Conduction Laser Welding

JOSEFINE SVENUNGSSON Licentiate Thesis Production Technology 2019 No. 27

Conduction laser welding modelling of melt pool with free surface deformation Conduction laser welding Laser welding is commonly used in the automotive-, steel- and aerospace industry.

However, deeper knowledge is still needed to better control this process, improve prod- CONDUCTION LASER WELDING - MODELLING OF MELT POOL WITH FREE SURFACE DEFORMATION modelling of melt pool with free surface deformation uct quality, produce components with less material, reduce production errors and thus contribute to sustainable manufacturing production. Process knowledge can be gained through modelling and experimental observation. The present study therefore aimed at developing and testing a simulation model dedicated to the thermal ﬂow and free surface deformation of the melt pool formed during laser welding. The physics implemented in the model includes the thermocapillary force that accounts Joseﬁne Svenungsson for the effect of temperature gradients on surface tension, the solid-liquid phase change, and the convection of fusion enthalpy. From the computed test cases, it was found that the convection of fusion enthalpy should not be neglected. It was also found that the numerical implementation of the thermocapillary force can lead to unphysical solutions. Therefore, it is recommended to select an approach consistent for all the surface forces of the problem. Finally, free surface oscillations known from experiments to occur are also computed outputs of the model. However, it remains to investigate whether these oscillations are, or not, disturbed by numerical noise. 2019 NO.27

ISBN 978-91-88847-35-5 (Printed version) ISBN 978-91-88847-34-8 (Electronic version)

118526_HV_Josefine_Svenungsson_omslag_190516_v2.indd 1 2019-05-16 09:32:15

Tryck: BrandFactory, maj 2019. Licentiate Thesis Production Technology 2019 No. 27

Conduction laser welding modelling of melt pool with free surface deformation

Josefine Svenungsson Department of Engineering Science University West SE-461 86 Trollhättan Sweden Telephone +46 (0)52 – 022 3000 www.hv.se

c Joseﬁne Svenungsson, 2019. ISBN 978-91-88847-35-5 (Printed version) ISBN 978-91-88847-34-8 (Electronic version)

Typeset by the author using LATEX.

Trollhättan, Sweden 2019 to my family

Sammanfattning

Titel: Laser konduktionssvets - modellering av smältpöl med deformation på ytan Nyckelord: Lasersvetsning, modellering, strömningslära, CFD, OpenFOAM, ytdeformation, numerisk validering ISBN 978-91-88847-35-5 (Printed version) ISBN 978-91-88847-34-8 (Electronic version)

Lasersvetsning är en vanlig produktionsmetod inom t.ex. bil-, stål- och fly- gindustrin. Det är en icke-linjär och starkt kopplad process där svetsgeometrin påverkas av flödet i smältan. Experimentella observationer är utmanande då smältpölen under ytan inte är tillgänglig under svetsning. Förbättrad pro- cesstyrning skulle möjliggöra att bibehålla, eller förbättra, produktkvaliteten med mindre material och bidra vidare till hållbarhet genom minskade produk- tionsfel. Numerisk modellering med, Computational Fluid Dynamics, CFD, ger en djupare förståelse med tillgång till processegenskaper som ännu inte kan nås vid experimentell observation. De befintliga numeriska modellerna brister dock i förutsägbarhet. Arbetet som presenteras här består av utveckling av en numerisk modell för konduktionssvetsning. Genom arbetet som presenteras här har en modell för lasersvetsning utveck- lats baserat på en existerande modell. Därefter tillämpas modellen på olika testfall för att undersöka specifika delar av processfysiken som implementerats. Två fall fokuserar på termokapillär konvektion i tvåfas- och trefasflöden med ytdeformation. Slutligen betraktar det tredje testfallet smältan och flödet i denna under konduktionssvetsning. Slutsatsen av studien är att konvektion av entalpin, försummad i tidigare studier, ska inkluderas i modellen. Implementering av termokapillärverkanbör vara konsekvent med de andra ytkrafterna för att undvika icke-fysikaliska lös- ningar. Instabiliteter på den fria ytan, kända från experimentella observationer, beräknas också numeriskt. Ytterligare studier behövs för att kontrollera att dessa oscillationer inte påverkas av numeriska instabiliteter.

i Abstract

Title: Conduction laser welding - modelling of melt pool with free surface deformation Language: English Keywords: Conduction laser welding, numerical modelling, Computational Fluid Dynamics, OpenFOAM, free surface deformation, melt pool ISBN 978-91-88847-35-5 (Printed version) ISBN 978-91-88847-34-8 (Electronic version) Laser welding is commonly used in the automotive-, steel- and aerospace industry. It is a highly non-linear and coupled process where the weld geometry is strongly affected by the flow pattern in the melt pool. Experimental observations are challenging since the melt pool and melt flow below the surface are not yet accessible during welding. Improved process control would allow maintaining, or improving, product quality with less material and contribute further to sustainability by reducing production errors. Numerical modelling with Computational Fluid Dynamics, CFD, provides complementary understanding with access to process properties that are not yet reachable with experimental observation. However, the existing numerical models lack predictability when considering the weld shape. The work presented here is the development of a model for conduction laser welding. The solver upon which the model is based is first described in detail. Then different validation cases are applied in order to test specific parts of the physics implemented. Two cases focus on thermocapillary convection in two-phase and three-phase flows with surface deformation. Finally, a third case considers the melt pool flow during conduction mode welding. It is concluded that the convection of fusion enthalpy, which was neglected in former studies, should be included in the model. The implementation of the thermocapillary force is recommended to be consistent with the other surface forces to avoid unphysical solution. Free surface oscillations, known from experimental observations, are also computed numerically. However, further investigation is needed to check that these oscillations are not disturbed by numerical oscillations.

ii Nomenclature

αC omax maximum alpha Courant number α thermal diffusivity

β thermal expansion coefﬁcient

∆H 0 standard heat of adsoprtion

∆T temperature difference

εn stabilization factor

dσ dT thermocapillary coefﬁcient Γ thermocapillary coefﬁcient

µ dynamic viscosity

ν kinematic viscosity

φf face volume ﬂux ρ density

ρ∗ dimensionless density

σ capillary (or surface tension) coefﬁcient

τ stress tensor c interface curvature ~g gravitational acceleration n~f interface normal vector u~ single ﬂuid velocity

Cα compression factor

iii NOMENCLATURE

C omax maximum Courant number fL liquid fraction h∗ dimensionless enthalpy k1 entropy factor Ma Mach number

SL∗ source term for latent heat t ∗ dimensionless time

Tm melting temperature u∗ dimensionless velocity C permeability coefﬁcient

Cp speciﬁc heat Co Courant number h heat transfer coefﬁcient k thermal conductivity

Lf latent heat of fusion p static pressure

T temperature

V characteristic velocity of ﬂow

iv Contents

Sammanfattning i

Abstract ii

1 Introduction 1 1.1 Background and motivation ...... 1 1.2 Problem and aim of work ...... 3 1.3 Objective and research questions ...... 5 1.4 Limitations ...... 5 1.5 Methodology/Strategy ...... 5 1.6 Outline ...... 6

2 Laser welding manufacturing process 7 2.1 Fusion welding ...... 8 2.1.1 Arc welding ...... 8 2.1.2 Power beam welding ...... 9 2.1.3 Hybrid welding ...... 10 2.2 Laser beam welding ...... 10 2.2.1 Laser energy sources applied to welding ...... 11 2.2.2 Laser welding process ...... 12

3 State of the art of laser welding models 21 3.1 Melt pool models neglecting fluid flow ...... 21 3.2 CFD melt pool models for conduction mode laser ...... 22 3.3 Coupled models from other fields than laser welding ...... 29 3.4 Interface capturing ...... 29 3.5 Implementation of the forces applied at the free surface ...... 33

4 The interFoam solver 37 4.1 MULES ...... 38 4.2 Pressure equation ...... 41 4.3 Pressure-velocity coupling ...... 43 4.4 InterFoam solution algorithm ...... 43

v CONTENTS

5 The melt pool model for laser welding in conduction mode 46 5.1 Laser energy source model ...... 46 5.2 Thermo-ﬂuid model ...... 50 5.2.1 Mass conservation equation ...... 50 5.2.2 Momentum conservation equation ...... 51 5.2.3 Energy conservation equation ...... 54 5.3 New developments made in interFoam ...... 55

6 Numerical applications - results and discussion 57 6.1 Two-phase flow driven by thermocapillary force ...... 57 6.1.1 Test case description ...... 58 6.1.2 Numerical setting ...... 58 6.1.3 Results and discussion ...... 59 6.1.4 Conclusion ...... 64 6.2 Three-phase flow driven by thermocapillary force ...... 64 6.2.1 Test case description ...... 65 6.2.2 Numerical setting ...... 65 6.2.3 Results and discussion ...... 67 6.2.4 Conclusion ...... 68 6.3 Melt flow with free surface deformation in conduction laser welding . . . . 69 6.3.1 Test case description ...... 70 6.3.2 Numerical setting ...... 71 6.3.3 Results and discussion ...... 72 6.3.4 Conclusion ...... 76

7 Conclusions 96

I Appendix

References 109

vi Chapter 1

Introduction

Welding is one of the most commonly applied methods for joining metal parts in manufacturing. It is found in the automotive industry, aerospace industry, shipbuilding, construction of buildings and bridges as well as joining of pipes in pipelines and offshore industry. Other application areas include medical and electronic devices. Around 100 different types of welding processes exist [1, 2]. In manufacturing the most commonly used are fusion welding processes, which include e.g. laser welding. This thesis is related to laser welding problems met in aerospace industry. It addresses the modelling and simulation of the melt pool in fusion welding when using a laser heat source.

1.1 Background and motivation

Today aerospace engines are produced using super-alloys resistant to high temperature in order to reach high engine efficiency. Producing engines with these materials requires advanced technologies such as laser welding which imply high production costs. Aerospace industry has a long term aim to reduce the weight of the aerospace engines. It would reduce the fuel consumption, and thus be beneficial for reducing greenhouse effect. It would also improve sustainability by reducing the waste due to materials processing. In order to achieve these goals light weight metals with high strength and good corrosion resistance can be used combined with welding process optimization to guarantee product quality. Aerospace industry has indeed high demands on quality. Improved process control would allow maintaining, or improving, product quality with less material and contribute further to sustainability by reducing production errors. Product quality is mainly governed by the microstructure, hardness and other metallurgical properties of the final weld. These properties are deeply influenced by the weld geometry and the evolution of the temperature field in the workpiece along the welding process (and the post treatment if any) [3, 4, 5]. For

1 CHAPTER 1.INTRODUCTION instance GKN Aerospace is facing problems with welds with too narrow waist in keyhole laser welding of Haynes282 and Alloy718. One problem is related to instabilities at the free surface of the liquid metal that can lead to gas entrapment and resultinpore formation. The secondproblem is related to the geometry of the melt front. This geometry presents a waist. It is thus broader on the top and bottom surface of the workpiece than in the center. Then non-destructive testing, conducted inspecting theworkpiece surface, does not allow detecting if theweld is too weak as the most narrow section is not observable. Figure 1.1 show a cross-section of a keyholeweld with a narrow waist andporosity. These problems were the motivation for starting this investigation.

Figure 1.1: Porosity and waistinaweld cross-section

Weld geometry and thermalhistory are in turn governedbythe material properties, the surrounding atmosphere and the process parameters such as the welding speed, the heat source power and its angle of incidence with the workpiece. Other process parameters of importance that are specific to laser welding are the focallength and the focal spot size (beam diameter). As an illustration, the depth of the melt pool is known to increase when decreasing thewelding speed or when increasing the laser power up to some limits above which the penetration depth does no longer increase [6].Ifthe beam diameteris decreased the beam intensity (or power per unit area) is increased and the depth of themelt pool increases [7, 8, 9].Adifficulty in predicting product quality comes from the non-linear influence of most of these parameters. In laser welding, thetemperature field is strongly coupled to convection in themelt pool.Themelt pool flow is driven by e.g. surface tension variations. Surface tension variations appear due to temperature gradients at the free surface and to local concentrations in surfactants that can comefrom both sides of the free surface [10, 11, 12]. Local concentrations in surfactants are governed bythetemperature gradients, by diffusion andby convection. Diffusion and convection involve transport properties such as diffusivity and viscosity that are material and temperature dependent. This exampleofphysics acting in the

2 1.2. PROBLEMANDAIMOFWORK melt pool along the process illustrates how tightly and non-linearly coupled the problem is. Experimental observation of the laser welding process can be challenging. The melt pool below the free surface is still not possbile to observe during welding for instance. The flow pattern, which is known to strongly influence the weld geometry and the thermal history cannot easily be measured nor monitored in-situ today. To optimize a process such as laser welding, without costly "trail and error", a thorough understanding of the process and how the process parameters do govern the process is still needed today. Numerical modelling provides complementary understanding with access to process properties not yet reachable to experimental observation. At low laser energy density per unit area (conduction mode) the numerical models address the melt pool. At high energy density per unit area (keyhole mode) they need to address both the melt pool and the keyhole physics. Based on the literature of laser welding it is clear that the physics in the melt pool is the same in keyhole mode as in the conduction mode. Modelling the melt pool is thus the first step to tackle; laser welding applications in conduction mode can then be used to test the model.

1.2 Problem and aim of work

Models for computing the melt pool flow in laser welding can be grouped in two categories: the models neglecting free surface deformation and the models accounting for free surface deformation. Most applications available in the literature belong to the first category [12, 13, 14, 15, 16, 17, 18, 19]. Neglecting free surface deformation allows running much quicker calculations than when tracking the free surface deformation. However this approach does not yet allow predicting the melt front location. Authors such as Pitscheneder et al. [12] have suggested that this problem is related to the assumption of laminar flow in the melt pool, while a turbulent approach should instead be used. The transient and unstable nature of the melt flows was experimentally observed at the free surface by various authors such as Zhao et al. [20]. Since the work by Pitscheneder et al. [12] many authors have chosen to mimic a supposed effect of turbulence. In that aim they artificially and uniformly enhance the viscosity [21, 22] or the thermal conductivity [23] or both of them [24, 25]. The level of enhancement is determined case by case to adjust the calculated melt front topography to experimental measurements. A large span of enhancement factors, f, can be found in the literature, such as f=4 for thermal conductivity enhancement alone in Mishra et al. [23], f=7 for both the viscosity and thermal conductivity in Pitscheneder et al. [12], up to f=17 in De and DebRoy [24, 25], or f=30 for viscosity enhancement alone in Anderson et al.

3 CHAPTER 1. INTRODUCTION

[22]. Choo et al. [26] also used f=30 but suggested the use of f=100 up to f=1000. A RANS approach was applied by Chakraborty [16]. The closure model for turbulence was the high Reynolds k-ε model with model parameters taken from aerodynamics applications. This author compared the results computed assuming laminar flow and turbulent flow. He found very similar results for the velocity and temperature fields, although the maximum values were lower in the turbulence computations. From these results it was concluded that turbulence might not really improve the prediction of the melt pool shape. However, Chakraborty did not explicitly compare the topography of the calculated melt front with experimental results. More recently, Kidess et al. [18] addressed this controversial question using direct numerical simulation (DNS) to avoid relying on questionable closure relations (since developed in a different context than melt pool flow). They simulated the melt pool produced during laser welding of a steel containing enough sulfur (150ppm) to cause a non-monotonic thermocapillary force at the free surface. The deformation of the free surface was neglected. They observed instabilities resulting from the interaction of two counter rotating flows induced by the change in sign of the thermocapillary (or Marangoni) force in the presence of temperature gradients. These instabilities turned out to act as non-uniform enhancements of the effective viscosity and thermal conductivity, which contradicts the former assumption of uniform enhancement. Moreover these instabilities were not yet sufficient to compute melt pool topography that reproduce experimental observation. In parallel, Saldi et al. [27] developed a 3D model for conduction laser welding including the solidification step after switching off the laser. The predicted melt shape was slightly improved when assuming laminar flow (f=1) compared to the case without solidification step. However, a discrepancy compared to experimental measurements was still observed. By applying an enhancement f=4 for viscosity and thermal conductivity the reproduced weld pool shape agreed better with the experimental observations made with 3850 W laser power and steel with 150 ppm sulfur content. It was concluded that the model might lack some important physics. As a further development Saldi [10] investigated the influence of surface deformation on the weld shape. In this case the enhancement factor set to 4 in the previous example could be lowered down to f=2, showing that there is still a problem. Today there is not yet a satisfying model for predicting the melt dynamics. The existing simulation studies do not address the instabilities observed experimentally at the free surface. Also they do not lead to the geometry of the melt pool observed experimentally. These two issues are important for the manufacturing applications that initiated this study. The aim of this work is

4 1.3. OBJECTIVEANDRESEARCHQUESTIONS thus to develop and test a numerical model for the melt pool in conduction laser welding that would agree better with experimental observations.

1.3 Objective and research questions

Based on the identified lack of predictability of the melt pool models in conduction laser welding, the main objective of this work is: To understand what physical phenomena not yet included in the melt pool model, could prove beneficial for an improved predictability of the laser welding process. With time this objective was refined into the following sub questions: RQ 1: Which numerical approach is suited for computing the thermocapillary force? RQ 2: What is the effect of changing numerical approach for computing the thermocapillary force compared to [10]? RQ 3: Which approach should be applied to model the deformation of the free surface? RQ 4: Which approach should be used to capture melt flow instabilities? RQ 5: What are the other model improvements that could have an effect on the prediction of the melt front topography?

1.4 Limitations

The modelling of the laser heat source is not made explicitly. It is simpliﬁed to an imposed input energy distribution. Metal vaporization is assumed negligible. The metal alloy is assumed to always have a uniform composition. The diffusion of its constitutive species is neglected.

1.5 Methodology/Strategy

The open source software OpenFOAM was the computational ﬂuid dynamic software used in this project to model melt pool ﬂow in laser welding. The work has been divided into the following main steps: Understand the basic physics governing the melt pool. • Analyze the state of the art of melt pool modelling and detect modelling • gap(s).

Understand the interFoam solver available in OpenFOAM (3.0.1) that • was used as starting point for developing the model. InterFoam is for 2

5 CHAPTER 1. INTRODUCTION

incompressible, isothermal, immiscible ﬂuids and uses an algebraic Volume of Fluid, VOF, method to capture the free surface.

Establish the developments needed to extend interFoam for computing • melt pool thermal ﬂow.

Understand the research work conducted by Saldi [10] for modelling • melt ﬂow using OpenFOAM and possible gaps in the approach used for numerical modelling.

Choose in which order to develop the new model components needed to • go from interFoam to the melt pool solver. It was decided to successively add:

1. Energy conservation equation and thermocapillary force applying the Continuum Surface Force, CSF, approach as in [10] 2. Solid phase and solid-liquid phase change 3. Sharper Surface Force, SSF, approach for the thermocapillary force 4. Geometric VOF to capture the free surface

Select, run and analyze documented cases to test the model along its • development, and draw conclusions.

1.6 Outline

This thesis is organized as follows. The fusion welding processes and speciﬁcally laser beam welding are described in chapter 2. Then the state of the art of laser welding models is presented in chapter 3. It is supplemented with a summary of numerical methods for free surface deformation and force balance at the free surface. The interFoam solver used as basis for developing the melt pool model is described in chapter 4. The general governing equations upon which the model is based are given in chapter 5. Chapter 6 presents the numerical applications. Finally, in chapter 7, the conclusions are drawn and the work is summarized.

6 Chapter 2

Laser welding manufacturingprocess

The main welding processes available can be seen in Figure 2.1. They include solid state welding, gas welding, resistance welding, thermo-chemical welding andfusion welding. Fusion welding is a category of welding processes that joins through melting. It can be made with two types of heat sources: electric arc and power beam [1,2,28].

Figure 2.1: Main welding processes

This chapter gives an introduction to the fusion welding processes, (section 2.1) followedby a more detaileddescription of the laser beam welding process (section 2.2) including different types oflaser sources (section 2.2.1). Then the two welding regimes (conduction andkeyholemode) are described in section 2.2.

7 CHAPTER 2. LASER WELDING MANUFACTURING PROCESS

2.1 Fusion welding

Common to all fusion welding processes is that a heat source interacts with metal workpieces and generates a molten pool. A joint is then produced when cooling and solidiﬁcation have taken place. Fusion welding processes include arc welding, power beam welding, and hybrid welding that combines the two previous types of heat sources. The fusion welding processes are further detailed in the following subsections.

2.1.1 Arc welding Arc welding methods are characterized by the heat source which is an electric arc generated by an electric current conducted between an electrode and a workpiece through an ionized gas. This process is often referred to as gas metal arc welding, GMAW, or gas tungsten arc welding, GTAW. GMAW utilizes a consumable electrode while GTAW has a non-consumable electrode [2, 28, 29]. In the case of a consumable electrode, the electrode melts and transfers droplets that mix with the metal in the molten pool. A shielding gas is used to protect the molten metal from atmospheric gases that might cause oxidation or contamination of the weld. The shielding gas can either be an inert gas (the process is then called Metal Inert Gas, MIG) or an active gas (Metal Active Gas, MAG). Typical inert gases are argon, helium or mixtures of those, and typical active gases are CO2 or mixtures of CO2 and argon. However, several other possible gas mixture compositions exist. Contrary to the monoatomic shielding gases often used in GTAW, active shielding gases always involve multiatomic species, thus holding energy in the molecular bindings. They can thus deliver more energy compared to monoatomic shielding gases. Advantages with the MIG and MAG processes are the stable welding performance and the relatively low occurrence of spatter formation [28]. Disadvantages are the relatively high heat input and large Heat Affected Zone, HAZ. These disadvantages are also shared by the next joining method. In the case of a non-consumable electrode, the electrode is usually made of tungsten alloy, which has a high melting temperature. Then the electrode is melted only locally at the electrode surface in the arc attachment area. There is no metal transfer from the electrode but a ﬁller material can be used if needed. Gas Tungsten Arc Welding, GTAW, also referred to as Tungsten Inert Gas, TIG, is the typical process with non-consumable electrode. It can be used with a broader range of material thicknesses compared to GMAW. GTAW is a stable and ﬂexible process but a drawback is the large spread of the heat source eventually decreasing the process control. The process is especially useful for thinner sections since it allows welding with low currents. However, it is relatively slow compared to the GMAW process [28].

8 2.1. FUSIONWELDING

Other processes with a non-consumable electrode are Submerged Arc Weld- ing, SAW, and Plasma Arc Welding, PAW. The latter is characterized by a con- centrated and stable arc, high welding speed (compared to TIG) and a small deformation of the workpiece. A wide range of material thicknesses can be welded using plasma arc. The plasma arc process generates deeper welds at higher welding speeds compared to the GTAW process. In SAW the arc is submerged in granular flux particles shielding both the arc and the workpiece from the ambient atmosphere. It has the benefits of a high deposition rate, deep penetration and a high weld quality but is limited to horizontal welds due to gravity effects on the shielding flux [28].

2.1.2 Power beam welding

Electron Beam Welding, EBW, and Laser Beam Welding, LBW, are power beam joining processes with a more powerful heat source compared to conventional arc welding methods. EBW can generate deep and narrow welds with a depth- to-width ratio of up to 25:1 when focus and process parameters are optimized. A drawback however, is that the process is limited to welding in vacuum. The complexity and size of possible structures to be welded is therefore limited by the size of the vacuum chamber. The energy content of EBW is higher than for 2 laser beam welding with up to 10 MW/mm [28, 29]. Thick materials up to 300 mm can be welded but beam tracking along the joint is essential due to the small spot diameter [28, 30]. EBW also holds the beneﬁt of welding without physical contact between the energy source and the workpiece [6].

Laser beam welding can be categorized depending on the state of the active medium as solid state lasers or gas lasers. Laser light is in all cases monochromatic and coherent. The wavelength of the generated light beam depends on the type of laser source. LBW holds the same beneﬁts as EBW but does not have the limitation of welding in vacuum. However, the melt pool needs to be protected by a shielding gas when welding materials that can be damaged by the atmosphere (e.g. oxidation). LBW permits also high welding speed and low distortion of the workpiece [6, 28]. A comparison of weld shapes generated with different welding sources when used in their high power density range is shown in Fig. 2.2. Even though laser beams and electric arcs differ as energy sources both processes can operate under shielding gas and at ambient pressure. It is thus possible to combine them (hybrid welding) and take advantages of their respective advantages.

9 CHAPTER 2.LASER WELDINGMANUFACTURING PROCESS

Figure 2.2: Schematic of weld geometries (cross-sections - bottom) using different highpower density fusion welding sources (top)

2.1.3 Hybrid welding

Laser-arc hybrid welding is one of the most establishedhybridprocesses. Thearc provides additionalheat and also, in some cases, filler material.The simultaneous application of a focused laser beam and an arc generates a single melt pool [6]. Hybrid welding benefits from high welding speed withhighprecision and is applicable to deeppenetration welding. Large welding gapsup to3mmwide can then be joined with the use of filler material [31, 32]. More detailsregarding laser beam welding are given in the next section.

2.2 Laser beam welding

The main application areas of laser welding today are found in the automotive industry, the aerospace industry, thesteel industry (such as pipeand vehicle superstructure), theelectronic industry and themedical industry. Laser welding is governed by the absorption of the laser beam energy into the workpiece, giving rise to two welding modes: conduction (in the low absorption range) and keyhole (in the high absorption range). An introduction to the laser heat sources can be found in appendix 1. Subsec- tion 2.2.1 gives a description of the different types oflaser sources available for welding,aswell as their different characteristics. Then the laser energy sources

10 2.2.LASER BEAM WELDING applied to welding are further described.

2.2.1 Laser energy sources applied to welding The most commonly usedlaser weldingheat sources in manufacturing industry today are ﬁbre lasers, Nd:YAG lasers, direct diode lasers and CO2 lasers.The

Figure 2.3: Transversal Electro Magnetic (TEM) mode beam produced at a laser exit is in Transverse Electro Magnetic (TEM) mode since theelectric and magnetic fields are orthogonal to each other and to the direction of propagation of the beam, see Fig. 2.3. Its power distribution in a plane perpendicular to the direction ofbeam propagation varies with theradial and angular position according to the laser mode. Equipment exists that can be used to filter or shape the beam according to the preferred beam mode. In the particular case commonly used in robotisedlaser welding, the laser lightis transportedinanoptical fibre [31, 33, 34, 6]. For high power laser applications in materials processing it is usually desirabletoutilize a beam with aTEM00 mode, which implies that the power distribution in a plane perpendicular to the direction of the beam obeys a Gaussian distribution. However, the beam mode may bealtered along its transportation towardstheworkpiece. Measurements (see for instance Pitscheneder et al [12]) can indeed show some deviation from theideal case. The quality of the beam is evaluatedby a non-dimensionalbeam quality factor or propagation factor M2,whichdescribes the focus strength of the beam. For an idealbeam the factor M2 equals1and it increases with increasing spot size. The power of a laser beam used in welding can be delivered in the form ofa continuous wave, cw, or a pulsed wave, pw. When welding with acwthe laser

11 CHAPTER 2. LASER WELDING MANUFACTURING PROCESS beam is emitted steadily. It can be used to weld a wide variety of metals and workpiece thicknesses. Pulsed welding, pw, is used when low energy input is required, for example when welding close to electronics or when welding small parts. The pw mode corresponds to a minimal amount of heat transfer and more control of the weld compared to cw. The solidiﬁcation of the melt pool can then begin after each pulse while for cw the melt pool solidiﬁes only when the laser beam travels away along the workpiece. Cw lasers are considered in this work.

2.2.2 Laser welding process

The power density (also called intensity) of lasers applied to welding is in the 4 7 2 range of 5 10 to 10 W/cm at the spot on the workpiece. The process behaves differently· in the low intensity range than the high intensity range. For a laser beam irradiating a workpiece with an intensity below some material dependent threshold only a small fraction of the laser beam energy is absorbed at the workpiece surface (Fresnel absorption). For instance for a CO2 laser and a steel 6 2 workpiece the threshold intensity is about 10 W/cm and the fraction of laser energy absorbed is less than 30 %. The heating of the workpiece by the laser results in the formation of a molten pool, weak vaporization and thus shallow weld penetration. The operating mode is thus called conduction welding mode. The aspect ratio (depth/width) of the molten pool is generally smaller than 0.5 (see Fig. 2.4 left). An advantage with the conduction laser welding mode is the stability of the process, and as a result the small occurrence of porosity and spatter formation. However, the process is relatively slow with high heat input and relatively low efficiency [6]. The main application areas are in welding of aluminium alloys or when welding dissimilar metals, for example aluminium to steel. When the laser intensity is increased above the intensity threshold a transition from conduction mode to keyhole mode takes place due to intense metal vaporization. The vapor pressure, often referred to as recoil pressure, then increases. This increase gives rise to the formation of a cavity, or keyhole that enables the laser beam to reach deeper into the metal [6]. The melt pool aspect ratio can be greater than 1.5 (see Fig. 2.4 right). This means that narrow welds with a narrow heat affected zone, HAZ, can be generated [6]. As a beam ray reaches the keyhole surface it is partially absorbed and partially reflected towards a new point of interaction. Absorption and reflection take place at each new point of interaction. Figure 2.5 shows a schematic view of the multiple reflections within the keyhole. This succession of partial absorptions increase the overall energy absorption and is called multiple Fresnel absorption. The absorption of the beam energy depends on the wavelength of the laser,

12 2.2.LASER BEAM WELDING

Figure 2.4: Conduction vs. keyhole mode

Figure 2.5: Schematic of multiple reﬂections within a keyhole as well as the surface material andquality [6, 31, 35].The amount oflaser energy absorbed in keyholemodeislarge compared to the conduction modeand the absorption can reach up to 90%.Theadvantages withkeyhole laser welding

13 CHAPTER 2. LASER WELDING MANUFACTURING PROCESS are the relatively small heat affected zone (HAZ), the narrow weld, and the deep penetration. The process is fast compared to the conduction laser welding mode and the intensity of the laser beam is high. The process is well suited for welding of thick parts due to the high aspect ratio of the weld. However, defects such as porosity may occur caused by insufﬁcient degassing due to the rapid solidiﬁcation. The physics of the conduction laser welding mode is now further described.

Conduction laser welding mode - process physics and modelling assumptions The main physical phenomena involved in the workpiece in conduction laser welding mode are: heat transfer by conduction • metal phase change from solid to liquid (and reverse solidification) thus • liquid metal flow in the presence of temperature gradients leading to – heat transfer by convection, – liquid metal inertia, – viscous friction, – gravitational and buoyancy force, – capillary force (or surface tension acting along the normal to the fluid surface) and thermocapillary force (or Marangoni force acting tangentially to the free surface), – species diffusion (not investigated here). The surface tension is a leading order force. Its physical meaning is thus recalled now. The surface tension applies between liquid and gas. It is caused by the attractive forces between molecules/atoms present on both sides of the liquid surface. The difference between the cohesive (molecules of same type) and adhesive (molecules of different types) forces generates an imbalance at the surface causing a tension. The liquid adjusts to have the smallest possible free surface area. Figure 2.6 illustrates the molecules in the liquid and in the gas including the bonding between them. Depending on if the cohesive or adhesive force is dominating the surface shape is convex or concave. If there are surfactants at the surface the bonding forces are affected which might change the force balance and in turn the surface tension effect. The relative importance of the physical phenomena can be seen expressing the conservation equations in dimensionless form. The conservation equation for mass expressed in the workpiece in dimensionless form is given by

∂t ρ∗ + Ma (ρ∗ u∗) = 0 (2.1) ∗ ∇ ·

14 2.2.LASER BEAM WELDING

Figure 2.6: Surface tension effects on the shape of the free surface due to imbalance in attraction forces between molecules and atoms of the liquid (blue) and gas (green) phases.

wheret∗, ρ∗ andu∗ are the dimensionless time, density and velocity respectively. Ma is the Mach number deﬁned in table 2.1. In fusion welding applications the velocity of the liquid in themelt pool is relativelymoderate. For instance a maximum velocity of about 0.5 m/s was obtained in the conductionmode melt pool studied in [12], for a maximum temperature of about 2100 K. The speed of sound is 5800 m/sand 3400 m/s for solid and liquid steel respectively [36]. Then Ma ≈ 10−4 <<0.3 implying that themedia is mechanically incompressible. It should be noticed that the liquid metal in the melt pool is however thermally compressible since its density dependsonthetemperature and this last one is not uniform. However, it will be assumed that the density variation is only important in the buoyancy force (Boussinesq approximation). This simplifying assumption applies since the density variation in themelt pool uses to beatleast one order of magnitude lower than the density at the melting point [37]. The Navier-Stokes equation in theworkpiece when expressed in dimension-

15 CHAPTER 2. LASER WELDING MANUFACTURING PROCESS less form reads 1 1 ∂t (ρ∗ u∗) + Ma (ρ∗ u∗) = Fg∗ + Ri Fb∗ + Eu Fp∗ + Fv∗ (2.2) ∗ ∇ · Fr Re where Fg∗, Fb∗, Fp∗, Fv∗ are dimensionless; they respectively denote the gravitational acceleration, buoyancy-, pressure- and friction force. At the free surface boundary of the melt pool the friction force balances the thermocapillary force,

Fth, along the tangential direction while the pressure force balances the capillary force, Fc , along the normal direction. In dimensionless form these conditions express, along the free surface tangential direction 1 Mg Fv∗ = F ∗ (2.3) Re Pr Re2 th and along its normal direction, 1 1 Eu Fp∗ = Fc∗ = Fc∗ (2.4) We Re Ca where Fc∗ is the dimensionless capillary force and Ft∗ h is the dimensionless thermocapillary force. The Froude (Fr ), Richardson (Ri), Euler (Eu), Reynolds (Re), Weber (We), Prandtl (Pr ), Marangoni (Mg) and Capillary (Ca) numbers are deﬁned in table 2.1. Equation 2.3 implies that the Reynolds, Prandtl and Marangoni numbers are dependent, Pr Re = Mg. The energy equation in dimensionless× form is given by

1 ∂ 2T ∂ ρ h ∗ S Q (2.5) t ∗ ∗ = 2 + L∗ + laser∗ ∗ Pe ∂ (x∗) where SL∗ is the source term for latent heat and Qlaser∗ represents the heat source. The Peclet (Pe) number is related to the Reynolds (Re) and Prandtl (Pr ) numbers according to Pe = Re Pr , see table 2.1. × 2 1 2 1 1 In table 2.1 Cp [kgm K− s− ] is the specific heat, µ [kgm− s− ] the viscos- 1 1 ity, k [Wm− K− ] the thermal conductivity, L [m] a characteristic length, u 1 2 1 3 [ms− ] the velocity, ν [m s− ] the kinematic viscosity, ∆ρ [kgm− ] the density 1 1 difference between two fluids separated by the interface, γ [Nm− K− ] the thermocapillary coefficient (or surface tension gradient), ∆T [K] the temperature difference, d [m] an initial melt depth which is also characteristic length in 2 1 2 1 the problem, α [m s− ] the thermal diffusivity, h [Wm− K− ] the heat trans- 2 1 fer coefficient, g [ms− ] the gravitational acceleration, σ [Nm− ] the capillary 1 (or surface tension) coefficient, β [K− ] the thermal expansion coefficient, Lf 1 1 [Jkg− ] the latent heat of fusion, V [ms− ] a characteristic velocity of the flow 1 2 and ∆p [kgm− s− ] the pressure difference between upstream and downstream flow.

16 2.2. LASERBEAMWELDING

It should be noted that in the literature these dimensionless numbers can be calculated based on different definitions of the characteristic length. For instance Limmaneevichitr et al. [8] defined the characteristic length, L, as the weld pool radius for the case without surface active elements while Pitscheneder et al. [12] defined it as the depth of the weld pool, when including surface active elements. The relative order of magnitude of the different forces in the melt pool can be evaluated based on the above dimensionless numbers. Their order of magnitude 1 for documented welding applications [4, 29, 35] is of 10− for the Prandtl number, 2 5 2 3 4 10 -10 for the Marangoni number, 10-10− for the Peclet number, 10− -10− for 1 3 the Bond number, 10− for the Capillary number, 10 for the Grashof number 3 and up to 10 for the Reynolds number [4, 29, 35]. A small Peclet number indicates a Marangoni driven flow. The Capillary number, Ca is important in surface tension driven flows and free-surface flows since it is related to the degree of surface deformation. A Capillary number ap- proaching zero indicates large surface tension effects. A flat or concave weld pool bottom indicates that Marangoni convection is dominating over gravity induced buoyancy convection [4, 29]. Fig. 2.7 shows the effect of the thermocapillary force on the flow in the melt pool. It acts along the free surface, inducing vortices in the melt pool. The direction and number of vortices depends on the material composition and the temperature gradient along the free surface. The effect of the buoyancy force on the melt pool flow is described by the dimensionless Bond number. It is also shown in Fig. 2.8 for temperature dependent density. Also the moving heat source has an influence on the flow. It was investigated by Mundra et al. [14]. Using a 3D transient model neglecting surface deformation and vaporization (but considering different constant values for solid and liquid material) it was shown that the temperature fields and temperature gradients were affected by the moving heat source. These in turn affected the flow in the melt pool as schematically viewed in Fig. 2.9. In keyhole laser welding mode, the forces driving the melt pool flow are the same as in conduction laser welding mode.

17 CHAPTER 2. LASER WELDING MANUFACTURING PROCESS

Name Deﬁnition Interpretation Example of application

Cp µ viscous diffusion rate Prandtl, Pr k thermal diffusion rate heat transfer C ∆T Stefan, St p sensible heat thermodynamics Lf latent heat of fusion uρCp L convective transport Peclet, Pe k diffusive transport rate heat transfer uL inertia force Reynolds, Re ν viscous force fluid dynamics 3 gβ(Ts T )L buoyancy force Grashof, Gr −ν2 ∞ viscous force fluid dynamics, heat transfer ρu2L inertia force Weber, We σ capillary force free surface flow µ∆T viscous force Capillary, Ca σ capillary force free surface flows γ∆T d thermocapillary force Marangoni, Mg αµ viscous force thermocapillary driven flows ∆ρg d 2 gravitational force Bond, Bo or Eövöts, Eo σ capillary force free surface flow 3 gβ(Ts T )L buoyancy force Rayleigh, Ra να− ∞ viscous force buoyancy driven flows ∆p pressure force Euler, Eu ρV 2 inertia force hydrodynamics gβTh Tr e f L potential energy − Froude, Fr U 2 kinetic energy buoyancy driven flows U 2 inertia force Richardson, Ri g L gravitational force free surface flow U flow velocity Mach, Ma c speed of sound gas dynamics

Table 2.1: Dimensionless numbers

18 2.2.LASER BEAM WELDING

Figure 2.7: Schematic of surface tension and thermocapillary effect on melt pool ﬂow for a) pure metal or an alloy withlow concentration in surfactant b) metal alloy with moderate concentration in surfactant c) metal alloy withhigh concentrationinsurfactant

Figure 2.8: Schematic of theeffect ofbuoyancy force on melt pool ﬂow for temperature dependent density

19 CHAPTER 2.LASER WELDINGMANUFACTURING PROCESS

Figure 2.9: Schematic of theeffect of moving heat source on melt pool ﬂow

20 Chapter 3

State of the art of laser welding models

This section aims at giving an overview of the existing numerical models for laser welding. The first models, which neglected melt flow, are presented in section 3.1. The more recent generation of models takes into account fluid convection which generally implies numerical resolution based on Computational Fluid Dynamics, CFD, techniques. This is the focus of sections 3.2 and 3.3. The existing CFD models are divided into two sub-groups to distinguish the welding mode (see Fig. 2.4). The first group addresses the conduction laser welding mode (section 3.2) and the second the keyhole laser welding mode. Different methods of accounting for deformations of the surface exist and are discussed here in section 3.4.

3.1 Melt pool models neglecting ﬂuid ﬂow

The first models developed to improve the physical understanding of laser welding addressed only the heat conduction in the workpiece. The fluid flow in the melt pool was neglected to allow performing an analytic resolution of the problem assuming simple geometries. Rosenthal [38] presented the first analytical solution of the heat conduction equation applied to welding. Different heat source models, point source, line source and moving line source, were proposed. Based on these analytical solutions several models for both conduction and keyhole welding were developed. The book by Carslaw and Jaeger [39] describes the analytical solutions upon which most of the early models are based. These models were improved step by step over a period of about 20 years in order to provide a more comprehensive description of the physics taking place due to the interaction between laser beam and material surface. A review of these analytic and numerical models was presented by Mackwood and Crafer [40]. The resolution of the heat conduction equation while neglecting the melt flow is still commonly used with finite element modelling (FEM) to

21 CHAPTER 3. STATEOFTHEARTOFLASERWELDINGMODELS study the thermomechanical properties of the welded component. This field of study, named computational welding mechanics (CWM), is reviewed in the three part review by Lindgren et al. [3, 4, 5] for laser beam welding, as well as other welding processes (e.g. arc and electron beam welding). The convective heat transfer in the melt pool is known to have a significant effect on both the weld geometry and the temperature field in the workpiece [41]. For this reason the parameters entering the Rosenthal double ellipsoid model used for reproducing the combined effect of moving heat source and weld pool convection need to be adjusted to reach simulation results that fit experimental measurements. A clear drawback is the lack of predictability of this approach on weld geometry.

3.2 CFD melt pool models for conduction mode laser

Several studies investigating the conduction mode laser welding exist and only a few key developments are reviewed here. Some existing models accounting for both beam-material interaction and fluid flow are listed in Table 3.1. The references are listed according to the physics taken into account, e.g. if the deformable free surface is considered or not. Chan et al. [41] were among the first to present a two-dimensional transient model for the prediction of melt pool flow. Their model included a heat source with constant power distribution irradiating a body of finite thickness but infinite surface area. The material properties were considered to be constant, except for the surface tension which was a linear function of temperature as for pure metals. For solid and liquid material the same thermal conductivity was used. The surface was considered to be flat (i.e. no surface deformation). Four dimensionless numbers (Prandtl number, Peclet number, surface tension number and dimensionless melting temperature) were used to evaluate the temperature- and velocity fields. It was concluded that the convective heat transfer significantly affects the fluid flow. It was also observed that a high velocity jet induced two counter rotating vortices in the melt. The surface tension gradient obtained a maximum value at the edges which generated an outward flow. The flow velocity was about 1 to 2 orders of magnitudes larger than the welding speed. As a 2D model is not able to capture the flow in the radial direction thus Chan et al. [42] further extended the model to a three-dimensional model. The heat source, now a Gaussian distribution, was also improved compared to the constant power distribution in the 2D model. It was concluded that the recirculating flow, which appear due to surface tension gradients, influences the melt pool width and depth. An increase of aspect ratio with 150% for convective flow was observed compared to pure conduction case. Also a non-uniform temperature gradient along the solid-liquid interface with a maximum at the edges and minimum at the bottom

22 3.2. CFD MELTPOOLMODELSFORCONDUCTIONMODELASER was computed. A study by Peng et al. [43] investigated the role of initial melt pool shape with focus on Marangoni convection. By first evaluating the time for melting, generating an initial melt pool and then including fluid flow they calculate the velocity and temperature fields of the melt pool. Their study consisted of three cases with different initial melt pool sizes. The effect of Marangoni convection on the melt pool shape was investigated. It was observed that the Marangoni convection plays a much more significant role than natural convection in the melt pool flow, thus it is important to include in numerical models of processes with solid-liquid phase change. It was also concluded that the Marangoni convection influences the temperature distribution and fluid flow, thus affecting the shape of the weld pool. Some models assume that Marangoni convection is negligible in the keyhole, this can be true for the keyhole but when modelling the complete process the melt pool (and not only the keyhole) is of interest. The thermocapillary convection influences the melt pool flow and the solidification process. It is indeed affected by gas flow, for example the shielding gas. Tan et al. [15] presented a model for fluid flow without surface deformation. The model was applied to pulsed fibre laser welding of stainless steel 304 samples. The numerically obtained results were compared to experimental measurements. They found that high power density increases the melting efficiency to a larger content than the pulse energy density. It was also concluded that the cooling rates are higher at the surface of the molten pool and lower at the pool bottom. Akbari et al. [44] investigated the influence of process parameters on weld shape. The 3D model does not consider surface deformation. A moving heat source was applied. The welding speed was varied between 3 and 9 mm/s. Experimental measurements were used for comparison. The width of the melt pool was shown to decrease with increasing welding speed. The prediction was approximated to 2-17 % error. The first model to consider surface deformation in convection driven melt flow was the study by Paul and DebRoy [45]. They assumed a constant Marangoni force, and included solidification of the melt after laser switch off and predicted the weld geometry for different welding powers and speeds. The numerical results were compared to experimental measurements. A surface elevation at the edge and at the centre of the weld pool was computed. During solidification this surface deformation decreased. However, the final weld shape after solidification was still elevated compared to the case assuming a flat surface. Han et al. [46, 47] developed a 3D model for different laser beam modes: TEM00, TEM01 and TEM10. The model included vaporization for recoil pressure, mixture properties, surface tension and thermocapillary convection. They considered a moving laser heat source. Free surface deformation was included and the geometry of the melt pool when applying a laser power of 1000 W was

23 CHAPTER 3. STATEOFTHEARTOFLASERWELDINGMODELS predicted. It was concluded that the deformation of the melt pool is essential, in the prediction of the weld geometry. Ha and Kim [48] investigated the weld pool geometry accounting for deformation of the free surface. In the numerical model a finite difference solver was developed to model the tophat distribution of the CO2-laser source applied on a workpiece of steel alloy. It was concluded that the deformation of the free surface generates oscillations and the convective heat transfer in the weld pool. By comparing calculations with and without surface deformation they concluded that the weld pool width and depth are about 2-4% larger when including surface deformation. However, considering the surface deformation was not enough to generate good predictions of the weld shape. He and DebRoy [17] used a 3D model predicting temperature and velocity distribution to investigate the melt pool shape. Laminar flow and a flat surface were assumed. The behaviour of the mushy zone (region where both liquid and solid phase co-exist during phase change) for Nd:YAG spot laser welded stainless steel 304 samples were studied. Fine grids and small time steps were used. By the use of dimensional analysis the relative importance of heat transfer by conduction and convection was investigated. The results were compared to experimental measurements. A large mushy zone, unique for conduction welding, was observed. The mushy zone grows significantly during solidification step and reaches a maximum when the pure liquid region disappears. The cooling and solidification rates in the weld pool were high compared to GTA welding processes. Kim and Kim [49] developed a 2D axi-symmetric model to investigate the thermocapillary effect on the fluid flow in a cylinder with fixed volume. The model included surface deformation and a predefined, curved, non-deformable surfaces were also considered for comparison. Surface tension was considered as a linear function of temperature. Three different power intensities were applied. It was concluded that the surface deformation increases as the melt pool volume decreases, due to the increased heat on the top surface close to the heat source generating increased outward molten material flow. A significant difference in velocity between a flat (non-deformable) surface and a free surface assumption was found by Shah et al. [50]. They investigated laser spot welding of stainless steel 304 with a laser power of 1000 W. The weld shape (depth and width) prediction was improved by approximately 3% when surface deformation was considered. It was also shown that the laser power strongly influences the melt pool geometry. Pitscheneder et al. [12] performed a study of the weld geometry in conduction mode spot laser welded Böhler S705 steels. They investigated the role of sulfur as surface active element on weld pool shape with a sulfur concentration of 20 ppm and 150 ppm respectively. The model included surface tension as a

24 3.2. CFD MELTPOOLMODELSFORCONDUCTIONMODELASER non-linear function, but surface deformation was neglected. Three different laser powers, 1900 W, 3850 W and 5200 W, were applied. The numerically obtained temperature distributions and flow fields were compared to experimental data. It was shown in the study by Pitscheneder et al. [12] that the concentration of surface active elements has an effect on the flow direction, as illustrated in Fig. 3.1. The computational results show that at low concentration (20 ppm sulfur [12]) the temperature field on the melt pool free surface leads to a uniformly negative Marangoni coefficient dσ/dT , thus an outward Marangoni force. This leading order force induces the vortices sketched in Fig. 3.1a, and results in a shallow weld pool. At higher concentration in surface active elemtns (150 ppm sulfur [12]) the temperature field on the melt pool free surface results instead in a non-uniform sign of the Marangoni coefficient. On the outer edge dσ/dT is positive, implying an inward force, and the formation of additional vortices rotating inwards that result in a deepening of the melt pool. An enhancement factor, i.e. an artificial increase of the dynamic viscosity and thermal conductivity of liquid steel by a factor f=7 was applied in order to adapt the weld shape to better agree with the experiments. However, this enhancement factor has no sound physical basis and different values have been used in different studies [27]. The occurrence of turbulence in the melt pool has been assumed by several authors for example by Sudnik et al [21]. They presented a coupled model for prediction of thermocapillary driven flow where an assumed effect of turbulence was somehow considered by increasing the viscosity using an enhancement factor f=16 in the melt pool and f=2.5 in the metal vapor inside the cavity. They studied the correlation between the weld pool geometry (depth and length) and the laser power. Turbulence was considered. It was concluded that there is an approximately linear correlation between the depth and length of the weld pool. Chakraborty [16] developed a 3D model with moving heat source to compared the temperature fields obtained with and without the influence of turbulence for a weld pool. Turbulence was modelled with a k " model. The constants needed to close the model were taken from calibration− studies performed in a different concept [16] as such studies are not available for welding melt pool applications. It was concluded that the temperature fields in both cases were similar and that the effect of turbulence on temperature was marginal. However, its effect on momentum diffusion was more significant. Saldi [10, 27] developed a 3D model for conduction spot laser welding including three phases, surface tension as non-linear function of temperature and Marangoni convection. The surface was assumed flat. He applied the same test case as Pitscheneder et al. [12] to study the effect of enhancement factors on the predictability of the weld shape. Laser powers of 1900W, 3850W and 5200W were applied on Böhler steel S705 workpieces. Using an enhancement factor f=7 (same as Pitscheneder et al. [12]) generated results with fair agreement to

25 CHAPTER 3. STATEOFTHE ART OF LASER WELDING MODELS experiments. When including thesolidification step (after laser switch off)the enhancement factor could bereduceddown to f=4 in the case oflow surfactant concentration, i.e. 20 ppmsulfur. For 150 ppmsulfur content however, theweld depth was increased significantlywhen considering thesolidification stage, compared to the case without solidification. As a further evaluation Saldi studied the influence of free surface deformation on weld pool simulations. The predictability was improvedbut the use of an enhancement factor f=2was still needed in order to reach good agreement with experimental measurements. It was concluded that the maximum temperature in the free surface case was lower for 20 ppmsulfur andhigher for 150 ppmsulfur compared to the case with no surface deformation. Since surface tension and surface tension gradients are temperature dependent theflowdirection is affectedbythe surfactant concentration. Kidess et al. [18, 51] presented a direct numerical simulation, DNS, model for laser welding. The model was implemented in the open source software OpenFOAM. A flat surface was assumed. The solidification step was included. They applied their model to laser spot welding of asteel alloy with 150 ppmsulfur, leading to a Marangoni flow similar to Fig.3.1b. It was concluded that fluid flow during solidification step significantly affects the weld shape prediction. However, discrepancy with experimental results was still present. The discrepancy in predictability of weld shape was assumed to be due to a non-uniform distribution of surfactant andpossibly due to the lack of mass transport for the surfactant.

Figure 3.1: Surface active elements effect on ﬂow direction in melt pool cross section. a) low surfactant concentration, thus negative Marangoni convection, b) increased surfactant concentration generating a new outer area with a positive Marangoni convection, c) larger surfactant concentration for which the Marangoni convection becomes positive everywhere.

Table 3.1 lists some of the existing CFD models starting from the assumption of a non-deformable free surface followedbydeformable free surface and the assumption of tubulent ﬂow. Thetable is used to categorize themodels based on physics included, for example number of space dimensions, surface deformation,

26 3.2. CFD MELTPOOLMODELSFORCONDUCTIONMODELASER surface tension and thermocapillary. Most of the models make use of different assumptions and the problem of accurately reproducing the melt shape remains.

27 CHAPTER 3. STATEOFTHEARTOFLASERWELDINGMODELS - heat source in H 7 1,2 = = - surface tension modelled σ ] 52 ) [ − ) or not ( × Speciﬁc properties *surface def. MAC CSF approach Heaviside moving heat source surface def.* with moving mesh ALE method surface active elements, enhancement f Gaussian distribution OpenFOAM surface active elements, enhancement f − − − × × × − × × × × r s s s s r r s s c c MHB c c c c c - c σ l l nl l l l nl nc s Main properties − ∗ − ∗ − − − − − DS 2 2 2 V nl3 nc L s nl3 nc r 2 2 3 3 3 3 2 V nl - buoyancy modelled ( B - Marangoni included as constant (c), non-constant (nc) M - surface deformation using Level Set (L) or VOF (V), S 3 laser, steel laser, steel laser, steel, iron laser, steel 2 2 2 2 - number of space dimensions, D Author Domain of study Han et al. (2005)He and DebRoy (2003)Shah et al. (2018)Pitscheneder et Nd:YAG al. laser spot (1996) welding, steel cw, steel cw CO laserKim spot and welding, Kim steel (2008, 2012) laser spot welding,Saldi steel (2013) cw CO Paul and DebRoy (1988) cw CO Tan et al. (2012) Nd:YAG pulsed laser, 304 steel Ha et al. (2005) cw CO Chan et al. (1984) steel, Al and NaNO Kidess et al. (2015, 2016) laser spot welding, steel Chakraborty (2009) dissimilar metals Cu-Ni as: constant (c), linear fnrelative (l), motion non-linear with base fn metal (nl), (r) or stationary (s), Table 3.1:

28 3.3. COUPLEDMODELSFROMOTHERFIELDSTHANLASERWELDING

3.3 Coupled models from other ﬁelds than laser welding

Chen and Wang [19, 53] developed a 3D model to study the metal vapor characteristics above the keyhole. The model was applied to CO2 laser welding of pure iron with argon shielding gas. Temperature and vapor velocity fields were investigated. It was concluded that the vapor temperature decreases rapidly with increasing distance to the workpiece. The maximum temperature of the (laser induced plasma) vapor plume was around 14500 K inside, or just above, the keyhole. Cho et al. [54] used a 3D model including surface deformation, thermocapillary convection, recoil pressure and assuming a laminar fluid. Pore generation during keyhole collapse is modelled. The effect of sulfur content, laser beam profile and vapor on the melt pool were studied. The results were compared to fibre laser welded carbon steel workpieces. It was concluded that the thermocapillary convection inside the keyhole (the shear stress on the keyhole wall) induced by metal vapor was insignificant. They also claimed that Marangoni force in keyhole welding is insignificant. This is true only on the keyhole surface but not when considering the complete weld pool and fluid flow. Some previous studies, [55, 56], focused on investigating the Marangoni convection in other applications than laser welding for example in a cavity. Other studies on cavities are for example the study by Tan et al. [57]. They developed a test case for validating the Marangoni driven free surface flow for models aiming at studying solidification and crystal growth. This test case has been applied in this work as part of the validation of the implemented model and will be further described in chapter 6.

3.4 Interface capturing

Predicting the evolution of the deformable free surface is a major part of the multiphase problem. It can be done using methods including interface fitting, interface tracking and interface capturing. Interface fitting is based on a La- grangian approach with a moving mesh. Points at the interface are tracked as they move with time, see Fig. 3.2a. Interface tracking uses a combination of a fixed and moving grid since the velocity field is solved on a fixed grid while the interface tracking is performed on a moving mesh, Fig. 3.2b. These methods are demanding in terms of computational resources since they treat the free surface as a sharp interface and follow its motion; the grid has to be deformable which has a high computational cost. These approaches do not seem to be used for laser welding simulation. Interface capturing approaches allow calculating complex surface deformations on a fixed grid. These methods are less accurate than the interface fitting

29 CHAPTER 3. STATEOFTHE ART OF LASER WELDING MODELS and interface tracking methods, but have a lower computational cost since they convert the problem to the resolution ofpartialdifferential equations. The interface capturing methodsareabletocapture large surface deformations and are thus appropriate for laser welding applications, see Fig. 3.2c. The interface capturing method provides a way to

• compute the evolution in time and space of the indicator function,

• reconstruct the interface, in particular compute locally its normal vector n and its curvature .

Figure 3.2: Interface captured using a) interface ﬁtting, b) interface tracking and c) interface capturing

Interface capturing methods can be divided into sub-categories depending on the indicatorfunction used to track the interface.Thetwo main families of interface capturing methods used in CFD for welding are theVolume of Fluid, (VOF) and theLevel Set, (LS). The VOF method, introduced by Hirt and Nichols [58], is based on the donor-acceptor formulation. It satisfies mass conservation. It uses an indicator function α rangingfrom0to1.Theindicator function used in the VOF method isthe volume fraction α defined as ⎧ ⎨⎪1 for cells completelyfilled with fluid1 α = 0 < α < 1 for cells containingboth fluids (3.1) ⎩⎪ 0 for cells completely filled with fluid 2.

The general transport equation for thevolume fraction α is given by

∂t α + ∇ · (αU)=0. (3.2)

30 3.4. INTERFACE CAPTURING

The main drawback is its diffusivity as it can affect the accuracy of the solution. It has been shown that this approach can deform the shape of the interface, [59]. The VOF method in its original version was not appropriate for problems with high surface tension effects. Due to its non-smooth nature the VOF method (see Fig. 3.3) has a limited capability of handling the large curvature gradients that appear for example in the case of small gas bubbles (porosity) and generates a less accurate interface compared to the LS method [60]. First the indicator function α is calculated, then the curvature c and the unit normal vector n~ are reconstructed. The curvature is calculated from α = ∇ . (3.3) c −∇ · α | ∇ | The Level Set method was introduced by Osher and Sethian (1988). The evolution of the function φ, with the zero level set always corresponding to the position of the interface, is followed. The method uses a continuous distance function to distinguish between the fluids. The sign of the distance, or indicator, function is negative in cells filled with one fluid and positive in cells filled with the other fluid. The LS method in its original version does not satisfy mass conservation. It is restricted to a uniform mesh size. For two-phase flows the LS transport equation is written as

∂t φ + U φ = 0 (3.4) · ∇ where φ is the smooth level set function. LS method was further developed by Sussman et al. [61] to handle compressible multiphase flows. Figure 3.3 shows a comparison of the interface captured using VOF in original version as presented by Hirt and Nichols [58], a more recent variant of VOF such as for example the isoAdvector and smooth LS-function. Several improved versions of the common interface capturing methods exist. Adalsteinsson et al. [62] introduced the narrow band level set in which a thin band of adaptive mesh is used around the interface. This method has been applied for example by Ki et al. [63]. Also methods combining VOF and LS, for example the Coupled Level Set and Volume of Fluid, CLSVOF, and Simple Coupled Level Set and Volume of Fluid, S-CLSVOF, exist. These methods combine the benefits of VOF and LS into a more accurate, but still mass conserving, method. By using VOF to calculate the advection of the interface and LS method for the interface normal the two methods are coupled. The CLSVOF method was first introduced by Bourlioux [64] and further developed by Sussman and Puckett [65] to handle incompressible two-phase flows in 3D. Using OpenFOAM Kunkelmann and Stephan [66] implemented a coupled version of VOF and LS. Their work

31 CHAPTER 3. STATEOFTHE ART OF LASER WELDING MODELS

Figure 3.3: Interface reconstructed using a) theoriginal VOF b) isoAdvector and c) LS was extended by Albadawi [60] to Simple-CLSVOF, or S-CLSVOF. Recently Haghshenas et al. [67] presented an improved version called Algebraic-CLSVOF, A-CLSVOF. In the case of a discontinuous function, such as VOF, the indicator function has to be smoothed in order to calculate thecurvature [68], i.e: gradient cannot becalculated accuratelywhichleadstospurious currents at the interface [59]. This significantly affects surface tension driven flows. Different solutions for avoiding spurious currents have been proposed, for example mesh refinement anddecreased time step [69, 70].However, this does not seem to decrease the magnitude of the spurious currents. Other approaches such as Sharp Surface tension Force, SSF, has been presented. Theadvection of theindicator function can be done either algebraically,as done in the VOF method applied in this work,orby geometric reconstruction of the interface. Geometric reconstruction captures the interface explicitly by reconstructing the fluid interface insideacell and then approximate theflux over the face. Then a Lagrangian method for advection of the indicator function and computation of thevolume fluxes is applied.Itcaptures the interface more accurately compared to algebraic methodsand the numericaldiffusion can be prevented [71]. Simple Line Interface Calculation, SLIC, Piecewise Linear Inter- face Calculation, PLIC and isoAdvector are exampleof geometric reconstruction methods. The first two methodsarehowever computationally demanding, com- plicated to implement and in most cases limited to structured meshes [60].The isoAdvector method is a sharp interface method implemented in OpenFOAM by Roenby et al. [72, 73]. It uses a continuous function to capture the interface and thevolume fraction is interpolatedfrom thecell centre to the surface. An isosurface is calculated insidethecell,splitting thecell into two parts, each containing only fluid 1orfluid 2. Thecalculation of the iso-surface is done

32 3.5.IMPLEMENTATION OF THE FORCES APPLIED ATTHE FREE SURFACE iteratively. The combined iso-surface generatedfor the interface is however not a continuous surface anymore but thevolume fractions are correct. Themethod is applicable to unstructured meshes and is more stablethan thealgebraic MULES scheme applied in the interFoam solver and commonly used in OpenFOAM. A Courant number of around 0.5 can be used to generate accurate solutions, compared to the MULES scheme which is limited to a Courant number around 0.1. The isoAdvector method is 2-4 times faster than MULES (described in section 4.1) and slightly faster compared to CICSAM and HRIC [72].The method is evaluated in this work through the test cases discussed in sections 6.1 and 6.2.

3.5 Implementation of the forces applied at the free surface

An accurate numerical modelling of the interfacialforces e.g. surface tension, is necessary in order to obtain a correct shape and deformation of the interface. In turn the prediction of the surface tension force is dependent on the accuracy of the predicted interface curvature and is thus difﬁcult to predict accurately [74]. Due to the fact that the ﬂuids (i.e. gas and liquid) are separated by an interface

Figure 3.4: Schematic view of interface normal and tangential stress components which is a boundary between thephases, interface conditions for the surface forces are required. Fig. 3.4 showsaschematic view of the interface with the normal and tangential components. Applied at the interface these boundary conditions read along tangentialdirection τ

∂ u ∂σ dσ ∂ T μ = = , (3.5) ∂ n ∂ T dT ∂τ

33 CHAPTER 3. STATEOFTHEARTOFLASERWELDINGMODELS and along normal direction n~

∂ p = σ, (3.6) −∂ n where T is the temperature. However, in this thesis a single-fluid model is used and it is then more con- venient to express the surface tension force components, n~ and τ~, as volume forces according to the Continuum Surface Force, CSF, method developed by ~ Brackbill [75]. By the use of CSF the surface force f is converted to a body force ~ F by multiplying f with a function δs (Dirac delta function) which represents ~ ~ ~ the interface thickness. The total body force is Fσ =Fsn + Fs t where the first term is the normal component and the second term is the tangential component. The definition of the delta function depends on the numerical method used for following the interface and is thus defined differently for VOF, LS and S-CLSVOF method. The normal component of the body force in general form is given by

~ Fsn = σ(x)cδs n~ (3.7) and tangential component

~ Fs t = s σ(x)δs , (3.8) ∇ where c is the curvature, n~ is the unit normal vector, δs is the projection on the surface and s σ is the surface gradient of σ. It is given by the following ∇ expression [76].

s σ = σ n~(n~ σ). (3.9) ∇ ∇ − · ∇ Different methods for calculating the surface force components exists. Ya- mamoto et al. [70] used a density-scaled balanced CSF approach proposed by Yokoi et al. [77] for calculation of the normal and tangential surface tension components. With this approach the equations are evaluated at the cell faces. The normal component is scaled using a smoothed non-symmetrical Heaviside function while the tangential component is scaled using a density-scaled balanced delta function. Table 3.2 lists the different interface delta functions for original

VOF (a), VOF (b) with an additional factor, 2ρ/(ρm +ρg ), to redistribute the large density- and viscosity differences between ﬂuid 1 and ﬂuid 2, ordinary LS (a) and LS with density-scaled balance, DSB LS(b). The indicator function α used in the VOF method is dimensionless while Ψ, used in the LS method, has the dimensional unit length [m]. Care has to be taken when multiplying the surface tension with the gradient of the indicator function or the indicator

34 3.5. IMPLEMENTATION OF THE FORCES APPLIED AT THE FREE SURFACE

Table 3.2: Interface delta function used for different interface capturing methods

VOF(a) VOF(b) LS(a) LS(b)

δs α α Factor Ψδ H | ∇ | | ∇ | ∇ ∇ function itself. In reality the thickness of an interface is zero but numerically it has a thickness that is different depending on which method is used. This thickness affects the accuracy of the interface. Four fundamental aspects to consider that affects the accuracy of the interface [71], when using either of the above described methods, are

accurate interface representation and advection using either geometrical • reconstruction or algebraic manipulation,

mass conservation has to be fulﬁlled (VOF and CLSVOF methods fulﬁll • this requirement),

spurious, or parasitic currents occur (solely numerical) due to inaccurate • interface capturing and lack of force balance (due to inconsistent formulation of force equations),

ability to handle large density ratios. • For example in the work by Saldi [10] the surface tension force was implemented as a surface force using geometrical reconstruct while thermocapillary force was implemented as a volumetric force. With this kind of implementation the force balance is not fulfilled and is not consistent, which might lead to problems with spurious currents. Therefore, in this work all surface forces are implemented consistently using the reconstruction approach. In the case of conduction welding, as investigated by Pitscheneder et al. [12] and Saldi [10], and in the case of keyhole laser welding, the density ratio between molten metal and the gas can reach up to 1:7000. With such high density ratios the smeared volume force at the interface generates spurious currents in the gas phase. A solution is thus to use a redistribution term, or factor, that re- distributes the forces to the phase with the higher density. This approach has been applied in several studies but was first proposed by Brackbill [75]. The method is applicable in the case of an indicator function that depends on density such as α in VOF. In other cases, such as LS method, where the indicator function is a distance the factor cannot be applied. The factor is defined as ρ factor = (3.10) ρl + ρg

35 CHAPTER 3. STATEOFTHEARTOFLASERWELDINGMODELS

In this work the factor is applied in combination with the VOF method. In the case of small density- and viscosity differences, such as in the study by Yamamoto [70], there is no need of using a redistribution term. This is not the same as the DSB-CSF which uses a Heaviside function.

36 Chapter 4

The interFoam solver

The CFD melt pool model for laser welding in conduction mode was implemented based on the OpenFOAM solver interFoam. InterFoam originates from the work done by Ubbink [68]. It is a solver for two isothermal, immiscible and incompressible Newtonian fluids, modeled with a single-fluid approach. The general set of equations to be solved in interFoam is the single fluid continuity equation

∂ ρ + (ρu~) = 0, (4.1) ∂ t ∇ · and the momentum conservation equation Z ∂ (ρu~) + (ρu~ u~) = p + τ + ρ~g + σcδnd~ Γ . (4.2) ∂ t ∇ · ⊗ −∇ ∇ · Γ ρ denotes the one fluid density and u~ the velocity vector. p is the static pressure T 2 and τ = µ u~ + ( u~ ) µ( u~)I the stress tensor for a Newtonian fluid. ∇ ∇ − 3 ∇ · ~g is the gravitational acceleration and the last term on the RHS of Eq. 4.2 is the surface tension. The thermophysical properties are assumed constant in each region, i.e gas and liquid region. At the interface, one cell can contain both gas and liquid and the different phases in the domain have to be distinguished. The deformation of the interface between the fluids is tracked by an algebraic volume of fluid, VOF, method with MULES developed by Weller [78]. The transport equation for the volume fraction is given by ∂t α + αu~ + C (1 α)αu~r = 0, (4.3) ∇ · α∇ · −

where Cα the compression factor introduced to ensure the sharpness of the interface and u~r the compression velocity [78]. This VOF method was also

37 CHAPTER 4. THEINTERFOAM SOLVER

used in, for example, the work by Otto et al. [81], Saldi [10] and Koch et al. [82, 83]. Example of other algebraic methods that compresses the interface and are applicable for unstructured mesh are CICSAM [68] and HRIC [68, 70, 71, 78]. This chapter describes the MULES solver applied to compute advection of liquid fraction, the derivation of pressure equation and the pressure-velocity coupling of the interFoam solver.

4.1 MULES

The solution of the transport equation for the volume fraction has to be bounded. To satisfy this requirement the explicit Multidimensional Universal Limiter with Explicit Solution, MULES solver is applied. MULES, and the semi-implicit MULESCorr, compute the ﬂuxes applying suited schemes to maintain stability and guarantee boundedness of the volume fraction α. The MULES algorithm can be performed with sub-iterations in time and/or ﬁxed point iterations. It is described by the following steps.

s t 1) Compute the low order flux FL of the second term of Eq. 4.2 using a 1 order upwind scheme. This scheme is set by default. The flux is computed n˜l 1 n 1 ~ as FL = α − u~ − Sf . The subscriptsãnd l refer to sub-iteration and fixed f f · point iteration, respectively. By subcycling the α-eq nAlphaSubCycle number of times the stability increases without decreasing the time step.

2) Compute the high order flux FH of the second and third term of Eq. 4.2. The schemes are selected by the user through system/fvSchemes directory. In interFoam tutorials these schemes are commonly set to the van Leer scheme and the 2nd order linear central differencing scheme. However, several different discretization schemes are available in interFoam. The high order flux is calcu- n˜l 1 n 1 ~ n˜l 1 n˜l 1 n 1 ~ lated from FH = α − u~ − Sf + α − (1 α − )u~ − Sf , where the volume f f · f − f r f · n 1 ~ flux φr f = u~ − Sf is given by Eq. 4.4. The compression velocity at the face r f · n 1 ~ center u~r f is not calculated but the volume flux φr f = u~ − Sf , according to r f · φ φ f f ~ φrf = min C | |,max | | (n~f Sf ) (4.4) α ~ ~ Sf Sf · | | | |

where Cα controls the compression of the interface and ensures its sharpness. Cα can be set to either 0, meaning no compression, 1, which is conservative compression, or higher than 1. The recommended value is between 0 and 2. In ~ this study it is set equal to 1. φf is the face volume ﬂux and Sf the face area vector pointing outwards. The interface normal vector on the cell face, n~f , is

38 4.1. MULES deﬁned as

n 1 ( α − ) ∇ f n~f = n 1 . (4.5) ( α − ) + εn | ∇ f |

εn is a stabilization factor to avoid division by zero accounting for non-uniform grids in regions where α approaches zero [71]. | ∇ | 3) The anti-diffusive flux, computed as FA = FL FH , has to be controlled with the limiter in order to avoid unboundedness. − 4) The next step is to compute the MULES limiter λ which is based on the Zalesak’s limiter. It is thus bounded between 0 and 1 and is iteratively determined. The number of iterations is by default set to 3 in interFoam. The value of λ is specific for each cell. The limiter ensures that the flux does not generate any new extrema. This is done in 4 steps. 4a) The local maximum and minimum are defined as

n˜l 1 max n˜l 1 n˜l 1 (αi − ) = max(αi − ,αj − ) (4.6) and

n˜l 1 min n˜l 1 n˜l 1 (αi − ) = min(αi − ,αj − ) (4.7) where the index j are the neighbours for the i-th cell. The in- and outﬂow contributions of the anti-diffusive ﬂux FA through the faces of each cell are calculated as

X n˜l l 1 − FA± = ( ) (Ff )A±. (4.8) ∓ f

4b) The local extrema of α are corrected using the global limits n˜l 1 max n˜l 1 max max (αi − ) = max (αi − ) ,αglobal (4.9) n˜l 1 min n˜l 1 min min (αi − ) = min (αi − ) ,αglobal , (4.10) where the global limits are set in the alphaEqn.H. 4c) The upper and lower bound of the minimum and maximum ﬂux is computed as

Ω fi + n˜l l 1 + i n˜l 1 min n˜l 1 X F = (F − ) = | | (α − ) α − + FL (4.11) L f L ∆t i i − f =1

39 CHAPTER 4. THEINTERFOAM SOLVER and

fi n˜l l 1 Ωi n˜l 1 n˜l 1 max X F − = (F − )− = | | α − (α − ) FL (4.12) L f L ∆t i i − − f =1 where ∆t denotes the time step and Ωi deﬁnes the control volume. | | v=1 4d) Now set λf = 1 for all faces and loop to ﬁnd the limiter λf

Pfi k n˜l l 1 h λf (F − )A± i ,k+1 ± f =1 f λ±i = max min ,1 ,0 . (4.13) FA±

The limiters λf for each face f of the cell are derived from Eq. 4.12 as

( +,k+1 ,k+1 n˜l 1 min(λ ,λ− ) for (F − )A 0 λv+1 i j f ≥ (4.14) f = ,k+1 +,k+1 n˜l 1 − min(λ−i ,λj ) for (Ff )A < 0 where i and j denote the owner and neighbour cells respectively.

n˜l 1 − 5) Finally the corrected ﬂux FC = (Ff )C = FL +λf FA for each face is calculated.

n 6) αi is solved so that no new extrema appears compared to the solution in the n previous time step. In the case of explicit MULES the αi is given by

fi n˜ n˜ 1 ∆t X n˜ 1 αi = αi − (Ff − )C , (4.15) − Ωi | | f =1 and for semi-implicit MULESCorr

fi n˜l n˜l 1 ∆t X n˜l 1 − − αi = αi (Ff )C (4.16) − Ωi | | f =1 where subscript l indicates the iterations within each time step defined by nAlphaCorr. With MULES the Courant number is generally considered to have an upper limit of about 0.25 while with MULESCorr there is no upper limit for the Courant number in terms of stability [79]. Due to reconstruction of the interface the interface normal vector is changed. Therefore the contact angles between solid wall and fluid, have to be corrected in order to fulfill the definition of the contact angle. This correction is done by comparing the current, or local, normal vector of the reconstructed face to the target normal vector of the cell. Any difference between these normal vectors lead to a local surface tension force which adjusts the interface shape until the target contact angle is reached [73, 75, 80].

40 4.2. PRESSURE EQUATION

4.2 Pressure equation

Based on the continuity and momentum equations (Eq. 4.1 and 4.2) the un- knowns to be deterimined are the velocity vector and the pressure field. In order to simplify the setting of boundary conditions in buoyant flow and increase stability a modified pressure is introduced in interFoam subtracting the hydrostatic pressure, to the pressure p. This modified pressure, prgh is thus defined as [78]

prgh = p ρg h (4.17) − ~g where g is the gravitational acceleration. The elevation h is deﬁned as x~ g href · | | − where href is the reference elevation, set by the user. ρ is the weighted average density deﬁned as

ρ = αρl + (1 α)ρg . (4.18) − The pressure gradient then becomes

prgh = p ρ~g g h ρ. (4.19) ∇ ∇ − − ∇ The momentum equation, including surface tension and modiﬁed pressure, is then written as

∂t ρu~ + (ρu~ u~) = prgh g h ρ + τ~ + σ α (4.20) ∇ · ⊗ −∇ − ∇ ∇ · c∇ T 2 where τ is the viscous stress τ = µ u~ + ( u~ ) µ( u~)I . The last ∇ ∇ − 3 ∇ · term of Eq. 4.20 is the surface tension. In interFoam the surface tension in the momentum equation is assumed constant, i.e. the thermocapillary force is neglected. The Continuum Surface Force, CSF, technique introduced by Brackbill et al. [75] to calculate the surface tension as a smooth volume force instead of a surface force is employed. With the original CSF approach the surface tension is represented by a continuous volume force acting within the interface region and given by

~ fCSF = σcn~δ (4.21) where σ is the surface tension coefﬁcient, n~ is the unit normal vector and δ is the Dirac delta function. The local interface curvature c obtained from the gradient of α (which is zero everywhere except across the interface) includes a stabilization factor εn to avoid division by zero. The curvature is computed as α = ∇ εn . (4.22) c −∇ · α | ∇ |

41 CHAPTER 4. THEINTERFOAM SOLVER

The divergence operator is computed applying Gauss integral theorem and sum- ming over the values at the face centers. To be consistent with the computation of the pressure gradient evaluated on a pseudo staggered grid through a Rhie- Chow type approach (see section 4.2) the surface tension is evaluated at the cell faces as

(fCSF )f = σ f ( α)f . (4.23) c ∇ Based on the single-ﬂuid momentum equation an equation for pressure is derived. In (semi) discretized form the momentum equation is written as n X n n 1 ai u~i + aj u~j = prgh g h ρ + σc α + ai u~i − (4.24) j=i −∇ − ∇ ∇ 6 where the index i indicates a cell centre value of the control volume Ωi and j a cell centre value of the neighbour control volume Ωj . The superscripts n and n 1 are related to the time discretization. The transport part including the matrix− coefﬁcients of all the neighbouring cells (the off-diagonal components) and the source part including the transient terms and all source terms (e.g. surface tension and gravitation) except from the pressure gradient are grouped into H~(u~) ~ n 1 X n Hi (u~) = ai u~ − aj u~j (4.25) i − Eq. 4.24 and 4.25 then lead to the expression for u~i as 1 ~ u~i = Hi (u~) prgh g h ρ + σc α (4.26) ai − ∇ − ∇ ∇ In accordance to the variant of the Rhie-Chow procedure applied in openFOAM, the velocities are estimated at the cell faces and thus have to be interpolated from the cell centred values. The face centred velocity is given by 1 ~ (u~i )f = Hi (u~) prgh g h ρ + σc α . (4.27) ai − ∇ − ∇ ∇ f Thus the pressure, and consequently surface tension, are also estimated at the cell faces. By substituting Eq. 4.24 into the discretized continuity equation X ~ X (ρu~) = Sf (ρu~) = F, (4.28) ∇ · f · f an equation for pressure is obtained as X 1 X 1 ~ ρ prgh Sf = ρ H(u~i ) g h ρ + σ α Sf (4.29) a ∇ a − ∇ c∇ f i f f i f The governing equations are solved in a segregated way. Due to the interde- pendence between velocity and pressure a coupling method is needed in order to couple the transport of velocity and pressure. The following section describes the pressure-velocity coupling employed in the interFoam solver.

42 4.3. PRESSURE-VELOCITYCOUPLING

4.3 Pressure-velocity coupling

The pressure-velocity coupling employed in interFoam is the Pressure-Implicit Method for Pressure Linked Equations, PIMPLE. It is a combination of the steady-state Semi-Implicit Method for Pressure Linked Equations, SIMPLE, method [84] and the transient Pressure Implicit with Splitting of Operators, PISO, method and is commonly used for unsteady ﬂow calculations. It is an iterative method that splits the solving of the momentum and pressure equations into an implicit momentum predictor step and explicit pressure corrector steps. The PISO-algorithm consists of the following steps:

The momentum equation, Eq. 4.20, can be solved with the pressure from • the previous time step. This is called the momentum predictor and is applied if requested by the user. It gives an approximation of the new velocity ﬁeld, otherwise the velocity ﬁeld is initialized using the velocity from the previous time step.

The face volume ﬂuxes, φf , are then calculated using the velocities ini- • tialized previously. At this stage the ﬂux does not take into account the pressure contribution.

Based on the predicted velocities the term H(u~), of Eq. 4.25, is assembled • and the pressure equation, Eq. 4.29, is solved generating the ﬁrst estimation of the new pressure ﬁeld.

The conservative ﬂuxes are updated according to the new pressure. • The velocity ﬁeld is corrected in two steps. First the explicit correction • 1 due to the change in pressure gradient, prgh , which is calculated from ai ∇ the face values of pressure (this is Rhie-Chow correction). The second part is an implicit corrector step to update H(u~) according to the new pressure. This last step can be repeated (at the request of the user) nCorrectors times to better capture the non-linearities and increase the computation stability.

4.4 InterFoam solution algorithm

An iterative solution method is utilized with the interFoam solver. It starts from an initial solution based on initial conditions and ﬁeld values at an initial time n 1. The solution is then iteratively improved until the user deﬁned tolerances are− met. The overall solution algorithm for interFoam consists of the following steps.

43 CHAPTER 4. THEINTERFOAM SOLVER

Start by setting the initial conditions for all field values. • Determine the time step when using self-adjusting time step control or • apply a fixed time step. The self-adjusting time step is either global time stepping, GTS, or local time stepping, LTS. This is a use- defined feature.

– GTS can be used both for transient and steady state solutions. The maximum local Courant number is calculated with the previous time step. The maximum Courant number is computed based on the usual Courant number as well as the interface Courant number.

The ﬁrst is computed based on the local cell centre velocity u~i . The latter is called the alpha Courant number and is computed from the cell face velocities. The maximum local Courant number is then deﬁned as

Co˜ h Co αCo i max min max , max (4.30) ˜ = Co maxi Coi maxΓ ,i CoΓ ,i

where C oΓ ,i is the interface Courant number, Comax is the maximum Courant number and αComaxis the maximum alpha Courant number set by the user. The new time step can thus increase or decrease compared to the previous time step if the estimated maximum local Courant number exceedes or not the maximum Courant number imposed by the user. The initial time step, set by the user in the systems/controlDict dictionary is only used for the ﬁrst iteration. The maximum time step allowed is also set by the user in the same dictionary. – LTS for steady state solutions. The local time step for each cell is then calculated as large as possible based on the local Courant number.

The option with subcycling enables solving the α-equation in several • subcycles which can increase stability and convergence rate for the solution without increasing the overall time step. The MULES algorithm is then computed at each subcycle. Based on the volume fraction obtained the interface unit normal vector and interface local curvature are updated. The mixture properties such as ρ and µ are also updated. The ﬁrst option in the PISO algorithm explained in section 4.3 is the one • applied in the present work. For the solution including under-relaxation the velocity is under-relaxed implicitly before the momentum predictor step. The under-relaxation factors are user deﬁned in the range of 0-1. By default the relaxation factors are set to 0, meaning no relaxation. The pressure relaxation is done explicitly. It is important when running transient

44 4.4. INTERFOAM SOLUTION ALGORITHM

calculations to set a number of iterations large enough to converge in time, or to set the relaxation factor for the ﬁnal iteration to unity. However, this second option might give imbalance between the equations. A common

recommendation for determination of relaxation factors is αu~ + αP = 1, where αu~ is the velocity relaxation factor and αP the pressure relaxation factor.

Continue to the next time level by determination of the new time step • until end time is reached.

45 Chapter 5

The melt pool model for laser welding in conduction mode

The physical model aims at describing the thermo-fluid evolution of the melt pool resulting from metal heating by a laser. It describes three phases (solid, liquid and atmosphere) including phase changes, for topologically separated flow (i.e. a continuous and deforming interface between liquid and gas). The first part of this chapter, section 5.1, presents the modelling of the laser heat source, indicating the modelling assumptions. The second part of this chapter, section 5.2, describes the thermo-fluid model including the equations specific for the melt pool.

5.1 Laser energy source model

A laser beam heat source used in welding can be described as an electromagnetic wave propagating through atmosphere, a shielding gas, a plasma plume and a workpiece skin layer using physical wave optics. This approach is based on the system of Maxwell equations [85]. For laser beam applications it can usually be assumed that i) the magnetic field intensity H~ and the electric field E~ are sufficiently weak, so that non-linear effects are negligible, ii) the optical media is homogeneous, isotropic and non-dispersive, so that the closure relations of linear optics can be applied. At atmospheric conditions and at the macroscopic scale, it can also be assumed that iii) the charge density can be set to zero since the space scale of interest for melt pool modeling is larger than the Debye length. Furthermore, if it is also assumed that iv) the laser light is harmonic with a frequency ω,

46 5.1.LASER ENERGY SOURCE MODEL

v) the geometrical optics regime can beapplied (i.e thecharacteristic length- scale for changes is large compared to thewavelength [86]). Laser light propagation is then based on theraydescription and thewaveissimpliﬁed to a plane wave. In laser welding the assumption ofplane wave is valid in the focalplane x = x (located in 0 in Fig. 5.1). Deviation from the plane wave assumption increases as moving away from the focal plane, such as when entering deeper into a keyhole.

Figure 5.1: Schematic form oflaser beam propagation

The problem is now further simplified assuming that vi) the electric conductivity of the media can be neglected. With the assumptions (i-vi) the Maxwell equations reduce to the Eikonal equation. The Eikonal equation was applied by Courtois et al. [87] to model a laser beam in keyholewelding mode. Thecharacteristic length of this problem is the laser wavelength. It necessitates solving the equations with a spatial resolution even smaller than the laser wavelength (at least 1/5 according to [87]).Itlimits therangeof wavelength that can bestudied with today’s standard computers as the equations need to be solved in reasonable computation time. To circumvent this difficulty Courtois et al. [87] assumed 2D axi-symmetry anddid scale the problem. They changed the 1.06 μmwavelength of their laser to a model wavelength of 50 μm and rescaled the results. In this way they could apply the same resolution length (mesh size) to both theelectromagnetic model and the thermalflowmodel. A laser beam as used in welding applications can also be described as propagating rays using geometrical wave optics and the ray tracing method rather than theEikonal equation. However, it should be noticed that geometrical wave optics with theEikonal equation is not strictlyequivalent to geometrical wave optics with the ray tracing method.The former takes into account the possible damping or amplification of the incident wave by interaction with the reflected waves whilethe latter does not. Ray tracing is the most commonly used methodfor modelling the laser beam raysinthefield ofkeyhole laser

47 CHAPTER 5.THE MELT POOL MODEL FOR LASER WELDINGIN CONDUCTION... welding when thewelding speed is large enough to lead to keyhole bending. It was introducedbyKi et al. [63].Inthis approach the beam is decomposed in a finite set of rays. The location of intersection of eachbeam ray with the workpiece surface is determined and the fraction of ray energy transmitted and reflected are determined using the Fresnel’s law, see e.g. [7, 63, 88]. The process is iterated with the reflected rays (multiple Fresnel reflection) until the reflected energy becomes lower than a pre-set cut off value. In all the former approaches themodelling of the laser energy transfer can be coupled with themodelling of thethermal flowintheweld pool (in particular the free surface deformation). For laser welding in the conduction mode, and laser welding in the keyhole mode at low welding speed (where the bending of the keyhole against thewelding speed is not significant) laser and thermal flow models use to be decoupled. A semi-empirical source term of heat flux at themetal surface including adjustable parameters is then imposed.Even with this simplified approach several different models of the laser source heat energy distribution exist. These models can be 3-dimensional for laser welding applications in the keyholemodewhilethey are generally2-dimensionalfor applications in the conduction mode. Xu et al. [89] represented the laser energy source by an adaptive volumetric heat source with the 3-dimensional Gaussian distribution. A schematic of the heat source with theadjustable parameters zi andze defining the height of the heat source, ri andre defining the diameter of the heat source is seen in Fig. 5.2. The heat input is then distributed in a

Figure 5.2: Schematic of volumetric heat source with adjustable parameters volume deﬁnedby a rotated conical shape. Heightand radius of thevolume are setinthemodel to represent thespot size of the laser beam. To consider the peakpower differences in the keyholeaproportion factor is included which is a

48 5.1. LASER ENERGY SOURCE MODEL

ratio between the power at the top (in ze ) and bottom (in zi ) surface of the heat ˙ source. The laser heat source term Qlaser is then given by

3η Q˙ 1 χ χ z z 3r 2 Q˙ L L z e i e x p (5.1) laser = 3 − + − 2 π(1 e )(E + F ) × ze zi ze zi − r0 (z) − − − − ˙ with ηL denoting the power efﬁciency. QL is the laser power and χ the propor- tion factor. The terms E and F are respectively given by

1 χ 1 z6 s z4 1 z6 s z4 s 2 E e e z2 i i z2 , (5.2) = − 2 + e 2 + + i ze zi p 6 p 2 − p 6 p 2 2 − and χ z z 1 z5 s z3 1 z5 s z3 s 2 F e i e 2 e s 2 z i i . (5.3) = − 2 + e 2 + + ze zi p 5 p 3 − p 5 p 3 z i −

The height ze and zi of the heat source volume deﬁne p and s according to

2 2 ze zi p = − , (5.4) re ri − and

r z2 r z2 s i e e i , (5.5) = 2 − 2 ze z − i where re and ri are the radii of the top and bottom surfaces of the heat source volume. This formulation was used in e.g. [90, 91] to model keyhole laser welding. As far as conduction welding is concerned, the laser energy source distribution uses to be assumed invariant by translation along the direction of beam propagation since then the melt does not deform deep into the workpiece. Both a top-hat and a Gaussian distribution for the laser beam heat input are commonly employed, see e.g. Solana and Negro [92]. The latter is given by

ηQ˙ ηr 2 Q˙ exp (5.6) laser = 2 2 πrL − rL where η is the efﬁciency and η=3 indicates that 95% of the laser power is in the area of rL, which is the effective radius.

49 CHAPTER 5. THEMELTPOOLMODELFORLASERWELDINGINCONDUCTION...

5.2 Thermo-ﬂuid model

The thermo-fluid problem involves three phases: solid metal, liquid metal and atmosphere. Each of these phases is assumed incompressible, see subsection Conduction laser welding mode - process physics and modelling assumptions in section 2.2.3. Notice that this assumption would not apply to the gas phase in keyhole welding mode. The possible transfer of vaporized metal into the atmosphere is not considered here. The two fluids are Newtonian; they are separated and the topology of their interface is deformable. The phases are thus treated as continuum using an Euler approach and a single fluid model representing averaged conservation equations for all the phases present. The averaging causes some information loss that needs to be compensated. In particular, as the boundaries between phases are embedded into the single fluid model, a new variable representing the phase fraction needs to be tracked. Also the transfer of momentum between the liquid and the gas phase now needs to be included into the single fluid momentum equation so as to keep satisfying the force balance along the melt free surface (see Eq. 2.3 and 2.4). The melt free surface is an interface with no thickness (at least theoretically). Concerning the solid-liquid interface in the metal, its thickness depends on the type of material. For pure metal the solid-liquid phase change is isothermal implying a sharp transition and a solidification (or melting) front with zero thickness (at least theoretically). However for metal alloys, which are more commonly used in welding than pure metals, the situation is different since the phase change is not isothermal. It takes place over a temperature range spanning from the solidus temperature Ts to the liquidus temperature Tl > Ts . This temperature range defines a transition region of non-negligible thickness that contains both solid and liquid phases. This region is comparable to a porous media where the liquid metal velocity is damped. It is thus modelled as a mushy zone based on the enthalpy formulation developed by Voller and Prakash [93]. This general approach is the standard one used in welding for modelling the melt pool flow when both free surface deformation and atmospheric gas are taken into account.

5.2.1 Mass conservation equation The mass conservation equation for the single-ﬂuid model is given by

~ ∂t ρ + (ρU) = 0 (5.7) ∇ · where ρ is the density of the mixture and U~ the ﬂuid velocity. The mass density of the mixture is deﬁned from the mass densities ρm and ρg of the metal (m) and gas (g) phases and their volume fraction according to ρ = αmρm + αg ρg . The mushy zone model is applied implying that the interface between the solid

50 5.2. THERMO-FLUIDMODEL and liquid metal is not sharp, and it is not captured as for the metal/atmosphere interface. Instead the fraction of liquid metal ρl modelled as a function of temperature so that αl = αm fL and αs = αm(1 fL) respectively deﬁne the liquid and the solid. Then only one closure equation− is needed to determine the volume fraction αm, since the volume conservation (incompressible phases) implies that αm = 1 αg . This closure is provided through the transport equation solved for capturing− the free surface; see section 3.4.1 Interface capturing and section 4.1 MULES.

The liquid fraction in a metal alloy fL is generally modelled as a linear function of temperature according to  0 T < T  s T Ts f − T < T < T (5.8) L = Tl Ts s l  − 1 T > Tl where Tl and Ts are the liquidus and solidus temperatures. This method is commonly applied in multiphase flow models including metal alloys with phase change, for example in [9, 10, 63, 94, 95, 96, 97]. A drawback of this function is to introduce discontinuities in the energy conservation equation through the melting enthalpy, see section 4.2.3. To circumvent this problem, Rössler and Brüggeman [37] introduced a continuous function for defining the liquid fraction 4(T Tm) fL = 0.5 erf − + 0.5 (5.9) · (Tl Ts ) − where erf denotes the error function. The melting temperature Tm is defined as Tm = (Tl + Ts )/2. A comparison between the two functions are shown in Fig. 5.3 applied to Gallium as in the work by Rössler and Brüggeman [37].

5.2.2 Momentum conservation equation The viscous momentum conservation equation for the single-ﬂuid model is given by

~ ~ ~ ~ ∂t (ρu~) + (ρu~ u~) = p + fvisc + ρ~g + fdamp + fbuoy + fsurf (5.10) ∇ · ⊗ − 5 where p is the pressure and ~g the gravitational acceleration. The viscous friction force for Newtonian ﬂuid is h i ~ T 2 fvisc = µ( u~ + u~ ) µ( u~)I (5.11) ∇ · ∇ ∇ − 3 ∇ ·

51 CHAPTER 5.THE MELT POOL MODEL FOR LASER WELDINGIN CONDUCTION...

Figure 5.3: Liquid fraction using the linear and continuous function [37] where μ the viscous friction averaged over the ﬂuid species and I istheidentity tensor. Two types of average can be found in the literature. Thevolume averaging or the density averaging [78, 98, 99]. Here the same averaging procedure was used as in OpenFOAM (in the original solver interFoam), with avolume fraction averaging, leading to

μ = αl μl + αg μg = αm fl μl + αg μg (5.12) where μl is the viscosity of the liquid metal and μg the viscosity of the atmospheric gas. The buoyancyforce fbuoy isapplied to the liquid metal only. Due to the low gas density (that is about three orders of magnitude lower than the density of liquid metal) the buoyancy force is of significantly lower order of magnitude inthe gas phase, and thus neglected.Thethermalbuoyancy force entering Eq. 5.10 is then given by fbuoy = αm fl ρl β(T − Tm)g (5.13) where β is the thermal expansion coefficient of the liquid metal. A numerical damping force, fdamp,isimposed in the mushy zone to prevent the solid metal from flowing while using a one-fluid model.Itisderivedfrom Darcy’s law fdamp = Au (5.14) where A is given by Carman-Kozeny equation for flow through a porous media [37, 100]

2 (1 − fL ) A = C . (5.15) (f 3 + ) L

52 5.2. THERMO-FLUIDMODEL

C is the permeability coefﬁcient and " is a small constant used to avoid division by zero. In this work the values used for C and " are respectively 1.0 106 m2 and 3 × 10− (dimensionless). These are the values used by most author modelling the melt pool ﬂow in conduction laser welding. It should however be noticed that changing these parameters, in particular C can change the solution. However this aspect was not further investigated in this study. ~ The force applied at the free surface, fsurf , includes a normal component ~ along n~ denoted fsn (see Fig. 3.4.) and a tangential component along τ~, denoted ~ fst. It is given by

~ dσ fsurf = σ n~ + s σ = σ n~ + [ T n~(n~ T )] (5.16) c ∇ c dT ∇ − · ∇ where c is the local curvature of the free surface defined by Eq. 4.17. The first term in the right hand side of Eq. 5.16 is the capillary force, also called surface tension. Surface tension results from the difference in interaction between the atoms present in the gas phase and the atoms in the liquid phase [101]. In general it depends on both temperature and the types of atoms present (so gas and liquid composition). In this study gradients in material composition are not considered. The right hand side of Eq. 5.16 then reduces to the tangential surface force induced by temperature gradients that is the thermo-capillary force (also called Marangoni force). The modelling level of σ in the literature devoted to the study of welding melt flow can vary depending on the author.

At the first level of modelling σ is assumed constant, see e.g. [81, 88, 94, • 96, 102, 103]. At a second level of modelling, the surface tension coefficient is assumed a • linear function of temperature, as for pure metals, see e.g. [7, 9, 54, 63, 69, 95, 97, 104, 105, 106]. Its expression is then dσ σ = σ0 + (T T0) (5.17) dT 0 − This expression is not systematically oversimplified to metal alloys, see e.g. [107] A third level of modelling consists in considering the main metal element • and, in addition, selecting among the alloying elements the one influencing the most the surface tension; usually this is a surfactant. Then a surface tension for binary metal, as developed by Sahoo et al. [101] and Su et al. [108] can be used

d 0 0 σ (∆H /RT ) σ = σm (T Tm) RT Γ1ln[1 + k1a1e− ] (5.18) − dT 0 − −

53 CHAPTER 5. THEMELTPOOLMODELFORLASERWELDINGINCONDUCTION...

0 where σm is the surface tension at melting point of a pure metal, dσ/dT0 is the negative temperature coefﬁcient of surface tension; i.e. Marangoni

coefﬁcient, of a pure metal and Tm is the melting point of the material. The last term on the right hand side of the Eq. 5.18 is the inﬂuence of

the alloying elements. R is the ideal gas constant, Γ1 the surface excess at saturation, k1 the entropy factor, a1 the weight percentage of active 0 element and ∆H the standard heat of adsorption. Sahoo et al. [101] applied this approach to steel and copper as primary metal, with sulfur, oxygen, selenium and tellurium as dominant metal alloy element, and pure argon as atmospheric gas. An advantage of this approach is to have an equation for σ that can be applied to various metal alloys for a given primary metal. A drawback is that the inﬂuence of the neglected alloying elements and of the atmosphere (if it differs from pure argon, e.g. [109]) might not be negligible. Su et al. [108] applied it to stainless steels.

At a fourth level, the surface tension model is designed for a speciﬁc • metal alloy conducting enough experiments varying over a temperature range to measure the surface tension. The measurement results are then interpolated to obtain a semi-empirical equation for the alloy surface tension as a function of temperature. This approach has been applied to some alloys such as the titanium alloy Ti6Al4V [107], stainless steels [108].

Different solutions for avoiding spurious currents have been proposed, for example mesh reﬁnement and decreased time step [69, 70]. However, this does not seem to decrease the magnitude of the spurious currents. Other approaches such as Sharp Surface tension Force, SSF, has been presented.

5.2.3 Energy conservation equation

The energy conservation equation for the single-ﬂuid model, expressed as a function of the enthalpy ρh to account for the phase changes, is given by

˙ ∂t (ρh) + (ρhu~) = (k T ) + Qlaser (5.19) ∇ · ∇ · ∇ ˙ where Qlase r is the laser energy rate source term, as discussed in section 5.1. The thermal conductivity k of the mixture is given by

k = αm(fLkl + (1 fL))ks + αg kg (5.20) − where kl and ks is the thermal conductivity of the metal in liquid and solid state and kg is the thermal conductivity of the gas. The speciﬁc enthalpy h(T) of the

54 5.3. NEWDEVELOPMENTSMADEININTERFOAM single ﬂuid is deﬁned as

R T  Cp dT T Ts Z T  Tref s ≤ R Ts ρ∆h T α ρ C dT α ρ Cp dT fLLTs < T < Tl ( ) = g g pg + m m Tref l + T  ref R T R T  Cp dT + fLL + Cp dT T Tl Tref s Tl l ≥ (5.21) with ∆h(T ) = h(T ) h(Tref ), Cp , Cp and Cp denote the specific heat capacity − s l g in solid, liquid and gas state respectively and L is the latent heat of fusion. Since the energy equation is solved for temperature, T, it is then written as ˙ ∂t (ρCp T ) + (ρCp T u~) = (kf T ) SL + Qlaser (5.22) ∇ · ∇ · ∇ − where ρCp defined for the single fluid is given by ρC α ρ C α ρ C (5.23) p = m m pm + g g pg with ρm and ρg being the density of the metal and gas respectively. k is given by Eq. 5.20 and C is the specific heat of the solid and liquid metal 110 given by pm [ ]

Cp = fLCp + (1 fL)Cp . (5.24) m l − s

The source term for latent heat, SL, in Eq. 5.22 is given by ¨ ∂t (ρfLL) + (ρfLLu~) T > Ts SL = ∇ · (5.25) 0 T Ts ≤

The convection of latent heat (second term in the above expression for SL) is included in several models for example [63, 81, 87, 94, 95, 96, 102, 104, 111, 112], while neglected in others for example [9, 10].

5.3 New developments made in interFoam

The developments made to implement the model for melt pool in conduction mode laser welding are now summarized. The implementation was made starting from the solver interFoam (3.0.1) for two isothermal and immiscible ﬂuids, • with constant surface tension, • and algebraic VOF with sharp surface tension force for capturing the free • surface.

55 CHAPTER 5. THEMELTPOOLMODELFORLASERWELDINGINCONDUCTION...

The new elements implemented thus include:

the solid metal state, that is • – the thermophysical properties for this solid state – the mushy zone model with Darcy damping force and Rösler and Brüggeman model for the liquid fraction of metal – the solid liquid phase change (fusion enthalpy)

the temperature dependence of the surface tension for both the pure metal • Eq. 5.17 and the binary metal, Eq. 5.18

the thermocapillary (or Marangoni) force for both pure metal and binary • metal. This was done on two ways: with the CSF approach of Brackbill et al. [75], as applied previously by Saldi [10] and with the SSF approach applied in interFoam to e.g. surface tension and pressure force [113]. the energy conservation equation for two phases (atmosphere, metal) and • three states (gas, liquid, solid) with solid/liquid phase changes, and a source term for modelling the laser energy heat input (either tophat or gaussian).

Besides, another variant of the numerical method for capturing the free surface, to provide a less diffusive computation of the free surface compared to the original VOF model of interFoam, were applied with this model:

the geometric variant of VOF named isoAdvector, using the source module • developed by Roenby et al. [72].

56 Chapter 6

Numerical applications - results and discussion

In order to test the simulation model, problems have been chosen from the literature. The two first test cases were intended to examine a specific part of the physics implemented. The first test case in section 6.1 addresses a thermocapillary driven flow and its effect on the free surface while retaining two states (gas, liquid and no solid). The second test case in section 6.2 includes the three states; the focus is on the effect of the thermocapillary flow on the solid/liquid melt front. For this, the intensity of the gravitational acceleration is scaled down. Finally the third test case in section 6.3 is about melt pool flow in conduction mode laser welding. Each of these test cases are also used to evaluate the effect on the computational results of

the CSF and SSF approach used for the thermocapillary force • the algebraic VOF (as interFoam), and the geometric VOF named isoAd- • vector. The later was designed to provide a sharper interface than the former.

6.1 Two-phase ﬂow driven by thermocapillary force

This time-independent test case was designed by Sen and Davis [114] based on dimensionless numbers set to impose a relative order of magnitude of the forces such that

0 = Fbuoy << Fst < Fvisc << Fsn.

Fbuoy is the buoyancy force, Fvisc the viscous force, Fst and Fsn are the tangential and normal components of the surface tension, i.e. thermocapillary and capillary forces, respectively. The capillary force, that can induce a deformation of the

57 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTSAND DISCUSSION

Table 6.1: Dimensionless numbers Number values Marangoni,Ma 0.2 Capillary, Ca 0.008 Prandtl, Pr 0.2 Reynolds, Re 1 Rayleigh, Ra 0 Grashof, Gr 0 free surface is thus dominant.The values of the dimensionless numbersare given in table 6.1 and their deﬁnitions in table 2.1.

6.1.1 Test case description The compuational domain consists of a two-dimensional cavity, as shown in Fig. 6.1. The aspect ratio d/L is set to 0.2. The temperature difference ΔT = 1K between the hot left wall at TH and thecold rightwall at TC induces a gradient in surface tension with temperature and in turn the thermocapillary ﬂow. The surface tension is the linear function of temperature given byEq. 5.17. The liquid and gas designed to satisfy the dimensionless numbers set in table 6.1 thus have thethermodynamic and transport properties reported in table 6.2.

Figure 6.1: Schematic view of the 2D-domain.

6.1.2 Numerical setting The computationaldomain, ofdimensions H=0.4m and L=1m,has been discretized with uniform quadratic cells as recommendedfor the VOF method

58 6.1. TWO-PHASE FLOW DRIVEN BY THERMOCAPILLARY FORCE

Table 6.2: Thermodynamic and transport data Properties Liquid Gas Units 3 Density 5000 1.185 [kgm− ] 10 1 1 Dynamic viscosity 0.14 1.185e− [kgm− s − ] 10 1 1 Thermal conductivity 70 1e− [W m− K − ] 1 1 Specific heat 100 1007 [J kg− K − ] TH 299 299 [K] TC 298 298 [K] 2 Surface tension, σ0 0.0025 [kgs − ] dσ 5 2 1 Marangoni coef., dT -9.8e− [kgs − K − ] 5 1 Thermal expansion coef. 1.34e− [K − ] from interFoam. A mesh of 300 120 cells was used as a reference. More refined meshes were also considered (see× table 6.4). A no-slip velocity and zero gradient pressure boundary condition was imposed on each of the walls enclosing the cavity. A reference pressure, set to 0, is imposed at a point close to the top right corner where very low flow velocities are expected. Adiabatic boundary conditions are imposed for the temperature on the top and bottom wall while

TH and TC are set to 299K and 298K respectively. The boundary conditions for the liquid volume fraction are set to zero gradient on the top and bottom wall while a constant contact angle of 90◦ is imposed on the side walls. Initially the ﬂuids are at rest, at temperature TH , the interface is ﬂat and the liquid layer has a uniform thickness d=0.2 m. The numerical schemes applied to compute this test case are given in table 6.3. The PISO solver is used to solve the pressure-velocity coupling with two inner 8 loops. Convergence is reached when the residuals are 10− for α, U, T and prgh. 12 An adjustable time step with an initial value of 10− and maximum Courant numbers of 0.2 are applied. Different maximum time steps (∆tmax) between 100s down to 0.1s are evaluated. With ∆tmax =100s as applied by Saldi [10], the free surface oscillated, and the temperature in the gas phase increased. The calculations results obtained by Saldi [10] for the gas phase were not reported in [10]. In this study, when the maximum time step is reduced down to 0.1s the oscillations are still noticeable but smaller and the temperature of the gas phase is no longer affected. Thus, ∆t=0.1s is the largest ∆tmax allowed to compute the calculation results presented in the next section.

6.1.3 Results and discussion

Mesh study It is found that a uniform mesh with size of 300 x 120 quadratic cells is appropriate

59 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTS AND DISCUSSION

Table 6.3: Numerical schemes Mathematical term Numerical scheme ddtScheme Euler gradScheme Gauss linear divScheme (ρφU~) Gauss linear Upwind divScheme (φα) Gauss vanLeer divScheme (ρν) Gauss linear laplacianScheme Gauss linear corrected interpolationScheme linear snGradScheme corrected

Table 6.4: Interface position

Free surface Fst mesh ∆tmax Case VOF CSF 300 120 0.1 a VOF SSF 300×120 0.1 b VOF SSF 300×120 0.01 c VOF SSF 400×160 0.1 d × in the case of VOF. The isoAdvector approach generated un-physical results and could not reach convergence. It resulted in a large wave that developed in the domain. In order to find out if this wave formation is due to the mesh, a mesh convergence study is performed as isoAdvector is known to be more sensitive to the mesh size than VOF. The number of cells of the initial mesh, 300 120 cells, is multiplied by 2, 3, 4, 5 and 6. The results of the mesh times 2, 4× and 6 are shown in Fig. 6.2 a, b and c respectively. It can be seen that the wave is less pronounced when increasing the mesh resolution. Fig. 6.2 d presents the results with the mesh times 6 after another 1000 iterations. It shows that the wave still develops even for the finer mesh. This problem might be due to numerical errors (noise) when computing the curvature of the free surface. The isoAdvector method is expected to provide a sharp interface. However, sharpening can lead to inaccuracte computation of the curvature. Thus, it leads to an imbalance between surface tension and pressure force at the interface. This imbalance generates parastic currents. A problem specific of low capillary number problems (as here) is that parasitic currents can dominate the physics [115]. It is suspected that this is the problem occuring here. Smoothing when calculating the curvature should improve this problem. Due to the lack of time this is not further investigated here. Thus, the computed test cases discussed in the sequel are those listed in table 6.4. Even when convergence is reached for temperature, velocity and pressure the interface oscillates, and no steady state solution is found. This problem, also

60 6.1.TWO-PHASE FLOW DRIVEN BY THERMOCAPILLARY FORCE

Figure 6.2: Temperature distribution and interface positions obtained for the different mesh resolutions with isoAdvector approach. discussedbyYamamoto et al. [70], is also assumed to be due to the occurrence of spurious currents. Thus, the evaluation of the interface position is not straight forward. Each case is also evaluated based on the temperature distribution, interface position on leftand rightwall,the maximum velocity in the liquid phase and the maximum total velocity. Reference case The so-called reference test case is the case a) since it was also computedbySaldi with the same method for the free surface tracking and the thermocapillary force Fst. Fig. 6.3 shows the results for this case. Thewhite horizontalline represents the interface. Figure 6.3a shows the isoline α=0.5 andthevalue of its elevation on the left and right walls. These measurements can be used to compare the calculation results to the analytical solution obtainedbySen and Davis [114] and to former numerical studies. For the computations made in this study, the measurement ranges (due to oscillation of the surface) are given for the leftand right positions. The oscillations are more pronounced on theright position. It is also there thewaveformed when applying isoAdvector. In order to compare the interface capturing methodsthethickness of the interface is measuredby plotting the isolines α = 0.1 and α = 0.9 and measuring the distance between these lines. Thecalculated maximum thickness is reported in table 6.5. This data was not provided in the earlier studies listed in table 6.5. Figure 6.3c is a plot of thevelocity magnitudeinthe liquidphase. Again theamplitudeof this velocity was not reportedby Sen and Davis [114] nor Saldi [10].Fig. 6.3d shows thevelocity magnitudeinthe gas phase and thevelocity vectors in the liquid phase. These last ones were also plottedbySaldi (see Fig. 3.12 in [10])and are

61 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTSAND DISCUSSION

Table 6.5: Interface position Case Left Right Thickness a) VOF-CSF (mesh 300×120) 0.188 - 0.189 0.218 - 0.219 0.0285 b) VOF-SSF (mesh 300×120) 0.187-0.188 0.219 - 0.220 0.0280 c) VOF-SSF (mesh 300×120) 0.187-0.188 0.217 - 0.218 0.0215 d) VOF-SSF (mesh 400×160) 0.187-0.188 0.215-0.216 0.0150 Hirt (mesh 21 × 10) [116] 0.192 0.207 Sasmal & Hochstein (mesh 34 × 31) [56] 0.174 0.224 Francois et al. (mesh 100 × 32) [117] 0.187 0.209 Saldi (mesh 100 × 120) [10] 0.187 0.208 Sen & Davis (analytical) [114] 0.188 0.213 very similar to 6.3d.

Figure 6.3: VOF with Marangoni CSF a)Temperature plot w. interface position, b) interface thickness, c) velocity in liquid Uliq, d)velocity Umax and ﬂow pattern.

Interface Figure 6.4 shows a comparison of the interface position computedfor the different cases a-d of table 6.4. The corresponding free surface heights and thicknesses are reported in table 6.5. As alreadyobserved in [10] the numerical results agree better with the analytical results when reﬁning the mesh size. This agreement is better on the leftsidethan on therightside. The largest oscillations of the surface are indeed on therightside. Theamplitudeof these oscillations reduces further when reducing the size of the mesh cells (cases b andd)and when reduc-

62 6.1.TWO-PHASE FLOW DRIVEN BY THERMOCAPILLARY FORCE

Figure 6.4: Temperature distribution and interface positions obtained for the different cases a,b,canddas denoted in table 6.4.

Figure 6.5: Velocity fieldsobtainedfor the different cases a, b,canddas denoted in table 6.4. ing the time step (cases b and c). The maximum thickness of the free surface reduces accordingly. Temperature and velocity fields Fig. 6.4 and Fig. 6.5 respectivelyshow a comparison of thetemperature fields

63 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTS AND DISCUSSION and the velocity field (in both phases and vectors in the liquid phase) computed for the test cases a-d of table 6.4. It can be seen that the flow observed in the gas phase reduces when reducing the time step (cases b and c). It suggests that the quality of the convergence in time and space might be further improved for the gas phase. No significant difference is found between VOF with CSF and SSF approach.

6.1.4 Conclusion Based on the results obtained it is concluded that,

good agreement with the reference case is obtained, • no signiﬁcant difference between CSF and SSF approach is found. • The oscillations observed at the interface, and assumed to be caused by • spurious currents according to Yamamoto [70], are much larger for the sharper interface capturing method isoAdvector than for the sharp VOF taken from interFoam,

further investigation is needed in order to reach a converged solution with • the isoAdvector approach. For this it is proposed (in a forthcoming study) to smooth the calculation of the free surface curvature c. This smoothing would also be of interest to implement in the sharp VOF of interFoam, to study if the non-physical oscillations observed at the free surface could be damped.

6.2 Three-phase ﬂow driven by thermocapillary force

This time independent test case was introduced by Tan et al. [57] for a gas and a pure metal (bismuth) undergoing fusion in a 2D container. It is associated with the dimensionless numbers of table 6.7 (see table 2.1 for their deﬁnitions), thus leading to a different ordering of the magnitude of the forces, compared to the previous test case,

0 < Fbuoy < Fvisc Fsn Fst. The thermocapillary force is now the leading order force. A temperature difference between the left and right wall of the container induces a temperature gra- 4 dient which generates a thermocapillary ﬂow. Microgravity with g = 4.145 10− 2 · m/s , is applied in order to enhance the effect of the thermocapillary force since this problem would be dominated by the buoyancy effect under normal gravitational conditions.

64 6.2.THREE-PHASE FLOW DRIVEN BY THERMOCAPILLARY FORCE

Table 6.6: Dimensionless numbers Number values Marangoni,Ma 244 Capillary, Ca 0.0022 Bond, Bo 0.000188 Rayleigh, Ra 0.031 Prandtl, Pr 0.019 Stefan, St 0.033

6.2.1 Test case description The computationaldomain, shown in Fig. 6.6, is enclosed into 4 walls. The gas above the bismuth is assumed to be argon (as for Saldi [10]). The material properties usedfor this test case are given in table 6.8. The surface tension is given by a linear function of temperature, Eq. 5.17.

Figure 6.6: Schematic view of the2Dcalculation domain

6.2.2 Numerical setting The dimensions of the computational domain are reported in Fig. 6.6. The non-uniform mesh is the same as in previous studies [10, 57] with a mesh sizeof 140 × 67 cells, and smaller cellsalong thesolid-liquid transition region in x = 10 mm and close to the free surface y= 4 mm. A second meshwith a number of cells increasedby 20% is also applied. At each wall enclosing the cavity thevelocity boundary condition is a no-slip condition and the pressure boundary condition is a Neumann condition with zero gradient. A reference pressure, set to 1 atmosphere, is imposed at a point close to thetop right corner where very low ﬂow velocities are expected and no

65 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTS AND DISCUSSION

Table 6.7: Material properties Properties Bismuth Argon Units 3 Density 9780 1.6337 [kg m− ] 3 5 1 1 Dynamic viscosity 1.6e− 2.26e− [kg m− s− ] 1 1 Thermal conductivity 10.35 0.0177 [Wm− K− ] 1 1 Speciﬁc heat 123 520 [J kg− K− ] 1 Latent heat of fusion 44600 [J kg− ] Melting temperature 544.55 [K] 2 Surface tension 0.07 [kg s− ] 5 2 1 Marangoni coef. -7.0e− [kg s− K− ] 4 1 Thermal expansion coef. 1.37e− [K− ]

Table 6.8: Case description

Free surface Fst mesh Case VOF CSF 140 67 a VOF SSF 140×67 b isoAdvector SSF 140×67 c isoAdvector SSF 168×81 d ×

phase change. A uniform temperature set to Thot=552.55K and to Tcold=540.55K is imposed on the left and right side wall, respectively. On the top and bottom boundaries, a linearly distributed temperature ranging from Thot to Tcold is imposed. The volume fraction α of bismuth is set to a ﬁxed value of 0 at the top boundary and 1 at the bottom boundary indicating only gas or metal respectively. At the left and right boundary walls a constant contact angle condition is applied setting the contact angle to 90◦. Initially the bismuth is in solid state. It has a uniform thickness of 4 mm so that the interface between bismuth and argon is horizontal. Both bismuth and argon are at Tcold and at rest. The numerical schemes applied, which are the same as in test case 1, are listed in table 6.3. A time step of ∆t=0.1s is used together with maximum Courant numbers of 0.2 for the VOF solver. The recommended Courant number of 0.1 for isoAdvector [72] is not low enough to converge the calculation, thus Courant numbers of 0.05 are used in this case. The calculations are considered to have reached convergence when the residuals for temperature and velocity are 10 8 below 10− , and below 10− for pressure. The cases discussed in the sequel are computed combining CSF, SSF with VOF and isoAdvector as listed in table 6.8.

66 6.2.THREE-PHASE FLOW DRIVEN BY THERMOCAPILLARY FORCE

6.2.3 Results anddiscussion Mesh study A comparison of the results computed with the two meshes described in section 6.2.2 is performed for both the VOF and the isoAdvector case. The position of thesolidification front differedbyless than 1 % between the two meshes. Therefore the coarsermesh with 140x67cells is used here.It should be noticed that this result is in agreement with the mesh convergence study presentedby Tan et al. [57], which showed only 0.6% difference on the predicted position of thesolidification front computed using meshes with 110 x 60 cells compared to 150x80cells. Velocity field A comparison of melt front position and streamlines between the isoAdvector resultand thereference case by Tanetal. [57] isshowninFig. 6.7. Streamlines are generatedfrom reference points and these points are chosen arbitrarily since they are not known for the reference case. Nonetheless the flow pattern computed in this study agrees well with thereference case [57].

Figure 6.7: isoAdvector (top) vs. reference case (bottom) [57]

67 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTS AND DISCUSSION

Free surface Oscillations at the liquid-vapor free surface are observed in all cases indicating that the solution obtained is not truly steady state. This can be visualized by plotting results at consecutive iterations. Fig. 6.8 shows the streamlines and the interface obtained at 5 consecutive iterations during the solution after convergence is reached. Small changes of the isocontours and free surface oscillations can be observed. In Fig. 6.9 small oscillations observed in the velocity ﬁeld are plotted along the y-direction around the interface.

Melt front The position of the solidiﬁcation front is plotted in Fig. 6.10 for the different cases. The x-position of the melt front in y=0 mm is the same for all cases as it is imposed through the temperature boundary condition. The x-position of the melt front at the elevation y=4 mm is reported in table 6.9 for the different cases computed. It can be seen from Fig. 6.10 that the VOF-CSF case leads to the largest discrepancy compared to the reference case (the black dotted line) [10] while the isoAdvector case shows better agreement. It should however be noticed that the reference case of Tan et al. [57] was not solved analytically but numerically (with Fluent). So the reference solution cannot be considered as exact. Besides, the minimum cell size for cases a) to c) is 0.03 mm. Thus, the difference in x-position obtained for the cases a) and b) is too small to be representative. A similar remark applies to the test cases c) and d). The largest discreapancy in melt front x-position, which is between case a) and the reference case, is less than 1 %, and thus negligible.

Table 6.9: Melt front position in y=4 mm Case x position a) VOF-CSF mesh 140 67 10.81 b) VOF-SSF mesh 140 ×67 10.80 c) isoAdvector-SSF mesh× 140 67 10.76 d) isoAdvector-SSF 168 82 × 10.75 × e) Tan et al. [57] mesh 140 62 10.7 ×

6.2.4 Conclusion

Based on the results obtained it is concluded that,

the solution obtained cannot be considered to be steady-state due to oscil- • lations of the free surface, however, the melt front is not affected by these oscillations.

68 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING

The position of the melt front is in good agreement with the reference • case, no signiﬁcant difference between the CSF and the SSF approach for the thermocapillary force, nor the so-called VOF and isoAdvector method for the free surface are found with this test case.

6.3 Melt ﬂow with free surface deformation in conduction laser welding

This third test case is time dependent and related to conduction laser welding. It is based on the experimental (and simulation) study by Pitscheneder et al. [12]. This test case considers the conduction welding of steel S705 and includes 3 phases, i.e. solid and liquid metal and argon shielding gas. In the experimental study conducted by Pitscheneder et al. [12] the steel sample was a large rectangular plate, of non-specified length and width, and a thickness of about 15 mm. Shielding was used, blowing argon gas through a pipe at a rate of 20 l/min. However, the pipe outlet location and orientation were not specified. The energy source was a CO2 laser and the intensity profile of the laser beam was measured. Its plot, provided in [12], shows a profile close to a Gaussian distribution, but the data were not provided so that the efficiency η (see Eq. 5.21) cannot be known. Three different laser powers of 1900, 3850 and 5200 W were applied, and steel samples with 20 and 150 ppm content in sulfur were used. Pitscheneder et al. [12] also simulated the melting process taking into account the melt flow and neglecting the deformation of the free surface. They assumed axisymmetry, and their computational domain had a radius of 15 mm. A tophat laser energy intensity was set. The shielding gas was not simulated. This test case has since been used as a reference by many authors including Ha and Kin [48], Saldi [10] and Kidess [18]. These authors model a cooling effect through boundary conditions imposing radiation, or convection, or combined radiation and convection heat transfer. The argon shielding gas flow is not taken into account. A difficulty is that each author simulating this problem selects different combinations of process parameters applied in the experiments. Ha and Kim [48] simulated the melt (with free surface deformation) for 20 ppm sulfur and 5200 W laser power. As can be seen in Fig. 6.11 and 6.12 Saldi [10] did it (without and with free surface deformation) for both 20 and 150 ppm sulfur and 3850 W laser power. Kidess et al. [18] did it for 150 ppm and 5200 W laser power. In this study the test case with free surface deformation studied by Saldi [10] is used as reference since it was also computed with a melt pool model developed from interFoam.

69 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTS AND DISCUSSION

Table 6.10: Thermodynamic and transport data Properties Fe-S Ar Units 3 Density 8100 1.6337 [kg m− ] 1 1 Dynamic viscosity 0.006 1.38e−5 [kg m− s− ] 1 1 1 Thermal conductivity solid 22.9 [J m− s− K− ] 1 1 1 Thermal conductivity liquid 22.9 [J m− s− K− ] 1 1 1 Thermal conductivity gas 0.0177 [J m− s− K− ] 1 1 Specific heat solid 627 [J kg− K− ] 1 1 Specific heat liquid 723.14 [J kg− K− ] 1 1 Specific heat gas 520 [J kg− K− ] Melting temperature 1620 [K] 6 1 1 Thermal expansion coef. 10e− [kg m− K− ] 1 Latent heat of fusion 250800 [J kg− ] 5 2 2 1 Standard heat of adsorption -1.66e [kg m s− mol− ] 2 Surface tension coef. 1.943 [kg s− ] 4 2 1 Surface tension temp. coef. -5.0e− [kg s− K− ]

6.3.1 Test case description

In the present study a 2D-axisymmetric computational domain of radius 15 mm is used. The solid steel sample has a thickness of 15 mm, and is covered by a 3.5 mm thick layer of argon gas, see Fig. 6.13. The thermodynamic and transport properties are given in table 6.10. The steel is Böhler S705 alloy that can contain a surfactant, namely sulfur. Concentrations of 20 ppm and 150 ppm sulfur content are investigated. The surface tension is thus modelled applying the model developed by Sahoo et al. [101] for binary metal and recalled in Eq. 5.18. Figures 6.14 and 6.15 show the coefficient and the temperature coefficient of surface tension for 20 ppm and 150 ppm, respectively. The solidus and the liquidus temperature of this alloy are difficult to find. The melting range

Tliq Tsol applied in the literature can vary from 0.1K in [10] to 10K [18] or more.− For consistency, the value used in the reference case is seleceted. Thus

Tliq = Tm + ∆Tm, Tsol = Tm ∆Tm. Tm denotes the melting temperature given − in table 6.10 and ∆Tm is set to 0.1K.

A tophat heat distribution is applied, as done in former simulation studies of this test case, e.g. [10, 12, 18]. It is assumed uniform along the beam axis and the heat transfer from the laser is only effective at the upper surface of the steel. The laser beam power of 3850 W and the beam radius of 1.4 mm are applied.

70 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING

6.3.2 Numerical setting

Figure 6.13 shows the computational domain. It is defined in OpenFOAM as a wedge in the 3D Cartesian coordinate system. It consists of metal (15 15 mm) and ambient gas (15 3.5 mm). Two meshes are used. The coarse mesh× × is as for Saldi [10] for the same test case. In the steel sub-domain the mesh has a resolution of 160 110 cells along the x-and y-direction. In the argon sub-domain it has a resolution× of 70 110 cells. In the x-direction the cells of smallest size, 0.02 mm, are at the free× surface. In the y-direction the cells of smallest size, 0.035 mm, are close to the symmetry axis. For the second mesh the cells in the sub-domain containing the melt pool are square with a side length 5 less than the Kolmogorov length scale estimated for this problem (i.e. 2 10− × m [18]). A mesh with a resolution of 320 232 cells in x- and y-direction is applied, as shown in Fig. 6.16. × No-slip velocity and zero gradient pressure conditions are imposed at the side and bottom wall of the sample. Notice that in this work the zero gradient conditions are set in OpenFOAM using fixedFluxPressure. At the top and side atmosphere an inlet-outlet condition is set for the velocity (with pressureIn- letOutletVelocity) and the total pressure is prescribed. Adiabatic conditions are imposed at the side and bottom of the sample and at the atmosphere side. As in [10], radiative cooling with an emissivity ε = 0.5 and T = 300K is set (with groovyBC via the library swak4Foam) at the atmosphere-top.∞ The boundary conditions for the liquid volume fraction are set to a fixed value at the bottom of the sample, a zero gradient at its side and an inletOutlet boundary condition for the atmosphere (both top and side), see Fig. 6.13. Initially the steel is everywhere solid, both steel and argon are at room temperature (300K), at rest, and the pressure is the atmospheric pressure (101325 Pa). The numerical schemes applied for the case are listed in table 6.11. An adjustable time step is applied. The initial and the maximum time steps are 9 6 set to 10− s and 10− respectively.The maximum Courant and alphaCourant numbers are set to 0.02 and 0.1 respectively, combined with 5 subcycles for the α-equation (see section 4.1). The convergence was difficult to reach with the solvers available in OpenFOAM (version 3.0.1) and used in [10]. Instead the solver BiCGStab (which is a fast and smooth variant of BiCG solver, accessed via the library fftw3.3.8) is used for solving the volume fraction, momentum and energy equations. For the pressure equation the GAMG solver is used. For pressure-velocity coupling (see section 4.3), PIMPLE is operated in the PISO mode with several correction steps for the velocity field (nCorrectors = 5). No under-relaxation is applied. The calculations are considered converged when 12 8 the residuals for T, U, and the volume fraction α are 10− , and 10− for prgh, as illustrated by the final residuals plotted in Fig. 6.17.

71 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTS AND DISCUSSION

Table 6.11: Numerical schemes Mathematical term Numerical scheme ddtScheme Euler gradScheme Gauss linear divScheme (ρφU~) Gauss linear Upwind grad(U) divScheme (φα) Gauss vanLeer

divScheme (φrbα) Gauss linear divScheme (ρφCp f ,T ) Gauss Minmod

divScheme (ρνEff ) Gauss linear laplacianScheme Gauss linear corrected interpolationScheme linear snGradScheme corrected

6.3.3 Results and discussion

The reference calculations are made as in [10] applying the CSF approach for implementing the thermocapillary force, and neglecting as Saldi [10] the convection of fusion enthalpy (see Eq. (5.22)-(5.25)). Also the mesh is set as in [10]. Next, different elements of the model are modiﬁed one by one (all other elements being maintained unchanged),

1. the convection of fusion enthalpy is included,

2. the implementation of the thermocapillary force is changed from the CSF approach to the SSF approach,

3. the mesh is reﬁned down to the Kolmogorov lenght scale everywhere within the melt pool (however, the computational domain remains 2- dimensional).

The calculation results of the present study are presented in radial cross sections (z=0mm). Reference case The computational results obtained by Saldi [10] for the test case used here as reference are reported in Fig. 6.11 for 20 ppm sulfur and Fig. 6.12 for 150 ppm. In [10] the CSF approach was applied by Saldi for implementing the thermocapillary force while surface tension, pressure force, weight and buoyancy were implemented with the SSF approach (as in interFoam). The computational results obtained in this study doing the same modelling assumptions as Saldi, applying the same approach (CSF) for the thermocapillary force and using the same mesh are reported on the left hand side of Fig. 6.18 (20 ppm) and Fig. 6.19

72 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING

(150 ppm). The outer contour of the melt pool (i.e. liquid steel) is plotted using the liquid volume fraction isoline fL = 0.5. It is recalled that all these models assume that both metal alloy and atmosphere are incompressible. However, the calculation results at 20 ppm sulfur, Fig. 6.11 and 6.18, show a deficit in metal alloy volume while at 150 ppm sulfur an excess is observed. The deficit at 20 ppm is more prononced in Fig. 6.11 than in Fig. 6.18. It is important to underline that Saldi did check that his calculations satisfy mass conservation. The same check was done in this study. The results of Fig. 6.18, with a depression in the weld pool center and a bump on the outer edge is closer to the free surface profile computed at 20 ppm by Ha and Kim [48], although with a larger laser power. It seems also in [48] that there is some problem with volume conservation (e.g. a deficit for the finer mesh 50 60). However, these authors did also check mass conservation and computed× that for their finer mesh (50 60) the mass loss was -0.0002%. In fact, there is no contradiction: mass and volume× are both conserved or almost. The deformation of the free surface in the melt pool (about 2.5 mm broad) is balanced by a slight deformation of the solid steel surface (the remaining 12.5mm). From a physical point of view this solid deformation is not expected to occur with the modelling approaches of these studies. Convection of fusion enthalpy Computation results from the reference case that neglects the convection of the fusion enthalpy, are now compared with results computed while accounting for the convection of the fusion enthalpy. The plots for times up to 1 s are compared in Fig. 6.18 and 6.19 for 20 and 150 ppm, respectively. Calculation results obtained for large time (up to 5s) are reported in Fig. 6.21 and 6.22. The computed flow, the position of the melt front and the position of the free surface show in all cases some oscillations. The oscillations of the computed free surface was also reported by Saldi [10]. The oscillations are discussed later on. The evolution of the melt pool with and without convection of fusion enthalpy shows only minor differences at the beginning of the melting of the steel alloy, in particular for 150 ppm sulfur (see Fig. 6.19). For 20 ppm sulfur (Fig. 6.18) some irregularity of the melt front becomes visible from time t = 0.5 s. These irregularities, are more visible in Fig. 6.20 when plotting only the contour of the melt pool and no velocity vectors. The irregularities start developing at the outer side of the melt pool and expand inwards at larger time, as can be seen in Fig. 6.21 at 20 ppm and Fig. 6.22 at 150 ppm. These irregularities remain during the cooling stage computed switching off the laser at time t=5 s, and reported in Fig. 6.23 for 150 ppm. It should be noticed that during cooling the liquid steel flow is reversed, as observed in [10]. As the convection of fusion enthalpy is observed to have an effect on the melt pool flow, in particular the melt front, the forthcoming cases take into account the convection of fusion

73 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTS AND DISCUSSION enthalpy.

CSF and SSF approach for the thermocapillary force The approach used for the thermocapillary force is now considered. The CSF approach was applied by Saldi [10] while the SSF approach was applied to the other forces such as the capillary and pressure force. In this study, for consistency purpose, the thermocapillary force is also implemented applying the SSF approach. Figures 6.24 and 6.25 show the melt pool and temperature field at different welding times. The plots on the left hand side are obtained with the CSF approach and the mirrored plots on the right hand side are compute with the SSF approach. It can be seen (at t=1s for instance) that the consistent SSF approach leads to a better balance of the melt pool volume deficit (when the free surface is below the dotted line) and the excendent (when the free surface is above the dotted line) compared to the case with CSF approach. An inconsistent computation of the forces applied at the free surface as done here when using the CSF approach for the thermocapillary force alone, generate spurious current due to deviations from the force balance at the free surface (see Eq. 3.3 and 3.4). If the spurious currents are large, it can affect the solution, as observed here. It is thus important to apply the SSF approach for the thermocapillary force in this study. Coarse and refined mesh The finest mesh used in [10], which here is the coarse mesh, has a non-uniform cell size distribution. Along the radial direction y the smallest cells are close to the symmetry axis and have a width of 0.035 mm. At the melt pool edge the cells are much larger with a width of about 0.1 mm in y=2.5mm. The refined mesh has cells of small width (0.03 mm) everywhere in a region that contains all the melt pool. The model with convection of latent heat and thermocapillary force implemented with the SSF approach was used to compute the solution with the refined mesh and 20 ppm sulfur. Figure 6.26 shows the computational results obtained at time t=0.5 s with the coarse mesh (upper figure) and the refined mesh (lower figure). The right hand side of Fig. 6.26 (which was mirrored) shows the velocity distribution both in steel and argon, and the velocity vectors in the melt pool. The outer contour of the melt pool (i.e. liquid steel) is plotted using the liquid volume fraction isoline fL=0.5. The left hand side of Fig. 6.26 shows the temperature distribution in both steel and argon, as well as temperature isolines (from 1800K to 2400K with an increment of 2000 K) in the melt pool. The maximum velocity in the melt pool is 0.35 m/s when computed with the coarse mesh and 0.81 m/s with the refined mesh. The maximum velocity in the gas phase reaches 0.68 m/s with the coarse mesh. A similar value was reported by Saldi [10]. The refined mesh lead to a maximum velocity almost

74 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING twice larger, with 1.17 m/s. Similarly, the maximum temperature is larger in the gas phase than in the melt pool. It reaches respectively 2592 K and 2467 K for the coarse mesh, and 2495 K and 2408 K for the refined mesh. It can be seen in Fig. 6.26 that the coarse mesh is not sufficient to capture the swirl at the melt pool edge, and the resultant deformation of the temperature isolines. The velocity in the melt pool is underestimated while the melt temperature is overestimated in the vicinity of the free surface. It should be noticed that convergence in mesh is not yet proved to be reached with the refined mesh. Anyway, it can be seen in Fig. 6.27 that the swirl plays an important role convecting the thermal energy from the free surface towards the melt front. The too weak prediction of this swirl when using the coarse mesh contributes to a deficit in thermal energy supply that, combined with fusion and solidification process, contributes to the irregularities observed at the melt front. These irregularities are almost absent in the case of the refined mesh, except close to the symmetry axis. Several possible reasons are foreseen:

convergence in mesh might not yet be reached, • the tophat laser energy distribution might be oversimpliﬁed, and a Gaus- • sian distribution might lead to different results,

the physics included in the model might be incomplete (see next section), • the numerical method might need improvement (see next section), which • might also be related to the non-desired oscillations observed with the former test cases of paragraph 6.1 and 6.2.

These issues remain to be investigated.

Free surface oscillation Free surface oscillations were observed by Saldi [10] but not dissuced. A plot of the time evolution of the free surface at the melt pool center made in [10] clearly shows free surface oscillations with an amplitude about 0.1 mm. It was also observed in [10] that the amplitude of these oscillations is significantly reduced when increassing artificially the thermal conductivity and viscosity by a factor f=2 to mimic a supposed effect of turbulence (see section 1.2). In the present study Fig. 6.28 shows zooms of the free surface in the vicinity of the symmetry axis (y=0) for time ranging from 0.49 to 0.509 s with the refined mesh. In Fig. 6.28 the mesh is also visible (the diagonals are only artefacts from the post-processor used for visualization). It shows that the oscillations take place over several cells. It also shows that the free surface oscillates with time when the mesh has melt pool cells size that do not exceed the Kolmogorov length scale. Looking at the velocity vectors in the melt pool, as in Figure 6.27,

75 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTS AND DISCUSSION it can be noticed that the oscillations of the free surface are combined with local oscillations of the velocity amplitude below the free surface. This in turn results in fluctuations in the intensity of the vortex located at the melt edge (see e.g. time 0.495, 0.496 and 0.497 s). This vortex transports hot liquid metal from the top surface to the colder bottom surface (the melt front). The intensity of this flow oscillates with time due to the above mentioned fluctuations in vortex intensity. Thus, the amount of thermal energy available at the melt front for promoting fusion varies with time. As a result, the position of the melt front oscillates at short time scale (see Fig. 6.28 at 0.495, 0.496 and 0.497 s) while moving deeper. Oscillations of the free surface are known to occur from experimental observation. However, they do not use to be documented in terms of oscillation amplitude and frequency. Melt pool flow is characterized by a small capillary 1 number, of the order of 10− (see section 2.2.2). The former test cases presented in section 6.1 and 6.2 are also characterized by small capillary numbers (indeed significantly smaller). They lead to surface oscillations that are known to be unphysical most probably due to spurious current generated by numerical errors made when computing the curvature of the free surface. According to Raeini et 2 al. [115], below a characteristic capillary number Ca=10− the spurious current becomes dominant over the physical flow. Numerical smoothing is then necessary to compute the free surface curvature with an acceptable level of numerical error so that the computed solution remains physically meaningful. The characteristic capillary number indicated by Raeini et al. [115] was established for another type of capillary flow (a stationary droplet) that differs from melt pool problem. It does not differ much from the capillary number characteristic of melt pool flow, although slightly smaller. It might thus happen that the free surface oscillations observed in this study are not only due to physics but also to numerical noise. This point needs to be further investigated.

6.3.4 Conclusion Based on the results obtained it is concluded that:

The mesh size used in former studies (e.g. [10]) was not sufﬁcient to • capture the melt swirl at the edge of the pool at 20 ppm sulfur. The check at 150 ppm remains to be made computing the solution on the reﬁned mesh.

The convection of fusion enthalpy should be included in the model since • it has a non negligable effect on the computed solution.

The thermocapillary force implemented as in [10] with the CSF approach • does not lead to the same results as when the SSF approach is applied. In

76 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING

particular, it results in a non-physical deformation of the metal (so that mass conservation is perserved).

The SSF approach is recommended for the thermocapillary force to be • consistent with the approach of interFoam, to satisfy the force balance at the free surface, to avoid generating spurious currents that could take over the physics of the ﬂow. The observed free surface oscillations, that are known to occur experimentally, need to be further investsigated to check that they are not affected by numerical noise.

Moreover, the relevance of simplifying the laser energy distribution to a • tophat might need to be questioned.

The effect of numerical smoothing when computing the surface curvature • should be investigated.

The above presented results are all with simplified modelling of the cooling. • In reality a shielding gas flow is applied as cooling. The shielding gas flow induces a pressure force on the melt pool thus affecting the melt pool shape, and its cooling effect might differ from the simplified cooling applied in the simulation. Its effect on the solution could be investigated as it might be non-negligible.

77 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTSAND DISCUSSION

Figure 6.8: Streamlines computed with isoAdvector at successive time steps

78 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING

Figure 6.9: Velocity values at different time step 79 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTSAND DISCUSSION

Figure 6.10: Comparison of solidiﬁcation front position 80 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING

Figure 6.11: Comparison of weld pool temperature (left) and velocity (right) obtainedbySaldi [10] neglectingfree surface deformation (left) or taking it into account (right) with no enhancement (f=1). laser power 3850 W, 20 ppmsulfur, 1and 5swelding time.

Figure 6.12: Comparison of weld pool temperature (left) and velocity (right) obtainedbySaldi [10] neglectingfree surface deformation (left) or taking it into account (right) with no enhancement (f=1). laser power 3850 W, 150 ppmsulfur, 1and 5swelding time.

81 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTSAND DISCUSSION

Figure 6.13: Computationaldomain

Figure 6.14: Surface tension cofﬁcient and surface tension temperature cofﬁcient (dottedline) for 20 ppmsulfur plotedfor temperature up to 2600 K.

82 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING

Figure 6.15: Surface tension cofﬁcient and surface tension temperature cofﬁcient (dotted line) for 150 ppm sulfur ploted for temperature up to 2600 K.

Figure 6.16: Calculation domain and mesh with 320 x 232 cells

83 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTSAND DISCUSSION

Figure 6.17: Residuals as function of time

84 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING

Figure 6.18: Comparison of 20 ppm without (left) and with (right) convection of fusion enthalpy

85 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTSAND DISCUSSION

Figure 6.19: Comparison of 150 ppm without (left) and with (right) convection of fusion enthalpy

86 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING

Figure 6.20: Melt pool contour obtained for 20 ppm without (left) and with (right) convection offusion enthalpy at time t=1s.

87 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTSAND DISCUSSION

Figure 6.21: Temperature contour (left) and velocity vectors (right) obtained using the CSF-approachfor 20 ppmsulfur up to time t=5s

88 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING

Figure 6.22: Temperature contour (left) and velocity vectors (right) obtained using the CSF-approachfor 150 ppmsulfur up to time t=5s.

89 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTSAND DISCUSSION

Figure 6.23: Melt pool contour and velocity vectors obtained using the CSF- approachfor 150 ppmsulfur during cooling up to time t=5.05s.

90 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING

Figure 6.24: Melt pool and temperature ﬁeld obtained for 20 ppm sulfur using CSF (left) and SSF (right) approach for time t=0.1 up to1sfor ΔT = 0.1K.

91 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTSAND DISCUSSION

Figure 6.25: Melt pool and temperature ﬁeld obtainedfor 150 ppmsulfur using CSF (left) and SSF (right) approach for time t=0.1 up to1sfor ΔT = 0.1K.

92 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING

Figure 6.26: Temperature contours and velocity vectors for VOF-SSF 20 ppm coarse mesh (top)and reﬁned mesh (bottom)

93 CHAPTER 6. NUMERICAL APPLICATIONS - RESULTSAND DISCUSSION

Figure 6.27: Reﬁned mesh 20 ppmvelocity vectors for consecutive time steps. 94 6.3. MELT FLOW WITH FREE SURFACE DEFORMATION IN CONDUCTION LASER WELDING

Figure 6.28: Free surface oscillation for VOF-SSF approachfor consecutive time steps. The dottedline is the originallocal of the sample surface.

95 Chapter 7

Conclusions

The objective of this study is to understand what physical phenomena not yet included in melt pool modelling, could prove beneﬁcial for an improved predictability of the laser welding process. In this Licentiate thesis, a numerical model for simulating conduction mode laser welding has been developed in the open source software OpenFOAM. The CSF and the SSF approach for computing the thermocapillary force and the algebraic VOF and geometrical VOF (isoAdvector) methods for capturing of the free surface deformtion have been applied to test cases focussing on separate parts of the physics. The complete model was also tested using a published case for conduction laser welding of S705 steel alloy. The results are summarize below and presented in relation to the reasearch questions.

Q1. Which numerical approach is suited for computing the thermo- • capillary force? The approach suited for computing the thermocapillary force should aim at satisfying the force balance at the interface (see Eq. 3.3, 3.4). In this study (as in [10]) the sharp surface force (SSF) approach used in interFoam for the pressure and the capillary force was kept. Consistency then imply that the thermocapillary force should also be computed with the SSF approach. The CSF approach, applied in [10], should not be retained with this framework as it would lead to numerical inconsistency that can generate spurious currents, and unphysical solution.

Q2. What is the effect of changing the numerical approach for com- • puting the thermocapillary force? When comparing computations that differ only through the approach used for implementing the thermocapillary force (CSF and SSF), the two ﬁrst test cases show very little difference in results. When considering the test case related to laser welding, differences were observed (Fig. 6.24

96 and 6.25). In particular, when implementing the thermocapillary force according to CSF, the deformation of the free surface was not consistent with the volume conservation of the incompressible liquid phase, implying a deformation of the solid phase to verify mass conservation. However, this deformation is not consistent with the physics of the model. The SSF approach did provide more consistent results.

Q3. Which approach should be applied to model the deformation of • the free surface? For the free surface deformation, in theory the isoAdvector method should be preferable to the algebraic VOF method of OpenFOAM. It is more accurate, it is claimed to run faster by its authors. This last point is important when addressing welding applications, to save CPU time. However, problems were faced to converge the computations made with the isoAd- vector (see case1) and these problems need to be solved before considering applying the method to melt ﬂow computations. It is believed that these problems could be due to numerical noise when computing the curvature of the free surface. This issue needs to be further investigated.

Q4. Which approach should be used to capture melt flow instabili- • ties? The models based on a RANS approach cannot yet be trustable as they rely on closure parameters established in contents quite different from melt pool flow. The advantage of DNS is to be independent of any closure parameters. However, a drawback is the high computational time due to the small cell size imposed by the Kolmogorov scale. Anyway, before running the weld pool case of section 6.3 with the refined mesh extended to 3 space dimensions and determining wether the melt flow is laminar or turbulent, it is first necessary to check that the oscillations already observed in 2 space dimensions are not cause by numerical noise. Besides the quantification of the instability observed at the free surface was not provided with the published test case. New test cases with more complete measurements, including the evolution in time and space of the instabilities observable at the free surface, are needed. Also experimental set up would require better descriptions such as quantitative characterisa- tion of the laser energy distribution, location and orientation of shielding system.

97 CHAPTER 7. CONCLUSIONS

Q5. What are the other model improvements that could have an • effect on the prediction of the melt front topography? The computational results obtained for the test case with melt pool show that the convection of fusion enthalpy should be included in the model, while it was neglected in former studies such as [10, 48].

Future work The irregularities of the melt front that were not expected and seem to • be related to irregularities in the fusion process should also be futher studied. The heat source energy distribution simpliﬁed to a top hat might contribute to these irregularities. A more realistic laser energy distribution could thus be of interest to investigate.

The surface tension and thermocapillary forces are temperature depen- • dent and thus affected by the cooling applied in the model. Cooling by shielding gas ﬂow applied in the experiment is thus important to consider in the numerical model. This eventually leads to a need of increased documentation of how the cooling is achieved during the experimental measurements.

The effect of smoothing when computing the free surface curvature is • foreseen as an important point to investigate.

The model can also be further developed to include the physics of keyhole, such as vaporisation. Furthermore, for complex components, as for example aerospace engine parts, structural deformations due to manufacturing, e.g welding, is an important aspect to consider during designing of the component. The problem is the computationally demanding calculations that are needed in order to take both fluid flow and structural mechanics in consideration at the same time. Thus, as a future work the model developed and implemented through this work can be used to obtain temperature fields that in turn can be used as input data for modelling of the structural deformations.

98 Part I

Appendix Laser energy sources

The precursor to the laser is the maser, Microwave Amplification by Stimulated Emission of Radiation. The first maser was built in 1953 at Columbia University. The maser and the laser are both based on the theory of stimulated emission by interaction of atoms with photons presented by Einstein in 1917. The main difference between the maser and laser is that the former uses microwaves in the wavelength range of 1 -1000 mm while a laser light is in the range of 0.1-10µm. The laser was first called an "optical maser". The next acronym used was Light Oscillation by Stimulated Emission of Radiation, loser, but this acronym was soon changed to laser [6, 33]. Laser is an acronym for Light Amplification by Stimulated Emission of Radiation. The laser light is coherent and monochromatic which differentiates it clearly from other types of light sources. Since its introduction, the laser power and efficiency have been continuously improved [6]. Fig 7.1 shows key steps in the development and use of the laser energy sources during the past 100 years. A laser light beam is generated by the stimulated emission of photons in an active medium excited to a higher energy level by an energy input. The energy input can be from an electric discharge or a flash lamp for instance. The latter method is called optical pumping and was first realized by Alfred Kastler in 1950. The optical pumping led to the first laser emission which was realised 10 years later by Theodore Maiman using a solid media. One year later the first laser using a gas media (helium-neon) was developed by Ali Javan. This was also the first continuous wave, CW, laser [6]. Lasers are classified depending on the state of the active medium (solid, liquid or gas), but also the wavelength range (visible, infrared, ultraviolet), the pumping method (optical, electrical) and the emission mode. In welding solid and gas active medium are commonly used. The mode of emission can be either continuous wave, CW, or pulsed wave, PW. The lasers applied to welding are

high power lasers (with a minimum of 1 kW power for CO2-lasers) [6]. The wavelengths met in welding applications are of the order of 10 µm and 1 µm. The 1 µm wavelength of the exiting beam can be transmitted through a fibre which makes it appropriate for robot applications. Benefits of robotised laser welding is the high efficiency/productivity and repeatability. Another benefit is improved weld quality [6, 33].

CO2-laser is the most common gas laser used in welding applications. It is a molecular gas laser, in fact its active medium is a mixture of three gases: CO2, N2 and He. The N2 enhances the pumping mechanism of the active molecule CO2. The He atoms de-excitate the lower energy level through collision energy exchanges. The stimulation is accomplished by a direct current and the beam is reﬂected by mirrors and transmitted. The emitted wavelength is approximately 10 µm and it can be operated in both CW and PW mode [33]. A drawback is

100 Figure 7.1: Historygraph of use oflaser energy sources

that a CO2 laser cannot be transmitted through a fibre which means that robot applications are complex and suffer from limitations. Robot applications are suitable if e.g. moving the workpiece rather than the laser and using an optical chain with mirrors to avoid disturbing the alignments. Solid state lasers include disk lasers, rod lasers, diode lasers and fibre lasers. In a rod laser arod shapedhost, which is the active medium, is doped with doping elements. Severaldifferent combinations ofhosts anddoping elements exist. One commonly used is an yttrium aluminium garnet (YAG) crystal as the host medium doped with neodymium (Nd3+)orytterbium (Yb3+).These are then called Nd:YAG laser and Yb:YAG respectively. The emitting wavelength ofrod lasers isaround1μm and the power capability depends on the volume of the rod. The Nd:YAG lasers were for a long period of time the only competitor [ ] to CO2-lasers until the development of fibre anddisklasers 6 .The benefit of Nd:YAG lasers is the wavelength which is 1064 nm making it suitable for welding ofhighly reflective metals. Another advantage compared toaCO2 laser is that, at this wavelength, the beam can be transported in a fibre which makes it suitable for robotised welding. As it is also less prone of generating laser-inducedplasma compared to CO2-lasers, Nd:YAG lasers are often found in high precision applications [118]. In a disk laser the active medium is still acrystaldoped with Yb or Nd,

101 but instead of a rod, its shape is a thin disk with a large cooling surface area compared to its active volume. The disk is pumped by a diode laser. Disk lasers can generate short- and ultra short pulses with high beam quality and high efficiency [6, 33]. The beam parameter product (BPP), which is a measure of the laser beam quality, is low compared to lamp pumped Nd:YAG and CO2 lasers. A low BBP indicates a small focus spot, i.e. high beam quality. The diode laser, also referred to as semiconductor laser, started to appear in materials processing rather recently. Initially semiconductor lasers were the drivers for the solid state laser market since they were used as pumping sources. Semiconductor lasers are characterized by a high efficiency at a low cost. They emit light at a wavelength of 405 -1100 nm [6]. They are found in different industrial applications ranging from fibre optic communications, laser pointers and barcode readers to laser printing. Diode lasers with improved beam quality are increasingly directly used for materials processing, and, for this reason, they are called direct diode lasers. Fiber lasers were initially developed for the telecommunication purposes in the 1960’s. The emitting wavelength is in the µm scale. They consist of a double cladding fibre (first it was a single cladding fiber with low power output) pumped with a high power source, usually a diode laser. The pumped light is reflected inside the fibre. Energy is absorbed by the core, which is designed with a higher refractive index than the rest of the fibre. The core is doped with an active medium, most commonly ytterbium (Yb3+). Several single mode fibres can be bundled together to generate multi-mode power output with a high beam quality. The main benefit with fibre lasers compared to other solid state lasers is the flexibility, compactness and lower need of cooling. The different industrial applications of fibre lasers in material processing include metal cutting, drilling, welding and additive manufacturing.

Finite volume discretization

In order to solve the governing equations, continuity equation, Eq.4.22, momentum conservation equation, Eq. 4.25, energy conservation equation, Eq. 4.38 and the volume fraction equation (Eq. 5.13), finite volume discretization is utilized. The discretization generates a set of algebraic expressions and consists of two parts. The first part is the discretization of the computational domain to generate the control volumes where the solution is sought for. The computational domain is divided into a finite number of control volumes, CV, where the computational point P is at the centre of the CV. Each face, denoted f, of the CV is shared with one neighbouring CV with centre N, S, E or W. The arrangement is shown in Fig. 7.2. The second part of the discretization is the discretization of the transport

102 Figure 7.2: CV with centre node P and neighbouring CV’s with centre N, S, E and W. equations which are solvedfor each CV and assembled on the form n n aP φp + anb φnb = Sφ, (7.1) where nb includes the neighbouring cells denoted N, S, W and E. The system of equations is transferred into a matrix system [A][φ]=[S].The coefﬁcients aP and anb are stored on the diagonal and off-diagonal respectivelyof thesparse matrix [A]. φ is a vectorfor all CV’sandSφ is the source term. Terms consisting of non-linear contributionscannot be written as convection or diffusion terms [119] and thus has to be linearised with respect to φ as done with the source term Sφ

Sφ(φ) = Su + Sp φ. (7.2)

In order to guarantee convergence iterative solvers require diagonal dominance such that | aP |> n | anb | in at least one row of the matrix. When the term SP isnegative the diagonaldominance is increased and thus convergence is increased. The discretization schemes affect the diagonal dominance, for examplethe convective term is diagonally dominate if an upwinddifferencing scheme is used. (Diagonal dominance can also be artificially created by using under-relaxation.) Each solution of thesystemof equations generates a new set of φ-values. Each term, temporal, convective anddiffusive, in the momentum equation is discretized separately using different numerical schemes. The first term of momentum equation, thetemporalderivative, capturing therateof changeof U,isdiscretized using a first order Euler scheme. OpenFOAM includes three alternative temporal schemes namely Euler, backward and Crank- Nicholson scheme. However in interFoam, Euler is theonlyapplicablescheme.

φ(t + Δt)=φ(t)+Δt∂t φ(t). (7.3)

103 n n 1 By deﬁning φ and φ − as

n φ = φ(tol d + ∆t), (7.4) and

n 1 φ − = φ(tol d ) (7.5)

The temporal derivative is then given as

n n 1 φ φ − ∂t φ = − V (7.6) ∆t where V is the cell volume. The time step size is correlated to the Courant number, which is a measure of the distance a ﬂuid particle travels during one time step compared to the cell size. It restricts the mesh size and/or the time step. In OpenFOAM the time step can be set to either a ﬁxed value or restricted by the maximum Courant number adjusting the time step such that the value of the Courant number is always smaller than the maximum value. The Courant number is calculated by

~ ~ Uf Sf Co = | · |∆t (7.7) ~ dPN Sf · ~ where Uf is the face velocity, dPN is the distance between node P and N, and ∆t the time step. Since each CV is confined by several faces the convective term is the sum of the integrals over all the faces for each cell and given by Z ~ X ~ ~ X (ρUφ)dV = S (ρU)f φf = F φf (7.8) V ∇ · f · f where f is each of the cell faces, S~ the cell face vector and F the mass flux through the cell face defined as

~ ~ F = S (ρU)f . (7.9) ·

The face value, φf is obtained from the cell centre values by an upwind differ- entiating scheme where φf takes the value of the upstream node, e.g. φf = φP for mass flux F 0 where the direction of F is the flow direction. The discretization schemes are≥ defined by the user in the system/fvSchemes dictionary. The upwind scheme guarantees boundedness and is utilized for the convective term (ρUφ) while a van Leer scheme is applied for the convection of the volume ∇

104 fraction ∇(φα).The vanLeer scheme is a secondorder bounded schemecom- monly usedfor volume fraction in multiphase solvers [120]. The diffusion terms are discretized using a linear corrected scheme with a non-orthogonal correction term to account for non-orthogonal mesh.In a non-orthogonal mesh thecell face vector Sf is not parallel to the vector d between thecell centres.Thecorrectionmaintainsthesamecontribution of φP and φE as with an orthogonal mesh and thus maintains the diagonaldominance [119].Fig. 7.3 shows the non-orthogonal mesh schematically with the vector Δ parallel to d.Inthe case of orthogonal mesh,asinthis thesis, the diffusion term

Figure 7.3: Schematic view of non-orthogonal mesh. with linearvariation of φ is given by S · (ρΓφ∇φ)f = (ρΓφ)f S · (∇φ)f (7.10) f f where

φN − φP S · ∇(φ) =| S | (7.11) f | d |

The face gradient (∇φ)f is calculated from the two points surrounding the face. In order to fulfil conservation of mass, thecalculation of the face fluxes must be consistent with thecalculation of the pressure field. In interFoam an approach with collocated variables (i.e. scalars and vectors stored at the same location) on a, so called, staggered grid arrangement, developedbyWeller [78],isemployed. It was developed in order to avoidhaving two separate meshes (as in staggered grid arrangement) but still avoiding thechecker-boardpressure field which can appear on collocated grids. Thenovelty with this method compared to original collocated arrangement, Fig. 7.4 a, is that thevelocities are obtainedfrom the corrected flux fields using a reconstruction procedure inspiredbytheRhie Chow procedure. Thus, the pressure and velocity fields are stored at the same location and thevelocity field is interpolatedfrom thecell centres to thecell faces in order

105 Figure 7.4: Collocated (a) vs. staggeredgrid (b)arrangement where arrows denote the velocities to obtain the face flux. By doing this the drawback of a collocated arrangement is avoided. The method however, requires that the surface- and volumetric forces are treated consistently. The pressure is solved at thecell centres and the pressure gradient is calculated from the face centred pressure values [119, 121]. Theindicator function and velocities has to be interpolatedfrom thecell centres to the cell faces. This is obtained by reconstructing the face volume flux. When the pressure field is known the corrected face field φf iscalculated.The face velocity Uf is then obtained by a point-to-point linear interpolation taking into account theadjacent cellsaswell as the distance between thecell centres [71].

Derivation of mass conservation equation-asD.Hill

Local continuity equation is written as: ∂ρ + ∇·(ρu)=0 (7.12) ∂ t

Multiplying equation 7.12 by phase indicator function χk :

∂ρ χ + χ ∇·(ρu)=0 (7.13) k ∂ t k

From derivation of product we get:

∂ρ ∂ ∂ χ = χ ρ − ρ χ (7.14) k ∂ t ∂ t k ∂ t k

χk ∇·(ρu) = ∇·(χk ρu) − ρu∇χk (7.15) Now rearranging equation 7.13 using equation 7.14 and 7.15:

106 ∂ χk ρ ∂ χk + (χk ρu~) = ρ + ρu~ χk (7.16) ∂ t ∇ · ∂ t · ∇ The temporal derivative of the phase indicator function is related to the motion of the interface, so

∂ χk = ν~i χk (7.17) ∂ t − · ∇ can be used to re-write equation 7.16:

∂ χk ρ + (χk ρu~) = ρnu~ i χk + ρu~ χk (7.18) ∂ t ∇ · − · · ∇ And rearranging:

∂k χk ρ + (χk ρu~) = ρ(u~ ν~i ) χk (7.19) ∇ · − ·

Now by implementing phase function αk in equation 7.19 we get:

∂k αk ρ¯k + (αk ρk u~) = ρki (u~ki ν~i ) nk Σ (7.20) ∇ · − ·

Including νi = u~ki +Sk nk ,where Sk is the surface propagation speed, in equation 9:

∂t αk ρ¯k + (αk ρk u~k ) = ρki (u~ki u~ki ) nk Σ ρki Sk nk Σ (7.21) ∇ · − · − · Then:

∂ α ρ¯ α ρ u~ ρ (7.22) t k k + ( k k k ) = ki Sk nk Σ ∇ · − · By assuimg non-reactive interfaces or incompressible phases the equation simpli- ﬁes further to:

∂k αk ρ¯k + (αk ρk u~k ) = 0 (7.23) ∇ ·

Momentum conservation and UEqn.H in OpenFOAM - as D.Hill Derivation of momentum conservation equation, local conservation of momentum

∂t ρu + (ρuu) = σ + ρg. (7.24) ∇ · ∇ ·

Multiply by phase indicator function χk

107 χk ∂t ρu + χk ρuu = χk σ + χk ρg. (7.25) ∇ ∇ · Using product rule and simplifying equation

∂t χk ρu + (χk ρuu)+σ χk = (χk σ)+ρu∂t χk +ρuu χk +χk ρg, ∇· · ∇ ∇ · · ∇ (7.26) and implementing temporal derivative of the phase indicator function, which is related to the motion of the interface, as

∂t χ k0 νi χk . (7.27) − · ∇ Then Eq. 7.26 is written as

∂t χk ρu+ (χk ρuu)+σ χk = (χk σ) ρuνi χk +ρuu χk +χk ρg. ∇· · ∇ ∇ · − · ∇ · ∇ (7.28) By rearranging equation we get

∂t χk ρu + (χk ρuu) = (χk σ)+χk ρg +ρuu χk ρuνi χk σ χk . ∇· ∇ · · ∇ − · ∇ − ∇ (7.29)

Conditioning the above equation with phase fraction αk

∂t αk ρk uk + αk ρk uk uk = αk σk +αk ρk g+ρk uk uk nk Σ ρk uk νi nk σki nk Σ, ∇· ∇· · − · · − · (7.30) which simpliﬁes to

∂k αk ρk uk + αk ρk uk uk = αk σ¯k +αk ρ¯k g ρki uki Sk nk Σ σki nk Σ. (7.31) ∇· ∇· − − By assuming incompressible phases and using Reynolds decomposition for turbulent stress the Eq. 7.31 is given by

∂k αk ρk u¯k + αk ρk u¯k u¯k = αk (σ¯k + σk ) + αk ρk g σki nk Σ. (7.32) ∇ · ∇ · − ·

Then by applying σk = - ρk uk0 uk0

∂k αk ρk u¯k + αk ρk u¯k u¯k = αk p¯k + αk (σk +τ¯k )+αk ρk g+pki nk Σ τki nk Σ. ∇· −∇· ∇· − · (7.33)

Finally by applying nk Σ = αk the momentum equation is written as ∇

∂t αk ρk u¯k + αk ρk u¯k u¯k = αk p¯k + αk τ¯k + αk σk +αk ρk g p¯k αk +pki nk Σ τki nk Σ. ∇· − ∇ ∇· ∇· − ∇ − · (7.34)

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JOSEFINE SVENUNGSSON Licentiate Thesis Production Technology 2019 No. 27

Conduction laser welding modelling of melt pool with free surface deformation Conduction laser welding Laser welding is commonly used in the automotive-, steel- and aerospace industry.

ISBN 978-91-88847-35-5 (Printed version) ISBN 978-91-88847-34-8 (Electronic version)

118526_HV_Josefine_Svenungsson_omslag_190516_v2.indd 1 2019-05-16 09:32:15