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
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
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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
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.
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.
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.
Samenvatting
iSummary
iii1 Introduction
11.1 Outline . . . 5
2 Preferred location of droplet collisions in turbulent flows
7 2.1 Introduction . . . 82.2 Background . . . 10
2.2.1 Droplet dynamics . . . 10
2.2.2 Collision statistics . . . 11
2.2.3 Flow field statistics . . . 11
2.3 Numerical setup . . . 12
2.4 Results . . . 15
2.4.1 Preferred location of droplet collisions . . . 15
2.4.2 Reynolds number and Stokes number effects . . . 18
2.4.3 Local versus mean view on collisions . . . 20
2.5 Occurrence of caustics . . . 22
2.6 Conclusions . . . 24 v
3 Relative velocity distribution of inertial particles in turbulence
253.1 Introduction . . . 26
3.2 Predicting the relative velocities . . . 27
3.3 Numerical setup . . . 29
3.4 Results . . . 31
3.5 Conclusion . . . 32
4 Effect of the eigenvalues of the velocity gradient tensor on particle
collisions
35 4.1 Introduction . . . 364.2 DNS framework . . . 38
4.2.1 Numerical details of the DNS . . . 40
4.2.2 DNS results . . . 41
4.3 Conceptual framework . . . 43
4.3.1 Motion of heavy particles and collisions statistics . . . 44
4.3.2 Model implementation and validation . . . 47
4.3.3 Results atoms model . . . 47
4.3.4 Effect of dissipation and enstrophy . . . 51
4.4 Predicting turbulent flow statistics using the atoms model . . 51
4.5 Concluding remarks . . . 52
5 Lagrangian droplet dynamics in the subsiding shell of a cloud
55 5.1 Introduction . . . 565.2 Background . . . 58
5.2.1 Droplet growth and condensation rate . . . 58
5.2.2 Droplet dynamics and collision statistics . . . 59
5.2.3 Flow field . . . 60
5.2.4 Transition length scale . . . 61
5.3 Numerical setup . . . 61
5.4 Results . . . 64
5.4.1 Effect of the droplet size distribution . . . 70
5.5 Discussion . . . 72
5.6 Conclusions . . . 76
6 Conclusions and outlook
77 6.1 Concluding remarks . . . 77Bibliography
83A Resolution dependence in isotropic turbulence
97Acknowledgements
99About the author
101Introduction
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
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.
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
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
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 hi ≈ 0.05m2s−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]
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.
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
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 hi, 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 hi is sufficiently able to represent the collision process. Or, phrased differently, how large should an averaging length l be for lto 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 hi 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.
(a)
4
ε
(b)9
ε
(c)Figure 2.1: Three domains with the same mean dissipation rate hi 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
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/(9ρfν) 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].
2.2.2
Collision statistics
The average number of collisions ˙N12per 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 N2are 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)2h|wr|ig(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 = 12 ∂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 ∇2p. 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]:
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= u0λ ν ; λ = 15νu02 1/2 (2.7)
where λ is the Taylor microscale and u0 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∂ 2u i ∂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 τf orc = 0.5pν/t
[89] to add kinetic energy to the largest scales. This energy has been set to (0.25tL)2/3; tdenotes 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.
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 10−1 100 101 10−10 10−5 100 105 κη E (κ )/ ǫ 2 / 3η 5 / 3 k−5/3 10−2 10−1 100 101 0 2 4 6 8 10 12 14 16 18 κ D (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( 27N
2 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+r2apart, a greater number of samples
Run L (m) Nx Reλ hi (m2/s3) u0(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/s3and 0.1m2/s3, respectively. The spatial resolution
limit of the simulations kmaxη, where kmax= Nx/2 and η = (ν3/hi)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.
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 ∇2P [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 posiDissipa-tion of the droplets. Droplets also tend to cluster where ∇2P ' 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 ∇2P (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 hi = hΩi which implies that the point (hi, hΩi) must be located right on the black line that represents ∇2P = 0. The graph shows that, on average,
10-1 100 101 102 0 0.2 0.4 0.6 0.8 1 ǫ/hǫf lowi ǫp ( ǫ) Flow Droplets Collisions 10-1 100 101 102 0 0.2 0.4 0.6 0.8 1 Ω/hΩf lowi Ω p (Ω ) Flow Droplets Collisions -0.040 -0.02 0 0.02 0.04 50 100 150 c ∇2P p( ∇ 2P ) Flow Droplets Collisions 0 1 2 3 0 1 2 3 d hǫi/hǫf lowi hΩ i/ hΩ f lo w i Flow Droplets Collisions ∇2P=0 10-410-2100 102 104 10-8 10-6 10-4 10-2 100 ǫ/hǫf lowi ǫp ( ǫ) 10-410-2 100 102 104 10-8 10-6 10-4 10-2 100 Ω/hΩf lowi Ω p (Ω ) h∇2P i = 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 ∇2P . The result is shown in
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 lo g10 [Ω / hΩ f lo w i] log10[ǫ/hǫf lowi] a -5 -4 -4 -3-2 -1-0.3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 b lo g10 [Ω / hΩ f lo w i] log10[ǫ/hǫf lowi] -5-4 -5 -3-2 -1-0.3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 c lo g10 [Ω / hΩ f lo w i] log10[ǫ/hǫf lowi] -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
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/τ η < ε>/< εflow >,<|v p |>/<u 0 > -1.5 -1 -0.5 0 0.5 1 1.50.9 1 1.1 1.2 < Ω >/< Ωflow > <ε|d>/<ε flow> <Ω|d>/<Ωflow> <|v p|>/<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-0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 hǫi/hǫf lowi hΩ i/ hΩ f lo w i Flow Droplets Collisions ∇2P=0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 hǫi/hǫf lowi hΩ i/ hΩ f lo w i Flow Droplets Collisions ∇2P=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 (0.6hΩflowi and hf lowi) 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
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 hΓi since they are related via hΓi =
Z ∞
0
ΓP()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 hi =R 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
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
hΩl|ci and hl|ci for collisions and hΩl|di and hl|di 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,
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 hL|ci & hi).
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
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 hǫ |c i/ hǫf lo w i 0 0.4 0.8 1.2 1.6 2 10−3 10−2 10−1 100 t/τη -3 -2 -1 0 hΩ |c i/ hΩ f lo w i 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.
2.6
Conclusions
Our key finding is that droplets statistically do not collide where they pref-erentially concentrate. Droplets prefpref-erentially cluster in regions of low en-strophy (≈ 0.6hΩi) and dissipation (≈ hi), whereas collisions favor regions with appreciable dissipation rates (up to ≈ 2.5hi) 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.
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
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 predicexcep-tions 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/(9ρfν) and the Kolmogorov time scale of the flow τη =
(ν/hi)1/2), the mean dissipation rate hi 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,
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 = u0τη/η, where u0 is the
root mean square of the velocity field and η = (ν3/hi)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
