3. Colliding droplets in turbulent flows: a
numerical study
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op vrijdag 19 juni 2015 om 12:30 uur
door
Vincent Emile PERRIN
natuurkundig ingenieur
geboren te Thonon-les-Bains, Frankrijk.
4. Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. H.J.J. Jonker
Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof. dr. H.J.J. Jonker, Technische Universiteit Delft, promotor
Onafhankelijke leden:
Prof. dr. A.P. Siebesma, Technische Universiteit Delft / KNMI
Prof. dr. ir. B.J. Boersma, Technische Universiteit Delft
Prof. dr. H.J.H. Clercx, Technische Universiteit Eindhoven
Prof. dr. B. Mehlig, G¨oteborgs Universitet
Prof. dr. D. H. Richter, University of Notre Dame
Dr. J.P. Mellado, Max-Planck-Institut f¨ur Meteorologie
Dit werk maakt deel uit van het onderzoekprogramma van de Stichting
voor Fundamenteel Onderzoek der Materie (FOM), die deel uitmaakt van
de Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)
Bij dit onderzoek is gebruik gemaakt van de supercomputer-faciliteiten van
SURFsara
Printed by: Uitgeverij BOXPress, ’s-Hertogenbosch
ISBN 978-94-6295-208-9
Een electronische versie van deze scriptie is verkrijgbaar via:
http://repository.tudelft.nl/.
5. Samenvatting
Wolkendruppels en de manier waarom ze botsen staan aan de basis van
de vorming van wolken en het onstaan van regen. De evolutie van een
wolkendruppel in een regendruppel beslaat drie stadia. In elk stadium zorgt
een ander mechanisme voor de groei van de druppel. In het eerste stadium
is condensatie het enige effectieve groeimechanisme. In het tweede stadium
zijn zowel condensatie als door zwaartekracht veroorzaakte botsingen niet
effectief en moeten druppels voorbij de zogenaamde condensatie-coalescentie
bottleneck zien te groeien om het derde stadium te bereiken. In het derde
stadium zijn druppels groot genoeg dat ze beginnen te vallen onder het ef-
fect van zwaartekracht en daarbij kleinere, nog zwevende, druppels met zich
meenemen. Dit vergroot de kans aanzienlijk op een botsing waardoor ze snel
kunnen doorgroeien tot een regendruppel.
Het meest aannemelijk mechanisme om voorbij deze bottleneck te ko-
men in het tweede stadium is de interactie tussen turbulentie en druppels,
welke de dynamiek van de druppels aanzienlijk kan veranderen, daarbij de
botsingskansen doet toenemen en de vorming van regen versnelt. In dit
proefschrift concentreren we ons op het beter begrijpen van het effect van
turbulentie op het botsen van druppels (of algemener: van deeltjes) gebruik-
makend van direct numerieke simulaties (DNS). DNS lost het stromingsveld
op tot de kleinste Kolmogorov schalen van de stroming. Het combineren van
DNS met een Lagrangiaanse particle tracking algoritme stelt ons in staat om
de banen van individuele druppels te volgen en op deze manier de interactie
tussen stroming en deeltjes te onderzoeken. Ook stelt het ons in staat om
6. ii Samenvatting
individuele botsingen te detecteren om zo het onstaan van een botsing beter
te begrijpen.
Een van de belangrijkste mechanismen ge¨ıdentificeerd in dit proefschrift
om botsingen tussen deeltjes met gelijk grootte te faciliteren is dissipatie.
Dissipatie kan geaccocieerd worden met snelheidsgradienten in de stroming.
Turbulentie heeft de neiging om deeltjes te laten clusteren in gebieden met
een lage vorticiteit. Dit clusteren brengt de deeltjes dichter naar elkaar toe.
Hierdoor, ervaren ze echter dezelfde stroming, wat hun relative snelheid ver-
laagd evenals hun botsingskansen. Dissipatieve gebeurtenissen ontkoppelen
de deeltjes van het onderliggende stromingsveld en decorreleert hun bewegin-
gen. Grote snelheidsverschillen tussen deeltjes kunnen dan gevonden worden
op kleine afstanden wat hun botsingskansen doet toenemen. Dissipatie kan
ook geaccocieerd worden met convergerende gebieden in de stroming (i.e. met
negatieve eigenwaarden van de snelheidsgradienttensor), wat deeltjes dichter
bij elkaar brengt en botsingen bevorderd. Ondanks dat dissipatie een minder
grote rol speelt dan vorticiteit in het beinvloeden van de ruimtelijke verdel-
ing van de deeltjes, heeft het dus wel een prominente rol in het initieren van
botsingen.
Verder hebben we laten zien dat de volledige distributie van relatieve
snelheiden tussen deeltjes in een turbulente stroming nauwkeurig voorspeld
kan worden met het theoretisch model van Gustavsson and Mehlig [52]. Dit
model is gebaseerd op twee asymptotische limieten, ´e´en waarin paar diffusie
domineert (i.e. veel coherentie tussen de beweging van deeltjes) en ´e´en waarin
caustics domineren (i.e. grote snelheidsverschillen tussen deeltjes dichtbij
elkaar). De distributie van relatieve snelheden geeft niet alleen waardevolle
informatie over de botsingsfrequentie, maar ook over de relatieve snelheid
tussen de deeltjes bij impact.
Als laatste hebben we in dit proefschrift de dynamiek van wolkendruppels
aan de rand van de wolk onderzocht, waar substantiele menging plaatsvindt
tussen vochtige stijgende wolkenlucht en droge stilstaande omgevingslucht.
Druppels worden uit de wolk gemengd en verdampen, wat de lucht eromheen
afkoelt. Hierdoor ontstaat er een neergaande wolkenschil aan lucht, die de
intensiteit van de turbulentie aan de wolkenrand drastisch doet toenemen
en nog meer menging veroorzaakt. Door een complexe wisselwerking tussen
turbulentie, verdamping en zwaartekracht lijkt de wolkenrand een gunstige
plek te zijn om druppels snel te laten groeien door coalescentie. Verdamping
verbreedt significant de distributie van de druppelgroottes, wat de botsings-
kansen doet toenemen. Zwaartekracht zorgt ervoor dat druppels langer in
ongesatureerde lucht verblijven. Hiervoor verbreedt de distributie van drup-
pelgroottes nog verder en worden de botsingskansen verder vergroot.
7. Summary
Droplets and the way they collide are at the very base of the formation of
clouds and the initiation of warm rain. The evolution of a cloud droplet into
a rain droplet can be classified into three stages. For each stage different
growth mechanisms can be identified. In the first stage condensation is the
only effective mechanism. In the second stage, neither condensation nor
gravity induced coalescence are effective, and droplets have to grow past
this condensation-coalescence bottleneck to reach the third stage. In the
third stage droplets are large enough that they start to fall under the effect
of gravity, and thereby collect smaller droplets which are still hovering. This
increases greatly the collision chances and allows the droplets to grow very
rapidly into raindrops.
The most plausible mechanism to bridge this bottleneck in the second
stage is turbulence-droplet interaction, which may significantly alter droplet
dynamics, increases the collision probability and therefore accelerates rain
formation. In this thesis we have focused on better understanding the ef-
fect of turbulence on droplet (or more generally on particle) collisions us-
ing direct numerical simulation (DNS). DNS solves the flow field up to the
smallest Kolmogorov scales of the flow. Combing DNS with a Lagrangian
particle tracking algorithm allows us to identify the trajectories of individual
droplets, and to investigate the interaction between particles and flow struc-
tures. It also allows us the detect individual collisions to better understand
the mechanisms behind a collision.
One of the key mechanisms identified in this thesis to favor collisions
8. iv Summary
between same-sized particles is dissipation. Dissipation can be associated
with velocity gradients in the flow. Turbulence tends to make particles
preferentially concentrate in regions of low vorticity. This clustering brings
particles closer to each other. Thereby they experience the same fluid flow
which reduces their relative velocities and collision rate. Dissipative events
detach the particles from the underlying flow field and decorrelate their mo-
tion. Large velocity differences can then be found at small separations which
increases the collision rate. Dissipation is also associated with converging re-
gions in the flow (i.e. negative eigenvalues of the velocity gradient tensor),
which bring particles closer together and favors collisions. While dissipation
does not seem to play a role as important as vorticity in influencing the
spatial distribution of the particle field, its role is prominent in initiating
collisions.
We have also shown that the full distribution of relative velocities between
particles in turbulent flows can be accurately predicted using the theoret-
ical model of Gustavsson and Mehlig [52]. This model is based on two
asymptotic regimes, one where pair diffusion dominates (i.e. large coherence
between particle motion) and one where caustics dominate (i.e. large velocity
differences between particles at small separations). Knowledge of the distri-
bution of relative velocities between particles provides not only invaluable
information on for example the collision rate but also on the particle relative
velocities at impact.
In this thesis we have also investigated the dynamics of cloud droplets
at the edge of a cloud, where substantial mixing occurs between moist and
positively buoyant cloudy air and the unsaturated and neutrally buoyant
environmental air. Droplets are detrained out of the cloud and evaporate,
which cools the surrounding air. As a result, this evaporative cooling creates
a descending cloud shell which increases the turbulent intensity at the cloud
edge and results in even more mixing. Through a complex interplay between
turbulence, evaporation and gravity, the cloud edge appears to be a very fa-
vorable location for a fast droplet growth through coalescence. Evaporation
significantly broadens the droplet size distribution and thereby increases the
collision rate. Under the effect of gravity, droplets remain longer in unsatur-
ated air at the cloud edge which allows evaporation to broaden the droplet
size distribution even further and increases the collision rate even more.
11. CONTENTS vii
Bibliography 83
A Resolution dependence in isotropic turbulence 97
Acknowledgements 99
About the author 101
List of journal publications 103
13. CHAPTER 1
Introduction
Without clouds, life on earth would be profoundly different. Clouds can
produce a wide range of different weather conditions, such as rain, snow and
hail, all having a direct impact on our daily lives. But the role of clouds goes
much further than only affecting the weather. Clouds play a major role in
Earth’s hydrological cycle by producing precipitation, which is the primary
route for water to return to the Earth’s surface. Clouds also influence the
Earth’s climate by reflecting radiation. Low clouds in general have a net
cooling effect by reflecting solar radiation back to space, while high clouds
trap some of the outgoing infrared radiation emitted by the Earth and have
a net warming effect on the surface of the Earth. They mitigate extreme
temperature changes, by reflecting a part of the solar radiation during the
day and preventing all the day’s heat to leave during the night.
The radiative response of clouds to global warming is a major source
of uncertainty in present days climate models [21]. One of the difficulties
understanding a cloud, is that all scales are intrinsically linked, and complex
interactions occur from the largest to the smallest scales. Small scale cloud
properties such as the droplet size distribution affect the cloud albedo, the
formation of rain and the lifetime of a cloud. The large scale dynamics in
turn affect the local level of turbulence, the energy dissipation rate (which
influences the collision rate) and the local thermodynamic properties of the
14. 2 Chapter 1. Introduction
air.
To understand the relation between the large and small scales of a cloud,
we need to understand more about the evolution of a cloud droplet and the
formation of rain. The growth from cloud droplets to raindrops occurs in
three consecutive stages. For the sake of simplicity we assume that no ice is
formed during the formation of rain and that we are dealing with so-called
warm clouds. The life of a cloud droplet typically starts when supersaturated
water vapor condenses on a cloud condensation nucleus (CCN), a process
known as heterogeneous nucleation. In the first stage cloud droplets typically
have a radius of a few micrometers, which is much smaller than the smallest
scales of the turbulent flow in which they reside, which is of the order of
1 mm. The droplets are so light that they almost float around as tracer
particles and follow the air flow, unaffected by gravity. The chance for two
droplets to collide at this point is very low, and even if it would occur the
hydrodynamic forces prevent droplets to coalesce[96]. According to K¨ohler
theory, if enough water vapor is present and the supersaturation is sufficiently
high, the droplets will start to grow by condensation. The surface to mass
ratio of the droplets is very large and the condensation process can therefore
occur relatively fast. However, the rate at which the droplet radius increases
is inversely proportional to the radius itself, making the condensation process
slower when the droplets are growing. In realistic cloud conditions, growth
by water-vapor diffusion seldom produces droplets with radii up to 20µm
[49] because of the low magnitude of the supersaturation field and the time
available for the growth (≈ 103
s).
Let us go directly go to the third stage and for now skip the second
stage. For droplets with a radius larger than 40µm the gravitational force
is sufficiently larger than the drag force and the droplets start to fall. This
allows them to overtake (and thereby coalesce) with floating smaller droplets,
grow and fall even faster. Once this self-collection process starts, cloud
droplets can very quickly grow to the size of raindrops.
In the second stage droplets are too large to effectively growth through
condensation, but insufficiently large for self-collection to be effective. The-
ory predicts [59, 105] that with condensation and gravity only, it takes around
40 minutes for droplets to grow from 10 to 50µm, while the lifetime of a
cumulus cloud is approximately 30 minutes [128]. Theory therefore fails to
predict how droplets can rapidly grow during stage 2 past this condensation-
coalescence bottleneck or ’size gap’ for which neither condensational growth
nor the gravitational collision-coalescence mechanism is effective [140]. From
the three stages of the evolution of droplets, we can understand that the
droplet size and the available water sensitively influences the growth rate.
15. 3
In the case of more CCN, more but smaller droplets will form, which in-
creases the size gap. A larger size gap generally requires more time to be
bridged, delaying or even postponing precipitation. More pollution, for ex-
ample, which consists of aerosols leads to more but smaller droplets. In
polluted areas clouds have a higher albedo and a longer lifetime, effects
known as the first and second indirect aerosol effect, after Twomey [130] and
Albrecht [3], respectively. See Lohmann and Feichter [78] for a review article
on the indirect effects of aerosols.
The most plausible mechanism explaining how droplets can rapidly grow
past the size gap in the second stage is the effect of cloud turbulence (Shaw
[122], Khain et al. [65], Devenish et al. [31], Grabowski and Wang [49] and
references within). The idea is that turbulence may significantly alter droplet
dynamics, increases the collision probability and therefore accelerates rain
formation. While small scale turbulence increases the collision rate, it is not
sufficient to explain the fast growth of droplets across the size gap [132, 122,
74, 31, 49]. All turbulent scales, from the smallest to the largest have to
be taken into account. Lanotte et al. [74] for example found a systematic
increase in the broadening of the droplet size distribution by condensation
with increasing Reynolds number (where the Reynolds number is a non-
dimensional measure for the separation of turbulence length scales), which
is consistent with the idea that all scales are intrinsically linked. A larger
scale separation (i.e. a higher Reynolds number) is associated with higher
levels of intermittency [152]. Intermittency is the notion that the spatial and
temporal distribution of a quantity is far from uniform, with large regions
of low intensity interspersed by localized bursts of very high intensity, which
for example can lead to a very large local supersaturation [121] and has a
direct effect on the dynamics of droplets and the occurrence of collisions.
While enormous progress has been made in the field of cloud micro-
physics, still a lot of unanswered questions remain when trying to understand
the effect of turbulence on droplet dynamics. Understanding how droplets
collide in stage 2 will be the overarching theme of this thesis. Cloud physics
is not the only field of study, where particles (e.g. droplets) in turbulent
flow are studied. Other research areas dealing with particles in turbulent
flows are for example fuel sprays in combustion engines, dust particles in
flue gases, pneumatic transport of grains in agriculture, sedimentation in
rivers and estuaries, dust/sand storms and protoplanetary disks. To address
not only the cloud micro-physics community, in this thesis we use the more
general term particle instead of droplet, unless additional physical properties
on for example the density and radius of the particle have to be set. In all
cases we confine ourselves to small and heavy particles occupying a very low
16. 4 Chapter 1. Introduction
volume fraction in a carrier flow.
But what is the role of turbulence in enhancing the collision rate? The
amount of collisions depends in general on five individual contributions [142].
The first two contributions are the amount of particles and the radius of the
particles, the more particles and the larger they are, the more collisions will
occur. The third contribution is the spatial distribution of particles. The
more particles cluster, the higher the chances are they will collide as com-
pared to a uniform distribution. The fourth contribution to particle collisions
is the relative velocity between particles. The higher this relative velocity,
the more (and more violent) collisions will occur. The fifth contribution is
the collision efficiency or hydrodynamic interaction between particles, which
is relevant in the case of droplets [96, 97, 101, 142, 102, 141]. Particles alter
the local flow field, which can prevent them to collide with other particles. In
this thesis we focus on the spatial clustering and relative velocity distribution
of particles. The collision efficiency will not be investigated.
Decades of numerical, experimental and theoretical studies have identified
different mechanisms for influencing the spatial distribution of the particle
field in a turbulent flow. Maxey [80] introduced the concept of preferential
concentration, making particles cluster at the smallest scales in regions of
low vorticity due to finite momentum effects. The sling effect [37, 38] or
caustics [146, 148] is a more general concept describing the detachment of the
particles from the underlying flow field. As a consequence, this detachment
allows particles to cluster onto a network of caustic lines, bringing them
locally closer to each other. Wilkinson et al. [147] introduced the concept of
multiplicative amplification, which is clustering due to a series of independent
small kicks. Larger and heavier particles tend to form vertically aligned
curtain-like manifolds in the presence of gravity, profoundly altering the
spatial distribution of the particles [138, 150, 53, 91, 17].
Turbulent fluctuations in a flow lead to continuously varying drag forces
on a particle, which leads to large variations in the relative velocities between
particle pairs and results in more collisions [99]. Numerous studies have
been dedicated to better understand the effect of turbulence on the relative
velocities of particles [111, 2, 134, 149, 71, 154, 12, 90, 22, 51, 52]. When
particles have a finite inertia the formation of caustics allows large relative
velocities at small separation and increases their chances on a collision. The
clustering of particles and their relative velocities are not independent of each
other. Clustering brings particles closer to each other, thereby experiencing
the same fluid flow and reduces their relative velocities.
A lot of experimental research has also been performed to better under-
stand particle collisions. Tracking particles in turbulence, however, requires
17. 1.1 Outline 5
sub-Kolmogorov scale temporal accuracy, and therefore very fast tracking
devices. The first successful studies were performed by Porta et al. [104]
and Voth et al. [136] using a detector adapted from high-energy physics to
track particles in a laboratory. Preferential concentration was quantified
during several studies by Salazar et al. [112], Saw et al. [113], Wood et al.
[151], Saw et al. [114], Monchaux et al. [84]. An overview and possibilities of
experimental results are given by Warhaft [144] and Stratmann et al. [126].
Observations have also been used extensively to understand the evolution of
cloud droplets (e.g. Shaw [122], Siebert et al. [123], Katzwinkel et al. [63])
1.1 Outline
In this thesis we address the effect of turbulence on particle collisions using
direct numerical simulation (DNS). DNS solves the flow field down to the
smallest Kolmogorov scales of the flow (∝ 1mm), making it computationally
very expensive. Combing DNS with a Lagrangian particle tracking algorithm
allows us to track the trajectories of individual particles, and to investigate
the interaction between particles and flow structures. It also allows us the
detect individual collisions and to better understand the mechanisms behind
a collision. This thesis is outlined in the following way.
In chapter 2, we investigate what makes particles collide. We use homo-
geneous and isotropic turbulence (which is believed to be representative for
the cloud interior) and investigate the behavior of particles with a Stokes
number of unity, where the Stokes number is the ratio between the charac-
teristic time scale of the flow and the particle response time. For a Stokes
number of unity, particles tend to resonate with the flow and a high amount
of clustering can be found. For cloud-like conditions (i.e. for a typical cumu-
lus cloud where the mean dissipation rate ≈ 0.05m2
s−3
[119]), a Stokes
value of unity corresponds to droplets with a radius of 30µm, and thus in-
side the condensation-coalescence bottleneck. By conditionally sampling the
flow field on the particle position and collision locations, we aim to better
understand the role of turbulence on the collision process. We also track
the path of particles before a collision, and thereby try to reconstruct the
physical picture of a typical collision and attempt to identify what makes
particles collide.
We investigate the distribution of relative velocities at small separations
in chapter 3. The distribution of relative velocities not only influences the
collision rate of particles, but influences also the intensity at which particles
collide and the characteristics of the collision. Gustavsson and Mehlig [52]
18. 6 Chapter 1. Introduction
proposed a model describing the full distribution of the relative velocities of
neighboring identical inertial particles as a function of their separation using
only the fractal correlation dimension of the particle distribution. The cor-
relation dimension is a measure for the fractal dimensionality of the particle
distribution. The model of Gustavsson and Mehlig [52] has been validated
for randomly mixing flow. We investigate to what extent this model is valid
for particles in turbulent flows.
In chapter 4, we investigate the effect of the local flow field on the collision
probability. We use the eigenvalues of the local velocity gradient tensor to
categorize the local flow structure into different types of saddle nodes and
vortices. We use both DNS and a conceptual framework to better understand
the effect of individual flow structures on particle collisions.
In chapter 5, we investigate droplet dynamics at the edge of a cloud.
Inside a cloud, moist air possesses positive buoyancy, resulting in updrafts
which promote heavy mixing at the cloud edge with the dryer environmental
air. In this undersaturated environment, droplets evaporate, which gives rise
to the formation of a subsiding cloud shell [56, 61, 57, 63]. In this chapter we
investigate the role of evaporation, coalescence and gravity on the intensity
of the mixing-layer and on the evolution of the droplet size distribution.
We end this thesis with concluding remarks and recommendations for
future work in chapter 6.
19. CHAPTER 2
Preferred location of droplet collisions in turbulent flows
This study investigates the local flow characteristics near droplet-
droplet collisions by means of direct numerical simulation (DNS)
of isotropic cloud-like turbulence. The key finding is that, gen-
erally, droplets do not collide where they preferentially concen-
trate. Preferential concentration is found to happen as expected
in regions of low enstrophy (vorticity magnitude), but collisions
tend to take place in regions with significantly higher dissipation
rates (up to a factor of 2.5 for Stokes unity droplets). Investiga-
tion of the droplet history reveals that collisions are consistently
preceded by dissipative events. Based on the droplet history
data, the following physical picture of a collision can be con-
structed: enstrophy makes droplets preferentially concentrate in
quiescent flow regions, thereby increasing the droplet velocity
coherence, i.e. decreasing relative velocities between droplets.
Strongly clustered droplets thus have a low collision probability,
until a dissipative event accelerates the droplets towards each
other. We study the relation between the local dissipation rate
and the local collision kernel and vary the averaging scale to
relate the results to the globally averaged collision and dissipa-
tion rates. It is noted that, unlike enstrophy, there is a positive
20. 8 Chapter 2. Preferred location of droplet collisions in turbulent flows
correlation between the dissipation rate and collision efficiency
that extends from the largest to the smallest scales of the flow.
1
2.1 Introduction
In cloud physics, droplets and the way they collide, are at the very base of
the formation of clouds and the initiation of warm rain. This stage in which
droplet growth is dominated by collision and coalescence occurs after the
condensation stage, in which condensation is the leading process. The third
and last stage of rain formation is the sedimentating stage in which gravity
plays a crucial role (e.g. Shaw [122]).
Previous studies [13, 41, 119] show that the collision efficiency can be
expressed as a function of the mean dissipation rate , to the extent that
increasing the mean dissipation rate increases the collision efficiency. Yet, for
high Reynolds number flows it is well known that the spatial and temporal
distribution of dissipation rates is far from uniform, with large regions of
low dissipation interspersed by localized bursts of very high dissipation rates
[43]. This particular characteristic of turbulence, referred to as intermittency,
makes one wonder whether the mean value is sufficiently able to represent
the collision process. Or, phrased differently, how large should an averaging
length l be for l to be a meaningful proxy for collision efficiency.
To further explore this issue, let us consider the following gedanken ex-
periment. Consider three domains with the same mean dissipation rate
and the same number of droplets, but with different spatial arrangements
of (see Fig. 2.1). We divide domain 2 into four subdomains, and domain
3 in nine subdomains, and vary the intermittency by setting some subdo-
mains to zero dissipation while increasing in others. Pursuing the notion
that the mean collision rate in a subvolume depends on the mean dissipation
rate of that subvolume, one concludes that no collisions take place in the
(white) subdomains with zero dissipation rate. So to maintain the mean
collision rate averaged over the entire volume, the subdomains of domain 2
and 3 must locally produce collisions rates that are 4 and 9 times higher,
respectively. This can only hold if the collision rate in a subdomain depends
linearly on the value of in that subdomain. However, even if this linear
dependence were true for large enough subdomains, one might further refine
the subdomains until one reaches a scale where a high dissipation rate might
1Published as: V. E Perrin and H. J. J Jonker. Preferred location of droplet collisions in
turbulent flows. Phys. Rev. E, 2014. Section 2.5 has been added to the original manuscript.
21. 2.1 Introduction 9
(a)
4 ε
(b)
9 ε
(c)
Figure 2.1: Three domains with the same mean dissipation rate with different
spatial distributions of . As a simplistic representation of intermittency, the white
subdomains have zero dissipation rate.
be associated with ejection of the droplets, making them cluster in more
quiescent regions. In other words, at some (small) scale it is not unthinkable
that a local high dissipation rate suppresses rather than enhances the local
droplet collision rate.
The above cartoon of intermittency is obviously much too simplistic, but
it underlines the importance of the non-uniformity of the dissipation field
for collisions. After all, the flow field of a cumulus cloud is far from uni-
form, in particular near the cloud edges [109, 123, 56]. It also underlines
the importance of understanding not only where droplets collide on average
in a turbulent flow, but also the circumstances preceding a collision. The
aim of this study is therefore to gain a better physical understanding of the
processes surrounding droplet collisions in turbulent flows. To this end, we
study isotropic turbulence with direct numerical simulation (DNS), tracking
droplets and collisions in a Lagrangian framework. To get a better under-
standing of collisions, we investigate the role of the local flow field conditions
of dissipation and enstrophy in this process. Conditional sampling allows us
to find which flow conditions favor collisions and to investigate whether the
positive correlation between dissipation and collisions still holds at small
scales. We will also look at mean droplet trajectories just before a collision
to investigate the flow conditions a droplet has traveled through.
The effect of both the Stokes number and the Reynolds number on the
collision preferences will be studied as well.
Furthermore, the role of nonuniformity is addressed by determining the
relation between the local dissipation rate and the local collision kernel and
22. 10 Chapter 2. Preferred location of droplet collisions in turbulent flows
by comparing the results with their mean counterparts. Finally we make an
estimate of the relevant scales involved in droplet dynamics and the scales
involved in collision dynamics.
2.2 Background
2.2.1 Droplet dynamics
Every droplet in a turbulent flow is to some extent influenced by turbulence.
The full equations of motion have been described by Maxey and Riley [81].
Many terms in these equations can be neglected when considering cloud
droplets, since the density of cloud droplets ρp is high compared to the
density of air ρf and since the radius r of droplets is small compared to the
Kolmogorov scale η of the flow. These assumptions reduce the equations to:
dvi(t)
dt
=
ui[xi(t), t] − vi(t)
τp
(2.1)
dxi(t)
dt
= vi(t) (2.2)
Gravity is omitted in this paper since it adds complexity to the problem in
a delicate way. The combined effect of turbulence and gravity is not merely
an addition of separate phenomena (see Woittiez et al. [150]). Under the
assumption of Stokes drag, τp = 2ρpr2
/(9ρf ν) is the droplet relaxation time
with ν the kinematic viscosity of the carrier fluid. The interaction between
the flow and the droplet can be described with the use of τp by the Stokes
number:
St =
τp
τη
(2.3)
where τη = (ν/ )
1/2
is the Kolmogorov time scale. In the limit of St →
0, droplets follow the flow, and in the limit of St → ∞ droplets are not
influenced by the flow. For St ≈ 1, droplets resonate with the flow and
cluster in regions of low enstrophy and become preferentially concentrated
[138].
23. 2.2 Background 11
2.2.2 Collision statistics
The average number of collisions ˙N12 per unit volume and unit time between
two groups of droplets with radii r1 and r2, is given by:
˙N12 = N1N2Γ12 (2.4)
where N1 and N2 are number concentrations of the two different groups and
Γ12 is the collision kernel. The collision kernel can be expressed as follows
[127]:
Γ12 = 2π(r1 + r2)2
|wr| g(r1 + r2) (2.5)
where |wr| is the magnitude of the radial relative velocity and g(r1 + r2)
the radial distribution function (RDF) at contact describing the spatial non-
uniformity of the droplet concentration. A value of g(r1 +r2) = 1 indicates a
uniform droplet concentration, whereas higher values are indicative of clus-
tering. Eq. (2.5) shows that the chance of colliding proportionally increases
to the relative velocity between the droplets and proportionally to the droplet
RDF.
2.2.3 Flow field statistics
Both the local dissipation rate and the local enstrophy Ω turn out to play
an important role in the spatial distribution of the droplets. The dissipation
has been computed using its formal definition = 2νSijSij, where Sij =
1
2
∂ui
∂xj
+
∂uj
∂xi
is the symmetric part of the deformation tensor of the flow.
Since we are focusing on local values of the dissipation rate, it is important
to precisely specify the employed definition of dissipation rate. For example,
using ˜ = ν ∂ui
∂xj
∂ui
∂xj
(termed pseudo-dissipation by e.g. Pope [103], p132) will
yield the same volume averaged values but can differ locally. While enstrophy
is often defined as the vorticity magnitude (i.e. Ω = |ω|2
), in this study we
define the enstrophy analogously to the dissipation rate as Ω = 2νAijAij,
where Aij = 1
2
∂ui
∂xj
−
∂uj
∂xi
is the anti-symmetric part of the deformation
tensor of the flow.
Luo et al. [79] showed that the instantaneous spatial distribution of in-
ertial droplets correlates well with the Laplacian of pressure 2
p. This was
also shown for very light particles [29]. For isotropic turbulence a direct
relation can be established between the enstrophy, the dissipation rate and
the Laplacian of pressure [43]:
Ω − = 2
P (2.6)
24. 12 Chapter 2. Preferred location of droplet collisions in turbulent flows
where P is a rescaled pressure P = (2ν/ρf )p.
A useful dimensionless number in isotropic turbulence is the Taylor based
Reynolds number Reλ, which can be calculated as follows:
Reλ =
u λ
ν
; λ =
15νu 2 1/2
(2.7)
where λ is the Taylor microscale and u the root-mean-square of the velocity
fluctuations.
2.3 Numerical setup
In order to explicitly simulate the turbulence, we use a direct simulation
code [133] to solve the incompressible Navier-Stokes equations on a uniform
staggered grid:
∂ui
∂xj
= 0 (2.8)
∂ui
∂t
+ uj
∂ui
∂xj
= −
1
ρf
∂p
∂xi
+ ν
∂2
ui
∂x2
j
(2.9)
where ui are the three velocity components, p is the pressure field, ν is the
kinematic viscosity and ρf is the fluid density. The Navier-Stokes equations
are discretized by the finite-volume method, with second-order central differ-
ences in space and Adams-Bashforth in time. We use a triple-periodic com-
putational domain. Time stepping is restricted by the Courant-Friedrich-
Lewy criterion using a Courant number C of 0.25. The code also makes
use of the MPI communication protocol as it is parallelized by domain de-
composition in two dimensions, making the code highly scalable and fit for
modern supercomputers.
Since a turbulent system is inherently dissipative, energy is injected at the
lowest wavenumber. To this end, we employ a forcing scheme similar to that
used by Woittiez et al. [150], using a nudging time scale τforc = 0.5 ν/ t
[89] to add kinetic energy to the largest scales. This energy has been set to
(0.25 tL)2/3
; t denotes the target mean dissipation rate of the simulation,
which is the mean dissipation rate we aim for (it is not necessarily exactly
equal to the mean dissipation rate of the actual simulation), and L denotes
the physical size of the computational domain. The use of DNS limits the
range of scales that can be resolved. As a result, the domain size and the
25. 2.3 Numerical setup 13
Reynolds number are limited and several orders of magnitude lower than in
real convective clouds.
Typical energy and dissipation spectra of such a simulation (in this case
R3; see Table 2.1 for more details) are shown in Fig. 2.2. It can be seen that
the DNS properly resolves the flow down to the smallest scales. In addition
we performed several resolution dependence tests to ascertain that flow and
droplet features were both properly resolved (see the Appendix A).
10
−2
10
−1
10
0
10
1
10
−10
10
−5
10
0
10
5
κη
E(κ)/ǫ2/3
η5/3
k−5/3
10
−2
10
−1
10
0
10
1
0
2
4
6
8
10
12
14
16
18
κD(κ)/ǫ
Energy spectrum
Dissipation spectrum
Figure 2.2: Energy and dissipation spectra for simulation R3.
The equations of motion of the droplets Eq. (2.1) and Eq. (2.2) are solved
using a second-order Runge-Kutta scheme. The velocity of the flow field at
the droplet position is computed using trilinear interpolation.
The collision routine checks the number of collisions and computes the
collision kernel both dynamically using Eq. (2.4) and kinematically using
Eq. (2.5). The algorithm of Chen et al. [27] is used to detect collisions,
which uses cell indexing and linked lists to check only droplet pairs that
could collide within one time step. The cost of this algorithm is O(
27N2
p
2NxNyNz
),
where Nx, Ny and Nz define the size of the computation domain in the x,
y, and z direction, respectively and Np is the number of droplets present in
the computational domain. To ensure that all collisions are detected, the
maximum travel distance of the droplets is restricted to half a grid distance
by using a dynamically adaptive timestep for the droplets.
Since the radial distribution function in Eq. Eq. (2.5) is defined at the
contact of two droplets and is therefore theoretically determined only by
droplets that are exactly a distance r1+r2 apart, a greater number of samples
26. 14 Chapter 2. Preferred location of droplet collisions in turbulent flows
Run L (m) Nx Reλ (m2/s3) u (m/s) r (µm) St Np/106
R1 0.2 1603 110 4.5x10−2 0.15 30 0.64 0.5
R2 0.4 3203 190 4.7x10−2 0.20 30 0.64 2
R3 0.6 5123 230 3.9x10−2 0.22 30 0.64 15
St1 0.2 1603 110 4.5x10−2 0.15 10 0.07 3
St2 0.2 1603 110 4.5x10−2 0.15 20 0.29 1
St3 0.2 1603 110 4.5x10−2 0.15 30 0.64 0.5
St4 0.2 1603 110 4.5x10−2 0.15 40 1.14 0.4
St5 0.2 1603 110 4.5x10−2 0.15 50 1.78 0.4
St6 0.2 1603 110 4.5x10−2 0.15 60 2.57 0.4
E1 0.6 5123 199 8.5x10−3 0.14 30 0.29 15
E2 0.6 5123 268 3.9x10−2 0.20 30 0.64 15
E3 0.6 5123 292 1.0x10−1 0.31 30 0.91 15
Table 2.1: Overview of the simulations. The R simulations study the impact of the
Reynolds number, the St simulations have been performed to investigate the Stokes
number effect, and the E simulations explore the effect of the mean dissipation
rate. Each simulation is shown together with the dimensions of the domain L, the
number of grid points Nx, the Taylor-based Reynolds number Reλ, the radius of
the droplets r, the Stokes number St, and the number of droplets Np. Note that
simulations R1 and St3 are identical and R3 and E2 are also identical and that all
simulations are monodisperse.
is acquired by considering all droplet pairs that are separated by a distance
of r1 + r2 ± δ. The same value as that used by Wang et al. [139] has been
used (i.e. δ = (r1 + r2)/100).
Three different sets of simulations have been performed to investigate the
effect of the Reynolds number (marked as R runs in Table 2.1), the effect of
the Stokes number (marked as St runs), and the similarities between local
and mean collision kernels (marked as E runs). All simulations (except the
E marked runs in Table 2.1) have been performed using a target dissipation
rate of t = 0.05m2
/s3
. The runs E1, E2 and E3 have been run with t set
to 0.01m2
/s3
, 0.05m2
/s3
and 0.1m2
/s3
, respectively. The spatial resolution
limit of the simulations kmaxη, where kmax = Nx/2 and η = (ν3
/ )1/4
, has
a value in between 1.2 and 2.1 for all simulations. More on resolution effects
can be found in the Appendix A.
In all simulations, droplets are released after 2.0s of simulation time and
the collision routine starts after 5.0s.
27. 2.4 Results 15
2.4 Results
2.4.1 Preferred location of droplet collisions
This section investigates the flow field characteristics favorable to droplets
(St 1) and collisions. The results are obtained from the R3 simulation
(see Table 2.1). Fig. 2.3 shows PDF’s of the dissipation rate [Fig2.3(a)],
the enstrophy Ω [Fig2.3(b)], and the Laplacian of the (rescaled) pressure
2
P [Fig2.3(c)]. Black lines represent the flow field, red lines the field when
conditionally sampled on positions of the droplets, and blue lines the field
conditionally sampled on locations of the collisions. Comparing the PDFs
of the dissipation and the enstrophy of the flow field, it can be observed
that enstrophy has a broader PDF and is therefore more intermittent, which
has been reported previously in literature for low Reynolds numbers; see,
e.g. Nelkin [85].
Fig. 2.3 shows that droplets concentrate in regions characterized by a
lower Ω value than the flow. This is clearly consistent with the theory of
preferential concentration [138]: Enstrophy swings out the droplets. Dissipa-
tion does not seem to influence the spatial position of the droplets. Droplets
also tend to cluster where 2
P 0, which has previously also been found
by Luo et al. [79].
Fig. 2.4 shows the joint PDF of the dissipation and the vorticity of the
flow field (a), of the field field conditionally sampled on the droplet positions
(b) and of the field field conditionally sampled on the collision locations (c)
of simulation R3. The joint PDF of the flow field agrees very well with
results previously found by Yeung et al. [153], both performed at a Reynolds
number Reλ ≈ 230.
It is tempting to presume that collisions occur where the droplet con-
centration is highest, implying that the statistics conditioned on collision
positions would yield similar results as the statistics conditioned on droplet
positions, but Fig. 2.3 and Fig. 2.4 prove otherwise. Both figures shows that
collisions occur at places where the dissipation rate is significantly higher
than where droplets reside (up to a factor of 2.5). Also the enstrophy at
collision locations is higher than at droplet locations. Both effects can also
be seen in the PDF of 2
P (Fig. 2.3c) which for collisions is strongly skewed
towards negative values. Fig. 2.3d concisely summarizes the main message as
it shows the averages conditioned on droplet locations and collision locations,
respectively, in comparison to the mean flow properties. The latter obeys
= Ω which implies that the point ( , Ω ) must be located right on
the black line that represents 2
P = 0. The graph shows that, on average,
28. 16 Chapter 2. Preferred location of droplet collisions in turbulent flows
10
-1
10
0
10
1
10
2
0
0.2
0.4
0.6
0.8
1
ǫ/ ǫflow
ǫp(ǫ)
Flow
Droplets
Collisions
10
-1
10
0
10
1
10
2
0
0.2
0.4
0.6
0.8
1
Ω/ Ωflow
Ωp(Ω)
Flow
Droplets
Collisions
-0.04 -0.02 0 0.02 0.04
0
50
100
150
c
∇2
P
p(∇2
P)
Flow
Droplets
Collisions
0 1 2 3
0
1
2
3
d
ǫ / ǫflow
Ω/Ωflow
Flow
Droplets
Collisions
∇2
P=0
10
-4
10
-2
10
0
10
2
10
410
-8
10
-6
10
-4
10
-2
10
0
ǫ/ ǫflow
ǫp(ǫ)
10
-4
10
-2
10
0
10
2
10
410
-8
10
-6
10
-4
10
-2
10
0
Ω/ Ωflow
Ωp(Ω)
2
P = 0
Figure 2.3: (Color) Fig. (a), (b) and (c) show PDF’s of respectively , Ω and
2
P, sampled over the flow field (black line), sampled over the locations of the
droplets and sampled over the locations of the collisions. The insets show the tails
of the distributions. Fig. (d) shows the mean of Fig.s (a) and (b).
droplets reside in regions of low enstrophy and average dissipation, whereas
collisions favor regions with appreciable dissipation rates and with enstrophy
values that are comparable to or slightly higher than the flow average value.
The figure leads to the conclusion that collisions do not tend to occur where
most of the droplets are located. This makes sense to the extent that prefer-
ential concentration favors droplet clustering, thereby increasing the velocity
coherence of droplets within a cluster. This increase in coherence implies a
decrease in the relative velocity between droplets and reduces the collision
probability. Recalling Eq. (2.5), it can be concluded that the gain via the
increased radial distribution function is outweighed by the loss in relative
velocity.
To get a better view on the flow conditions surrounding droplet collisions,
we have sampled the local flow field around every collision so as to obtain
radial profiles of dissipation, enstrophy, and 2
P. The result is shown in
Fig. 2.5. A local maximum can be observed in the dissipation, whereas
29. 2.4 Results 17
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
log10[Ω/Ωflow]
log10[ǫ/ ǫflow ]
a
-5-4
-4
-3
-2
-1
-0.3
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
b
log10[Ω/Ωflow]
log10[ǫ/ ǫflow ]
-5
-5
-4
-3
-2
-1-0.3
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
c
log10[Ω/Ωflow]
log10[ǫ/ ǫflow ]
-2
-2
-2
-1
-1
-0.3
Figure 2.4: Fig. (a), (b) and (c) show the joint PDF’s of the dissipation
and the enstrophy sampled on the flow field, the droplets positions and the col-
lision locations, respectively. Both the x-axis and the y-axis are on a logarithmic
scale. The isolines are also scaled logarithmically and correspond to the values
10−0.3
, 10−1
, 10−2
, 10−3
, 10−4
and 10−5
, respectively.
a minimum can be observed in the enstrophy as well as in the pressure
Laplacian. Note that the local enstrophy is maximum near r/η ≈ 15, where
η = ν3
/
1/4
is the Kolmogorov microscale. Apart from a spatial analysis,
it is also interesting to conduct a temporal analysis. For each collision, a
mean droplet trajectory is determined for 100 time steps before and after
a collision. Fig. 2.5 shows the mean droplet trajectory prior to a collision
in terms of dissipation rate, enstrophy and, absolute velocity when averaged
over all collisions, where t = 0 represents the actual collision (dashed line).
It is interesting to note that prior to a collision the enstrophy profile is still
increasing. From this, one can infer that collisions are generally not the
result of droplets that are swung out of a vortex, which excludes vortices
as being the primary source of relative velocities and initiators of collisions.
However, a significant dissipation peak can be found prior to a collision,
which emphasizes the important role of dissipative events for the collision
process. The time scale involved is of the order of τη, which is of the same
order as the droplet relaxation time τp. Absolute velocity is only marginally
increased.
Based on these results the following physical picture emerges of the col-
lision process in turbulent flows for droplets with St 1. Droplets preferen-
tially cluster under the influence of enstrophy; however they do not collide
yet since the increase in velocity coherence yields a decrease in relative ve-
locity. Once clustered, dissipative events are found to precede collisions,
because they appear vital for decorrelating the droplet velocities. A possible
30. 18 Chapter 2. Preferred location of droplet collisions in turbulent flows
0 10 20 30 40
-1
-0.5
0
0.5
1
1.5
2
2.5
3
r/η
<ε|r>/<εflow
>
<Ω|r>/<Ωflow
>
<∇2
P|r>/<εflow
>
-1.5 -1 -0.5 0 0.5 1 1.5
1.4
1.9
2.4
2.9
t/τη
<ε>/<εflow
>,<|vp
|>/<u0
> -1.5 -1 -0.5 0 0.5 1 1.5
0.9
1
1.1
1.2
<Ω>/<Ωflow
>
<ε|d>/<εflow
>
<Ω|d>/<Ωflow
>
<|vp
|>/<u0
>
Figure 2.5: (Left) Radial profile of the sampled flow field around collisions. Shown
are the dissipation rate (solid line), the enstrophy (dashed line) and the Laplacian
of the pressure (dotted line).(Right) Conditionally sampled dissipation rate (solid
line), enstrophy (dashed line) and, velocity magnitude (dash-dotted line) of droplets
before and after they collide. The x-axis shows the time prior and after a collision.
source for these dissipative events could be the iteraction between vortex
filaments [116].
The need for dissipation to collide is in agreement with the existence of
’caustics’. Velocity gradients make droplets detach from the flow field which
allows large relative velocities at small separations. More research is needed
to investigate this need for dissipation.
It should be noted that this physical picture of the collision process is
reconstructed for a monodisperse droplet distribution. In real clouds, this
is by far not the case [23]. According to Celani et al. [26], Lanotte et al.
[74], turbulent velocity fluctuations cause a large spread of the droplet size
distribution. It is very interesting to investigate to what extent the physical
picture sketched for monodisperse droplet collisions is valid for a polydisperse
droplet size distribution. Collisions in such a distribution depend less on the
decorrelation of the droplet velocities by a dissipative event since significantly
less coherence is found between droplets of different size [150].
2.4.2 Reynolds number and Stokes number effects
This section examines the effects of the Reynolds number and the Stokes
number on the preferred droplet and collision locations. First three sim-
31. 2.4 Results 19
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
3
ǫ / ǫflow
Ω/Ωflow
Flow
Droplets
Collisions
∇2
P=0
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
3
ǫ / ǫflow
Ω/Ωflow
Flow
Droplets
Collisions
∇2
P=0
Figure 2.6: (Left) Effect of the Reynolds number on the preferred droplet positions
(red dots) and collision locations (blue dots). The larger the dot, the higher the
Reynolds number. The axes have been scaled with the mean flow dissipation and
enstrophy. (Right) Effect of the Stokes number on the preferred droplet positions
(red dots) and collision locations (blue dots). The larger the dot, the higher the
Stokes number. The axes have been scaled with the mean flow dissipation and
enstrophy. Simulation details can be found in Table 2.1.
ulations have been performed with varying Reynolds numbers (R1-R3; see
Table 2.1). From the data we construct a figure comparable to Fig. 2.3d in
order to show the impact of the Reynolds number, see Fig. 2.6. Interestingly
the droplet preferred location (0.6 Ωflow and flow ) is not influenced by
the Reynolds number. The collisions tend to occur at even larger dissipation
rates and enstrophy when increasing the Reynolds number.
It is interesting to note that for high Renolds numbers, Nelkin [85] ana-
lytically shows that the anomalous scaling exponent of dissipation and en-
strophy are equal. Recent studies by Schumacher et al. [116] and [153] sug-
gest that for high Reynolds number, not only do dissipation and enstrophy
scale in the same manner, but extreme events in both would also tend to
spatially occur together. The effect this spatial correlation in extreme events
would have on mean droplets and collision statistics would be interesting to
study, but is presently computationally very demanding.
A similar plot can be made to show the impact of the Stokes number. In
simulations St1-St6, droplets with a radius between 10 µm (St = 0.07) and
60 µm(St = 2.45) have been used (see Table 2.1). Results are presented in
32. 20 Chapter 2. Preferred location of droplet collisions in turbulent flows
Fig. 2.6. In the limit cases of St → 0 and St → ∞, both droplet and collision
preferred locations are statistically identical to the flow field mean values
(black dot). Droplets with zero Stokes number follow the streamlines and
will not preferentially concentrate. Stokes infinity droplets are not influenced
by the flow field at all and will also not preferentially concentrate. The
radial distribution function for a simulation with t = 0.05m2
/s3
peaks at
a droplet radius of 30 µm [150] since they become resonant with the small
scale vortices. The higher this resonance, the easier the droplets can reach
quiescent regions of the flow (i.e. with low enstrophy). This can be seen in
Fig. 2.6, where the red dot corresponding to 30 µm droplets has the lowest
enstrophy value. Droplets with a larger or smaller radius are all located
closer to the flow mean (black dot). For the collisions, we see that the blue
dots follows an elliptical path, with the 20 µm droplets at the extremity.
The reason why heavy droplets prefer a lower enstrophy is not yet entirely
clear.
2.4.3 Local versus mean view on collisions
In this section we return to the thought experiment formulated in the Intro-
duction, where the issue of intermittency was addressed. The question was
whether a mean collision kernel can be properly represented by the mean
dissipation rate or whether a more local approach is needed. We concluded
that a mean approach might work only if the collision kernel is locally a
linear function of the dissipation rate; otherwise the nonuniformity of the
dissipation field has to be taken into account.
By conditionally sampling the local dissipation rate at each collision
(i.e. |c), it is possible to determine the number of collisions between and
+ d . This enables one to link the collision statistics to and hence to
calculate within a single simulation the collision efficiency Γ as a function
of the (local) dissipation rate . One can regard Γ as a decomposition of
the volume mean collision efficiency Γ since they are related via
Γ =
∞
0
Γ P( )d (2.10)
where P( ) represents the PDF of of the flow field [see e.g. the black line in
Fig. 2.3 (a)]. We have conducted such an analysis for three different target
dissipation rates t, leading to three different values of = P( )d ; see
runs E1-E3 in Table 2.1. The results have been plotted in Fig. 2.7. In the
same plot, the mean values of the collision kernel and the mean dissipation
33. 2.4 Results 21
rate are indicated as dots. It is interesting to note that all three local collision
kernels are monotonically rising functions of . However, one also notes
that the dependence of the local collision kernels on the local values of
is essentially nonlinear. For the simulation E3, the collision rate seems to
saturate for larger values of the dissipation rate. We can therefore conclude
that intermittency and the nonuniformity of the flow field make it nontrivial
to derive a general relationship between the mean collision kernel and the
mean dissipation rate if no reference to the averaging length scale is specified.
0 0.02 0.04 0.06 0.08 0.1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
ε/<εflow
>
Γε
9
10
Γε
@ εt
=0.01
Γε
@ εt
=0.05
Γε
@ εt
=0.1
<Γ>
1 2 8 32 128 512
0.5
1
1.5
2
2.5
l/∆
<εl
|d>/<εflow
>
<εl
|c>/<εflow
>
<Ωl
|d>/<Ωflow
>
<Ωl
|c>/<Ωflow
>
Figure 2.7: (Left) Relation between the local dissipation rate and the local collision
kernel Γ . The local collision kernel is related to the mean collision kernel (indicated
as dots with corresponding linestyle) via Eq. (2.10). (Right) Conditionally sampled
statistics of coarse grained values of dissipation rate and enstrophy. The sampling
is based on droplet positions (red lines) or collision locations (blue lines). The
averaging length scale varies from grid size l = ∆ to domain size l = L = 512∆.
.
To better understand the influence of the averaging size, we repeat the
analysis where we conditionally sample enstrophy and dissipation rate based
on the occurrence of droplets and collisions; however, now, instead of using
the local values of Ω and , we first determine coarse-grained values Ωl and
l when a collision is detected. This analysis provides conditional averages
Ωl|c and l|c for collisions and Ωl|d and l|d for droplet locations.
Here l denotes the linear size of the averaging volume, which we vary from
l = ∆ (grid scale) to l = L (domain size) by factors of 2. For l = ∆, one
exactly retrieves the results of the previous section based on the local values,
34. 22 Chapter 2. Preferred location of droplet collisions in turbulent flows
whereas for l = L the conditionally sampled averages should get very close
to the flow-field averages (there is still a small difference due to temporal
correlations that make L|c ).
Fig. 2.7 is obtained from simulation R3 and shows the coarse-grained
dissipation rate and enstrophy when conditionally sampled on the droplet
positions (red lines) and collision locations (blue lines) for different length
scales l. As expected, one observes for larger averaging volumes a gradual
convergence to the mean flow statistics. It is interesting to note that the
differences between and Ω disappear at l/∆ ≥ 16, a scale that perhaps
could be associated with the value of the Taylor scale (see Eq. Eq. (2.7)).
Apparently at this scale the average of the pressure Laplacian effectively
vanishes, implying that enstrophy and dissipation are interchangeable from
this scale onward. However, at smaller scales it is important to retain the
distinction between and Ω. Indeed, one notices a consistent positive cor-
relation between collisions and l all the way to the smallest scale of the
flow. However, this is not the case for Ωl, which, after a maximum, exhibits
a reduced correlation at smaller scales. This finding is consistent with Fig.
2.5, which revealed that the enstrophy field increases further away from the
collision location; so if one increases the averaging area, one includes the
distant enstrophy contributions in the average.
Finally, Fig. 2.7 provides an estimate of the relevant scales involved in
preferential concentration and collisions. The effect of preferential concen-
tration becomes small at a scale l/∆ 16. The collision-based statistics,
however, exhibit a scale dependence up to much larger scales, i.e. almost as
large as domain size.
2.5 Occurrence of caustics
The previous results, and especially Fig. 2.3 and 2.5, are very insightful in
how a mean collision occurs. Recent studies however suggest that it could
be difficult to define a mean collision. The formation of caustics [146, 148]
(also known as the sling effect [37, 38]) makes droplets detach from the
flow field and allows very large relative velocities at small separation (see
chapter 3 for more details). This implies that two regimes of collisions can
be distinguished: one where diffusion driven, ’mild’ collisions occur, and one
where caustic induced violent collisions occur.
The contribution of both type of collisions can statistically be estimated
from the scaling of the moments [148, 34, 51]. Caustic contributions to
collisions has been found to be around 50% for droplets with a Stokes number
35. 2.5 Occurrence of caustics 23
around unity [135]. Although the contribution of caustics to the collision
kernel can statistically be estimated, no method exist to identify the type
of a single collision. In this paragraph we therefore investigate if both type
of collisions are initiated in a similar fashion. Fig. 2.8 shows the history
of a droplet prior to a collision sampled not only on the dissipation rate
(left) and enstrophy (right), but also on the relative velocities at impact.
Data is obtained from simulation R1. The bins of the relative velocity are
logarithmically distributed and their values are shown in the legend. Note
that the relative velocity has been scale with the Kolmogorov velocity scale
uη = (ν )
1/4
. This figure clearly shows that large relative velocities at impact
are a result of dissipative events. In the case of small relative velocities, no
dissipation peak is observed.
Fig. 2.8 (right) shows the enstrophy prior to a collision sampled on the
relative velocity at impact. Droplets participating in a violent collision ex-
perience an increase in local enstrophy prior to a collision.
t/τη
-3 -2 -1 0
ǫ|c/ǫflow
0
0.4
0.8
1.2
1.6
2
10−3
10−2
10−1
100
t/τη
-3 -2 -1 0
Ω|c/Ωflow
0.1
0.2
0.3
0.4
0.5
0.6
10−3
10−2
10−1
100
Figure 2.8: Conditionally sampled dissipation rate (left) and enstrophy (right) of
droplets before they collide, conditionally sampled on the relative velocity at impact.
The legends shows the relative velocity bins (logarithmically distributed), where the
relative velocities have been scaled with the Kolmogorov scale.
36. 24 Chapter 2. Preferred location of droplet collisions in turbulent flows
2.6 Conclusions
Our key finding is that droplets statistically do not collide where they pref-
erentially concentrate. Droplets preferentially cluster in regions of low en-
strophy (≈ 0.6 Ω ) and dissipation (≈ ), whereas collisions favor regions
with appreciable dissipation rates (up to ≈ 2.5 ) and with enstrophy values
that are comparable to or slightly higher than the flow average value. The
higher dissipation values serve to enhance the collision rate by decorrelating
the motion of nearby droplets. By studying the history data of droplets, one
can observe a distinct dissipation peak preceding a collision. In particular
at small scales it is important to make a distinction between enstrophy and
dissipation as they both play a different role. Enstrophy causes droplets
to cluster, but the resulting preferential concentration decreases the relative
velocities between droplets, making them less likely to collide. A dissipative
event provides them with the necessary acceleration towards each other. The
larger this event, the larger the relative velocity at impact will be.
The Reynolds number does not have a large effect on the dissipation and
enstrophy levels at which droplets reside in the flow. Collisions, however,
tend to occur in regions of higher dissipation and enstrophy for higher Reyn-
olds numbers. The Stokes number exerts a large influence on the collision
conditioned flow statistics. For very low and very high Stokes numbers, these
statistics resemble that of the mean flow, but for moderate Stokes numbers
the differences are pronounced.
We have studied the relation between the local dissipation rate and the
local collision kernel. A consistent positive correlation exists between both,
but the relation is non-linear. To understand the relation between local and
global collision kernels, we determined coarse-grained values of enstrophy and
dissipation rate. Unlike enstrophy, a consistent positive correlation between
the dissipation and collision rate was found from the largest to the smallest
scales of the flow.
37. CHAPTER 3
Relative velocity distribution of inertial particles in
turbulence
The distribution of relative velocities between particles provides
invaluable information on for example the rate and character-
istics of particle collisions. We show that the theoretical model
of Gustavsson and Mehlig [52], within its anticipated limits of
validity, can predict the joint distribution of relative velocities
and separations of identical inertial particles in isotropic tur-
bulent flows with remarkable accuracy. We also quantify the
validity range of the model. The model matches two limits (or
two types) of relative motion between particles: one where pair
diffusion dominates (i.e. large coherence between particle mo-
tion) and one where caustics dominate (i.e. large velocity dif-
ferences between particles at small separations). By using direct
numerical simulation combined with Lagrangian particle track-
ing, we asses the model prediction in homogeneous and isotropic
turbulence. We demonstrate that when sufficient caustics are
present and the particle response time is significantly smaller
than the integral time scales of the flow, the distribution exhib-
its the same universal power-law form as dictated by the correl-
ation dimension as predicted by the model of Gustavsson and
38. 26 Chapter 3. Relative velocity distribution of inertial particles in turbulence
Mehlig [52]. Compared to the direct numerical simulations, the
model yields accurate results up to a separation distance of one
tenth of the Kolmogorov scale. In agreement with the model, no
strong dependency on the Reynolds number is observed. 1
.
3.1 Introduction
Turbulence is a phenomenological field of study, where analytically based
predictions are very rare. The four-fifths law [67] is one of the few excep-
tions where predictions can be made from first principles. In more complex
situations such as suspended particles in turbulent flows, the nature of tur-
bulence makes it difficult to make analytically based quantitative predictions
of their dynamics, let alone the occurrence of particle collisions. The idea
about turbulence-particle interactions is that turbulence may significantly
affect the particle dynamics and increases the collision probability. While
analytically predicting non-trivial quantities such as the collision rate re-
mains problematic, the collision rate of particles suspended in a turbulent
flow is of interest to many research areas, such as rain formation in clouds
[122, 31, 49], turbulent spray combustion, residue deposition in rivers, ag-
glomeration of fine powders in gas flow, air filtration equipment, fast fluidized
beds and dust grain dynamics in astrophysical environments.
Wang et al. [139] showed that the frequency at which inertial particles
collide in turbulent flows is determined by two different contributions. The
first contribution is the radial distribution function, which is the probability
of finding particles at contact, and quantifies the non-homogeneity of the
particle distribution. The second contribution is the mean radial relative
velocities of the particles. In this chapter we will focus on quantifying the
(radial) relative velocity between particles.
The radial relative velocity between particles (also known as the collision
velocities) is of crucial importance not only to understand the collision rate
but also the collision characteristics. Whether the collision will be elastic,
non-elastic, sticky, or such that break-up of the particles will occur depends
sensitively on the speed at which the particles collide [31]. Factors influencing
the relative velocities are the Stokes number St (ratio of particle relaxation
time τp = 2ρpr2
/(9ρf ν) and the Kolmogorov time scale of the flow τη =
(ν/ )1/2
), the mean dissipation rate and the Reynolds number Re [111,
1Under review as: V. E Perrin and H. J. J Jonker. Relative velocity distribution of
inertial particles in turbulence : a numerical study. Phys. Rev. E,
39. 3.2 Predicting the relative velocities 27
2, 134, 149, 71, 154, 12, 90, 22, 51, 52]. ν represents the viscosity of the
carrier fluid, ρp the density of the particles and ρf the density of the fluid.
The sensitive dependence on the Stokes number of the velocity at which
particles impact can be attributed to the formation of caustics [146, 148]
(also known as the sling effect [37, 38]). This phenomenon describes the
detachment of particles from the underlying flow field, allowing them to
have large relative velocities at small separation. Experimental evidence of
the caustic effect has been found by Bewley et al. [18]. The collision rate
of particles can therefore be decomposed into a smooth contribution due to
pair diffusion similar to the tracer limit of Saffman and Turner [111] and a
singular contribution due to caustics [148, 34, 135, 52], similar to the ballistic
limit of Abrahamson [2].
3.2 Predicting the relative velocities
In this chapter we will focus on the model of Gustavsson and Mehlig [52]
(from now on referred to as the GM model). The GM model is unique in its
capacity to make analytically based, very quantitative predictions about the
distribution of relative velocities ρ(∆v, R) of neighboring identical inertial
particles as a function of the separation R between the particles using only
the correlation dimension. This distribution is also independent of the Kubo
number Ku ; the Kubo number is defined as Ku = u τη/η, where u is the
root mean square of the velocity field and η = (ν3
/ )1/4
the Kolmogorov
length scale.
We will now briefly introduce the GM model using the original notation.
Let R = |∆x| be the magnitude of the non-dimensional spatial separation
vector ∆x = x2 − x1 between a particle pair, and V = |∆v| the magnitude
of the non-dimensional relative velocity vector ∆v = v2 − v1. The quant-
ities R and V have been made non-dimensional using the typical time and
length scale of the flow. The joint distribution of the relative velocities and
separations of particles suspended in randomly mixing or turbulent flow can
be divided in three regimes (see Fig. 3.1 for a graphical representation). In
region 1 and 2 two different types of relative motion can be distinguished.
In region 1 pair diffusion is dominant. Close-by particles move in a very
coherent manner which implies that V R. Since R does not change much
in this region, the distribution is independent of V . In region 2 caustics
are dominant. Particles can detach from the flow field which allows large
velocity differences at short separations. This implies that V R and that
the distribution becomes independent of R. Clustering of the particles in
40. 28 Chapter 3. Relative velocity distribution of inertial particles in turbulence
combination with caustics results in a power-law distribution for the relative
velocities. The exponent equals D2 − 2d where D2 is the phase-space correl-
ation dimension of the particle field and d the embedding spatial dimension
(in our case 3). z∗
is the matching length scale and represents the typical
value of the relative magnitude of velocity and separation between particles
(i.e. z∗
≈ V/R). In region 3 , where V > Vc, the scaling of the relative
velocities loses its universal aspect and becomes system specific. Vc is the
cut-off value, above which the distribution has been set to zero as a simple
assumption. Formally the joint distribution ρ(∆v, R) is given by:
ρ(∆v, R) ∼ Rd−1
RD2−2d
for region 1
|∆v/z∗
|D2−2d
for region 2
0 for region 3
(3.1)
where the regions 1 , 2 and 3 are defined as follows:
region 1 where V ≤ z∗
R, R ≤ 1 and V ≤ z∗
region 2 where V > z∗
R, R ≤ 1 and V ≤ z∗
region 3 where R > 1 or V > Vc = z∗
(3.2)
Figure 3.1: Graphical representation of the three regions of the GM model, after
Gustavsson and Mehlig [52].
The GM model provides both the distribution of the relative velocity and
the distribution of the radial relative velocities, the latter being needed to
41. 3.3 Numerical setup 29
compute the collision kernel and differs from the former only in the exponent
of the power-law in region 2 (i.e. D2−d−1 for the radial counterpart instead
of D2 − 2d). The distribution of radial relative velocities can be constructed
by projecting the relative velocities in Eq. Eq. (3.1) onto the unit vectors of
the corresponding spherical coordinate system and integrating over all angles
(see Gustavsson and Mehlig [52] for more details). The model is expected
to hold for any Stokes number if two conditions are met:
1 the Stokes number of the particles has to be sufficiently high to allow
caustics to form
2 the dynamics of the particles have to be insensitive to the nature of
the large scale forcing of the system.
It should be noted that the GM model does not explicitly depend on the
Kubo number. In isotropic turbulence Ku ∝
√
Reλ, where Reλ is the Taylor
based Reynolds number, and therefore the GM model does not explicitly de-
pend on the Reynolds number either. However, the GM model still can have
an implicit Reynolds number dependency through the correlation dimension
D2, although previous studies have shown that the effect of the Reynolds
number on the correlation dimension is rather limited [39, 15].
3.3 Numerical setup
Our aim in this chapter is to compute the joint PDF of the relative velo-
cities at small separations of identical inertial particles in turbulent flows
using direct numerical simulations (DNS), and compare our results with the
predictions of the GM model. Since the GM model does not predict a direct
Kubo (or Reynolds) number dependency, we will also investigate whether
this is justifiable.
Under the assumption of small and heavy particles (i.e. r η and ρp
ρf ), the equations of motion for the particles are given by [81, 44]:
dvi
dt
=
u[xi, t] − vi
τp
(3.3)
dxi
dt
= vi (3.4)
where xi and vi are the position and velocity vectors of particle i and u is
the velocity of the flow field, governed by the Navier-Stokes equations. The
42. 30 Chapter 3. Relative velocity distribution of inertial particles in turbulence
in-house developed DNS we use solves the Navier-Stokes equations using
pseudo-spectral methods. Time integration is performed using a third order
Adams-Bashforth scheme. Both advection and diffusion are treated expli-
citly, and the 3/2-rule is used to fully deal with aliasing errors (see e.g. Ca-
nuto et al. [24]). Time stepping is restricted by the Courant-Friedrich-Lewy
criterion using a Courant number of 0.1.
The equations of motion of the particles Eq. (3.3) and Eq. (3.4) are
integrated using a fourth order Runge-Kutta scheme to be able to accurately
track particles with St 1. The velocity of the flow field at the particle
position is computed using trilinear interpolation. For more details, see
Perrin and Jonker [92].
In order to construct the PDF of the relative velocities, we use the al-
gorithm of Allen and Tildesley [5] to find neighboring particle pairs (see also
Chen et al. [27] and Perrin and Jonker [92]). This algorithm uses cell index-
ing and linked lists to check only particle pairs which could collide within
one time step, and therefore avoids the prohibitive O(N2
p ) scaling of the
computational cost. Using neighboring particle pairs we construct the joint
distribution of V and R at the smallest scales. We consider four separation
distances R = 10−4
, R = 10−3
, R = 10−2
and R = 10−1
. Results obtained
from the DNS have been made non-dimensional using the Kolmogorov length
scale η and the Kolmogorov velocity scale vη = (ν )1/4
. The DNS allows
us to construct ρDNS(V, R), from which ρ(∆v, R) can be computed using
isotropy: ρ(∆v, R) ∝ ρDNS(V, R)/V 2
. The phase-space correlation dimen-
sion D2 has been computed from the spatial correlation dimension d2 using
d2 = min(D2, d) [52]. The correlation dimension d2 quantifies the spatial
dimensionality of the particle distribution and is computed by taking mul-
tiple snapshots of the entire particle field and binning all the particle pairs
according to their separation distance R. Integrating this histogram yields
the correlation sum C(R), the number of particle pairs within a separation
distance R:
C(R) =
1
N2
p
Np
i,j=1
i=j
H(R − |xi − xj|) ∝ Rd2
(3.5)
with Np the number of particles in the domain, H(·) the Heaviside function
and d2 the correlation dimension.
Three simulations have been performed with different Reynolds numbers.
Per simulation three particle categories are considered (St 1, St ≈ 1 and
St > 1) to investigate the validity of the GM model. To ensure that all scales
of turbulence are sufficiently resolved and that the particle trajectories are
43. 3.4 Results 31
Run N3
x Reλ Ku kmaxη r(µm) St Np/106
S1 1283
60 3.9 3.4 10,40,70 0.045, 0.72, 2.20 0.6
S2 2563
105 5.2 3.2 10,40,70 0.050, 0.80, 2.45 1.2
S3 5123
178 6.8 2.8 10,40,70 0.049, 0.78, 2.39 2.4
Table 3.1: Overview of the simulations. Each simulation is shown together with
the number of grid points N3
x , the Taylor based Reynolds number Reλ, the Kubo
number Ku, the dimensionless resolution parameter kmaxη, the Stokes numbers St
considered and the number of particles Np. The particles are equally distributed
among the categories.
properly tracked, we use a value for the dimensionless resolution parameter
kmaxη of at least 2.8 for all simulations, where kmax is the highest resolvable
wavenumber (Nx/3 in our case). All dimensionless parameters are shown in
table 3.1.
3.4 Results
Figure 3.2 shows the distribution of the relative velocities at different separa-
tions computed with the DNS and compared with the GM model predictions
for a Stokes number around unity. The DNS confirms the three different re-
gimes predicted by the GM model. In region 1 where pair diffusion is
dominant, the plateaus in the distribution of the relative velocities scale ac-
cording to the predicted ρ(∆v, R)/R2
∝ RD2−2d
for all separation distances
R. In region 2 the distribution behaves universally and collapses onto the
single power-law ρ(∆v, R)/R2
∝ |∆v/z∗
|D2−2d
. For the highest separation
distance considered (R = 0.1) the slope of the power-law is well predicted
but the distribution starts to depend on R which indicates that the universal
aspect is lost. As expected the GM model only holds for small separations
and R = 0.1 is on the boundary of its validity. But for practical applications
such as atmospheric clouds, a separation distance R > 0.1 has limited use
anyway, since cloud droplets that are large (i.e. r ≈ 100µm) do not obey
linear drag as assumed in Eq. Eq. (3.3); in addition gravity needs to be
accounted for. In region 3 , the distribution falls off in a similar way as the
numerical results by Gustavsson and Mehlig [52].
Fig. 3.3 and 3.4 show to what extent the GM model loses its validity if
one of the two conditions for the model validity is not met. Fig. 3.3 shows the
joint distribution of the relative velocities at different separations for a Stokes
number St 1. As expected, the model cannot predict the distribution
44. 32 Chapter 3. Relative velocity distribution of inertial particles in turbulence
(and especially the power-law) since caustics are not sufficiently present; see
Voßkuhle et al. [135] for an estimate of the amount of caustics present for
different Stokes number. The particles do not possess enough momentum to
be able to detach from the flow field and large relative velocities at small
separations are very rare. Fig. 3.4 shows the joint distribution of the relative
velocities at different separations for a Stokes number St > 1. Although the
slope of the power-law is fairly accurately predicted, the power-law does not
collapse onto a single line as in Fig 3.2. For St > 1, the particle relaxation
time is of the order of the system turnover-time, violating the assumption
that the dynamics of the particles have to be insensitive to the nature of the
large scale forcing of the system. As a results, the universal aspect of the
power-law is lost.
From our previous results we can expect that the GM model can accur-
ately predict the distribution of (radial) relative velocity within the bound-
aries of its validity for any Stokes number if two conditions are met: (1) the
Stokes number of the particles has to be sufficiently high to allow caustics
to form and (2) the dynamics of the particles have to be insensitive to the
nature of the large scale forcing of the system. One could speculate that for
higher Reynolds numbers the range of validity of the model in terms of Stokes
number can only improve. For very light particles, the larger the Reynolds
number becomes the more extreme events occur in the flow(e.g. Yeung et al.
[153]) which promotes the formation of caustics, required by the model. For
heavier particles, the larger the Reynolds number becomes the larger the
separation between the particle response time and the integral timescale of
the system (∝ Re1/2
) becomes which reduces the impact of the large scale
forcings on the particle dynamics.
It is to be noted that the results presented here were obtained in absence
of gravity (consistent with the GM model). For some applications, this limits
the applicability of the model. Under the effect of gravity, particles can form
curtain-like manifolds, which profoundly affects both the spatial distribution
and the relative velocity of the particles [150, 53, 91, 17, 138, 13, 12, 42]).
Gravity reduces the formation of caustics [53] and the power-law prediction
is expected to fail.
3.5 Conclusion
Our key conclusion is that the joint distribution of relative velocities and
separations of particles in the absence of gravity in a turbulent flow can be
accurately predicted by the model of Gustavsson and Mehlig [52], up to a
45. 3.5 Conclusion 33
separation distance of one tenth of the Kolmogorov scale (dependent on the
Reynolds number). This model matches two asymptotic limits of the dis-
tribution: One where pair diffusion dominates (i.e. large coherence between
particle motion) and one where caustics dominate (i.e. large velocity differ-
ences between particles at small separations). The model is expected to hold
for any Stokes number if two conditions are met: (1) sufficient caustics have
to be present to allow large velocity differences at small separation, otherwise
the power-law prediction breaks. (2) A sufficiently large scale separation has
to exist between the particle time scales and the system integral time scales
to ensure that the nature of the driving of the system does not influence
the dynamics of the particles. No significant Reynolds number effect has
been found on the joint distribution of relative velocities, in accordance with
the model of Gustavsson and Mehlig [52], although the range of Reynolds
numbers studied here is limited. Since the model is capable of accurately
predicting the distribution of radial relative velocities at contact, it paves
the way to not only a better prediction of the collision kernel, but also to a
better understanding of the collision characteristics.
|∆v|
10-4
10-3
10-2
10-1
100
101
102
ρ(∆v,R)/R2
10
-11
10
-7
10
-3
10
1
105
109
1013
1017
blue : S3
magenta : S2
∝ |∆v|D2−2d
St ≈ 1
black : S1
Figure 3.2: The relative velocity distribution for different separations ((×) R =
10−1
, (◦) R = 10−2
, ( ) R = 10−3
and (♦)R = 10−4
) and different Reynolds
numbers (S1 (black), S2 (magenta), S3 (blue)) for the particles with St ≈ 1. The
red lines shows the GM model prediction. The dotted line shows the expected power-
law based on Eq. 3.1. See table 3.1 for an overview of the simulations
46. 34 Chapter 3. Relative velocity distribution of inertial particles in turbulence
|∆v|
10-4
10-3
10-2
10-1
100
101
102
ρ(∆v,R)/R2
10
-11
10
-7
10
-3
10
1
105
109
1013
1017
∝ |∆v|D2−2d
St ≪ 1
black : S1
blue : S3
magenta : S2
Figure 3.3: Same as Fig. 3.2 but with St 1, which implies that model assump-
tion 1 is violated. The red lines shows the GM model prediction. The dotted line
shows the expected power-law based on Eq. 3.1. See table 3.1 for an overview of
the simulations
|∆v|
10-4
10-3
10-2
10-1
100
101
102
ρ(∆v,R)/R2
10
-11
10
-7
10
-3
10
1
105
109
1013
1017
St > 1
black : S1
blue : S3
magenta : S2
∝ |∆v|D2−2d
Figure 3.4: Same as Fig. 3.2 but with St > 1, which implies that model assump-
tion 2 is violated.The dotted line shows the expected power-law based on Eq. 3.1.
See table 3.1 for an overview of the simulations
47. CHAPTER 4
Effect of the eigenvalues of the velocity gradient tensor
on particle collisions
This study uses the eigenvalues of the local velocity gradient
tensor to categorize the local flow structure in incompressible
turbulent flows into different types of saddle nodes and vortices
and investigates their effect on the local collision kernel of heavy
particles. Direct numerical simulation results show that most of
the collisions occur in converging regions with real and negat-
ive eigenvalues. Those regions are associated not only with a
stronger preferential clustering of particles, but also with a rel-
atively higher collision kernel.
To better understand the direct numerical simulation results,
a conceptual framework is developed to compute the collision
kernel of individual flow structures. Converging regions, where
two out of three eigenvalues are negative, posses a very high
collision kernel, as long as a critical amount of rotation is not
exceeded. Diverging regions, where two out of three eigenvalues
are positive, have a very low collision kernel, which is governed
by the third and negative eigenvalue. Due to oversimplification
of the model, however, the effect of enstrophy on collisions is
48. 36 Chapter 4. Effect of the eigenvalues of the velocity gradient tensor on particle collisions
overestimated. 1
.
4.1 Introduction
The collision rate of particles suspended in a turbulent flow is of interest to
many research areas, such as rain formation in clouds [120] or dust grain
dynamics in astrophysical environments. One contribution to the collision
rate is the radial distribution function [127, 139], which is a measure of the
non-homogeneity or local concentration of the particle field. Maxey [80]
showed that particles tend to cluster in regions of low vorticity and high
strain as a result of their inertia, a mechanism later called preferential con-
centration. Preferential concentration is most pronounced when the particles
have a Stokes number St around unity, which represents the non-dimensional
response of the particle to the flow field. The Stokes number is defined as
St = τp/τη, where τη = (ν/ )1/2
is the Kolmogorov time scale of the flow
and τp = 2ρpr2
/(9ρf ν) the particle relaxation time; ν represents the viscos-
ity of the carrier fluid, r the particle radius and is the mean dissipation
of the flow. ρp is the density of the particles and ρf of the carrier fluid.
Squires and Eaton [125] used the flow classification of Hunt et al. [58],
to divide the flow into eddies, streams and convergence zones, and showed
that particles tend to move through the flow via streaming zones and tend
to avoid eddies. Streaming zones typically end in convergence zones, where
an increase in particle concentration can be found [125, 35].
Chong et al. [28] introduced a different classification of the local flow
based on the PQR invariants, i.e. the first, second and third invariants of
the velocity gradient tensor of the flow. In incompressible flows, mass con-
servation implies that P equals zero. Q is a measure of the relative intensity
between strain and enstrophy and R is helpful in differentiating between re-
gions of stretching and compression. Several studies have investigated the
role of local flow topology on particle behavior [110, 19] in wall bounded
flows and show that particles with Stokes numbers of the order of unity
preferentially concentrate into stream-wise flowing low-speed streaks.
By investigating the average flow pattern in local coordinate systems,
Elsinga and Marusic [36] identify the average flow patterns responsible for
some apparently universal aspects of small-scale turbulent motion such as
a preferential alignment of the vorticity vector with the eigenvector corres-
ponding to the intermediate eigenvalue of the strain rate tensor as well as
1Under review as: V. E Perrin and H. J. J Jonker. Effect of the eigenvalues of the
velocity gradient tensor on particle collision. Journal of Fluid Mechanics,
49. 4.1 Introduction 37
the characteristic teardrop shape of the joint probability density function of
Q and R. Those aspects have been observed in several turbulent flows such
as turbulent boundary layers, turbulent channel flows and homogeneous and
isotropic turbulence.
Collisions in turbulent flows occur in regions of the flow where the dissipa-
tion rate is higher than flow averaged. Perrin and Jonker [92] constructed the
following physical picture of a collision in cloud-like turbulence using direct
numerical simulation: enstrophy makes particles preferentially concentrate
in quiescent flow regions, thereby increasing the particle velocity coherence,
i.e. decreasing relative velocities between particles, needed to approach each
other. Strongly clustered particles thus have a low collision probability, until
a dissipative event accelerates the particles towards each other.
In this chapter we will use an approach comparable to Elsinga and
Marusic [36] to identify the effect of the local flow topology on particles
and on particle collisions. The local flow topology can be categorized based
on the eigenvalues λ1, λ2 and λ3 of the local velocity gradient tensor of the
velocity field u, i.e. J = ∂u
∂x (see figure 4.1). x represents the position vector.
Physically, complex eigenvalues can be associated with vortices [156, 155],
whereas real eigenvalues correspond to convergence/divergence zones. Here
we choose the following classification for incompressible flows. When the
eigenvalues of the velocity gradient tensor are complex (two out of the three
eigenvalues form a conjugate pair) rotation is present in the local flow and
the following holds:
R(λ1) = R(λ2) (4.1)
I(λ1) = −I(λ2) (4.2)
λ3 = −2R(λ1) (4.3)
Following the nomenclature of Chong et al. [28], complex eigenvalues rep-
resent a focus. The real part of the eigenvalues λ1 and λ2 in such a case
determines whether the focus is inward spiraling (a) or outward spiraling
(b). By looking only at the eigenvectors e1 and e2, we could also classify
the focus as stable (a) or unstable (b). In the case of only real eigenvalues
of the velocity gradient tensor, rotation is absent in the local flow structure
and the eigenvalues can always be rearranged such that λ1 < λ2 < λ3. The
local flow structure forms a saddle node and the sign of the eigenvalue λ2
determines if the node is stable and converging along the eigenvectors e1
and e2 (c) or unstable and diverging along the eigenvectors e2 and e3(d).
Blackburn et al. [20] show a graphical representation in the QR plane of the
50. 38 Chapter 4. Effect of the eigenvalues of the velocity gradient tensor on particle collisions
four categories (a), (b), (c) and (d) as presented in Fig 4.1.
Defining the local flow categories is based on a point-wise evaluation
of the velocity gradient tensor; we interpret these four categories as fun-
damental building blocks of the flow, and refer to them as atoms of flow.
This categorization is closely related to the commonly used categorization
of Chong et al. [28] which uses the PQR invariants. Be it a matter of taste,
the use of eigenvalues can be somewhat more intuitive since an eigenvalue
directly represents the local flow direction along its eigenvector. The eigen-
values λ1, λ2 and λ3 for incompressible flows can directly be translated into
the PQR invariants with the following identities:
P = λ1 + λ2 + λ3 = 0
Q = λ1λ2 + λ1λ3 + λ2λ3
R = λ1λ2λ3 (4.4)
This chapter is outlined in the following way. Using direct numerical
simulation (DNS) of homogeneous and isotropic turbulence, we will first
categorize the flow into the four atoms of flow and investigate the effect
of the local flow topology on the presence of particles and the occurrence
of collision (section 4.2). We will use this information to find in the DNS
the collision kernel associated with each of these atoms of flow. In section
4.3 we develop a simplified framework based on the atoms of flow to gain
more conceptual insight in the effect of an individual flow structure on the
occurrence of collisions.
4.2 DNS framework
The dynamics of the turbulent flow field are governed by the Navier-Stokes
equations. If the density of the particles ρp is high compared to the density
of the carrier fluid ρf and if the particle radius r is small compared to the
Kolmogorov scale η = (ν3
/ )1/4
of the flow, the full equations of motion of
particles in turbulence [81, 44] can be reduced to:
dv(t)
dt
=
u[x(t), t] − v(t)
τp
(4.5)
dx(t)
dt
= v(t) (4.6)
where v(t) is the particle velocity vector, x(t) the particle position vector and
u[x(t), t] the flow velocity field at the particle position. Gravity is omitted
51. 4.2 DNS framework 39
(a) (b)
Stable focus Unstable focus
(c) (d)
Stable saddle node Unstable saddle node
Figure 4.1: Graphical representation of the four atoms of flow.
52. 40 Chapter 4. Effect of the eigenvalues of the velocity gradient tensor on particle collisions
in this chapter since it adds complexity to the problem in a delicate way.
The combined effect of turbulence and gravity is not merely an addition of
separate phenomena [150, 17, 53].
4.2.1 Numerical details of the DNS
In this chapter, an in-house developed direct numerical simulation code has
been used, which solves the Navier-Stokes equations using pseudo-spectral
methods. We use a triple-periodic computational domain, in which time
stepping is restricted by the Courant-Friedrich-Lewy criterion using a Cour-
ant number C of 0.1. Since a turbulent system is inherently dissipative,
energy is injected at the lowest wavenumber. To this end, we employ a
forcing scheme similar to that used by Woittiez et al. [150] to add kinetic
energy to the largest scales. This energy has been set to (0.25 tL)2/3
; t de-
notes the target mean dissipation rate of the simulation, which is the mean
dissipation rate we aim for (it is not necessarily exactly equal to the mean
dissipation rate of the actual simulation), and L denotes the physical size of
the computational domain. The use of DNS limits the range of scales that
can be resolved. As a result, the domain size and the Reynolds number are
limited and several orders of magnitude lower than in real convective clouds.
Time integration is performed using a third order Adams-Bashforth scheme.
Both advection and diffusion are treated explicitly, and the 3/2-rule is used
to fully deal with aliasing errors (see e.g. Canuto et al. [24]).
The equations of motion (Eq. (4.5)) and (Eq. (4.6)) are updated using
a second order Runge-Kutta scheme. The velocity of the flow field at the
particle position is computed using trilinear interpolation.
A collision routine checks for collisions using the algorithm of Chen et al.
[27], which uses cell indexing and linked lists to check only droplet pairs that
could collide within one time step. The cost of this algorithm is O(
27N2
p
2NxNyNz
),
where Nx, Ny and Nz define the size of the computation domain in the x,
y, and z direction, respectively and Np is the number of particles present
in the computational domain. To ensure that all collisions are detected, the
maximum travel distance of the particles is restricted to half a grid distance
by using a dynamically adaptive timestep for the particles. See Perrin and
Jonker [92] for more information on the numerical aspects.
