Scenarios of drop deformation and breakup in sprays

by

T´ımeaK´ekesi

September 2017 Technical Report Royal Institute of Technology Department of Mechanics SE-100 44 Stockholm, Sweden Akademisk avhandling som med tillst˚andav Kungliga Tekniska H¨ogskolan i Stockholm framl¨aggestill offentlig granskning f¨oravl¨aggandeav teknologie dok- torsexamen fredagen den 15 september 2017 kl 10.15 i D3, Lindstedtsv¨agen5, Kungliga Tekniska H¨ogskolan, Stockholm.

TRITA-MEK Technical report 2017:10 ISSN 0348-467X ISRN KTH/MEK/TR-2017/10-SE ISBN 978-91-7729-500-6

c T. K´ekesi 2017 Tryckt av Universitetsservice US-AB, Stockholm 2017 Scenarios of drop deformation and breakup in sprays

T´ımeaK´ekesi Linn´eFLOW Centre, KTH Mechanics, The Royal Institute of Technology SE-100 44 Stockholm, Sweden

Abstract Sprays are used in a wide range of engineering applications, in the food and pharmaceutical industry in order to produce certain materials in the desired powder-form, or in internal combustion engines where liquid fuel is injected and atomized in order to obtain the required air/fuel mixture for ideal combustion. The optimization of such processes requires the detailed understanding of the breakup of liquid structures. In this work, we focus on the secondary breakup of medium size liquid drops that are the result of primary breakup at earlier stages of the breakup process, and that are subject to further breakup. The fragmentation of such drops is determined by the competing disruptive (pressure and viscous) and cohesive (surface tension) forces. In order to gain a deeper understanding on the dynamics of the deformation and breakup of such drops, numerical simulations on single drops in uniform and shear flows, and on dual drops in uniform flows have been performed employing a Volume of Fluid (VOF) method. The studied parameter range corresponds to an intermediate Weber number of W e = 20, sufficiently high so that breakup occurs, but still much lower than the limit for abrupt catastrophic breakup, and a range of Reynolds numbers covering the steady wake regime for liquid drops, Re = 20 − 200. In order to account for varying materials of the liquid in different applications, a set of different density and viscosity ratios are considered, ρ∗ = 20 − 80, and µ∗ = 0.5 − 50 respectively. Single drop simulations show that depending on the Reynolds number, as well as on the density and viscosity ratios, various breakup modes besides the classical bag and shear breakup may be observed at a constant Weber number. The characteristics of the deformation process and the time required for breakup are significantly different for these breakup modes. A criterion on the expected breakup mode in the form of a regime map, and a formula estimating the breakup time is suggested, based on the Reynolds number and material properties of the flow. This breakup time is significantly decreased by gradients in the flow surrounding the drop, for which another empirical model has been suggested. Dual drop simulations show that the interaction scenario between two drops is determined by the above parameters (Reynolds number, density and viscosity ratios), and additionally, the initial relative position of the two drops. It is found that the interaction behaviour of drops in tandem arrangement may be predicted based on data obtained for single drops, such as breakup time and the development of the wake region behind the drop. Furthermore, the region where drops behind another drop are likely to collide

iii with this aforementioned drop is identified as a streak two diameters wide and eight diameters long behind the drop, however, weaker forms of interaction may occur up to distances of twenty diameters between the drops. Results presented in this thesis may be applied to formulate enhanced breakup models regarding the deformation, breakup, and interaction of liquid drops employed in spray simulations.

Descriptors: Drop, deformation, breakup, interaction, breakup time, regime map, uniform flow, shear flow, Volume of Fluid (VOF)

iv Deformation och uppbrytning av sprejdroppar

T´ımeaK´ekesi Linn´eFLOW Centre, KTH Mekanik, Kungliga Tekniska H¨ogskolan SE-100 44 Stockholm, Sverige

Sammanfattning Sprejer anv¨andsi flera ingenj¨orstill¨ampningar;inom livsmedels- och farma- ceutindustrin, men ˚aterfinnsocks˚ai till¨ampningars˚asomf¨orbr¨anningsmotorer d¨arbr¨anslei flytande form sprutas in och bryts upp f¨oratt optimera luft- br¨ansleblandningensom i sin tur leder till f¨orb¨attradf¨orbr¨anning. F¨oratt lyckas med en s˚adanoptimeringsprocess s˚abeh¨ovsen grundl¨aggandef¨orst˚aelse f¨orhur v¨atskestrukturer bryts upp. I denna avhandling ligger fokus p˚aatt studera den s˚akallade sekund¨ara uppbrytningen av mellanstora v¨atskedroppar som uppst˚atti ett tidigare skede i uppbrytningsprocessen, den prim¨arauppbrytningsfasen. S¨attetsom dessa droppar bryts upp p˚a/ fragmenteras, styrs av motverkande krafter d¨artryck- och visk¨osakrafter verkar f¨oratt bryta upp droppen medan ytsp¨anningskrafter verkar f¨oratt h˚alladroppen samman. F¨oratt f¨orb¨attrav˚arf¨orst˚aelse f¨or dynamiken i droppdeformationen och uppbrytningen har numeriska fl¨odes- simuleringar anv¨ands. Grundl¨aggandestudier av b˚adeenkla droppar plac- erade i station¨araoch skjuvfl¨oden har genomf¨ortssamt f¨ordroppar som plac- eras i n¨arhetenav varandra. Ytsp¨anningenhar definierats s˚aatt droppupp- brytning uppst˚ar(We = 20) och f¨orh˚allandetmellan konvektiva och visk¨osa krafterna (Reynolds tal) har varit s˚aatt vaken som bildas bakom droppen ¨ar station¨ar.F¨oratt studera hur uppbrytningen varierar mellan olika v¨atskor s˚a har olika densitets- och viskositetsf¨orh˚allanden mellan droppe och omgivande gas anv¨ants i simuleringarna. Resultaten visar att, f¨orutomde klassiska ”bag” och skjuv-uppbrytnings- processerna, s˚auppst˚arandra uppbrytningsscenarier d˚aviskositets och den- sitetsf¨orh˚allandetvarierats. S¨arskilttiden tills dess att uppbrytning uppst˚ar p˚averkas av dessa parametrar. Detta resultat redovisas ¨aven i formen av en formel som beskriver hur tiden till uppbrytning p˚averkas av fl¨odes samman- s¨attning(Reynolds tal samt viskositet- och densitetsf¨orh˚allande). Uppbryt- nignstiden minskar d˚agradienter i fl¨odetsom omger dropparna introduceras. Simuleringar av tv˚adroppar placerade i n¨arheten av varandra visar p˚aatt interaktionen mellan dropparna styrs av ovann¨amndaparametrar samt av hur dropparna ¨arplacerade relativt varandra. Resultaten visar att interaktionen s˚asomuppbrytningstiden och vakutveckling mellan droppar placerade i tandem kan predikteras av data fr˚anensamma droppar. Regionen i vilken en droppe placerad bakom en annan leder till kollision mellan dropparna identifieras av ett omr˚adebakom den fr¨amredroppen som ¨artv˚adroppdiametrar bredd och ˚attadroppdiametrar l˚angt. Det finns dock mindre situationer med svagare interaktion d¨ardenna region kan vara s˚al˚angsom 20 droppdiametrar.

v Resultaten i denna avhandling kan anv¨ands f¨oratt f¨orb¨attrauppbryt- ningsmodeller som anv¨andsi sprejsimuleringar.

Nyckelord: Droppar, deformation, uppbrytning, uppbrytningstid, station¨ar str¨omning,skjuvstr¨omning,Volume of Fluid (VOF)

vi Preface This thesis focuses on numerical investigations on the deformation and breakup of liquid drops in different scenarios. The thesis is a compilation thesis, divided into two parts. Part I contains seven chapters including an introduction to mul- tiphase flows and sprays, the concepts of liquid breakup, an overview of the numerical methods developed for multiphase flows, followed by the description of the method used in this study. Thereafter, the cases studied along with results are presented. Finally, conclusions and a future outlook followed by papers and authors contribution is provided. Part II consists of four journal articles: two are published and two are submitted for publication. The layout of these papers has been adjusted to fit the format of this thesis, but their con- tents have not been changed as compared to the original/submitted versions.

The included papers are:

Paper 1. Drop deformation and breakup. T. Kekesi,´ G. Amberg, L. Prahl Wittberg Int. J. Multiphase Flow 66, 1 – 10, 2014

Corrigendum to: ”Drop deformation and breakup”. Int. J. Multiphase Flow, 66, (2014) 1-10 T. Kekesi,´ G. Amberg, L. Prahl Wittberg Int. J. Multiphase Flow 93, 213 – 215, 2017

Paper 2. Drop deformation and breakup in flows with shear T. Kekesi,´ G. Amberg, L. Prahl Wittberg Chemical Engineering Science 140, 319 – 329, 2015

Paper 3. Interaction between two deforming liquid drops in tandem subject to uniform flow T. Kekesi,´ M. Altimira, G. Amberg, L. Prahl Wittberg Submitted to Phys. Fluids

Paper 4. Interaction between two deforming liquid drops in tandem and various off-axis arrangements subject to uniform flow T. Kekesi,´ M. Altimira, G. Amberg, L. Prahl Wittberg Submitted to Int. J. Multiphase Flow

vii viii You don’t have to be great to start, but you have to start to be great. — Zig Ziglar x Contents

Abstract iii

Sammanfattning v

Preface vii

Part I - Overview & Summary 1

Chapter 1. Introduction 3

Chapter 2. The breakup of liquid jets and drops 9 2.1. Non-dimensional parameters 10 2.2. Primary breakup 12 2.3. Secondary breakup 12 2.4. Interaction between drops 16

Chapter 3. Numerical handling of multiphase flows 18 3.1. Interface handling methods 20

Chapter 4. Numerical method 25 4.1. Governing equations for multiphase flows 25 4.2. Volume of Fluid (VOF) 26

Chapter 5. Case setup and results 30 5.1. Mesh refinement study 32 5.2. Single drop simulations 34 5.3. Dual drop simulations 40

Chapter 6. Concluding remarks and future work 47

Chapter 7. Papers and Author contribution 50

Bibliography 53

xi Acknowledgements 60

Part II - Papers 63

Paper 1a. Drop deformation and breakup

Paper 1b. Corrigendum to: ”Drop deformation and breakup”

Paper 2. Drop deformation and breakup in flows with shear

Paper 3. Interaction between two deforming liquid drops in tandem subject to uniform flow

Paper 4. Interaction between two deforming liquid drops in tandem and various off-axis arrangements subject to uniform flow

xii Part I

Overview & Summary

CHAPTER 1

Introduction

Multiphase flows are systems in which different states of matter (or commonly referred to as phases), gas, liquid, or solid are present simultaneously. Such flows surround us in all areas of life, various examples are found in nature, at our households, and in industrial applications, a few of which are shown in Figure 1.1 and described below. Sandstorms and volcanic eruptions are natural phenomena where solid par- ticles are transported in gas-, whereas in rivers and coastal areas solid-state sediments are carried in liquid flows. When water condensates in the gaseous atmosphere, a liquid-gas flow is represented by the forming clouds and the rain- fall that follows. In more unpleasant hails, where part of the water freezes and rain mixes with snow, all three phases, gaseous, liquid, and solid phases, may be present simultaneously. The opposite phenomenon occurs in geysers, where the water heated up by hot rocks evaporates and the resulting steam shoots up hot liquid in the air. We can easily find multiphase flows in our own households without having to look for too long. Very simple examples are the stream of water flowing from our taps, or the various cleaning detergents that we often use in spray form. It is perhaps not surprising that over half of all the things produced by modern industry are estimated to encounter multiphase flows at some stage of the production process (Prosperetti & Tryggvason (2006)). Multiphase flows may be used to transport solid materials, either in gas (pneumatic transport) or in liquid (slurry transport) flows, whereas in case of pollution control sys- tems, solid particles and liquid droplets are removed from industrial effluents. Material processing and manufacturing applications often involve sprays, where liquid is injected through a nozzle and is atomized into small droplets that can then be further processed depending on the application. In spray casting, semi- solid drops are sprayed onto substrates in order to obtain products of various shapes, or to produce metal matrix composites that are difficult to produce by other methods. Sprays can also be used for high quality surface coating. Another wide area of spray applications is that of the food and pharmaceutical industry, where spray drying is used to produce various materials in powder form; small drops resulting from the atomization of the liquid spray are dried in hot gases resulting in the desired powder. Aerosol sprays are also efficient means of drug delivery in case or respiratory illnesses. Furthermore, high speed water jets, with or without abrasive materials in the liquid, may be used to

3 4 1. INTRODUCTION

a) b)

c) d)

Figure 1.1: Examples of multiphase flows: a)1the eruption of a geyser, b) rainclouds and rainfall above the sea, c) cleaning detergent sprays, and d) aerosols. cut materials. In case of pulverized-coal-fired furnaces, solid propellant rockets and internal combustion engines, multiphase flows are utilized for energy con- version in propulsion systems. Another kind of energy conversion systems is represented by fire suppression systems, where the vaporizing liquid drops cool the hot gases. In this work we turn our focus towards multiphase processes occurring in various sprays, such as the ones in aerosol sprays, spray drying, or sprays in energy conversion systems. Although the method and tools employed in this work may be applied to a wide range of applications involving different material properties and flow conditions, parameters in this work are chosen to fit those in internal combustion engines, (Battostoni & Grimaldi (2012); Kim et al. (2013); Kourmatzis et al. (2013); Moradi et al. (2013); Presser & Gupta (1993); Rantanen et al. (1999); Sirignano (1983) and Park et al. (2013)). Therefore, the latter is described in more detail below. The main purpose of an internal combustion engine (ICE) is to convert the energy stored in the fuel to power in the form of motion. This is realized by injecting the liquid fuel in the combustion chamber, where it forms a spray, mixes with hot and compressed gases, evaporates, and ignites. Such ignition, which is due to compression in case of diesel, or by a spark in case of gasoline, increases the pressure that moves the piston and therefore performs mechanical

1courtesy of: @paulkporter 1. INTRODUCTION 5 work. Even though the development of alternative propulsion systems has been the subject of intensive research for decades now, liquid fuels are still preferred over those in gaseous phase for two main reasons. Firstly, they usually have a higher energy density per unit volume, and second, they are both easier and safer to store and to transfer. Gasoline and diesel are still the most commonly used fuels in internal combustion engines today (Stan (2016)). The process involving multiphase flows is where the fuel is injected in the cylinder through a nozzle at the injector tip, and forms a spray consisting of small liquid drops. A sketch of a nozzle and the injection process is shown in Figure 1.2. As it will be described in Chapter 2, a liquid jet may disintegrate in various ways depending on the flow conditions. In the procedure illustrated in Figure 1.2, the jet first breaks up into larger ligaments of liquid (primary breakup), which then continue to break down to smaller and smaller drops (secondary breakup) as the spray evolves. The atomization of the fuel jet and further breakup of liquid drops is a crucial factor regarding the efficiency of the process. This is due to the fact that smaller drops have larger surface area per volume, are heated up and evaporate more quickly, resulting in better mixing. An optimal air/fuel mixture leads to the highest efficiency and minimal levels of emissions of CO, NOx, unburned hydrocarbons, and soot (Shi et al. (2011)). Fuel combustion is a complicated physical problem and depends on many factors, such as injection scheme and velocity as well as local flow conditions, determining the atomization and breakup of the fluid, , the transport of droplets by the carrier gas, droplet evaporation, and the mixing of the gaseous fuel with the ambient gas. Furthermore, all the above processes depend on the physical properties of the fuel, such as its density, viscosity, surface tension, and heat- conductivity. With emission laws becoming more and more strict in the past decades (Di- rective 2009/28/EC and (EC)No 443/2009 of the European Parliament and of the Council, Renewable Fuel Standard Program (USA), Annual Energy Review 2011), the development of combustion engines has taken new turns. Studies have shown that using alternative fuels may be a possible way to reduce both exhaust gas and particulate matter emissions (Richenhagen et al. (2015)), and thereby to reduce the proportion of fossil fuel used. As follows, the efficient use of alternative fuels from biomass in combustion engines has become a core development area. However, the application of renewable fuels imposes new challenges on engines in use, which should be able to utilize fuels of signifi- cantly different properties without major changes in performance or emission production. Biofuels may have significantly different material properties such as viscosity, evaporation- and boiling behaviour, and self-ignition for example, which have a direct effect on their atomization and breakup properties, and are therefore crucial regarding mixing, the efficiency of the combustion process, and thereby emission. Thus, it is of crucial importance to study the altered spray characteristics and breakup dynamics of liquid fuels with different properties 6 1. INTRODUCTION

Figure 1.2: A schematics of the injection and atomization of liquid fuel.2 in the injection system.

A framework project was formed at the Mechanics department of The Royal Institute of Technology (Kungliga Tekniska H¨ogskolan, KTH), supported by the Swedish Research Council (Vetenskapsr˚adet),with the aim to establish validated computational tools for the investigation of energy conversion systems using fuels of various sources, such as conventional gasoline or diesel, alterna- tive bio-fuels, or a mixture of these. This joint effort had four sub-projects with their respective focus areas covering all stages and scales, from the flow inside the nozzle to the evaporation of secondary droplets. These projects can shortly be described as the following: (I) simulations of the liquid fuel flow in narrow nozzles, focusing on cavitation and its effect on the breakup of the liquid in later stages (Altimira & Fuchs (2014, 2015)), (II) Large eddy simulations (LES) of liquid jets, investigating the primary breakup of the liquid fuel jet and the effect of various injection schemes (Nyg˚ard(2016); Nygaard et al. (2016)) (III) Direct numerical simulations employing a Lattice-Boltzmann method in order to investigate evaporating drops in idealized conditions, with and with- out turbulence in the flow (Albernaz et al. (2016, 2017); Albernaz (2016)), and lastly, (IV) the development of new sub-grid scale models designed for LES simulations of reacting and turbulent flows (Grigoriev (2016); Grigoriev et al. (2016, 2013)). Although the present work has formally not been a part of this framework, it is closely associated and is therefore mentioned in the same con- text. While project (I) in the framework investigates the very first stage of the injection process, projects, (III), (IV), and the present work provide additional

2Reprinted from Mixture Formation in Internal Combustion Engine, Chapter 2 Fundamen- tals of Mixture Formation in Engines, p10, Springer-Verlag Berlin Heidelberg (Baumgarten (2006)), with permission of Springer. 1. INTRODUCTION 7 data for the further development of models applied in spray simulations, such as those in project (II).

Scope and main objectives of the thesis The scope of the work is to study the secondary breakup of liquid drops pro- duced in the earlier stages of the breakup process, primary breakup or atom- ization, in the context of fuel sprays used in internal combustion engines. As mentioned above, the details of the breakup process, the number and size of resulting drops, and the time required to obtain that given state is essential for an efficient combustion process, and thereby has a great importance regarding emission control and environmental effects. In the work presented in this thesis, numerical simulations are performed employing a Volume of Fluid (VOF) method in order to investigate the dy- namics of liquid drops in continuous gas flows under different flow conditions. A relatively wide range of material properties, density- and viscosity ratios of the phases, is considered, accounting for altered properties of bio- as compared to conventional fuels. The main aims of the study can be summarised in the following points:

• to understand the details of deformation leading to breakup, and the process that determines the way liquid drops break up, i.e. the breakup mode

• to gather data on the time required for breakup and the history of force coefficients, mainly the drag coefficient

• to study the interaction between drops and its effect on the deformation and breakup process

• to provide data that can be used to formulate new or enhanced breakup- and drop interaction models for spray simulations, such as simulations in (II), or also independent Lagrangian Particle Tracking (LPT) simu- lations

The thesis is organized in the following manner. Chapter 2 describes the physical aspects of liquid jet and drop breakup, furthermore, discusses the in- teraction between two liquid drops. The numerical methods available for the simulations of multiphase flows in general are reviewed in Chapter 3, with a fo- cus towards interface handling methods (Section 3.1). The detailed description of the method used in this study is found in Chapter 4 along with grid strat- egy, discretization and solution algorithm. Chapter 5 presents the case setup considered for this work along with a short summary of the results. Conclud- ing remarks and suggestions for future works are discussed in Chapter 6, and 8 1. INTRODUCTION

finally, a short summary of the papers and author contributions is provided in Chapter 7. CHAPTER 2

The breakup of liquid jets and drops

In this chapter, the breakup characteristics of liquid jets and drops are dis- cussed. As described in Chapter 1, the behaviour of liquid structures of dif- ferent scales plays a major role in the performance of devices in various spray applications. Therefore, the main stages and the parameters influencing the disintegration of liquid jets from the initial liquid core to the smallest drops are reviewed in the following. Taking the example of an internal combustion engine (ICE), fuel is injected to the combustion chamber in a way that the initial coherent liquid structure breaks up into a large number of small drops. This is necessary for the efficient mixing of the fuel with the hot gases in the chamber, so that an efficient combustion cycle with minimal emission production is obtained. The breakup of the liquid in this way, however, is no straightforward matter, the outcome is determined by many parameters. The breakup process is highly dependent on both geometrical and physical factors, such as the geometry of the nozzle (e.g. different nozzle designs, the length of the injection orifice, the curvature of the orifice inlet, or the shape of the orifice outlet) that determines the flow inside and exiting the nozzle and may cause turbulence or cavitation (see Fig. 1.2), affecting downstream regions of the flow. Other factors are the injection velocity and material properties, such as the densities and viscosities of the phases and interfacial surface tension (Arai et al. (1991); Hiroyasu et al. (1991); Schugger & Renz (2001, 2003); Soteriou et al. (1995); Tamaki et al. (2000); Wu & Faeth (1995)). For a general case when a stream of liquid is injected into a quiescent gas, instabilities emerge on the liquid surface as the result of the competing desrup- tive and cohesive forces, as described in the preceding section. Depending on the flow conditions, these instabilities may then grow and lead to the disin- tegration of the jet. The breakup of a liquid jet consists of two main stages, primary breakup closer to the nozzle exit, where larger ligaments and drops detach from the core of the jet, and secondary breakup farther downstream, where these larger liquid elements deform and continue to break down into larger and then to smaller drops, until a stable drop size is obtained, as seen in Figure 1.2.

9 10 2. THE BREAKUP OF LIQUID JETS AND DROPS

2.1. Non-dimensional parameters As described above, the deformation and breakup of the liquid structures are determined by the competition of desruptive and cohesive forces. While aero- dynamic (inertial and viscous) forces tend to deform the liquid interface, surface tension forces act to retain the shape requiring minimum surface energy; spheri- cal shape in case of a liquid drop for example. Combinations of the above forces are often used to characterise the flow in the form of dimensionless numbers: the Reynolds number is defined as the ratio of inertial versus viscous forces, the Weber number as inertial compared to surface tension forces, whereas the Ohnesorge number relates all three forces. Depending on the problem, one may choose if the density and viscosity of the liquid or of the gas should be used when formulating the Reynolds and Weber numbers. Furthermore, a charac- teristic length and velocity need to be appointed. It is common in the literature on this topic to use liquid properties, the diameter of the injector nozzle Dnoz, and the injection velocity Uinj in case of jets, whereas it is more practical to use the gas properties, the diameter of the drop Ddrop, and the relative velocity between the drop and the gas phase Urel in case of drop flows. As a conse- quence, the Ohnesorge number can be expressed as a function of the Weber and Reynolds numbers in case of liquid jets, whereas for drop flows the density and viscosity ratios of the phases enter the expression in addition. A summary of the described non-dimensional parameters and their definitions are listed in Table 2.1. 2.1. NON-DIMENSIONAL PARAMETERS 11 W e Re √ ∗ ∗ ρ µ drop drop √ D D g = σ µ 2 rel rel U U noz g g D l ρ ρ σ µ l ρ l g l g √ µ ρ ρ µ = = ∗ ∗ ρ µ W e Re √ noz noz = D D l σ µ 2 inj inj noz U D U l l g σ µ l ρ ρ ρ √ inertial forces viscous forces inertial forces surface tension forces viscous forces · = surface tension forces = Re inertia √ W e = Oh Table 2.1: Non-dimensional parameters characterising multiphase flows Weber nr. NameReynolds nr. Ohnesorge Physical nr. meaning.Density ratio Definition for jetsViscosity ratio Definition for drops 12 2. THE BREAKUP OF LIQUID JETS AND DROPS

2.2. Primary breakup Primary and secondary breakup have been vastly studied by numerous authors, investigating underlying phenomena leading to breakup, stability criteria, as well as identifying different breakup modes and regimes where they occur. Four distinguished modes of primary breakup of a liquid stream were ob- served in the studies of Ohnesorge (1936) and Reitz (1978), namely Rayleigh, first and second wind induced breakup, and atomization regimes. Photographs representing each mode are shown in Figure 2.1. In the Rayleigh regime, the liquid jet is disrupted by capillary instability, due to the growth of axisymmetric perturbations with a wavelength of the same order of magnitude as the jet diameter. As a result, drops of equal size that are also comparable to or even larger than the diameter of the jet, are obtained. In the first wind-induced regime perturbations evolving on the jet surface are still present, and are rather axisymmetric. However, the resulting drops are slightly smaller and their size distribution becomes wider. Satellite drops may also appear between the main drops. In case of the second wind-induced regime the liquid jet is perturbed at the nozzle exit already, and due to the growing perturbations its shape becomes much more chaotic as compared to the first two modes described above. Drops show an even wider distribution in size. Close to the nozzle exit, smaller drops are peeled off the interface, while farther downstream the remaining liquid core breaks up into larger ligaments subject to secondary breakup in later stages. While liquid inertia and surface tension are the main driving forces responsible for instabilities in the Rayleigh regime, these forces are amplified by aerodynamics forces due to the interaction with the surrounding gas in the first wind-induced regime. In case of the second wind- induced regime, initial perturbations are caused by turbulence inside the nozzle, which, similarly to the first wind-induced regime, are enhanced by aerodynamic forces. When the complete jet breaks up abruptly right at the nozzle exit, we talk about the atomization regime. In this case, the resulting drops have diameters much smaller than that of the jet. Criteria for the above regimes may be defined using the Reynolds and Ohnesorge numbers, based on the works of Miesse (1955); Ohnesorge (1936); Reitz & Bracco (1982), and Torda (1973). However, as shown by Torda (1973) and Hiroyasu & Arai (1990), atomization is enhanced by increasing the density of the gas. Therefore, Reitz (1978) suggested an extended regime map in the form of a three-dimensional Ohnesorge diagram, as shown in Figure 2.2, where the dependence on the density ratio of the phases is taken into account.

2.3. Secondary breakup As we move further downstream along the jet, ligaments and larger drops de- form and eventually break up as a result of the interaction with the surrounding flow. The list of studies investigating drop deformation and breakup in uniform flows is rather extensive, from the early works of Giffen & Muraszew (1953) and Hinze (1955), to more recent works of Cao et al. (2007); Chou et al. (1997); Dai 2.3. SECONDARY BREAKUP 13

a) b) c) d)

Figure 2.1: Primary breakup regimes.1a) Rayleigh regime b) first wind-induced regime c) second wind-induced regime d) atomization regime.

Figure 2.2: Regimes for primary breakup in terms of the Reynolds number Rel, ρ 2 Ohnesorge number (here Z), and the inverse of the density ratio, g/ρl.

1Reprinted from On the experimental investigation on primary atomization of liquid streams, Exp. Fluids 45:371–422, Dumouchel (2008), with permission of Springer. 2Reprinted from Mixture Formation in Internal Combustion Engine, Chapter 2 Fundamen- tals of Mixture Formation in Engines, p6, Springer-Verlag Berlin Heidelberg (Baumgarten (2006)), with permission of Springer. 14 2. THE BREAKUP OF LIQUID JETS AND DROPS

& Faeth (2001); Faeth et al. (1995); Gel’fand et al. (1974); Guildenbecher et al. (2009); Han & Tryggvason (1999); Helenbrook & Edwards (2002); Hsiang et al. (1995); Hsiang & Faeth (1992, 1995); Krzeczkowski (1980); Pilch & Erdman (1987), and Wierzba & Takayama (1988). When inertial and viscous forces exceed that of surface tension, an initially spherical drop will deform into an oblate or prolate spheroid in such a way, that the larger the Weber number the larger the aspect ratio of the deformed drop becomes. Increasing the Weber number, this deformation becomes unstable and the drop shows an oscillatory deformation, i.e. oscillates between spheroids of different aspect ratios. At sufficiently high Weber numbers deformation exceeds stable shapes, leading to the breakup of the drop. The most commonly known breakup modes are illustrated in Figure 2.4 and are described bellow. For Weber numbers below the critical value, W e ≈ 12, drops undergo an oscillating deformation, and if such oscillations become unstable they might break up into a few relatively large drops. This is called the vibrational breakup mode. For Weber numbers above the critical value the drop experiences bag breakup. Drops in this range typically deform to a spherical cap and then to a flat disk shape. This disk continues to stretch in the radial direction while its middle part becomes thinner. Pressure forced between the front and back sides of the drop act to indent this thin middle region, resulting in a so called bag. This bag is further inflated by the flow, as a consequence its circumference becomes thinner, until small holes eventually appear on its surface. Finally, the bag breaks up into many small drops, whereas the remaining rim produces a smaller number but larger drops. For even higher Weber numbers shear breakup is observed, which regarding the shape of the drop during the deformation phase, can be thought of as the inverse of bag breakup. Drops in this case deform to an oblate spheroid or spherical cap shape, then viscous forces from the gas first sweep the rim of the drop downstream, and break off small droplets from the stretched rim later on. Shear breakup is typically much faster than bag breakup. In flows with very high Weber numbers, the drop abruptly breaks up to a large number of small drops. This phenomena is referred to as catastrophic breakup. One should note that there is no sharp line between the observed breakup modes; mixed breakup modes have also been observed for flows with Weber numbers close to the transitional Weber numbers between two defined breakup modes. A few of the intermediate or mixed breakup modes mentioned in the literature are bag/plume, plume/shear, multimode, bag and stamen and bag- jet breakup. However, some of these modes describe the same or very similar phenomena, as it is the case for the bag-jet, bag and stamen, and bag/plume modes, for example. Additionally, critical and transitional Weber numbers defining the limits between breakup regimes show variations between various works, as shown in Figure 2.3 for the most commonly citied studies. While only slight differences 2.3. SECONDARY BREAKUP 15

Figure 2.3: Breakup modes in terms of the Weber number according to several sources, adapted from K´ekesi et al. (2014).

Vibrational breakup

Bag breakup

A mixed breakup mode

Shear breakup

Catastrophic breakup

Figure 2.4: Illustrations of various drop breakup modes.3 appear regarding the onset of breakup, much larger differences are found for other modes. The most widely used set of transitional Weber numbers are the ones settled by Chou et al. (1997), Dai and Faeth (2001), and Faeth et al. (1995); W e = 13 for bag, W e = 35 for multimode, W e = 80 for shear, and W e = 350 for catastrophic breakup.

3Reprinted from Mixture Formation in Internal Combustion Engine, Chapter 2 Fundamen- tals of Mixture Formation in Engines, p9, Springer-Verlag Berlin Heidelberg (Baumgarten (2006)), with permission of Springer. 16 2. THE BREAKUP OF LIQUID JETS AND DROPS

Hsiang and Faeth Faeth et al. (1995) summarised the results of these works and formed a regime map based on the Ohnesorge and Weber numbers of the flow, as shown in Figure 2.5. The first outstanding observation from this map is that for flows with low viscosities of the liquid, Oh < 0.1, breakup is determined solely by the Weber number. However, higher Weber numbers are needed in order to achieve breakup in more viscous flows. The green area in Figure 2.5 represents the parameter range considered in this thesis. As it will be shown, even though based on the Weber- and Ohnesorge numbers bag breakup is expected for most cases, various breakup modes of the drops are observed. This suggests that the density- and viscosity ratios of the phases may also alter the expected breakup modes compared to the regimes in Figure 2.5. The deformation and breakup of drops may be altered by velocity gradients in the flow, however. Three main scenarios are described in the literature for shear flows by Rumscheidt & Mason (1961). Tipstreaming, where the drop ob- tains a sigmoidal shape with pointed ends from which small drops break up. In other cases, the central part of the droplet forms a progressively thinning neck in the middle, resulting in two larger daughter drops, and small satellite drops in between. In the third case, the drop extends to a long thread that disrupts after a long time. As concluded by Rumscheidt & Mason (1961) and Karam & Bellinger (1968), viscosity ratio plays an important role in determining the outcome of sheared drop configurations.

2.4. Interaction between drops Within the evolving spray, drops may collide and coalesce, or break up, and therefore show a significantly different behaviour as compared to single drops regarding the time required for breakup, aerodynamic force coefficients, result- ing drop size distribution, etc. When two drops are in the vicinity of each other, the drop behind another drop is no longer subject to the free-stream flow, but to a flow altered by the wake behind the drop in front of it. As a consequence, this drop deforms differently, and obtains a drag coefficient different than that of the front drop, thus, may accelerate and collide into the front drop. If the drops are sufficiently far from each other, the front drop has no or only a weak effect on the back drop. In such a case the drops do not collide and behave similarly to single drops. The interaction between two liquid drops has been studied in various ar- rangements. Tandem formulations were investigated in detail by Raju & Sirig- nano (1987, 1990) and Chiang & Sirignano (1987, 1993); side-by-side config- urations were considered in the works of Kim et al. (1991, 1992) and Kim et al. (1993). Additionally, a few authors investigated interaction phenomena for drops in staggered arrangements (Shuen (1987); Temkin & Ecker (1989); Wu & Sirignano (2011)), or drops where the secondary drops is arbitrarily positioned anywhere around the primary drop(Prahl et al. (2006)). 2.4. INTERACTION BETWEEN DROPS 17

ρl/ρg = 20 − 80

µl/µg = 0.5 − 50

Figure 2.5: Secondary breakup regimes in terms of the Weber and Ohnesorge numbers. The green area represents the parameter range considered in this thesis.4

The relative positions of the two drops and the ratio of their diameters have been reported as the two main factors determining the interaction scenario. Works of Sirignano and co-authors, Raju & Sirignano (1987, 1990) and Chiang & Sirignano (1987, 1993) have shown that the collision of drops in tandem arrangement is likely even if the drops are 8 diameters apart initially, although as the diameter of the back drop decreases collision becomes less likely. Temkin & Ecker (1989) suggested that the region where the second drop is influenced by the first one, is a narrow paraboloid wake with a width of 1D and length of 15D drop diameters. Additionally, the drag coefficient of not only the back drop, but also of the front drop is reduced, however, not as significantly as of the second drop. In this work, it is shown that besides the above mentioned initial distance, additional parameters such as the Reynolds number of the flow and the density- and viscosity ratios are relevant to determine the interaction between deforming drops.

4Reprinted from Drop deformation and breakup due to shock wave and steady disturbances, International Journal of Multiphase Flow, vol 21(4):545-560 Hsiang & Faeth (1995), with permission from Elsevier. CHAPTER 3

Numerical handling of multiphase flows

Multiphase flows are present in all areas of life, in nature and industrial applica- tions, have many different forms and can be categorised by various aspects, As discussed in Chapter 1. Depending on which phases constitute the flow for ex- ample, we may talk about gas-liquid, gas-solid, liquid-solid, or even three-phase flows. Based on the distribution of the phases we can differentiate between tran- sient, separated and dispersed multiphase flows. Examples of the first, transient flows, are flows with a transition from pure liquid to a vapour flow due to ex- ternal heating. Stratified, slug, or film flows, are separated flows, where both phases are considered continuous, between which there is only one continuous interface. Lastly, in case of dispersed multiphase flows, the focus of this work, one of the phases is present in the form of liquid drops, solid particles, or gas bubbles, whereas the other phase is continuous. By definition, dispersed flows are such that one can pass from one point to any other points in the continuous phase while remaining in the same medium. It is, however, not possible to pass from one drop to another for example without going through the continuous phase. In the general description of flow regimes and numerical methods below, the term ”particles” may refer to liquid drops, solid particles, or gas bubbles. Dispersed two-phase flows can be further divided into two regimes, namely dense and dilute systems. In dense systems inter-particle spacing is low, and the transport of particles is dominated by collisions between them. It is therefore appropriate to consider the particles to behave as a continuum with properties analogous to those of the corresponding fluid. A common approach with such flows is to handle the two phases as interpenetrating continua, i.e. assume that both phases are present at the same place at the same time, and keep track of the relative amount of each phase. The local instantaneous equations describ- ing the properties and the behaviour of the flow are averaged over a control volume, then the averaged quantities are treated as continuous variables. In order for the continuum assumption to be valid, averaging is performed in a volume that is much larger as compared to both the particle size and the mean spacing between the particles, but also much smaller than the domain itself. The outcome of the method is a double set of equations, one for each phase, and an additional variable, volume fraction, determining the amount of each phase. Furthermore, appropriate boundary conditions need to be provided at

18 3. NUMERICAL HANDLING OF MULTIPHASE FLOWS 19 the interface. The above described method is the two-fluid approach, or also referred to as the Eulerian-Eulerian approach. When the two fluids are combined in a single set of equations and the interface is handled by a method specifically designed for that task (by a step- or also known as marker function), the one-fluid approach is followed. Such methods together with other interface handling methods are described in more detail in section 3.1. As opposed to dense systems, the spacing between particles in dilute sys- tems is rather large, direct interaction between the particles is rare, and fluid dynamic forces are governing particle transport. Additionally, interaction be- tween the dispersed particles and the fluid, the particles and walls, as well as body forces, become important. In this case it is more practical to track every particle of the dispersed phase, and compute the mass, velocity, and temper- ature history of each particle in the cloud, while the continuous phase is still computed in an Eulerian way. Consequently, this method is referred to as the Eulerian-Lagrangian approach, or also called Lagrangian Particle Tracking (LPT) method. The mathematical formulation of Eulerian-Eulerian and Eulerian-Lagran- gian methods are reviewed in the article by van Wachem & Almstedt (2003). Multiphase problems exhibit complexities of numerous sources; phenomena occurring at a wide range of time and length scales simultaneously due to turbulence, the presence as well as advection of an interface or contact line, or perturbations on the interface. As a consequence, numerical simulation has become essential, and sometimes the only available, tool to perform detailed investigations of the flow. While analytical methods are strongly limited to simple geometries and low Reynolds numbers, visualization and the realizable degree of control brings the main challenges to laboratory measurements and experiments (Dec (1997)). On the contrary, numerical simulations enable the choice of nearly arbitrary values to any physical parameter (density, viscosity, surface tension, etc.), or switch gravity effect on or off for example, which would be difficult or even impossible in an experiment. Depending on the size and purpose of the problem one may resolve the flow down to the smallest length scales, or simplify the description and em- ploy models and averaged equations. Three options are available from this aspect: Direct Numerical Simulation (DNS), Large Eddy Simulation (LES), and Reynolds Averaged Navier Stokes (RANS). While RANS provides an av- erage description of the field only, where all scales are modelled, the most energetic eddies are resolved in LES simulations, and only the smallest scales are modelled (Gorokhovski & Herrmann (2007); Jiang et al. (2010)). In DNS all length and time scales are fully resolved, there is no modelling beyond the continuum hypothesis. Naturally, DNS provides the most detailed description of the flow but at the same time is also the most computationally expensive method. On the other extreme, RANS simulations are much cheaper but also less accurate, yet, they often provide a good estimation for the given problem, 20 3. NUMERICAL HANDLING OF MULTIPHASE FLOWS especially for large and complex industrial systems where time and computa- tional capacity is limited. LES is a good compromise between the two, and therefore has become more and more widely used.

3.1. Interface handling methods In academy, it is a common practice to use well refined simulations such as LES and DNS, as they provide detailed solutions and enable us to study the fun- damental mechanisms governing multiphase flows, such as topological changes occurring at the interface for example. Problems involving different types of multiphase flows may require different solution methods, thus, various numer- ical algorithms have been developed over the past few decades, which in turn also contributed to a more widespread application of numerical simulations. The physics of interfaces, as well as a description of the main concepts regarding the various interface handling methods are reviewed in the article by Scardovelli & Zaleski (1999). A selection of the most widely used interface handling methods available are also listed in Table 3.1, and briefly described below.

Table 3.1: Interface handling methods available for simulating multiphase flows

Marker point meth. Fixed grid meth. Moving grid meth.

volume tracking marker function meth.: body fitted grid

front tracking VOF, LS, CIP unstructured grid

Immersed Boundary phase field sharp interface meth.

marker particle

As shown in Table 3.1, three main categories of interface handling methods have been developed, namely, marker point methods, fixed grid methods, and moving grid methods. In marker point methods, the two phases and/or the interface is represented and tracked by a selection of points. Methods where the phases and the interface are identified by the numerical value of a marker function are referred to as marker function methods, and constitute a group of methods with fixed grids. Lastly, in case of moving grid methods, the grid is constructed in a way that elements of the grid always coincide with the interface. 3.1. INTERFACE HANDLING METHODS 21

Marker point methods Marker point methods can be further divided into two groups, namely, volume tracking and front tracking methods. The marker-and-cell (MAC) approach by Harlow & Welch (1965) within volume tracking (also called marker particle methods) has the historical importance of being the first algorithm specifi- cally developed to solve multiphase problems. In volume tracking methods each phase is represented by a series of disconnected marker points distributed uniformly in the region they occupy, and the instantaneous positions of these particles are used to retrieve the Eulerian properties of the fluid. Volume track- ing methods are relatively easy to implement. However, the mentioned method by Harlow & Welch (1965) was not very successful in representing the inter- face, therefore, further improvements of this approach followed, resulting in marker function methods, described later. The second type of marker point methods are front tracking methods, where connected marker points along the interface are advected with the flow (Unverdi & Tryggvason (1992a)). Such marker points are used to track the interface, while the governing equations for the continuous phase are solved in an Eulerian manner on a fixed grid. As follows, information between the moving ”front grid” and the stationary ”fixed grid” needs to be transferred by interpolation. As the interface is stored on a separate mesh, it is inherently very well-defined. However, as a consequence, merging and breakup requires separate modelling. The main advantages of these methods are accuracy and robustness. On the other hand, the moving front may require remeshing, resulting in complex implementation. A review of front tracking methods and its applications is provided in Tryggvason et al. (2001). Comparing volume and surface marker methods, it can be concluded, that while volume marker methods are easier to implement, as the markers become distorted together with the fluids, surface marker methods provide a more accurate representation of the interface, since markers are tracking the interface directly (Scardovelli & Zaleski (1999)).

Immersed Boundary (IB) Marker point methods were further developed first by Peskin (1977) and then by Unverdi & Tryggvason (1992b), who used connected marker points to advect the boundary between two different fluids, and computed surface tension from the geometry of the interface. The above procedure resulted in the Immersed Boundary method. Similarly to marker methods, the interface is represented in a Lagrangian way, while a fixed Eulerian grid is used to solve the governing equations for the continuous phase. The advantage of the IB method is that the interface is well-defined and tracked in time, however, special algorithms are needed to handle coalescence and breakup phenomena. Further description of the method is found in the reviews by Peskin (2002) and Mittal & Iaccarino (2005). 22 3. NUMERICAL HANDLING OF MULTIPHASE FLOWS

Marker function methods Marker function methods are realizations of the one-fluid formulation, where one set of Navier-Stokes equations is solved for the whole system including the interface. In these methods, the distribution of each phase is known and is stored in the so called marker (or colour) function. If the densities and viscosities of each fluid are constant, fluid properties can be computed by these constant properties and the heaviside function at each point. Following initial works based on marker points, the first version of marker function methods was implemented by Harlow & Welch (1965), which later developed into the Volume of Fluid (VOF) method. However, this early VOF method involved difficulties and encouraged the development of other marker function methods, such as Level Set (LS), Constrained Interpolated Propagation (CIP) (Takewaki et al. (1985); Takewaki & Yabe (1987)), and phase field (Jacqmin (1999)) methods. In case of the Volume of Fluid method (Ferrari & Lunati (2013, 2014); Hirt & Nichols (1981); Jofrea et al. (2014)), the marker function is the volume fraction representing the amount of each fluid in the given cell. Volume fraction has a constant value of 1 and 0 in each phase. Thus, cells where only one of the phases is present obtain a volume fraction 0 or 1, whereas cells shared by the two phases, therefore containing the interface, have a volume fraction between 0 and 1. The reconstruction of the interface constitutes a key step in improving the accuracy of VOF methods, which in itself has become the focus of numerous studies. The simplest reconstruction methods available are the simple line interface calculation (SLIC) by Noh & Woodward (1976), and the SOLA-VOF algorithm by Hirt & Nichols (1981). A more accurate technique today is the piecewise linear interface contruction (PLIC) method Gueyffier et al. (1999); Rider & Kothe (1995); Youngs (1982)). The aforementioned reconstruction methods are reviewed in the article by Scardovelli & Zaleski (1999). The Level Set method builds on a similar idea as VOF, however, a distance function is used instead of the volume fraction, which expresses the closest distance to the interface at any given cell. The value of the distance function is positive in one phase, and negative in the other. A marker function similar to that used for VOF is produced by smoothing the distance function (Osher & Sethian (1988); Sussman et al. (1994)). Both methods (VOF and LS) are similar in the sense that after the ge- ometry of the interface is computed, the motion of the interface caused by the velocity field can be determined by geometric considerations. While LS is less difficult to implement and has the advantage of a more accurate computation of the normal and curvature of the surface, and therefore, a smoother interface is obtained, it is not mass-conservative in its original form. On the contrary, mass conservation is more easily obtained by employing the VOF method, whereas the main difficulty remains the accurate reconstruction of the interface. The VOF and LS methods are often combined in coupled VOF-LS methods, as the main disadvantages associated with each method can be handled by the other, 3.1. INTERFACE HANDLING METHODS 23 at the same time maintaining their respective advantages. (Ferrari (2000); Sussman & Puckett (2000)).

Moving grid methods Although being relatively simple to implement and computationally effective at the same time, it is difficult to obtain an accurate topology representation using fixed grid methods. This inspired the development of moving grid methods, where elements of the grid are always aligned with the interface. Moving grid methods fall under three main categories, namely, body fitted, and unstructured grid methods, and sharp interface methods. In body fitted grid methods, a structured grid is adjusted to the interface and is deformed in such a way that its edges always remain attached to the interface (Hirt et al. (1970); Ryskin & Leal (1983, 1984)). Naturally, these methods are very limited to simple geometries and small deformations. To avoid this limitation, unstructured grid methods were considered, where cells move in such a way that an edge of the cell always coincides with the interface. The third, sharp interface methods aim to combine the advantages of the one-fluid approach with those representing the interface more accurately. In these methods, a regular structured grid is used but the interface is treated in a special way, either by using difference formulas specifically taking into account the jump across the interface (Lee & LeVeque (2003)), by using ghost points (Fedkiw et al. (1999)), or reconstructing the control volumes in a way that one of their edges is always aligned with the interface (Udaykumar et al. (1997)). The main advantage of moving grid methods is that since cells are not divided by the interface, an accurate representation of the interface is eas- ily obtained. The main disadvantage is that since remeshing is required the method becomes complicated and expensive.

Advantages and disadvantages of the methods The main advantages and disadvantages of the above described methods are summarized in Table 3.2. As listed for front tracking, IB, and VOF methods, the inherent occurrence or non-occurrence of merging and breakup can be con- sidered both advantageous and disadvantageous depending on the nature of the problem. Methods where merging and breakup are inherently handled, such as in case of the VOF and LS methods, do not need sub-grid models to capture these topological changes. In problems where merging or breakup is expected, this naturally acts as an advantage. However, due to the nature of these meth- ods, merging and breakup may occur incorrectly when two interfaces approach each other closer than one computational cell. In such cases, coalescence and breakup are referred to as artificial or numerical coalescence and breakup, and are certainly not wanted. As follows, this property of the above methods is listed below both the advantage and disadvantage column. 24 3. NUMERICAL HANDLING OF MULTIPHASE FLOWS

Table 3.2: Advantages and disadvantages of the most widely used interface handling methods.

Advantages Disadvantages

Marker point methods

volume tracking easy to implement does not represent the accounts for interface very well topological changes

front tracking accurate, robust complex to implement accounts for (remeshing, transfer topological changes between grids) merging and breakup do NOT occur automatically

Immersed Boundary accurate transfer between grids merging and breakup requires sub-grid models merging and breakup do NOT occur automatically

Marker function methods

VOF mass conservative accurate interface needs accounts for extra treatment topological changes

merging and breakup DO occur automatically

LS smooth interface mass conservation issues

Moving grid methods

accurate interface expensive easily obtained requires remeshing CHAPTER 4

Numerical method

4.1. Governing equations for multiphase flows The Navier-Stokes equations express the conservation of mass, momentum and energy of Newtonian fluid flows. For isothermal flows the energy equation does not need to be considered. The governing equations for incompressible and isothermal multiphase flows without phase change can thus be written as;

∂u˜ i = 0 (4.1a) ∂x˜i      ∂u˜i ∂u˜i ∂p˜ ∂ ∂u˜i ∂u˜j ˜ ρ˜k +u ˜j = − + µ˜k + + σκn˜ iδ (4.1b) ∂t˜ ∂x˜j ∂x˜i ∂x˜j ∂x˜j ∂x˜i in dimensional form, where subscript k represents the phase and may be sub- stituted with l for liquid and g for gas;u ˜i is the velocity vector of fluid k;ρ ˜, µ˜, andp ˜ are the density, viscosity, and pressure respectively. Furthermore, σ is the constant surface tension coefficient that acts only on the interface, and therefore enters the momentum equation as a singular interface term by using Dirac’s distribution function δ. κ and ni are the curvature and unit normal of the interface respectively. As shown in Chapter 2, non-dimensional analysis can be of great practical use when describing different flows. Dimensionless parameters and govern- ing equations are obtained by identifying the characteristic properties of the flow, such as characteristic length, time, velocity, and pressure. Considering a spherical liquid drop of diameter D in a uniform flow of velocity U, where the ambient gas has a density ofρ ˜g and viscosityµ ˜g, the following dimensionless quantities may be defined:

u˜ x˜ ˜ U p˜ u = x = t = t p = 2 U D D ρgU (4.2) ρ˜k µ˜k ˜ ρk = µk = κ =κ ˜ D δ = δ D ρg µg Substituting these into Eqs. 4.1 and rearranging the equations, the non- dimensional form of the Navier-Stokes equations are obtained;

25 26 4. NUMERICAL METHOD

∂u i = 0 (4.3a) ∂xi      ∂ui ∂ui ∂p 1 ∂ ∂ui ∂uj κniδ ρk + uj = − + µk + + . (4.3b) ∂t ∂xj ∂xi Re ∂xj ∂xj ∂xi W e Equation 4.3b contains all relevant dimensionless parameters needed in order to characterise the flow around a liquid drop in an intermediate-range inertial flow. As introduced in Chapter 2, the drop Reynolds and Weber numbers and the density and viscosity ratios of the phases for drop flows are defined as

ρ UD ρ U 2D ρ µ Re = g , W e = g , ρ∗ = l , µ∗ = l . (4.4) µg σ ρg µg An additional commonly used dimensionless group is the Ohnesorge number, which for drop flows becomes √ µ µ∗ W e √ l √ Oh = = ∗ . ρlDσ ρ Re

4.2. Volume of Fluid (VOF) In this work, the VOF approach is used to handle the interface. The employed methodology and parts of the numerical code were used to study bubble and drop flows in the works of (L¨orstad(2003); Prahl (2007)), as well as for spray simulations by Großhans (2013) and Nyg˚ard(2016). As described in Chapter 3, the volume fraction, commonly denoted by α, is used to expresses how much of each phase - liquid and gas - the given cell contains. Cells that have values of α = 0 and α = 1 contain only one phase, while both phases, and thus the interface, are present in cells where 0 < α < 1. Properties, such as density and viscosity, are computed as weighted averages of the two phases;

ρk = ρl + (ρg − ρl)α ˜ (4.5a)

µk = µl + (µg − µl)α ˜ . (4.5b)

Hence, ρk in Eq. 4.3 becomes 1 in the gas phase, and ρl/ρg, i.e. the density ratio, in the liquid phase. Similarly, µk is 1 in the gas, and µl/µg in the liquid phase. Furthermore,α ˜ in Eq. 4.5 is the α field obtained by applying the cubic kernel described by Rudman (1997).

Surface tension According to the Continuum Surface Force model (CSF) by Brackbill et al. (1992), the surface tension term in 4.3b can be modelled as; 4.2. VOLUME OF FLUID (VOF) 27

κn δ κ ∂α i = . (4.6) W e W e ∂xi However, as surface forces are directly proportional to the curvature, which in turn is obtained from the α field, the accuracy of surface tension computations is highly sensitive to errors both in α and in the curvature κ. Following the method described in L¨orstad & Fuchs (2004) and L¨orstad et al. (2004), the Direction Averaged Normal (DAN) and Direction Averaged Curvature (DAC) methods are used to compute the normal and the curvature of the interface in each cell containing the interface;

    ∂z nx − /∂x   ∂α   n =   = ∂z ·  ∂z  (4.7a)  ny  ∂α  − /∂y    | ∂z |   nz 1

  nz z,ii z,iz,jz,ij κ = − 3 (4.7b) |nz| |n| |n| for a case where z is the direction of the longest normal component.

Phase transport Along with the continuity and momentum equations in 4.3, an additional equa- tion for the transport of the volume fraction,

∂α ∂u α + i = 0 , (4.8) ∂t ∂xi is solved using a 3D direction split method (Youngs (1982)); three 1D problems are solved in a sequence. First, the normal vector of the interface is determined in each cell using the DAN method. This normal vector together with the volume fraction α determines the plane - the interface - dividing the cell into two parts. Thereafter, this reconstructed interface is translated in the outflow direction, and outgoing fluxes are computed based on the geometrical area of the α-field protruding outside the cell, as illustrated in Figure 4.1.

4.2.1. Grid & discretization Discretization Governing equations are solved on a Cartesian staggered grid; velocity compo- nents are defined on cell faces, whereas scalar variables (density and pressure) are stored in the centre of the cell, as illustrated in Figure 4.2. The advantage of such a grid is that continuity is satisfied on the cell level, thus, mass conser- vation is globally fulfilled. However, as velocity components are not collocated, special treatment at the boundaries is required. 28 4. NUMERICAL METHOD

Figure 4.1: A 2D example of phase transport using Youngs’ model.

Figure 4.2: A 2D example of a Cartesian staggered grid.

Except for the convective terms, for which a hybrid first- or second order scheme is used depending on the Peclet number, all terms of the Navier-Stokes equations are discretized using central differences of second order. Higher order terms are introduced as single-step defect corrections in order to achieve high numerical accuracy as well as a robust and efficient solver, Gullbrand et al. (2001). A three-level second-order implicit scheme is used for temporal dis- cretization and is solved by a multi-grid method in each time step for a faster 4.2. VOLUME OF FLUID (VOF) 29 convergence rate. The pressure-velocity coupling is based on a simultaneous update of the dependent variables (Fuchs & Zhao (1984)).

Multigrid The basic underlying idea behind using multi-grid iterations is that high fre- quency errors, with wave-lengths comparable to the cell-size of a given mesh, are damped quickly, whereas low frequency errors are slow to remove. As low frequency errors on a finer mesh become high frequency errors on a coarser mesh with larger cell-size, iterations between meshes of different resolutions result in damped errors of all frequencies. An initial approximation of the so- lution is obtained on the finest mesh, which is then sequentially interpolated to the coarser meshes, where high frequency errors are reduced. Thereafter, the solution is successively prolongated back to the finer meshes. Iterations may be performed according to a simpler V-cycle scheme, or according to schemes in- volving repeated steps between the coarser mesh levels, as illustrated in Figure 4.3. The procedure is repeated until convergence criteria are fulfilled.

Figure 4.3: An illustration of two different multi-grid cycle schemes, a V-scheme (left), and a W-scheme (right). CHAPTER 5

Case setup and results

In this work, the secondary breakup of liquid drops produced in earlier stages of the atomization process in sprays is studied in two main steps. First, we look at single liquid drops in continuous gas flows in order to study the dynamics of deformation and breakup induced by steady disturbances. In the second part, the interaction between two drops is investigated, representing denser regions of sprays where drops cannot be assumed to behave independently of each other. Both for the single and the dual drop simulations, a variety of Reynolds numbers and density and viscosity ratios are considered, accounting for the various flow conditions in complex spray flows, as well as for varying material properties found in actual spray applications.

Computational setup In order to study the behaviour of liquid drops, a large cubical box is chosen as the computational domain, where one or two drops are placed in the y−z plane halving the domain. A side view of this mid-plane is shown in Figure 5.1. The size of the box has been chosen such that the boundaries of the domain have no influence on the solution. Prahl (2007) showed that under the very same flow setup as the one used in the present study, a domain 32 times the initial drop diameter is sufficient. In this study we work with smaller drops in order to be able to handle highly deformed drops without significantly increasing the size of the domain, and the computational cost. Thus, the edges of the domain become 128 times the initial drop diameter. Boundary conditions are defined as velocity inlet on the inlet face, zero gradient conditions at the outlet, and symmetry conditions on the four side planes of the domain. Two flow cases, uniform and non-symmetric shear flow are considered, defined by the inlet boundary condition. The inlet velocity profiles for the two flow cases are illustrated in Figure 5.1, and an example of the flow surrounding a single drop in the beginning of the simulations is shown in Figure 5.2 for both setups. The drops are followed in a moving reference frame fixed to the centre of the drop. For dual drop arrangements, the reference frame is fixed to the primary - or front - drop. In dual drop simulations, the secondary drop is allowed to move with respect to the primary drop. This is realized by applying

30 5. CASE SETUP AND RESULTS 31

Figure 5.1: Schematic 2D side-views of the computational domain and the initial and boundary conditions (streamwise velocity profile) applied for the uniform flow case (left) and for the shear flow case (right). Black circles repre- sent single- or primary drops, whereas unfilled circles represent the secondary drop at different positions in case of dual drop arrangements.

2 1.6 2 1.6

1.4 1.4 1 1 1.2 1.2

)/D 1 )/D 1 0 0 drop 0.8 drop 0.8 (Y-Y (Y-Y 0.6 0.6 -1 -1 0.4 0.4

0.2 0.2 -2 -2 0 0 -2 -1 0 1 2 -2 -1 0 1 2 (Z-Z /D (Z-Z /D drop drop

Figure 5.2: The flow around the drop at the beginning of simulations for the uniform flow case (left) and for a shear flow case (right), coloured by velocity magnitude (after K´ekesi et al. (2014, 2016)).

an artificial gravity force on the drops based on the computed acceleration of the primary drop at every instance. As the farfield of the drop does not need to be as resolved as the region close to the drop, a multigrid technique is employed as described in Chapter 4. This is realized by adding several local refinement layers on top of the global mesh, gradually decreasing in expanse and halving the size of the cells in each new layer. This way the mesh becomes finer as approaching the drops from the outer boundaries, as illustrated in Figure 5.3. 32 5. CASE SETUP AND RESULTS

GLOBAL MESHES - level 1,2

LOCAL REFINEMENTS

Ug - level 4 - - - - r b level 3

y 6 -z

Figure 5.3: Schematic 2D side-view of the computational domain and mesh structure.

5.1. Mesh refinement study A mesh refinement study with 4 different resolutions was performed for a single drop case in uniform flow, where the coarsest mesh had cell sizes of h = D/8 and the finest h = D/64. In Figure 5.4 velocity profiles of the streamwise and spanwise components are showed for the four different resolutions along two vertical lines, one through the centre of the drop and the other 1.5D behind the drop, for a case with a Weber number of W e = 1, Reynolds number of Re = 100, and density and viscosity ratios of ρ∗ = 80, µ∗ = 55.56. Additionally, the relative error of the drag coefficient of the drop, CD, changing with the resolution of the mesh is shown in Figure 5.5. As it is seen, the velocity profiles for the h = D/32 and h = D/64 meshes nearly overlap and the values of CD differ only by 4.6%. Therefore, and since the computational cost is much higher for the h = D/64 mesh than for the h = D/32 mesh (the number of cells increases from approximately 5 million to 25 million cells), the mesh with h = D/32 cell size is considered sufficient for the purpose of this work. All further simulations are thus performed on a mesh with cell sizes of h = D/32, except for two setups in the dualdrop configuration, where the distance between the drops is so large that keeping this resolution would be unfeasible. A coarser, h = D/16, mesh is then used for those cases. Since the focus is not on the details of drop shape and deformation, but on the relative movement of the drops, a lower resolution is considered acceptable. The expanse of the innermost local mesh is adjusted to ensure that its edges are at least 2.25D away from the centre of any drop, this way the drops never leave this mesh level while deforming. This is necessary due to the smoothing kernel applied through the interface mentioned in Chapter 4. The 5.1. MESH REFINEMENT STUDY 33

0.3 1.2

0.2 1

0.8 0.1

0.6 0

0.4 -0.1 0.2 h = D/8 h = D/8 Vertical velocity, V [-] h = D/16 h = D/16

Streamwise velocity, W [-] -0.2 0 h = D/32 h = D/32 h = D/64 h = D/64 -0.2 -0.3 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 Vertical position, (y-y )/D [-] Vertical position, (y-y )/D [-] drop drop

1.2 0.08 h = D/8 1 0.06 h = D/16 h = D/32 0.8 0.04 h = D/64

0.6 0.02

0.4 0

0.2 -0.02

0 h = D/8 -0.04 Vertical velocity, V [-] h = D/16 Streamwise velocity, W [-] -0.2 h = D/32 -0.06 h = D/64 -0.4 -0.08 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 Vertical position, (y-y )/D [-] Vertical position, (y-y )/D [-] drop drop

Figure 5.4: Velocity profiles for the streamwise and vertical components along a vertical line at the drop centre (upper row) and 1.5D behind the drop (lower row) for meshes of four different resolutions. W e = 1, Re = 100, ρ∗ = 80, µ∗ = 55.56.

40

35

30

25

20

15

10 Relative error (%) 5

0

-5 0 8 16 32 64 D/h [-]

Figure 5.5: The relative error of drag coefficient of the drop, CD, changing with the resolution of the mesh. 34 5. CASE SETUP AND RESULTS same strategy is used for the dual drop simulations, although the expanse of levels 5-8 is increased in order to maintain the necessary distance between the edges of the finest mesh level and the centre of any drop. The final size of the mesh is approximately 5million cells for single drop-, and 7.5 − 15 million cells for dual drop simulations.

5.2. Single drop simulations The aims of single drop simulations are to determine the onset of breakup, the type and characteristics of the breakup that the drop experiences, and to gain information on the time required for breakup under different conditions. The shear flow imposed on top of the uniform flow means to represent velocity gradients in the flow around the drop in a spray. Results of these studies are published in Paper 1 for uniform flow, and Paper 2 for shear flow. After a sufficient mesh has been set, a short study aiming to find the critical Weber number is performed, followed by an investigation on a wider range of parameters for a constant Weber number. Cases considered for single drop simulations are listed in Table 5.1 for uniform flow, and Tables 5.2 & 5.3 for shear flow.

5.2.1. Single drop, uniform flow As a first step, simulations for a set of increasing Weber numbers are performed until breakup is observed. The purpose of this short study is to validate our results by observing the critical Weber number for breakup found in the liter- ature, and to observe the increasing rate of deformation as the Weber number increases. The parameters used for these simulations are listed in Table 5.1 under ”Steady deformation”. Examples of the observed drop shapes are shown in Figure 5.6.

Figure 5.6: Deformation increasing with Weber number. From left to right: W e = 0.1, 1, 5, 10 and 12. Onset of breakup at W e = 12. (K´ekesi et al. (2014))

As shown in the images, the drop remains spherical for W e = 0.1, and only slight deformation is visible for W e = 1. However, further increasing the Weber number, the drop takes the shape of an oblate spheroid, as shown for W e = 5. At W e = 10 the drop is highly deformed, and breakup is observed for W e = 12, in agreement with the Weber number often referred to as the critical Weber number for breakup in the literature. 5.2. SINGLE DROP SIMULATIONS 35 0.1, 1, 5, 10, 1210080 20 55.6 20, 50, 80, 100, 200 20, 40, 80 0.5, 5, 20, 35, 50 D 2 g σ U UD µ l g l g g g ρ µ ρ ρ ρ µ ∗ ∗ W e Re ρ µ Table 5.1: Cases for single drop simulations in uniform flow ParameterWeber Abbr. number Definition Steady deformation Breakup Reynolds number Density ratio Viscosity ratio 36 5. CASE SETUP AND RESULTS

In the next step, the Weber number is kept fixed at W e = 20, which is higher than the critical number for breakup but still relatively low, and a parameter study for a wide range of Reynolds numbers as well as density and viscosity ratios is performed. The parameters used for these simulations are found in Table 5.1 under ”Breakup”. All combinations of the above parameters are considered for this study.

a)

t = 0.5 t = 5.0 t = 20.5 t = 21.5

b)

t = 0.5 t = 2.5 t = 5.0 t = 7.5

Figure 5.7: Examples of a single drop experiencing (a) bag breakup and b) shear breakup in uniform flows (images from K´ekesi et al. (2016).

Beside the classical and well-known shear- and bag breakup (Figure 5.7), various intermediate breakup modes were observed. One of the main outcome of these simulations is a regime map predicting the breakup mode depending on the Reynolds number and material properties of the flow included in an µ pρ inherited parameter A = C l/µg g/ρl, as shown in Figure 5.8. Besides this regime map, data on the time required for breakup is gathered, that can serve as input for the development of enhanced breakup models, a summary of which is shown in Figure 5.9. Additionally, breakup times have been related to a the- oretical expression of the time required for shear breakup (Fig. 5.10), derived in (K´ekesi et al. (2014, 2017)). This relation may be used to predict breakup time for single drops. Based on the above results, a representative case of each observed breakup mode has been chosen as baseline cases for follow-up studies. These cases and their parameters are listed in Table 5.2 5.2. SINGLE DROP SIMULATIONS 37

JS TS TB 1 10 S RS M B

A 0 10

−1 10

20 50 80 100 200 Re g

Figure 5.8: A new regime map for drop breakup in the bag and shear breakup regime. JS - jellyfish shear, TS - thick rim shear, TB - thick rim bag, S - shear, RS - rim shear, M - mixed, B - bag breakup mode (K´ekesi et al. (2014)).

Figure 5.9: The time required for breakup for a single drop as a function of the Reynolds number Re and material properties A, based on results in K´ekesi et al. (2014, 2017). 38 5. CASE SETUP AND RESULTS

4

3.5

3

2.5 sh

/t 2 bu t 1.5

1

0.5

0 0 5 10 15 A

Figure 5.10: Comparison of breakup times obtained from simulations and the- ory (K´ekesi et al. (2014, 2017)). 5.2. SINGLE DROP SIMULATIONS 39 1875 . 2 , 875 . 1 , 5625 . 1 , 25 . 1 , 9375 . shear shear bag 0 , 625 . 0 , 3125 . 50 5020 4050 20 0.520 80 20 20 5 80 20 20 20 20 20 50 20 0 Bag Shear Jellyfish Thick rim Thick rim Set1 Set2 Set3 Set4 Set5 , 0 D D 2 0 0 g σ U U µ l g l g g g ρ µ ρ µ ρ ρ 0 U U ∆ ∗ ∗ ρ µ Re W e G Parameter Abbr.Breakup mode Def. Density ratio Viscosity ratio Weber nr. Reynolds nr. Table 5.3: Non-dimensional sear rate and its values for single drop simulations with shear flow ParameterShear Abbr. rate Def. Values Table 5.2: Baseline cases for single drop simulations with shear flow and for off-axis dual drop simulations 40 5. CASE SETUP AND RESULTS

5.2.2. Single drop, shear flow Next, simulations on single drops in non-symmetric shear flows are performed for a wide range of shear rates of the flow. The purpose of this study is to more closely imitate flow conditions observed in real sprays, where drops may be surrounded by complicated turbulent flows rather than the academic case of uniform flows. The shear flow around the drop represents velocity gradients around the drop in real flows.

∆U Figure 5.11: Parameters defining the non-dimensional shear rate, G = /U0.

The definition of the non-dimensional shear rate used in this study is il- lustrated in Figure 5.11, and its values used in the simulations are shown in Table 5.3. All listed shear rate values are considered for the five baseline cases, Set1-5 presented in Table 5.2. The main achievement of this study is to demonstrate the effect of shear flow on the breakup mode and time as compared to the baseline cases in uniform flow, as shown in Figure 5.12. With increasing velocity gradients in the flow, breakup time decreases significantly, and the breakup mode of a specific case becomes more similar to shear breakup.

5.3. Dual drop simulations In the second part of the thesis, the interaction between two drops in various arrangements is studied. As illustrated in Figure 5.13, the relative position of the drops is determined by two parameters, namely the distance ` between the centres of the drops, and the angle β that the line connecting the centre of the drops closes with the horizontal direction. The distance and the angle between the drops is varied for a selection of cases representing each breakup mode observed in single drop simulations, Set1-5 in Table 5.2. The aim of this study is to investigate the interaction between the drops regarding collision or independent behaviour, and the time for breakup. Results 5.3. DUAL DROP SIMULATIONS 41

10 2 Classical bag Thick rim bag Small tilting bag Stretching sheet Tilting shear Jellyfish bag, thick rim Jellyfish bag/shear Drop tilts, dual shear K1*G m1 K2*G m2 10 1 bu t µ∗ 6

Re ?

0 10 G-

10 -1 10 0 10 1 G

Figure 5.12: Breakup times and breakup modes changing with shear rate (K´ekesi et al. (2016)).

of this work are presented in Paper 3 for tandem arrangements (β = 0◦) and in Paper 4 for off-axis configurations (β 6= 0◦), and summarised briefly below.

Figure 5.13: Definition of the parameters determining the relative position of the drops: the separation distance `, and the angle β.

For tandem arrangement, a selection of cases from Table 5.1 are chosen such that the points representing each case in Figure 5.8 are well distributed in the shown Re − A frame. Four Reynolds numbers, Re = 20, 50, 100 and 200, four combinations of density and viscosity ratios resulting in A ≈ 0.14 − 19.5, and six initial separation distances, `/D = 1.5, 3, 5, 8, 12, and 20 are considered. Additionally, off-axis configurations for three values of the initial separation 42 5. CASE SETUP AND RESULTS distance, `/D = 1.5, 3 and 5, and three angles β = 30◦, 45◦, and 90◦, beside the corresponding tandem cases are considered for Set1-5 in Table 5.2.

merge shoot through shoot through, connected by tail accelerate, breakup before collision accelerate, abrupt breakup no collision Re = 20 Re = 100 10 2 10 2

10 1 10 1 A A

10 0 10 0

10 -1 10 -1 0 5 10 15 20 0 5 10 15 20 `/D `/D Re = 50 Re = 200 10 2 10 2

10 1 10 1 A A

10 0 10 0

10 -1 10 -1 0 5 10 15 20 0 5 10 15 20 `/D `/D

Figure 5.14: Regime maps for interaction scenarios as a function of initial separation distance `/D and material properties A for Reynolds numbers of Re = 20, 50, 100 and 200 (Paper 3).

5 2.5 a) 5 merge b) shoot through ` = 5D no collision 4 2 4 ` = 3D 3 3 1.5 )/D )/D single bu drop drop / t bu 2 t

(Y-Y ` = 1.5D (Y-Y 2 1

1 1 0.5

0

0 0 0 1 2 3 4 5 1 2 3 4 5 (Z-Z )/D (Z-Z )/D drop drop

Figure 5.15: a) The behaviour of the Bag breakup baseline case, Set1 in Table 5.2. b) The breakup time of the secondary drop relative to that of a single drop. (Paper 4) 5.3. DUAL DROP SIMULATIONS 43

The outcome of these simulations are regime maps displaying which are the regions where the drops interact, and where they deform and break up independently of each other, i.e. behave as single drops. The regime map obtained from the tandem study is shown in Figure 5.14, and an example for the behaviour of an off-axis configuration is shown in Figure 5.15a for Set1, Bag breakup in Table 5.2. Furthermore, the breakup time of the secondary drop relative to that of a single drop under the same circumstances is reported, such as in Figure 5.15b. Sequential images illustrating the various interaction scenarios observed are provided, such as the one in Figure 5.16 for a case where the drops merge, in Figure 5.17a where the secondary drop shoots through the primary drop, or as in Figure 5.18 where the drops are independent of each other. Additionally, the velocity profile in the wake around the drops is analysed for a few cases in order to reveal underlying phenomena determining the conditions causing collision. One such example is shown in Figure 5.17b.

t = 0.5 t = 24.5

t = 5.5 t = 30.5

t = 13.5 t = 33

t = 16 t = 35

t = 17.5 t = 42.5

Figure 5.16: An example for merging drops. Re = 50, ρ∗ = 20, µ∗ = 50, and `/D = 3. (Paper 3) 44 5. CASE SETUP AND RESULTS

a)

t = 0.5 t = 10

t = 3.5 t = 13

t = 6.5 t = 15

t = 8 t = 17

1 1

b) 0.5 0.5

0 0

W (-) -0.5 -0.5 t = 0.5 t = 6.5 -1 -1

-2 0 2 4 6 -2 0 2 4 6

1 1

0.5 0.5

0 0

W (-) -0.5 -0.5

-1 t = 3.5 -1 t = 7

-2 0 2 4 6 -2 0 2 4 6 (Z-Z )/D (-) (Z-Z )/D (-) drop drop

Figure 5.17: a) An example where the back drop shoots through the front one. b) The development of the wake profile. Blue dashed lines indicate the position of the drops. Re = 200, ρ∗ = 20, µ∗ = 50, and `/D = 1.5. (Paper 3)

Lastly, a criterion for drop interaction/collision is suggested based on the length of the wake behind a single drop as it develops with time, and on the breakup time observed for a single drops, presented in the form of a regime drop, as shown in Figure 5.19. 5.3. DUAL DROP SIMULATIONS 45

t = 0.5 t = 2.5 t = 4.5 t = 7

Figure 5.18: An example for independent drops in a side-by-side arrangement. Set2, Shear breakup: Re = 50, ρ∗ = 40, µ∗ = 0.5, β = 90◦, and ` = 3D. (Paper 4) 46 5. CASE SETUP AND RESULTS

25

20 20

INDEPENDENT

ACCELERATE 15 )/D g

12 `/D (W<0.9U

10 wake L

8 STRONG INTERACTION 5 5

3

1.5

0 0 5 10 15 20 25 30 35 40 45 t (-)

Figure 5.19: Datapoints for dual drop (tandem) simulations in a time-length co- ordinate system, where values on the horizontal axis correspond to the breakup time of a single drop under the same circumstances as the given datapoint, and the length on the vertical axis represents the initial separation distance be- tween the drops for the given point. The figure relates data from single drop simulations to the outcome of the interaction of two drops. Thin black lines show the length of the wake in time behind a single drop. CHAPTER 6

Concluding remarks and future work

In this thesis, the dynamics of deformation and breakup of liquid drops in gaseous flows is investigated in detail. The main motivation of the work is the need for improved breakup models employed in simulations of complex spray flows that cannot be resolved to the scale of small drops, and therefore, the behaviour of the smallest scales needs to be modelled. A more detailed understanding on the deformation and breakup, the lifetime (breakup time), and aerodynamic coefficients of the drops, such as lift and drag, is needed. The work completed in this thesis can be thought of as a series of numeri- cal experiments. Given the experimental setup (a numerical code in this case), experiments on the system (liquid drops) are performed for a selection of pa- rameters and flow setups. Based on the observations from these experiments, the aim is to describe underlying phenomena in a simple way, as well as create models predicting the behaviour of the system as a function of a small number of parameters. The main questions sought for are the following; how is the deformation and breakup of liquid drops effected by;

• altering material properties, such as the density (ρ∗) and viscosity (µ∗) ratios of the phases

• velocity gradients present in the flow surrounding the drops

• the interaction between two drops in the vicinity of each other

In order to seek answers to the above questions, the dynamics of deforma- tion and breakup of liquid drops is studied in three main steps. In the first part of the work, single liquid drops are studied in uniform flows, with the aim of gaining a better understanding of the deformation and breakup of drops in an intermediate Reynolds- and Weber number range. The main achievments of this study are the detailed time histories of the defor- mation process leading to breakup; the observation of various intermediate breakup modes at a constant Weber number that have not been reported in earlier studies; and the physical analysis of the forces acting on a deforming drop resulting in a regime map and a criterion for the outcome of the breakup

47 48 6. CONCLUDING REMARKS AND FUTURE WORK process, bag/shear breakup mode. Furthermore, an estimate for the time re- quired for breakup is expressed as a function of material properties of the flow. As a second step, shear flows with various velocity gradients are considered, with the aim to move towards more realistic flow configurations that are present in real applications. It is found that velocity gradients present in the flow can significantly alter the breakup process, both the breakup mode, and the time required for breakup. The main contribution to the field of this study is the empirical model suggested for breakup time, expressing the dependence of breakup time on velocity gradients present in the flow. In the last part of the work, simulations on two liquid drops in different arrangements are performed. Two parameters, namely the initial separation distance, and the angle between the line connecting the centres of the drops and the freestream direction are varied. It is found that in tandem arrangement, the wake of the primary drop has a significant influence on the behaviour of the secondary drop even for large initial separation distances, the drops collide and merge in most cases. However, for the cases where the two drops are in off-axis arrangements, the drops deform independently of each other for most cases, except the ones with the shortest initial distances (in tandem or off-axis). Naturally, the breakup time is close to those observed in case of single drops for the cases that behave independently. However, in cases where the drops merge, a much longer breakup time is experienced.

Limitations, suggested improvements and future work:

• Gravitational forces are neglected in the simulations. This is reasonable for drops in internal combustion engines, where the velocity of the drops is typically much higher than the velocity difference caused by gravity. However, gravity effects may need to be considered in case of performing simulations for other applications.

• The resolution of the simulations is limited in the sense that it is not able to capture the precise geometrical details of breakup and coales- cence. However, since this work is primarily focused on the dynamics of the deformation leading to breakup, and the time it takes for the drops to reach breakup, the chosen resolution is reasonable and sufficient, as shown by the mesh refinement study described in Section 5.1.

• An interesting and necessary completion of the work is the extension of the parameter range to higher Weber numbers. As known from the literature, breakup scenarios at higher Weber numbers are significantly different than those in low Weber number flows. It would be interest- ing to investigate to what extent the density and viscosity ratios alter breakup phenomena in higher Weber number ranges. 6. CONCLUDING REMARKS AND FUTURE WORK 49

• Other canonical flows, such as extensional flow, would provide further insight into the characteristics of drop dynamics in flows significantly different from uniform flows.

• Data obtained for the interaction between drops may directly be im- plemented in spray simulations. One suggestion for implementation is to measure the region behind the primary drop where the secondary drop is strongly effected, and employ as an if-condition for drops close behind. Any secondary drop that is inside the noted region of influence would collide into the primary drop. The final outcome and parameters of the resulting drop(s) may be concluded from regime maps, such as those presented in Figures 5.14 and 5.15.

• Results of the present work may be combined with those obtained for evaporating drops in the studies of Albernaz et al. (2016); Albernaz (2016), and Albernaz et al. (2017). This combined effort could be used for advanced models regarding the deformation and breakup of evapo- rating drops employed in spray simulations, such as the works in Nyg˚ard (2016); Nygaard et al. (2016). CHAPTER 7

Papers and Author contribution

Paper 1 Drop deformation and breakup. T. Kekesi´ (TK), G. Amberg (GA), L. Prahl Wittberg (LPW) Int. J. Multiphase Flow 66, 1 – 10, 2014

Corrigendum to: Drop deformation and breakup. Int. J. Multiphase Flow, 66, (2014) 1-10 T. Kekesi´ (TK), G. Amberg (GA), L. Prahl Wittberg (LPW) Int. J. Multiphase Flow 93, 213 – 215, 2017

In this study, an in-house Volume of Fluid (VOF) method is applied to in- vestigate the deformation and breakup of an initially spherical drop induced by steady disturbances. Different fluid properties (density- and viscosity ratios) were studied for a wide range of drop Reynolds numbers in the steady wake regime. Interesting mixed breakup modes were observed for a Weber number of 20 when varying the fluid properties and Reynolds numbers. The results are presented in the form of a new regime map, and a criterion for the transition between bag- and shear breakup was defined relating the competing inertial and shear forces appearing in the flow. All the numerical simulations and anal- ysis of the results were carried out by TK under the supervision of GA and LPW. The manuscript was prepared by TK, with feedback of GA and LPW. The Corrigendum and the work associated with it was carried out by TK, with feedback from GA and LPW.

Part of this work has been published in or presented at: • 8th International Conference on Multiphase Flow (ICMF 2013), Jeju, Republic of Korea, May 26 – 31, 2013 • 68th Annual Meeting of the APS Division of Fluid Dynamics, Boston, MA, USA, Nov 22 – 24, 2015

Paper 2 Drop deformation and breakup in flows with shear T. Kekesi´ (TK), G. Amberg (GA), L. Prahl Wittberg (LPW) Chemical Engineering Science 140, 319 – 329, 2015

50 7. PAPERS AND AUTHOR CONTRIBUTION 51

In this study, the deformation and breakup of a single liquid drop is stud- ied in shear flows superimposed on a uniform flow using an in-house Volume of Fluid (VOF) method. The effect of shearing on the breakup mechanism is investigated as a function of the shear rate. The parameter range studied is obtained from our previous study (Paper 1) along with the non-dimensional flow shear rate. The characteristics of the shear breakup were found to remain similar independent of shear rate whereas the other breakup modes were sig- nificantly altered when changing the shear rate. In the work, a breakup time model as a function of shear rate was formulated. Modifications on the bound- ary conditions were carried out by TK and TK performed all the computations. TK analysed the data under the supervision of GA and LPW. The manuscript was written by TK, with feedback provided by GA and LPW.

Part of this work has been published in or presented at:

• Swedish Industrial Association for Multiphase Flows (SIAMUF) semi- nar, Lund, Sweden, May 28 – 29, 2015 • 68th Annual Meeting of the APS Division of Fluid Dynamics, Boston, MA, USA, Nov 22 – 24, 2015 • 9th International Conference on Multiphase Flow (ICMF 2016), Flo- rence, Italy, May 22 – 27, 2016

Paper 3 Interaction between two deforming liquid drops in tandem subject to uniform flow T. Kekesi´ (TK), Mireia Altimira (MA), G. Amberg (GA), L. Prahl Wittberg (LPW) Submitted to Phys. Fluids

In this study, an in-house Volume of Fluid (VOF) method is employed to in- vestigate the interaction of two liquid drops placed in tandem. The parameter range studied is obtained from our previous study (Paper 1). The results are presented in the form of sequential images illustrating the various interaction scenarios in the applied parameter range, regime maps regarding the relative movement of the drops are presented. Furthermore, the streamwise velocity in the wake around the two drops is analysed. By correlating data from the present study and from single drop simulations in an earlier work, criterions for weak and strong interaction of the drops is defined. Additionally, data for the length of the wake region behind a single drops is provided. TK carried out the changes needed in the code to setup the new cases, and carried out all simulations under the supervision of GA, LPW, and MA. The manuscript was prepared by TK, with feedback of GA, LPW, and MA. 52 7. PAPERS AND AUTHOR CONTRIBUTION

Paper 4 Interaction between two deforming liquid drops in tandem and various off-axis arrangements subject to uniform flow T. Kekesi´ (TK), Mireia Altimira (MA), G. Amberg (GA), L. Prahl Wittberg (LPW) Submitted to Int. J. Multiphase Flow

In this study, the interaction of two liquid drops placed in tandem and var- ious off-axis configurations is investigated using an in-house Volume of Fluid (VOF) method. The parameter range studied is obtained from our previous study (Paper 1), and is the same as in Paper 2. The results are presented in the form of sequential images illustrating the various interaction scenarios for various angles and initial separation distances for all five baseline cases. Regime maps are presented summarising the relative movement of the drops. Furthermore, data on the breakup time of the secondary drop as compared to the breakup time experienced by a single drop under the same circumstances is provided. TK carried out the changes needed in the code to setup the new cases, and carried out all simulations and analysis of the results under the su- pervision of GA, LPW, and MA. The manuscript was prepared by TK, with feedback of GA, LPW, and MA. Bibliography

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This work was supported by the Linn´eFLOW Centre and by the Swedish Research Council (VR) and is gratefully acknowledged.

First of all, I would like to thank my supervisors Prof. Gustav Amberg and Dr. Lisa Prahl Wittberg for offering me the opportunity to work on this project. It has been a long ride with ups and downs of all kind, but I can confidently say that I have always enjoyed your full support in every sense, at all times. I am very thankful for all the time you dedicated on our meetings all these years despite your busy schedules, and for the endless discussions and feedback. Thank you for your constant optimism, encouragement, and not least, patience. I would also like to thank Dr. Mireia Altimira, my third supervisor who joined our project during this last year, for bringing fresh ideas from a more applied area in the field to our discussions, for keeping up with meetings and massive amount of feedback, including smileys at times :), even though you have been busy taking care of the youngest member of your family. Carolina, Heide, Jonna, Malin, P¨ar,and Stefan, thank you for taking care of everything and making everyday technicalities almost unnoticeable. Thanks to all my office-mates during these years for the good discussions, fun times, and most important, a happy environment to spend our everydays in: Abdul-Malik, Amer, Andreas, Anthony, Arash, Daniel, Yuli, and Zeinab. A special thanks goes to the guys on 5th floor for accepting me as an honorary member of your crazy family: Alex, Asuka, Bernhard, Feng, Lucas, Niclas, Romain, and Stevin. A good laugh is always guaranteed there! I am very fortunate and happy to have enjoyed the company of many others at the de- partment, I hope to keep in touch with many of you in the future: Armin, Ellinor, Enrico, Erik, Igor, Iman, Jacopo, Nima, Priti, Ruoli, Taras, Tomas, Uˇgis,Werner, and Yang. I also want to thank friends and team-mates at L˚angholmenFC and KTH FC, for making sure football-time is always fun and refreshing. After all, we need to compensate for all that sitting! A special thanks to Teresa, I am very happy to have shared this bumpy journey of being a PhD-student with you, it’s comforting to remember we are all in the same shoes.

60 And lastly, I would like to send a special thanks to my family, for mak- ing sure I have all the possibilities to reach my goals, for understanding the sometimes too seldom visits home, and to cope with my brief communication at busy times. K¨osszencs!=)