at very small scales remain to be understood.

[SUMMARY POINTS]

  1. 1.

Statistical models provide a qualitative understanding of the phase-space
dynamics of particles in turbulence, allowing to single out aspects of the
dynamics that are sensitive to details of the turbulent flow. In certain
cases, statistical models can be analysed systematically using dynamical-
systems and perturbation theories, making it possible to gain valuable insight
into key mechanisms and parameters.

  2. 2.

Statistical models explain that dissipative-range spatial clustering of
particles in turbulence is determined by the history of the flow encountered
by the particles. To predict small-scale clustering in a quantitative fashion,
one must therefore take into account that this history is biased by
preferential sampling of certain flow regions. Additionally, settling alters
the way in which the particles sample the flow. Specifically, settling
decreases spatial clustering for small particle inertia, and increases
clustering for large inertia.

  3. 3.

Caustic singularities result in multi-valued particle velocities, causing
continuum configuration-space descriptions to fail. Caustic formation depends
sensitively on the Stokes number, at small particle inertia, as only extreme
fluctuations of turbulent strain allow particles to detach from the flow. The
models show that this sensitivity exists in any flow, but the specifics depend
on the non-universal tails of the distribution of fluid-velocity gradients.
The models also explain that caustics are responsible for the anomalous
scaling of the particle-velocity structure functions, describing large
collision velocities.

  4. 4.

Statistical models predict which properties of the inertial-particle dynamics
depend on the Reynolds number of the turbulent flow and which do not. For
instance, fluctuations of particle separations and relative velocities depend
only weakly on the Reynolds number. This prediction is borne out by DNS and
experiments, when settling speed and particle inertia are sufficiently small
to ignore inertial-range fluctuations.

[FUTURE QUESTIONS]

  1. 1.

Many uncertainties remain regarding the hydrodynamic forces and torques on
small particles in turbulence, concerning the effect of convective fluid
inertia, in particular upon added-mass, history, and lift forces.

  2. 2.

While the moments of relative velocities of heavy particles in turbulence are
well understood at small spatial separations, their distribution is not. To
make progress, it is necessary to formulate and analyse non-Gaussian
statistical models.

  3. 3.

The relative dynamics of strongly inertial particles explores the inertial
range of turbulence. This can give rise to intricate dependencies of
preferential sampling and relative particle velocities on the Reynolds number
of the carrier flow, that remain to be understood. Formulating statistical
models that consistently describe how the large-scale fluid motion affects
smaller scales remains a challenge.

  4. 4.

Natural flows tend to be inhomogeneous and anisotropic, simply because they
contain boundaries. It is an open question how to model the effect of large-
scale structures – such as mean flow gradients – on the small-scale dynamics.

  5. 5.

To predict particle collision rates in turbulence, it is necessary to
incorporate interactions into statistical models, including not only
hydrodynamic but also electrostatic interactions. The challenge is to find
reliable models for the collision efficiency that account for particle and
fluid inertia, the breakdown of hydrodynamics below the mean free path,
droplet deformation, and van-der-Waals interactions.

  6. 6.

For non-spherical particles, translation and rotation are coupled. Developing
statistical models for their collisions requires extending the position-
velocity phase space to orientations and angular velocities, and to model
hydrodynamic interactions between non-spherical inertial particles.

## DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or
financial holdings that might be perceived as affecting the objectivity of
this review.

## ACKNOWLEDGMENTS

JB acknowledges support from French Investments for the Future (project
UCAJEDI ANR-15-IDEX-01), from PRACE (project PRA031), and GENCI (grants TGCC
t2016-2as027 and IDRIS 2019-A0062A10800). KG was supported by a grant from
Vetenskapsrådet (no. 2018-03974). BM was supported by Vetenskapsrådet (grant
no. 2021-4452), and acknowledges a Mary Shepard B. Upson Visiting
Professorship with the Sibley School of Mechanical and Aerospace Engineering
at Cornell. We acknowledge the hospitality of the KITP in Santa Barbara (USA)
where part of this review was written during the program Multiphase Flows in
Geophysics and the Environment. Statistical-model simulations were performed
on resources provided by the Swedish National Infrastructure for Computing
(SNIC). We thank A. Bragg for calculating the KE-model predictions shown in
Figures 3 and 4, and E. Bodenschatz for providing Figure 2c.

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