Renée El

Melhem Bat

Blaise Pascal

Insa : R Gourdon

M Stéphane

M Gérard Scorletti
email: gerard.scorletti@ec-lyon.fr

Je Remercie Chaleureusement Eric

M Amara

M Boutaous

Keywords: Two-phase incompressible ow simulation, interface capturing, level-set method, volume-of-uid method, moment-of-uid method, coupled level-set and volume-of-uid, ghost Simulation d'écoulement diphasique incompressible, suivi d'interface, méthode level-set, méthode VOF, méthode MOF, méthodes couplées level-set et VOF, ghost uid method

This thesis is devoted to the development and comparison of interface methods for incompressible two-phase ows. It focuses on the selection of robust interface capturing methods, then on the manner of their coupling with the Navier-stokes solver. The level-set method is rst investigated, in particular the inuence of the advection scheme and the reinitialization step on the accuracy of the interface capturing. It is shown that the volume constraint method for reinitialization is robust and accurate in combination with the conservative fth-order WENO schemes for the advection. It is found that interface errors increase drastically when the CFL number is very small. As a remedy, reinitializing the level-set eld less often reduces the amount of numerical diusion and non-physical interface displacement. Mass conservation is, however, not guaranteed with the level-set methods. The volume-of-uid (VOF) method is then investigated, which naturally conserves the mass of the reference uid. A geometrical consistent and conservative scheme is adopted, then an alternative technique more easily extended to 3D. It is found that both methods give very similar results. The moment-of-uid (MOF) method, which reconstructs the interface using the reference uid centroid, is found to be more accurate than the VOF methods. Dierent coupled level-set and VOF methods are then investigated, namely: CLSVOF, MCLS, VOSET and CLSMOF. It is observed that the level-set method tends to thicken thin laments, whereas the VOF and coupled methods break up thin structures in small uid particles. Finally, we coupled the level-set and volume-of-uid methods with the incompressible Navier-Stokes solver. We compared dierent manners (sharp and smoothed) of treating the interface jump conditions. It is shown that the VOF methods are more robust, and provide excellent results for almost all the performed simulations. Two level-set methods are also identied that give very good results, comparable to those obtained with the VOF methods.

Résumé

Cette thèse est consacrée au développement et à la comparaison des méthodes de suivi d'interface pour les écoulements diphasiques incompressibles. Elle s'intéresse à la sélection de méthodes robustes de suivi d'interface, puis à leur couplage avec le solveur des équations de Navier-Stokes. La méthode level-set est en premier lieu étudiée, en particulier l'inuence du schéma d'advection et de l'étape de réinitialisation sur la qualité des résultats du suivi d'interface. Il a été montré que la méthode de réinitialisation avec contrainte de volume est robuste et précise en combinaison avec des schémas conservatifs WENO d'ordre 5 pour l'advection. Il a été constaté que les erreurs du suivi d'interface augmentent de manière abrupte lorsque la condition CFL est trop petite. Comme remède, la réinitialisation du champ level-set eectuée moins souvent réduit la diusion numérique et le déplacement non-physique de l'interface. La conservation de la masse n'est pas assurée avec les méthodes level-set. Les méthodes VOF (volume-of-uid) qui conservent naturellement la masse du uide de référence sont alors étudiées. Une résolution géométrique avec un schéma consistent et conservatif est alors adoptée, ainsi qu'une autre technique alternative plus aisément extensible en 3D. Il a été trouvé que ces deux dernières méthodes donnent des résultats très proches. La méthode MOF (moment-of-uid), qui reconstruit l'interface en utilisant le centre de masse du uide de référence, est plus précise que les méthodes VOF. Diérentes méthodes couplées entre level-set et VOF sont alors étudiées, notamment: CLSVOF, MCLS, VOSET et CLSMOF. Il a été observé que la méthode level-set tend à épaissir les laments minces, tandis que VOF et les méthodes couplées les fragmentent en petites particules. Finalement, on a couplé les méthodes level-set et VOF avec le solveur incompressible des équations de Navier-Stokes.

On a comparé diérentes manières de prise en compte des conditions de saut à l'interface (lisse et raide). Il a été montré que les méthodes VOF sont plus robustes, et donnent d'excellents résultats pour quasiment toutes les simulations. Deux méthodes level-set donnant de très bons résultats, comparables à ceux de VOF, sont aussi identiées.

Résumé étendu Introduction

Les écoulements diphasiques concernent les écoulements avec deux uides non-miscibles. Ils font intervenir diérentes phases et ils sont souvent caractérisés par des rapports de densité et de viscosité élevés. Ces écoulements sont omniprésents dans la nature et dans les procédés industriels. Les vagues marines et les rivières sont des exemples concrets qui mettent en jeu l'écoulement de l'eau en interaction avec une surface libre. Dans les procédés industriels, ces écoulements sont beaucoup plus complexes. Les écoulements à bulles sont souvent rencontrés dans les échangeurs de chaleur. Les jets de combustible dans les moteurs à combustion sont atomisés an de créer un mélange parfait d'air et de combustible. Cela représente une liste non-exhaustive des écoulements diphasiques. Des phénomènes de changement de phase peuvent avoir lieu dans ces écoulements sous l'eet d'un échange d'énergie. L'étude expérimentale de ces phénomènes est généralement compliquée et donne seulement des corrélations valides dans certaines conditions. La simulation numérique apparait comme une bonne alternative -moins coûteuse -si les outils numériques sont testés, validés et issus de méthodes robustes.

La modélisation de ces écoulements nécessite l'utilisation d'une méthode able permettant de localiser avec précision l'interface (frontière physique) qui sépare les deux uides. Diérentes méthodes de suivi d'interface sont proposées dans la littérature. Elles permettent une approche à un uide lorsqu'elles sont couplées avec le solveur des équations de Navier-Stokes où un seul système d'équations est résolu (les deux uides sont considérés comme étant un seul uide avec des propriétés physiques variables). La simulation numérique des écoulements diphasiques dépend donc fortement de la capacité de ces méthodes à localiser l'interface.

L'objectif à long terme de ce travail est l'étude numérique des écoulements diphasiques avec changement de phase qui ont déjà fait l'objet de nombreuses études expérimentales au laboratoire. Dans une cellule mixte qui contient les deux uides, la fraction a une valeur comprise entre 0 et 1. Le transport de ces fractions est régit par une équation d'advection similaire à celle de levelset. Cependant, les méthodes de résolutions ne sont pas les mêmes. La résolution de l'équation d'advection de VOF est souvent réalisée par des méthodes géométriques.

La position de l'interface n'est pas connue avec précision, et dépend de l'algorithme utilisé pour sa reconstruction. La majorité de ces algorithmes de reconstruction de l'interface prédisent une interface discontinue sur la face entre deux cellules voisines. Dans le cadre de ce travail, on utilise la méthode ELVIRA (Ecient Least-squares Vof Interface Reconstruction Algorithm).

Dans une cellule mixte, l'interface est reconstruite en se basant sur les informations dans un bloc de 3 × 3 cellules autour de la cellule en question.

On résout l'équation d'advection géométriquement, direction par direction, tout en inversant la direction par laquelle on commence après chaque pas de temps. En deux dimensions par exemple, on commence au premier pas de temps par la direction x puis y, et dans le second pas de temps par la direction y puis x et ainsi de suite. Cette inversion de directions (de commencement dans chaque deux pas de temps successifs) garantie un ordre 2 de la discrétisation en temps.

On a décrit et implémenté un algorithme consistant (qui permet de garder les fractions C entre 0 et 1) et conservatif (qui conserve la masse totale,

C n = C n+1 ) similaire à celui de Scardovelli and Zaleski (2003) et [START_REF] Tryggvason | Direct numerical simulations of gas-liquid multiphase ows[END_REF]. La complexité de sa mise en ÷uvre vient du fait que cette advection utilise un mapping Lagrangien implicite dans la première direction et un mapping Lagrangien explicite suivant l'autre direction en deux dimensions. Ses extensions actuelles à 3D ne permettent malheureusement pas de satisfaire les conditions de consistance et de conservation de masse. En revanche, l'algorithme alternatif développé par [START_REF] Weymouth | Conservative volume-of-uid method for free-surface simulations on cartesian-grids[END_REF] (que l'on note VOF-WY) est consistant et conservatif, et est aisément extensible en 3D. La version 2D des deux algorithmes a été intégrée dans notre code. [START_REF] Dyadechko | Moment-of-uid interface reconstruction[END_REF], 2006, 2008) ont proposé la méthode MOF (moment-of-uid ) qui permet de reconstruire l'interface seulement en se basant sur les informations connues sur la cellule en question, à savoir le centre de masse du uide de référence et sa fraction. Il est à noter que la position des centres de masse doit être recalculée et mise à jour après chaque advection, étape qu'on réalise suivant une manière similaire à celle pour le transport des fractions.

On a présenté les algorithmes de chacune des méthodes, ainsi que ceux de quelques fonctions utiles pour l'élaboration de ces méthodes en 2D. La comparaison entre ces méthodes (VOF, VOF-WY et MOF) a montré qu'elles sont robustes et moins sensibles à la condition CFL que level-set. La méthode VOF [START_REF] Weymouth | Conservative volume-of-uid method for free-surface simulations on cartesian-grids[END_REF] est identique à celle de Scardovelli and Zaleski (2003) dans le cas du cercle de Zalesak du moment que le champ de vitesse imposé ne fait subir aux cellules ni compression ni expansion. Dans le cas du serpentin et des laments minces, à l'inverse de level-set qui les rend plus épais, les méthodes VOF ne gardent pas une interface continue mais causent la fragmentation de ces laments. La méthode MOF au contraire, du fait qu'elle n'utilise que les informations de la cellule en question pour reconstruire l'interface, garde une structure continue des laments. Cependant, on a remarqué quelques problèmes de stabilité numérique (détachement non-physique de petites particules uides) de la méthode MOF lorsque le CFL imposé est trop petit. Ces problèmes pourront probablement être résolus dans une future implémentation plus soignée de MOF.

Chapitre 3 : Méthodes couplées level-set et VOF (volume-of-uid)

Les avantages et inconvénients de chacune des méthodes level-set et VOF sont bien connus. [START_REF] Bourlioux | A coupled level-set volume-of-uid algorithm for tracking material interfaces[END_REF] a eu l'idée de coupler level-set avec VOF an de rassembler dans une même xi méthode les avantages de chacune, à savoir la conservation de masse de VOF et la bonne représentation de l'interface de level-set. [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF] a concrétisé ce couplage en publiant le premier article, suivi par un autre article de [START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF]. La méthode a été appelée CLSVOF (Coupled level-set and volume-of-uid ). Dans cette méthode, il y a interaction entre les champs de level-set et des fractions en plusieurs étapes. D'une part, l'advection de VOF nécessite la connaissance de la normale à l'interface qui est déterminée par le biais de level-set. D'autre part, le champ de level-set doit conserver la masse, et ainsi il est réinitialisé par level-set et VOF. Dans ce travail, on a distingué deux versions de CLSVOF : CLSVOF-Sussman [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF] et CLSVOF-Son [START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF]. Deux étapes sont réalisées diéremment : (1) l'advection de level-set et (2) la méthode de détermination de la normale à l'interface de VOF.

Quelques années plus tard, van Sun and Tao (2010) ont proposé le couplage appelé VOSET (Coupled volume-of-uid and level-set). Dans ce dernier, il s'agit simplement de la détermination de la fonction de distance signée à partir de VOF via un processus géométrique itératif. Il est à noter que pour ce couplage, le champ level-set n'est pas transporté mais déduit à partir des fractions. La précision de VOSET dépend alors fortement de celle de l'algorithme utilisé pour la détermination du champ level-set.

Pour cette méthode, on a considéré diérentes variantes, chacune avec un nombre d'itérations géométriques donné.

Enn, après l'apparition de la méthode MOF, elle a été couplée avec level-set par [START_REF] Jemison | A coupled level set-moment of uid method for incompressible two-phase ows[END_REF] dans une méthode appelée CLSMOF (Coupled level-set and moment-of-uid ). Cette dernière propose d'utiliser soit la reconstruction de MOF, soit celle de CLSVOF. Dans son principe, ce couplage est très semblable à celui de la méthode CLSVOF-Sussman.

Les fractions volumiques dans ces diérents couplages sont advectées avec un schéma consistant et conservatif des fractions volumiques. Ceci nous a permis de ne pas utiliser des algorithmes Chapitre 4 : Méthodes numériques pour les écoulements diphasiques incompressibles Dans les chapitres précédents, les méthodes de suivi d'interface développées sont appliquées à des cas tests académiques avec un champ de vitesse imposé (connu). Lorsque ces méthodes sont couplées avec un solveur des équations de Navier-Stokes, elles permettent une approche à un uide (les deux uides sont considérés comme un seul uide avec des propriétés variables).

Dans les simulations d'écoulements diphasiques, c'est par le biais du solveur des équations de Navier-Stokes que le champ de vitesse est déterminé et sert ensuite à transporter l'interface. Les propriétés du uide sont déterminées à partir de la position de l'interface prédite par la méthode de suivi d'interface. Les conditions sur l'interface sont ainsi aisément appliquées du moment que la position de l'interface est connue.

Diérentes manières de détermination des propriétés du uide sont proposées dans la littérature, certaines considèrent une variation lisse au passage à l'interface de la propriété du uide 1 et celle du uide 2. D'autres proposent un saut brusque de ces propriétés. La manière de déterminer la courbure de l'interface aussi varie d'une méthode à l'autre.

On a utilisé un maillage décalé (MAC, Marker-And-Cell ) où les scalaires (pression, propriétés xiii du uide, level-set, VOF) sont déterminés au centre de la cellule, et les vitesses sont au centre des faces. La discrétisation temporelle des équations de Navier-Stokes est réalisée par le schéma prédicteur-correcteur Crank-Nicolson. Les termes visqueux sont traités de manière implicite an de permettre d'utiliser des pas temps plus élevés. La méthode de projection non-incrémentale est utilisée et les termes convectifs sont discrétisés par le schéma ENO d'ordre 2. Nous avons alors gardé cette structure identique, et utilisé diérentes approches pour : le suivi de l'interface, la dénition des propriétés du uide et la prise en compte de la tension de surface.

On a ensuite construit 6 solveurs basés sur level-set et 4 sur VOF. Les solveurs basés sur level-set dièrent principalement dans la manière suivant laquelle les conditions de l'interface sont traitées, à savoir :

• Variation lisse de la pression (CSF) et des propriétés physiques à travers l'interface et un traitement lisse de la force de tension de surface en considérant une interface épaisse. Le solveur LS-CSF utilise cette approche avec une courbure donnée par [START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF].

• Saut raide de la pression (GFM) et des propriétés physiques à travers l'interface, et un traitement raide de la tension de surface en utilisant la méthode GFM. Le solveur LS-GFM1 utilise cette approche et est similaire au travail de [START_REF] Son | A level set method for analysis of lm boiling on an immersed solid surface[END_REF].

• Saut raide de pression (GFM) et de la densité, et une viscosité lissée à travers l'interface.

Deux solveurs utilisent cette approche : LS-GFM (avec une courbure donnée par [START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF]) et LS-GFMk (avec une courbure calculée selon [START_REF] Marchandise | A stabilized nite element method using a discontinuous level set approach for the computation of bubble dynamics[END_REF])

• Saut raide de la pression (GFM) à travers l'interface avec des propriétés physiques localement moyennées. Les solveurs LS-LA1 (avec une courbure donnée par Vigneaux ( 2008)) et LS-LA2 (où la courbure est calculée selon Sussman et al. (1999)) utilisent cette approche.

On a aussi construit 4 diérentes méthodes VOF (dont une basée sur MOF) avec des propriétés pondérées et déterminées à partir de la fraction volumique. Deux méthodes de calcul de la courbure ont été utilisées : la fonction de hauteur (Height Function) de Popinet (2009), notée HF1, et celle de Hernández et al. (2008) notée HF2. De plus, nous avons implémenté un solveur des équations de Navier-Stokes basé sur un maillage centré, similaire à celui du code de calcul Basilisk (Popinet et al.), noté VOF-WYC, an de pouvoir déterminer si la disposition des variables sur le maillage joue un rôle important sur la qualité des résultats. Il est à noter que le suivi d'interface de cette dernière méthode est réalisé avec la méthode VOF [START_REF] Weymouth | Conservative volume-of-uid method for free-surface simulations on cartesian-grids[END_REF].

La confrontation de ces diérentes méthodes est réalisée sur diverses simulations : les courants parasites (se développant dans une bulle stationnaire en l'absence de gravité), bulle qui oscille dans un champ de gravité nul, onde capillaire stationnaire, instabilité de Rayleigh-Taylor et la bulle ascendante dans un uide au repos.

Les résultats obtenus ont montré que les solveurs diphasiques basés sur les méthodes VOF sont beaucoup plus stables et robustes à l'exception du solveur qui utilise VOF avec la fonction de hauteur de Hernández et al. (2008) pour l'estimation de la courbure de l'interface. Les solveurs basés sur level-set sont plus ou moins sensibles au cas traité. On a identié deux solveurs basés sur level-set (LS-LA2 et LS-GFM ), qui donnent approximativement de bons résultats pour toutes les congurations. On a également noté que le solveur basé sur la méthode MOF est beaucoup plus sensible et donne de mauvais résultats lorsque le pas de temps (qui est xé par la physique qui régit la simulation) est petit.

On présente les résultats de deux diérentes simulations : l'onde capillaire stationnaire et la bulle ascendante. Dans la première, une interface sinusoïdale sépare deux uides 1 et 2 de densité et viscosité uniformes. La gure 2 trace l'évolution de la hauteur de l'interface pour xiv LS-GFMk LS-GFM1 LS-GFM Le deuxième exemple concerne l'ascension d'une bulle initialement au repos (benchmark 2, [START_REF] Hysing | Quantitative benchmark computations of two-dimensional bubble dynamics[END_REF]). L'interface au temps t=3 est présentée sur la gure 3 pour 6 diérentes méthodes. On voit que la forme de la bulle dière d'une méthode à une autre, et c'est ce qui est aussi observé par [START_REF] Hysing | Quantitative benchmark computations of two-dimensional bubble dynamics[END_REF]. Pour ce dernier cas, aucune solution analytique n'est disponible, mais nous avons utilisé le résultat de référence de (Featow). xvi

Conclusion

Dans ce travail, on a pu identier une méthode level-set qui est stable et donne de bons résultats sur les cas tests académiques avec un champ de vitesse imposé en 2D cartésiennes. Nous l'avons comparée avec les méthodes VOF, MOF et les méthodes couplées.

La méthode MOF et la méthode couplée CLSMOF ont donné d'excellents résultats pour les problèmes avec champ de vitesse imposé. Cependant, de petites particules uides se détachent de l'interface lorsque la méthode MOF est utilisée pour suivre l'interface en champ de vitesse donné pour des CFL petits. Ces problèmes ont aecté la stabilité du solveur diphasique dans lequel la méthode MOF est intégrée, ce qui a fortement impacté les résultats. L'utilisation de la méthode CLSMOF à la place de MOF dans ce solveur garantira ainsi de bons résultats vu que dans cette approche, level-set corrige le champ des fractions volumiques et supprime les particules uides se détachant du uide de référence. Ceci n'est néanmoins valable que si la prise en compte de l'interface dans le solveur des équations de Navier-Stokes est able et précise.

Les travaux futurs incluront la validation du solveur de Navier-Stokes avec les méthodes couplées en 2D, puis par la validation du solveur axisymétrique avec les méthodes VOF et levelset. Ceci va permettre d'étudier, par exemple, les bulles, les gouttes et les écoulements dans des canaux circulaires. L'extension du code de calcul en 3D ouvrira de nouvelles perspectives et permettra de traiter une large gamme de congurations.

Inclure l'équation de l'énergie et les conditions appropriées (pour la prise en compte de la variation de masse du uide de référence due au changement de phase) va permettre de simuler la croissance d'une bulle lors du changement de phase. D'autres phénomènes tels que l'ébullition en lm peuvent être ainsi prédits sur des surfaces plates. La prise en compte du changement de phase sur la surface irrégulière d'un solide nécessitera l'application de bonnes conditions aux limites de vitesse (à l'aide de la méthode de la frontière immergée par exemple).

Les simulations numériques en 3D sont très couteuses, surtout si les calculs sont séquentiels (ce qui est le cas du code développé actuellement). Deux techniques largement utilisées permettent de réduire signicativement le coût de calcul : la première consiste à paralléliser le code via MPI; la seconde consiste à utiliser un maillage adaptatif avec ranement local. Dans ce contexte, l'utilisation des codes open-source tels que Gerris (Popinet) et Basilisk (Popinet et al.) pourrait être une voie intéressante. Ceci permettra d'aborder des problèmes réels et aidera à la compréhension des phénomènes physiques liés au changement de phase.

xvii

General introduction

Two-phase ows concerning ows with two immiscible uids are omnipresent in nature and in industrial processes. These ows often involve uids with dierent phases (liquid-gas) characterized by high density and viscosity ratios. Sea waves and rivers are concrete examples involving water ow in interaction with a free surface. In the industrial processes and engineering applications, these ows are much more complicated and manifest under dierent forms. Bubbly ows, for example, are often encountered in heat exchangers. Inkjet printers are jetting devices in which ink is spread through a nozzle. This latter is very similar to what happens in combustion engines, except that the liquid jet is atomized in order to ensure perfect fuel-air mixing. This is a non-exhaustive list of examples. Phase change phenomena can occur under the inuence of energy exchange. The experimental study of these phenomena is often fastidious and provides only correlations valid in some conditions. Numerical simulation seems to be a good alternative if the numerical tools are tested, validated and based on robust methods.

The immiscible nature of the involved uids implies that they are separated by an interface.

Many numerical methods devoted to follow the evolution of an arbitrary interface advected by the uid ow are proposed in literature, called interface capturing/tracking methods. These methods allow a one uid approach when coupled with the Navier-Stokes solver, i.e. only one set of equations is solved with variable properties (density, viscosity, ...). The numerical simulation of two-phase ows therefore depends strongly on the ability of the interface method to localize the interface.

The interface methods belong to one of these two categories: Lagrangian or Eulerian techniques. Lagrangian methods consist of seeding the interface with (massless) markers [START_REF] Unverdi | A front-tracking method for viscous, incompressible, multiuid ows[END_REF], which are then advected with the local velocity. In this case we talk about interface tracking or front-tracking. On the other hand, Eulerian methods are based on the resolution of a scalar transport on a xed Eulerian mesh and in this case we talk about interface capturing or front-capturing. The Lagrangian methods are generally more accurate than the Eulerian methods. However, many diculties arise especially if the interface separating the two uids undergoes major topological changes (coalescence and/or breakup). Sometimes, the interface is deformed to such a point that it is compulsory to add more markers in some regions in order to guarantee the accuracy of interface transport. On the other hand, the Eulerian methods naturally take into account the collapse and coalescence of the interface.

In this work, we focused on Eulerian techniques, in particular the level-set and the volume-ofuid methods. In the level-set (LS) approach, the scalar is the algebraic distance to the interface.

This scalar is the volume fraction of a reference uid (liquid or gas) inside each computational cell in the volume-of-uid (VOF) approach. The level set method permits to access easily to the geometrical properties of the interface, while the mass of the reference uid is not fully conserved.

From the other hand, the volume-of-uid method conserves the mass but sometimes small bodies General introduction detach from the reference uid (often called otsam and jetsam).

The ultimate objective of this work is to study two-phase ows with phase change. This thesis focuses on the study and comparison of interface capturing methods for two-phase ows, which is split it into two steps. The rst one aims to select robust methods able to accurately predict interface transport in imposed velocity eld. The second one deals with the investigation of the dierent manners of coupling the interface methods with the incompressible Navier-Stokes solver.

Many dierent implementations of the level-set method are proposed in the literature. We recall them in Chapter 1, and compare them in accurately transporting interfaces in xed velocity elds. The second chapter is devoted to the volume-of-uid techniques, where a literature review is presented. The moment-of-uid method (MOF), which in addition to the fraction uses the centroid information is also detailed. Finally, similar test cases to those of the rst chapter are performed to evaluate the performances of each method.

Since the level-set method suers from mass loss/gain, the volume-of-uid method from the interface representation, [START_REF] Bourlioux | A coupled level-set volume-of-uid algorithm for tracking material interfaces[END_REF] had the idea to couple both methods in one method.

Coupled techniques became widely used after the pioneering work of [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF] then [START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF]. Later, many methods are proposed in the literature such as the mass conserving level-set (MCLS) method, the coupled volume-of-uid and level-set (VOSET) method and coupled level set and moment-of-uid (CLSMOF) method. These coupling techniques are presented in Chapter 3, and dierences between them are highlighted. A comparison is performed with the standalone methods of Chapters 1 and 2 in order to show the benets of the coupling.

Chapter 4 deals with the two-phase ow solver. It is based on the incompressible Navier-Stokes equations coupled with the interface method. The uids are considered Newtonian and immiscible. Dierent manners of including the eect of the dierent physical properties and phenomena such as viscosity and surface tension are available in the literature, and the most popular formulations are implemented in our solver. In total, we have tested 11 methods: 4 are coupled with the volume-of-uid method, 1 with the moment-of-uid method and 6 with the level-set method. We compared the dierent implementations in accurately reproducing well known test cases in two-dimensional cartesian grids.

Chapter 1

Level-set method

Introduction

In the level-set approach, an interface is dened by the zero level-set of a continuous function.

This function should change its sign when crossing the interface, i.e. it should be positive (or negative) inside the domain containing the reference uid and negative (or positive) outside it.

Although an innity of functions satisfy the previous condition, the algebraic distance function is generally selected for its regularity and other interesting properties. The signed-distance function allows us for example to dene an innity of iso-levels, each one corresponding to a constant distance from the interface.

Given an interface dened by a level-set function φ under velocity eld u = (u, v) (or u = (u, v, w) in 3D), the evolution of the interface Γ is governed by the advection equation:

∂φ ∂t + u • ∇φ = 0 (1.1)
The level-set function permits the easy access to interface geometrical properties such as the normal to the interface n = ∇φ |∇φ| and the curvature κ = ∇ • n

The geometrical properties are easily obtained from the level-set. However, the precision in determining these geometrical properties depends on the accuracy of localizing the zero level-set.

As any other numerical method, the level-set suers from drawbacks. The main disadvantage of the level-set is that the numerical methods that solve Eq. (1.1) do not conserve exactly the mass of the reference uid. On the contrary, it automatically handles the topological changes that an interface can undergo such as the coalescence and/or the break-up.

In 1.2, we will see how to advect an arbitrary interface Γ dened by the zero level-set. The velocity eld is often non-uniform, which disturbs the signed-distance property and introduces irregularities in the level-set function when only Eq. (1.1) is solved. The remedy to this is to solve in regular intervals a supplementary equation called the reinitialization equation, which reset φ as the signed-distance function without moving the zero level-set. This step will be detailed in 1.3. Solving the advection and reinitialization equations on the entire domain is numerically expensive, we present therefore in 1.4 a method to solve these equations on a narrow band around the interface. Finally, in 1.5 and 1.6 the dierent numerical methods for advection and 1. Level-set method reinitialization are evaluated and compared.

In fact, we choose the staggered (MAC) grid in which the scalars (the level-set value, φ) are stored at the cell centers and the velocity vectors at the center of cell-faces as shown on Fig. 1.1.

This grid conguration oers simplicity and easiness of using high-order schemes.

i -

1 i i + 1 i -1 2 i + 1 2 j -1 j j + 1 j -1 2 j + 1

Numerical resolution of the advection equation 1.2.1 Time discretization

The third-order Runge-Kutta TVD method [START_REF] Shu | Ecient implementation of essentially non-oscillatory shock-capturing schemes, ii[END_REF]) is a popular choice in combination with ENO and WENO schemes, we use it in the current work. Let φ n be the solution of Eq. (1.1) after n time steps. The solution at the next step is computed from:

φ (1) = φ n + ∆tL(φ) (1.2) φ (2) = 3 4 φ n + 1 4 φ (1) + 1 4 ∆tL(φ (1) ) (1.3) φ n+1 = 1 3 φ n + 2 3 φ (2) + 2 3 ∆tL(φ (2) ) (1.4)
where φ (1) and φ (2) are intermediate values, φ n+1 is the solution at time t n+1 and L is the spatial discretization of the term (-u • ∇φ) in Eq. (1.1) which is described in the next sections.

Spatial discretization

The level-set advection equation can be written in the form of Eq. (1.1), which is non-conservative since its resolution is based on point values. In 2D, the discrete form looks like:

∂φ i,j ∂t + u i,j ∂φ ∂x i,j + v i,j ∂φ ∂y i,j = 0

(1.5)

where () i,j denotes a quantity dened at the grid point (x i , y j ).

For incompressible uids, a conservative formulation can be derived by re-writing Eq. (1.1) in a nite-volume sense as follows:

∂φ ∂t + ∇ • ( uφ) = 0

(1.6)

Numerical resolution of the advection equation

We can solve this equation directly in a nite-dierence method, or in the weak sense using a nite-volume method. The solution φi,j in a nite-volume sense is the integral value of the function on the cell of area Ω i,j . The Gauss-Ostrogradsky theorem allows the transformation of the integral of the second term into a surface integral representing the net ux of the conserved quantity across cell faces:

Ω i,j ∂φ ∂t dΩ + Ω i,j ∇ • ( uφ)dΩ = Ω i,j d dt φ i,j + ∂Ω i,j ( uφ)
• n dS = 0

(1.7)

where Ω i,j is the volume of the grid cell and ∂Ω i,j its boundary. In the 2D cartesian grid setting, the discretization of this equation can be written as:

d dt φ i,j + F i+ 1 2 ,j -F i-1 2 ,j ∆x + G i,j+ 1 2 -G i,j-1 2 ∆y = 0
where F are the numerical uxes on the left and right boundaries of Ω i,j , and G on the upper and lower boundaries. For example, the upper and right uxes are dened as:

         F i+ 1 2 ,j = y i,j+ 1 2 y i,j-1 2 u(x i+ 1 2 , y)φ(x i+ 1 2 , y) dy G i,j+ 1 2 = x i+ 1 2 ,j x i-1 2 ,j v(x, y j+ 1 2 )φ(x, y j+ 1 2 ) dx (1.8)
The advection equation (1.1) is a Hamilton-Jacobi type dierential equation. Centered nitedierence schemes are not suitable for Hamilton-Jacobi equations because they do not capture information from the correct direction of propagation, and tend to introduce oscillations in the solution. In the next section, we will explain the Essentially Non-Oscillatory (ENO) and Weighted ENO (WENO) schemes for the level-set advection equation (1.1). Both are commonly used in level-set solvers. ENO schemes were rst developed to treat shock waves and discontinuities by [START_REF] Harten | Uniformly high order accurate essentially non-oscillatory schemes, iii[END_REF], and have been extended to Hamilton-Jacobi equations later in [START_REF] Osher | Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations[END_REF] and [START_REF] Osher | High order essentially non-oscillatory schemes for Hamilton-Jacobi equations[END_REF].

The schemes we are going to detail hereafter will give approximations to the derivative φ x = ∂φ/∂x which is computed on a stencil biased to the left or the right, noted as φ -

x and φ +

x , respectively. The choice of which estimation to use depends on the direction of propagation of the characteristic functions of the Hamilton-Jacobi equation. In the case of the level-set advection equation, the characteristic speed is simply the velocity u. In the reinitialization equation, the expressions for the direction of propagation are slightly more complex. Once the direction is known, upwinding consists in selecting the estimation for φ x which is based on a stencil biased in the opposite direction of the characteristics (in the direction where the information comes from).

For the advection equation, we have:

∂φ ∂x = φ x = φ - x if u > 0 φ + x if u < 0 (1.9)
There are other ways to estimate φ x from φ - x and φ +

x , which are not used in this work however.

The Lax-Friedrichs schemes use the average between the left and the right derivatives and adds some numerical viscosity. The Roe-Fix scheme chooses between the left and the right derivatives by an upwinding based method and whilst taking into account coecients for entropy-x (for the Navier-Stokes equations). Neglecting the entropy-x coecients, the Roe-Fix algorithm is reduced also to upwinding. The Godunov's scheme yields the same choice procedure in the 1. Level-set method particular case of the level-set advection. For more information on these schemes applied to the level-set equations, see [START_REF] Osher | Level set methods and dynamic implicit surfaces[END_REF].

The level-set is treated in a direction-by-direction fashion (as in the method of lines), which means that the above scheme is applied in the same manner to each spatial derivative in Eq. (1.5).

Only the expressions for the x-direction are shown in the following, for the sake of brevity, and we drop the subscript j in order to alleviate notations.

1.2.2.1 First-order upwind scheme

It gives a rst-order approximation of the term φ x . The derivatives at the grid point i based on the left-and right-biased stencils are given by:

(φ - x ) i = φ i -φ i-1 ∆x (1.10) (φ + x ) i = φ i+1 -φ i ∆x (1.11) 1.2.2.2 Hamilton-Jacobi ENO schemes
First, let us introduce the general kth-order ENO scheme, for k = 1, 2 or 3. The values of φ -

x and φ +

x are calculated using the ENO scheme proposed by Osher and Sethian (1988) then generalized by [START_REF] Osher | High order essentially non-oscillatory schemes for Hamilton-Jacobi equations[END_REF], which proceeds as follows:

Step 1. We rst compute the so-called undivided dierences of degree j ≤ k, noted V j i . Starting with the value φ i on a grid cell i, the undivided dierence of degree zero is given by

V 0 i ≡ φ i (1.12)
and the undivided dierences of degree j ≥ 1 are subsequently computed from the recurrent relation

V j i ≡ V j-1 i+1 -V j-1 i (1.13)
The j-th undivided dierence may be interpreted as an approximation of the j-th derivative such that V j i (∆x) j (∂ j φ/∂x j ), based on a stencil of j + 1 points with x i as the left-most point.

Step 2. For a given grid point i, the selection procedure for the ENO stencil starts with the rst-order polynomial approximation of φ based on the values of φ on the two-point stencil [x i , x i+1 ]:

P 1 (x) = V 0 i + V 1 i (x -x i )/∆x (1.14)
An approximation of φ x is then calculated from

(φ + x ) i ∂P 1 ∂x = 1 ∆x V 1 i = φ i+1 -φ i ∆x (1.15)
Step 3. The order of accuracy is then progressively increased by adding one data point on the left or right side of the stencil. Let us denote k the order of the new polynomial function P k (x) and m the rst (left-most) grid point of the new stencil. We started the procedure in Eq. (1.14) with m = i and k = 1. The choice of the new stencil is based on the minimization of the absolute value of the undivided dierence that will be added to the approximation:

k = k + 1 if |V k m | > |V k m-1 | then m = m -1 1.2.

Numerical resolution of the advection equation

In this way, the approximation will always be based on the stencil containing the smoothest data set.

Step 4. At the level k, the stencil contains k + 1 grid points [x m , ..., x m+k ] on which a unique polynomial P k (x) of order k can be tted, from which a new approximation (φ + x ) i = (∂P k /∂x) i of order k can be computed. Several expressions for (φ +

x ) i can be found in the literature. Since P k (x) is unique, they all lead to the same results. The formulation used in this work is based on a linear combination of the rst-degree undivided dierences [START_REF] Shu | Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws[END_REF]:

(φ + x ) i = 1 ∆x k-1 j=0 C r-1,j V 1 i-r+j (1.16) (φ - x ) i+1 = 1 ∆x k-1 j=0 C rj V 1 i-r+j
(1.17) where r = mi and the coecients C rj are given by

C rj = k m=j+1 k l=0 l =m k q=0 q =m,l (r -q + 1) m+k-1 l=m l =s (m -l) (1.18)
An expression for C rj for a non-uniform grid, based on the rst-degree divided dierences, can be found in [START_REF] Shu | Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws[END_REF].

Note that, even though the higher-degree undivided dierences do not appear in Eqs. (1.16) and (1.17), they are still needed in the ENO selection procedure. The order of accuracy is reduced near the boundary points (i = 1 and i = N ) to avoid the use of non-existing grid points:

k = min(k max , min(i, N -i))
where k max is the order of accuracy for inner grid points and N is the number of cells. The quantities (φ -

x ) 1 and (φ + x ) N are not computed and set to zero. This will generally provide a smooth treatment of the level-set near the boundary and avoid generation of unwanted oscilla-

Hamilton-Jacobi WENO schemes

The Weighted ENO schemes are due to [START_REF] Liu | Weighted essentially non-oscillatory schemes[END_REF] then improved by [START_REF] Jiang | Ecient implementation of weighted eno schemes[END_REF]. It has been later extended to Hamilton-Jacobi equations by [START_REF] Jiang | Weighted ENO schemes for Hamilton-Jacobi equations[END_REF]. The procedure starts with the computation of the divided dierences, although only those of degree one are required here:

V 1 i = φ i+1 -φ i ∆x (1.19)
The derivatives φ x are evaluated in a cell-by-cell manner. For the i-th cell, the WENO approximation uses the symmetric stencil [x i-(k-1) , ..., x i+(k-1) ] where k = 2 for the third-order scheme 1. Level-set method and k = 3 for the fth-order scheme. The left-and right approximations of φ x are computed from a linear combination of a set of kth-order polynomial approximations:

(φ + x ) i = k r=1 ω + r (φ + x ) r i (1.20) (φ - x ) i+1 = k r=1 ω - r (φ - x ) r i (1.21)
where (φ ± x ) r i for r = 0...k -1 are the kth-order polynomial approximations, based on dierent stencils, and ω ± r are the weights which provide a nal order of accuracy of 2k -1 for smoothly varying functions. For k = 2, the polynomials are:

(φ + x ) 1 i = - 1 2 V 1 i+1 + 3 2 V 1 i with stencil [x i , ..., x i+2 ] (φ + x ) 2 i = 1 2 V 1 i + 1 2 V 1 i-1 with stencil [x i-1 , ..., x i+1 ]
We only provide the expressions for (φ + x ) i . Those for (φ - x ) i can be obtained from symmetry considerations around the grid point i. For k = 3, the polynomials are:

(φ + x ) 1 i = 1 3 V 1 i+2 - 7 6 V 1 i+1 + 11 6 V 1 i with stencil [x i , ..., x i+3 ] (φ + x ) 2 i = - 1 6 V 1 i+1 + 5 6 V 1 i + 1 3 V 1 i-1 with stencil [x i-1 , ..., x i+2 ] (φ + x ) 3 i = 1 3 V 1 i + 5 6 V 1 i-1 - 1 6 V 1 i-2 with stencil [x i-2 , ..., x i+1 ]
Figure 1.2b shows the three sub-stencils used to compute these third-order accurate derivatives.

u i+1/2 u i+1/2 i -3 i -2 i -1 i i + 1 i + 2 i -2 i -1 i i + 1 i + 2 i + 3 sub-stencil 2 sub-stencil 2 sub-stencil 1 sub-stencil 1 sub-stencil 3 sub-stencil 3 (b) (a)
(φ + x ) i .
Next, the weights are dened by :

ω + r = ω + r k s=1 ω + s with ω + r = d r (ε + β r ) 2 (1.22)
We verify that k r=1 ω + r = 1. The coecients β r are smoothness indicators and the small parameter ε avoids division by zero and is set to

ε = 10 -6 max{(V 1 i-2 ) 2 , (V 1 i-1 ) 2 , (V 1 i ) 2 , (V 1 i+1 ) 2 , (V 1 i+2 ) 2 } + 10 -16
1.2. Numerical resolution of the advection equation as recommended by [START_REF] Fedkiw | Simplied discretization of systems of hyperbolic conservation laws containing advection equations[END_REF]. The weights d r are determined such that the highest order of accuracy (2k -1) for smoothly varying functions is obtained, leading to

d 1 = 1 3 , d 2 = 2 3 for k = 2 and d 1 = 1 10 , d 2 = 3 5 , d 3 = 3 10
for k = 3. The smoothness indicators β r allow the scheme to decrease the weights of those stencils where the function shows sharp variations. For k = 2, they are given by :

β 1 = (V 1 i+1 -V 1 i ) 2 β 2 = (V 1 i -V 1 i-1 ) 2 (1.23)
For k = 3, they are written :

β 1 = 13 12 (V 1 i+2 -2V 1 i+1 + V 1 i ) 2 + 1 4 (V 1 i+2 -4V 1 i+1 + 3V 1 i ) 2 β 2 = 13 12 (V 1 i+1 -2V 1 i + V 1 i-1 ) 2 + 1 4 (V 1 i+1 -V 1 i-1 ) 2
(1.24)

β 3 = 13 12 (V 1 i -2V 1 i-1 + V 1 i-2 ) 2 + 1 4 (3V 1 i -4V 1 i-1 + V 1 i-2 ) 2 1.2.2.4
High-order upstream central (HOUC) schemes [START_REF] Nourgaliev | High-delity interface tracking in compressible ows: Unlimited anchored adaptive level set[END_REF] have used high-order upwind nite-dierences schemes to compute the left and the right derivatives in the level-set equations, and called it the HOUC schemes. The choice of the xed stencil and coecients (without weighting coecients and smoothness indicators) was motivated by the reduction in computational time and increase in order of accuracy with respect to the WENO schemes. To test the HOUC schemes we use the upwinding as in Eq. (1.9) to choose between the left and the right derivatives. The derivatives based on the left-biased stencil are given by:

(φ - x ) i | HOU C3 = 1 6 φ i-2 -φ i-1 + 1 2 φ i + 1 3 φ i+1 + O(∆x 3 ) (1.25) (φ - x ) i | HOU C5 = - 1 30 φ i-3 + 1 4 φ i-2 -φ i-1 + 1 3 φ i + 1 2 φ i+1 - 1 20 φ i+2 + O(∆x 5 ) (1.26) (φ - x ) i | HOU C7 = 1 140 φ i-4 - 1 15 φ i-3 + 3 10 φ i-2 -φ i-1 + 1 4 φ i + 3 5 φ i+1 - 1 10 φ i+2 + 1 105 φ i+3 + O(∆x 7 ) (1.27) (φ - x ) i | HOU C9 = - 1 630 φ i-5 + 1 56 φ i-4 - 2 21 φ i-3 + 1 3 φ i-2 -φ i-1 + 1 5 φ i + 2 3 φ i+1 - 1 7 φ i+2 + 1 42 φ i+3 - 1 504 φ i+4 + O(∆x 9 ) (1.28) 1. Level-set method (φ - x ) i | HOU C11 = 1 2772 φ i-6 - 1 210 φ i-5 + 5 168 φ i-4 - 5 42 φ i-3 + 5 14 φ i-2 -φ i-1 + 1 6 φ i + 5 7 φ i+1 - 5 28 φ i+2 + 5 126 φ i+3 - 1 168 φ i+4 + 1 2310 φ i+5 + O(∆x 11 ) (1.29)
where the stencil length depends on the considered order. The right-biased stencil based derivatives are obtained by symmetry. Note that the coecients for the 11th-order scheme are derived here from Taylor expansions and are slightly dierent from those provided by [START_REF] Nourgaliev | High-delity interface tracking in compressible ows: Unlimited anchored adaptive level set[END_REF].

ENO schemes for conservation equations

The ENO schemes were initially proposed for hyperbolic conservation equations [START_REF] Harten | Uniformly high order accurate essentially non-oscillatory schemes, iii[END_REF][START_REF] Shu | Ecient implementation of essentially non-oscillatory shock-capturing schemes, ii[END_REF] in order to compute the numerical uxes at the cell boundaries.

Both nite-volume and nite-dierence schemes have been proposed [START_REF] Shu | Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws[END_REF]. The level-set advection equation (1.6), although not derived from a conservation law, can be written as a conservation equation. In one-dimensional space we have for example

∂φ ∂t + ∂f ∂x = 0 (1.30)
where f = uφ is the numerical ux. The main idea of the nite-dierence ENO scheme for this type of conservation equations is to start from point values at cell centers (φ i , f i = u i φ i , etc.) and to express the numerical uxes at the left and at the right cell faces ( f i-1/2 and f i+1/2 ) in such a way that its rst-order divided dierences represents a high-order approximation of the ux spatial derivative. Eq. (1.30) is therefore approximated by the semi-discrete equation

1 ∂φ i (t) ∂t + f i+1/2 -f i-1/2 ∆x = 0 (1.31)
In the Hamilton-Jacobi ENO scheme, we have expressed the left and the right derivatives as a function of the rst-order undivided dierences. In the conservative formulation (1.31), we express the right and the left uxes by the same formulas as in Eqs. (1.16-1.17) by replacing the rst-order undivided dierences with the zero-th-order undivided dierences. The starting stencil would be then

x i-1 2 , x i+ 1 2
, and if we want to increase the order of polynomials we add either the left point x i-3 2 or the right one x i+ 3 2 to the stencil by choosing the smoothest resulting stencil as explained in the " Step 3 " of standard ENO.

Given a grid point i, the expression of the numerical uxes based on the right/left stencils are respectively:

f - i+ 1 2 = k-1 j=0 C rj f i-r+j (1.32) f + i+ 1 2 = k-1 j=0 C (r-1)j f i-(r-1)+j (1.33)
where the constants C rj are given by Eq. (1.18). For the advection equation (1.31), simple 1 This scheme guarantees primary conservation of the variable φ, since the numerical uxes in the interior of the domain cancel out when the cell values φi are summed, i.e.

∂ ∂t

( N i=1 ∆xφi) = f 1/2 -f N +1/2
1.2. Numerical resolution of the advection equation upwinding is still used to choose between the numerical uxes based on left-and right-handed stencils:

f i+ 1 2 =    f - i+ 1 2 if u i+ 1 2 > 0 f + i+ 1 2 if u i+ 1 2 < 0
Note that the cell-centered velocity, u i , is used to construct the numerical uxes, whereas values on cell faces, u i+ 1

2

, are needed to select the upwind direction. In case the cell-centered values are not available (staggered arrangement), we use averaging, ūi = (u i-

1 2 + u i+ 1 2 )/2.
The level-set advection equation is not derived from a conservation law, and therefore the ENO scheme described above can be implemented in dierent conservative and non-conservative ways. The three formulations that have been tested in this work (with their corresponding abbreviations in brackets) are:

[Jia ] The rst one corresponds to the original conservative formulation, which expresses the advection equation in semi-discrete form as [START_REF] Jiang | Ecient implementation of weighted eno schemes[END_REF]:

∂φ i (t) ∂t + (φū) i+ 1 2 -(φū) i-1 2 ∆x = 0 (1.34)
where the velocity ū is the interpolated velocity at the cell center. For example, the ux (φū) i+ 1 2 is obtained by taking f i = (ūφ) i in Eqs. (1.32) and (1.33).

[Est ] [START_REF] Estivalezes | Une formulation conservative du schéma weno pour améliorer la résolution de la méthode level-set[END_REF] proposed a slightly dierent formulation for the ux derivative:

∂φ i (t) ∂t + u i+ 1 2 (φ) i+ 1 2 -u i-1 2 (φ) i-1 2 ∆x = 0 (1.35)
which does not require the interpolation of the velocity. This scheme is still conservative, but the accuracy is reduced to second-order since the velocity is not included in the highorder reconstruction step.

[Adv ] In the current study, we add a third non-conservative (advective) formulation which is written as:

∂φ i (t) ∂t + u i (φ) i+ 1 2 -(φ) i-1 2 ∆x = 0 (1.36)
where the velocity should again be interpolated from the cell faces to the cell centers. Here, we have used the advection equation (1.1) as a starting point (rather than the conservative formulation (1.6)) and the ENO scheme is merely used to compute the spatial derivative of the level-set function. The accuracy of the ENO scheme is preserved in the calculation of the spatial derivative, but the cell averaging of the velocity introduces a second-order error.

WENO schemes for conservation equations

The conservative WENO is a convex combination of the approximations of conservative ENO on possible stencils and it is derived following the same manner as the non-conservative WENO scheme in 1.2.2.3. The main dierence is that the approximation of the derivatives becomes an approximation of the uxes, and the zeroth-order undivided dierences are used instead of the rst-order undivided dierences. After having expressed the numerical uxes, the derivative comes from the rst-order undivided dierences of uxes. Figure 1.3 below shows how the stencils 1. Level-set method are used to approximate the level-set ux at grid points (i + 1

2 ) where velocity is positive (Fig.

1.3a) and negative (Fig. 1.3b).

u i+1/2 u i+1/2 φ- i+1/2 φ+ i+1/2 i -2 i -2 i -1 i -1 i i i + 1 i + 1 i + 2 i + 2 sub-stencil 2 sub-stencil 2 sub-stencil 1 sub-stencil 1 sub-stencil 3 sub-stencil 3 (b) (a)
Figure 1.3: Stencils of the conservative 5th-order WENO scheme for the ux approximation.

As for the conservative ENO schemes, we have tested three dierent formulations for the 5th-order WENO scheme (Eq. 1.34,1.35 and 1.36). The second formulation (1.35) has been used by [START_REF] Tanguy | Développement d'une méthode de suivi d'interface. application aux écoulements diphasiques[END_REF], [START_REF] Zuzio | Direct numerical simulation of two phase ows with adaptive mesh renement[END_REF] and [START_REF] Voronetska | Simulation numérique directe des écoulements à phases dispersées[END_REF] in combination with the 5th-order WENO scheme.

HOUC schemes for conservation equations

The conservative HOUC scheme is built here in the same manner as the conservative ENO scheme. The dierence is that the ENO scheme chooses the smoothest stencil, while the HOUC scheme uses a xed stencil. For a given grid cell (i, j), the numerical ux at (i + 1/2, j) is computed using a stencil centered at that grid point. These uxes based on the left and the right stencils are given, as in the conservative ENO schemes, by Eqs. (1.32) and (1.33) where the coecients C r,j are given by Eq. (1.18). For the fth-order reconstruction, for example, we set r = 2 and the ux at i + 1/2 of a given quantity f is:

f - i+ 1 2 = 3 128 f i-2 - 20 128 f i-1 + 90 128 f i + 60 128 f i+1 - 5 128 f i+2 (1.37) f + i+ 1 2 = 3 128 f i+3 - 20 128 f i+2 + 90 128 f i+1 + 60 128 f i - 5 128 f i-1 (1.38)
where f could be either φ or φū. The derivative is then computed using one of the formulations (1.34), (1.35) and (1.36).

Semi-Lagrangian advection

The semi-Lagrangian discretization of the level-set advection, proposed by (Wang et al., 2012a), consists in determining the interpolating point associated with each grid point (i, j). It is obtained by tracing it back along the characteristics from time t n+1 to t n . The interpolating point is obtained using a two-step procedure (Wang et al., 2012a):

       x (1) = x i,j -u(x i,j , t n+1 )∆t x * = x i,j + x (1) 2 x (2) = x i,j -u(x * , t n+1/2 )∆t (1.39) 1.3. Level-set reinitialization
Then, by interpolating the level-set φ n on the point x (2) we obtain the level-set value which would be on the grid point x i,j at time t n+1 . The interpolation of the level-set is performed using the bilinear or the bicubic interpolation as detailed in A.1. Near the domain boundaries, the bilinear interpolation is preferred since the stencil is not large enough to perform a bicubic interpolation. The bicubic interpolation could also generate a non-physical high curvature of the interface. As in Wang et al. (2012a), we use the bicubic interpolation only if:

-β < (φ bicubicφ bilinear ) < β with β = ∆x/20. The velocity is also interpolated using the bilinear interpolation.

Level-set reinitialization

After having advected the level-set, the signed-distance property is disturbed. In order to maintain φ( x, t) close to a signed-distance function, the following reinitialization equation is solved:

∂φ ∂τ + S(φ 0 )(|∇φ| -1) = 0 (1.40)
where τ is a pseudo-time, φ 0 = φ(τ = 0) is the initial level-set eld before reinitialization and S is the sign function. The exact signed-distance function is recovered when this equation is solved up to steady state, but in practice it is sucient to stop the computation at some pseudotime τ max which is of the order of n∆x, where n depends on the stencils used. [START_REF] Mukherjee | Numerical study of single bubbles with dynamic contact angle during nucleate pool boiling[END_REF] for example considered 10 iterations for the reinitialization equation with the third-order TVD Runge-Kutta method. The ctitious time step suggested by the same authors is ∆τ = ∆x/2. They reinitialized the level-set after each advection step.

In order to avoid numerical diculties, a smoothed version of the sign function is generally used. We use the denition proposed by [START_REF] Sussman | A level set approach for computing solutions to incompressible two-phase ow[END_REF]:

S(φ 0 ) = φ 0 φ 2 0 + δ 2 (1.41)
where δ is the thickness of smoothing region, generally taken as the grid size δ = ∆x. Other formulations of the sign function exist, in particular we may integrate in this function the so-called sub-cell x proposed by [START_REF] Russo | A remark on computing distance functions[END_REF] (see 1.3.1), which reduces the displacement of the interface by the reinitialization procedure and thereby improves mass conservation.

In this work, the third-order TVD Runge-Kutta scheme is also employed for reinitialization, except for the case where the alternative semi-Lagrangian advection is used. For this latter, a semi-Lagrangian reinitialization without volume enforcement is used according to Wang et al. (2012a). For the Eulerian approach, let φ n be the solution of Eq. (1.40) after n pseudo-time steps, then the solution at the next step is computed from:

φ (1) = φ n + ∆τ S(φ 0 )(1 -|∇φ n |) (1.42) φ (2) = 3 4 φ n + 1 4 φ (1) + 1 4 ∆τ S(φ 0 )(1 -|∇φ (1) |) (1.43) φ n+1 = 1 3 φ n + 2 3 φ (2) + 2 3 ∆τ S(φ 0 )(1 -|∇φ (3) |) (1.44)
where φ (1) and φ (2) are intermediate values and φ n+1 is the solution at time τ n+1 . The term |∇φ| in Eq. (1.40) is determined from |∇φ| = (φ 2 x + φ 2 y ) 1/2 where the values of φ 2

x 1. Level-set method and φ 2 y are computed using the Godunov method (Osher and Fedkiw, 2003), resulting in:

φ 2 x = max max(φ - x , 0) 2 , min(φ + x , 0) 2 for S(φ 0 ) > 0 (1.45) φ 2 x = max min(φ - x , 0) 2 , max(φ + x , 0) 2 for S(φ 0 ) < 0 (1.46)
where φ - x and φ + x are approximations of the derivative ∂φ/∂x based on left-and right-hand sided stencils. Similar expressions are used for φ 2 y . The spatial discretization, i.e.: the determination of the spatial derivatives, is performed usually using the Hamilton-Jacobi schemes. We have tested the following schemes:

• 1st-order Upwind.

• 2nd-, 3rd-and 4th-order HJ-ENO.

• 3rd-and 5th-order HJ-WENO.

One of the main disadvantages of the reinitialization procedure is that it generally causes a slight displacement of the interface and thereby a change of the mass of the uids on either side.

Several xes have been proposed in literature that try to limit the movement of the interface.

The sub-cell x method for example by [START_REF] Russo | A remark on computing distance functions[END_REF] uses a rst-order discretization for cells near the interface. Later, the volume constraint method by [START_REF] Sussman | An ecient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible uid ow[END_REF] adds a supplementary term to the reinitialization equation in order to correct the mass errors during the reinitialization step. Solving this latter is a two step procedure: rst, an intermediate level-set is computed, then the mass correction is applied. A global mass correction scheme was proposed by [START_REF] Yap | A global mass correction scheme for the level-set method[END_REF], then enhanced by [START_REF] Zhang | Numerical simulation of free-surface ow using the level-set method with global mass correction[END_REF], is in principle similar to the volume constraint method. In this context, other methods could be found for example in [START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF] and [START_REF] Zhang | Numerical methods for partial dierential equations[END_REF] in which authors proposed to add a supplementary term for the mass correction which takes into account the interface curvature.

In the current work, we have tested only the sub-cell x and the volume constraint methods.

Sub-cell x

The reinitialization should make the level-set as a signed-distance function. However, it moves slightly the zero level which causes non-physical change of the reference uid mass. To overcome this drawback, [START_REF] Russo | A remark on computing distance functions[END_REF] proposed a solution called the sub-cell x. The remedy to the interface displacement is the use of the sign function for the reinitialization on the whole domain, then for the grid cells near the interface a truly upwind discretization is employed near the interface based on the distance to the interface. A modication of the sub-cell x reinitialization is proposed in Sun et al. (2010), but is not used in this work. Instead, we have used the original method by [START_REF] Russo | A remark on computing distance functions[END_REF].

First of all, we should locate cells cut by the interface. More specically, we say that the interface passes through a cell (i, j)

∈ ∆x , if φ 0 i,j φ 0 i-1,j < 0 or φ 0 i,j φ 0 i+1,j < 0 or φ 0 i,j φ 0 i,j-1 < 0 or φ 0 i,j φ 0 i,j+1 < 0.
Then, the minimum distance separating the interface from the grid node (i, j), at these interface neighboring grid cells is computed by:

D i,j = φ 0 i,j |∇φ 0 i,j | with |∇φ 0 i,j | = φ 2 x + φ 2 y
where the gradient norm is evaluated by

φ x = max |φ 0 i+1,j -φ 0 i-1,j | 2∆x , |φ 0 i+1,j -φ 0 i,j | ∆x , |φ 0 i,j -φ 0 i-1,j | ∆x 1.3. Level-set reinitialization and φ y = max |φ 0 i,j+1 -φ 0 i,j-1 | 2∆y , |φ 0 i,j+1 -φ 0 i,j | ∆y , |φ 0 i,j -φ 0 i,j-1 |

∆y

The corresponding smoothed signed-distance in this region is a function of the distance D i,j :

S(φ 0 ) = D i,j ∆x
Let us consider the rst sub-step of RK3 time discretization of Eq. (1.42) which is the Euler discretization. The spatial discretization of the reinitialization Eq. (1.40) should be modied on the region near the interface, and it takes the form:

φ n+1 = φ n - ∆τ ∆x [sign(φ 0 )|φ n | -D i,j ]
(1.47)

One should note that sign(S(φ 0 )) = sign(φ 0 ).

Finally, we resume the sub-cell x reinitialization in these two steps:

1. Compute the smoothed signed-distance function

S(φ 0 ) =        D i,j ∆x if (i, j) ∈ ∆x φ 0 φ 2 0 + ∆x 2 otherwise (1.48)
2. Advance the reinitialization equation with a pseudo time step ∆τ

φ n+1 =      φ n -∆τ sign(φ 0 ) |φ n | ∆x -S(φ 0 ) if (i, j) ∈ ∆x φ n -∆τ S(φ 0 )(|∇φ n | -1) otherwise (1.49)
In the above reinitialization equation, the spatial discretization is only rst-order near the interface and we are free to use higher-order schemes for grids not on the interface.

If time advancement is done by a multi-step scheme, sub-cell x should be applied to each sub-time-step.

Based on the fact that reinitialization at each physical time step introduces articial displacement of the interface, sub-cell x reinitialization is not performed after each advection but after many advection steps [START_REF] Russo | A remark on computing distance functions[END_REF]. This would introduce less errors. Instead of solving the reinitialization equation for a ctitious time τ max after each advection, we can solve it on a ctitious time of cst × τ max where cst > 1 after k advection steps.

Volume constraint

A dierent approach to improve mass conservation during the reinitialization procedure is proposed by [START_REF] Sussman | An ecient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible uid ow[END_REF]. This modication is mainly based on adding a new term to the reinitialization Eq. (1.40). This term is a mass correction, and it preserves the zero levelset whilst redistancing. The so-called reinitialization with constraint could be expressed by the equation:

∂φ ∂τ + sign (φ 0 ) (|∇φ| -1) -λf (φ) = 0 (1.50) where f (φ) is chosen as : f (φ) ≡ H (φ 0 )|∇φ 0 |.
H ∆x (φ) =              1 if φ > ∆x 0 if φ < -∆x 1 2 + φ 2∆x + 1 2π sin πφ ∆x otherwise (1.51)
The resulting smooth sign function is then:

sign ∆x (φ) = 2(H ∆x (φ) -1/2)
The main idea of this method is that when discretized, the area of each computational cell should remain unchanged. The area is the integral of the Heaviside function in space:

A i,j = Ω i,j H ∆x (φ)d x
where Ω i,j includes all the points (x, y) delimited by x i-1/2 < x < x i+1/2 and y i-1/2 < y < y i+1/2 . The constraint that A i,j remains unchanged is equivalent to:

Ω i,j H ∆x (φ 0 ) ∂φ ∂τ d x = 0 (1.52)
So to apply the local constraint to the reinitialization equation, one should rst calculate the intermediate reinitialized level-set eld at the time n + 1, denoted as φn+1 , from the eld at the step n, i.e.: φ n . This is done by solving the Eq. (1.50) while ignoring the added term, and this comes to solving the standard reinitialization equation given by Eq. (1.40). Now, let us nd an expression of λ. By replacing ∂φ/∂τ with its expression given by Eq. (1.50) in Eq. (1.52), and by the use of the intermediate level-set φn+1 denition we get :

Ω i,j H ∆x (φ 0 ) φn+1 -φ 0 ∆τ + λ i,j f (φ 0 ) d x = 0
and thus, it follows that:

λ i,j = -Ω i,j H (φ 0 ) φn+1 -φ 0 ∆τ d x Ω i,j H 2 (φ 0 )|∇φ 0 |d x (1.53)
These integrals are computed using a nine-point quadrature in 2D, i.e.: we use the information of the level-set on the grid point of interest and we take into account the information of the neighboring grids. For a grid point not on the edges of the discretized domain, the volume integral is computed by means of:

Ω i,j g d x ≈ h 2 24   16g i,j + 1 m,n=-1;(m,n) =(0,0) g i+m,j+n

  ∆x∆y

If we want to implement a 3D or an axisymmetric version of this constraint, one should rst adapt the volume integral given by the previous quadrature.

Local level-set method

Since we have already evaluated φn+1 , the gradient projection step during which the con- straint is applied is simply

φ n+1 = φn+1 + ∆τ λH ∆x (φ 0 )|∇φ 0 | (1.54)
When the third-order Runge-Kutta scheme is used for temporal discretization of the reinitialization equation, the gradient projection step is performed only once after the last sub-time step.

In the sub-cell x approach, however, the correction is applied after each sub-time step.

Local level-set method

The level-set advection and reinitialization equations are expensive from a computational eort point of view when solved on the whole computational domain. [START_REF] Peng | A PDE-based fast local level set method[END_REF] have developed a method to solve these equations in a narrow band around the interface.

First, let β and γ be two constants comparable to ∆x with γ > β > 0. The largest constant, γ is used to dene two tubes (narrow bands) N and T as follows:

T = {(x, y) : |φ(x, y)| ≤ γ} (1.55) N = {(x, y) : |φ(x ± δx, y ± δy)| ≤ γ} for δx < ∆x and δy < ∆y
Assuming that the level-set function is approximately a signed-distance function, the tube T contains all cells whose centers are within a distance of γ to the interface. The tube N contains the tube T , but also the neighboring cells.

Then, let us dene the cut-o function [START_REF] Peng | A PDE-based fast local level set method[END_REF]:

c(φ) =          1 if |φ| ≤ β (2|φ| + γ -3β) (|φ| -γ) 2 (γ -β) 3 if β < |φ| ≤ γ 0 if |φ| > γ (1.56)
This cut-o function is equal to 1 in the narrow band of distance β on each side of the interface, and then smoothly decreases to zero towards the outer boundary of the tube T . This function is integrated into the advection equation (1.1) which is rewritten as

∂φ ∂t + c(φ) u • ∇φ = 0 (1.57)
The objective is to smoothly damp out the advection near the edges of the tube T to avoid generation of disturbances at this location. It is clear that the advection equation now only needs to be solved in the tube T , since the advection velocity is zero outside. Note that the exact advection equation is solved for all points with |φ| ≤ β.

The reinitialization equation is solved in the tube N . Since this tube is slightly larger than tube T , the disturbances near the outer boundary of the latter are reduced to a minimum. Once the reinitialization step is completed, the values of the signed-distance function in the tube T are conserved, while they are set to a constant value outside the tube. In other words, the following correction is applied:

φ(x, y) =    -γ if φ(x, y) < -γ φ(x, y) if |φ(x, y)| ≤ γ γ if φ(x, y) > γ (1.58)
In order to preserve the accuracy of the numerical schemes near the interface, the values of β 1. Level-set method and γ need to be dened as a function of the stencil size used for the approximation of the spatial derivatives of Eq. (1.57). [START_REF] Peng | A PDE-based fast local level set method[END_REF] recommanded β = 3∆x and γ = 6∆x for the third-order ENO and the fth-order WENO schemes. In our computation, these two constants are set to β = 6∆x and γ = 9∆x to allow the use of high-order schemes.

Testing the level-set method

As shown in the previous sections, many variants of the level-set method can be built depending on the numerical schemes employed for the spatial discretization of the advection and the reinitialization equations. Dierent reinitialization methods can also be obtained by selecting one of the already described methods to reduce the interface displacement while reinitializing. The general algorithm of the level-set method is given by Algorithm 1.1, where Nreinit is the number of advection steps to perform before reinitializing the level-set.

Algorithm 1.1 Level-set method Input: φ n , u n+1/2 , ∆t, τ max , ∆τ , Nreinit φ n+1 ← Advect level-set Eq. (1.1) using 3rd-order TVD Runge-Kutta (φ n , u n+1/2 , ∆t)

if (mod(n, Nreinit) = 0) then

φ n+1 ← Reinitialize level-set (φ n+1 , τ max , ∆τ ) end if u n+3/2 ← Advance ow equations (φ n+1 , u n+1/2 , ∆t) return φ n+1 , u n+3/2
This section is presented as follows: we start rst by providing denitions for the interface error, then we describe some reference test cases in order to compare and validate the methods. The idea then is to rst validate our implementation of the level-set method with the results provided by [START_REF] Enright | A hybrid particle level set method for improved interface capturing[END_REF]. This level-set is given by Algorithm 1.1 with these specications:

Level-set: The reinitialization is performed using Eq. (1.40) with the denition of the sign function given by Eq. (1.41). This equation is solved on a band of τ max = 10∆x around the interface, with a ctitious time step ∆τ = ∆x. The level-set is reinitialized at the end of each advection step (Nreinit = 1). The fth-order HJ-WENO scheme is used for the spatial discretization of the advection and the reinitialization equations.

Next, we vary the scheme for the advection of the level-set, and we compare it to the level-set without reinitialization in order to show the benets/drawbacks of the reinitialization procedure.

Then, we test the level-set with the sub-cell x reinitialization. For this we vary the advection scheme, then the CFL number in order to investigate its stability. Some of the suited schemes for the advection of the level-set with the sub-cell x will be identied. For these identied schemes, we investigate the inuence of the reinitialization scheme on the behavior of the method. The sub-cell x is given by Algorithm 1.1 with the corresponding specications:

Sub-cell x : The reinitialization is performed using the sub-cell x method Eq.(1.49) with the denition of the sign function in Eq. (1.48). The ctitious time is set to τ max = 10∆x with a time step ∆τ = ∆x/2. The reinitialization is performed after the rst level-set advection step, then after each Nreinit = 10 advection steps.

1.5. Testing the level-set method

The study of the level-set with volume constraint reinitialization is performed in the same manner as for the sub-cell x. The level-set with volume constraint reinitialization is also given by Algorithm 1.1, with:

Volume constraint: The reinitialization is performed using Eq. (1.40) with the denition of the sign function in Eq. (1.41). The volume constraint Eq. (1.54) is then solved to enforce the level-set mass conservation. The ctitious time is τ max = 2∆x with ∆τ = ∆x/2. The reinitialization is performed after each advection step (Nreinit = 1).

For the temporal discretization of the advection and the reinitialization equations, we have used the third-order TVD Runge-Kutta scheme. Concerning the spatial discretization, all the dierent schemes detailed in the previous sections were tested in order to identify the advantages and disadvantages of each scheme. The schemes and abbreviations used to advect the level-set equation are:

• 1st-order UPWIND (UPWIND1) 1.2.2.1

• 2nd-order Hamilton-Jacobi ENO (HJ-ENO2) 1.2.2.2

• 3rd-and 5th-order Hamilton-Jacobi WENO (HJ-WENOk, where k is the order) 1.2.2.3

• 3rd-, 5th-, 7th-, 9th-and 11th-order HOUC (HOUCk, where k is the order) 1.2.2.4

• 2nd-, 3rd-and 4th-order conservative ENO (ENOk-(Jia, Adv and Est), where k is the order) 1.2.2.5

• 5th-order conservative WENO-(WENO5-(Jia, Adv and Est)) 1.2.2.6

• 5th-order conservative HOUC-(HOUC5-(Jia, Adv and Est)) 1.2.2.7

and for the reinitialization equation:

• 1st-order UPWIND (UPWIND1)

• 2nd-, 3rd-and 4th-order Hamilton-Jacobi ENO (HJ-ENO2, HJ-ENO3, HJ-ENO4)

• 3rd-and 5th-order Hamilton-Jacobi WENO (HJ-WENO3, HJ-WENO5)

In addition to these schemes, the semi-Lagrangian advection (Wang et al., 2012a) detailed on 1.2.3 is also tested, always in combination with the semi-Lagrangian reinitialization.

Denition of interface errors

The accuracy of the interface location and the contained area of the reference uid are two important comparison criteria. The contained area (directly related to the total mass) is dened by the integral of the Heaviside over the whole domain. The mass loss/gain error is then given by:

E M = mass -Exact mass Exact mass = Ω H(φ C )dΩ -Exact mass Exact mass (1.59)
The exact mass may be computed either analytically from the initial condition or numerically from the initial level-set eld, while the mass at a given time of the simulation is the integral of 1. Level-set method the Heaviside over all the domain. The Heaviside integral is estimated using the bilinear or the bicubic interpolation as described in Appendix A.1.

The positioning error E 1 is expressed as:

E 1 = Ω |H (φ E ) -H (φ C )| dΩ Ω H(φ E )dΩ (1.60)
where subscripts E and C refer to exact and computed solutions. Note that the exact solution is dened either by the exact level-set eld if this latter is available, or by a reference solution.

The positioning error E 1 is computed as described in Appendix A.1. Note that E 1 and E M are computed using the bicubic interpolation for mixed cells which are subdivided into m 2 sub-cells with m = 1000, except otherwise specied.

The convergence rate based on the positioning error, E 1 , is dened as:

O = log E (r) 1 /E (c) 1 log ∆x (r) /∆x (c) (1.61)
where superscripts (r) and (c) mean that the quantity is computed for rened and coarse grid, respectively.

We dene also the mean absolute deviation of the level-set from a signed-distance function as:

E |∇φ| = N i k=1 ||∇φ| k -1| N i with |∇φ| = φ 2 x + φ 2 y (1.62)
where N i is the number of cells cut by the interface and the xand y-derivatives are computed using the 2nd-order central dierences.

Zalesak disk problem

Zalesak's rotating disk [START_REF] Zalesak | Fully multidimensional ux-corrected transport algorithms for uids[END_REF] test evaluates the ability of interface methods in transporting rigid objects. It is dierent from the circle translation/rotation test case [START_REF] Pilliod | Second-order accurate volume-of-uid algorithms for tracking material interfaces[END_REF] due to the presence of sharp angles; each interface capturing method will handle dierently these discontinuities. The Zalesak disk simulation is therefore a good measure of the numerical diusivity of interface methods.

The slotted disk (Fig. 1.4) is dened by a radius R = 0.15, a slot width w = 0.05 and of a distance h = 0.05. This disk is placed in a 1 × 1 domain and initially centered at (0.5, 0.25). The imposed velocity eld is that of a solid rotation around the origin (0.5, 0.5) :

     u = (y -0.5) π 3.14 v = (0.5 -x) π 3.14
The CFL value is based on the maximum velocity given by: CFL = max (|u|, |v|) ∆t ∆x = π∆t 6.28∆x

A full rotation of the disk is accomplished after 6.28 time units and the exact position of the interface is known at any time by rotating the axis since it undergoes a solid rotation.

Shearing vortex problem

While the Zalesak's disk problem is a good indicator of diusion errors in an interface capturing method, it does not test the ability of an Eulerian scheme to accurately resolve thin laments which can occur in stretching and tearing ows [START_REF] Bell | A second-order projection method for the incompressible navier-stokes equations[END_REF]. In this shearing vortex given by

       u = - ∂Ψ ∂y v = ∂Ψ ∂x 1. Level-set method
The use of the analytical formula of the velocity in the discrete domain leads to a non zero discrete divergence. We rather use the central dierences to numerically derive the velocity eld as follows:

       u i+1/2,j = - Ψ i+1/2,j+1/2 -Ψ i+1/2,j-1/2 ∆y v i,j+1/2 = Ψ i+1/2,j+1/2 -Ψ i-1/2,j+1/2 ∆x
from which it is easily shown that the discrete velocity divergence, (u i+1/2,ju i-1/2,j )/∆x + (v i,j+1/2v i,j-1/2 )/∆y is equally zero.

Computations are performed up to a time t = T /2 = 3 at which point the interface reaches maximum stretching; then, the velocity sign is reversed which causes the return to the initial position which would be a circle at time t = T = 6 in the ideal case.

For the computation of the mass error E M and the positioning error E 1 at the beginning and the end of the simulation, the exact interface is known which is a circle. However at an intermediate time step, no analytic formula gives the exact interface of the vortex. In the current work, we use an accurate Lagrangian method to estimate the interface position at a given time as detailed in Appendix A.2.

Time-reversed single vortex problem

The time-reversed single vortex is similar to the shearing vortex problem. The initial interface form and conguration are identical. What changes is the stream function from which the velocity eld is derived, and the period T during which the interface is transported. The stream function given by the Eq. (1.63) is multiplied by cos(πt/T ), which gives: Ψ = 1 π sin 2 (πx) sin 2 (πy) cos πt T

(1.64)

In order to conserve mass, the velocity eld should be divergence free, hence the discrete velocity elds are derived in the same manner as explained for the shearing vortex problem. The ow inversion at t = T /2 is directly ensured by the cosine term. Due to this cosine term, the velocity magnitude is also decreasing in time until cos(πt/T ) = 0, then it increases again until cos(πt/T ) = 1. The period T is set to 2, which make this test less severe since the interface undergoes less stretching. The exact position of the interface is determined in the same manner as for the shearing vortex problem, using a Lagrangian method.

Results and discussions

First, we start by comparing our level-set implementation with results found in literature, more specically with those of [START_REF] Enright | A hybrid particle level set method for improved interface capturing[END_REF] for the Zalesak disk (Fig. 1.4). In this work, the level-set advection Eq. (1.1) and reinitialization Eq. (1.40) are discretized in time using TVD-RK3 and in space using HJ-WENO5 scheme. For advection, the CFL is 0.5. The only dierence between the present implementation (with the same advection and reinitialization schemes) and the one presented in [START_REF] Enright | A hybrid particle level set method for improved interface capturing[END_REF] is that in our case the level-set reinitialization Eq. (1.40) is solved on the entire domain, and in the work of [START_REF] Enright | A hybrid particle level set method for improved interface capturing[END_REF] on the narrow band with the use of a fast marching method [START_REF] Chopp | Some improvements of the fast marching method[END_REF][START_REF] Sethian | Fast marching methods[END_REF] on the rest of the domain.

The mass error E M and positioning error E 1 for dierent grid sizes are summarized on Table 1.1. In the work of [START_REF] Enright | A hybrid particle level set method for improved interface capturing[END_REF], these errors were computed using the bilinear interpolation with 1000 2 sub-cells. We have used the same method for this comparison.

1.6. Results and discussions Table 1.1: Zalesak disk: comparison of level-set method with the results of [START_REF] Enright | A hybrid particle level set method for improved interface capturing[END_REF].

The HJ-WENO5 scheme is used for both advection and reinitialization with Nreinit = 1 and CFL = 0.5. We remark that, although we observe the same order of magnitude and tendency for the errors, the current implementation provides slightly more accurate results, especially for the grid 50 2 where the interface vanishes in the implementation of [START_REF] Enright | A hybrid particle level set method for improved interface capturing[END_REF]. These dierences are most probably due to the reinitialization method which is slightly dierent.

In the next subsection, we vary the scheme for the advection of the level-set and we compare the resulting interface from each advection with its corresponding when we do not perform reinitialization.

Eect of advection scheme and reinitialization

We now investigate the eect of the advection scheme and compare at the same time the level-set with and without reinitialization. The results with reinitialization are obtained with Nreinit = 1, τ max = 10∆x and ∆τ = ∆x. No sub-cell x or volume constraint is applied. The HJ-WENO5 scheme is used for the spatial discretization of the reinitialization Eq. (1.40) except for the semi-Lagrangian level-set where the reinitialization is of the semi-Lagrangian type. The CLF number for advection is set to 0.5.

Zalesak disk

In the Zalesak disk test case, the velocity u depends only on the y-coordinate (u k,j = u l,j , ∀(k, l, j) ∈ [1, n] 3 ). Similarly, the velocity in the y-direction depends only on the x-coordinate.

Therefore, the three formulations of the advection scheme (see 1.2.2.5) are strictly equivalent:

u i+ 1 2 ,j φ i+ 1 2 ,j -u i-1 2 ,j φ i-1 2 ,j ∆x = u i,j φ i+ 1 2 ,j -φ i-1 2 ,j ∆x = φu i+ 1 2 ,j -φu i-1 2 ,j ∆x
Hence, there is no need to specify which formulation is used in this test case.

The resulting interface is presented in Fig. 1.6 for both the level-set with (in blue) and without reinitialization (in red), for the Zalesak disk after a complete revolution (at time t = 6.28). We observe that reinitialization prevents the interface from vanishing when the advection of the level-set is ensured by the rst-order upwind scheme, and it does not smooth the entire slab for the HJ-ENO2, ENO2 and the HJ-WENO3 schemes. However, for higher-order schemes, the reinitialization causes a deterioration of the interface representation with respect to the method without reinitialization. This latter is due to:

1. Level-set method • to relatively large ctitious time step (∆τ = ∆x).

• to the use of the fth-order HJ-WENO scheme in the reinitialization process.

• to the relatively long reinitialization time (τ max = 10∆x).

• to the fact that the level-set is reinitialized after each advection.

The mass and positioning errors corresponding to each advection scheme are reported on Table 1.2 for the level-set with and without reinitialization. In this test case, the interface positioning error, E 1 (%), is probably the most relevant. The mass error E M (%) sometimes show very small values when positive and negative error contributions compensate (for the HJ-ENO2 and WENO5 schemes for example). When considering the error E 1 (%), the main contribution for the cases without reinitialization is due the rounding of the interface near the sharp corners.

The use of high-order schemes, in particular the HOUCk scheme with k ≥ 7, seems to reduce this error. However, high-order schemes use larger stencils and are much more sensitive to small perturbations in the level-set eld. The eect of reinitialization is therefore greater, in particular for the HOUC schemes with xed stencil. We can almost imagine that the level-set without reinitialization is better than the level-set with reinitialization. However, without reinitialization the deviation of the level-set from the signed-distance function, E |∇φ| , is much larger.

1.6. Results and discussions

Shearing vortex

The interface at times t = T /2 = 3 for the shearing vortex test case are plotted in Fig. 1.7 and the corresponding interface errors are summarized on Table 1.3. Note that since the u-velocity now depends on x, and the v-component on y, the various formulations of the advection equation (noted Adv, Est and Jia, see 1.2.2.5) do not lead to the same results, although the dierences here are rather small.

Similar to the Zalesak disk, we remark that the interface vanishes before t = T /2 without reinitialization when the 1st or 2nd-order advection schemes are used. It vanishes also for the 3rd-order HJ-WENO or the semi-Lagrangian schemes. Reinitialization of the level-set clearly improves the results for these schemes.

Higher-order schemes perform well without reinitialization, in particular the HOUC schemes, but reinitialization now tends to decrease the accuracy of these methods, as it is in the case for the Zalesak disk. Reinitialization tends to thicken the laments at t = T /2, thereby increases the positioning error and the total mass.

The HJ-schemes with and without reinitialization generally show poor accuracy in this test case, possible due to the slightly larger stencils which do not capture the thin laments well.

In Fig. 1.8, the results for the interface at time t = T = 6 are shown, obtained after having inverted the ow direction in order to come back to the initial conguration. We obtain generally the same conclusions as for the results at t = T /2 = 3.

1. Level-set method Advection: CFL = 0.5. Reinitialization: HJ-WENO5 with ∆τ = ∆x, τ max = 10∆x and Nreinit = 1. Reinitialization

Advection scheme

E M (%) E 1 (%) E |∇φ| E M (%) E 1 (%) E |∇φ| E M (%) E 1 (%) E |∇φ| E M (%) E 1 (%) E |∇φ|

UPWIND1

-100.00 100.00

-

Time-reversed single vortex

The results for the time-reversed single vortex at t = T /2 = 1 are presented in Fig. 1.9 and Table 1.4. The interface is not as deformed as in the shearing vortex problem, and the numerical results are closer to the reference solution. The interface obtained is almost independent from the advection scheme, except for the UPWIND1 for which the mass loss is quite important. Fig. 1.10

shows that for t = T the UPWIND1 in combination with the non-reinitialized level-set causes the loss of the total mass of the vortex. For high-order schemes, the non-reinitialized level-set ts the reference interface better than when reinitializing the level-set. We can also remark on Advection: CFL = 0.5. Reinitialization: HJ-WENO5 with ∆τ = ∆x, τ max = 10∆x and Nreinit = 1. Reinitialization

Advection scheme

E M (%) E 1 (%) E |∇φ| E M (%) E 1 (%) E |∇φ| E M (%) E 1 (%) E |∇φ| E M (%) E 1 (%)

Level-set method

As a conclusion, for the high-order spatial discretization of the advection equation, the interface representation and advection is more accurate when the level-set is not reinitialized.

However, the level-set then diverges from the signed-distance function (|∇φ| = 1). The reinitialization step helps in keeping the level-set as close as possible to a signed-distance function, and improves the accuracy for low-order schemes. Nevertheless, it deteriorates the accuracy for high-order schemes, in particular the HOUCk schemes with k ≥ 7. There are two possible contributions to this deterioration: displacement of the interface and introduction of perturbations in the level-set eld during reinitialization. The sub-cell x and the volume constraint reinitializations are two methods helping to prevent the interface displacement while reinitializing. In the next subsections, we study the inuence of the dierent schemes for the advection and the reinitialization on both methods, the sub-cell x and the volume constraint, on dierent grid sizes and CFL numbers.

Sub-cell x

We now include in our tests the sub-cell x, described in 1.3.1. First, we investigate the inuence of the spatial discretization scheme for advection on this method. Then, we verify the sensibility of the method to the CFL number variations. Finally, we study the convergence rate of the method as a function of the grid size. The inuence of the reinitialization scheme will be also investigated. In this section, the reinitialization equation is solved during a ctitious time τ max = 10∆x with a step of ∆τ = ∆x/2.

Eect of the advection scheme

The reinitialization equation is discretized in space using the HJ-ENO2 scheme and is solved after each Nreinit = 10 advection steps. The CFL number for advection based on the maximum velocity is set to 0.5. A 100 2 grid is adopted for both test cases, the Zalesak disk and the shearing vortex, while a 64 2 grid is used for the time-reversed single vortex problem.

The resulting interfaces are plotted on

• Fig. 1.11 for the Zalesak disk after a full rotation

• Figs. 1.12 and 1.13 for the single vortex at times t = T /2 = 3 and t = T = 6, respectively.

• Figs. 1.14 and 1.15 for the time-reversed single vortex at t = T /2 = 1 and t = T = 2, respectively.

and the errors for all cases are summarized on Table 1.5.

For the Zalesak disk, we remark that the slot has been entirely smoothed when the method is advected using the UPWIND1 scheme. On the contrary, HJ-ENO2 and HJ-WENO3 keeps the circular form but the slot is almost half its exact height. The HOUC and conservative ENO/WENO schemes lead to a deformation of the circular part of the disk and the slab. This is due to the displacement of the interface during the reinitialization procedure. The HJ-WENO5 scheme deforms less the interface and predicts its position better than the other schemes. We see that the Zalesak disk mass is conserved up to ±3% for all advection schemes except for the 1st-and 2nd-order schemes and the HJ-WENO3 scheme.

In Fig. 1.12, the shearing vortex result at t = T /2 = 3 shows important deformations: the lament thickness at time T /2 is too small and the level-set method has the tendency either to make it thicker (in this case we gain mass) or to let it vanish (and in this case we loose mass).

The errors are much larger than for the Zalesak disk, the mass gain/loss is 35% at best for the HOUC11 advection scheme. The Hamilton-Jacobi schemes have the tendency of making the 1.6. Results and discussions vortex loosing mass, while the other schemes increase the mass of the vortex. High-order HOUC schemes seem to be more accurate for this simulation, at time T /2 as well as at time T (Fig. 1.13).

The interface mass increases between the start of the simulation and time T /2, then decreases until the end of the simulation for HOUC schemes of 5th-order or higher; this is due to the fact that these schemes use a xed stencil, therefore the same stencils are used when advecting the vortex forward and backward in time. At time T , the interface form when advected with one of the HOUC schemes is very close to what presented by [START_REF] Nourgaliev | High-delity interface tracking in compressible ows: Unlimited anchored adaptive level set[END_REF].

The time-reversed single vortex simulation is similar to the shearing vortex with a small period T . The mass errors at t = T /2 = 1 (Fig. 1.14) are lower than 1% for all the advection schemes except the Hamilton-Jacobi type. At the end of computation (Fig. 1.15), only the fth-order HJ-WENO scheme ts better the interface.

In Table 1.5, the mass error E M is always smaller than the positioning errors except for the shearing vortex simulation at T /2 with some advection schemes. This is due only to the fact that if the mass increases/decreases, it will be taken into account in the positioning error. The HJ-WENO5 scheme produces the smallest mass and positioning errors for all the simulations, but it does not reproduce thin structures very well.

Level-set method

t = 6.28 t = T/2 = 3 t = T = 6 t = T/2 = 1 t = T = 2 E M (%) E 1 (%) E M (%) E 1 (%) E M (%) E 1 (%) E M (%) E 1 (%) E M (%) E 1 (%)
Error of the reference eld Sub-cell x method with the HJ-ENO2 reinitialization scheme for dierent advection schemes.

∆τ = ∆x/2, τ max = 10∆x and Nreinit = 10. Green: Exact interface ; Blue: Sub-cell x method.

1. Level-set method 1.6. Results and discussions 1.6.2.2 Sensibility to the CFL number

0.2 0.4 UPWIND1 HJ-ENO2 HJ-WENO3 HJ-WENO5 ENO2-Adv ENO3-Adv 0.2 0.4 ENO4-Adv WENO5-Adv HOUC5-Adv ENO2-Est ENO3-Est ENO4-Est 0.2 0.4 WENO5-Est HOUC5-Est ENO2-Jia ENO3-Jia ENO4-Jia WENO5-
The inuence of the CFL number on the sub-cell x method will be investigated for three advection schemes, which are the HJ-WENO5, the WENO5-Jia and the HOUC5 schemes. We use the HJ-ENO2 scheme for the reinitialization. Also, we dene a CFL number related to the reinitialization, that we name CFLreinit, dened as:

CFLreinit = CFL × Nreinit (1.65)
where Nreinit is the number of advection steps before each reinitialization of the level-set. The CFLreinit should not be confused with the CFL number for the reinitialization equation, which is kept constant equal to 0.5 since ∆τ = ∆x/2. The CFLreinit represents a link between the reinitialization and the advection steps. A dierent approach was investigated in [START_REF] Gómez | On the reinitialization procedure in a narrow-band locally rened level set method for interfacial ows[END_REF] and later used by [START_REF] Herrmann | A balanced force rened level set grid method for two-phase ows on unstructured ow solver grids[END_REF], which consist of reinitializing the level-set if its deviation from the signed-distance function E |∇φ| is greater than a certain value. This latter is not investigated in the current work. Our approach is interesting in the sense where in practical simulations, the deviation of the level-set from the signed-distance is not computed after each advection step, contrary to the approach of [START_REF] Gómez | On the reinitialization procedure in a narrow-band locally rened level set method for interfacial ows[END_REF].

Fig. 1.16 presents in solid lines (for Nreinit = 10) the positioning error E 1 , the mass error E M and the mean absolute deviation of the level-set from a signed-distance function E |∇φ| as a function of the CFL number for the three test cases. For the three advection schemes, the errors are strongly dependent on the CFL number. Near CFL = 1.0, the errors seem to stabilize and are in general very small. However, when decreasing the CFL number these errors increase to unacceptable levels. This is principally due to the fact that for small CFL numbers, the reinitialization frequency during the simulation is very high and each reinitialization deteriorates the solution. The changes in the level-set eld for small time steps are also very limited and does not justify a reinitialization every 10 time steps. In conclusion, for all cases and advection schemes, the errors are minimal only for CFL numbers in the range [0.1, 1] when the Nreinit is kept equal to 10.

When on the other hand the CFLreinit is kept constant, the interface errors remain constant and the simulations do not diverge even for very small CFL values (dash-dot lines with cross markers on Fig. 1.16). The errors are plotted for the CFLreinit leading to the smallest errors for all the simulations for each advection scheme. Our choice is CFLreinit = 2 for the HJ-WENO5 scheme, 8 for the WENO5-Jia scheme and 9 for the HOUC5 scheme. We remark also that the signed-distance property is not altered when keeping CFLreinit constant, since E |∇φ| remains constant.

Fig. 1.17 shows the dependency of the interface errors on the CFLreinit value in the range [10 -3 , 10]. The plotted errors correspond to two dierent CFL numbers: 10 -3 (symbols) and 10 -2 (solid lines). In general, a CFLreinit in the range [1,10] seems to be a good choice.

Inuence of the reinitialization scheme

Now, we test the dierent reinitialization schemes coupled with the sub-x method. The advection schemes that we employ here are the HJ-WENO5 and WENO5-Jia schemes. The HJ-WENO5 scheme is the one giving good results for almost all the simulations. Table 1.6 summarizes the errors E M and E 1 for the Zalesak disk and the shearing vortex simulations on a 100 2 grid and for the time-reversed single vortex on a 64 2 grid. The CFL number and CFLreinit value are 0.5 and 5.0, respectively.

The interface errors remain in the same range when changing the reinitialization scheme. This is due to the rst-order accuracy of the sub-cell x method at the interface. From these results, 1. Level-set method we may conclude that the HJ-ENO2 scheme seems to be the most suited for the HJ-WENO5 advection, while the HJ-ENO4 scheme seems to be more suited for the WENO5-Jia advection.

Convergence rate

The results presented previously have been computed on a 100 2 grid. We have shown the interface errors depend not only on the CFL number itself, but also on the frequency of reinitialization. This latter has led to the introduction of the CFLreinit. Now, we study the inuence of the grid size and the CFL number on the sub-cell x method. Only the Zalesak disk after a full revolution and the shearing vortex problems are treated. The results for the time-reversed single vortex problem are similar to the shearing vortex. The reinitialization equation is discretized in space using the HJ-ENO2 scheme. Fig. 1.18 shows the positioning error, E 1 , after a complete rotation for dierent CFLreinit numbers. We present also the curve for CFLreinit = ∞, which corresponds to a unique reinitialization of the level-set function. The reinitialization in this case is performed after the rst advection step. For all high-order schemes, a nearly second-order behavior is observed except Table 1.6: Sub-cell x: eect of the reinitialization scheme on the HJ-WENO5 and the WENO5 advection schemes. Errors are expressed in %.

Simulation Zalesak disk

Shearing vortex Time-reversed single vortex 1. Level-set method for very ne grids and for small values of CFLreinit. The smallest errors and a genuine secondorder accuracy is obtained for the HOUC schemes without reinitialization, but the same schemes show very poor results when the sub-cell x reinitialization is used. This is also illustrated in Figure 1.20: Sub-cell x: the 7th-order HOUC scheme for the advection with CFL = 0.25 and Nreinit=10. left: Zalesak disk after a full revolution on a 200 2 grid and right: shearing vortex at t = T /2 = 3 on a 800 2 grid. The HJ-ENO2 scheme is used for the reinitialization. Green: Exact interface; red: Sub-cell x advection.

t = 6.28, Grid 100 2 t = T /2 = 3, Grid 100 2 t = T /2 = 1, Grid 64 2 Adv. sch. HJ-WENO5 WENO5 HJ-WENO5 WENO5 HJ-WENO5 WENO5 Reinit. sch. E M E 1 E M E 1 E M E 1 E M E 1 E M E 1 E M E 1 UPWIND1 -0.
1. Level-set method 1.6.3 Volume constraint

In the previous sections, we have investigated the inuence of the dierent parameters such as the choice of the advection and reinitialization schemes and the CFL number on the sub-cell x method. In the current section, we will study their inuence on the volume constraint method, which was described in 1.3.2.

Eect of the advection scheme

The reinitialization equation is solved here after each advection step (Nreinit= 1) during a ctitious time of τ max = 2∆x with a step size of ∆τ = 0.5∆x. The HJ-ENO2 scheme is used for the spatial discretization of the reinitialization equation. The CFL number for advection is kept constant equal to 0.5. We present the resulting interfaces obtained with dierent advection scheme for the following cases:

• Zalesak disk after a full rotation, in Fig. 1.21, on a 100 2 grid.

• Shearing vortex at t = T /2 = 3 and t = T = 6, in Figs. 1.22 and 1.23, on a 100 2 grid.

• Time-reversed single vortex at t = T /2 = 1 and t = T = 2, in Figs. 1.24 and 1.25, on a 64 2 grid.

Table 1.7 summarizes the positioning and the mass errors.

Results and discussions

We observe for the Zalesak disk in Fig. 1.21 a great improvement for most schemes when compared to the results obtained with reinitialization but without the volume constraint in The HOUC advection schemes with fth-or higher order are the only schemes that show a deterioration of the interface, as was the case with the sub-cell x. The high-order Hamilton-Jacobi schemes do a very good job on the other hand. The HJ-WENO5 scheme gives the best interface transport for the sub-cell x method, while the slab's height becomes smaller than what it is initially with the volume constraint method. For the conservative schemes, we remark that the ENO2 scheme causes the thickening of the slab. For higher order schemes, the disk ts almost the exact interface and we can hardly distinguish the ENO3, ENO4 and WENO5 results. The results for the time-reversed single vortex (Figs. 1.24 and 1.25) are, as in the previous cases, very similar to those obtained for the shearing vortex, with the dierence that the interface undergoes less deformation. The best results are obtained with the ENO3, ENO4 and WENO5 schemes at T /2 as well as at T , while the interface vanishes at time T with the UPWIND1 scheme.

Now, if we compare the results in Table 1.7 with those obtained with the sub-cell x method in Table 1.5, we remark that the volume constraint generally performs better than the sub-cell

x for the Zalesak disk and the shearing vortex for all advection schemes. However, the sub-cell

x method seems to be more accurate for the time-reversed single vortex for all schemes except the ENO4 scheme in its three versions. The WENO5 scheme in its three versions (Jia, Adv and Est) is also more accurate than the HJ-WENO5 scheme for all test problems.

1.6. Results and discussions 1. Level-set method 1.6. Results and discussions

0.2 0.4 0.6 UPWIND1 HJ-ENO2 HJ-WENO3 HJ-WENO5 ENO2-Adv ENO3-Adv 0.2 0.4 0.6 ENO4-Adv WENO5-Adv HOUC5-Adv ENO2-Est ENO3-Est ENO4-Est 0.2 0.4 0.6 WENO5-Est HOUC5-Est ENO2-Jia ENO3-Jia ENO4-Jia WENO5-
0.2 0.4 UPWIND1 HJ-ENO2 HJ-WENO3 HJ-WENO5 ENO2-Adv ENO3-Adv 0.2 0.4 ENO4-Adv WENO5-Adv HOUC5-Adv ENO2-Est ENO3-Est ENO4-Est 0.2 0.4 WENO5-Est HOUC5-Est ENO2-Jia ENO3-Jia ENO4-Jia WENO5-
t = 6.28 t = T/2 = 3 t = T = 6 t = T/2 = 1 t = T = 2 E M (%) E 1 (%) E M (%) E 1 (%) E M (%) E 1 (%) E M (%) E 1 (%) E M (%) E 1 (%)
Error of the reference eld We now investigate the CFL eect on the interface positioning and mass errors for the level-set method with volume constraint. The reinitialization equation is again discretized in space using the HJ-ENO2 scheme. The reinitialization equation is solved at each physical time step for a ctitious time of τ max = 2∆x with a time step of ∆x/2. We will use the WENO5 advection scheme, which gave relatively good results (see Table 1.7) and we add to this study the HJ-WENO5 and HOUC5 advection schemes. = 1 ; dash-dot lines with cross markers: constant CFLreinit (= 0.2 for HJ-WENO5 ; = 0.2 for WENO5 and = 0.9 for HOUC5).

Results and discussions

To have an idea of the best values for CFLreinit, Fig. 1.27 plots the interface errors as a function of the CFLreinit for two CFL numbers, 10 -2 (solid lines) and 10 -3 (symbols). It is seen that the CFL number has no inuence on the accuracy when the CFLreinit is kept constant. In most cases, a large range of suitable values for CFLreinit can be chosen (roughly between 0.01 and 1) without degrading the solution. We remark also that the WENO5 scheme performs again slightly better than the two other presented schemes.

Eect of the reinitialization scheme

We now study the inuence of the reinitialization scheme on the level-set method with volume constraint for CFL = CFLreinit = 0.5. The ctitious time for the reinitialization is again τ max = 2∆x and the ctitious time step ∆τ = ∆x/2. We investigate the inuence of the reinitialization schemes on the volume constraint method combined with two advection schemes:

WENO5 and HJ-WENO5.

Table 1.8 summaries the mass and the positioning errors for the Zalesak disk after a full rotation, the shearing vortex and the time-reversed single vortex at the half period. We remark that the UPWIND1 reinitialization leads to the highest errors, while the remaining schemes give 1. Level-set method relatively similar results for the shearing and time-reversed single vortex problems. The HJ-ENO2 and the HJ-WENO3 schemes seem to give the best results and at the same time they are faster than the higher-order schemes (HJ-ENO4, HJ-WENO5).

Table 1.8: Volume constraint: eect of the reinitialization scheme on the HJ-WENO5 and the WENO5 advection. Errors are expressed in %.

Simulation

Zalesak disk at t = 6.28 Shearing vortex at t = 3 Time-reversed at t = 1 Adv. scheme HJ-WENO5 WENO5 HJ-WENO5 WENO5 HJ-WENO5 WENO5 Reinit. scheme We use the HJ-ENO2 for reinitialization and investigate the spatial order of each advection scheme in combination with the volume constraint method by varying the grid size and the CFLreinit number. The positioning errors, E 1 , are presented on Fig. 1.28. In contrast to the sub-cell x method, the volume constraint is less sensitive to the CFLreinit number in the range

E M E 1 E M E 1 E M E 1 E M E 1 E M E 1 E M E 1 UPWIND1 -
[0.1, 1].
The volume constraint method is almost second-order for all advection schemes for which the order of accuracy is greater than 1. It is also more accurate than the sub-cell x method. 1. Level-set method

-2 -1 0 UPWIND1 HJ-ENO2 HJ-WENO3 HJ-WENO5 HOUC5 ENO2-Adv ENO3-Adv -2 -1 0 ENO4-Adv WENO5-Adv HOUC5-Adv ENO2-Jia ENO3-Jia ENO4-Jia WENO5-Jia -2 -1 0 HOUC5-Jia ENO2-Est ENO3-Est

Conclusions

In this chapter, we have investigated the implementation of the level-set method. We started rst by a validation of the method with the results provided by [START_REF] Enright | A hybrid particle level set method for improved interface capturing[END_REF]; our implementation is slightly more accurate. Then, we compared the level-set with and without reinitialization. The reinitialization disturbs the zero level-set, while it conserves the signeddistance property. When we do not reinitialize the level-set, the interface transport is more accurate for high-order advection schemes, in particular the HOUC schemes. However, without reinitialization the level-set will deviate progressively from the signed-distance function, and eventually become highly inaccurate.

Many attempts to enhance the reinitialization procedure of the level-set method were proposed in the literature. Two popular methods were tested in the current work, namely the sub-cell x and the volume constraint. The level-set with sub-cell x seem to be less robust and sensitive to the choice of parameters and schemes. For each given ow problem, we should nd the appropriate advection scheme for optimal results. For the Zalesak disk, the fth-order HJ-WENO scheme is a good choice, but this was not the case for the shearing vortex problem.

The reinitialization with volume constraint is generally more accurate and robust. We have seen that it gives good results in combination with the conservative schemes, in particular the WENO5 scheme, and it is less inuenced by variations of the CFL number. Our tests have also shown that the reinitialization scheme has not much inuence on the results for both methods.

The 2nd-order HJ-ENO scheme combines small interface errors and reduced computational eort with respect to the higher-order reinitialization schemes.

In realistic simulations, the CFL number is probably limited by the physical phenomena taken into account (surface tension, viscous diusion, etc.). The CFL number required for the stability of the computation may therefore be very small. The performed tests shows that the interface errors may increase drastically for very small CFL numbers, since in this case the interface is advected less and reinitialized more often. The current results show a relevant relation between the frequency of the reinitialization and the interface errors. This led to the introduction of the CFLreinit number, which is the product of the CFL number and the number of advection steps to perform before each reinitialization of the level-set. Keeping the CFLreinit constant leads almost to constant errors for all CFL numbers (provided CFL 1). Results have shown that the sub-cell

x method is nonetheless more sensitive to the CFLreinit number variations; it diverges when CFLreinit is smaller. The volume constraint method is less inuenced by CFLreinit variations for values in the range [0.1 -1.0].

The CPU-time has not been taken into account in this study, which has been focused mainly on the identication of the level-set method which combines both accuracy and robustness. On the one hand, in realistic simulations the level-set method will run in parallel with the ow solver, which will certainly be more CPU-time consuming. On the other hand, a local resolution of the level-set has been implemented in our code (see 1.4) and can be applied in order to reduce the computational time. It has not been used in the results shown in this chapter, but when used it roughly gives the same results.

In the light of the presented results on this chapter, the level-set with volume constraint seems to be a good choice. In addition, we will use in the reminder of this work:

• the 5th-order WENO-Adv scheme for advection

• reinitialization after each advection step (Nreinit = 1) using the 2nd-order HJ-ENO scheme. This choice oers the advantage of being robust (works ne on all the performed simulations) and less sensitive to the CFL number.

Chapter 2

Volume-Of-Fluid methods

Introduction

The Volume-Of-Fluid method (VOF) is an Eulerian method for capturing the interfaces separating immiscible uids. Let f be the characteristic function with value 1 in the reference uid and zero elsewhere. The volume-of-uid (VOF) function C i is then dened as the volume fraction of the reference uid in the grid cell Ω i :

C i = 1 V i Ω i f dv (2.1)
where V i = vol(Ω i ) is the volume of the grid cell. In a ow without phase change, the phase indicator function f is constant on the trajectory of a uid particle (in a Lagrangian reference system) so that we can write

Df Dt = ∂f ∂t + u • ∇f = 0
In incompressible ows, we have also ∇ • u = 0 so that we can write

∂f ∂t + u • ∇f + f ∇ • u = ∂f ∂t + ∇ • (f u) = 0 (2.2)
Combining Eqs.(2.1) and (2.2) and making use of the divergence theorem, we can write an evolution equation for C:

∂C i ∂t + 1 V i Ω i ∇ • (f u) dv = ∂C i ∂t + 1 V i ∂Ω i f u • n dS = 0 (2.3)
where ∂Ω i is the boundary of the grid cell Ω i and n is the unit normal vector on ∂Ω i pointing outwards of Ω i . The volume-of-uid is therefore a conserved quantity in an incompressible uid.

In the literature, Eq. (2.3) is often expressed simply as an advection equation:

∂C i ∂t + ∇ • ( uC) = 0 (2.4)
Two dierent approaches are proposed in the literature in order to compute the evolution of C i (t). The rst one consists of numerically solving the advection equation (2.4) using the so-called compressive scheme [START_REF] Weller | The development of a new ame area combustion model using conditional averaging[END_REF][START_REF] Ubbink | Numerical prediction of two uid systems with sharp interfaces[END_REF][START_REF] Ubbink | A method for capturing sharp uid interfaces on arbitrary meshes[END_REF][START_REF] Weller | A new approach to vof-based interface capturing methods for incompressible and compressible ow, opencfd ltd[END_REF]Denner et al., 2014b), which are based on the donor-acceptor scheme by [START_REF] Hirt | Volume of uid (vof ) method for the dynamics of free boundaries[END_REF] and does not necessitate interface reconstruction in the advection process. The second one uses 2. Volume-Of-Fluid methods a geometrical evaluation of the uxes in Eq. ( 2.3) through the cell faces [START_REF] Youngs | Time-dependent multi-material ow with large uid distortion[END_REF][START_REF] Pilliod | An analysis of piecewise linear interface reconstruction algorithms for volume-ofuid methods[END_REF][START_REF] Rider | Approximate projection methods for incompressible ows: Implementation, variants and robustness[END_REF][START_REF] Rider | Reconstructing volume tracking[END_REF][START_REF] Scardovelli | Interface reconstruction with least-square t and split eulerian lagrangian advection[END_REF], which indeed needs the knowledge of the interface form. The use of the rst or the second approach depends strongly on the considered grid type. If the grid is cartesian and uniform, the geometrical resolution is the most suited since it is relatively more accurate. On the counterpart, the compressive schemes could be used for arbitrary mesh [START_REF] Ubbink | A method for capturing sharp uid interfaces on arbitrary meshes[END_REF] and is more easily implemented in existing codes with unstructured grids.

In the current work, the geometrical approach is adopted. Therefore, in order to compute the evolution of C i (t) we need to reconstruct the scalar eld of f in the neighborhood of the grid cell which serves to compute the numerical ux f u • n on the boundary. Volume-of-uid methods based on the geometrical approach are therefore always build as a two-step procedure. In the so-called reconstruction step, a subset of C i in the neighborhood of Ω i is used to estimate the phase indicator f in this grid cell. From this representation of the interface, Eq. ( 2.3) is solved by evaluating the uxes in a geometrical way.

Interface reconstruction

The volume-of-uid (VOF) method was originally developed in the 1970s. [START_REF] Rider | Reconstructing volume tracking[END_REF] have presented a historical overview of the earlier papers on VOF. It was soon discovered that in order to preserve a sharp interface, standard nite-volume upwind methods are not well suited since they tend to introduce smearing of the interface. Instead, a geometrical reconstruction of the interface followed by an advection step which preserves the discontinuous uid properties seems to be a much better approach. The most simple geometrical reconstruction method is the simple line interface calculation (SLIC) method [START_REF] Noh | Slic (simple line interface calculation)[END_REF], which describes the interface as a horizontal or vertical line cutting the grid cell in two parts. Although very simple to implement, the method is not very accurate and the interface reconstruction is not smooth. For a more realistic description of the interface, the piece-wise linear interface calculation (PLIC) method was proposed [START_REF] Youngs | Time-dependent multi-material ow with large uid distortion[END_REF]. The interface is still described by a straight line (or plane surface in 3D), but the normal of the interface is not necessarily aligned with the vertical or horizontal axis but can take on any direction. In PLIC methods, the interface described by a plane surface is most easily dened in the Hessian normal form

x • n + d = 0
where n is the unit normal vector. The parameter d represents the distance of the interface to the origin which is chosen here arbitrary as the center of the grid cell under consideration. Once the normal vector n is known, the distance d follows from the consideration that the volume fraction dened by the interface reconstruction equals the target value (see A.3.2), for example in 2D:

C i,j = 1 V i,j Ω i,j H( x • n i,j + d i,j ) dv
PLIC methods dier in the way the normal vector is dened (see for example [START_REF] Scardovelli | Interface reconstruction with least-square t and split eulerian lagrangian advection[END_REF] and [START_REF] Pilliod | Second-order accurate volume-of-uid algorithms for tracking material interfaces[END_REF]). In the least-squares volume-of-uid interface reconstruction algorithm (LVIRA) by [START_REF] Puckett | A volume-of-uid interface tracking algorithm with applications to computing shock wave refraction[END_REF], the linear interface reconstruction in a grid cell is extended to the neighboring grid cells (as shown by the dashed line on Fig. 2.1).

The dierence between the reconstructed and actual volume fraction is then minimized in the least-squares sense by adjusting the normal to the interface. In 2D, the least-squares error is 2.2. Interface reconstruction dened as:

E i,j = i+1 k=i-1 j+1 l=j-1 C k,l ( n i,j ) -C i,j ) 2 (2.5)
where C k,l is the reconstructed volume fraction, which depends on the orientation of the interface dened through n: Although this method has shown to be second-order accurate for smooth surfaces, its main drawback is that the optimization algorithm tends to be time-consuming, especially in 3D. The ecient least-squares volume-of-uid interface reconstruction algorithm (ELVIRA) by [START_REF] Pilliod | An analysis of piecewise linear interface reconstruction algorithms for volume-ofuid methods[END_REF] reduces the costs by limiting the optimization procedure to a set of 6 candidate normals in 2D (9 in 3D) constructed from simple nite-dierencing. The rst three candidates are constructed from backward, central and forward nite-dierence approximations in the x-direction:

C k,l = 1 V k,l Ω k,l f ( x • n i,j + d i,j ) dv
n b x = j+1 l=j-1 (C i,l -C i-1,l ) n c x = 1 2 j+1 l=j-1 (C i+1,l -C i-1,l ) n f x = j+1 l=j-1 (C i+1,l -C i,l )                         
with n y = 1.

Another 3 candidates are obtained by dierentiation in the y-direction:

n b y = j+1 l=j-1 (C k,j -C k,j-1 ) n c y = 1 2 j+1 l=j-1 (C k,j+1 -C k,j-1 ) n f y = j+1 l=j-1 (C k,j+1 -C k,j )                          with n x = 1.
The normal vector is then selected among these 6 candidates as the one that gives the smallest value for the least-squares error, E i,j . We use the ELVRA reconstruction except otherwise 2. Volume-Of-Fluid methods specied.

Many other reconstruction techniques were proposed in literature, that we do not implement in the current work, among which we cite

• The parabolic reconstruction of the interface developed by [START_REF] Renardy | Prost: A parabolic reconstruction of surface tension for the volumeof-uid method[END_REF] • The linear and quadratic least-squares ts of the interface in 2D [START_REF] Scardovelli | Interface reconstruction with least-square t and split eulerian lagrangian advection[END_REF] which are extended to three-dimensions by [START_REF] Aulisa | Interface reconstruction with leastsquares t and split advection in three-dimensional cartesian geometry[END_REF]. This tting is slightly more accurate than the ELVIRA, however it needs more computational resources.

• Later, [START_REF] Vignesh | Noniterative interface reconstruction algorithms for volume of uid method[END_REF] suggested to reconstruct a continuous interface from the knowledge of one endpoint of the interface which is determined from the already reconstructed interface on the neighboring cell.

A comparison of the earlier reconstruction methods can be found in Rider and Kothe (1998), [START_REF] Scardovelli | Interface reconstruction with least-square t and split eulerian lagrangian advection[END_REF] and [START_REF] Pilliod | Second-order accurate volume-of-uid algorithms for tracking material interfaces[END_REF].

Interface advection

Although the transport equation ( 2.3) is conservative by construction, a numerical scheme that conserves the total volume of the reference uid is not so easily obtained. The main problem resides in the fact that the volume fraction C is limited between 0 and 1. Many standard nite-volume schemes will cause small over-or undershoots of C, and subsequent clipping of the volume fractions will inevitably lead to loss of the conservation properties. This section describes several geometrical methods developed in the literature that are both consistent (0 ≤ C ≤ 1)

and conservative.

Conservative operator splitting

We will start by describing a splitting method that allows us to solve the transport equation independently for each spatial direction. A conservative split method [START_REF] Puckett | A high-order projection method for tracking uid interfaces in variable density incompressible ows[END_REF] is

given on a 2D Cartesian grid by:

C * i,j = C n i,j + (δt/∆x) F i+1/2,j -F i-1/2,j 1 -(δt/∆x) u i+1/2,j -u i-1/2,j (2.6) C n+1 i,j = C * i,j 1 + δt ∆y v i,j+1/2 -v i,j-1/2 + δt ∆y G i,j+1/2 -G i,j-1/2 (2.7)
where F and G are the mass uxes across the cell faces in the xand y-direction. Note that this is a numerical approximation of the following advection equation:

∂C i ∂t + ∇ • ( uC) = C(∇ • u)
rather than the conservation equation (2.4). This scheme will nonetheless conserve the mass if the ow eld is incompressible, i.e.

(u i+1/2,ju i-1/2,j )/∆x + (v i,j+1/2v i,j-1/2 )/∆y = 0 which can easily be shown by inserting Eq. (2.6) into Eq. (2.7).

However, despite the conservation of mass, it is still possible to obtain non-physical values for C n+1 i,j , below zero or above one. In order to maintain C n+1 i,j within the interval [0, 1], it is necessary that the computation of the numerical uxes F i+1/2,j and G i,j+1/2 are based on a geometrical approach, as explained in detail in 2.3.2 and 2.3.3. This operator splitting scheme is made second-order in time by using Strang-splitting. The simplest implementation is to alternate the order of spatial directions on subsequent time steps. [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF] have extended this scheme to 2D cylindrical coordinates and 3D cartesian coordinate systems. For cylindrical coordinates, we may use:

C * i,j = C n i,j + (δt/r i ∆r) r i+1/2 F i+1/2,j -r i-1/2 F i-1/2,j 1 -(δt/r i ∆r) r i+1/2 u i+1/2,j -r i-1/2 u i-1/2,j
(2.8)

C n+1 i,j = C * i,j 1 + δt ∆y v i,j+1/2 -v i,j-1/2 + δt ∆y G i,j+1/2 -G i,j-1/2
(2.9)

and nally for 3D equations:

C * i,j,k = C n i,j,k + (δt/∆x) F i+1/2,j,k -F i-1/2,j,k 1 -(δt/∆x) u i+1/2,j,k -u i-1/2,j,k (2.10) C * * i,j,k = C * i,j,k + (δt/∆y) G i,j+1/2,k -G i,j-1/2,k 1 -(δt/∆y) v i,j+1/2,k -v i,j-1/2,k (2.11) C * * * i,j,k = C * * i,j,k + (δt/∆z) H i,j,k+1/2 -H i,j,k-1/2 1 -(δt/∆z) w i,j,k+1/2 -w i,j,k-1/2
(2.12)

C n+1 i,j,k =C * * * i,j,k -δt C * i,j,k ∆x u i+1/2,j,k -u i-1/2,j,k + C * * i,j,k ∆y v i,j+1/2,k -v i,j-1/2,k + C * * * i,j,k ∆z w i,j,k+1/2 -w i,j,k-1/2 (2.13)
To show that this scheme preserves the total mass, one needs to substitute the expression of

C * i,j,k , C * * i,j
,k and C * * * i,j,k given by the Eqs. (2.10-2.12) in Eq. (2.13), then to sum over all the computational cells. This does not, however, guarantee that C n+1 i,j,k is within the interval [0, 1].

One-dimensional explicit Lagrangian scheme

This section describes how the numerical ux is computed in the second step Eq. (2.7) of the 2D directional split scheme. We will rst focus on the one-dimensional Eulerian cell E i dened by the interval [x i-1/2 , x i+1/2 ], as proposed by [START_REF] Tryggvason | Direct numerical simulations of gas-liquid multiphase ows[END_REF]. Consider the explicit Lagrangian advection formula for a point x given by

x n+1 = x n + u(x n )∆t
The velocity is now described by the linear interpolation formula:

u(x) = u i-1/2 + (u i+1/2 -u i-1/2 )(x -x i-1/2 )/∆x (2.14)
The advection equation describes how each uid particle in the Eulerian cell E i is moved from t n to t n+1 . The set of particles positions x n+1 denes a new interval called the Lagrangian cell L i . From the above formulas, we can easily calculate the locations of the boundaries of the 2. Volume-Of-Fluid methods Lagrangian cell, noted x L,n+1 i+1/2 :

x L,n+1 i+1/2 = x i+1/2 + u i+1/2 ∆t
The compression (or extension) coecient of the cell, dened as the size ratio of the Lagrangian and Eulerian cells, is now given by

a i = x L,n+1 i+1/2 -x L,n+1 i-1/2 ∆x = 1 + (u i+1/2 -u i-1/2 )∆t ∆x (2.15)
where ∆x = x i+1/2x i-1/2 . It is also possible to trace backward the initial locations of the uid particles x E,n i+1/2 which end up at the Eulerian boundary x i+1/2 at t n+1 :

x i+1/2 = x E,n i+1/2 + u x E,n i+1/2 ∆t
Supposing that u i+1/2 is positive as shown in Fig. 2.2, we obtain

x E,n i+1/2 = x i+1/2 -u i+1/2 ∆t /a i
where a i is the compression coecients for the donor cell E i . If the sign of u i+1/2 is negative, the indices of a should be changed so that it always points to the donor cell i.e. E i+1 . It is therefore convenient to dene

a i+1/2 = a i if u i+1/2 ≥ 0 a i+1 if u i+1/2 ≤ 0 (2.16)
In Fig. 2.2, the dashed zones indicate the volume that is uxed through the cell boundaries. The numerical ux F i+1/2 is therefore dened as

F i+1/2 = S u i+1/2 ∆x∆y y j+1/2 y j-1/2
x i+1/2 -min(0,u i+1/2 )∆t/ai+1

x i+1/2 -max(0,u i+1/2 )∆t/ai f (x, y) dx dy

(2.17)

where f (x, y) is the phase indicator function obtained from the reconstructed interface indicated by the red line in Fig. 2.2 and S() is the sign function. The integration limits are written in a way they work for both positive and negative values of u i+1/2 . The one-dimensional explicit Lagrangian scheme can also be written as an algebraic equation as:

u i-1/2 ∆t a i-1 u i+1/2 ∆t a i F i-1/2 F i+1/2 i -1 i i + 1 Advection n n + 1 III II I u i+1/2 ∆t u i-1/2 ∆t
C n+1 i = a i C n i -a i+1/2 F i+1/2 + a i-1/2 F i-1/2 (2.18)
where the rst term on the right-hand side corresponds to the sum of the areas (II) and (III), the second term subtracts the area (III) and the third term is equal to the area (I). If u i+1/2 or u i-1/2 changes sign, the gray areas in Fig. 2.2 should be adapted accordingly, but Eq. (2.18) remains valid. Note that the multiplication factor a i+1/2 is indexed according to the sign of u i+1/2 and not to the cell under consideration (it is attached to the cell face, like the numerical ux F i+1/2 ), it is given by Eq. (2.16). We can therefore write:

N i=1 C n+1 i = N i=1 a i C n i -a N F N +1/2 + a 0 F 1/2
On its own, this scheme is not conservative, due to the coecients a i in the rst term, but we see that the uxes at the inner cell boundaries cancel out, in a manner similar to classical nite volume scheme. We will see in 2.3.4 that the combination with the implicit Lagrangian scheme described in 2.3.3 will make the splitting method conservative. It is important to note here that the present scheme is consistent due to the geometrical reconstruction of the reference uid in the grid cell at the next time step.

Finally, Eq. (2.18) can also be written as

C n+1 i -C n i ∆t + u i+1/2 C i+1/2 -u i-1/2 C i-1/2 ∆x = C n i u i+1/2 -u i-1/2 ∆x (2.19)
if we dene C i+1/2 = a i+1/2 F i+1/2 ∆x/(u i+1/2 ∆t). This equation can be seen as a discrete approximation of the advection equation ∂C/∂t + ∂(uC)/∂x = C∂u/∂x with the right-hand side term in the explicit formulation (with respect to C).

One-dimensional implicit Lagrangian scheme

This section describes how the numerical ux is computed in the rst step, Eq. (2.6), of the 2D directional split scheme. We will again focus on the Eulerian cell E i . Consider now the implicit Lagrangian advection formula given by

x n+1 = x n + u(x n+1 )∆t (2.20)
The velocity is again described by the linear interpolation, Eq. (2.14). The advection equation denes the Lagrangian cell L i with boundaries x L,n+1 i+1/2 given by:

x L,n+1 i+1/2 = x i+1/2 + u x L,n+1 i+1/2 ∆t
Supposing again that u i+1/2 is positive, the combination of Eqs. (2.14) and (2.20) leads to the following expressions:

x L,n+1 i+1/2 = x i+1/2 + b i+1 u i+1/2 ∆t
where b i is the compression (or extension) ratio of the cell given by

b i = 1 1 -u i+1/2 -u i-1/2 ∆t/∆x (2.21)
The index of b i refers to the receiving cell and should be modied when u i+1/2 changes sign.

Tracing backwards the location of the cell boundary is relatively simple and gives:

x i+1/2 = x E,n+1 i+1/2 + u i+1/2 ∆t
F i+1/2,j = S u i+1/2 ∆x∆y y j+1/2 y j-1/2
x i+1/2 -min(0,u i+1/2 )∆t

x i+1/2 -max(0,u i+1/2 )∆t f (x, y) dx dy (2.22)
The volume fraction C n+1 is dened as the area of the two grayed surfaces (I) and (II), whereas the area (III) contributes to the fraction C n+1 i+1 of the neighboring cell E i+1 .

The equivalent algebraic equation for the one-dimensional implicit Lagrangian scheme is:

C n+1 i = b i C n i -F i+1/2 + F i-1/2 (2.23)
where the rst term on the right-hand side corresponds to the area (II) and the second term is equal to the area (I). Note that the multiplication factors b i for the numerical uxes F i-1/2 and F i+1/2 are indexed according to the cell under consideration. Therefore, the numerical uxes do not cancel out if we sum up the values of C n+1 i Finally, Eq. (2.23) can also be written as

for i = 1, • • • , N . u i-1/2 ∆t u i+1/2 ∆t F i-1/2 F i+1/2 i -1 i i + 1 Advection n n + 1 III II I b i+1 u i+1/2 ∆t b i u i-1/2 ∆t
C n+1 i -C n i ∆t + u i+1/2 C i+1/2 -u i-1/2 C i-1/2 ∆x = C n+1 i u i+1/2 -u i-1/2 ∆x (2.24)
if we dene C i+1/2 = F i+1/2 ∆x/(u i+1/2 ∆t). This equation can be seen as a discrete approximation of the advection equation ∂C/∂t + ∂(uC)/∂x = C∂u/∂x with the right-hand side term in

Conservative and consistent ux splitting

It is clear that the geometrical construction of the volume fraction always leads to consistent values of C: the cell is either empty, C = 0, completely lled, C = 1, or partially lled, 0 < C < 1.

This can only be obtained by a geometrical construction in which the set of Lagrangian cells do not overlap (with areas counted twice) and do not present gaps, as is the case for the explicit and implicit schemes described above. Both schemes are not conservative, but for a two-dimensional incompressible ow, the application of the implicit scheme in one direction, followed by the explicit scheme in the other direction leads to the desired conservation properties. Let us take the x-direction for the implicit scheme, so that we can write the following equation for the provisional volume fraction C * i,j :

C * i,j = b i,j C n i,j -F i+1/2,j + F i-1/2,j (2.25)
where F i+1/2,j is dened by Eq. (2.22). The explicit scheme is used for the y-direction:

C n+1 i,j = a i,j C * i,j -a i,j+1/2 F * i,j+1/2 + a i,j-1/2 F * i,j-1/2 (2.26)
where F i,j+1/2 is dened by Eq. (2.17), with f computed from the interface reconstruction based on C * i,j . The coecient a i,j+1/2 is dened by a i,j or a i,j+1 depending on whether v i,j+1/2 is positive or negative.

The coecients a i,j and b i,j are dened by

a i,j = 1 + v i,j+1/2 -v i,j-1/2 ∆t/∆y b i,j = 1 -u i+1/2,j -u i-1/2,j ∆t/∆x -1
and due to the incompressibility condition,

u i+1/2,j -u i-1/2,j ∆x + v i,j+1/2 -v i,j-1/2 ∆y = 0 the product a i,j b i,j is equal to unity, a i,j b i,j = 1 + v i,j+1/2 -v i,j-1/2 ∆t/∆y 1 -u i+1/2,j -u i-1/2,j ∆t/∆x = 1
The combined implicit/explicit ux splitting scheme nally leads to the following global equation:

C n+1 i,j = a i,j b i,j C n i,j -F i+1/2,j + F i-1/2,j -a i,j+1/2 F i,j+1/2 + a i,j-1/2 F i,j-1/2 = C n i,j -F i+1/2,j + F i-1/2,j -a i,j+1/2 F i,j+1/2 + a i,j-1/2 F i,j-1/2
It is already shown that the uxes of the explicit scheme cancel out on inner cell boundaries.

Due to the incompressibility condition, the uxes in the x-direction now also cancel out and we can write:

N i=1 M j=1 C n+1 i,j = N i=1 M j=1 C n i,j + M j=1 F 1/2,j -F N +1/2,j + N i=1 a i,1/2 F i,1/2 + a i,M +1/2 F i,M +1/2
This ux-splitting scheme is therefore both conservative and consistent. To obtain second-order accuracy in time, Strang-splitting should be used which consists of alternating the xand ydirections for successive time steps. This also avoids the development of ow structures in privileged directions due to the use of dierent schemes in each direction.

Finally, the conservation properties can also be evaluated from the following equivalent discrete approximations of the advection equations:

C * i,j -C n i,j ∆t + u i+1/2,j C i+1/2,j -u i-1/2,j C i-1/2,j ∆x = C * i,j u i+1/2,j -u i-1/2,j ∆x (2.27) C n+1 i,j -C * i,j ∆t + v i,j+1/2 C * i,j+1/2 -v i,j-1/2 C * i,j-1/2 ∆y = C * i,j v i,j+1/2 -v i,j-1/2 ∆y (2.28)
It is now easily seen that due to the combination of the implicit and explicit treatment of the compression terms on the right-hand side, they cancel out in an incompressible ow. The global equation reads:

C n+1 i,j -C n i,j ∆t + u i+1/2,j C i+1/2,j -u i-1/2,j C i-1/2,j ∆x + v i,j+1/2 C * i,j+1/2 -v i,j-1/2 C * i,j-1/2 ∆y = 0
and is in conservative form.

An alternative simple conservative and consistent ux splitting

A dierent simpler conservative and consistent ux splitting, working in both two and three dimensions, is recently proposed by [START_REF] Weymouth | Conservative volume-of-uid method for free-surface simulations on cartesian-grids[END_REF].

C * i,j -C n i,j ∆t + u i+1/2,j C i+1/2,j -u i-1/2,j C i-1/2,j ∆x = f i,j u i+1/2,j -u i-1/2,j ∆x (2.29) C n+1 i,j -C * i,j ∆t + v i,j+1/2 C * i,j+1/2 -v i,j-1/2 C * i,j-1/2 ∆y = f i,j v i,j+1/2 -v i,j-1/2 ∆y (2.30)
where f i,j is the characteristic function evaluated at the cell center, given by:

f i,j = 1 if C n i,j > 1/2 0 Otherwise.
which is treated fully explicitly. Note that if there is no cell compression/expansion, Eqs. (2.29-2.30) and Eqs. (2.27-2.28) are identical. The update of the volume fractions is now performed as follows:

C * i,j = C n i,j -F i+1/2,j + F i-1/2,j + f i,j (u i+1/2,j -u i-1/2,j )∆t/∆x (2.31) C n+1 i,j = C * i,j -G i,j+1/2 + G i,j-1/2 + f i,j (v i,j+1/2 -v i,j-1/2 )∆t/∆y (2.32)
where the uxes F and G on the cell faces are computed using the implicit Lagrangian mapping in Eq. (2.22). It is easily seen that this scheme is conservative. [START_REF] Weymouth | Conservative volume-of-uid method for free-surface simulations on cartesian-grids[END_REF] presented a proof of the consistency, which for brevity sake is omitted here. This alternative conservative and consistent ux splitting is also implemented in our VOF solver, combined with the ELVIRA reconstruction of the interface, and will be referred to as VOF-WY.

2.4. Moment-Of-Fluid (MOF) method 2.4 Moment-Of-Fluid (MOF) method

The Moment-of-Fluid method is an improvement of the VOF method developed initially by [START_REF] Dyadechko | Moment-of-uid interface reconstruction[END_REF], 2006, 2008). The general idea is to store not only the volume fraction, but also the mass centroid of the reference uid in each grid cell. A more accurate estimation of the interface normal can be obtained in this manner. An additional advantage is that information of the neighboring cells is not necessary, making it possible to capture small scale interface structures.

The method is based on moments from which we can deduce the volume fraction and centroid of the reference phase. Given a grid cell Ω and the phase indicator function f , the rst two moments of the volume occupied by the reference uid are dened as:

M 0 ≡ Ω f dv (2.33) M 1 ≡ Ω xf dv (2.34) If V ≡ Ω
dv is the volume of the grid cell, the volume fraction and centroid are subsequently dened as:

C ≡ M 0 V (2.35) x c ≡ M 1 M 0 (2.36)
In the MOF method, a piecewise linear interface (PLIC) reconstruction is used to describe the interface as in most second-order VOF methods. Each interface is therefore dened by its normal n and distance from the cell center d. It is not possible in the general case to nd a linear interface which exactly matches both the volume fraction and centroid. The priority is given to the volume fraction, which is matched exactly, whereas a best approximation is sought for the centroid. For a given normal vector n, the distance d is therefore dened by the condition that the volume occupied by the reference uid is exactly equal to M 0 = C • V , which in turn also denes the centroid x * c of the reconstructed cell. The interface reconstruction procedure therefore consists in nding the normal vector n that gives the best match for x * c by minimizing the function:

E M OF = || x * c ( n) -x c || 2 (2.37)
The original MOF scheme is implemented using a multi-dimensional Lagrangian remapping advection scheme. However, this scheme does not exactly conserve the cell volume and a local correction is added to avoid inconsistency. Since the idea of the MOF method can be applied to any VOF advection scheme, we will use here the conservative and consistent directional-split advection scheme described in 2.3.4.

Interface reconstruction

The procedure for the interface reconstruction in 2D is explained in detail by [START_REF] Dyadechko | Moment-of-uid interface reconstruction[END_REF]. We write the linear interface representation in the Hessian normal form x • n + d = 0 where the origin for x is the center of the grid cell under consideration and n is the unit normal vector. In a two-dimensional setting, the interface is entirely dened by the polar angle φ, since the normal vector can be expressed as n (φ) = (cos (φ) , sin (φ)) and the distance d is 2. Volume-Of-Fluid methods determined from the target volume fraction C. This also denes the centroid of the reconstructed reference phase as a function of φ, i.e. x * c (φ). A rst estimation of the normal vector is given by n 0 =x ref c , from which we can deduce the initial estimate of the angle φ 0 . We will use the Gauss-Newton algorithm to minimize the centroid error E M OF . A dierent method is used by [START_REF] Dyadechko | Moment-of-uid interface reconstruction[END_REF] for general polyhedral grid cells.

The derivative of Eq. (2.37) with respect to φ gives

E M OF = 2 x c (φ) -x ref c • x c (2.38)
where x c is given by

x c = - |Γ(φ)| 3 12C|Ω| t (2.39)
where |Γ(φ)| is the length of the interface, |Ω| is the cell volume and t = (-sin(φ), cos(φ)) is the tangent unit vector. A minimum error E M OF is obtained when

( x c (φ) -x ref c ) • t = 0, which implies that ∆ x c = x c (φ) -x ref c
should be aligned with n. Eq. (2.39) suggest that a good new estimate for φ is then

φ k+1 = φ k + ∆ x c • t |Γ(φ)| 3 12C|Ω|
which corresponds to the Gauss-Newton method. Iterations are repeated until ∆φ falls below a small value, say 10 -6 . A slightly more robust estimate of φ k+1 is obtained if we consider the function g

(φ) = ( x c (φ) -x ref c ) • t. The derivative of g is then g (φ) = x c • t + ∆ x c • ∂ t ∂φ = - 12|Γ(φ)| 3 C|Ω| -∆ x c • n
The Newton method to obtain g(φ) = 0 (condition for a minimum of E M OF ) gives

φ k+1 = φ k - g(φ k ) g (φ k ) = ∆ x c • t |Γ(φ)| 3 12C|Ω| + ∆ x c • n
This method converges in most cases except when the grid cell is almost empty C < 0.01 or entirely lled C > 0.99. For these cases, we use the ELVIRA method to estimate the interface normal. After having determined the normal to the interface, the distance d follows from enforcing the volume fraction.

The one-dimensional explicit Lagrangian scheme revisited

First-order moments of the reference phase uxed through the left and right cell boundaries are also computed:

M 1,i+1/2,j = S(u i+1/2,j ) ∆x∆y

y j+1/2 y j-1/2 x i+1/2 -min(0,u i+1/2 )∆t/a i+1 x i+1/2 -max(0,u i+1/2 )∆t/a i xf (x, y)dxdy (2.40)
from which we can easily deduce the centroid of the uxed uid x c,i+1/2,j = M 1,i+1/2,j /F i+1/2,j .

From the denitions in Eqs. (2.33) and (2.34), it is clear that when the cell is divided in parts, the moments can be obtained by addition of the partial moments, which is exactly what is done for the volume fraction update:

C n+1 i = a i C n i -a i+1/2 F i+1/2 + a i-1/2 F i-1/2
The update of the centroid can also be written as the sum of partial rst-order moments:

x n+1 c,i C n+1 i = x c,II a i C n i -x c,III a i+1/2 F i+1/2 + x c,I a i-1/2 F i-1/2
(2.41) where x c,I , x c,II and x c,III denote the centroids of the cell parts in Fig. 2.2.

Since the velocity is described by linear interpolation Eq. (2.14), for a given volume-of-uid, the location of the centroid does not change relative to the (moving) boundaries. It is therefore convenient to compute rst the non-dimensional centroid, 0 < x c < 1, which remains constant when moving from t n and t n+1 . The centroids of the uxed uid (explicit or implicit scheme)

can then be computed easily from the new locations of boundaries.

The one-dimensional implicit Lagrangian scheme revisited

In addition to the uxed uid fraction F i+1/2,j dened by Eq. (2.22), the rst moments of the reference phase uxed through the left and right cell boundaries are also computed:

M 1,i+1/2,j = S(u i+1/2,j ) ∆x∆y

x i+1/2 -min(0,u i+1/2 )∆t x i+1/2 -max(0,u i+1/2 )∆t xf (x, y)dxdy (2.42)
from which we can deduce the centroids x c,i+1/2 = M 1,i+1/2,j /F i+1/2,j . The update of the volume fraction and the centroid is therefore according to

C n+1 i = b i C n i -F i+1/2 + F i-1/2 x n+1 c,i C n+1 i = b i x c,II C n i -x c,III F i+1/2 + x c,I F i-1/2
(2.43)

Overview of tested methods

The volume-of-uid methods conserve the total mass. The relative mass error, E M , given by Eq. (1.59) is therefore extremely small and mainly due to the initialization of the fractions. In the absence of initialization errors, E M is of the order of 10 -14 . The positioning error, E 1 , of the interface for the volume-of-uid methods, however, is not negligible. It is dened in the same sense as for the level-set method, as:

E 1 = Ω |f R -f C |dv Exact volume (2.44)
where the superscripts R and C refer to the reference and computed solutions and f is the phase indicator function.

For the Zalesak disk (1.5.2), the exact signed-distance to the interface is known. We note this signed-distance φ E , which is dened on the cell vertices. The positioning error, E 1 , in Eq. (2.44) is computed by:

E 1 = (i,j)∈Ω Ω i,j |H(φ E i,j ) -f C i,j |dΩ Exact volume (2.45)
Using the same procedure as for the level-set on A.1, we compute the positioning error, E 1 , by subdividing the cell in 1000 2 sub-cells, and we determine the fraction provided by the reference solution inside each sub-cell using bicubic interpolation if the cell is mixed. For the computed 2. Volume-Of-Fluid methods solution, we use the linear interpolation since the reconstruction algorithm provides a linear interface inside mixed cells, or the fraction for one uid cells.

In the case of the shearing vortex (1.5.3) and the time-reversed single vortex (1.5.4), we use the Lagrangian markers as a reference solution as in 1.5.1. The same treatment as described for the Zalesak disk is adopted for the error computation at the end of the simulation. On a given (coarse) grid at an intermediate time, we use a slightly dierent formulation than that in Eq (2.45). The procedure is:

1 Then, the positioning error, E 1 , is computed from:

E 1 = (i,j)∈Ω |C R i,j -C C i,j |∆x∆y Exact volume (2.46)
The volume-of-uid methods we have implemented are purely geometric as described before.

We have tested the three methods :

VOF

The VOF method that we implemented in this work is given by Algorithm 2.1. Note that the direction x and y are alternated after each time step.

(C * , n * , d * , v n+1/2 ) u n+3/2 ← Advance ow equations (C n+1 , u n+1/2 , ∆t) return C n+1 , u n+3/2

VOF-WY

The same algorithm as VOF (given on Algorithm 2.1), except that the volume fraction is advected as recommended by [START_REF] Weymouth | Conservative volume-of-uid method for free-surface simulations on cartesian-grids[END_REF]

n * , d * , v n+1/2 ) u n+3/2 ← Advance ow equations (C n+1 , u n+1/2 , ∆t) return C n+1 , x n+1 c , u n+3/2

Results and discussion

We start rst by validating our implementation of the VOF method by comparing it with the results provided by [START_REF] Scardovelli | Interface reconstruction with least-square t and split eulerian lagrangian advection[END_REF] for the time-reversed single vortex of 1.5.4 at the nal time t = T = 2. The CFL number based on the maximum velocity is kept constant for all the simulations and equal to unity. The only dierence between our simulations and that of [START_REF] Scardovelli | Interface reconstruction with least-square t and split eulerian lagrangian advection[END_REF] is the manner in which the positioning errors are computed. In our computation, the error E 1 is computed using the interpolation inside mixed cells, while the average fraction over the cell is considered in [START_REF] Scardovelli | Interface reconstruction with least-square t and split eulerian lagrangian advection[END_REF].

Table 2.1 below shows the interface positioning errors, E 1 . The method named "ELVIRA/EI-LE" is identical to our VOF implementation and the error levels compare well. For the VOF-WY, the positioning error is slightly higher compared to the VOF method, which is perhaps due to the large CFL number. The MOF method is more accurate than the volume-of-uid methods.

Table 2.1: Time-reversed single vortex: comparison of our VOF implementation with the results of [START_REF] Scardovelli | Interface reconstruction with least-square t and split eulerian lagrangian advection[END_REF]. [START_REF] Scardovelli | Interface reconstruction with least-square t and split eulerian lagrangian advection[END_REF] Our implementation In the next section, the inuence of the CFL number on the accuracy of the interface transport will be investigated. The spatial convergence rate is presented on 3.6.3.

ELVIRA/EILE VOF VOF-WY MOF Grid E 1 (%) O E 1 (%) O E 1 (%) O E 1 (%) O 32 2 3.

Zalesak disk

The Zalesak disk interface is plotted after a full revolution when its interface is advected using the VOF and MOF method on a 100 2 grid on Fig. 2.4 for three dierent CFL numbers. The VOF-WY gives exactly the same interface as the VOF method since the term (u i+1/2,ju i-1/2,j )/∆x = (v i,j+1/2v i,j-1/2 )/∆y = 0 which means that there is no compression/expansion of the cells (see 2.3.5). The gure shows that the MOF method is less sensitive to the CFL number than the VOF method.

Exact

Results and discussion

is observed when the CFL number is decreased. This is due to the use of the information of the centroids that makes the interface reconstruction and advection more accurate than in the standard VOF methods.

In comparison with the level-set method with volume constraint (Figs. 1.21 and 1.26 in 1.6.3), it seems that the level-set method advected with WENO5 scheme is better (if we do not take into consideration the mass loss/gain) than the VOF method, and is comparable to the MOF method.

2.6.2 Time-reversed single vortex

In the time-reversed single vortex, the interface is only moderately stretched at t=T/2 since the velocity magnitude is decaying in time and the period is relatively small (T=2). The obtained interface at maximum stretching when advected with the three tested methods is represented on Again, we see that the MOF method is more precise than the VOF and VOF-WY methods. The positioning error at T /2 is less than that at T , which is due to the accumulation of errors. This error does not vary for the CFL number in the range [10 -4 , 10 -1 ]. In this range, the VOF and the VOF-WY methods provide exactly the same interface error. However, for CFL in [0.1, 1],

the VOF-WY method seems to be slightly more accurate than the VOF method. In this range, the positioning error decreases than increases. The MOF method remains more accurate than the VOF and VOF-WY methods. The volume-of-uid and MOF methods predict better this test case than the level-set method with volume constraint (Figs. 1.24 and 1.26 on 1.6.3).

Shearing Vortex

In the shearing vortex simulation, the interface undergoes more stretching and deformation than for the previous test case. Fig. 2.8 plots the interface of the shearing vortex at time t = T /2 = 3 for the two VOF methods and MOF method. We choose again the three CFL values: 1.0, 0.5 and 0.1. We see that when rening the time step, the VOF and VOF-WY methods loose in precision, i.e. the vortex is more fragmented and the non-fragmented part of the lament becomes shorter.

These two methods also have a larger error in the prediction of the starting point of the lament.

The MOF method seems to predict it more accurately and generates a less fragmented lament. Its particularity when rening the time step is to make the lament tail slightly displaced from the exact solution.

The fragmentation of thin laments seems to be a result and counterpart of the mass conservation property of VOF methods. As shown in the previous chapter, the level-set methods tend to either thicken or shorten the lament, gaining or losing mass in the process, the VOF methods break up the laments in small parts of thickness roughly the size of several grid cells.

The overall mass is conserved, but the fragmentation introduces non-physical droplets in the simulation.

Fig. 2.9 shows the results when the velocity is inverted at t = T /2 and the interface should return to the initial position (circle) at t = T . Again, we remark that the interface position prediction is better for high CFL numbers. When the CFL number is too small, a part of the lament is not advected back into the circle. The MOF method seems to be more aected by this phenomena as shown also on Fig. 2.10. This is perhaps due to the very thin lament obtained at t = T /2, which is not advected accurately backward until t = T .

In comparison to the level-set method with volume constraint (Figs. 1.22, 1.23 and 1.26 in 1.6.3), the volume-of-uid methods tend to break the lament (at t = T /2 = 3) while the level-set method makes it thicker. The MOF method is clearly better than the level-set method. 2. Volume-Of-Fluid methods

VOF VOF-WY MOF

Conclusions

In this chapter, we have investigated the geometrical resolution of the volume-of-uid advection equation in two-dimensional uniform grids. The selected directional-splitting methods for temporal integration are consistent (keeps C within the limits of 0 and 1) and conservative (the total mass of the reference uid is conserved). We have implemented the ELVIRA and moment-ofuid method to reconstruct the piecewise linear interface. We investigated the accuracy of these methods for solving two-dimensional interface transport problems using the well know test cases of the Zalesak disk, the shearing vortex and the time-reversed single vortex.

In general, both the volume-of-uid and moment-of-uid methods are shown to be more accurate and robust than the level-set methods. In particular, the moment-of-uid method is more accurate and less sensitive to the time step variations than the ELVIRA method. On the contrary, the moment-of-uid method is much more dicult to implement, in particular when cylindrical or unstructured grids need to be used. The ELVIRA volume-of-uid methods are relatively simple to implement, in particular on Cartesian grids and using the basic operations dened in Appendix A. While not explicitly shown, the volume-of-uid methods require less computational time than the level-set methods due the localized approach (the reconstruction algorithm is applied only on grid cells containing the interface). However, the volume-of-uid and moment-of-uid methods tend to break up thin laments into articial small droplets of the size of a few grid cells.

Chapter 3

Coupled Volume-Of-Fluid and Level-Set methods

Introduction

A coupled method combines two dierent interface methods, often Eulerian, although some techniques couple an Eulerian and a Lagrangian interface methods. The coupling between Eulerian methods was rst conceived by [START_REF] Bourlioux | A coupled level-set volume-of-uid algorithm for tracking material interfaces[END_REF], who proposed to combine the level-set and the volume-of-uid methods. The idea is to gather the advantages of each one in the same method:

the well-dened interface description of the level-set method and the mass conservation of the volume-of-uid methods. This rst coupling technique became well known by the work of Sussman and [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF], who proposed a detailed description of the algorithm and improved the coupling technique by reinitializing the level-set method by means of the level-set and the volume-of-uid method. They also described the implementation of the method for axisymmetric and three-dimensional coordinate systems and named it Coupled Level-Set and Volume-Of-Fluid (CLSVOF) method. Later, [START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF] have presented a slightly dierent implementation of the CLSVOF method in two dimensions and in axisymmetric coordinate systems, then in three dimensions in a subsequent paper [START_REF] Son | Ecient implementation of a coupled level-set and volume-of-uid method for threedimensional incompressible two-phase ows[END_REF]. The Hybrid Particle-Level-Set (PLS)

method which couples the level-set with a Marker method (Lagrangian) has been implemented

by [START_REF] Enright | A hybrid particle level set method for improved interface capturing[END_REF]. The markers in this case ensure the mass conservation of the level-set method and enhance the interface description in the under-resolved regions. However, these markers need to be reseeded when the interface undergoes topological variations. In the current work, we focus on the study of the coupling techniques issued from the Eulerian methods, mainly the level-set and the volume-of-uids. We recall that the reason of coupling the level-set and the volume-of-uid methods is to gather the advantages of both methods in the same coupled method. The volume-of-uid method conserves mass but the interface representation depends on the reconstruction algorithm, and is often discontinuous on the junction between two neighboring cells. On the contrary, the level-set method gives a good interface representation despite of not conserving the mass. We can distinguish two families of the coupled methods:

1. The level-set and the volume-of-uid are both advected in time. In this category, we have for example the CLSVOF method (Sussman et al., 1999), the CLSMOF method [START_REF] Jemison | A coupled level set-moment of uid method for incompressible two-phase ows[END_REF] and MCLS method (van der [START_REF] Van Der Pijl | A mass-conserving level-set method for modelling of multi-phase ows[END_REF]. The coupled methods dier in how each method interacts with the other.

2. Only VOF is advected in time, and the level-set eld is deduced from the volume fractions.

In this category, the normal to the interface is either computed by means of the PLIC In order to verify the benets of coupled methods with respect to the basic VOF and LS methods, we have selected, among the most popular coupled methods, the VOSET, MCLS, CLSVOF and CLSMOF approaches. In the next subsections, we will detail the main steps constituting each one of these coupled methods. Then, we test the ability of each method in transporting simple interfaces.

Coupled Volume-Of-uid and level-SET method (VOSET)

The coupled volume-of-uid and level-set (VOSET) method was rst introduced by Sun and Tao (2010). It consists only of advecting the volume fractions in time, and then computing by geometric iterations the signed-distance function to calculate the surface tension term. It is relatively easy to implement in the sense that when the volume-of-uid method is already used, it is easy to integrate the computation of the signed-distance function.

Few works are related to the VOSET method in the literature since it is a recent technique.

We can for example mention the work of [START_REF] Guo | Phase change heat transfer simulation for boiling bubbles arising from a vapor lm by the voset method[END_REF] and [START_REF] Ling | A direct numerical simulation for nucleate boiling by the voset method[END_REF], where it is used to study phase change phenomena. Recently, it has been extended to adaptive grid systems in [START_REF] Wang | A coupled volume-of-uid and level set (voset) method on dynamically adaptive quadtree grids[END_REF][START_REF] Wang | Numerical simulation of two-phase ows using 3d-voset method on dynamically adaptive octree grids[END_REF], then applied to study the bubble dynamics in the presence of electric elds [START_REF] Wang | Numerical simulation of bubble dynamics in a uniform electric eld by the adaptive 3d-voset method[END_REF][START_REF] Wang | Three-dimensional numerical simulation of bubble dynamics in microgravity under the inuence of nonuniform electric elds[END_REF]. In another study, [START_REF] Shi | Numerical simulation of a falling ferrouid droplet in a uniform magnetic eld by the voset method[END_REF] has applied the VOSET method to simulate a falling ferrouid droplet in a uniform magnetic eld. The VOSET method is also presented for 3D cartesian grids in [START_REF] Ling | A three-dimensional volume of uid & level set (voset) method for incompressible two-phase ow[END_REF].

The main steps constituting the VOSET method are summarized in 2D in Algorithm 3.1.

Note that for a pure interface advection problem (without the Navier-Stokes equations), the 3.2. Coupled Volume-Of-uid and level-SET method (VOSET)

VOSET is identical to the VOF method.

return φ n+1 , u n+3/2 , C n+1
In this work, the VOSET is implemented as follows. The volume-of-uid is rst advected using the conservative-consistent EI/LE operator splitting of 2.3.4. In the advection step, the interface is reconstructed using the ELVIRA algorithm, whereas Sun and Tao (2010) used the Youngs-PLIC method for this purpose. The reconstruction algorithm gives in the case of the VOSET method also a rst estimation of the interface for the signed-distance reconstruction.

Once the VOF function is advected and the interface is reconstructed, an iterative procedure is used to determine the signed-distance function, with the following steps:

Step 1. Set the initial eld for the signed-distance function φ in the whole computational domain such as:

φ 0 i,j = -β if C i,j ≥ 0.5 β if C i,j < 0.5 (3.1)
where β is a distance of few grid sizes. It's value will be given in the next step.

Step 2. Flag cells near the interface. The region of agging is of width of 3∆x on each side of the interface. This region is large enough to smooth the discontinuities and to determine accurately the curvature. With the agging region equal to 3∆x, β should be taken as 4∆x.

Step 3. Compute the signed-distance function φ in the agged region around the interface. This is done by computing the minimal distance d separating the cell center (i, j) from the interface on the region of 3∆x around the interface. Then, set the sign of the distance according to the volume fraction inside the cell (i, j) as follows:

φ i,j =    -d if C i,j ≥ 0.5 0 if C i,j = 0.5 d if C i,j < 0.5 (3.2)
Step 4. In each cell containing the interface (0 < C < 1), reconstruct the piecewise-linear interface by rst determining the normal from the level-set function as n = ∇φ/|∇φ| using second-order central dierencing. The distance of this interface to the cell center is then the only free parameter that we should enforce to satisfy the volume fraction.

Step 5. Once the reconstruction in each interface cell is completed, compute a new signeddistance function by repeating steps 1 to 4 if the maximum number of iterations is not reached.

3. Coupled Volume-Of-Fluid and Level-Set methods [START_REF] Sun | A coupled volume-of-uid and level set (voset) method for computing incompressible two-phase ows[END_REF] recommended to repeat steps 1 to 4 one time if the grid is coarse, and three times for ne grids. In our computations, we have used 1, 3, 5 and 10 iterations with a xed grid size.

Mass Conserving Level-Set method (MCLS)

The mass conserving level-set (MCLS) was rst introduced in two-dimensional cartesian grids In this work, inside the reference uid the level-set is (arbitrary) dened negative and the volume fraction is equal to unity. In the following, we drop the index (i, j). In 2D, the volume fraction is given by:

C = f (φ, ∇φ) =                        1 if φ ≤ -φ max 1 - 1 2 (φ max + φ) 2 φ 2 max -φ 2 mid if -φ max < φ < -φ mid 1 2 - φ φ max + φ mid if -φ mid ≤ φ ≤ φ mid 1 2 (φ max -φ) 2 φ 2 max -φ 2 mid if φ mid < φ < φ max 0 if φ ≥ φ max (3.4)
Note that the gradient ∇φ is included in the function arguments in order to emphasize that the interface normal is computed from the derivatives of φ prior to evoking Eq. (3.4). This The level-set represents the signed distance to the interface. Correcting this distance consists in moving the reconstructed interface in the normal direction to this interface. So, we need rst to compute φ max and φ mid for each mixed cell, which is given by Eq. (3.3). We dene an intermediate fraction C mid :

C mid = φ max + 3φ mid 2(φ max + φ mid )
The fractions C mid and 1-C mid correspond to the cases where the interface cuts through corners of the cell (the level-set equals φ mid and -φ mid respectively, see Fig. 3.1. When the estimated volume fraction inside the cell (Eq. (3.4)) is not equal to the volume fraction C n+1 , we correct the level-set. The new signed-distance function in this cell is then given by:

φ = g(C, ∇φ) =              φ max if C ≤ 0 ζ if 0 < C < 1 -C mid γ if 1 -C mid ≤ C ≤ C mid β if C mid < C < 1 -φ max if C ≥ 1 (3.5) β = 2(1 -C) φ 2 max -φ 2 mid -φ max γ = -C - 1 2 (φ max + φ mid ) ζ = -2C φ 2 max -φ 2 mid + φ max
This function computes the distance from a given volume fraction and normal n. Note that a similar function (Algorithm A.2) with a slightly dierent implementation is also used in the VOF method to compute the distance d.

Volume-of-uid advection and ux computation

The advection of the volume fractions is realized through the conservative and consistent directional splitting as in 2.3.4, with some modications in the manner of determining the uxes through the cell faces. The ux at the cell-face (i + 1/2, j), for example, is given by

F i+1/2,j =            u i+1/2,j ∆t f φ i+1/2,j , -u i+1/2,j ∆t ∆x n x,i+1,j , n y,i+1,j t if u i+1/2,j < 0 u i+1/2,j ∆t f φ i+1/2,j , u i+1/2,j ∆t ∆x n x,i,j
, n y,i,j t elsewhere.

(3.6) in the implicit step Eq. (2.27), and

F i+1/2,j =           
u i+1/2,j ∆t f φ i+1/2,j , -u i+1/2,j ∆t a i+1,j ∆x n x,i+1,j , n y,i+1,j t if u i+1/2,j < 0 u i+1/2,j ∆t f φ i+1/2,j , u i+1/2,j ∆t a i,j ∆x n x,i,j , n y,i,j t elsewhere.

(3.7)

in the explicit step in Eq. (2.28) where f () permits to determine the volume fraction and is given by the Eq. (3.4). φ i+1/2,j is the level-set extrapolated in both, space and time, given in the implicit step by:

φ i+1/2,j =        φ i+1,j - ∆x 2 1 + u i+1/2,j ∆t ∆x n x,i+1,j if u i+1/2,j < 0 φ i,j + ∆x 2 1 -u i+1/2,j ∆t ∆x n x,i,j if u i+1/2,j > 0 and φ i+1/2,j =        φ i+1,j - ∆x 2 1 + u i+1/2,j ∆t a i+1,j ∆x n x,i+1,j if u i+1/2,j < 0 φ i,j + ∆x 2 1 -u i+1/2,j ∆t a i,j ∆x n x,i,j if u i+1/2,j > 0
in the explicit step.

In the original MCLS method (van der [START_REF] Van Der Pijl | A mass-conserving level-set method for modelling of multi-phase ows[END_REF], the authors used the scheme of [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF] given by Eq. 2.27 and 2.28 for the advection of the volume fractions with the uxes computed according to Eq. (3.6) in both the implicit and the explicit steps, which leads to a non-consistency of the fractions. A ux redistribution step is therefore proposed in which the over-/undershoots of the volume fraction after advection are redistributed to neigh-3. Coupled Volume-Of-Fluid and Level-Set methods boring cells to conserve mass. This ux redistribution is not needed in our implementation since the volume fractions are advected using the consistent and conservative scheme.

In this chapter, we compare 4 dierent implementations of the MCLS method where only the level-set advection and reinitialization dier.

3.4 Coupled Level-Set and Volume-of-Fluid method (CLSVOF)

The coupled level-set and volume-of-uid (CLSVOF) method is the most well-known coupling technique. The pioneering work is from [START_REF] Bourlioux | A coupled level-set volume-of-uid algorithm for tracking material interfaces[END_REF], but it became popular after the work of [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF]. Later, [START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF] have proposed a slightly dierent implementation for 2D cartesian and 2D-axisymmetric grid systems, then a fully 3D implementation in a subsequent paper [START_REF] Son | Ecient implementation of a coupled level-set and volume-of-uid method for threedimensional incompressible two-phase ows[END_REF]. A multidirectional implementation of the CLSFVOF on non-uniform grids is proposed by [START_REF] Ningegowda | A coupled level set and volume of uid method with multi-directional advection algorithms for two-phase ows with and without phase change[END_REF], in which the level-set advection terms are discretized using second-order ENO scheme and the volume-ofuid is advected using the Edge Matched Flux Polygon Advection (EMFPA) algorithm [START_REF] López | A volume of uid method based on multidimensional advection and spline interface reconstruction[END_REF].

The CLSVOF has been the subject of comparison with the level-set method and with volumeof-uid methods for example in [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF]; Wang et al. (2012b); [START_REF] Gerlach | Comparison of volume-of-uid methods for surface tension-dominant two-phase ows[END_REF] and Wang et al. (2012a). In addition to the classical tests of validation of interface methods, the CLSVOF method has been used for example in the study of bubble formation and/or bubble rising [START_REF] Ohta | A computational study of the eect of initial bubble conditions on the motion of a gas bubble rising in viscous liquids[END_REF][START_REF] Wang | Deformation and oscillations of a single gas bubble rising in a narrow vertical tube[END_REF][START_REF] Gerlach | Numerical simulation of periodic bubble formation at a submerged orice with constant gas ow rate[END_REF][START_REF] Buwa | Numerical simulations of bubble formation on submerged orices: Period-1 and period-2 bubbling regimes[END_REF][START_REF] Chakraborty | Bubble generation in quiescent and coowing liquids[END_REF][START_REF] Chakraborty | A coupled level-set and volume-of-uid method for the buoyant rise of gas bubbles in liquids[END_REF][START_REF] Albadawi | Inuence of surface tension implementation in volume of uid and coupled volume of uid with level set methods for bubble growth and detachment[END_REF], jet atomization [START_REF] Menard | Coupling level set/vof/ghost uid methods: Validation and application to 3d simulation of the primary break-up of a liquid jet[END_REF], boiling [START_REF] Tomar | Numerical simulation of bubble growth in lm boiling using a coupled level-set and volume-of-uid method[END_REF][START_REF] Tomar | Multimode analysis of bubble growth in saturated lm boiling[END_REF], coalescence and breakup processes [START_REF] Kwakkel | Extension of a clsvof method for droplet-laden ows with a coalescence/breakup model[END_REF], impact of a droplet on a dry surface [START_REF] Yokoi | Numerical studies of the inuence of the dynamic contact angle on a droplet impacting on a dry surface[END_REF]. [START_REF] Gerlach | Comparison of volume-of-uid methods for surface tension-dominant two-phase ows[END_REF] have compared the two surface tension models: Continuous Surface Force (CSF) and Continuous Surface Stress (CSS) on the "spurious currents" test case and concluded that the CLSVOF-CSF behaves slightly better than CLSVOF-CSS. In a subsequent paper, they extended their code to axisymmetric geometries and simulated periodic bubble formation at a submerged orice [START_REF] Gerlach | Numerical simulation of periodic bubble formation at a submerged orice with constant gas ow rate[END_REF].

The CLSVOF algorithm has been detailed in [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF] for axisymmetric and cartesian grid systems. Algorithm 3.4 summarizes the dierent steps of this method.

Computation of the interface normal

We seek the normal to the interface of the cell (i, j), which can be computed either by solving a minimization problem of the error between the level-set and a linear approximation [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF][START_REF] Menard | Coupling level set/vof/ghost uid methods: Validation and application to 3d simulation of the primary break-up of a liquid jet[END_REF], or by using the second-order central dierences of the level-set [START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF], or using the local averaging [START_REF] Vigneaux | Méthode level set pour des problèmes d'interfaces en microuidique[END_REF].

• In [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF] and [START_REF] Menard | Coupling level set/vof/ghost uid methods: Validation and application to 3d simulation of the primary break-up of a liquid jet[END_REF], the normal vector is determined by minimizing the error between φ and a reconstructed bilinear level-set function φ R i,j

E i,j = x i+1/2 x i-1/2 y j+1/2 y j-1/2 δ(φ) φ -φ R i,j 2
dxdy
where δ is the Dirac function and φ R i,j is dened by φ R i,j = n x,i,j (xx i ) + n y,i,j (yy j ) + d i,j

(3.8)

φ n+1 ← Reinitialize the level-set (φ n+1 , C n+1 ) u n+3/2 ← Advance ow equations (φ n+1 or C n+1 , u n+1/2 ) return φ n+1 , C n+1 , u n+3/2
where n x,i,j , n y,i,j are respectively the normal components following the xand the y-direction and d i,j the distance from the cell center (x i , y j ) to the interface. The superscript R refers to the reconstructed interface. Using a nine-point stencil, the integral is discretized as [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF]:

E i,j = i+1 i =i-1 j+1 j =j-1 w i -i,j -j δ (φ i ,j ) φ i ,j -n x,i,j (x i -x i ) -n y,i,j (y i -y j ) -d i,j 2
where δ is the smoothed Dirac function dened by

δ (x) =    0, |x| > 1 2 1 + cos πx , |x| < with = √ 2∆x.
The discrete weights w i -i,j -j are chosen to be as w i -i,j -j = 16 if ii = jj = 0 and w i -i,j -j = 1 if ii = 0 or jj = 0. An expression of the weights in three dimensions is provided by [START_REF] Jemison | A coupled level set-moment of uid method for incompressible two-phase ows[END_REF].

The minimization of the error reads:

∂E i,j ∂n x,i,j = ∂E i,j ∂n y,i,j = ∂E i,j ∂d i,j = 0
and this leads to a resolution of the 3 × 3 linear system:

   
whX 2 whXY whX whXY whY 2 whY whX whY wh

      n x,i,j n y,i,j d i,j   =     whφX whφY whφ     (3.9)
with the following notations [START_REF] Menard | Coupling level set/vof/ghost uid methods: Validation and application to 3d simulation of the primary break-up of a liquid jet[END_REF]:

←→ i+1 i =i-1 j+1 j =j-1 wh ←→ w i -i,j -j δ (φ i ,j ) X ←→ x i -x i Y ←→ y i -y j φ ←→ φ i ,j
The resolution of this linear system in Eq. (3.9) gives the components of the normal to the interface and the distance. We should then normalize the normal components.

• [START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF] have proposed a simple method which consists of computing the normal to the interface directly from the level-set using the second-order central dierences scheme. where (δφ/δx) i = (φ i+1φ i-1 )/(2∆x).

Once the normal to the interface is computed, the distance from the interface to the cell center should satisfy the volume fraction inside the cell if this latter is available (in our case, we use Algorithm A.2). If the volume fractions are not available, especially in the rst time step where only the level-set eld is known, the volume fraction inside the mixed cell (i, j) is computed as:

C i,j = 1 ∆x∆y x i+1/2 x i-1/2 y j+1/2 y j-1/2 H φ R i,j dxdy = 1 ∆x∆y x i+1/2 x i-1/2 y j+1/2 y j-1/2
H n x,i,j (xx i ) + n y,i,j (yy j ) + φ i,j dxdy where H is the Heaviside function.

Note also that the latter equation, combined with the method to determine the interface normal, is yet another way to compute the volume fractions from the level-set. A similar function, Eq. (3.4), was proposed in the MCLS method and described in A.3.2. The integral can also be computed with the Algorithm A.1, given the normal from CLSVOF.

Volume-of-uid advection

For the advection of the volume-of-uid, we use the conservative and consistent ux splitting detailed on 2.3.4. For the x-direction:

C * i,j -C n i,j ∆t + u i+1/2,j C i+1/2,j -u i-1/2,j C i-1/2,j ∆x = C * i,j u i+1/2,j -u i-1/2,j ∆x (2.27)
and for the y-direction :

C n+1 i,j -C * i,j ∆t + v i,j+1/2 C * i,j+1/2 -v i,j-1/2 C * i,j-1/2 ∆y = C * i,j v i,j+1/2 -v i,j-1/2 ∆y (2.28)
This scheme is consistent and conservative, i.e. the fractions are never neither greater than unity nor negative and the total mass is conserved. However, the volume fraction inside a given cell could be slightly lower than unity or slightly greater than zero due to round-o errors. The remedy to this is to truncate the volume fractions using the level-set as follows: (3.11) This truncation may also eliminate the small non-physical uid particles (oatsam and jetsam).

C i,j =    0 if C i,j ≤ 0 or φ i,j > ∆x 1 if C i,j ≥ 1 or φ i,j < -∆x C i,j otherwise.
Of course, the truncation of the volume fractions causes unfortunately a little variation in the total mass [START_REF] Wang | A coupled level set and volume-of-uid method for sharp interface simulation of plunging breaking waves[END_REF][START_REF] Li | Numerical simulation of gas-assisted injection molding using clsvof method[END_REF]. However, this mass conservation error is considered to be much less than that induced by the level-set method.

Level-set advection

Concerning the level-set advection, we have several options:

• Using directional splitting as in [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF] as follows:

φ * i,j -φ n i,j ∆t + u i+1/2,j φ n i+1/2,j -u i-1/2,j φ n i-1/2,j ∆x = φ * i,j u i+1/2,j -u i-1/2,j ∆x (3.12)
For the y-direction:

φ n+1 i,j -φ * i,j ∆t + v i,j+1/2 φ * i,j+1/2 -v i,j-1/2 φ * i,j-1/2 ∆y = φ * i,j v i,j+1/2 -v i,j-1/2 ∆y (3.13)
where the level-set at the cell faces is determined by extrapolating the level-set in both space and time [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF] . For example, φ i+1/2,j is determined by upwinding:

φ i+1/2,j =        φ i,j + ∆x 2 1 -u i+1/2,j ∆t ∆x φ i+1,j -φ i-1,j ∆x if u i+1/2,j > 0 φ i,j - ∆x 2 1 + u i+1/2,j ∆t ∆x φ i+2,j -φ i,j ∆x if u i+1/2,j < 0 (3.14)
• [START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF] have used the directional splitting to solve the level-set advection equation as follows:

φ * i,j -φ n i,j ∆t + u i,j ∂φ n ∂x i,j = 0 φ n+1 i,j -φ * i,j ∆t + v i,j ∂φ * ∂y i,j = 0 (3.15)
where spatial derivatives are evaluated with 2nd-order ENO schemes using the minmod limiter:

∂φ n ∂x = φ i,j -φ i-1,j ∆x + minmod φ i+1,j + φ i-1,j -2φ i 2∆x , φ i,j + φ i-2,j -2φ i-1,j 2∆x ∂φ n ∂x = φ i+1,j -φ i,j ∆x -minmod φ i+1,j + φ i-1,j -2φ i,j 2∆x , φ i,j + φ i+2,j -2φ i+1,j 2∆x if u i+1/2,j < 0. with minmod(a, b) = sign(a) min(|a|, |b|) if ab > 0 0
elsewhere.

The advection of the level-set method in the CLSVOF by [START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF] is similar to the level-set that we already tested in the 1.6.1. In principle, any other advection scheme described before could be used.

Reinitialization of the level-set eld

After the advection step, the level-set needs to be reinitialized in order to keep the signed-distance property. The reinitialization in the CLSVOF method is performed using the information provided by the level-set itself and the volume-of-uid [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF][START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF]. It is performed on a bandwidth of k cells around the interface. An interface may pass through a given cell (i, j) if the level-set changes its sign between this given cell and one of its neighbors:

min(φ i,j × φ i ,j ) ≤ 0 for i ∈ [i -1, i, i + 1] and j ∈ [j -1, j, j + 1], (i , j ) = (i, j) (3.16)
The reinitialization procedure is shown on Algorithm 3.5 and explained in detail in [START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF]. (i,j)

← d if 0 < C < 1 Sign( 1 2 -C) × k∆x Otherwise.
for all computational cells (i, j) where Eq. (3.16) is veried do if (C i,j = 0 or C i,j = 1) then (i, j)-th cell is single phase in VOF for each neighbor cells (i , j

) do i ∈ [i -k, i + k] and j ∈ [j -k, j + k] φ temporary i ,j
← minimal distance from the (i , j ) cell center to the nearest point on the (i, j) cell faces if φ i,j φ i ,j ≤ 0 then

φ i ,j ← |φ temporary i ,j | × Sign(φ i ,j ) end if end for else (i, j)-th cell is mixed in VOF for each neighbor cells (i , j ) do i ∈ [i -k, i + k] and j ∈ [j -k, j + k] (x 1 , y 1 ) ∈ Ω i,j ← coordinates of the nearest point on cell (i, j) to (i , j ) cell center d 1 ← distance from (x 1 , y 1 ) to the linear interface.
if φ i ,j × d 1 < 0 then opposite sign of (i , j ) and nearest point φ temporary i ,j ← distance from the (i , j ) cell center to the point (x 1 , y 1 ) else same signs for cell (i , j ) and nearest point φ temporary i ,j ← distance from the (i , j ) cell center to the linear interface.

end if if |φ i ,j | ≥ |φ temporary i ,j | then φ i ,j ← |φ temporary i ,j | × Sign(φ i ,j )
end if end for end if end for return φ

Coupled Level-set and Moment-Of-Fluid (CLSMOF)

The coupled level-set and moment-of-uid (CLSMOF) method was rst introduced by [START_REF] Jemison | A coupled level set-moment of uid method for incompressible two-phase ows[END_REF]. Instead of coupling the level-set with the volume-of-uid as in the CLSVOF method, the authors coupled the level-set with the moment-of-uid method with some modications in the Algorithm 3.4. The main dierence consists in the manner the normal to the interface is computed.

The CLSMOF method of [START_REF] Jemison | A coupled level set-moment of uid method for incompressible two-phase ows[END_REF] combines the level-set slope reconstruction from CLSVOF and the MOF approach to reconstruct the interface:

Step 1. An initial estimate of the interface normal, n 0,1 , is rst determined by minimizing the error, Eq. (3.9), as in the CLSVOF method. The distance d is then adjusted so that the volume fraction of the reconstructed interface corresponds exactly to the target value C n of the cell under consideration. This is equivalent to the CLSVOF reconstruction.

Coupled Volume-Of-Fluid and Level-Set methods

Step 2. A second normal is computed from the target centroid, n 0,2 =x ref c , and both n 0,1 and n 0,2 are used to initialize the MOF algorithm. The normal which gives the smallest centroid error of Eq. (2.37) is used in the rst iteration of the next step.

Step 3. The MOF method is then used to compute a third normal vector, n 0,3 , by matching the target centroid x ref c as close as possible to the centroid of the reconstructed cell (see 2.4.1).

Step 4. The nal selection between the MOF or CLSVOF reconstruction ( n 0,1 and n 0,3 ) is based on the values of the centroid error E M OF , Eq. (2.37), and the curvature κ(φ) (the latter is computed from the levelset eld). When the actual centroid matches closely the target centroid, or in regions of high curvature, the MOF reconstruction is preferred. More specically, the MOF reconstruction is selected if any of the following equations holds:

κ(φ) > 1 β∆x , E M OF < 10 -8 ∆x , E M OF > ∆x/720
where β = 6. Otherwise the CLSVOF reconstruction is used.

Results and comparison between interface methods

In this section, the coupled level-set/volume-of-uid methods described above are compared with each other and with the non-coupled level-set and VOF methods using the following test problems: the Zalesak disk and the shearing vortex.

Overview of tested methods

We provide here rst the abbreviations used in this section and a brief description of the dierent methods:

VOF: The volume-of-uid fractions are advected using the conservative and consistent ux splitting scheme of 2.3.4. The normal to the interface is determined by ELVIRA, then the distance is computed according to the Algorithm A.2. This method is similar to that tested in Chapter 2 where it is given by Algorithm 2.1.

VOF-WY: It is given by Algorithm 2.1, but the volume fractions advection is realized through the fully explicit, conservative and consistent scheme of Weymouth and Yue (2010) detailed on 2.3.5. The interface is reconstructed by computing the normal using ELVIRA, and the distance using the Algorithm A.2. When there is no cell compression/expansion, this method is exactly the same as the VOF method. LS: The level-set method with reinitialization after each advection using the volume constraint reinitialization Eq. (1.50). This latter is discretized in space using the 2nd-order Hamilton-Jacobi ENO scheme while the spatial discretization of the advection equation is performed using the 5th-order WENO-Adv scheme. It is reinitialized after each advection on a bandwidth of 2∆x on each side of the interface with a ctitious time step of 1 2 ∆x.

MCLS/VC: In this coupled method, the level-set equation is solved in the same manner as for the LS method described above which includes the volume constraint (VC) for reinitialization. All other characteristics are detailed in Algorithm 3.2.

MCLS/STD: Similar to the previous one, but the reinitialization of the level-set function is performed without the volume constraint equation.

MCLS/STD/HJ-ENO2: Similar to the previous one, but the advection step for the level-set function is done using the 2nd-order HJ-ENO scheme instead of the 5th-order WENO-Adv scheme. Both the advection and the reinitialization equations are now solved using the faster 2nd-order HJ-ENO scheme.

MCLS/STD/UPWIND1: Similar to the previous one, but both the advection equation and the reinitialization equation are solved using the rst-order Upwind scheme.

VOSET-k The one-way coupled method, where the k = (1, 3, 5, or 10) represents the number of iterations performed to obtain the level-set eld. Note that in the interface advection test, the results are strictly identical to those obtained with the VOF method.

CLSVOF-Sussman: The coupled level-set/volume-of-uid method, in which the interface reconstruction is based on the resolution of the linear system of Eq. (3.9) and the level-set advection equation is solved using Eqs. (3.12-3.14), as recommended by [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF].

CLSVOF-Son: The coupled level-set/volume-of-uid method where the interface is reconstructed by computing the normal using central dierencing of the level-set, while the level-set is advected using the 2nd-order ENO scheme, as recommended by [START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF].

MOF:

The moment-of-uid method is advanced in time according to the Algorithm 2.2 given on 2.5.

CLSMOF: The coupled level-set and moment-of-uid is given by Algorithm 3.4, with the normal computed either by the MOF methods or as in the CLSVOF method as described in 3.5. The level-set is advected in time using Eqs. (3.12-3.14), as recommended by [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF]. The moment-of-uid advection is realized as already described for the MOF method.

In the next subsection, the quantitative and qualitative comparison will be shown.

Results and discussion

We have performed simulations for the three test problems already presented in 1.5.2-1.5.4, with the CFL number based on the maximum velocity set to 0.5 for all methods. Table 3.1 summarizes the global error measurements for mass conservation, E M , and interface location, E 1 , dened in 1.5.1. Note that in the following, the gures showing the interface results do not include those obtained with the VOSET method, since they are indiscernible from the VOF results (the interface advection is solved with the same method). However, the global errors for the VOSET methods are included in Table 3.1 since they are computed from the level-set eld (using bicubic interpolation with a 1000 2 sub-cells). The error for this method therefore only reects the accuracy of the algorithm that computes the level-set from the VOF, detailed in 3.2.

For comparison, the error induced by this algorithm when applied to the exact solution is also indicated in Table 3.1 (error of the reference eld).

Zalesak disk

Fig. 3.4 shows the results for the Zalesak disk after 1, 2, 4 and 8 complete revolutions on a 100 × 100 grid. All the methods conserve the general form of the Zalesak disk even after 8 revolutions, with smoothed angles. The VOSET, VOF, CLSVOF and CLSMOF methods 3. Coupled Volume-Of-Fluid and Level-Set methods produce a slight displacement of the top of the slab in the ow direction, while the level-set method displaces it slightly in the counter-ow direction. The MOF method is less sensitive to the number of the revolutions than the other methods.

The errors, E M , in Table 3.1 show that the VOF and MOF methods conserve, as expected, global mass within machine precision (the error is based here on the volume fractions). In the case of the VOSET methods, the interface is advanced with the VOF method, but the level-set is computed at each time step from the volume fractions and the global mass error is computed from the level-set eld. The global mass error induced by this conversion is, however, relatively small.

The CLSVOF methods have mass errors, computed from the level-set eld, comparable to VOSET and location errors comparable to both VOF and VOSET. Since the interfaces of CLSVOF in Fig. 3.4 are very close to that observed with the VOF methods, it seems that the interface evolution with these methods are mainly governed by the volume-of-uid advection schemes, rather than the level-set schemes.

Shearing vortex

The results for the single vortex problem at maximum stretch at t = T /2 = 3 are shown in Fig. 3.5. As already observed in chapters 1 and 2, the level-set method tends to thicken the vortex lament, while the VOF and MOF methods cause the formation of small bubbles due to the mass which needs to be conserved and the fact that the grid is coarse and more than one interface passes through reconstruction blocks of 3 × 3 cells. The coupled methods also show the breakup of the lament, with dierent degrees of regularity and length of the non-fractured lament. The MOF and CLSMOF methods produce results with the longest part of the nonfractured vortex. They produce also the smallest location errors. These methods clearly prot from the use of the centroid information, in addition to the volume fraction, for the interface reconstruction.

The results obtained with the CLSVOF-Son method are very close to that of the VOF method, and is slightly better than that given by the CLSVOF-Sussman method. The interface location errors, E 1 , are slightly worse than for the VOF methods, while the global mass is comparable to that of the VOSET method (since this error is calculated from the level-set). The global mass error for the VOF and MOF methods are again very small, since computed directly from the volume fractions.

Furthermore, the MCLS methods all produce very thin laments, close to the exact solution, but with very irregular interfaces. This may be a problem when for the computation of the surface tension term when combined with the Navier-Stokes equations. The global mass and location error are larger than all other methods, except the level-set method.

When inverting the velocity direction and coming back to the initial position, the interface methods should reproduce the initial interface, which is a circle. However, the interface obtained at time t = T = 6 deviates slightly from a circle as shown in Fig. 3.6. The worst results are given by the level-set method, mainly due to the important gain of mass, while the best results are obtained with the CLSMOF and MOF methods. Grid 64 2 t = 6.28

Results and comparison between interface methods

t = T/2 = 3 t = T = 6 t = T/2 = 1 t = T = 2 E M (%) E 1 (%) CPUt E M (%) E 1 (%) E M (%) E 1 (%) CPUt E M (%) E 1 (%) E M (%) E 1 (%) CPUt
Error of the reference eld

Rate of convergence

To determine the convergence rate, we have chosen only the methods that gave the best results in the previous section, namely:

• LS, VOF, MOF, CLSVOF-Son, CLSVOF-Sussman and CLSMOF

• VOSET with 3 geometric iterations

• MCLS with the standard level-set reinitialization, and the 2nd-order HJ-ENO scheme for the spatial discretization of the advection and the reinitialization equations. The level-set is reinitialized after each advection.

For the Zalesak disk problem, the location error is plotted on Fig. 3.7 for the CFL numbers varying in the range 0.25 to 1.0. The CFL variation has not much eect on the errors, except that for most methods the location error increases when reducing the CFL number. None of the methods show a second-order behavior on ne grids. This is mainly due to the presence of the sharp angles. The MOF method remains the method yielding the least location error in the range of tested grid sizes, closely followed by the CLSMOF method. The worst result is obtained with the LS and MCLS methods.

For the shearing vortex, the location error is plotted at maximum stretch on Fig. 3.8 . A nearly second-order convergence is now observed for nearly all methods and CFL numbers. The best results are again obtained with the MOF and CLSMOF method, while the LS method gives the highest errors. The MCLS method seems to do a fairly good job, but this is mainly due to the mean thickness of the lament, which is close to that of the exact solution. However, as mentioned above, the irregularity of the interface might be a problem when calculating the curvature (not included in the present tests).

VOF MOF VOSET

Conclusions

The mass error for the coupled methods, except the MCLS, are very small, and mainly due to the calculation of the mass from the level-set eld. Mass is therefore correctly conserved by these methods. However, the MCLS method does not seem to conserve the total mass very well, which is most probably due to the accumulation of errors while determining the volume fractions eld from the level-set eld at each time step.

The MOF and CLSMOF methods are fast and more accurate than all other methods for all test cases. The CLSVOF-Son method is also more accurate than the CLSVOF-Sussman method for all the presented test cases. In fact, in these interface advection tests, the CLSVOF and CLSMOF methods seem to produce results that are very similar to that obtained with their uncoupled versions, respectively the VOF and MOF methods. They have the advantage over the uncoupled methods to produce also the level-set eld. It remains to be seen, however, if this advantage can be exploited when combined with the Navier-Stokes equations.

The MCLS/STD/HJENO2 and the MCLS/STD/UPWIND1 methods are more accurate than the other MCLS tested variants. The overall results are, however, less accurate than the other methods. Only the level-set method seems to produce less accurate results. For the VOSET method, the interface is advected with the volume-of-uid method, and therefore the present results cannot evaluate the benets of this methods. It is shown, nonetheless, that only a few iterations for the level-set calculation seem to be sucient to obtain an accurate results.

The CPU-time for each method is also represented in Table 3.1. It shows that the level-set and the MCLS methods need more computational resources than the other methods, which is in our case due to the reinitialization and advection that are performed on the whole domain (we have not used the local level-set method).

Chapter 4

Incompressible two-phase ow solver

Introduction

In the previous chapters, we have investigated dierent implementations of interface capturing methods based on Eulerian approaches, and compared the performances of these methods in accurately transporting an interface in an imposed velocity eld.

In realistic ows, the velocity eld is not prescribed but determined by solving the Navier-Stokes equations. The interface methods (level-set and volume-of-uid) allow a one uid approach for non-miscible uids. The interface method predicts the location of the interface, and allows the evaluation of the uid properties and the application of the jump conditions at the interface for the solution of the Navier-Stokes equations. On the other hand, the Navier-Stokes solver will provide the velocity eld which is used to predict the evolution of the interface.

The aim of this chapter is to describe and test dierent implementations of the coupling between the interface methods and the Navier-Stokes solver for two-phase incompressible ows. Section 4.2 gives an overview of algorithms used to compute the ow equations in combination with the level-set and/or the VOF interface methods. The literature on this subject is vast and we do not claim to give an exhaustive overview. Instead, we focus on major issues, on the most popular methods and those who t best in our nite-dierence framework. Nonetheless, the number of possibilities to compose a numerical method for two-phase ows is practically unlimited. We have selected a limited number of methods, inspired largely from other studies and popular codes, which are described in detail in 4.3. Finally, in 4.4 we describe the numerical results obtained with these methods in some well-known test problems.

Navier-Stokes equations

The Navier-Stokes equations can be written in the conservative and non-conservative formulations. The latter is generally preferred in problems with non-miscible two-phase ows, probably since it avoids computation of gradients in the term ∇ • (ρ u u) with jumps in the density. For an incompressible uid, the Navier-Stokes equations read:

∂ u ∂t + u • ∇ u = - ∇p ρ + 1 ρ ∇ • µ (∇ u) + (∇ u) T + f σ + g (4.1) ∇ • u = 0 (4.2)
where p is the dynamic pressure, ρ and µ are the density and dynamic viscosity of the uid, f σ is the continuous surface tension force (see 4.2.2) and g is the acceleration due to gravity.

Incompressible two-phase ow solver

We have limited this work to 2D problems only and use a staggered grid, with grid lines aligned to the coordinate axes, where the scalars are dened on cell centers and the velocity vectors on the cell faces as shown on Fig. 4.1. A grid cell Ω i,j is dened by the region

[x i-1 2 , x i+ 1 2 ]× [y j-1 2 , y j+ 1 2
]. We will note the cell size in the xand y-direction by respectively ∆x = x i+ 1 2 -x i-1 2 and ∆y = y j+ 1 2 y j-1 2 . A uniform grid is used in this work, such that ∆x and ∆y are constants.

Computational domain boundaries are always dened on the cell boundaries (for example, the x-coordinates of the vertical boundaries are x 1 2 or x N + 1 2 with N the number of cells in this direction).

i -

1 i i + 1 i -1 2 i + 1 2 j -1 j j + 1 j -1 2 j + 1

Fluid properties

Let φ be the level-set function such that φ = 0 denes the liquid-gas interface and φ > 0 and φ < 0 dene respectively the liquid and gas phase. We will consider two immiscible uids with constant uid properties, ρ g and µ g in the gas phase and ρ l and µ l in the liquid phase. The density and viscosity of the uid may then be dened as

ρ(φ) = ρ g + (ρ l -ρ g )H(φ) (4.3) µ(φ) = µ g + (µ l -µ g )H(φ) (4.4)
where H(φ) is the Heaviside function. In the VOF-based methods, the uid properties are generally given by a volume-fraction weighted average [START_REF] Puckett | A high-order projection method for tracking uid interfaces in variable density incompressible ows[END_REF][START_REF] Popinet | Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries[END_REF]:

ρ(C) = ρ g + (ρ l -ρ g )C µ(C) = µ g + (µ l -µ g )C
The Heaviside function leads to a discontinuity in the uid properties at the interface, and thereby to many numerical diculties when using nite-dierence approximations in a one-uid approach. To avoid this problem, a smoothed Heaviside function is often employed to compute the uid properties. In volume-of-uid methods, regularization of the uid properties is obtained by smoothing the volume fractions C.

Navier-Stokes equations

Smoothed Heaviside and delta functions

A smoothed Heaviside function is sometimes used for the computation of ρ and/or µ (Eqs.4.3 and/ or 4.4). The smoothed Heaviside function for uid properties is often dened as [START_REF] Sussman | A level set approach for computing solutions to incompressible two-phase ow[END_REF][START_REF] Van Der Pijl | A mass-conserving level-set method for modelling of multi-phase ows[END_REF]:

H (x) =        0, x < - 1 2 1 + sin πx 2 , |x| < 1, x >
where most often = 3∆x/2. A slightly dierent function can also be used [START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF][START_REF] Sussman | An improved level set method for incompressible two-phase ows[END_REF][START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF][START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF]:

H (x) =        0, x < - 1 2 1 + x + 1 π sin πx , |x| < 1, x > (4.5) with
= 3∆x/2. Furthermore, to regularize the continuous surface tension force, a smoothed delta function is often used [START_REF] Peskin | Numerical analysis of blood ow in the heart[END_REF][START_REF] Unverdi | A front-tracking method for viscous, incompressible, multiuid ows[END_REF][START_REF] Sussman | A level set approach for computing solutions to incompressible two-phase ow[END_REF][START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF]:

δ (x) =    0, |x| > 1 2 1 + cos πx , |x| < (4.6)
which is in fact the derivative of Eq. (4.5), δ = dH /dx.

Smoothed volume fraction

It is generally preferable for both the accuracy and the stability of the method to use a smoothed version of the VOF function [START_REF] Denner | On the convolution of uid properties and surface force for interface capturing methods[END_REF]. A popular smoothing kernel is the K 8 -kernel, dened by

K 8 (r) = A 1 - r 2 2 4 for r ≤ (4.7)
and K 8 (r) = 0 for r > , where A is a normalization constant. On a 2D Cartesian uniform grid, this kernel would be implemented as:

C i,j = A M k=-M M l=-M 1 - (k∆x) 2 + (l∆y) 2 2 4 C i+k,j+l with M = ceiling( /h) and A is dened by A M k=-M M l=-M 1 - (k∆x) 2 + (l∆y) 2 2 4 = 1
Another possibility is to use vertex averaging, in which case the smoothing kernel is dened by:

C i,j = 1 16   1 2 1 2 4 2 1 2 1   C (4.8)
4. Incompressible two-phase ow solver

Alternatively, the cosine delta function of Eq. 4.6 could be used as a smoothing kernel with = 3∆x/2.

Smoothing by averaging

Vigneaux ( 2008) proposed a set of local averaging operations to compute uid properties at dierent grid locations. More details on how this is done are provided in 4.3.3 and 4.3.5.

The subsequent averaging steps can be regarded as a manner of smoothing the density and the viscosity.

Surface tension

In an inviscid ow with constant surface tension, the pressure jump condition at the interface (see for example [START_REF] Tryggvason | Direct numerical simulations of gas-liquid multiphase ows[END_REF] is reduced to the well-known Laplace formula, p lp g = σκ, which can be interpreted as a normal surface force per unit area given by σκ n. In viscous ows, additional terms have to be taken into account related to the jump in the viscosity [START_REF] Kang | A boundary condition capturing method for multiphase incompressible ow[END_REF]. Nonetheless, the surface tension term often represents the major contribution to the pressure jump. Therefore, many numerical methods only take this term into account, and we will adopt the same approach. The continuous surface force approach (CSF) [START_REF] Brackbill | A continuum method for modeling surface tension[END_REF] rewrites the interface surface force as a volume force:

f σ = σκ nδ s 1 2 (ρ l + ρ g )
where δ s is a delta function concentrated at the interface surface 1 . In the continuum limit, we can write nδ s = -∇c where c is a smoothed color function varying from 0 to 1, for example the smoothed Heaviside function H (φ).

Level-set approach

According to [START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF], since ∇H(φ) = δ(φ)∇φ, the surface tension force can be expressed as:

f σ = 1 ρ σκδ(φ)∇(φ)
The density is evaluated using one of the methods described in 4.2.1. However, some authors (for example [START_REF] Van Der Pijl | A mass-conserving level-set method for modelling of multi-phase ows[END_REF] prefer using the mean density, (ρ g + ρ l )/2. This approach tends to shift the surface tension force towards the heavier uid, thereby stabilizing the numerical method. However, it is incompatible with the so-called balanced force approach [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF], which states that the density in the pressure and surface tension terms should be the same in order to obtain a correct balance between these forces.

When φ denes the liquid and vapor phase as φ > 0 and φ < 0 respectively, the normal vector n and curvature of the interface κ may be calculated from the level-set eld using

n = ∇φ |∇φ| κ = -∇ • n (4.9)
1 The delta function δs can be interpreted as follows. Whenever a volume integral contains the Dirac function δs, it may be replaced by the surface integral, in other words

V f ( x)δsdV ≡ S f (x)dS
where S is the interface surface in V . In the continuum limit, δs designates the surface area per unit volume.

With these denitions, the normal vector points in the direction of the liquid phase. [START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF] used an expression for the curvature based on only rst-and second-order spatial derivatives of the level-set function

κ = - φ 2 y φ xx -2φ x φ y φ xy + φ 2 x φ yy (φ 2 x + φ 2 y ) 3/2
applicable to two dimensional cases. An expression in three dimensions can be found in Osher and Fedkiw (2002, p.12).

The advection and reinitialization steps introduce small perturbations on the level-set eld which tend to degrade the accuracy of curvature estimators based on nite dierences. When using central-dierence discretizations to compute the curvature from the normal divergence relation (Eq. 4.34), it is often preferable to add some smoothing in order to reduce the inuence of spurious oscillation on the level-set eld. In Sussman et al. (1999), the curvature is computed from Eq. (4.9) using a central dierence scheme based on a compact 3 × 3 stencil instead of the 9-point larger stencil of standard central discretization. [START_REF] Vigneaux | Méthode level set pour des problèmes d'interfaces en microuidique[END_REF] also used Eq. (4.9)

to compute the curvature at the cell faces. Smoothing of the level-set function is obtained by applying a series of averaging procedures, resulting in a rather large stencil of 4 × 4 for each velocity node.

In order to obtain a second-order accurate curvature on a slightly perturbed level-set eld, a least-square method is proposed by [START_REF] Marchandise | A stabilized nite element method using a discontinuous level set approach for the computation of bubble dynamics[END_REF]. A multi-dimensional linear or quadratic expansion of the level-set function is constructed at each grid point near the interface.

The coecients of this expansion are determined from an evaluation of the polynomial function on a stencil of neighboring grid points. The stencil size exceeds the number of coecients (resp.

3 and 6 for linear and quadratic expansions in 2D, and 4 and 10 in 3D) and the overdetermined system is solved using a least-squares method. On a two-dimensional uniform Cartesian grid, this method leads to the following convolution kernels for the spatial derivatives in the xand y-direction using a 3 × 3 stencil:

1 6   -1 0 1 -1 0 1 -1 0 1   ; 1 6   -1 -1 -1 0 0 0 1 1 1   ;
(4.10)

and with a 5 × 5-point stencil:

1 50

       -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2        ; 1 50        -2 -2 -2 -2 -2 -1 -1 -1 -1 -1 0 0 0 0 0 1 1 1 1 1 2 2 2 2 2       
;

These kernels are used to compute rst the normal vector n from the level-set eld and then a second time to evaluate the curvature from Eq. (4.9). The averaging eect of the larger stencil makes the least-squares method less sensitive to small perturbations. A truly second-order accurate curvature is then obtained, in contrast to other methods based on nite-dierence operators [START_REF] Marchandise | A stabilized nite element method using a discontinuous level set approach for the computation of bubble dynamics[END_REF][START_REF] Desjardins | An accurate conservative level set/ghost uid method for simulating turbulent atomization[END_REF].

Volume-of-uid approach

A second-order accurate curvature estimation from the VOF function can be obtained using the Height Function (HF) method, developed initially by [START_REF] Helmsen | Non-convex prole evolution in two dimensions using volume of uids[END_REF] (see for example 4. Incompressible two-phase ow solver [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF]; [START_REF] Puckett | On the second-order accuracy of volume-of-uid interface reconstruction algorithms: convergence in the max norm[END_REF][START_REF] Bellman | On the properties and limitations of the height function method in two-dimensional cartesian geometry[END_REF] for a summary of recent developments). In the method by [START_REF] Sussman | A second order coupled level set and volume-of-uid method for computing growth and collapse of vapor bubbles[END_REF], the curvature is computed from a height function reconstructed on a 3×7 stencil centered around the cell containing the interface. The orientation of the height function, h(x) or h(y), and the stencil is determined by the largest component of the interface normal. A discrete approximation of the height function is then obtained by a column-wise summation of the volume fractions on the stencil. The curvature is then computed

from κ = h (1 + h 2 ) 3/2
where the spatial derivatives are computed with standard central dierence schemes. For example when the columns are oriented in the y-direction we have h(x) who's derivatives are approximated by h = (h i+1 -h i-1 )/(2∆x) and h = (h i+1 -2h i +h i-1 )/(∆x 2 ). [START_REF] Cummins | Estimating curvature from volume fractions[END_REF] tested the Height Function method against two other approaches based on nite-dierence approximations, applied to either the smoothed volume-of-uid function or the reconstructed signed-distance function (although they did not calculate the latter as the exact signed-distance function to the reconstructed interface as in the CLSVOF method). The height function showed superior accuracy and approximately second-order grid convergence in static interface tests, except when the interface structures are not well resolved. The inaccuracy of the Height Function method in regions with small-scale interface structures or high curvatures has been mentioned by other authors. For example, [START_REF] Hernández | A new volume of uid method in three dimensions part i: Multidimensional advection method with face-matched ux polyhedra[END_REF] proposed an improvement by adapting the stencil according to the local volume fraction eld and introducing small local corrections of the volume fraction. Additional improvements of the Height Function method are proposed in subsequent papers [START_REF] López | An improved height function technique for computing interface curvature from volume fractions[END_REF][START_REF] López | On reducing interface curvature computation errors in the height function technique[END_REF].

In an independent study, [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF] also pointed out that the 3 × 7 stencil is not always an appropriate choice for under-resolved interface structures, for example when the radius is of the order of the mesh size or when the two interfaces are separated by only a few grid points.

He proposed an algorithm that adapts the stencil according to the local layout of the interface.

The sizes of the columns are reduced or increased in order to contain both sides of the interface, without crossing two interfaces. When a consistent stencil is not found, the method falls back to a parabolic curve tting approach. Numerical results show that the method, used in the CFD freeware Gerris (Popinet) and Basilisk (Popinet et al.), is robust and exhibits second-order behavior for a wide range of curvature scales. Finally, in order to reduce the so-called spurious currents in surface-tension dominated ows, the balanced-force algorithm was developed by [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF] in which the surface tension term is discretized in exactly the same manner as the pressure gradient. Their method is based on the VOF-CSF methodology in which the surface term is modeled as:

f σ = 1 ρ σκ∇ C
where C is either a smoothed or unsmoothed version of the volume-of-uid function C (VOF methods), or the smoothed Heaviside C = H (φ) (level-set methods). [START_REF] Denner | On the convolution of uid properties and surface force for interface capturing methods[END_REF] showed that in the VOF-based methods, smoothing of the volume fraction does not contribute to the accuracy and stability of the method and should be avoided. To obtain a balance by pressure and surface tension forces, it is necessary (in addition to an accurate curvature estimation) that, (1) the density at the cell face is the same as that used in the pressure gradient,

(2) the discretization of the gradient ∇ C is equivalent to that for the pressure gradient.

Yet another approach to compute the surface tension term is the so-called Continuous Surface

Navier-Stokes equations

Stress (CSS) [START_REF] Lafaurie | Modelling merging and fragmentation in multiphase ows with surfer[END_REF][START_REF] Gueyer | Volume-of-uid interface tracking with smoothed surface stress methods for three-dimensional ows[END_REF] in which the surface tension is modelled as an additional stress term: The advantage of this approach is that the surface tension term is now in conservative form and that variations of the surface tension σ are properly taken into account. Not many studies have adopted this approach, however, and we have not included the CSS method in our numerical tests.

f σ = - 1 ρ ∇ • T with T = -σ(I -n ⊗ n)|∇H|

Projection methods

One of the diculties in solving the Navier-Stokes equation (4.1) is the coupling between velocity and pressure by the incompressibility constraint (4.2). The projection method, initially proposed for uniform-density ows by [START_REF] Chorin | Numerical solution of the navierstokes equations[END_REF], eciently decouples the computation of the velocity and the pressure. For two-phase ows, the jump of the density and the surface tension force at the interface needs to be taken into account. This section describes several projection methods that have been employed for interface problems.

In [START_REF] Bell | A second-order projection method for variable-density ows[END_REF], a projection scheme is proposed for variable-density ows without interfaces in which a temporary velocity and density eld, u * and ρ * , is computed using the timelagged pressure gradient:

ρ * -ρ n ∆t + [ u n • ∇ρ] n+ 1 2 = 0 u * -u n ∆t + [ u n • ∇ u] n+ 1 2 = 1 ρ n+ 1 2 -∇p n-1 2 + ∇ • µ n+ 1 2 (∇ u * ) + (∇ u n ) T + g
where ρ n+ 1 2 = (ρ n + ρ n+1 )/2 and ∆t is the time step. Note that part of the viscous terms are treated implicitly, but these equations can be solved eciently using standard linear solvers. The temporary velocity u * does not fulll the incompressibility constraint (4.2). The projection step which corrects the velocity eld to obtain an incompressible ow is given by:

u n+1 -u * ∆t = - 1 ρ n+ 1 2 ∇ p n+ 1 2 -p n-1 2 ∇ • u n+1 = 0
Applying the divergence to the rst and inserting the second equation leads to the equation for 4. Incompressible two-phase ow solver the pressure:

∇ • 1 ρ n+ 1 2 ∇ p n+ 1 2 -p n-1 2 = 1 ∆t (∇ • u * )
This is a variable-coecient Poisson equation, which will consume a large portion of the computational time. This equation can be solved for example with Krylov-based solvers, multi-grid solvers or preconditioned conjugate gradient methods [START_REF] Golub | Matrix computations[END_REF]. This projection scheme is said incremental, since the pressure of the previous time step is used in the provisional velocity equation and an equation for the increment of the pressure is solved.

In [START_REF] Puckett | A high-order projection method for tracking uid interfaces in variable density incompressible ows[END_REF] and [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF], this scheme is adapted to twophase ows. First, the interface is advected from t n to t n+1 . Flow properties at t n+ 1 2 are then evaluated from

φ n+ 1 2 = (φ n + φ n+1 )/2 ρ n+ 1 2 = ρ(φ n+ 1 2 ) µ n+ 1 2 = µ(φ n+ 1 2 )
Surface tension is taken into account by adding f

n+ 1 2 σ
to the equation for the preliminary velocity.

A scheme staggered in both space and time, similar to the one proposed for low Mach number ows by [START_REF] Pierce | Progress-variable approach for large-eddy simulation of turbulent combustion[END_REF] and [START_REF] Wall | A semi-implicit method for resolution of acoustic waves in low mach number ows[END_REF], is used by several authors for two-phase ows [START_REF] Desjardins | An accurate conservative level set/ghost uid method for simulating turbulent atomization[END_REF][START_REF] Van Der Pijl | A mass-conserving level-set method for modelling of multi-phase ows[END_REF]. First, the interface is advanced from t n-1 2 to t n+ 1 2 . The density ρ n+ 1 2 , viscosity µ n+ 1 2 and curvature κ n+ 1 2 are then directly computed using the updated level-set function φ n+ 1 2 and no time interpolation is needed. Another major dierence is that the pressure gradient and the surface tension term are not included in the equation for the provisional velocity (non-incremental projection scheme):

u * -u n ∆t + ( u n • ∇ u) n+ 1 2 = 1 ρ n+ 1 2 ∇ • µ n+ 1 2 (∇ u * ) + (∇ u n ) T (4.11)
The projection step includes all "forgotten" terms and computes the new pressure eld and the nal incompressible ow eld:

u n+1 -u * ∆t = - 1 ρ n+ 1 2 ∇p n+ 1 2 + f σ + g (4.12) ∇ • u n+1 = 0
The equation for the pressure is then:

∇ • 1 ρ n+ 1 2 ∇p n+ 1 2 = ∇ • u * ∆t + f σ + g (4.13)
In several works [START_REF] Desjardins | An accurate conservative level set/ghost uid method for simulating turbulent atomization[END_REF][START_REF] Van Der Pijl | A mass-conserving level-set method for modelling of multi-phase ows[END_REF][START_REF] Son | A level set method for analysis of lm boiling on an immersed solid surface[END_REF], the density jump in the pressure equation is implemented using the Ghost Fluid Method (GFM) described in 4.2.4. This allows a sharp representation of the density. In [START_REF] Desjardins | An accurate conservative level set/ghost uid method for simulating turbulent atomization[END_REF], the surface tension term is also implemented directly as a sharp jump condition in the pressure equation using the Ghost Fluid Method, and, therefore, f σ is omitted in Eqs. (4.12) and (4.13). [START_REF] Son | A level set method for analysis of lm boiling on an immersed solid surface[END_REF] included f σ , but their method is strictly equivalent on the discrete level to the Ghost Fluid Method employed by [START_REF] Desjardins | An accurate conservative level set/ghost uid method for simulating turbulent atomization[END_REF] due to a particular choice of f σ (see 4.2.4). On the other hand, van der Pijl et al. ( 2005) used the CSF approach and f σ is computed from a smoothed delta-function.

Due to the strong coupling between the pressure gradient and the surface tension term, it seems important to always keep both terms together and at the same time level. For example, in an incremental projection method, both terms should be evaluated at the previous time step t n-1 2 and included in the provisional velocity equation (4.11). But in the projection step, the old surface tension term should be replaced by the new one for t n-1 2 , in order to compute the new pressure. This approach is used in a collocated ow solver, detailed in B.1, that is also tested.

Ghost Fluid Method

The Ghost Fluid Method (GFM) was rst developed by [START_REF] Fedkiw | A non-oscillatory Eulerian approach to interfaces in multimaterial ows (the ghost of uid method)[END_REF] for the application to shock-waves and the solution of the Euler equations for (inviscid) compressible ows. [START_REF] Liu | A boundary condition capturing method for poisson's equation on irregular domains[END_REF] applied this method to the variable coecient Poisson equation with jump conditions at the interface. In [START_REF] Gibou | A second-order-accurate symmetric discretization of the Poisson equation on irregular domains[END_REF], a similar Poisson equation is considered but with Dirichlet boundary conditions (instead of jump conditions), in which case the solution is not coupled across the interface. In [START_REF] Ng | Guidelines for Poisson solvers on irregular domains with Dirichlet boundary conditions using the Ghost Fluid Method[END_REF], an improved method for Dirichlet conditions is proposed that shows second-order behavior also for the gradients of the scalar eld. [START_REF] Kang | A boundary condition capturing method for multiphase incompressible ow[END_REF] applied the GFM to simulate ame surface propagation in an inviscid incompressible ow, taking into account the velocity jump at the interface.

The Ghost Fluid Method is basically an extrapolation technique which allows implementation of jump conditions at the interface in classical numerical methods, and which has gained most of its popularity in the eld of two-phase ows. Probably for historical and practical reasons, the Ghost Fluid Method is often combined with the level-set method, although in principle any other interface method could be used. The ability to implement actual jump conditions avoids numerical smearing of uid properties. In the Ghost Fluid Method, a robust scheme is obtained by applying standard spatial and temporal discretizations on grid points belonging only to one side of the interface. For stencils across the interface, additional ghost points are dened on the other side of the interface on which ow variables are extrapolated using the interface jump conditions. The technique consists of dening values at the ghost points across the interface as the sum of the local value and the jump condition at the interface.

Applied to the variable-coecient Poisson equation [START_REF] Liu | A boundary condition capturing method for poisson's equation on irregular domains[END_REF], it is possible to take into account jump conditions in the coecients, the normal gradient and the solution itself.

Interestingly, the associated linear system is not modied and produces the standard symmetric matrix, for which ecient readily-available variable-coecient solvers can be used. We will apply this method to the pressure equation, in which both the density and the pressure contains a jump at the interface. In the following, jump conditions across the interface Γ are noted as [p] Γ = p lp g . The model problem is then given by:

∇ • 1 ρ ∇p = 1 ∆t (∇ • u * ) ≡ f for x ∈ Ω (4.14) [p] Γ = σκ ≡ a for x ∈ Γ (4.15)
Furthermore, we have a density jump [ρ] Γ = ρ lρ g at the interface. In a standard second-order nite-dierence setting, the variable-coecient Poisson equation without jump conditions would

1 ∆x 1 ρ i+ 1 2 ,j p i+1,j -p i,j ∆x - 1 ρ i-1 2 ,j p i,j -p i-1,j ∆x + 1 ∆y 1 ρ i,j+ 1 2 p i,j+1 -p i,j ∆y - 1 ρ i,j-1 2 p i,j -p i,j-1 ∆y = f i,j (4.16)
If all grid points belonging to the 5-point stencil are on the same side of the interface, this equation needs no modication. If, however, there is a change of sign in the level-set function on two or more grid points, an additional term should be included in the RHS of this equation and the cell-face density should be modied. We have several possibilities:

• When φ i,j φ i-1,j < 0, then dene θ = |φ i-1,j | |φ i,j | + |φ i-1,j | ρ i-1 2 ,j = ρ i-1,j θ + ρ i,j (1 -θ) (4.17) a Γ = a i,j θ + a i-1,j (1 -θ)
where the densities ρ i-1,j and ρ i,j are determined in a sharp manner (either ρ l or ρ g depending on the value of φ). Note the dierence in the averaging for ρ i-1 2 ,j and a Γ . The value ρ i-1 2 ,j denotes the eective density along the path between x i-1,j and x i,j which replaces ρ i-1 2 ,j in Eq. (4.16), whereas a Γ denotes the interpolated value of a on the interface.

In order to take into account the pressure jump, the term

S (φ i-1,j ) a Γ ρ i-1 2 ,j (∆x) 2
is added to the RHS of Eq. (4.16), where S(φ i-1,j ) denotes the sign of φ i-1,j .

• When φ i,j φ i+1,j < 0, then dene

θ = |φ i+1,j | |φ i,j | + |φ i+1,j | ρ i+ 1 2 ,j = ρ i+1,j θ + ρ i,j (1 -θ) (4.18) a Γ = a i,j θ + a i+1,j (1 -θ)
Next, the term

S (φ i+1,j ) a Γ ρ i+ 1 2 ,j (∆x) 2 (4.19)
is added to the RHS of Eq. (4.16).

A similar treatment is used for the situations where φ i,j φ i,j-1 < 0 or φ i,j φ i,j+1 < 0. The resulting coecient matrix remains symmetric. If for example φ i,j φ i-1,j < 0, then ρ i-1 2 ,j is modied according to Eq. (4.17) in the row containing the discrete Poisson equation for the grid point (i, j). The corresponding entry on the other side of the diagonal, located on the row for the grid point (i -1, j), contains exactly the same coecient dened by Eq. (4.18). The advantage of a symmetric sparse matrix for the Poisson equation with respect to a non-symmetric matrix is the ability to use more ecient solvers. 4.3. Implementation of the Navier-Stokes solver [START_REF] Son | A level set method for analysis of lm boiling on an immersed solid surface[END_REF] have shown that the extra term in the RHS of the pressure equation can also be taken into account implicitly by including an additional term for the surface tension in the provisional velocity equation, in a manner similar to the CSF method. Suppose for example that u * is the provisional velocity that does not include the surface tension term and that φ i,j φ i+1,j < 0 so that the interface passes between the corresponding grid points. We may now dene

u * * -u * ∆t = 1 ρ σκ∇H(φ) (4.20)
For the velocity at the grid point x i+ 1 2 ,j this gives

u * * i+ 1 2 ,j -u * i+ 1 2 ,j ∆t = (σκ) i+ 1 2 ,j ρ i+ 1 2 ,j H(φ i+1,j ) -H(φ i,j ) ∆x = S(φ i+1,j )(σκ) i+ 1 2 ,j ρ i+ 1 2 ,j ∆x
In the pressure projection step, we replace u * by u * * in the RHS of Eq. (4.14) which gives for the grid point x i,j :

1 ∆t (∇ • u * * ) i,j = u * * i+ 1 2 ,j -u * * i-1 2 ,j ∆x + v * * i,j+ 1 2 -v * * i,j-1 2 ∆y = u * i+ 1 2 ,j -u * i-1 2 ,j ∆x + v * i,j+ 1 2 -v * i,j-1 2 ∆y + S(φ i+1,j )(σκ) i+ 1 2 ,j ρ i+ 1 2 ,j (∆x) 2
The additional term in the RHS is exactly the term in Eq. (4.19), supposing a Γ = (σκ) i+ 1 2 ,j .

Therefore, this term needs to be evaluated using the interpolation procedure stipulated in Eqs. (4.17) and (4.18), as well as the density ρ i+ 1 2 ,j dened according to Eq. (4.20). The addi- tional terms for other grid points (such as p i,j ) as well as other interface crossings on the same stencil are implicitly taken into account in a similar manner. In conclusion, the Ghost Fluid Method can be rewritten in terms of a continuous surface force f σ = (σκ) Γ ∇H(φ)/ ρ, which can be integrated in any of the aforementioned projection-type methods provided we make a proper choice of ρ and (σκ) Γ .

Implementation of the Navier-Stokes solver

There are many ways to implement a numerical method for the incompressible Navier-Stokes equations with liquid-gas interfaces. At dierent levels of the construction of a discrete model, several choices have to be made, each of which will have an eect, more or less known, on the accuracy and robustness of the simulations. In this work, we have focused on a limited number of numerical schemes. The selection of these schemes was guided on the one hand by the choices made by other studies in the literature and on the other hand from our numerical experience.

These numerical schemes are described more in detail in this section.

Time integration

For the decoupling of pressure and velocity, we choose the non-incremental projection method which consists of calculating rst a provisional velocity without the pressure gradient, followed by a correction step in which the velocity is projected onto a divergence-free vector space. A predictor-corrector time scheme has been used for the provisional velocity step. The predictor 4. Incompressible two-phase ow solver step is given by:

u * ,0 -u n ∆t + [ u • ∇ u] n = 1 ρ n+ 1 2 ∇ • µ n+ 1 2 (∇ u * ,0 ) + (∇ u n ) T
where the superscript 0 refers to the prediction step of the time scheme. Note that the rst part in the viscosity term is treated implicitly. The resulting system of linear equations is solved using an iterative ADI approach [START_REF] Knikker | Study of a staggered fourth-order compact scheme for unsteady incompressible viscous ows[END_REF]. At each iteration, the implicit term is solved separately in each spatial direction using a second-order accurate discretization. This leads to tridiagonal systems, which are solved eciently using the Thomas algorithm.

The projection step is constructed from the velocity correction:

u n+1,0 -u * ,0 ∆t = - 1 ρ n+ 1 2 ∇ δ p n+ 1 2 ,0 + g + f n+ 1 2 σ ∇ δ • u n+1,0 = 0
where ∇ δ = (δ/δx, δ/δy) and ∇ δ • u = δu/δx + δv/δy. The notation δ/δx and δ/δy is used here for the standard second-order centered dierence schemes. For example (δφ/δx) i+ 1 2 ,j = (φ i+1,jφ i,j )/∆x. The projection step requires the solution of the following Poisson equation for the pressure

∇ δ • 1 ρ n+ 1 2 ∇ δ p n+ 1 2 ,0 = ∇ δ • u * ,0 ∆t + g + f n+ 1 2 σ
In order to solve this equation, the preconditioned conjugate gradient method is used in combination with the modied incomplete Cholesky decomposition [START_REF] Golub | Matrix computations[END_REF]) using the NSPCG package [START_REF] Thomas C Oppe | Nspcg user's guide, version 1.0: A package for solving large sparse linear systems by various iterative methods[END_REF].

The corrector step of the overall time scheme consists of repeating the above steps, except that the explicit terms are replaced by a second-order approximation. The provisional velocity step is now:

u * ,1 -u n ∆t + 1 2 [ u • ∇ u] n+1,0 + [ u • ∇ u] n = + 1 ρ n+ 1 2 ∇ • µ n+ 1 2 (∇ u * ,1 ) + 1 2 (∇ u n+1,0 ) T + 1 2 (∇ u n ) T
The nal projection step is:

∇ δ • 1 ρ n+ 1 2 ∇p n+ 1 2 = ∇ δ • u * ,1 ∆t + g + f n+ 1 2 σ followed by u n+ 1 2 = u * * ,1 + ∆t g + f n+ 1 2 σ - 1 ρ n+ 1 2 ∇p n+ 1 2
The interface is dened at intermediate time levels, t n+ 1 2 , such that uid properties and the surface tension term can be expressed directly at this instant. For example, when the level-set method is used, φ is rst advanced from φ n-1 2 to φ n+ 1 2 using the velocity eld u n , for example

φ n+ 3 2 -φ n+ 1 2 ∆t + u n+1 • ∇φ n+ 1 2 = 0
A similar equation can be derived for the volume-of-uid methods. The interface advection step is done only once each time step. The density, viscosity and surface tension term at t n+ 1 2 is therefore the same for both the predictor and corrector step.

Time step restriction

Several references evoke time step calculation [START_REF] Kang | A boundary condition capturing method for multiphase incompressible ow[END_REF][START_REF] Sussman | An improved level set method for incompressible two-phase ows[END_REF][START_REF] Van Der Pijl | A mass-conserving level-set method for modelling of multi-phase ows[END_REF][START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF][START_REF] Sussman | A sharp interface method for incompressible two-phase ows[END_REF][START_REF] Denner | Numerical time-step restrictions as a result of capillary waves[END_REF].

In the current work, we rst apply the standard Courant-Friedrichs-Lewy (CFL) condition and viscous time step restriction:

∆t u ≤ CFL |u| max ∆x + |v| max ∆y + |w| max ∆z
where CFL = 0.5 (except otherwise specied) and

∆t ν ≤ Fo min(∆x, ∆y, ∆z) 2 max(µ g /ρ g , µ l /ρ l )
with Fo = 2.0.

According to [START_REF] Brackbill | A continuum method for modeling surface tension[END_REF], the explicit treatment of the surface tension term is stable when the time step is suciently small to resolve capillary wave phenomena, which leads to the condition:

∆t σ ≤ CFL σ (ρ l + ρ g ) min(∆x, ∆y, ∆z) 3 4πσ
When gravity forces are present, another time step restriction is added

∆t g ≤ ∆z |g|
Finally, the time step is set to ∆t = min(∆t u , ∆t ν , ∆t σ , ∆t g )

Density at the cell faces and centers

When the volume-of-uid method is used to capture the interface, the density is dened by the volume-fraction weighted average:

     ρ i,j = ρ g + (ρ l -ρ g )C i,j ρ i+ 1 2 ,j = ρ g + (ρ l -ρ g )C i+ 1 2 ,j ρ i,j+ 1 2 = ρ g + (ρ l -ρ g )C i,j+ 1 2 (4.21) where C i+ 1 2 ,j = (C i,j + C i+1,j )/2 and C i,j+ 1 2 = (C i,j + C i,j+1
)/2. When a smoothing kernel is used (see 4.2.1.2), the variable C is replaced by the smoothed fraction C in the above equations.

When the interface is described by the level-set method, we use one of the following methods to determine the density at the cell faces and centers:

• In order to obtain a sharp interface approach, the density is determined using the Ghost Fluid Method described in 4.2.4, in particular Eqs. (4.17) and (4.18). The density at the 4. Incompressible two-phase ow solver cell center is provided by the Heaviside function as:

ρ i,j = ρ g + (ρ l -ρ g )H(φ i,j )
• In the case of a smooth interface approach, the density is given by

     ρ i,j = ρ g + (ρ l -ρ g )H (φ i,j ) ρ i+ 1 2 ,j = ρ g + (ρ l -ρ g )H (φ i+ 1 2 ,j ) ρ i,j+ 1 2 = ρ g + (ρ l -ρ g )H (φ i,j+ 1 2 ) (4.22)
where H is the smoothed Heaviside function dened by Eq. (4.5) and φ i+ 1 2 ,j and φ i,j+ 1 2 are dened from the cell centered values of φ i,j using averaging.

• [START_REF] Vigneaux | Méthode level set pour des problèmes d'interfaces en microuidique[END_REF] proposed a set of averaging operations to compute uid properties at dierent grid locations. In order to determine the density at the cell faces, the value at the cell center is rst determined from

ρ i,j = ρ g + (ρ l -ρ g )H(φ i,j )
The density at the cell vertices are then computed from:

         ρ u i+ 1 2 ,j+ 1 2 = ρ i,j • ρ i,j+1 ρ i,j + ρ i,j+1 + ρ i+1,j • ρ i+1,j+1 ρ i+1,j + ρ i,j+1 ρ v i+ 1 2 ,j+ 1 2 = ρ i,j • ρ i+1,j ρ i,j + ρ i+1,j + ρ i,j+1 • ρ i+1,j+1 ρ i,j+1 + ρ i,j+1
The density at the cell faces ρ i+ 1 2 ,j and ρ i,j+ 1 2 then follow from averaging of values at cell centers and cell vertices:

         ρ i+ 1 2 ,j = 1 4 ρ u i+ 1 2 ,j+ 1 2 + ρ u i+ 1 2 ,j-1 2 + ρ i,j + ρ i+1,j ρ i,j+ 1 2 = 1 4 ρ v i+ 1 2 ,j+ 1 2 + ρ v i-1 2 ,j+ 1 2 + ρ i,j + ρ i,j+1 (4.23)
The subsequent averaging steps can be regarded as a manner of smoothing the density.

Advection terms

The advection terms in the momentum equations are discretized using the second-order ENO scheme, which seems to be a popular choice among other works on two-phase ows [START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF][START_REF] Son | A level set method for analysis of lm boiling on an immersed solid surface[END_REF]. We use:

[ u • ∇u] u i+ 1 2 ,j = u i+ 1 2 ,j ∂u ∂x ENO i+ 1 2 ,j + v i+ 1 2 ,j ∂u ∂y ENO i+ 1 2 ,j (4.24) [ u • ∇v] u i,j+ 1 2 = u i,j+ 1 2 ∂v ∂x ENO i,j+ 1 2 + v i,j+ 1 2 ∂v ∂y ENO i,j+ 1 2 (4.25) where v i+ 1 2 ,j = 1 4 v i,j+ 1 2 + v i,j-1 2 + v i+1,j+ 1 2 + v i+1,j-1 2 u i,j+ 1 2 = 1 4 u i+ 1 2 ,j + u i-1 2 ,j + u i+ 1 2 ,j+1 + u i-1 2 ,j+1
The advection terms are upwind approximations determined from:

u ∂φ ∂x ENO = uφ - x if u ≥ 0 uφ + x if u < 0
The spatial derivatives φ -

x and φ +

x are calculated using the second-order ENO scheme. Dening,

Dφ - i = φ i -φ i-1 Dφ + i = φ i+1 -φ i D 2 φ - i = φ i -2φ i-1 + φ i-2 D 2 φ 0 i = φ i+1 -2φ i + φ i-1 D 2 φ + i = φ i+2 -2φ i+1 + φ i the ENO scheme is formulated as φ - x = 1 h Dφ -+ 1 2h minmod D 2 φ 0 , D 2 φ - φ + x = 1 h Dφ + - 1 2h minmod D 2 φ 0 , D 2 φ +
1 ρ i+ 1 2 ,j [∇ • µ(∇u)] i+ 1 2 ,j = 1 ρ i+ 1 2 ,j ∆x µ i+1,j u i+ 3 2 ,j -u i+ 1 2 ,j ∆x -µ i,j u i+ 1 2 ,j -u i-1 2 ,j ∆x + 1 ρ i+ 1 2 ,j ∆y µ u i+ 1 2 ,j+ 1 2 u i+ 1 2 ,j+1 -u i+ 1 2 ,j ∆y -µ u i+ 1 2 ,j-1 2 u i+ 1 2 ,j -u i+ 1 2 ,j-1 ∆y 1 ρ i,j+ 1 2 [∇ • µ(∇v)] i,j+ 1 2 = 1 ρ i,j+ 1 2 ∆x µ v i+ 1 2 ,j+ 1 2 v i+1,j+ 1 2 -v i,j+ 1 2 ∆x -µ v i-1 2 ,j+ 1 2 v i,j+ 1 2 -v i-1,j+ 1 2 ∆x + 1 ρ i,j+ 1 2 ∆y µ i,j+1 v i,j+ 3 2 -v i,j+ 1 2 ∆y -µ i,j v i,j+ 1 2 -v i,j-1 2 ∆y 4.
3. Implementation of the Navier-Stokes solver complicated, however, and is not used in this work. Instead, we use the method proposed by [START_REF] Son | A level set method for analysis of lm boiling on an immersed solid surface[END_REF], which is a simplied version of the Ghost Fluid Method by [START_REF] Kang | A boundary condition capturing method for multiphase incompressible ow[END_REF]. In this approach, the viscous terms are computed from slightly dierent expressions:

1 ρ i+ 1 2 ,j [∇ • µ(∇u)] i+ 1 2 ,j = 1 ρ i+ 1 2 ,j ∆x µ x i+1,j u i+ 3 2 ,j -u i+ 1 2 ,j ∆x -µ x i,j u i+ 1 2 ,j -u i-1 2 ,j ∆x + 1 ρ i+ 1 2 ,j ∆y µ y i+ 1 2 ,j+ 1 2 u i+ 1 2 ,j+1 -u i+ 1 2 ,j ∆y -µ y i+ 1 2 ,j-1 2 u i+ 1 2 ,j -u i+ 1 2 ,j-1 ∆y 1 ρ i,j+ 1 2 [∇ • µ(∇v)] i,j+ 1 2 = 1 ρ i,j+ 1 2 ∆x µ x i+ 1 2 ,j+ 1 2 v i+1,j+ 1 2 -v i,j+ 1 2 ∆x -µ x i-1 2 ,j+ 1 2 v i,j+ 1 2 -v i-1,j+ 1 2 ∆x + 1 ρ i,j+ 1 2 ∆y µ y i,j+1 v i,j+ 3 2 -v i,j+ 1 2 ∆y -µ y i,j v i,j+ 1 2 -v i,j-1 2 ∆y 1 ρ i+ 1 2 ,j ∇ • µ(∇ u) T u i+ 1 2 ,j = 1 ρ i+ 1 2 ,j ∆x µ x i+1,j u i+ 3 2 ,j -u i+ 1 2 ,j ∆x -µ x i,j u i+ 1 2 ,j -u i-1 2 ,j ∆x + 1 ρ i+ 1 2 ,j ∆y µ x i+ 1 2 ,j+ 1 2 v i+1,j+ 1 2 -v i,j+ 1 2 ∆x -µ x i+ 1 2 ,j-1 2 v i+1,j-1 2 -v i,j- ∆x 1 ρ i,j+ 1 2 ∇ • µ(∇ u) T v i,j+ 1 2 = 1 ρ i,j+ 1 2 ∆x µ y i+ 1 2 ,j+ 1 2 u i+ 1 2 ,j+1 -u i+ 1 2 ,j ∆y -µ y i-1 2 ,j+ 1 2 u i-1 2 ,j+1 -u i- ,j ∆y + 1 ρ i,j+ 1 2 ∆y µ y i,j+1 v i,j+ 3 2 -v i,j+ 1 2 ∆y -µ y i,j v i,j+ 1 2 -v i,j-1 2 ∆y
The viscosity is evaluated using an interpolation method similar to the Ghost Fluid Method (Eqs. (4.17) and (4.18)):

                                   (µ x i+ 1 2 ,j+ 1 2 ) -1 = µ -1 i+1,j+ 1 2 |φ i+1,j+ 1 2 | + µ -1 i,j+ 1 2 |φ i,j+ 1 2 | |φ i+1,j+ 1 2 | + |φ i,j+ 1 2 | (µ y i+ 1 2 ,j+ 1 2 ) -1 = µ -1 i+ 1 2 ,j+1 |φ i+ 1 2 ,j+1 | + µ -1 i+ 1 2 ,j |φ i+ 1 2 ,j | |φ i+ 1 2 ,j+1 | + |φ i+ 1 2 ,j | (µ x i,j ) -1 = µ -1 i+ 1 2 ,j |φ i+ 1 2 ,j | + µ -1 i-1 2 ,j |φ i-1 2 ,j | |φ i+ 1 2 ,j | + |φ i-1 2 ,j | (µ y i,j ) -1 = µ -1 i,j+ 1 2 |φ i,j+ 1 2 | + µ -1 i,j-1 2 |φ i,j-1 2 | |φ i,j+ 1 2 | + |φ i,j-1 2 | (4.29) with µ i+ 1 2 ,j = µ g + (µ l -µ g )H(φ i+ 1 2 ,j ) µ i,j+ 1 2 = µ g + (µ l -µ g )H(φ i,j+ 1 2 ) and φ i+ 1 2 ,j = (φ i+1,j + φ i,j )/2 φ i,j+ 1 2 = (φ i,j+1 + φ i,j )/2

Surface tension

Following the review in 4.2.2 and the generalization of the Ghost Fluid Method in 4.2.4, the surface tension term can be expressed as

f σ = σ ρ κ∇H(φ) (4.30)
We suppose that σ is constant. The density in this term is dened as the value at the cell face, ρ = ρ i+ 1 2 ,j or ρ = ρ i,j+ 1 2

. Furthermore, the Heaviside function can be either the sharp interface representation (which is then equivalent to the Ghost Fluid Method, Eq. (4.20)), or the smoothed Heaviside function (4.5). A VOF-based CSF implementation is given by:

f σ = σ ρ κ∇C (4.31)
where the variable C is replaced by C when a smoothing kernel is used. What remains are the dierent choices for the computation of the curvature, κ. We have used the following algorithms:

• The curvature, κ = -∇• n, is computed directly at the cell face from normal vectors derived at the cell centers and vertices [START_REF] Vigneaux | Méthode level set pour des problèmes d'interfaces en microuidique[END_REF]. To achieve this, the level-set at the cell vertices are rst computed from:

φ i+ 1 2 ,j+ 1 2 = 1 4 (φ i,j + φ i+1,j + φ i,j+1 + φ i+1,j+1 ) followed by φ u i+ 1 2 ,j = 1 4 φ i+ 1 2 ,j+ 1 2 + φ i+ 1 2 ,j-1 2 + φ i,j + φ i+1,j = 1 16 (φ i,j+1 + φ i+1,j+1 + 5φ i,j + 5φ i+1,j + φ i,j-1 + φ i+1,j-1 ) φ v i,j+ 1 2 = 1 4 φ i+ 1 2 ,j+ 1 2 + φ i-1 2 ,j+ 1 2 + φ i,j + φ i,j+1 = 1 16 (φ i+1,j + φ i+1,j+1 + 5φ i,j + 5φ i,j+1 + φ i-1,j + φ i-1,j+1 )
The interface normal is then computed at the cell center using The surface terms in the momentum equation are then dened as:

(f u σ ) i+ 1 2 ,j = σ ρ i+ 1 2 ,j κ i+ 1 2 ,j δH δx (f v σ ) i,j+ 1 2 = σ ρ i,j+ 1 2 κ i,j+ 1 2 δH δy
where H(φ) is dened at the cell center.

• The curvature is computed at the cell center from κ = -∇ • n (4.34) using standard central dierencing. The values at the cell center are then extrapolated to the cell faces as dened for a Γ in Eq. (4.17).

• The curvature is computed using the alternative expression proposed by [START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF], which uses only rst-and second-order spatial derivatives of the level-set function:

κ = - φ 2 y φ xx -2φ x φ y φ xy + φ 2 x φ yy (φ 2 x + φ 2 y ) 3/2 (4.35)
The curvature is evaluated at the cell center using standard central dierence schemes.

The extrapolation to the cell faces is done using the standard interpolation.

• The spatial derivatives for the interface normal and curvature are computed using the kernels in Eq. (4.10) dened by the least-squares approach proposed by [START_REF] Marchandise | A stabilized nite element method using a discontinuous level set approach for the computation of bubble dynamics[END_REF], see 4.2.2. The extrapolation to the cell faces is then performed according to the Ghost Fluid Method in equation Eq. (4.17).

• The curvature is computed as κ = -∇ • n at the cell center using central dierence discretization with some extra smoothing (Sussman et al., 1999). The gradient of the level-set function is computed at the vertices using the formulas:

∂φ ∂x i+ 1 2 ,j+ 1 2 φ i+1,j+1 + φ i+1,j -φ i,j+1 -φ i,j 2∆x ∂φ ∂y i+ 1 2 ,j+ 1 2 φ i+1,j+1 + φ i,j+1 -φ i+1,j -φ i,j 2∆y
The unit normal vector is then obtained by normalization of the level-set gradient as in Eq. (4.32). The curvature at the cell center is then computed using the relations:

∂n x ∂x i,j n x i+ 1 2 ,j+ 1 2 + n x i+ 1 2 ,j-1 2 -n x i-1 2 ,j+ 1 2 -n x i-1 2 ,j-1 2 2∆x ∂n y ∂y i,j n y i+ 1 2 ,j+ 1 2 + n y i-1 2 ,j+ 1 2 -n y i+ 1 2 ,j-1 2 -n y i-1 2 ,j-1 2 2∆y (4.36)
The curvature is then interpolated to the cell faces similarly as for a Γ in Eq. (4.17).

• The curvature is computed from the volume-of-uid function using the Height Function method. In each cell containing a line interface (determined previously by the VOF-PLIC advection method), the largest component normal vector denes the orientation of the 4. Incompressible two-phase ow solver height function. In the following, we will assume (without restricting the method) that |n y | > |n x | and n y > 0 (empty cells at the top of the columns and lled cells at the bottom). For the interface cell (Ω i,j ) and its two neighbors (Ω i+1,j and Ω i-1,j ), the top and bottom cell of each column (i, j t ) and (i, j b ) are dened according to the method by [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF], within the limit of the usual 3 × 7 stencil. The discrete height functions h i-1,j , h i,j and h i+1,j are then obtained by summing up the values of the volume fraction in each column and correcting for the dierences in the coordinates of the bottom cells of each column, for example:

h i-1,j = ∆y   jt(i-1) j =j b (i-1) C i-1,j -j b (i -1) + j - 1 2  
The curvature is then dened by

κ = h (1 + h 2 ) 3/2 (4.37)
where h = (h i+1h i-1 )/(2∆x) and h = (h i+1 -2h i + h i-1 )/(∆x) 2 . The curvature is required at the cell faces, while in this example it is dened at (x i , y i + h i ∆y).

In case no valid stencil is found for one of the columns, the same method is applied in the other direction. If this does not work either, the method falls back to a parabolic tting method through the points that are validated in the above procedure. If the number of valid points (columns) is below 3, however, a parabolic t is tried through the centroids of the line elements in a 3×3 grid around the cell. The parabolic tting procedure is normally only used for under-resolved interface structures.

In addition to the method by [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF], which is referred to in this work by HF1, we have implemented the method by [START_REF] Hernández | A new volume of uid method in three dimensions part i: Multidimensional advection method with face-matched ux polyhedra[END_REF]. They proposed an improvement of the earlier Height Function methods by adapting the stencil according to the local volume fraction eld and by introducing small local corrections of the volume fraction. For more details on this method, noted here as HF2, the reader is referred to [START_REF] Hernández | A new volume of uid method in three dimensions part i: Multidimensional advection method with face-matched ux polyhedra[END_REF], and the subsequent papers [START_REF] López | An improved height function technique for computing interface curvature from volume fractions[END_REF][START_REF] López | On reducing interface curvature computation errors in the height function technique[END_REF].

The Height Function method gives an estimate for the curvature at the center of the cell indicated by the height function value h i (most of times this is simply the cell Ω i,j ). In order to obtain values at the cell face, we interpolate from the cell-centered values if the curvature is available in both adjoining cells. Otherwise, we just copy the value from the cell containing the curvature. The cell averaging is based on the expression by [START_REF] Renardy | Prost: A parabolic reconstruction of surface tension for the volumeof-uid method[END_REF], which puts the weight on cells containing large portions of the interface and rules out inaccurate tiny interface segments. For example, for the u-component we use:

σκ ∂C ∂x i+ 1 2 ,j = σ w 1 κ i,j + w 2 κ i+1,j w 1 + w 2 1 ∆x (C i+1,j -C i,j ) with w 1 = C i,j (1 -C i,j ) and w 2 = C i+1,j (1 -C i+1,j ).
The curvature computed this way is not precise, we therefore use the same formula as in the free CFD software Basilisk (Popinet et al.). The interface curvature at the cell face is given by: κ i+ 1 2 ,j = (κ i,j + κ i+1,j )/2 if a valid curvature is available on the two cell centers (i, j) and (i + 1, j), while we use the same value as curvature at the cell center if only κ i,j or κ i+1,j is valid. otherwise, the curvature is set to zero.

Overview of tested methods

In the previous chapters, we have veried the ability of dierent interface capturing methods in accurately predicting the transport of an interface placed in a known velocity eld. From now on, we consider only one level-set (LS) method, two volume-of-uid (VOF) methods and one moment-of-uid (MOF) method as in 3.6.1. Note that the local level-set approach of 1.4 is used here in combination with the LS parameters of 3.6.1. We couple each of these interface capturing methods with the Navier-Stokes solver described in the previous section.

The coupling between the interface method and the Navier-Stokes solver is established through the determination of the uid properties and surface tension force. Taking example from the numerical implementations available in the literature, we have implemented dierent methods that are described hereafter.

LS-LA: Similar to the method by [START_REF] Vigneaux | Méthode level set pour des problèmes d'interfaces en microuidique[END_REF], based on the level-set (LS) method, the density and viscosity are determined from local averaging (LA) of cell-centered values, Eqs. (4.23) and (4.26). Two methods for the surface tension are derived following the manner in which the curvature is computed: LS-LA1 Following the work by [START_REF] Vigneaux | Méthode level set pour des problèmes d'interfaces en microuidique[END_REF], the curvature is directly determined at the cell-face from normal vectors derived at cell centers and vertices, Eqs. (4.33).

LS-LA2

The curvature is determined at the cell centers using the central dierencing formula (4.36) proposed by Sussman et al. (1999), and then interpolated at the cellface using the formula (4.17) and (4.18).

In the absence of surface tension forces, these two variants are identical, and we note them as LS-LA.

LS-CSF:

This level-set implementation takes its features from the popular method by [START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF]. The smoothed Heaviside function (4.5) is used in the determination of uid properties. The surface tension is implemented using the continuous surface force (CSF) approach, Eq. (4.30). The curvature is computed by expression (4.35).

LS-GFM: This is the sharp-interface level-set method in which the density and surface tension at the cell face is determined using the Ghost Fluid Method (GFM), see 4.2.4. Depending on the manner of determining the viscosity, we have the following variants: LS-GFM1 Following the work by [START_REF] Son | A level set method for analysis of lm boiling on an immersed solid surface[END_REF], the viscosity is determined in a sharp-interface manner using the simplied expressions (4.29) of the GFM, while the curvature at the cell center is computed from κ = -∇ • n in a straightforward manner.

LS-GFM The viscosity is determined using the smoothed Heaviside function (4.5) and the curvature is computed directly as in [START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF], Eq. (4.35). [START_REF] Desjardins | An accurate conservative level set/ghost uid method for simulating turbulent atomization[END_REF], the viscosity is determined using the smoothed Heaviside function (4.5) and the curvature and level-set normals are computed using the least-squares method of [START_REF] Marchandise | A stabilized nite element method using a discontinuous level set approach for the computation of bubble dynamics[END_REF], Eq. (4.10).

LS-GFMk Following the work by

LS-GFM2

In the absence of surface tension, these two previous methods are identical and are simply denoted as LS-GFM2. In contrast to the method LS-GFM1, the viscosity is smoothed near the interface.

For the volume-of-uid methods, the density and viscosity at the cell-faces are determined using the volume-of-uid weighted average, Eqs. (4.21) and (4.28). No smoothing kernel is used, except when mentioned explicitly. We have used dierent methods depending on the volume-ofuid advection and reconstruction method and the curvature calculation:

VOF-HF2 HF2

VOF-WYC HF1 LS-LA if σ = 0, then LS-LA1 and LS-LA2 are identical.

LS-GFM2

if σ = 0, then LS-GFMk and LS-GFM are identical.

HF1

Height Function method by [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF].

HF2

Height Function method by [START_REF] Hernández | A new volume of uid method in three dimensions part i: Multidimensional advection method with face-matched ux polyhedra[END_REF]. The density at the cellfaces in the surface tension term (Eq. 4.31) is dened as in Eq. (4.38).

123 4. Incompressible two-phase ow solver 4.4 Results and discussions

Spurious currents in a stationary circular bubbles

In the absence of gravity, a circular bubble will reach a state of equilibrium between surface tension and pressure forces, with the ow eld at rest and a dierence between the pressure inside and outside of the bubble given by the Young-Laplace equation. In numerical simulations of this basic two-phase ow problem, the velocity will not be exactly zero, but show uctuations known as parasitic or spurious currents. These currents can be quite strong in certain cases and have a strong impact on the dynamic behavior of the bubbles when simulating two-phase ows in general. The evaluation of the magnitude of the spurious currents is therefore a well known test case and has been employed in many works involving the development of numerical schemes for two-phase ows.

We will adopt the ow parameters used in [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF], [START_REF] Herrmann | A balanced force rened level set grid method for two-phase ows on unstructured ow solver grids[END_REF] and [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF]. The computational domain is dened as the unit square with symmetry boundary conditions on all four boundaries. The bubble is located in the center of the domain and has a diameter of 0.4. The surface tension is set to σ = 73 and the density outside the bubble ρ 1 = 1. The density inside the bubble is dened by the density ratio ρ 2 /ρ 1 which can be varied between 0 and 1. One can generally identify two contributions to the production of the spurious currents: the imbalance at the discrete level between the surface tension and pressure forces, and the error committed when estimating the curvature.

4.4.1.1 Spurious currents due to unbalanced forces [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF] presented some results after one time step ∆t = 10 -6 with µ 1 = µ 2 = 0 in which the curvature in the computation is replaced by the exact value κ = 1/R. If the numerical scheme is able to reproduce an exact equilibrium, the resulting velocity eld will remain at rest and the pressure dierence will be exactly ∆p = 36.5. For this case, results obtained with the Ghost Fluid Methods (LS-GFM , LS-GFMk and LS-GFM1) are nearly identical. The same is true for the VOF methods, and also for those based on the local averaging (LS-LA1 and LS-LA2).

The results are given in Table 4.2.

All the methods reproduce the exact balance in which the velocity eld remains zero to machine precision. Indeed, in these schemes the surface tension term is discretized in the same manner as the pressure gradients, as described by the balanced-force approach [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF].

Methods based on the Ghost Fluid Method (LS-GFM) and the local averaging (LS-LA)

reproduce the exact pressure eld within machine precision. With the continuous surface force method, however, the pressure eld is smeared out and a signicant error remains. But these errors are concentrated near the interface and an exact pressure dierence is recovered far from the interface, although this cannot be observed in the presented results.

Spurious currents due to inaccurate curvature computation

The balanced-force approach allows an exact balance between the surface tension and pressure terms, but only when the curvature is constant. Errors in the estimation of the curvature also lead to spurious currents. The following results are therefore obtained with the full method in which the curvature is computed in dierent manners as shown on Table 4.1. We will rst present the results after one time step of ∆t = 10 -6 on a 40 × 40 grid, see Table 4.3. In this case, the initial level-set and volume-of-uid elds are exact and the spurious currents are mainly due to the error in the curvature estimation. It is seen that the LS-LA2, LS-GFM and the VOF methods give the most precise results in this case, nearly independent of the density ratio.

The results obtained with a smoothing kernel are presented on Table 4.4. For the VOF methods determining the curvature through the HF1 method, the use of the smoothed fraction

C

instead of C has not much inuence on the maximum velocity and the pressure dierence. On the other hand, with the VOF-HF2 method, the smoothing kernel slightly increases the amplitude of the spurious currents. This is probably due to the use of the mean density (ρ g + ρ l )/2 in the surface tension term, whereas Eq. (4.21) is used for the density in the pressure equation.

According to [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF], it is then not possible to obtain a correct balance between surface tension and pressure forces. (see Table 4.1).

Time evolution of the spurious currents

Now we turn our attention to the time evolution of the spurious currents in uids with non-zero viscosity. The additional relevant non-dimensional number is the Laplace number dened as:

La ≡ σD ρ 1 ν 2 1
We set µ 1 = µ 2 = 0.1 in our computations, in addition to D ≡ 0.4 and σ = 73, which leaves the density as the only free parameter to choose the Laplace number. Three characteristic time scales can be dened in this problem:

t σ = ρ 1 D 3 σ ; t ν = D 2 ν and t ref = µ 1 D σ

Ca ≡ |u| max µ σ

We will also adopt this denition to present our results.

Note that VOF-HF1, VOF-WYC and VOF-WY produce almost identical results, which is due to the fact that the curvature and uid properties are determined in the same manner. Only the results obtained with the VOF-HF1 method are therefore presented. Note also that the time step have only a minor inuence on the time evolution of the spurious currents. For brevity sake, these results are not shown here.

Whole domain versus quarter domain simulations

The results for the tested methods are shown on Fig. 4.2 for a density ratio equal to unity, CFL σ = 1 and La= 12000. For each method, we have performed a simulation on the whole domain and the quarter of the domain (assuming symmetry conditions at the boundaries).

The simulations over the whole domain (red lines) often show an increase of the spurious currents towards the end of the simulation. Only the LS-CSF and LS-LA1 methods seem to be able to maintain the equilibrium solution over a long time period. However, when the quarter of the domain is simulated, all the methods maintain a constant level of the spurious currents. These observations can be explained by the fact that the bubble tends to drift away from the origin when the whole domain is simulated. This eect is suppressed by computing only a quarter of the bubble, as is done in [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF], in which case the bubble center is xed by the symmetry conditions and cannot move around. The MOF-HF1 method shows cycles of increase and decrease in the spurious currents. In fact these spurious currents are due to round-o errors in the interface method (MOF) and occur at very small CFL numbers, as pointed out in 2.6.2.

These errors may be suppressed in a future implementation by more carefully programming the MOF method. for La = 12000 when the whole domain is simulated vs. the quarter of the bubble. The time is made non-dimensional by the viscous time scale t ν . The density ratio is ρ 1 /ρ 2 = 1. Grid size 32 × 32 (R/h = 6.4). No reinitialization of the level-set eld. The VOF-WY and VOF-WYC methods yield similar results as the VOF-HF1 method.

Inuence of the Laplace number and density ratio Now, we investigate the inuence of the Laplace number and the density ratio on the spurious currents. The density ratio has been varied from 1 to 1000, and the Laplace number from 120 to 120000.

We have noticed that for some methods the results are insensitive to variations of the density ratio, and the spurious currents are mainly dependent on the Laplace number. Fig. 4.3 plots the temporal evolution of the spurious currents produced with these methods for a unit density ratio. All methods, except the continuous surface force approach (LS-CSF), reach an equilibrium state with a very low amplitude of spurious currents over the entire range of Laplace numbers.

The remaining methods (MOF-HF1, LS-LA1, LS-LA2, LS-GFMk and LS-GFM1) are more or less aected by the density ratio and/or the Laplace number. suers from numerical instability at higher values. The VOF-HF1 method, which diers only in the way the interface normal is reconstructed, did not show this numerical instability (Fig. 4.3).

Again, a future improvement of the MOF implementation might solve this problem. The methods LS-LA1, LS-GFM1 and LS-GFMk also have a diculty in maintaining the equilibrium state at La=120000. Only the LS-LA1 gives a stable equilibrium state for all density ratios and Laplace numbers. In conclusion, all VOF methods and the LS-GFM and LS-LA2 level-set methods are able to produce a correct equilibrium state of a stationnary bubble in a wide range of parameters.

All other methods may produce signicant spurious currents depending on the values of the density ratio and Laplace number.

Inuence of the level-set reinitialization

One important parameter omitted in the previous results (and generally also in results shown in the literature) is the reinitialization of the level-set eld. Fig. 4.5 shows the more realistic case of the level-set method with reinitialization every Nreinit time step. It is seen that the reinitialization procedure introduces a perturbation of the equilibrium state and a sudden increase of the spurious currents. When the level-set is reinitialized each time step, the magnitude of the spurious currents reaches a constant level. Increasing Nreinit helps to diminish the spurious currents, but the levels are still much higher than that obtained without reinitialization or with the VOF methods. Among the level-set methods, LS-GFM and LS-LA2 still produce the best results, but with reinitialization the LS-GFMk and LS-GFM1 methods are not far behind.

MOF-HF1, La=120

MOF-HF1, La=1200 MOF-HF1, La=120000

ρ 1 /ρ 2 =1 ρ 1 /ρ 2 =10 ρ 1 /ρ 2 =100 ρ 1 /ρ 2 =1000 LS-LA1, La=120
LS-LA1, La=1200 LS-LA1, La=120000

ρ 1 /ρ 2 =1 ρ 1 /ρ 2 =10 ρ 1 /ρ 2 =100 ρ 1 /ρ 2 =1000 LS-LA2, La=120
LS-LA2, La=1200 LS-LA2, La=120000

ρ 1 /ρ 2 =1 ρ 1 /ρ 2 =10 ρ 1 /ρ 2 =100 ρ 1 /ρ 2 =1000

LS-GFM1, La=120

LS-GFM1, La=1200 LS-GFM1, La=120000 the level-set eld on the evolution of the maximum velocity for La = 120. The time is made non-dimensional by the viscous time scale t ν . Only a quarter of the bubble is simulated on a 16 × 16 grid size (R/h = 6.4) with CFL σ = 1.0. The density ratio ρ 1 /ρ 2 = 1.

Oscillating drops

Now, we study the oscillating drop problem. An initially elliptical drop is placed in a square domain with zero gravity. Due to the surface tension forces, the drop will oscillate around its equilibrium state (the perfectly circular drop at rest). This test problem has been widely used to validate the surface tension implementation in ow solvers. The case for two-dimensional cartesian grids can be found for example in [START_REF] Torres | The point-set method: front-tracking without connectivity[END_REF]; [START_REF] Shin | Accurate representation of surface tension using the level contour reconstruction method[END_REF]; [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF]; [START_REF] Tong | A numerical method for capillarity-dominant free surface ows[END_REF]; [START_REF] Herrmann | A balanced force rened level set grid method for two-phase ows on unstructured ow solver grids[END_REF] and [START_REF] Wang | A sharp surface tension modeling method for two-phase incompressible interfacial ows[END_REF] and for cylindrical coordinate systems in [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF]; [START_REF] Aulisa | A novel representation of the surface tension force for two-phase ow with reduced spurious currents[END_REF][START_REF] Sussman | A sharp interface method for incompressible two-phase ows[END_REF] and [START_REF] Jemison | A coupled level set-moment of uid method for incompressible two-phase ows[END_REF].

In the case of viscous uids, the drop oscillates with a decreasing amplitude due to the viscous dissipation. But, in the absence of the viscosity, there will be no damping. This can only be obtained, however, with non-diusive numerical methods. In the following, we will treat both cases: zero viscosity in 4.4.2.1 and viscous uids in 4.4.2.2. 4.4.2.1 Inviscid drop oscillations [START_REF] Torres | The point-set method: front-tracking without connectivity[END_REF] considered a drop with an initial radius given by: r(θ) = r 0 + α cos(nθ)

(4.39)
where θ runs from 0 to 2π. In the case of inviscid uids, the theoretical frequency of oscillations is [START_REF] Fyfe | Surface tension and viscosity with lagrangian hydrodynamics on a triangular mesh[END_REF]:

ω 2 n = (n 3 -n)σ (ρ d + ρ e )r 3 0 (4.40)
with ρ d the drop density and ρ e that of its exterior. [START_REF] Torres | The point-set method: front-tracking without connectivity[END_REF] used a density ratio of 100, which is relatively small. A similar test case with a density ratio of 1000 can be found in the test suite of the free CFD Software Gerris (Popinet). We use the same parameters: r 0 = 0.1, α = 0.005, n = 2. The initial interface is therefore an ellipse of semi-major and semi-minor axis 0.105 and 0.095, respectively. The domain is 2 × 2 and only a quarter is simulated with symmetric boundary conditions. The drop density is equal to unity while that of its surrounding uid is set to 0.001, and the surface tension coecient σ is set to unity. The simulations are performed until time t = 1.

Since the drop and its surrounding uid are inviscid and no energy enters the domain boundaries, the total energy is conserved. The kinetic energy should oscillate in time with constant amplitude and frequency in the ideal case. The peaks correspond to the maximum of the kinetic energy which is reached when the drop is circular (maximum velocity and minimum surface tension forces), and the minima are attained when the drop is an ellipse (minimum velocity and high curvature and surface tension). The total kinetic energy is computed as:

K E = 1 2 Ω ρ| u| 2 dv = ∆x∆y 2 i,j ρ i,j (u x i,j + v y i,j ) 2 (4.41)
where u i,j and v i,j are the velocity components at the cell center. In the case of a staggered grid, they are computed by averaging of the values at the cell faces, for example u i,j = (u i+ 1 2 ,j + u i-1 2 ,j )/2. The density at the cell center, ρ i,j , is determined according to the approach the method uses (see 4.3.3).

We tested the dierent methods in Table 4.1 with a CFL σ equal to 1.0 on dierent grid sizes ranging from 32 2 to 256 2 ). We have noticed that the CFL σ does not have a great inuence on the results in the range [0.2 -1].

Results and discussions

The MOF-HF1 method showed a non-physical increase of the kinetic energy for all tested grid sizes due to the spurious currents of high magnitude (see for example the corresponding results to this method on Fig. 4.4 for La=120000). For this reason, the results with this method are not presented here.

The temporal evolution of the total kinetic energy, K E , on two dierent grids (642 and 256 2 ) is plotted on Fig. 4.6. The ticks on the x-axis correspond to the theoretical period given by Eq. (4.40). The error in the frequency of oscillation is tabulated on Table 4.5. The errors in mass conservation are given in Table 4.6. Note that for the level-set methods, reinitalization is applied at each time step.

On the 64 2 grid, the amplitude of oscillations decreases in time for all VOF methods. It decreases even on the rened grid of 256 2 for the VOF-HF1, VOF-WY and VOF-HF2 methods, which is perhaps due to velocity interpolation from cell faces to cell centers. The VOF-WYC method seems to be less diusive than the other methods, probably to the centered framework that this method uses. On the other hand, the frequency of the oscillations is better predicted with the VOF-HF1, VOF-HF2 and VOF-WY methods than with the VOF-WYC method.

The results obtained with the free CFD software Gerris (Popinet) are also plotted on Fig. 4.6.

The kinetic energy decreases less than with our VOF methods, which is due probably to the cell centered framework of Gerris. However, concerning the frequency error 2 our volume-of-uid implementation seems to be more accurate on grid sizes ner than 64 2 .

A strong damping of the oscillations of the kinetic energy is observed on the 64 2 grid with the LS-CSF and LS-LA1 methods. For the LS-LA1 method, this could be due to the increase of the spurious currents (see the corresponding results on Fig. 4.4 for La=120000). For the LS-CSF method, we see on Table 4.6 that the mass gain is very important for coarse grids (around 42% and 21% on the 32 2 and 64 2 grids respectively). Knowing that the density of the drop is 1000 times that of the surrounding uid, this increases the kinetic energy until it stops from oscillating. The other level-set methods exhibit much smaller mass errors (around 10% on coarse grids). The LS-LA2 method is again more accurate than the LS-LA1 method for this inviscid problem. The Ghost Fluid Methods (LS-GFMk, LS-GFM and LS-GFM1) slightly overestimate the kinetic energy on the coarse grid of 64 2 , but give accurate results on ne grids.

Eect of the reinitialization

The perturbation introduced by the reinitialization of the level-set eld can be reduced by increasing Nreinit. This is done here by using a CFLreinit as is described in 1.6.2.2. The results are signicantly enhanced only for the LS-CSF method, for which the temporal evolution of the kinetic energy is plotted on Fig. 4.7. The mass error is now very small (see Table 4.6) compared to the case where the level-set is reinitialized after each advection. It does not exceed the 8% on the 32 2 grid. The error on the frequency (presented on Table 4.5) are also much better than when the level-set is reinitialized each time step.

Eect of the smoothing kernel

The kinetic energy prole obtained with the vertex-averaged smoothing kernel is plotted on Fig. 4.8 for the two grid sizes 64 2 and 256 2 . The frequency error is summarized on Table 4.7 for all the smoothing kernels. The smoothing kernel generally improves the accuracy of the VOF methods, both in terms of frequency error and when comparing the decrease of kinetic energy without smoothing kernel. For the VOF-WYC method, the best results are obtained using the Table 4.5: Inviscid oscillating drop: error in the oscillation frequency (in %) for the dierent methods and grid sizes. In the case of a uctuating frequency, the error has not been computed. The capillary wave test case veries the coupling between the surface tension, inertial eects and viscosity. In a rectangular domain with zero gravity, two uids A and B are separated by a horizontal interface. A sinusoidal perturbation of small amplitude λ 0 is imposed on the initial location of the interface. A damped oscillation of the interface will result with angular frequency given by the dispersion relation

ω 2 0 = σk 3 /(ρ 1 + ρ 2 )
where k is the wavenumber. In the case of small amplitudes of the wave and equal kinematic viscosity of the uids, [START_REF] Prosperetti | Motion of two superposed viscous uids[END_REF] and [START_REF] Cortelezzi | Small-ampltitude waves on the surface of a layer of a viscousliquid[END_REF] provided an analytical solution which can be written as [START_REF] Popinet | A front-tracking algorithm for accurate representation of surface tension[END_REF]:

λ(τ ) λ 0 = 4(1 -4β) 2 8(1 -4β) 2 + 1 erfc( √ τ ) + 4 i=1 z i Z i ω 2 0 z 2 i -ω 0 exp (z 2 i -ω 0 )τ ω 0 erfc z i τ ω 0 (4.42)
where λ(τ ) is the amplitude at the dimensionless time τ = tω 0 and = νk 2 /ω 0 is the nondimensional viscosity. The z i 's are the four roots of the algebraic equation

z 4 -4β( ω 0 ) 1 2 z 3 + 2(1 -6β) ω 0 z 2 + 4(1 -3β)( ω 0 ) 3/2 z + (1 -4β)( ω 0 ) 2 + ω 2 0 = 0 and Z i = 4 j=1,j =i (z j -z i ). The dimensionless parameter β is dened as β = ρ 1 ρ 2 /(ρ 1 + ρ 2 ) 2 .
In the following, we consider that the initial perturbation is a cosine wave of amplitude λ 0 = L/100 where L = 1 is the domain width. The height of the computational domain is three times its width, which reduces the boundary eects [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF]. The initial position of the interface is given by y = λ 0 cos(kx) + y 0 with y 0 = 3L/2. The gravity is zero and we perform the simulations until time τ = 25. Only the density and viscosity are varied in the next subsections.

We will start rst by treating the case of equal density and viscosity in 4.4.3.1 then the case of air-water capillary wave in 4.4.3.2.

The simulated interface location y S (τ ) at x = 0 is determined either from the PLIC reconstruction in the case of the volume-of-uid methods, or in the case of the level-set methods by subdividing the mixed cell into 1000 × 1000 sub-cells. The interface position y A (τ ) = λ(τ ) cos(ω 0 τ ) + y 0 is then determined from the analytical solution (4.42) at the same discrete time levels in order to evaluate the RMS error, dened as

E RM S = 1 λ 0 1 N N i=1 (y S (τ i ) -y A (τ i )) 2
where N is the number of time steps.

Fluids with equal densities and viscosities

The uid properties are set to the same values as in [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF]: ρ 1 = ρ 2 = 1, σ = 1 and ν 1 = ν 2 = 0.0182571749236, which gives a corresponding Laplace number of La≡ σL/ρν 2 = 3000. interface remains nearly at and the curvature is rather small. The Height Function method is particularly well suited to this problem, since the orientation of the interface is always in the positive y-direction. This explains also the similar results obtained with the two versions HF1 and HF2 of the Height Function method.

The VOF-HF1 and MOF-HF1 methods are very accurate and recover the frequency even on the coarse grid of 8 × 24. The VOF-WYC method shows an increase in the period on the coarse grid, which is most probably due to the cell-centered discretization of the Navier-Stokes equations. A similar behavior is observed with the free CFD software Gerris (Popinet), which also uses a cell-centered grid setting.

The increase of the oscillation period is also observed for the LS methods on the grids 8 × 24 and 16 × 48. With these methods, the amplitude is also overestimated on these coarse grids.

VOF-HF1

MOF-HF1 VOF-WYC

Incompressible two-phase ow solver

The RMS error E RM S between the analytical solution and the simulation is plotted on Fig. 4.12 3 . The VOF methods (in blue) are clearly more accurate than the LS methods. How- ever, the LS methods keep a constant second-order convergence, while the VOF-HF1 and MOF-HF1 methods fail to do so on the nest grids. The best results are obtained with the free CFD software Gerris (Popinet). Since the problem is surface-tension driven, the time step is limited by the CFL σ number. We will check here the inuence of CFL σ on the accuracy of the simulations. The RMS error E RM S is plotted on Fig. 4.13 for dierent grid sizes and CFL σ values for two methods. The inuence of CFL σ is in general weak. We see that the RMS error slightly decreases with the CFL σ for the VOF-HF1 method. For the level-set method, the RMS slightly increases for decreasing CFL σ number, which is most probably due to the reinitialization of the level-set which is performed more often.

Air-water capillary wave

We now set the density ratio equal to 850 and the dynamic viscosity ratio to 55.72, as in Popinet (2009). The analytical solution data are taken from the Gerris website (Popinet). We use cubic interpolation to interpolate our data on the same time steps for which the analytical solution is available, in order to compute the RMS error. The temporal evolution of the interface is not shown here since it is almost similar to those of the rst test case. We only plot the RMS error on Fig. 4.14 4 . The results of the Gerris code are obtained using the smoothed density (vertex- averaged smoothing kernel of Eq. (4.8)), while no smoothing kernel is used here for the VOF methods.

We see that on the coarse grid of 8 × 24, the LS-CSF method provides the smallest error, but for ner grids the VOF methods give the best results. The LS-GFMk and LS-GFM methods are only rst-order accurate, while the LS-GFM1 method shows a second-order convergence on ne grids. The LS-LA methods give good results on coarse grids, but are less accurate on the ner grids. The level-set is reinitialized after each advection step.

Eect of the smoothing kernel

The inuence of the smoothing kernel on the VOF-HF1 method is summarized on Table 4.8

.

The results are clearly improved, in particular on the coarse grids. The Vertex-Avg smoothing kernel seems to give the best results. Under the inuence of gravity forces, the heavy uid falls and forms the instability. The interface initially separating the two uids is given by: y = λ 0 cos(kx) + y 0 (4.43)

where λ 0 and k = 2π are the initial amplitude and wavenumber of the perturbation, respectively.

Many variants of this test case can be found in literature. The main dierence is in the physical phenomena that are taken into account (surface tension, viscosity). A brief overview is given here.

• [START_REF] Gómez | On the reinitialization procedure in a narrow-band locally rened level set method for interfacial ows[END_REF] studied the instability with viscous uids, both with surface tension forces (simulations up to the time t = 1.3) and without (until t = 0.9).

• [START_REF] Desjardins | A spectrally rened interface approach for simulating multiphase ows[END_REF] have studied the instability with viscous uids and surface tension until t = 1.2 on grids going from 32 × 128 to 512 × 2048. The authors computed the penetration depth and compared it to the results on the nest grid of 512 × 2048.

• [START_REF] Brackbill | A continuum method for modeling surface tension[END_REF] and [START_REF] Gerlach | Comparison of volume-of-uid methods for surface tension-dominant two-phase ows[END_REF] have studied the instability with inviscid uids and surface tension, and compared their results to the analytical solution available in this case.

In the current work, we will start rst by studying the case of viscous uids without surface tension in 4.4.4.1. Next, we add the surface tension in 4.4.4.2. Finally, the case of inviscid uids with surface tension is treated in 4.4.4.3.

Viscous uids without surface tension

The domain is [0, 1] × [0, 4]. The heavy (light) uid has a density ρ 1 = 1.225 (ρ 2 = 0.1694) and a viscosity µ 1 = 0.00313 (µ 2 = µ 1 ) [START_REF] Gómez | On the reinitialization procedure in a narrow-band locally rened level set method for interfacial ows[END_REF]. The initial perturbation is given by Eq. (4.43) with an amplitude of λ 0 = 0.05 and y 0 = 2. The acceleration due to gravity is g = 1.

Since there is no surface tension, the LS-GFM1 and LS-GFM methods are identical in this particular case. Similar for the LS-LA1 and LS-LA2 methods, and the VOF-HF1 and VOF-HF2 methods. In addition, results for the VOF-WY and VOF-HF1 methods are indistinguishable and only the latter is shown.

The interface is plotted on Fig. 4.15 for the times t = 0.6 and 1.0 on a 64 × 256 grid. The interface on the grid 128 × 512 is plotted at the nal time on Fig. 4 The level-set eld is reinitialized each time step.

On the ne grid, all methods again give similar results, although the spike is deeper with Gerris than with our VOF and LS methods. On the coarse grids, the interface structures are much thinner, with break-o for some methods. In the case without surface tension, the LS-LA method does not provide good results. However, for the present case with surface tension, the two methods LS-LA1 and LS-LA2 perform very well. The LS-LA1 gives results comparable to the other methods, while the LS-LA2 method produces similar results on both the ne and coarse grids. The results provided by the LS-GFM methods are slightly dierent. They resemble more to those obtained at the coarse grids, which might indicate poorer accuracy.

Inviscid uids with surface tension

The Rayleigh-Taylor instability simulated in the previous sections concerns the case of viscous uids with or without surface tension eects. The comparison between the dierent methods was based on the shape of the interface at a given time of the simulation. In the case of inviscid uids with surface tension, the spike grows exponentially as exp(nt) where n is the growth rate.

The two physical phenomena governing the growth of the instability are the acceleration due to gravity, g, and the surface tension, σ, which tends to delay the growth of the spike. An analytical expression of the growth rate, n, for this specic problem is provided by [START_REF] Bellman | On the properties and limitations of the height function method in two-dimensional cartesian geometry[END_REF]:

n 2 = kgA 1 - k 2 σ (ρ 1 -ρ 2 )g where A = ρ 1 -ρ 2 ρ 1 + ρ 2 (4.44)
with A the Atwood number, k the wavenumber and ρ 1 and ρ 2 the densities of the heavy and light uids, respectively.

The critical surface tension, σ C , is determined from Eq. (4.44) by setting n = 0. [START_REF] Daly | Numerical study of the eect of surface tension on interface instability[END_REF] has introduced a dimensionless parameter, which is the ratio of the surface tension coecient to the critical value:

ξ = σ σ c = σk 3 (ρ 1 -ρ 2 )g (4.45)
This dimensionless parameter is a measure of the importance of surface tension forces to gravity forces. If it is close to zero, the gravity forces are much greater than the surface tension forces.

If ξ is close to unity, the surface tension dominates the growth process of the spike.

In our simulations, the depth of the spike, λ, is determined by nding the location of the interface at the half-width of the domain. [START_REF] Brackbill | A continuum method for modeling surface tension[END_REF] and [START_REF] Gerlach | Comparison of volume-of-uid methods for surface tension-dominant two-phase ows[END_REF] have compared the growth rate to the analytical solution given by Eq. (4.44). In this work, we adopt the same parameters as [START_REF] Gerlach | Comparison of volume-of-uid methods for surface tension-dominant two-phase ows[END_REF]. The domain is rectangular, of width equal to 2π and wavenumber k = 1. The domain height is three times its width with slip lateral boundaries and no-slip top and bottom boundaries. The Atwood number is chosen equal to A = 0.6 and ξ is varying in the range [0.05, 0.9]. We dene a dimensionless time τ = t/ kgA. The initial perturbation (Eq. (4.43))

has an amplitude of λ 0 = 0.001. We set CFL= 0.5 and CFL σ =1. Furthermore, we take g = 1 and ρ 1 = 1. The simulations are performed until time τ = 10.

The temporal evolution of the penetration depth is plotted on Fig. 4.19 for dierent values 4. Incompressible two-phase ow solver of ξ. The VOF-HF2 method did not reach the nal time and is not presented. The VOF-HF1, MOF-HF1 and VOF-WY methods provide similar results. Therefore, only the former is shown in this section.

The perturbation grows slower with the VOF-WYC method for ξ = 0.9 while the perturbation vanishes with the VOF-HF1 and MOF-HF1 methods towards the end of the simulations.

With the LS-LA1 and LS-LA2 methods, the perturbation starts to oscillate with constant amplitude, even for larger grid sizes. These two methods are therefore not shown here.

When we rene the grid, LS-GFM1 and LS-GFMk methods give very similar results (see Fig. 4.20) even though the surface tension is computed dierently in both methods. The anomalies appear only when the grid is rened and ξ is close to unity (high surface tension). In these conditions, the value of the CFL σ number has little inuence on the evolution of the penetration depth. The parameter that has a much more inuence is the amplitude of the initial perturbation, λ 0 , as we can see it on No smoothing kernel is used with the VOF methods. The level-set eld is reinitialized each time step.

The interface at the nal time τ = 10 on the nest grid is plotted on Fig. 4.22 for dierent values of ξ. Little dierences appear between the dierent methods, although the interface shape with the VOF-WYC method is really dierent from the others for ξ = 0.9.

The growth rate, determined between τ = 3 and τ = 6, is plotted on Fig. where P a denotes the perimeter of a circle with diameter d a which has an area equal to that of the bubble with perimeter P b . The circularity is equal to unity for a perfectly circular bubble, and less than unity for non-circular shapes. The mean velocity is dened by

LS-GFMk

LS-GFM1 LS-GFM -3 -2 -1 0 1 2 3 -2 -1 0 1 2 3 4 ξ = 0.05 ξ = 0.25 ξ = 0.50 ξ = 0.75 ξ = 0.90 -3 -2 -1 0 1 2 3 -2 -1 0 1 2 3 4 ξ = 0.05 ξ = 0.25 ξ = 0.50 ξ = 0.75 ξ = 0.90 -3 -2 -1 0 1 2 3 -2 -1 0
U c = Ω 2 uds Ω 2 1ds (4.48)
In this section, the level-set eld is reinitialized after each time step. The eect of the smoothing kernel on the volume-of-uid methods is not investigated.

Benchmark 1

The rst benchmark is characterized by a low density and viscosity ratios (ρ l /ρ g = µ l /µ g = 10)

and high surface tension coecient as shown on Table 4.9. The bubble is initially circular at rest, and the bubble ascension is simulated with the dierent methods until time t = 3. The bubble shape at this particular time is plotted on Fig. 4.25 for grid sizes ranging from 32 × 64 to 256 × 512. The bubble does not undergo major topological changes due to the relatively low density and viscosity ratios and high surface tension. In fact, in this test case, the shape of the interface is surface tension controlled.

The comparison with the reference solution (Featow) is excellent, in particular for the VOF methods. For these methods, the result is almost independent from the grid size. The VOF methods are more accurate than the LS methods which seem to suer from mass loss on coarse grids. The MOF method is not shown since it gives very similar results as that obtained with the VOF-HF1 and VOF-WY methods. The results for the VOF-HF2 method on the grid 32 × 64 is also not shown, since it does not reach the end of the simulations due to a problem in the computation of the curvature. The LS-CSF and LS-GFM methods behave slightly better than the other level-set methods, which is probably due to the identical manner of computing curvature. The relative mass errors are, however, of the same magnitude for all LS methods 7 .

The instantaneous mean rising velocity is plotted on Fig. 4.26 for three methods. Similar proles are obtained with the other methods. The reference solution (red stars) is also reported on this gure for the sake of comparison. Initially, the bubble is stagnant with zero rising velocity, then under the inuence of buoyancy forces, it starts to move upward until it reaches a steadystate velocity. The rising velocity reaches rst a maximum value, then decreases towards the steady-state velocity. The correct velocity prole is recovered with all methods. Slight dierences are observed only on the coarsest grid. For this grid, the VOF-WYC method provides the most accurate solution.

Values of the maximum velocity and the time of occurrence are summarized on Table B.4.

The L2-norm of the error between the computed mean velocity prole and the one provided by the reference solution is shown on Fig. 4.28. For the LS-GFMk and LS-GFM methods, the error is very high on the ne grid due to the convergence of the maximum velocity to a value dierent from that of the reference solution. It conrms that the VOF methods are more precise than the LS methods for this particular case.

The bubble circularity is plotted on Fig. 4.27 8 . As with the mean rising velocity, the cir- 4.4. Results and discussions VOF-HF1 VOF-WYC 0.9 cularity also converges to the reference data for all methods on ne grids. The circularity is greater than unity for the VOF methods during the rst time steps, because the predicted bubble perimeter (obtained with the PLIC reconstruction) is smaller than the theoretical one. We assume that the PLIC reconstruction is also responsible for the erratic behavior of the circularity prole. The L2-norm of the error of the circularity prole is plotted on Fig. 4.28. It shows that the circularity prole is better predicted with the solvers using the volume-of-uid methods.

1 1.1 1.2 1.3 0.9 1 1.1 1.2 1.3 VOF-HF2 LS-CSF 0.9 1 1.1 1.2 1.3 0.9 1 1.1 1.2 1.3 LS-LA1 LS-LA2 0.9 1 1.1 1.2 1.3 0.9 1 1.1 1.2 1.3 LS-GFMk LS-GFM 0.9 1 1.1 1.2 1.3 0.9 1 1.1 1.2 1.3 LS-GFM1 0.9 1 1.1 1.2 1.3
The temporal evolution of the bubble's centroid y-position is presented on The grid convergence for the circularity, the rising velocity and centroid are plotted on Fig. 4.28. The VOF and MOF methods are clearly more accurate than the LS methods. Only the VOF-HF2 method seems to have some problems on the ne grids. Among the LS methods, the LS-CSF method shows the best results, followed by the GFM methods. 9 The nal positions of the center of mass are reported on

Benchmark 2

The second case, characterized by a high density and viscosity ratios, is a more dicult test problem. The interface at the nal time t = 3.0 is plotted on Fig. 4.29. The VOF-HF2 method is the only one not reaching the end of the simulation, due to a failure in the curvature computation. The reference solution (Featow) is also shown. However, in the absence of experimental results, it is not clear whether this numerical result represents correctly the actual solution of this problem, in particular with respect to the length and shape of the thin laments. This has to be taken into account in the comparison of the dierent methods.

The VOF-HF1, MOF-HF1 and VOF-WY methods give very similar results in which the thin lament breaks up in small bubbles. This is a typical behavior for VOF methods that is also evidenced in Chapter 2. On the other hand, the LS methods tend to prevent the formation of the thin laments (probably along with some loss of mass). The LS-CSF method, which uses smoothing of the uid properties, is very unstable and fails to compute the interface on the nest grid. Some diculties are also observed for the LS-GFM1 method. The VOF-WYC method converges best towards the reference solution. When rening the grid, the thin laments are well reproduced without fragmentation.

The instantaneous relative mass error for this benchmark is plotted on Fig. 4.30 10 . The mass error is much larger for the second benchmark, since the bubble undergoes more topological changes. The mass error decreases when increasing the grid size, except for the LS-CSF and LS-GFM1 methods.

The mean rising velocity is plotted on Fig. 4.31

11 . All the presented methods seem to repro- duce the reference values on coarse grids until t = 1.5, except the LS-LA1 and LS-LA2 methods for which the velocity prole diverges rapidly from the reference velocity.

The instantaneous bubble circularity is plotted on Fig. 4.32. The circularity for the reference solution (red stars) steadily decreases in time due to the formation of the tails. The tested methods, however, produce all very dierent proles. These large dierences between methods were also observed in the work of [START_REF] Hysing | Quantitative benchmark computations of two-dimensional bubble dynamics[END_REF]. The VOF and MOF methods follow the reference solution rather well, due to the formation of the fragments. The LS methods prevent the formation of the laments. Therefore, an increase of the circularity is observed after t = 2, except for the LS-CSF method which produces many non-physical fragments.

The center of mass of the bubble is mainly determined by the bulk shape of the bubble, which is correctly reproduced by nearly all methods. The results are therefore not shown here 12 .

The L2-norm of the error between the velocity prole and that of the reference solution is plotted on Fig. 4.33. Although it is dicult to make objective conclusions, due to the absence of a true reference solution, it seems that the tested methods are at best rst-order accurate.

None of the methods stick out in performance, but the VOF methods have at least the benet of showing a rst-order convergence for the mean rising velocity and the center of mass, while the LS methods show a rather erratic behavior. 4. Incompressible two-phase ow solver

Conclusions

In this chapter, we have described dierent existing methods for the coupling between the Navier-Stokes solver with an interface method (mainly, the level-set and the volume-of-uid) in two dimensional cartesien grids. The considered level-set and volume-of-uid methods have been already tested in congurations with imposed velocity eld in Chapter 3.

These coupling techniques are inspired by some well-known works in two-phase ows and are implemented in our in-house code on 2D cartesian (MAC) grids. A cell-centered implementation of the ow solver coupled with the volume-of-uid method, similar to that of the free CFD software Basilisk (Popinet et al.) has been also implemented. This latter served in the comparison between cell-centered and MAC grids.

The dierent coupling manners of the level-set method with the ow solver include: sharp interface representation (using the GFM method), smooth interface approach (using the smoothed Heaviside H ) and an intermediate approach which consists of using the GFM method with locally-averaged uid properties. Other variants which consider a sharp variation of density with a smooth viscosity are also implemented and tested. The volume-of-uid methods are based on the CSF approach for the surface tension. They dier either in the interface advection method or in the curvature estimation.

The comparison between these methods is performed through a set of well-known test cases.

Table 4.10 gives an overview of the performances of each method. It is clear that the volume-ofuid methods in the exception of the VOF-HF2 method have a good overall behavior. Indeed, the VOF-HF2 method does not reach the end of some simulations due to a failure in curvature computation. The smoothing kernel makes the results more accurate in some cases, but it make it worse in others. It is seen that the VOF-HF1 and VOF-WY methods, which dier only in the volume-of-uid advection scheme, give very similar results for all the simulations. We can conclude that the inuence of the advection schemes [START_REF] Weymouth | Conservative volume-of-uid method for free-surface simulations on cartesian-grids[END_REF] and [START_REF] Scardovelli | Interface reconstruction with least-square t and split eulerian lagrangian advection[END_REF]) is negligible here.

For the particular problem of spurious currents, the amplitude of the spurious currents decreases in time with the VOF methods as well as with the LS methods when the level-set eld is not reinitialized. However, when we reinitialize the level-set method after each advection step, the amplitude of these spurious currents is much higher.

In the level-set approach, the LS-GFM and LS-LA2 methods seem to be the most accurate.

These two methods perform well for most of the simulations, and both have performances comparable to those of the VOF methods. On the other hand, the LS-LA1 method gives poor results for many ow problems.

Conclusions and perspectives

The aim of this work was to develop a numerical method to simulate two-phase incompressible ows with non-miscible uids. It is a complex numerical problem that implies the knowledge of the theoretical background of uid mechanics, mathematics and numerical analysis. This thesis focused on the analysis and the comparison of several numerical methods, especially those dealing with the interface capturing and their coupling with the ow solver. This work is split into four parts. The rst three parts deal with the interface transport in an imposed velocity eld, and the last one with the coupling with the ow solver.

The rst part is devoted to the analysis and selection of the robust numerical tools for the level-set method. In particular, the reinitialization step and the advection schemes were rigorously investigated including a wide range of numerical schemes. The reinitialization step keeps the signed-distance property of the level-set eld, which is needed to accurately determine interface properties such as the curvature and the normal to the interface. However, this step tends to move the interface and thereby causes the change of the reference uid mass. We implemented two enhancements of the reinitialization step: the sub-cell x proposed by [START_REF] Russo | A remark on computing distance functions[END_REF] and the volume constraint by Sussman et al. (1999). The well-known test problems of the Zalesak disk, shearing vortex and time-reversed single vortex are used to verify the accuracy of these methods. The volume constraint is much more robust especially with conservative schemes, in particular the fth-order WENO scheme. In addition, it is less sensitive to the CFL number.

The second part is devoted to the volume-of-uid methods, for which it is much easier to conserve mass. The directional-splitting conservative (with respect to the mass of the reference uid) and consistent (which keeps the volume fraction C between 0 and 1) geometrical resolution [START_REF] Scardovelli | Interface reconstruction with least-square t and split eulerian lagrangian advection[END_REF][START_REF] Tryggvason | Direct numerical simulations of gas-liquid multiphase ows[END_REF] for the volume fraction advection is studied. We included in our comparison the alternative directional-splitting conservative and consistent scheme developped by [START_REF] Weymouth | Conservative volume-of-uid method for free-surface simulations on cartesian-grids[END_REF], which can be easily extended to 3D cartesian grids. In both methods, the piecewise linear interface reconstruction is performed using the ELVIRA algorithm which combines accuracy and less CPU resources. The results of these two VOF methods are almost identical in passive advection tests. In contrast to the level-set method which tends to thicken thin interface laments, the VOF methods break it into fragments in the case where more than one interface pass through the reconstruction blocks of 3×3 cells. The moment-of-uid method [START_REF] Dyadechko | Moment-of-uid interface reconstruction[END_REF], 2006, 2008) reconstructs the interface in each mixed cell using the center of mass of the reference uid in that cell. Thus, these centroids are also advected using a similar technique as for the volume fractions. The MOF method is more accurate than the VOF methods in the passive advection tests and is a promising method since it does not break the lament. However, some numerical diculties have been encountered especially for small CFL number, but this may be avoided by a careful implementation of this method.

The third part deals with coupled level-set and volume-of-uid methods. These coupled techniques are supposed to combine the advantages of both constituting methods, namely a good interface representation and the mass conservation. Some coupled methods were easy to implement such as the VOSET method [START_REF] Sun | A coupled volume-of-uid and level set (voset) method for computing incompressible two-phase ows[END_REF]. In fact, this latter needs only a supplementary routine for determining the signed-distance to the interface. The accuracy of the VOSET methods depends rst on the accuracy of the used VOF method, then on that of the algorithm computing the level-set eld. Other methods are more complex such as the CLSVOF [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF][START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF] and CLSMOF [START_REF] Jemison | A coupled level set-moment of uid method for incompressible two-phase ows[END_REF] methods. In our implementation of coupled approaches, the VOF fractions are advected using the conservative and consistent scheme. As a consequence, algorithms for the volume fraction redistribution (for cases where C > 1 or C < 0) are not required.

We investigated the VOSET, MCLS (van der [START_REF] Van Der Pijl | A mass-conserving level-set method for modelling of multi-phase ows[END_REF], CLSVOF and CLSMOF (which couples LS with MOF) methods. Some of these coupled methods use alternative approaches to overcome fastidious algorithms for the VOF method. In general, the behavior of the coupled methods is very close to that of the VOF methods. For example, the coupled techniques break thin laments. Only the degree of regularity and length of the non-fractured vortex lament dier from a method to another. In the MCLS coupling, the use of the standard reinitialization equation with 1st-or 2nd-order advection schemes provide more accurate results than with enhanced reinitialization methods and high-order advection schemes. The CLSVOF methods produce similar results as those of the VOF method, and the CLSMOF provides the most accurate results.

The last part concerns the coupling between the incompressible Navier-Stokes solver and the interface method. The ow solver is based on a 2D uniform staggered (MAC) grid. The convective terms are discretized in space using the 2nd-order ENO scheme. The diusion terms are treated implicitly and discretized in space using the 2nd-order central-dierence scheme. The pressure-velocity decoupling is realized through the projection method. We have implemented six dierent methods based on the level-set approach and ve on the VOF approach, inspired by some well-known works from the literature. In the level-set based ow solver, the treatment of the interface is realized either through a

• Smoothed pressure jump (CSF, [START_REF] Brackbill | A continuum method for modeling surface tension[END_REF]; [START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF]) with smoothed uid properties

• Sharp pressure jump (GFM) with jump in uid properties [START_REF] Son | A level set method for analysis of lm boiling on an immersed solid surface[END_REF] • Sharp pressure jump (GFM) with jump in density, and smoothed viscosity [START_REF] Desjardins | An accurate conservative level set/ghost uid method for simulating turbulent atomization[END_REF] • Sharp pressure jump (GFM) with locally averaged uid properties [START_REF] Vigneaux | Méthode level set pour des problèmes d'interfaces en microuidique[END_REF] In the VOF-based ow solver, the uid properties are computed using the volume-of-uid weighted-average. The main dierence is the interface method itself (VOF [START_REF] Scardovelli | Interface reconstruction with least-square t and split eulerian lagrangian advection[END_REF], VOF [START_REF] Weymouth | Conservative volume-of-uid method for free-surface simulations on cartesian-grids[END_REF] or MOF). Thus we have three VOF methods for which we add one coupled to a Navier-Stokes solver with a collocated grid setting. All of these methods use the Height Function by [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF] except one which uses the Height Function implementation of [START_REF] Hernández | A new volume of uid method in three dimensions part i: Multidimensional advection method with face-matched ux polyhedra[END_REF].

These 11 methods are compared in dierent ow congurations: spurious currents in a stationary circular bubble, oscillating drops, standing capillary wave, Rayleigh-Taylor instability and rising bubbles. The obtained results showed that the volume-of-uid methods using the Height Function method by [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF] give very good results for most simulations. Concerning the level-set based solvers, we found that two methods generally give very good results and are comparable to those obtained with VOF methods. The LS-GFM uses the GFM approach with a jump in density and a smoothed viscosity, while the interface curvature computed according to [START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF]. The LS-LA2 method uses also the GFM approach with locally-averaged uid properties [START_REF] Vigneaux | Méthode level set pour des problèmes d'interfaces en microuidique[END_REF], while the curvature is computed using a dierent method (Sussman et al., 1999). A very similar method to the LS-LA2, named LS-LA1, computes the interface curvature according to [START_REF] Vigneaux | Méthode level set pour des problèmes d'interfaces en microuidique[END_REF] fails in many simulations.

Future work includes the validation of the ow solver coupling with coupled volume-of-uid and level-set methods in 2D, then the axisymmetric ow solver with level-set and VOF methods.

This will allow to simulate, for example, bubbles, droplets and two-phase ows inside circular channels. Eventually, the extension of the code to fully three dimensional grids should be considered in order to be able to treat a wide range of congurations.

Including the energy equation and accurate interface conditions (to take into account the change in the mass of the reference uid due to the phase change) to the actual code will allow to simulate bubble growth during phase change, which is of a major interest to our laboratory.

Other phenomena such as lm boiling could be then predicted and studied only at solids. In order to take into account the phase change on irregular solid surfaces, accurate velocity boundary conditions should be applied. This can be performed using the immersed boundary method.

From a computational time point-of-view, it would be interesting either to extend the current sequential implementation to parallel computers or to use the local grid renement. In this context, the use of open-source CFD codes such as Gerris (Popinet) and Basilisk (Popinet et al.) is interesting. This will allow to study more realistic problems and help in the understanding of the physical phenomena behind phase change. The total area of the reference uid is the sum of its areas in each computational cell of the domain.

The mass error due to the approximation of the Heaviside Eq. (A.1) for a given grid and sub-cells number is bounded by:

L∆x ≥ Exact Mass - Ω H(φ E )dΩ ≥ L∆x m 2
where L is interface length. [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF] have used the bilinear interpolation which is 2nd-order. We have also implemented bicubic interpolation based on fourth-order Lagrange polynomials.

If the staggered cell delimited by the grid points (i, j), (i, j + 1), (i + 1, j) and (i + 1, j + 1) contains both uids, we subdivide it into m × m cells and the bilinear interpolation yields the values of the level-set on the center of sub-cells (x k , y l ): φ(k, l) = 1 ∆x∆y φ i,j (x i+1x k )(y j+1y l ) + φ i+1,j (x kx i )(y j+1y l )+ φ i,j+1 (x i+1x k )(y ly j ) + φ i+1,j+1 (x kx i )(y ly j )

where

x k = x i + k -1 2 m ∆x, y l = y i + l -1 2 m
∆y and (k, l) ∈ [1, m] 2 . φ(k, l) refers to interpolated level-set into the position (x k , y l ).

Fourth-order Lagrange polynomials are based on a stencil of 4 × 4 points. We interpolate the level-set between i and i + 1 rst in the x-direction. For the j-th column for example, the interpolation formula reads: φ(k, j) = 1 6(∆x) 3φ i-1,j (x kx i )(x kx i+1 )(x kx i+2 )+ 3φ i,j (x kx i-1 )(x kx i+1 )(x kx i+2 )-3φ i+1,j (x kx i-1 )(x kx i )(x kx i+2 )+ φ i+2,j (x kx i-1 )(x kx i )(x kx i+1 )

A.1. Mass and positioning error computation Then, we interpolate these level-set values between j and j + 1 following the y-direction on positions x k with k ∈ [1, m] as follows:

φ(k, l) = 1 6(∆y) 3φ k,j-1 (y ly j )(y ly j+1 )(y ly j+2 )+ 3 φ k,j (y ly j-1 )(y ly j+1 )(y ly j+2 )-3 φ k,j+1 (y ly j-1 )(y ly j )(y ly j+2 )+ φ k,j+2 (y ly j-1 )(y ly j )(y ly j+1 )

The knowledge of the reference eld and the computed level-set eld on each computational staggered cell is needed in the positioning error E 1 estimation. We apply then the same treatment as for the determination of Heaviside integral.

E 1 = Ω |H (φ E ) -H (φ C )| dΩ Ω H(φ E )dΩ = (i,j)∈Ω Ω ij |H (φ E (i, j)) -H (φ C (i, j))| dΩ
Exact mass

where Ω ij H(φ E (i, j))dΩ is the Heaviside integral on the cell (i, j).

The procedure we used to compute the numerator for each staggered cell is described below with the use of the following notation: In order to evaluate the accuracy of this approach, we compute the exact signed-distance function (φ E ) to a circle of radius r = 0.15 centered at (0.25, 0.5) in a 1 × 1 domain (see Fig. 1.5). Its exact area is πr 2 . Grid sizes are varied in the range of 1/8 up to 1/1024 and the number of sub-cells from 1 (no sub-cell) to 2000 2 . second-order and the error is almost independent from the number of sub-cells except for m = 1.

For the bicubic interpolation, error decreases by increasing the number of sub-cells. We see also that for a given sub-cell number, the error is uctuating depending on the grid; this is due to the position of the interface on the grid. For m ≥ 1000, the bicubic interpolation is 4th-order accurate. We remark also that the 4th-order Lagrange polynomials provides a better measure of mass than the bilinear interpolation. Now, we consider a more complicated interface with sharp angles which is the Zalesak notched disk (see 1.5.2). Its exact area is given by (πR 2w( R 2 -(w/2) 2 + 3R -2h)/2) where w is the notch width, (R + R 2 -(w/2) 2h) is the notch depth and R the radius. The order of the mass error, E M , is the same as the order ot the used interpolation. However, we can remark that the error depends mainly on the position of the angles and the notch on the grid. We remark also that the order of convergence is approximately identical to that of the used interpolation and that the bicubic interpolation remains more precise than the bilinear one.

As the bicubic interpolation provides a better estimation of mass, we use it to estimate the positioning error E 1 and the mass error E M , with a number of sub-cells m 2 = 1000 2 unless otherwise specied.

A.2 Determination of the reference solution of the single vortex

Shearing Vortex

The knowledge of the exact position of the interface is compulsory since it is needed to compute the positioning error E 1 . However, in this test case, interface position is not given by an analytical formula; only the initial and nal positions are exactly known. The exact interface is determined at any time of the simulation as follows:

1. At t = 0, seed the interface with equidistant massless markers (10 5 markers)

2. Advect these markers by solving the equation of motion for a particle u = d x dt using the fourth-order Runge-Kutta scheme

x (n+1) = x (n) + ∆t 6 u (1) + 2 u (2) + 2 u (3) + u (4)

A.2. Determination of the reference solution of the single vortex detailed in the previous section.

A.3 VOF: Distance and fraction computation

The objective here is to compute the fraction C inside a computational cell (2D or 2D axisymmetric) given the line equation

x • n + d = 0 (A.2)
where n is the normal and d the distance to the interface. We also consider the inverse problem in which the fraction C and the normal n to the interface are known and we want to deduce the distance d. These two procedures are needed to reconstruct the interface and to compute the mass uxes through the cell faces. This problem was already treated in [START_REF] Li | Calcul d'interface ane par morceaux[END_REF]; [START_REF] Gueyer | Volume-of-uid interface tracking with smoothed surface stress methods for three-dimensional ows[END_REF]; [START_REF] Scardovelli | Analytical relations connecting linear interfaces and volume fractions in rectangular grids[END_REF] for 2D and 3D congurations, while [START_REF] Son | A coupled level set and volume-of-uid method for the buoyancy-driven motion of uid particles[END_REF] have detailed the algorithm for 2D cartesian and axisymmetric coordinates. Other methods use an iterative procedures to nd the distance to the cell center [START_REF] Rider | Reconstructing volume tracking[END_REF]. The main challenge when designing robust algorithms is to take into consideration round-o errors in limiting cases, which tend to produce non-physical results. Examples are when the interface is nearly horizontal or vertical, and/or when the fraction is close to zero or one.

Distance between a line and a point

We consider the 2D cell plotted on Of course, this can also be obtained directly from Eq. (A.2) by setting x = ( -∆x 2 , -∆y 2 ).

A.3.1 Fraction computation

We want to compute the fraction of the reference uid inside a 2D computational cell given a linear interface with a unit normal vector n = (n x , n y ) and a distance d to the cell center O. The A.3. VOF: Distance and fraction computation line equation is then n x x + n y y + d = 0. Several cases need to be considered depending on the normal orientation and the distance to the cell center. In order to reduce the number of possible cases, and without loss of generality, we rotate the interface in a way that the normal always points to the positive xand y-directions.

The convention for the distance from the cell center to the interface is as follows:

• The distance is zero if the linear interface passes through the cell center, in which case the fraction is equal to 1 /2 in 2D/3D cartesian setting.

• If the cell center is inside the reference uid as in Fig. A.7, the distance is dened negative, in which case the fraction is greater than 1 /2. The cell center is then below the line.

• Finally, if the cell center is outside the reference phase then the distance is positive and therefore the fraction is smaller than 1 /2. The cell center is then above the line.

This way, the normal vector is pointing in the outward direction of the reference uid. The fraction C inside the cell is computed following Algorithm A.1. Round-o errors that may appear if X or Y are much greater than unity or close to zero are avoided in the current implementation. Note that in the case where ∆x = ∆y, the algorithm could be made more compact since the number of cases is reduced to 5.

O O O O O O O O x y X 1 1 X X 1 X 1 1 X 1 1 1 Y Y Y Y Y 1 X 2 Y 2 O 2
(1) (2) (3) (4)

(5) (6) (7) (8)

d U R d LR 0 d U L d LL Distance C U R C LR 1 2 C U L C LL Fraction d U R d U L 0 d LL d LR Distance C U R C U L 1 2 C LR C LL

Fraction

A.3.2 Distance computation

Now, we suppose that the volume fraction C inside the cell is known, the normalized components of the normal to the interface n = (n x , n y ) are also known. We want to nd the distance from this interface to the cell center that satises the fraction C inside this cell. This procedure is the inverse of what we have already seen on A.3.1. Again, in order to reduce the number of possible cases, we rotate the interface in a way that the normal points out in the positive xand y-directions. One of the major advantages of the collocated arrangement, where all variables including the velocity are dened at the cell center, is that it greatly facilitates the implementation of adaptive grid renement [START_REF] Almgren | A conservative adaptive projection method for the variable density incompressible navier-stokes equations[END_REF][START_REF] Popinet | Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries[END_REF]. In order to avoid the well-known checkerboard problem associated with the discretization of the pressure equation on collocated grids, [START_REF] Puckett | A high-order projection method for tracking uid interfaces in variable density incompressible ows[END_REF] developed an approximate projection method for two-phase ows (coupled to the VOF method), based on the initial version by [START_REF] Almgren | A numerical method for the incompressible navier-stokes equations based on an approximate projection[END_REF] for low mach number ows. Many variants of this approach have been used since, including non-incremental projection methods [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF], and incremental pressure correction [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF] in which the pressure term of the previous time step is included in the computation of the provisional velocity.

B.1.1 Time integration

For the decoupling of pressure and velocity, we choose the approximate incremental projection method which consists of calculating rst a provisional velocity using the previous pressure gradient, followed by a correction step in which the velocity is projected onto a divergence-free vector space. The overall time scheme is a corrector-predictor method.

In the method by [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF], the provisional velocity is rst computed at the cell center according to 1 u * ,0 c -

u n c ∆t + [ u • ∇ u] n c = g + f σ - 1 ρ f ∇p n-1 2 f →c + 1 ρ n+ 1 2 c ∇ • µ n+ 1 2 (∇ u * ,0 ) + (∇ u n ) T c
where the superscript 0 refers to the corrector step of the time scheme, and the subscripts c and f denote respectively cell-centered and face-centered quantities. The pressure gradient and the surface tension term are computed at the cell faces (as in the staggered grid arrangement). These terms are kept together and computed with the same gradient operator, such as described by the balanced-force approach [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF]. The transfer from face-centered to cell-centered vector quantities is noted by [.] f →c , which is done by simple averaging in the direction of the respective components 2 . Next, the pressure and surface tension terms are substracted from the provisional velocity, which is then interpolated at the cell faces:

u * * c = u * c -∆t g + f σ - 1 ρ f ∇p n-1 2 f →c u * * f = [ u * * c ] c→f
The projection step is then dened by:

∇ •   1 ρ n+ 1 2 f ∇p n+ 1 2 ,0   = ∇ • u * * ,0 f ∆t + g + f n+ 1 2 σ u n+ 1 2 ,0 c = u * * ,0 c + ∆t   g + f n+ 1 2 σ - 1 ρ n+ 1 2 f ∇p n+ 1 2 ,0   f →c u n+ 1 2 ,0 f = u * * ,0 f + ∆t   g + f n+ 1 2 σ - 1 ρ n+ 1 2 f ∇p n+ 1 2 ,0   f
This approach is called an approximate projection method, since the collocated velocity eld u n+1/2,0 c is not exactly divergence free. However, the divergence of the staggered velocity eld u n+1/2,0 f is forced to zero at each time step, and this keeps the continuity error to a minimum.

The corrector step of the overall time scheme consists of repeating the above steps, except that the updated pressure gradient is used and the explicit terms are replaced by a second-order approximation. The provisional velocity step is now: The interface is dened at intermediate time levels, t n+ 1 2 , such that uid properties and the interface curvature can be expressed directly at this instant. The volume fraction C is advanced rst from C n-1 2 to C n+ 1 2 using the velocity eld u n f . It is important at this point to use the divergence-free cell-face velocity, in order to guarantee conservation of the volume fractions. The interface advection step is done only once each time step. The density, viscosity and surface tension term at t n+ 1 2 is therefore the same for both the predictor and corrector step.

The pressure gradient and velocity divergence are computed on the staggered grid. Therefore, the same standard discrete gradient and divergence operators can be used as those given in 4.3.1.

Also, the surface tension term is needed at the cell face, as is the case for the staggered ow solver. For a description of this term, the reader is referred to 4.3.6.

B.1.2 Advection terms

The advection terms in the momentum equations are discretized using the second-order ENO scheme:

[ u • ∇u] u i,j = u i,j ∂u ∂x

ENO i,j + v i,j ∂u ∂y ENO i,j [ u • ∇v] v i,j = u i,j ∂v ∂x ENO i,j + v i,j ∂v ∂y ENO i,j
The advection terms are upwind approximations determined from :

u ∂φ ∂x ENO = uφ - x if u ≥ 0 uφ + x if u < 0
The spatial derivatives φ -

x and φ +

x are calculated using the second-order ENO scheme. Dening,

Dφ - i = φ i -φ i-1 Dφ + i = φ i+1 -φ i D 2 φ - i = φ i -2φ i-1 + φ i-2 D 2 φ 0 i = φ i+1 -2φ i + φ i-1 D 2 φ + i = φ i+2 -2φ i+1 + φ i B.

  Cette thèse se focalise sur l'étude et la comparaison des méthodes Eulériennes de suivi d'interface pour les écoulements diphasiques. Ce dernier est réalisé en deux étapes : la première vise à sélectionner des méthodes robustes et ables capables de prédire avec précision l'évolution de l'interface dans un champ de vitesse imposé ; et la seconde examine et traite le couplage entre les méthodes de suivi d'interface et le solveur de Navier-Stokes. Dans ce travail, nous nous intéressons aux méthodes Eulériennes de suivi d'interface, plus particulièrement level-set, volume-of-uid (VOF) et les couplages level-set/VOF. Chapitre 1 : La méthode level-set La méthode level-set est une méthode Eulérienne de suivi d'interface et son transport est régit par une équation d'advection d'un scalaire. L'interface est dénie par la courbe de niveau zéro ix chaque cellule, ainsi elle vaut 1 si une cellule est pleine et elle est nulle si la cellule est vide.

  de redistribution des fractions (algorithmes qui permettent de redistribuer les fractions an que avec reinit. standard et UPWIND d'ordre 1 pour l'adv.

Figure 1 :

 1 Figure 1: Interface (niveau zéro de level-set) du serpentin à t = T /2 = 3 (déformation maximale) sur un maillage de 100 × 100 avec CFL = 0.5. L'interface en vert représente la solution exacte.

Figure 2 :

 2 Figure 2: Onde capillaire stationnaire avec une densité et viscosité uniformes : évolution de la hauteur de l'interface à x = 0 avec CFL=CFL σ = 0.5.

Figure 3 :

 3 Figure 3: Assension d'une bulle (benchmark 2) : forme de la bulle à t = 3.0. Les couleurs font référence au maillage : Bleu (32 × 64), vert (64 × 128), rouge (128 × 256), noir (256 × 512) et cyan (solution de reference (Featow)).

Figure 1

 1 Figure 1.1: Staggered grid for the spatial discretization.

  tions. Note that in the examples shown in this work, the interface never reaches the boundary of the domain. Correct implementation of boundary points and interfaces connecting with the boundary are postponed to future work.

Figure 1

 1 Figure 1.2: 5-th order Hamilton-Jacobi WENO scheme stencils for the derivative approximation at the grid point i, (a) (φ - x ) i and (b) (φ + x ) i .

  Figure 1.4: Zalesak disk initial conguration.

Figure 1 . 5 :

 15 Figure 1.5: Single vortex initial conguration.

Figure 1

 1 Figure 1.6: Level-set with and without reinitialization: Zalesak disk after a full rotation for CFL = 0.5 and a 100 2 grid. Results obtained with dierent advection schemes: Green: Exact interface ; Red: without reinitialization ; Blue: HJ-WENO5 reinitialization with ∆τ = ∆x, τ max = 10∆x and Nreinit = 1.

Figure 1 .

 1 Figure 1.10: Level-set with and without reinitialization: time-reversed single vortex interface at t = T = 2 with CFL = 0.5, a 64 2 grid and dierent advection schemes. Green: Exact interface; Red: without reinitialization; Blue: HJ-WENO5 reinitialization with ∆τ = ∆x, τ max = 10∆x and Nreinit = 1.

Figure 1

 1 Figure1.11: Zalesak disk after a full rotation on a 100 2 grid with CFL = 0.5 and dierent advection schemes. Sub-cell x method with HJ-ENO2 reinitialization. ∆τ = ∆x/2, τ max = 10∆x and Nreinit = 10 Green: Exact interface ; Blue: Sub-cell x method.

Figure 1 .Figure 1 Figure 1

 111 Figure1.12: Shearing vortex at t = 3 on a 100 2 grid with CFL = 0.5. Sub-cell x method with the HJ-ENO2 reinitialization scheme for dierent advection schemes. ∆τ = ∆x/2, τ max = 10∆x and Nreinit = 10. Green: Exact interface ; Blue: Sub-cell x method.

Figure 1

 1 Figure1.15: Time-reversed single vortex at t = T = 2 on a 64 2 grid with CFL = 0.5. Sub-cell x method with the HJ-ENO2 reinitialization scheme for dierent advection schemes. ∆τ = ∆x/2, τ max = 10∆x and Nreinit = 10. Green: Exact interface ; Blue: Sub-cell x method.

Figure 1 Figure 1

 11 Figure 1.16: Sub-cell x: interface errors E M (Blue), E 1 (Red) and E |∇φ| (Black) as a function of the CFL number. Grid : 100 2 for the Zalesak disk and the shearing vortex and 64 2 for the time-reversed single vortex. HJ-ENO2 scheme is used for the reinitialization. Solid lines : Nreinit = 10; dashed-dotted lines with cross markers: constant CFLreinit (= 2 for HJ-WENO5; = 8 for WENO5 and = 9 for HOUC5).

Fig. 1 .Figure 1 Figure 1

 111 Fig. 1.20. The frequency of reinitialization has again an important inuence on the accuracy of the interface method. For ne grids, higher values of CFLreinit seem to give better results.

Figure 1

 1 Figure 1.21: Volume constraint: interface of the Zalesak disk after a complete rotation with the dierent advection schemes; HJ-ENO2 scheme for the reinitialization; CFL = 0.5; Grid 100 2 Green: Exact interface ; Blue: Volume constraint.

Fig. 1 .

 1 Fig. 1.6 and those obtained with the sub-cell x in Fig. 1.11. Only the results obtained without reinitialization in Fig. 1.6 provide better results, but this is mainly due to the solid rotation of the level-set eld which in principle does not require reinitialization.

Figure 1 Figure 1

 11 Figure1.22: Volume constraint: shearing vortex interface at time t = T /2 = 3 with the dierent advection schemes on a 100 2 grid with CFL = 0.5. the HJ-ENO2 scheme for the reinitialization. ∆τ = ∆x/2, τ max = 2∆x and Nreinit = 1. Green: Exact interface ; Blue: Volume constraint.

Figure 1

 1 Figure1.24: Volume constraint: time-reversed single vortex interface at time t = T /2 = 1 for the dierent advection schemes on a 64 2 grid with CFL = 0.5. the HJ-ENO2 scheme for the reinitialization . ∆τ = ∆x/2, τ max = 2∆x and Nreinit = 1. Green: Exact interface ; Blue: Volume constraint.

Figure 1

 1 Figure 1.25: Volume constraint: time-reversed single vortex interface at time t = T = 2 for the dierent advection schemes on a 64 2 grid with CFL = 0.5. the HJ-ENO2 scheme is used for the reinitialization. ∆τ = ∆x/2, τ max = 2∆x and Nreinit = 1. Green: Exact interface ; Blue: Volume constraint.

Fig. 1 .

 1 Fig.1.26 shows the evolution of the errors when varying the CFL number for the three test cases. As for the sub-cell x method, the best results are obtained when the CFLreinit is kept xed (thereby increasing Nreinit as CFL decreases). When Nreinit is kept constant, the interface error is larger when the time step is small, since the high number of reinitializations tends to deteriorate the interface position.

Figure 1

 1 Figure 1.26: Volume constraint: interface errors E M (Blue), E 1 (Red) and E |∇φ| (Black) for dierent CFL numbers. Grid : 100 2 for the Zalesak disk and the shearing vortex and 64 2 for the time-reversed single vortex. HJ-ENO2 scheme for the reinitialization. Solid lines : Nreinit

Figure 1

 1 Figure1.27: Volume constraint: errors E M (Blue), E 1 (Red) and E |∇φ| (Black) for dierent CFL numbers.Grid : 100 2 for the Zalesak disk and the shearing vortex and 64 2 for the time-reversed single vortex. HJ-ENO2 scheme for the reinitialization. Solid lines: CFL = 10 -2 ; symbols: CFL = 10 -3 .

Fig. 1 .Figure 1

 11 Fig.1.29 shows the positioning error of the shearing vortex interface for dierent CFLreinit values. Again, the volume constraint method is less sensitive to the CFLreinit variations. Nevertheless, the errors are of the same order as for the sub-cell x method. The HOUC schemes also show nearly second-order behavior, although the error levels for all grid sizes are higher than for ENO3, ENO4 and WENO5.

Figure 1

 1 Figure 1.29: Volume constraint: shearing vortex interface positioning error, E 1 , at t = T /2 = 3 for dierent CFL numbers as a function of the grid size. Nreinit = 1 and HJ-ENO2 scheme for the reinitialization.

Figure 2

 2 Figure 2.1: Illustration of the least-squares error (which is the sum of the squares of the coloured area in each cell) in a 3 × 3 block around a central cell in the LVIRA and the ELVIRA techniques. The red line represents the desired (or exact) interface and the blue solid line the interface inside the central cell. the blue dashed line represents the extension of the normal in the 3 × 3 block of cells.

Figure 2

 2 Figure 2.2: Illustration of the one-dimensional explicit Lagrangian time scheme.

Fig. 2 .

 2 Fig. 2.3 illustrates the advection of cell boundaries in the case both u i+1/2 and u i-1/2 are positive. The dashed zones indicate the volume that is uxed at the cell boundaries, F i-1/2 and F i+1/2 , still dened by

Figure 2

 2 Figure 2.3: Illustration of the one-dimensional implicit Lagrangian time scheme.

  . Compute the level-set on a 2000 × 2000 grid for the shearing vortex (or 1024 × 1024 grid for the time-reversed single vortex) (see A.2) 2. Determine the volume fraction using the Heaviside integral on each of the 2000 2 (1024 2 , respectively) cells. 3. Determine C R i,j on each cell of the actual coarse grid by summing up the fractions on (2000/N ) 2 corresponding sub-cells ((1024/N ) 2 sub-cells for the time-reversed single vortex).

Figure 2

 2 Figure 2.4: Zalesak disk interface after a full revolution for dierent CFL numbers using the VOF (left) and MOF (right) methods. The grid size is 100 2 .

Fig. 2 .

 2 Fig. 2.6 for three dierent CFL numbers. The gure shows that the CFL number has almost no inuence on the quality of the obtained interface.

Figure 2 . 7 :

 27 Figure 2.7: Positioning error of the time-reversed single vortex interface at t=T/2=1 and t=T=2 as a function of the CFL number. The grid size is 64 2 .

Figure 2 . 9 :Figure 2

 292 Figure 2.9: Positioning error of the shearing vortex interface at t = T = 6 as a function of the CFL number. The grid size is 100 2 . Green: exact solution, Black: CFL= 1.0 ; Blue: CFL= 0.5 and Red: CFL= 0.1

  Other coupled methods have been proposed in the literature, such as the Mass Conserving Level-Set (MCLS) method by[START_REF] Van Der Pijl | A mass-conserving level-set method for modelling of multi-phase ows[END_REF], in which a slightly dierent coupling algorithm was proposed for two-dimensional congurations, and three dimensions in van der[START_REF] Van Der Pijl | Computing three-dimensional two-phase ows with a mass-conserving level set method[END_REF]. The MCLS method is mainly based on the level-set method. It is not as widely used as the CLSVOF methods. A comparison between the volume-of-uid and the MCLS methods, coupled with the Navier-Stokes solver, can be found inDenner et al. (2014b). Their results shows that the MCLS method is slightly less accurate than the VOF methods, which is most probably due to the implementation of the surface tension term.[START_REF] Sun | A coupled volume-of-uid and level set (voset) method for computing incompressible two-phase ows[END_REF] have proposed another coupled method called the coupled Volume-Of-uid and level-SET (VOSET) method in which only the volume-of-uid is advanced in time. The level-set is determined as a signed-distance function through a geometric iterative method. Later,[START_REF] Albadawi | Inuence of surface tension implementation in volume of uid and coupled volume of uid with level set methods for bubble growth and detachment[END_REF] proposed a simplied coupling between the volume-of-uid and the level-set methods named S-CLSVOF that the authors implemented into the free CFD software OpenFOAM. In reality, 3. Coupled Volume-Of-Fluid and Level-Set methods this last one is similar to the VOSET technique since only the volume-of-uid is advected in time and the level-set is rst deduced from the volume fractions and then reinitialized using the standard reinitialization equation. The new volume-of-uid method byWang et al. (2012b) is somehow another VOSET alternative, but its authors have not categorized it among the coupled methods. Finally,[START_REF] Jemison | A coupled level set-moment of uid method for incompressible two-phase ows[END_REF] proposed to couple the Moment-Of-Fluid method, a few years after its apparition, with the level-set method. The method is called Coupled Level-Set and Moment-Of-Fluid (CLSMOF) method. A literature review on the interface capturing methods (level-set, VOF, CLSVOF, VOSET, phase eld methods, surface tension models) may be found in[START_REF] Wörner | Numerical modeling of multiphase ows in microuidics and micro process engineering: a review of methods and applications[END_REF].

  algorithm (VOSET method by[START_REF] Sun | A coupled volume-of-uid and level set (voset) method for computing incompressible two-phase ows[END_REF], reconstructed distance function by[START_REF] Cummins | Estimating curvature from volume fractions[END_REF]), or from the deduced level-set (new VOF method byWang et al. (2012b)).

  Algorithm 3.1 VOSET method Input: C n , u n+1/2 , ∆t n, d ← Reconstruct the interface using a PLIC Algorithm (C n ) C * ← Advect the volume-of-uid in the x-(or y-) direction (C n , u n+1/2 , n, d) n, d ← Reconstruct the interface using a PLIC Algorithm (C * ) C n+1 ← Advect the volume-of-uid in the y-(or x-) direction (C * , v n+1/2 , n, d) n, d ← Reconstruct the interface using a PLIC Algorithm (C n+1 ) φ n+1 , n, d ← Determine the signed-distance function iteratively (C n+1 , n, d) u n+3/2 ← Advance ow equations (φ n+1 , u n+1/2 )

n

  Figure 3.1: Characteristic distances in 2D cartesian coordinates.

  function computes the exact volume fraction, considering a piecewise-linear interface with normal computed from the gradient and distance from the level-set. It is a dierent manner of writing Algorithm A.1 on A.3.1. It is much faster to execute Eq. (3.4) than the Algorithm A.1.The total mass of the reference uid dened by the level-set eld can now be determined by rst using Eq. (3.4) to compute the volume fraction inside each computational cell, then sum over all cells. The mass error of the initial interface of the Zalesak disk and a circle are plotted on Fig.3.2. This method has a second-order accuracy behaviour in the determination of the total mass of the Zalesak disk and a circle and it is slightly more accurate than the bilinear interpolation (Appendix A).

Figure 3 . 2 :

 32 Figure 3.2: Mass error when the mass of a circle (left) or a Zalesak disk (right) is estimated using Eq. (3.4) vs bilinear interpolation with 1000 2 sub-cells.

  It is based only on the volume fraction eld and the linear interface reconstruction in each mixed cell. Examples of the computation of the minimal distance between the cell center and the interface is shown on Fig. 3.3, where the distance is represented by a line segment and the reference uid by the blue color.

Figure 3 . 3 :

 33 Figure 3.3: Minimal distance between the cell boundaries and a cell center.

Figure 3 . 4 :

 34 Figure 3.4: Interface reconstruction from the level-set contour for the Zalesak disk calculated on a 100 × 100 grid with CFL = 0.5. Dierent colors correspond to the number of complete revolutions: 1 (blue), 2 (cyan), 4 (black), 8 (red). The green line indicates the exact solution.

Figure 3

 3 Figure 3.7: Zalesak disk after one full revolution. Location error E 1 as a function of the grid size (in one direction). Dierent colors correspond to dierent values of the CFL number. The pink line indicates the slope of a second-order method.

Figure 3 . 8 :

 38 Figure 3.8: Single vortex at t = T /2 = 3. Location error E 1 as a function of the grid size (in one direction). Dierent colors correspond to dierent values of the CFL number. The pink line indicates the slope of a second-order method.

Figure 4

 4 Figure 4.1: Staggered (MAC) grid for the ow solver

  This method does not require the explicit computation of the curvature. When applied as a part of the VOF method, the delta function |∇H| = |∇ C| should be computed from the smoothed function C. Typical smoothing kernels of size 3 × 3 and 5 × 5 are

a

  if |a| < |b| and ab > 0 b if |b| < |a| and ab > 0 0 otherwise 4.3.5 Viscous terms In a staggered grid arrangement, discretization of the viscous terms is straight-forward. Despite the dierent locations of the velocity components, additional interpolations are not needed. The implicit viscous terms are discretized as:

  Figure 4.2: Spurious currents of a stationary circular bubble: evolution of the maximum velocity

  Fig. 4.4 shows the results for dierent density ratios and for three dierent Laplace numbers: 120 (left column), 1200 (central column) and 120000 (right column).The MOF-HF1 method gives excellent results at low values to the Laplace number, but clearly 4

Figure 4

 4 Figure 4.3: Spurious currents of a stationary circular bubble: inuence of the Laplace numbers on the evolution of the maximum velocity for a uniform density. The time is made non-dimensional by the viscous time scale t ν . Only a quarter of the bubble is simulated. The grid size is 16 × 16 (R/h = 6.4) and CFL σ = 1.0. No reinitialization of the level-set eld.

Figure 4 Figure 4

 44 Figure 4.4: Spurious currents of a stationary circular bubble: inuence of density ratio on the evolution of the maximum velocity for La = 120. The time is made non-dimensional by the viscous time scale t ν . Only a quarter of the bubble is simulated. The grid size is 16 × 16 (R/h = 6.4) and CFL σ = 1.0. No reinitialization of the level-set eld.

Figure 4

 4 Figure 4.7: Inviscid oscillating drop: kinetic energy versus time for dierent grid sizes using LS-CSF method with CFL σ = 1 and CFLreinit= 0.2.

  Figure4.9: Temporal evolution of the total kinetic energy (blue) for the viscous droplet oscillations test case for the dierent methods on a 64 2 grid. CFL σ = 1.0 and CFL= 0.5. The level-set is reinitialised after each advection step, while no kernel smoothing is used for the volume-of-uid based methods. In red, the exponential envelope of the kinetic energy versus time using a front tracking method according toTorres and Brackbill (2000, data with the unconnected points).

Figure 4 .

 4 Figure 4.10: Relative mass error of the viscous drop as a function of time for the dierent level-set methods on a 64 2 grid. CFL σ = 1.0. The level-set is reinitialised after each advection step

Fig. 4 .

 4 Fig. 4.11 shows the relative height, yy 0 , of the interface at x = 0 as a function of the nondimensional time, τ , for dierent grid sizes. The analytical solution given by Eq. (4.42) is represented by the blue symbols. It is found that the VOF-HF1 and VOF-HF2 methods give similar results, therefore only the result of the rst is shown in Fig. 4.11. Due to the small amplitude of the wave, the

Figure 4 .

 4 Figure 4.11: Capillary standing wave for a uniform density and viscosity: temporal evolution of the interface height at x = 0 for CFL=CFL σ = 0.5.

Figure 4 .

 4 Figure 4.12: Capillary standing wave for uids with uniform density and viscosity: RMS error of the temporal evolution of the interface height as a function of grid size. CFL σ = 0.5. The dashed lines denote 1st-and 2nd-order convergence. The level-set is reinitialized after each advection step.

Figure 4

 4 Figure 4.13: Capillary standing wave for a uniform density and viscosity: inuence of the time step on the RMS error E RM S . The level-set is reinitialized after each advection step.

Figure 4 .

 4 Figure 4.14: Capillary standing wave for water-air: RMS error on the interface height evolution as a function of grid size. CFL σ = 0.5. The red-dashed lines denote 1st-and 2nd-order convergence.

FigFigure 4

 4 Figure 4.19: Rayleigh-Taylor instability with inviscid uids: evolution of the penetration depth(solid lines). The analytical slopes of the growth are also plotted (dashed lines). Grid size 20×60.

Figure 4 Figure 4

 44 Figure 4.20: Rayleigh-Taylor instability with inviscid uids: evolution of the penetration depth (solid lines) as function of ξ. The analytical slopes of the growth are also plotted (dashed lines).No smoothing kernel is used with the VOF methods. The level-set eld is reinitialized each time step.

Figure 4 Figure 4

 44 Figure 4.22: Rayleigh-Taylor instability with inviscid uids: interface at time τ = 10 obtained on a 80 × 240 grid for dierent values of ξ. λ 0 = 0.001, CFL= 0.5 and CFL σ = 1.0. No smoothing kernel is used with the VOF methods. The level-set eld is reinitialized each time step.

Figure 4 .Figure 4

 44 Figure 4.25: Rising bubble benchmark 1: Bubble shape at the nal time t = 3.0 for dierent grid sizes: blue (32 × 64), green (64 × 128), red (128 × 256), black (256 × 512) and cyan (reference solution).

Fig

  the proles obtained are similar and are superposed on that of the reference solution.

Figure 4

 4 Figure 4.28: Rising bubble benchmark 1: L2-error of the temporal evolution of the dierent benchmark quantities as a function of the grid size in the x-direction.

Figure 4

 4 Figure 4.29: Rising bubble benchmark 2: Bubble shape at the nal time t = 3.0. Blue (32 × 64), green (64 × 128), red (128 × 256), black (256 × 512) and cyan (reference solution).

Figure 4

 4 Figure 4.33: Rising bubble benchmark 2: L2-norm of the error on the evolution of the circularity, the rising velocity and the centroid position as a function of the grid size in the x-direction.

Figure A. 1 :

 1 Figure A.1: Staggered grid for mass estimation. Interpolation is performed within the dashed cells through which interface passes. The exact and the computed level-set are dened at the vertices of the dashed cells represented by red circles.

  where x k = x i + k -1 2 m ∆x and k ∈ [1, m].

  φ E (i : i + 1, j : j + 1)] min E = min [φ E (i : i + 1, j : j + 1)] max C = max [φ C (i : i + 1, j : j + 1)] min C= min [φ C (i : i + 1, j : j + 1)] then the cell is lled or empty of reference uid in both solutions. Hence,Ω ij H(φ E (i, j)) -H(φ C (i, j)) dΩ = 0 then the cell is mixed in both cases. Hence, we subdivide it into m × m sub-cells, andΩ ij H(φ E (i, j)) -H(φ C (i, j)then the cell contains both uids in the exact solution and only one uid in the computed one. Hence, we interpolate the exact level-set at the centers of the m × m sub-cells, andΩ ij H(φ E (i, j)) -H(φ C (i, j)) dΩ = ∆x∆y m E (k, l)) -Ω ij H(φ C (i, j))dΩ • Else if min C × max C < 0,then the cell contains both uids in the computed solution and one uid in the exact one. Hence, we interpolate the computed level-set at the centers of the m × m sub-cells, and Ω ij H(φ E (i, j)) -H(φ C (i, j)) dΩ = Ω ij H(φ E (i, j))dΩ -

  Figure A.2: Mass error E M of the circle of radius 0.15 dened by the exact signed-distance function, as a function of grid and sub-grid sizes. left: Bilinear interpolation; right: Bicubic interpolation.

  Figure A.3: Mass error E M of the Zalesak disk dened by the exact signed-distance function on dierent grid and sub-grid sizes. left: Bilinear interpolation; right: Bicubic interpolation.

Fig

  Fig. A.5 shows E M of the reference solution for grid sizes in the range [1/512, 1/32] at time t = T /2 = 1. The positioning error of the interpolated signed-distance function is of the same range as the mass error, and this error is too small comparing to the mass error and the positioning error of the computed level-set, it represents approximately 1% of the level-set error.

Figure A. 5 :

 5 Figure A.5: Mass error E M due to the bicubic interpolation of the reference solution dening the time-reversed single vortex interface at time t = T /2 = 1.

  Figure A.6: Distance between a line and a point.

  Figure A.7: Linear interface segment and the characteristic distances.

Algorithm A. 1

 1 Fraction computation in 2D cartesian grid system (for the dierent cases, refer to Fig. A.8).Input: n = (n x , n y ), d, ∆x, ∆yd LL ← (|n x | ∆x + |n y | ∆y) /2 if (d ≥ d LL ) then the cell is empty (case 1) C ← 0 else if (d ≤ -d LL ) then the cell is full (case 2) C ← 1 else if (|n y | < 10 -8 ) then the line is horizontal (case 3) if (|n x | < 10 -8 ) thenthe line is vertical (case 4) d LL + d |n x |∆x Intersection point with the x-axis Y 2 ← d LL + d |n y |∆y Intersection point with the y-axis C ← 1 -X 2 Y 2 /2 else if (X ≤ 1) x |∆x -|n y |∆y)d |n x |∆x C ← (X + X 1 )|n x |∆x + |n y |∆y)d |n y |∆y C ← (Y + Y 1 )

Figure A. 8 :

 8 Figure A.8: Possible cases of the linear interface in 2D cartesian grid setting.

Figure A. 9 :

 9 Figure A.9: Fraction of the reference uid as a function of the distance for a given normal to the interface. The left gure corresponds to |n x | > |n y |, and the right one to |n x | < |n y |.

  Figure A.10: Possible cases of the linear interface and the reference distances in 2D cartesian grid system.

  n+ 1 2 (∇ u * ,1 ) + 1 2 (∇ u n+1,0 ) T + 1 2 (∇ u n )

  Figure B.1: Rising bubble benchmark 1: Position of the bubble centroid as a function of time on dierent grid sizes. Blue (32 × 64), green (64 × 128), red (128 × 256), black (256 × 512) and red stars (reference solution).

  der Pijl et al. (2005) ont proposé un couplage entre levelset et VOF qu'ils ont nommé MCLS (mass conserving level-set). Le champ de level-set est avancé et réinitialisé par les mêmes schémas numériques que level-set classique (φ

n → φ n+1 ) mais avec un ordre moins élevé an de garantir le caractère lisse de la solution. Le champ VOF (C n ) est déterminé à partir de level-set (φ n ) avant l'étape d'advection et avancé dans le temps (C n → C n+1 ). Il est ensuite utilisé pour forcer la conservation de masse du champ level-set à la n de l'itération (φ n+1 ). Pour cette méthode, on a comparé diérentes variantes avec des schémas de diérents ordres, avec une réinitialisation volume constraint ou une réinitialisation standard.

  interface. λ is a constant. The Heaviside function is a step function taking either zero or unity. In this method, we rather use the smeared-out Heaviside on a narrow band of thickness ∆x on each side of the interface, denoted H ∆x (φ) and dened as:

The derivative of the Heaviside function is used in the added term to correct the zero level-set without disturbing the distance function property away from the

Table 1 .

 1 2: Comparison of level-set errors with and without reinitialization for the Zalesak disk after a full rotation on a 100 2 grid. Advection: CFL = 0.5. Reinitialization: HJ-WENO5 with ∆τ = ∆x, τ max = 10∆x and Nreinit = 1.

	Advection scheme	No reinitialization		Reinitialization
		E M (%)	E 1 (%)	E |∇φ|	E M (%)	E 1 (%)	E |∇φ|
	UPWIND1 HJ-ENO2 HJ-WENO3 HJ-WENO5 ENO2 ENO3 ENO4 WENO5 HOUC5-Adv HOUC3 HOUC5 HOUC7 HOUC9 HOUC11 Semi-Lagrangian	-100.00 -0.28 6.66 -0.87 14.71 3.59 0.22 -0.01 -1.23 3.37 -1.24 -0.80 -0.31 -0.18 3.27	100.00 36.73 29.84 2.95 31.89 6.53 4.04 4.13 3.27 5.97 3.26 1.83 1.10 0.88 5.60	-0.404 0.338 0.133 0.385 0.172 0.106 0.105 0.078 0.160 0.078 0.066 0.054 0.047 0.165	-53.47 4.99 7.29 5.12 -2.04 -0.30 1.13 1.58 10.00 0.37 10.00 7.02 2.91 0.56 6.97	80.21 23.26 19.69 13.70 22.21 10.52 10.71 10.44 18.28 10.13 18.28 18.13 15.24 13.31 10.08	2.26 × 10 -3 3.20 × 10 -3 3.23 × 10 -3 3.34 × 10 -3 3.43 × 10 -3 4.86 × 10 -3 3.71 × 10 -3 3.83 × 10 -3 5.20 × 10 -3 4.09 × 10 -3 5.20 × 10 -3 4.99 × 10 -3 4.66 × 10 -3 4.09 × 10 -3 8.95 × 10 -3

Table 1 .

 1 3: Comparison of level-set errors with and without reinitialization for the shearing vortex test case on a 100 2 grid.

Table 1 .

 1 4 that the level-set without reinitialization is characterized by a low E |∇φ| at time t = T since the interface has not been much stretched at t = T /2.

		UPWIND1	HJ-ENO2	HJ-WENO3	HJ-WENO5	ENO2-Adv	ENO3-Adv	ENO4-Adv
	0.2
	0.4
	0.6
	0.8
		WENO5-Adv	HOUC5-Adv	ENO2-Est	ENO3-Est	ENO4-Est	WENO5-Est	HOUC5-Est
	0.2
	0.4
	0.6
	0.8
		ENO2-Jia	ENO3-Jia	ENO4-Jia	WENO5-Jia	HOUC5-Jia	HOUC3	HOUC5
	0.2
	0.4
	0.6
	0.8	HOUC7	HOUC9	HOUC11	Semi-Lag	0.2 0.4 0.6	0.2 0.4 0.6	0.2 0.4 0.6
	0.2
	0.4
	0.6
	0.8	0.2 0.4 0.6	0.2 0.4 0.6	0.2 0.4 0.6	0.2 0.4 0.6
	Figure 1.9: Level-set with and without reinitialization: time-reversed single vortex interface at
	t = T /2 = 1 with CFL = 0.5, a 64 2 grid and dierent advection schemes. Green: Exact interface;
	Red: without reinitialization; Blue: HJ-WENO5 reinitialization with ∆τ = ∆x, τ max = 10∆x
	and Nreinit = 1.

Table 1 .

 1 5: Sub-cell x: Zalesak disk, shearing vortex and time-reversed single vortex interface errors.

	Adv. scheme	Zalesak disk Grid 100 2	Shearing vortex Grid 100 2	Time-reversed single vortex Grid 64 2

   72.22 -1.06 1.23 0.27 2.10 HJ-WENO5 -0.61 5.30 0.29 15.15 -60.11 62.69 74.24 72.51 -0.96 1.16 0.31 2.15

	99 5.19 1.07 15.09 -59.54 62.36 73.79 72.34 -1.23 1.43 0.48 HJ-ENO2 -0.58 4.47 0.16 15.27 -59.43 62.05 73.81 72.23 -1.04 1.21 0.29 HJ-ENO3 -1.01 5.34 0.35 15.18 -60.30 62.91 74.05 72.36 -1.05 1.24 0.37 HJ-ENO4 -2.44 6.39 0.59 14.86 -61.38 63.95 73.97 72.28 -1.27 1.53 0.45 HJ-	2.24 2.10 2.05 2.01

Table 1 .

 1 7: Volume constraint: Zalesak disk, shearing vortex and time-reversed single vortex interface errors.

	Adv. scheme	Zalesak disk Grid 100 2	Shearing vortex Grid 100 2	Time-reversed single vortex Grid 64 2

  10.90 29.48 -6.50 13.32 -64.40 74.44 54.91 59.09 -6.55 8.43 -2.31 4.23

	HJ-ENO2 HJ-ENO3 HJ-ENO4 HJ-WENO3 HJ-WENO5	1.21 0.86 -0.52 5.14 -3.42 5.66 -61.02 64.46 63.42 61.60 -2.40 4.04 -0.03 2.17 5.70 -0.84 2.54 -60.00 63.92 62.49 61.34 -2.50 4.14 -0.13 2.21 0.03 2.15 5.18 0.08 3.53 -60.29 63.89 63.40 61.59 -2.26 3.96 1.48 5.48 -0.68 2.42 -59.75 63.70 62.80 61.52 -2.40 4.07 -0.04 2.16 1.20 4.51 0.19 4.08 -59.89 63.41 64.20 62.39 -2.09 3.78 0.16 2.14
	1.6.3.4 Spatial convergence rate

←

  It is provided by the algorithm below Algorithm 2.2 MOF method Input: C n , x n c , u n+1/2 , ∆t for cells with 0 < C n < 1 do n n , d n ← Compute interface normal by minimizing Eq. (2.37) (C n , x n Advance and remap in x-(or y-) direction using using Eqs. (2.25) and (2.43) (C n , n n , d n , u n+1/2 ) for cells with 0 < C * < 1 do n * , d * ← Compute interface normal by minimizing Eq. (2.37) (C * , x * Advance and remap in y-(or x-) direction using Eqs. (2.26) and (2.41) (C * ,

	c )
	end for
	C * , x * c ← c )
	end for
	C n+1 , x n+1 c

. We use Eqs. (2.31) and (2.32) instead of Eqs. (2.25) and (2.26). MOF

  3.4. Coupled Level-Set and Volume-of-Fluid method (CLSVOF) Algorithm 3.4 CLSVOF method Input: φ n , C n , u n+1/2 , ∆t n ← Compute the normal to the interface from the level-set Eq. (3.8), or (3.10) (φ n ) d ← Enforce the distance to satisfy the volume fraction inside the cell (C n , n) C * ← Advect the volume-of-uid in the x-(or y-) direction Eq. (2.27) or (2.28) on 2.3.4 ( n, d, C n , u n+1/2 ) φ * ← Advect the level-set in the x-(or y-) direction using Eq. (3.12) or (3.13) (φ n , u n+1/2 ) C * ← Truncate the volume fraction Eq.(3.11) (C * , φ * ) n ← Compute the normal to the interface from the level-set Eq. (3.8), or (3.10) (φ * ) d ← Enforce the distance to satisfy the volume fraction inside the cell (C * , n) C n+1 ← Advect the volume-of-uid in the y-(or x-) direction Eq. (2.28) or (2.27) ( n, d, C * , u n+1/2 ) φ n+1 ← Advect the level-set in the y-(or x-) direction using Eq. (3.13) or (3.12) (φ * , u n+1/2 ) C n+1 ← Truncate the volume fraction Eq.(3.11) (C n+1 , φ n+1 )

Table 3 .

 3 1: Mass and location errors for the Zalesak disk, single vortex and time-reversed single vortex for all the interface methods

	Time-reversed single vortex
	Shearing vortex	Grid 100 2
	Zalesak disk	Grid 100 2
		Method

Table 4 .

 4 1: Description of the dierent numerical methods for the Navier-Stokes equations for two-phase ows: expressions used to compute the curvature κ, the density at the cell-face and the viscosity.

	Method	κ	ρ	µ	Conv. term	Time adv.
	LS-LA1 LS-LA2 LS-CSF LS-GFM LS-GFMk LS-GFM1 VOF-HF1 MOF-HF1 VOF-WY	Eq. (4.33) Eq. (4.36) Eq. (4.35) Eq. (4.10) Eq. (4.34) HF1	Local Averaging Eq. (4.23) Eq. (4.26) H Eq.(4.22) H Eq. (4.27) H Eq. (4.27) Eqs. (4.17) and (4.18) Eq. (4.29) Weighted-Average Eq. (4.21) Eq. (4.28)	2nd-order ENO 4.3.4	predictor-corrector Crank-Nicolson	4.3.1

Table 4 .

 4 2: Spurious currents of a stationary circular bubble at zero viscosity and curvature replaced by its exact value. Maximum absolute value of the velocity and maximum error on the pressure eld after one time step of ∆t = 10 -6 . Grid size 40 × 40 (R/h = 10).

	Method ρ 2 /ρ 1	|u| max	|p -p exact | max
	VOF-HF1 VOF-WYC LS-CSF LS-LA LS-GFM	1 10 -3 10 -5 1 10 -3 10 -5 1 10 -3 10 -5 1 10 -3 10 -5 1 10 -3 10 -5	1.71 × 10 -17 3.27 × 10 -16 9.88 × 10 -16 1.60 × 10 -17 3.16 × 10 -16 9.02 × 10 -16 1.62 × 10 -17 1.16 × 10 -15 2.26 × 10 -14 2.01 × 10 -17 6.28 × 10 -16 1.42 × 10 -14 2.01 × 10 -17 6.75 × 10 -16 5.53 × 10 -15	14.97 14.97 14.97 14.97 14.97 14.97 16.42 16.42 16.42 2.22 × 10 -11 3.02 × 10 -11 5.78 × 10 -11 2.22 × 10 -11 2.93 × 10 -11 5.88 × 10 -11

Table 4 .

 4 3: Spurious currents of a stationary circular bubble at zero viscosity. Maximum absolute value of the velocity and maximum error on the pressure eld after one time step of ∆t = 10 -6 . Grid size 40 × 40 (R/h = 10).

	Method ρ 2 /ρ 1	|u| max	|p -p exact | max
	VOF-HF1 and VOF-WY VOF-HF2 VOF-WYC LS-CSF LS-LA1 LS-LA2 LS-GFM1 LS-GFM LS-GFMk	1 10 -3 10 -5 1 10 -3 10 -5 1 10 -3 10 -5 1 10 -3 10 -5 1 10 -3 10 -5 1 10 -3 10 -5 1 10 -3 10 -5 1 10 -3 10 -5 1 10 -3 10 -5	1.02 × 10 -7 1.73 × 10 -7 1.73 × 10 -7 8.86 × 10 -8 1.87 × 10 -7 1.87 × 10 -7 9.97 × 10 -8 1.68 × 10 -7 1.68 × 10 -7 2.06 × 10 -7 1.64 × 10 -5 5.88 × 10 -4 3.33 × 10 -6 1.09 × 10 -5 1.10 × 10 -5 2.52 × 10 -7 5.45 × 10 -7 5.46 × 10 -7 3.48 × 10 -7 1.21 × 10 -6 1.22 × 10 -6 1.27 × 10 -7 4.79 × 10 -7 4.81 × 10 -7 4.50 × 10 -7 9.19 × 10 -7 9.23 × 10 -7	14.85 14.88 14.88 14.86 21.76 21.78 14.85 14.88 14.88 17.24 17.26 17.26 0.47 0.78 0.80 0.15 0.22 0.22 0.29 0.32 0.32 6.60 × 10 -2 4.07 × 10 -2 4.06 × 10 -2 0.28 0.39 0.39

Table 4 .

 4 4: Spurious currents of a stationary circular bubble at zero viscosity. Maximum absolute value of the velocity and maximum error on the pressure eld after one time step of ∆t = 10 -6 using dierent smoothing kernels with the VOF-HF2 method. Grid size 40 × 40 (R/h = 10). are related by t ν /t σ = √ La and t ν /t ref = La. Popinet (2009) presented results for a time period of t = t ν and we will use the same value for our simulations. Desjardins et al. (2008) used a time period of 250t ref , which is a shorter period for high values of the Laplace number (La > 250). The velocity is usually presented in terms of the capillary number,

	Kernel ρ 2 /ρ 1	|u| max	|p -p exact | max
	none K 8 δ Vertex-Avg	10 -3 10 -5 10 -3 10 -5 10 -3 10 -5 10 -3 10 -5	1.87 × 10 -7 1.87 × 10 -7 5.94 × 10 -6 6.02 × 10 -6 8.04 × 10 -6 8.10 × 10 -6 9.50 × 10 -6 9.56 × 10 -6	21.76 21.78 21.06 21.07 20.42 20.44 19.44 19.45

which

  The level-set eld is reinitialized at each time step except for the last row. Table4.6: Inviscid oscillating drop: Mass error E M (%) of the level-set methods at the nal time on dierent grid sizes with CFL σ = 1. The level-set eld is reinitialized at each time step except for the last row.

				Method/Grid	32 2	64 2	128 2	256 2
				VOF-HF1 VOF-WY VOF-HF2 VOF-WYC LS-CSF LS-LA1 LS-LA2	1.08 0.01 1.07 0.003 0.009 0.09 1.40 0.24 0.09 2.59 0.94 0.41 -1.65 -1.60 1.25 -5.95 1.95 0.66 ------0.93
				LS-GFMk LS-GFM LS-GFM1 Gerris	---4.71 3.43 1.84 2.28 -5.49 2.01 2.21 -	1.41 1.37 1.34 1.17
				Gerris-Vertex-Avg	2.87 1.79 0.70	0.59
				LS-CSF, CFLreinit=0.2	-	3.89 2.14	0.35
				Method/Grid	32 2	64 2	128 2	256 2
				LS-GFMk LS-GFM LS-GFM1 LS-LA1	9.64 -0.70 -1.77 -0.83 0.63 10.34 -1.50 -1.09 9.03 -1.13 -1.10 -0.56 5.91 6.85 9.58 1.83
				LS-LA2	10.71	2.35	2.17	0.12
				LS-CSF	42.31 20.85	8.33	4.89
		LS-CSFwith CFLreinit=0.2	9.01	7.37	4.88 -0.58
		10	-3
	Kinetic energy	10 10	0 -5 -4	2 2π/ω Grid : 256 2 Grid : 64 2	Time		1

Table 4 .

 4 8: Capillary standing wave for water-air: inuence of the smoothing kernel on the RMS error with the VOF-HF1 method on dierent grid sizes. CFL=CFL σ =0.5.The test case of the Rayleigh-Taylor instability is similar to the previous one (capillary wave) but with gravity forces included. It consists of considering a sinusoidal perturbation on a horizontal interface separating two immiscible uids, the lighter uid is placed beneath the heavier one.

	method	kernel /Grid	8	16	32	64	128
		none	0.1798 0.0531 0.0160 0.0060 0.0033
	VOF-HF1	K 8 δ	0.1670 0.0414 0.0153 0.0062 0.0034 0.1746 0.0501 0.0161 0.0062 0.0034
		Vertex-Avg	0.1496 0.0198 0.0134 0.0062 0.0034
	4.4.4 Rayleigh-Taylor instability

  Taylor instability with viscous uids without surface tension: interface at t = 1 on a 128 × 512 grid with CFL=0.2. No smoothing kernel is used with the VOF methods.

	4. Incompressible two-phase ow solver
	VOF-HF1 MOF-HF1 VOF-WYC	LS-CSF	LS-LA	LS-GFM2 LS-GFM1	Gerris
	Figure 4.16: Rayleigh-
				.16. The results obtained with

  Taylor instability with surface tension: interface at t = 1.3 on a 128 × 512 grid with CFL=0.2 and CFL σ = 1.0. No smoothing kernel is used with the VOF methods. The level-set eld is reinitialized each time step.

												4.4. Results and discussions
	VOF-HF1	VOF-WY	VOF-WYC	MOF-HF1	LS-CSF	LS-LA1	LS-LA2	LS-GFMk	LS-GFM	LS-GFM1	Gerris
	Figure 4.18: Rayleigh-
	VOF-HF1	VOF-WY	VOF-WYC	MOF-HF1	LS-CSF	LS-LA1	LS-LA2	LS-GFMk	LS-GFM	LS-GFM1	Gerris

Figure

4

.17: Rayleigh-Taylor instability with surface tension: interface at t = 1.1 (upper row) and t = 1.3 (lower row) on a 64 × 256 grid with CFL=0.2 and CFL σ = 1.0. No smoothing kernel is used with the VOF methods. The level-set eld is reinitialized each time step.

Table B

 B Circularity of the bubble as a function of time on dierent grid sizes. Colors correspond to the grid size: blue (32 × 64), green (64 × 128), red (128 × 256), black (256 × 512) and red stars (reference solution).

									4.4. Results and discussions
			VOF-HF1				MOF-HF1
		1.02					1.02
		1.00					1.00
	Circularity	0.96				Circularity	0.96
		0.92					0.92
		0 0.88	1	Time	2	3	0 0.88	1	Time	2	3
			VOF-WYC				LS-GFM
		1.02					1.02
		1.00					1.00
	Circularity	0.96				Circularity	0.96
		0.92					0.92
		0 0.88	1	Time	2	3	0 0.88	1	Time	2	3
	Figure 4.27: Rising bubble benchmark 1:
							.4.

Table B .

 B 2 Capillary wave B.2.1 Uniform viscosity and density Table B.1: Capillary standing wave for uids with uniform density and viscosity: RMS error of the temporal evolution of the interface height as a function of grid size. CFL σ = CFL = 0.5. 2: Capillary standing wave for water-air: RMS error on the interface height evolution as a function of grid size. CFL σ = CFL = 0.5. The level-set is reinitialized after each advection step. No smoothing kernel is used with the VOF methods. Table B.4: Rising bubble benchmark 1: minimum circularity c min , maximum velocity v max and their time of occurrence. Final centroid position Y c and relative mass error E m .

											B.3. Rising bubble benchmark
	B.3.1 Benchmark case 1
	The level-set is reinitialized after each advection step. Method Grid c min t| c=cmin v max	t| v=vmax	Yc	E m (%)
		32	0.8906		1.8430		0.2412	0.9659	1.0836	0
	VOF-HF1	64 128 256	Grid LS-LA1 0.8980 0.9002 0.9006	8 0.2251 1.8588 0.0966 16 0.2416 32 0.0242 1.8685 0.2417 1.8840 0.2417	64 0.0058 0.9266 0.9248 0.9230	128 1.0816 1.0815 0.0022 1.0816	0 0 0
		32	LS-LA2 0.8775	0.2179 2.1041 0.0838 0.2413 0.0207	0.0049 0.9607	0.0018 1.0835	0
	MOF-HF1	64 128 256	LS-GFM1 LS-CSF 0.8909 0.8932 0.9000	0.2307 0.2447 2.2686 0.1095 0.1295 0.2417 0.0287 0.0391 2.1387 0.2417 1.9082 0.2417	0.0070 0.0101 0.9266 0.9248 0.9225	0.0023 1.0816 1.0815 0.0030 1.0816	0 0 0
		32	LS-GFMk 0.8910	0.2326 1.8430 0.1098 0.2412 0.0288	0.0070 0.9659	0.0023 1.0836	0
	VOF-WY	64 128	LS-GFM 0.8981 0.9002	0.2171 1.8588 0.0835 0.2416 0.0206 1.8685 0.2417	0.0049 0.9266 0.9248	0.0018 1.0816 1.0815	0 0
		VOF-WYC 256 0.9006	0.1483 1.8840 0.0295 0.2417 0.0090	0.0040 0.9232	0.0019 1.0816	0
	VOF-WYC	VOF-HF1 0.8711 VOF-WY 0.8975 0.8999 VOF-HF1 32 64 128	0.0549 2.0988 0.0246 0.2407 0.0068 0.0549 0.0246 0.0068 1.9585 0.2415 1.8724 0.2417 MOF-HF1	0.0016 0.9659 0.0016 0.9340 0.9261 VOF-WYC 0.0011 1.0781 0.0011 1.0802 1.0813	0 0 0 LS-GFM
	VOF-HF2	256 32 64 128	MOF-HF1 0.9006 VOF-HF2 Gerris 0.8780 0.8977 0.9003	0.0509 1.8849 0.0282 0.2417 0.0090 0.0541 0.0253 0.0069 0.1591 0.0316 0.0077 1.9318 0.2411 1.9013 0.2414 1.8698 0.2416	0.0025 0.9230 0.0016 0.0022 0.9659 0.9285 0.9248	1.0817 0.0013 0.0011 0.0005 -1.0804 1.0806	0 0 0 0
		256	0.9006		1.8847		0.2416	0.9223	1.0809	0
		32	0.9214		1.8900		0.2391	0.9450	1.0901	4.34
	64 128 B.2.2 Air-water capillary wave 0.9075 1.8404 LS-CSF 0.9030 1.8828 256 0.9016 1.9100		0.2411 0.2416 0.2417	0.9285 0.9235 0.9220	1.0834 1.0818 1.0817	1.80 0.62 0.25
		32	0.9256		1.7386		0.2382	0.9241	1.0953	4.49
	LS-LA1	64 128	0.9115 0.9056		1.8367 1.8730		0.2410 0.2416	0.9230 0.9215	1.0860 1.0830	1.95 0.77
		256	0.9031		1.8881		0.2417	0.9202	1.0823	0.31
		32	0.9257		1.7386		0.2385	1.0129	1.0963	4.90
	LS-LA2	64 128	Grid 0.9113 0.9053	8	1.8385 1.8743	16	32 0.2409 0.2415	64 0.9119 0.9235	128 1.0863 1.0827	1.99 0.75
		VOF-HF1 256 0.9030	0.1798 1.8886 0.0531 0.2416 0.0160	0.0060 0.9181	0.0033 1.0820	0.30
	LS-GFM	VOF-WY 0.9194 VOF-HF2 32 64 0.9067 128 0.9026	0.1798 1.7647 0.0531 0.2387 0.0160 0.1827 0.0632 0.0285 1.8385 0.2415 1.8841 0.2422	0.0060 0.9346 0.0172 0.9211 0.9176	1.0879 0.0033 0.0105 1.0822 1.0813	4.51 1.83 0.61
		VOF-WYC 256 0.9011	0.2062 1.8692 0.1101 0.2424 0.0375	0.0132 0.9176	0.0058 1.0817	0.27
	LS-GFMk	MOF-HF1 32 0.9187 LS-CSF 64 0.9069 128 0.9026	21.8143 1.7751 0.0526 0.2390 1.4594 0.1570 0.0800 0.0451 1.8330 0.2417 1.8835 0.2423	0.0957 0.9346 0.0174 0.9211 0.9176	1.0931 0.0932 0.0039 1.0832 1.0812	4.86 2.01 0.71
		256 32	LS-LA1 0.9011 0.9250	0.2521 1.8514 0.0865 0.2424 0.0232 1.7386 0.2387	0.0104 0.9172 0.9398	1.0814 0.0056 1.0978	0.31 4.94
	LS-GFM1	LS-LA2 LS-GFMk 64 0.9098 128 0.9044 256 0.9023	0.2472 0.3090 1.8404 0.0655 0.1978 0.2410 0.0219 0.1125 1.8704 0.2416 1.8690 0.2417	0.0112 0.0627 0.9285 0.9202 0.9218	0.0061 1.0866 1.0835 0.0315 1.0827	1.87 0.66 0.25
		LS-GFM ref 0.9013	0.3037 1.8999 0.1811 0.2417 0.1035	0.0598 0.9239	0.0308 1.0819	0
		LS-GFM1	0.3002	0.1768	0.0600	0.0194	0.0049
			Gerris	0.2041	0.0795	0.0191	0.0069	0.00313

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2017LYSEI052/these.pdf © [N. Djati],[2017], INSA Lyon, tous droits réservés

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2017LYSEI052/these.pdf © [N. Djati], [2017], INSA Lyon, tous droits réservés Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2017LYSEI052/these.pdf © [N. Djati], [2017], INSA Lyon, tous droits réservés

Note that the frequency errors in Table4.6 are dierent from those presented on the Gerris website, since we have used a dierent method to compute the error.

The numerical values of the errors are presented in Appendix B on TableB.1.

The numerical values of the errors are presented in Appendix B on Table B.2.

The data for the other VOF methods are reported in Appendix B onTable B.3

Note that a larger time step of ∆tmax = 5 × 10 -3 is used in Popinet, which resulted however in a too large penetration depth.

Numerical values are available on Table B.4

Values of the minimum circularity and the time of occurrence are reported on TableB.4.

The mass errors at the end of the simulation are reported on TableB.5.

The maximum velocity and its time of occurrence are reported on TableB.5.

Results are shown on Fig. B.2 and Table B.5.

Note that in the method by[START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF], the conservative formulation of the momentum equation is used. As in[START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF], and in agreement with many other studies on two-phase interface ows, we have chosen to use the non-conservative formulation.

For example, for a variable q dened at the cell faces, [q] f →c is dened by qi,j = (q i+1/2,j + q i-1/2,j )/2 or qi,j = (q i,j+1/2 + q i,j-1/2 )/2. Likewise, the transfer from cell-centered to face-centered quantities is noted by[q] c→f , we use q i+1/2,j = (qi+1,j + qi,j)/2 or q i,j+1/2 = (qi,j+1 + qi,j)/2.

Remerciements

conseiller, m'orienter, m'aider, me soutenir et te rendre disponible! Je tiens également à remercier Eric ALBIN pour les corrections, les conseils, Je remercie aussi les doctorants/Postdoc/ingénieurs/stagiaires

The explicit viscous terms in the equations for u and v are calculated from respectively:

Dierent methods are used in this work to dene the viscosity at the cell center and vertices:

• In the work of [START_REF] Vigneaux | Méthode level set pour des problèmes d'interfaces en microuidique[END_REF], the viscosity is dened as:

• In a level-set based method, the viscosity can also be evaluated using the smoothed Heaviside function (4.5):

µ i,j = µ g + (µ lµ g )H (φ i,j ) µ i+ 1 2 ,j+ 1 2 = µ g + (µ lµ g )H (φ i+ 1 2 ,j+ 1 2 ) (4.27) where φ i+ 1 2 ,j+ 1 2 = (φ i,j + φ i+1,j + φ i,j+1 + φ i+1,j+1 )/4

Here, the superscripts () u and () v are dropped, since both terms are identical.

• In a volume-of-uid based method, the viscosity is evaluated using:

where

where C is replaced by C when a smoothing kernel is applied (see 4.2.1.2).

Note that when the viscosity is smeared out using one of the methods above, the rst derivative is continuous at the interface, and hence the second-derivatives can be determined using standard central dierencing to compute the viscous terms.

• When the viscosity is represented in a sharp fashion, jump conditions should be taken into account when evaluating the viscous terms [START_REF] Kang | A boundary condition capturing method for multiphase incompressible ow[END_REF]. This approach is rather VOF-HF1: The volume-of-uid is advected using the consistent and conservative implicit/explicit geometrical mapping given by Algorithm 2.1 in 2.5. The ELVIRA reconstruction by Pilliod and [START_REF] Pilliod | Second-order accurate volume-of-uid algorithms for tracking material interfaces[END_REF] is used. This interface method (VOF) has been already dened in 3.6.1. The curvature is computed using the Height Function method by [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF], named HF1, and we use the implementation from the Basilisk code (Popinet et al.).

MOF-HF1:

We use the moment-of-uid advection and reconstruction proposed by [START_REF] Dyadechko | Moment-of-uid interface reconstruction[END_REF], already detailed in 3.6.1. The curvature is determined using the Height Function method HF1 by [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF].

VOF-HF2: The curvature is computed using the slightly dierent Height Function method proposed by [START_REF] Hernández | A new volume of uid method in three dimensions part i: Multidimensional advection method with face-matched ux polyhedra[END_REF], named HF2. Furthermore, the density at cell-faces used in the surface tension term (Eq. 4.31) is computed as the mean density. For example, the density at the (i + 1/2, j)-th cell face is dened as:

The interface method (VOF) is identical to that of VOF-HF1.

VOF-WY:

The volume-of-uid is advected following the alternative conservative and consistent scheme by [START_REF] Weymouth | Conservative volume-of-uid method for free-surface simulations on cartesian-grids[END_REF] described in 3.6.1. As in VOF-HF1, the curvature is determined using the Height Function method HF1 by [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial ows[END_REF].

VOF-WYC: This method has been largely inspired by (and is as close as possible to) the Basilisk code (Popinet et al.). A collocated grid setting is used, in which all variables including the velocity are located at the cell center. This solver, based also on a predictorcorrector time scheme, is described in detail in B.1. The ow solver is coupled to the VOF method, which is identical to that used for the VOF-WY method.

The dierent methods are summarized on Table 4.1.

VOF-HF1

VOF-WY

Gerris

Gerris-Vertex-Avg vertex-averaged volume fraction. For the VOF-HF1 and VOF-WY methods, the δ -kernel gives the best result in terms of frequency error. The VOF-HF2 method results, on the other hand, are deteriorated with the use of the smoothing kernel, probably due to the usage of the mean density in the surface tension term.

Viscous drop oscillations

Now, we consider the same conguration as in [START_REF] Torres | The point-set method: front-tracking without connectivity[END_REF], where the initial interface is an ellipse given by x 2 /9 + y 2 /4 = 1. The drop of density ρ d = 1.0 and viscosity µ d = 0.01 is surrounded by a uid with density ρ e = 0.01 and viscosity µ e = 5 • 10 -5 . The surface tension coecient is σ = 1.0. The drop is centered in a [0, 20] 2 domain with symmetric boundary conditions. Under the inuence of the surface tension forces, the drop will oscillate.

Due to the viscous nature of both uids, the total kinetic energy will decrease.

The temporal evolution of the total kinetic energy is plotted on Fig. 4.9. The exponential envelope of the kinetic energy provided by [START_REF] Torres | The point-set method: front-tracking without connectivity[END_REF] using a Lagrangian marker method is also presented (solid red lines). The VOF based methods predict correctly the kinetic energy evolution at the onset of the simulation, except for the VOF-WYC method. For this method, viscous eects are somehow underestimated resulting in slowly vanishing oscillations.

The MOF-HF1 method is working better with this test case, except at the end of the simulation where some increase in the kinetic energy and an apparent variation in the frequency of the simulation could be remarked. The LS-CSF method underestimates the damping of the kinetic energy, which could be related to the mass gain of the drop. The LS-GFM1, LS-GFMk and LS-GFM methods overestimates the kinetic energy towards the end of the simulations. The LS-LA1 method is very diusive and the LS-LA2 method seems to work well for the inviscid case as well as for the viscous one.

The relative mass error of the level-set methods is plotted on Fig. 4.10. The dierence with the inviscid case is that the mass error is decreasing for most methods then increases, while it is monotonically increasing or decreasing in the inviscid case (not shown here).

the software Gerris (Popinet) with a maximum time step 6 of ∆t max = 5 × 10 -4 are also added to this comparison. At times t = 0.6, all methods give similar results except the LS-LA method which has a deformed interface. At the nal time, the VOF-WYC method and Gerris, both with a cell-centered grid setting, show very similar results. The interface obtained with the other methods are almost similar, except for the LS-LA method for which there is no break-o and the neck is thinner. If we compare the results on the ne grid (128 ×512) in Fig. 4.16, we remark that all methods give a similar interface shape, except the LS-LA method which seems less accurate for this case. On the other hand, Gerris and the VOF-WYC method already presented a rather accurate prediction on the coarse grid.

VOF-HF1 MOF-HF1 VOF-WYC LS-CSF LS-LA LS-GFM2 LS-GFM1 Gerris

Viscous uids with surface tension

When the surface tension is added to the previous case, the penetration depth will be smaller since the surface tension tends to retain the interface. We set σ = 0.1337 as in [START_REF] Gómez | On the reinitialization procedure in a narrow-band locally rened level set method for interfacial ows[END_REF] while CFL σ = 0.5 and CFL= 0.2. For Gerris, we use ∆t max = 5 × 10 -4 . The interface is plotted on the 64 × 256 grid at times t = 1.1 and 1.3 on Fig. 4.17 for all methods except the VOF-HF2 method which fails to calculate up to the nal time. The interface at the nal time t = 1.3 on the 128 × 512 grid is plotted on Fig. 4.18.

Rising bubble

The rising bubble is a classical test case to validate two-phase ow solvers. It tests the overall behavior of the solver, i.e. the accuracy of the viscous terms, the surface tension, for relatively high density and viscosity ratios. For axisymmetric and fully three dimensional ows, the nu- The initially spherical bubble of diameter D = 0.5 is centered at (0, 0.5) in a [-0.5, 0.5]×[0, 2]

domain. Symmetry conditions are applied to the lateral boundaries and no-slip conditions are used for the upper and lower walls. The uid properties for the two cases are summarized on Table 4.9 [START_REF] Hysing | Quantitative benchmark computations of two-dimensional bubble dynamics[END_REF][START_REF] Aland | Benchmark computations of diuse interface models for two-dimensional bubble dynamics[END_REF]. The bubble is simulated with CFL σ = CFL = 0.5. Our results are compared to those provided by the MooNMD group [START_REF] Hysing | Quantitative benchmark computations of two-dimensional bubble dynamics[END_REF][START_REF] Featow | [END_REF], we consider it as a reference solution since it is based on a Lagrangian interface method. The dierent benchmark quantities are the centroid, circularity and mean rising velocity of the bubble. The centroid of the bubble is given by:

Interface Methods

A.1 Mass and positioning error computation

The mass loss/gain error is given by:

Exact mass

(1.59)

The positioning error E 1 is expressed as:

The Heaviside integral in equations (1.59) and (1.60) can be computed by dierent methods.

However, only few authors tell how they evaluate it. [START_REF] Sussman | A coupled level set and volume-of-uid method for computing 3D and axisymmetric incompressible two-phase ows[END_REF] proposed the following procedure:

For each staggered cell [x i , x i+1 ] × [y j , y j+1 ], check:

1. if max[φ(i : i + 1, j : j + 1)] > 0, then the cell is entirely inside the reference uid. and so

H(φ i,j )dΩ = ∆x∆y.

2. if min[φ(i : i + 1, j : j + 1)] < 0, then the cell is entirely outside the reference. Hence,

where

The time-step used for the advection of markers is set to ∆t = min(h, 10 -2 ) where h is the time-step of the interface capturing method. Such value of ∆t ensures that between the two times t = 0 and t = T = 6, the maximum displacement of the markers is in the range of 10 -8 . By linking the set of markers at a given time, we obtain the exact interface location.

3. Reconstruct the signed-distance function, which is the distance from the center of each computational cell to the closest marker. We consider it as the reference solution. Both, the reference solution and the level-set are dened at the same grid points; the positioning error E 1 is therefore computed using the bicubic interpolation with 1000 2 sub-cells (see 1.5.1).

The mass error E M of the reference solution is also computed using the bicubic interpolation with 1000 2 sub-cells, and is plotted on Fig.

A.4 for grid sizes going from 1/50 2 downto 1/800 2 at time t = T /2 = 3. The positioning error of the reference solution is of the same range as the mass error E M . These errors are too small comparing to that of the transported level-set, it represents approximately 1%.

Time-reversed single vortex

The time-reversed single vortex problem is dened on 1.5.4. The reference solution at any time, except at the start and the end of the simulation, is determined using the Lagrangian method depending on the value of the target fraction, we determine the distance d satisfying this target fraction. The procedure of determining the distance in 2D cartesian grid systems is detailed on Algorithm A.2. Input: n = (n x , n y ), C, ∆x, ∆y if (|n y | < 10 -8 ) then

The line is horizontal

The line is vertical

Compute the reference distances Eq. A.3 and the corresponding fractions Eq. A.4

if

The line passes through the cell center

else Linearly interpolate the distance, case (2) or ( 5) The implicit viscous terms are discretized as:

The explicit terms in the equations for u and v are calculated from respectively:

The viscosity at the cell faces is dened as:

The density at the cell faces is dened in a similar manner:

Inuence of the smoothing kernel

The MOF-HF1 method does not work very well for this test case since the density ratio is large, and the smoothing kernel does not enhance the results signicantly. For the VOF-HF1 and VOF-WY methods, it enhances the results for coarse grids only there is a net enhancement with the VOF-HF2 and VOF-WYC methods. The Vertex-Avg smoothing kernel seems to be the one yielding the best results. In the Gerris software (Popinet), the smoothing of the density and viscosity is slightly more accurate on ne grids. This thesis is devoted to the development and comparison of interface methods for incompressible two-phase flows. It focuses on the selection of robust interface capturing methods, then on the manner of their coupling with the Navier-stokes solver. The level-set method is first investigated, in particular the influence of the advection scheme and the reinitialization step on the accuracy of the interface capturing. It is shown that the volume constraint method for reinitialization is robust and accurate in combination with the conservative fifth-order WENO schemes for the advection. It is found that interface errors increase drastically when the CFL number is very small. As a remedy, reinitializing the level-set field less often reduces the amount of numerical diffusion and non-physical interface displacement. Mass conservation is, however, not guaranteed with the level-set methods. The volume-of-fluid (VOF) method is then investigated, which naturally conserves the mass of the reference fluid. A geometrical consistent and conservative scheme is adopted, then an alternative technique more easily extended to 3D. It is found that both methods give very similar results. The moment-of-fluid (MOF) method, which reconstructs the interface using the reference fluid centroid, is found to be more accurate than the VOF methods. Different coupled level-set and VOF methods are then investigated, namely: CLSVOF, MCLS, VOSET and CLSMOF. It is observed that the level-set method tends to thicken thin filaments, whereas the VOF and coupled methods break up thin structures in small fluid particles. Finally, we coupled the level-set and volume-of-fluid methods with the incompressible Navier-Stokes solver. We compared different manners (sharp and smoothed) of treating the interface jump conditions. It is shown that the VOF methods are more robust, and provide excellent results for almost all the performed simulations. Two level-set methods are also identified that give very good results, comparable to those obtained with the VOF methods.

MOTS-CLÉS : Two-phase incompressible flow simulation, interface capturing, level-set method, volume-of-fluid method, moment-of-fluid method, coupled level-set and volume-of-fluid, ghost fluid method, continuous surface force.