averaging kernel . . . . . . . . . . . C.4 Integration region and nomenclature of the surface tension integral . . . . . . . . . . . . D.1 Definitions of points and vectors used for the derivation of reconstruction relations .

List of Figures

Preface and Acknowledgements

The study of complex flows of engineering interest using CFD software, which includes models to take into account intricate phenomena, is always an enjoyable task. First of all, as in any problem of physics, the understanding of phenomena of interest is of paramount importance. Before trying to gain any insight from CFD software, this first step enables us to approve or reject our modeling assumptions by comparing our results with results of experimental studies. Thus, we naturally move on to the second step. It consists of the theoretical formulation of the problem and a clear understanding of the modeling assumptions. From the theoretical formulation stems a numerical solution algorithm that we implement in a computer code. Due to the rapid development of innovative hardware and novelties on programming languages, modern scientific computing becomes more challenging but opens new horizons to CFD. Therefore, there are four significant aspects of a CFD problem:

1. Understand the phenomena in mind 2. Theoretically formulate the problem 3. Construct a numerical solution algorithm 4. Implement the numerical solution algorithm with modern programming I have tried to produce an authentic and distinguishable document, which explains some aspects of the above as they emerged during my work. Without the aid of the organizations and people that supported me, such task would have been impossible.

Firstly, I would like to express my appreciation to the reviewers and the members of the examination committee for their constructive and encouraging comments. Without the financial support of the Direction Général de l'Armement (DGA -French Ministry of Defence) and CNRS, the current study would not have been feasible. I completed this study as a research engineer of the LHEEA laboratory in Ecole Centrale de Nantes and a member of the METHRIC team that develop the ISIS-CFD code. This work proposes extensions to ISIS-CFD and therefore, without the efforts of the members of the METHRIC team, I would not have been able to produce a working final product. Specifically, I recognize the worth of the constructive discussion I had with Alban Leroyer and kindly thank him for the time he allocated. I also greatly appreciate the collaboration I had with the current head of the team, Patrick Queutey. He guided me through several parts of the code, and his valuable advice allowed me to construct upon ISIS-CFD. Last but not least, I express my deepest gratitude to my advisor, Michel Visonneau. We have spent a significant amount of time discussing the material presented next and without his counseling, I would not have been able to reach the depth of understanding I currently have for modern CFD software. More importantly, Michel Visonneau supported me during the darkest period of my life, when I was not able to efficiently produce scientific work, and I was urged to spend time for recovering physically and mentally, from a severe health issue during the first and second year of this study. It is due to his patience, guidance, and support, which I sincerely appreciate, that this work has reached completion. xiii Chapter 1

Air Lubrication Methods for Drag Reduction in

Hydrodynamic Flows

Experimental Results and Contributions to ISIS-CFD

Introduction

The steadily increasing computing power challenges modern CFD software to take into account phenomena whose modeling has encountered difficulties some years before. In naval engineering, a variety of physical phenomena is, in principle, connected with the accurate modeling of the air-water interface dynamics and two-phase flows. For example, accurate predictions for ship resistance are related to the prediction of spray/foam resistance and the air-water interface mechanisms that govern the generation of strong ventilated vortices emanating from the bow of the ship. The study of phenomena related to other issues of practical importance such as cavitation, which naturally occurs in flows around marine propellers or water-jet impellers, is clearly related to accurate numerical modeling of two-phase flows. Finally, in air lubrication methods the accurate mathematical modeling of both the air-water interface dynamics and two-phase flows becomes an indispensable component for the design of both air-lubricated hulls and air-lubricated underwater vessels.

The concept of the "air-lubricated" hull proposes a new generation of ships whose resistance is significantly reduced. For an "air-lubricated hull", a part of its surface is covered by a layer of air (in the form of a cavity, sheet or bubbles) that reduces the wetted surface and, consequently, the ship's resistance. Such approaches, commonly characterized as air lubrication methods, can notably contribute to the design of more efficient green ships in the near future. From a practical perspective, the design of an air-lubricated ship is directly related to the detailed modeling capabilities of a CFD code to predict the dynamics of air-water interfaces. The purpose of this work is to contribute to the accurate numerical modeling of the physical mechanisms that drive these phenomena, firstly by proposing physically concise surface tension models and secondly by developing the required tools for air entrainment modeling (i.e. models that account for the presence of bubbles in a sub-scale level).

In general, it is recognized by the scientific community that surface tension contributes to the evolution of free surface flows and governs the mechanisms through which the air-water interface breaks up. The significance of surface tension in the development of an interface (e.g. air-water interface or vapor-water interface), is directly proportional to its curvature. In real fluid flow problems, the evolution of the free boundary is not known beforehand. Thus, the correct modeling proposes the coupling of the dynamically resolved geometry of the interface with the Navier-Stokes equations, 1 even though surface tension effects are expected to be small in regions where the interfaces are flat.

Since surface tension is a property of the interface, a significant research effort has been undertaken to develop interface reconstruction techniques that provide explicit representations of the interface (i.e. as a surface grid). Through the explicit interface representation, the dynamics of the interface related to its shape can be accurately coupled even in a solution framework where explicit representations of the interfaces are not intrinsically available. The above is a description of the approach we adopted in this work using ISIS-CFD, a Navier-Stokes Volume Fluid Solver. Besides surface tension modeling, explicit interface representations can be used to introduce other modeling schemes such as air entrainment models. Through surface tension and air entrainment modeling, two important and required tools will be eventually available to study the whole range of physical phenomena observed in fluid flows related to air lubrication methods.

Air lubrication methods for hydrodynamic flows are a current research subjects with a rapidly growing literature. In this introductory chapter, we briefly present some recent results adapted from experimental studies. The purpose is to summarize the principal mechanisms governing the dynamic evolution of the air-water interface involved in air lubrication methods for hydrodynamic flows. More specifically, in the first section, we present some of the latest experimental research, focused on applying the partial air cavity method to reduce the resistance of a hull. Emphasis is given to the experimental findings regarding cavity control, a crucial subject for a successful design of an airlubricated ship. In the second section, we present, classic and modern, experimental results that

Partial Air Cavities for Drag Reduction of Ships

Introduction Economical and ecological reasons have led the scientific community to a systematic research focused on improving the overall propulsive efficiency of ships. The purpose is to reduce the fuel costs and promote novel green ship design concepts. One promising solution is suggested by "air lubrication methods." In principle, the wetted surface of the hull is injected with air to decrease the viscous drag and, as a result, reduce the resistance of the ship. Different versions of this concept have been proposed in the literature, figure 1 Air is continually injected into a properly designed chamber at the bottom of the hull. A cavity of air is generated inside the chamber and lubricates the hull. A major problem regarding the design of PCDR is the selection of air chamber geometry and the air influx (air injected into the chamber) so that the air cavity is attached to the hull for a broad range of design conditions. The speed of the vessel, its draft, maneuvering and sea keeping conditions, add up to the difficulty of the design problem.

After a brief revision of reported efficiency gains for ships using air cavities, we demonstrate several experiments which propose that the effectiveness of the air cavity is related to maintaining a stable air-water interface.

The testing of artificial cavities as a resistance reduction method is traced back to 1882, as noted by [START_REF] Latorre | Ship hull drag reduction using bottom air injection[END_REF] [START_REF] Latorre | Ship hull drag reduction using bottom air injection[END_REF]. Latorre reviewed several previous attempts where significant resistance reductions were reported, 15-18% for model scale and 10-12% in full scale, both for ships designed at low Froude numbers designed and high-speed/planning crafts. [START_REF] Sverchkov | Application of air cavities on high-speed ships in Russia[END_REF] [START_REF] Sverchkov | Application of air cavities on high-speed ships in Russia[END_REF], emphasized the efficiency of air injection as a resistance reduction method, based on results from Russian research studies. The maximum resistance reduction was 25%-30%. Sverchkov also noted that the power consumption to generate the air cavity is low, i.e. in the range of 2%-3% of the total power, while sea-keeping Figure 1.1: Different drag reduction methods concepts using air injection as illustrated by [START_REF] Makiharju | On the energy economics of air lubrication drag reduction[END_REF] [START_REF] Makiharju | On the energy economics of air lubrication drag reduction[END_REF]. A red arrow depicts the injection of air, and the figure indicates the principal air-water interfaces formed (also indicated by each method's name).

CHAPTER 1. AIR LUBRICATION METHODS FOR DRAG REDUCTION

performance is also improved. More recently, Gorbachev and Armomin (2012) [START_REF] Gorbachev | Ship Drag Reduction By Hull Ventilation From Laval To Near Future : Challenges and Successes[END_REF] reviewed a large number of examples of modern vessels that use artificial partial air cavities. A significant resistance reduction followed by an increase of the ship's speed was reported for every case. According to their review, resistance reductions up to 28% can be achieved at model scales. Among other studies, tests performed at SSPA, Allenström and Leer-Andersen (2010) [START_REF] Allenstrom | Model tests with air lubrication[END_REF] and TU Delft and Marin, Zverkhovskyi (2014) [START_REF] Zverkhovskyi | Ship drag reduction by air cavities[END_REF], confirm the previously reported efficiencies of resistance reduction.

Several examples of hull designs with their proper air chambers hosting artificial air cavities are shown in figure 1.2. A crucial problem of the air chamber design is to find its optimal geometry which is capable of conserving the air-water interface at multi-point design conditions, controlled by the amount of air-injected inside the cavity. In the remainder of this section, we briefly examine the connection between the air cavity form, Froude number and the influx of air in the cavity as demonstrated by experimental research studies.

(a) Hard chine hull, Gockay (2004) [START_REF] Gokcay | Revisiting artificial air cavity concept for high speed craft[END_REF] (b) Cargo ship, Krylov Ship Research Institute, Gorbachev (2012) [START_REF] Gorbachev | Ship Drag Reduction By Hull Ventilation From Laval To Near Future : Challenges and Successes[END_REF] (c) Landing Ship, David Taylor Model Basin, Gorbachev (2012) [START_REF] Gorbachev | Ship Drag Reduction By Hull Ventilation From Laval To Near Future : Challenges and Successes[END_REF] (d) Three artificial cavities beneath a hull [START_REF]Sustainable Methods for Optimal Design and Operation of ships with Air-lubricated Hulls[END_REF] Figure 1.2: Artificial cavity designs in different hulls

PARTIAL AIR CAVITIES FOR DRAG REDUCTION OF SHIPS

Air-water interface Formations and Evolution of Air Cavities

We present results of three recent experimental studies. We will use these to demonstrate characteristic formations (shapes) of the airwater interface and relate them to the stability of the cavity. In turn, the stability of the cavity is directly related to its effectiveness as a lubrication (drag reduction) mechanism.

The experimental data of Allenström and Leer-Andersen (2010) [START_REF] Allenstrom | Model tests with air lubrication[END_REF] demonstrate that off-design conditions can lead to the destruction of the air cavity, causing resistance to increase. The photographs on the right of figure 1.3 show the three air chambers which are "closed", as shown for PCDR in At the design point, the clear (and slightly perturbed) air-water interface denotes that the air cavity is well-maintained inside the air chamber. At off-design conditions, the air-water interface becomes opaque (and rough), which indicates significant transverse perturbations. Far downstream (from right to left), the air cavity breaks up to a bubbly cloud. The diagram on the left of figure 1.3, provides the measured force (resistance) for different velocities of the hull with: (i) air chambers filled with air (ii) air chambers without air and (iii) with a flat bottomed hull i.e. without air chambers (and thus no air cavities).

For the hull with air chamber but without air cavity the measured force is higher, as expected than the force exerted to the hull with a flat bottom. For the hull with the air chamber filled (air cavities are CHAPTER 1. AIR LUBRICATION METHODS FOR DRAG REDUCTION formed), a lubrication effect is observed at a range of hull speeds from 2 m/s to 3.5 m/s. At this range, a well-formed air cavity is observed inside the air chamber. As the speed increases further, the air cavity loses its form and along with it the gain in resistance. In this region, the resistance of the flat bottom hull (without an air chamber) is less in comparison to the case where air is injected.

In conclusion, the experiment demonstrates that stable cavities can be formed below the hull of a ship, and their performance is related to the speed of the vessel. As we move further away from the design point, the cavities become more disturbed and eventually the lubrication effect is lost.

To ensure that air lubricates the hull (and the ship's resistance drops), in both design and off-design conditions the control of the appropriate amount of air (and thus the estimation of the air influx) is a very important aspact of the problem. The next set of experimental results provides clear visualizations of the characteristic air-water interface shapes in a single wide air chamber before off-design states, as these described above, are reached. [START_REF] Matveev | Study of air-ventilated cavity under model hull on water surface[END_REF] [START_REF] Matveev | Study of air-ventilated cavity under model hull on water surface[END_REF] conducted experiment with a different configuration. For every case, the air cavity was bounded by the side walls of the air chamber. The air-water interface was flat as observed from bottom to top. Only the inclination of the flat air-water surface (the inclination relative to the chamber's bottom) and the shape of the cavity downwind (the cavity closure) changed for different Froude numbers. For the smallest Froude number (top figure) the flat part of the interface was almost parallel to the bottom of the air cavity. The cavity closure formed a "tongue".

The tongue's plane of symmetry is the hull's centerline. As the Froude number increased, the air-water interface inclination increased towards the bottom of the chamber. The cavity closure transformed to two tongues which touch the walls of the air chamber and the inclined flat part was located at the hull's centerline (middle figure). Finally, for the largest Froude number, the flat part of the interface inclined towards the bottom and covered the whole width of the chamber. From the bottom view, the air cavity appeared as rectangular. [START_REF] Matveev | Study of air-ventilated cavity under model hull on water surface[END_REF] do not mention how the amount of injected air affects the cavity and the interface's shape. However, as he describes, air escapes either in the form of air-patches or as small bubbles (entrained to water from the tongues or in a random fashion from cavity closure for the highest F r case) a constant supply of air is required. Therefore, for low Froude numbers, it appears that the conditions at cavity closure describe the principal mechanisms that air escapes the cavity. Nevertheless, it is clear that the above cases do not correspond to the turbulent bubbly flow that was formed in the experiments of Allenström and Leer-Andersen (the first experimental set of results we have previously reviewed).

The experiments of [START_REF] Arndt | Creation and Maintenance of Cavities Under Horizontal Surfaces in Steady and Gust Flows[END_REF] [4] bridge the gap regarding the passage from a clear For small Froude numbers, the cavity closure resembles the interface's cavity closure shapes obtained by [START_REF] Matveev | Study of air-ventilated cavity under model hull on water surface[END_REF]. [START_REF] Arndt | Creation and Maintenance of Cavities Under Horizontal Surfaces in Steady and Gust Flows[END_REF] [4] also commented on the air entrainment coefficient (a dimensionless air influx) for each case as follows. For Froude numbers up to 0.2, the cavity closure forms a single tongue (that mounts the beach) and the air entrainment coefficient to maintain the cavity is almost zero. As the Froude number increased, the air cavity passes from a stage where the wave crests move closer to the bottom of the cavity and at F r = 0.41 the cavity interface forms two tongues, from which air escapes the cavity. An air supply is required to maintain the air cavity. For higher Froude numbers, the air cavity gradually covers the whole air chamber. After a further increase of the free stream velocity, the cavity dissolves in the bubbly boundary layer shown in the top figure, due to turbulent mixing. [START_REF] Arndt | Creation and Maintenance of Cavities Under Horizontal Surfaces in Steady and Gust Flows[END_REF] noted that, for this case, mixing takes place in the exterior of the air cavity from the side of the interface where water is present and at the cavity closure.

In [4], Arndt et al. concluded that there are two values of the Froude number acts as thresholds to the underlying physical mechanisms. A Froude number up to which the length of the cavity is CHAPTER 1. AIR LUBRICATION METHODS FOR DRAG REDUCTION modulated by surface waves and a minimum Froude number for which the length of the cavity will be less than the provided length of the air chamber (the cavity does not entirely cover the air chamber).

For even higher Froude numbers and larger air entrainment coefficients, air is entrained from the cavity to water and the generation of bubbles becomes the principal mechanisms through which air escapes the air cavity. These mechanisms are modulated by turbulence.

Therefore the air entrainment coefficient required to maintain a steady cavity depends on the Froude number and a critical Reynolds number. For the low Froude numbers, as these used by [START_REF] Matveev | Study of air-ventilated cavity under model hull on water surface[END_REF] the air entrainment coefficient to maintain the cavity is small. For higher Froude numbers there is a specific Reynolds number which marks the entrainment of air to water from the air cavity to the ambient flow of water. Thus, both the cavity closure (the flow conditions at the downwind stagnation point where the air-water interface meets the hull) and turbulence, describe fundamental physical mechanisms that air escapes the cavity. Makihärju (2012) [START_REF] Makiharju | On the energy economics of air lubrication drag reduction[END_REF] also confirmed the above observations. He also emphasized that an understanding of the physical mechanisms occurring in cavity closure are critical for the design of the artificial partial cavities as a drag reduction mechanism.

Furthermore, his experiments showed that surface tension effects are important specifically for lower Reynolds and Froude numbers. Indeed, if these physical mechanisms are not well understood, not only at model scale but also at full scale (a nonlinear behavior is expected as suggested by the results of Makiharju,2012 [63, p.80]), then when passing from model scale to full scale significantly different demands of air might be required and eventually might lead to full-scale designs that do not fulfill their purpose as lubrication "devices".

Concluding Remarks Significant research efforts have been made worldwide for the study of partial artificial cavities especially at an experimental level. We have briefly examined only partial air cavities.

Ceccio (2010) [START_REF] Ceccio | A cost-benefit analysis for air layer drag reduction[END_REF] describes three basic air lubrication methods, namely, the Air Cavity Method for axisymmetric bodies, the Artificial Partial Air Cavity Method and the Bubbly Flow Drag Reduction method.

The air layer method or the bubble drag reduction approach will not be discussed here since bubbles are allowed to move freely (i.e. without a limiting compartment) and a large number of phenomena of entirely different type emerge. The latter are related to modulation of turbulence in multiphase flows as discussed by [START_REF] Michaelides | Particles , Bubbles and Drops : Their Motion , Heat and Mass Transfer[END_REF] [70, p.232-242]. Nevertheless, research studies are conducted in parallel with the methods shortly described here. Recent highlights include the multi-chambered air compartments tested by [START_REF] Zverkhovskyi | Ship drag reduction by air cavities[END_REF] [START_REF] Zverkhovskyi | Ship drag reduction by air cavities[END_REF], and the experimental serf-propulsion results by Samsung, Jang et al. (2014) [START_REF] Jang | Experimental investigation of frictional resistance reduction with air layer on the hull bottom of a ship[END_REF].

In this section, we have provided a brief description of phenomena related to artificial air cavities.

These are straightforward methods for reducing the drag of bodies which are immersed either partially or entirely in water. We have emphasized that air escapes the cavity either near the cavity closure and due to turbulent effects near the air-water interface. In the next section, we revisit a different cavity method for axisymmetric bodies, a lubrication method which is used for smaller underwater vessels.

Air/Vapor Cavities for Drag Reduction of Underwater Vehicles

Introduction The generation of vapor bubbles due the drop of pressure in some locations in the flow around an immersed body (and eventually their collapse near its surface) describes cavitation that occurs under natural circumstances. For example, common cavitation patterns encountered in flows around propellers include vortex cavitation, cloud cavitation, sheet cavitation, etc. In complex free surface flows, air cavities may begin from the free surface and grow naturally inside the region occupied mostly by water. The last case could be described as a ventilated flow, since there is no change of phase associated with water, however, it also occurs under natural circumstances. On the other hand, there are other applications where air is artificially injected into the flow of water using an external source, for example injecting air from a propeller's hub orifice to the propeller blades, to avoid cavitation damage.

In this section, we are interested to air injection as a drag reduction mechanism for underwater vehicles which are in principle axisymmetric, and their characteristic is that cavitation occurs naturally and is simultaneously assisted by injecting air till supercavitation takes place. This implies that, besides the naturally occurring cavitation, air is injected into the flow. The cavities that arise in this case are characterized as artificial cavities for axisymmetric bodies. Even though they are not directly related to the drag reduction of hulls, they fall into the general category of drag reduction using air cavities for hydrodynamic flows. We will shortly discuss some experimental results and identify the mechanisms through which the gas mixture (air and vapor) escapes the cavity, to arrive at an interesting conclusion by comparing the previous cases with the case described in this section.

Cavity Shapes Around Axisymmetric Bodies

Several examples of cavitation patterns that are either naturally generated or artificially assisted are demonstrated by the experiments performed by [START_REF] Brennen | Cavity Surface Wave Pattern and General Appearance[END_REF] [START_REF] Brennen | Cavity Surface Wave Pattern and General Appearance[END_REF]. For the sphere in figure 1.6a, a bubbly flow is clearly present in the wake of the body and a well-defined limit exists between the bubbly flow and water. For a larger freestream velocity, figure 1.6b, we observe the formation of a transparent cavity which is entirely filled by vapor. For the ogive body in figure 1.6c the wake is similar to the sphere of 1.6a. When air is artificially injected perpendicularly to the mounting rod (in a cross-stream sense) to the flow close to the sharp edge of the body, figure 1.6d, a well-defined cavity is clearly visible and slightly further downstream, a bubbly region is formed.

Therefore ventilation can be used to enhance the stability of natural cavities.

Disk heads are usually used to generate ventilated supercavities which surround a large part of the body. The body in these cases is called the cavitator. Brennen found that these could be stable for a broad range of free stream velocities, as the cavity shown for the disk in figure 1.7a. Axisymmetric bodies with sharp conical noses, disk noses, as shown figure 1.7b, and other configurations have been extensively studied in the past by both theoretically and experimentally, see for example [START_REF] Semenenko | Defense Technical Information Center Compilation Part Notice[END_REF] [START_REF] Semenenko | Defense Technical Information Center Compilation Part Notice[END_REF], Nesteruk (2012) [START_REF]Supercavitation[END_REF], Arndt and his colleagues [START_REF] Wosnik | Experimental study of a ventilated supercavitating vehicle[END_REF], [START_REF] Kawakami | Investigation of the Behavior of Ventilated Supercavities[END_REF], [START_REF] Lee | Investigation of the Behavior of Ventilated Supercavities in a Periodic Gust Flow[END_REF].

Following [START_REF] Ceccio | A cost-benefit analysis for air layer drag reduction[END_REF] [START_REF] Ceccio | A cost-benefit analysis for air layer drag reduction[END_REF], three basic non-dimensional parameters are used for the experimental study of artificial cavities. These are, the cavitation number σ c based on the cavity pressure, the Froude number F r based on the cavitator's diameter d c , the air entrainment coefficient C q (a dimensionless volumetric flux of gas entering the cavity):

σ c = 2(p ∞ -p c ) ρV ∞ 2 F r = V ∞ g d c C q = Q V ∞ d 2 c (1.3.1)
Other parameters are the stability parameter β and the cavity closure parameter γ:

β c = σ V σ c γ c = σ 3/2 c F r 2 where σ V = 2(p ∞ -p V ) ρV ∞ 2 (1.3.2)
where V ∞ and p ∞ are the freestream velocity and pressure respectively, ρ the density of the fluid and p V the vaporization pressure (for water at 20 o C , p V = 2348P a).

Semenenko (2001) [START_REF] Semenenko | Defense Technical Information Center Compilation Part Notice[END_REF] gives several correlation relations between the three primary dimensionless variables for the cavity and geometric characteristic of the cavity. The effectiveness of a cavitator as a lubrication device is related to the area of the body that is inside the cavity. Nesteruk (2012) [73, p.82-88] provides semi-empirical correlations for the drag coefficient based on the volume of the body [START_REF] Brennen | Cavity Surface Wave Pattern and General Appearance[END_REF] [START_REF] Brennen | Cavity Surface Wave Pattern and General Appearance[END_REF]. Incoming flow is from right to left.

for slender cavitators with a sharp nose. He demonstrates that larger drag reductions can be achieved with smaller cavitation numbers. This tendency is due to the enlargement of the cavity. At the same time, larger quantities of air have to be supplied to the cavity. Figure 1.8 shows two different cavities formed around the same cavitator for the same Froude number F r = 24.5 and freestream velocity V ∞ = 8.9 m/s and different cavitation numbers (σ c = 0.0334, upper figure, and σ c = 0.0644, lower figure). As the cavity occupies a larger region of the cavitator's wake, a larger part of the cavitator's surface is lubricated. [START_REF] Semenenko | Defense Technical Information Center Compilation Part Notice[END_REF] [92, p.14] and Nesteruk (2012) [73, see "Preface" and p. [86][START_REF] Renardy | PROST: A Parabolic Reconstruction of Surface Tension for the Volume-of-Fluid Method[END_REF][START_REF] Rider | Reconstructing Volume Tracking[END_REF][START_REF] Roenby | A Computational Method for Sharp Interface Advection[END_REF][START_REF] Sangeeth | Dynamics of Collapse of Free Surface Bubbles[END_REF][START_REF] Scardovelli | Analytical Relations Connecting Linear Interfaces and Volume Fractions in Rectangular Grids[END_REF],

notes that supercavitation allows slender bodies to move with velocities over 1 km/s, underwater.

Cavity Control

The cavity reaches a steady state when natural cavitation and artificial ventilation provide a sufficient gas influx to the cavity. To achieve a stable state, the volume of gas that enters the cavity must be equal to the volume of gas that exits the cavity. Therefore, the stability of the cavity depends on the flow characteristic in the downstream locations where the area of the cross section of the cavity becomes small, and eventually a volume of gas escapes the cavity. This is the cavity closure zone or simply cavity closure. Two flows patterns through which gas escapes the cavity or "modes" have been experimentally observed (for moderate air entrainment coefficients). These are nicely depicted in photographs taken by the experiments of [START_REF] Kawakami | Investigation of the Behavior of Ventilated Supercavities[END_REF] [START_REF] Kawakami | Investigation of the Behavior of Ventilated Supercavities[END_REF].

At the cavity closure, a re-entrant jet is formed. For the first mode, the re-entrant jet hits the mounting strut of the cavitator and a water/gas mixture enters the cavity which is rejected in the form of the foam clouds as observed in the first picture of figure 1.9. For the second mode as the air entrainment coefficient is increased, second picture of figure 1.9, the re-entrant jet hits the mounting strut, some mixing takes place and, with further increase of the air entrainment coefficient, principally the gas exit the cavity by forming two vortex tubes, which are also visible in figure 1.7b and both cases demonstrated in 1.8. [START_REF] Semenenko | Defense Technical Information Center Compilation Part Notice[END_REF] [START_REF] Semenenko | Defense Technical Information Center Compilation Part Notice[END_REF] identifies different regions based on γ c that are related to the importance of the vortex pair in the second regime. Specifically, when γ c 1.5 then the vortex pair is the principal mechanism that air escapes the cavity. He also notes that the parameter γ c is also an index of the perturbation of the vortex pairs, such perturbation were also observed by [START_REF] Kawakami | Investigation of the Behavior of Ventilated Supercavities[END_REF] [49] (second and third pictures of figure 1.9).

The experiments of [START_REF] Kawakami | Investigation of the Behavior of Ventilated Supercavities[END_REF] [START_REF] Kawakami | Investigation of the Behavior of Ventilated Supercavities[END_REF] demonstrated that when a cavity reaches a steady state (as in the figures 1.7b, 1.8), for a certain air entrainment coefficient C q , then from this point on, the cavity remains stable even for a smaller air influx. Thus lowering the air entrainment coefficient does not affect the stability of the cavity. Therefore, these stable cavities ("transparent" or "clear" cavities, as opposed to the opaque "foamy" cavities) can be maintained from this point on by lower values of the air entrainment coefficient while the mechanism that air escapes the cavity (the twin vortex regime) remains the same.

In conclusion, the control of the cavity is directly related to (i) the interactions of the interface with the wake of the body at the cavity closure (ii) the air entrainment coefficient. Even though it is clear that the above two observations can be made for the control of the air cavity for hulls, it is interesting to note that air injection for flows around axisymmetric bodies seems to act in a stabilizing manner. Flow from left to right, adapted from Semenenko (2001) [START_REF] Semenenko | Defense Technical Information Center Compilation Part Notice[END_REF] When sufficient air is continuously injected, a clear (transparent) interface is generated and, under certain conditions, it can be even maintained with smaller air entrainment coefficients. Thus, there is a significant difference between the flows of artificial air cavities of axisymmetric bodies and artificial air cavities for hulls; the vorticity generated upwind in water before reaching the cavity must play a significant role to its stability.

Closing In this section we have provided a short description of phenomena that appear to supercavitation assisted by artificial air ventilation, a method for reducing the drag of underwater vehicles.

Complex flow patterns arise in these cases that are related to several phenomena taking place simultaneously such as cavitation, bluff body wakes, generation of bubbly mixtures and vorticity dynamics at the far wake of the body. In connection to the previous drag reduction mechanism that considered a ship's hull, the cavity closure plays a significant role to the prediction of the air influx that is required to maintain the cavity. However, for the more complex flow of a ship's hull, the vorticity generated at the ships bow, before reaching the air cavity, should also greatly affect the stability of the cavity.

Contributions to ISIS-CFD for the Study of Air Injection

Methods for Drag Reduction

In the previous two sections, we have described phenomena related to air injection methods for drag reduction in hydrodynamic flows. The recurring physical mechanisms that would describe the overall state of the system are inertia driven and govern the splitting of the air-water interface. Air is entrained to water either in the formation of patches of air or as bubbly mixtures. Therefore, the experimental results presented in the previous section demonstrate that the stability of the cavity is regulated by the air-water interactions through the interface and, simultaneously, the generation of smaller airwater structures, bubbles, that appears as a dispersed state at the cavity closure or in general near the air-water interface.

The above observation suggests that we should distinguish between two different scales that have to be taken into account during a CFD simulation:

1. A large scale, where a single or multiple interacting air-water interfaces dynamically evolve and can be described as a system of free surfaces. These free surfaces can be tracked down during a CFD simulation by their direct numerical modeling using the one-fluid formulation.

2. A small scale, where multiple interacting air-water interfaces are present, and can be described as a fluid mixture where the injected air is dispersed in water or a bubbly mixture. Special modeling has to be introduced to take into account the bubbly mixture along with the one-fluid formulation.

The purpose of this work is to develop the set of required tools to study air lubrication method in hydrodynamic flows and the required models to take into account the physical mechanisms, occurring at different length scales, that govern the evolution of these phenomena, and incorporate them to ISIS-CFD. The LHEEA Laboratory is a joint research unit affiliated with the French National Centre for Scientific Research, CNRS (Centre National de la Recherche Scientifique). The code can be shortly described as follows:

ISIS-CFD

1. ISIS-CFD is a second-order accurate finite volume unsteady flow solver using collocated variables and unstructured grids equipped with accurate Volume of Fluid interface capturing schemes.

2. Through different grid manipulation strategies, such as moving/deforming grid, automatic grid refinement and sliding grids, the code models in detail complex flows around hulls including all of its appendages, or sail boats.

3. A wide variety of turbulence modeling approaches is implemented along with other modeling tools such as cavitation, mooring, and coupling with BEM codes, and so forth. Each item concerns the study of drag reduction with air-injection methods on different but interrelated levels, either by introducing appropriate discretization methods (for example the curvature calculation for surface tension) or associated techniques (for example, interface reconstruction) and, lastly, the details of the programming approach (feasibility of parallel computations).

Contributions to ISIS-CFD Firstly, without a general initialization scheme for the volume fraction, simulations that involve complex configurations with multiple cavities (air-water interfaces) cannot be studied. Secondly, as we have already noted, two different scales must be separated during a numerical simulation. A scale where the interface is represented by the volume fraction as a free surface and a "sub-scale" where the volume fraction would refer to bubbles. The distinction between scales can be performed by the proposed interface reconstruction algorithm, a mechanism absent for compressive discretization schemes in principle. The same mechanisms can be used for distinguishing multiple interfaces that are present in the computation domain at the same time.

The interface reconstruction method has to produce a surface grid (which is in general unstructured)

that can be used to perform approximations related to the surface. To that end, we have to construct connectivities that define two fundamental topological relations. The "relations" of a surface patch with its edges and nodes and the "relations" of the surface grid with the volume grid, since data must be interpolated from one grid to the other. As we have already noted, interface separation and merging take place frequently in the phenomena we wish to capture accurately and thus surface tension is a mechanism that can contribute significantly to details of these processes. In connection with the procedures introduced to reconstruct the interface, a new set of surface tensions methods resulted, which is based on explicit interface representations (surface grids) for the volume of fluid approach with compressive discretization schemes.

Specifically for surface tension, it was proven difficult to obtain converging normal vector and curvature calculations. To that end, we introduced higher-order differencing schemes that are used especially for these calculations. Higher order differencing schemes use variable stencils that can be best described as cell (or patch, for a surface grid) neighborhoods. Therefore, from a programming point of view, the classic parallel (multiblock) approach followed had to be extended as well.

A generalized multiblock approach resulted through these extensions. In the generalized multiblock approach, the main concept is to express all the required stencils used in a numerical scheme, as connectivities expressed locally (process or processor-wise). Therefore, there is no need to store global connectivities in all the processors simultaneously. Moreover, these connectivities have to be constructed in a dynamic manner because they are directly related to the moving surface grid. In contrast to the volume grid, parts of the surface grid are defined only for a certain set of processors, smaller than the set of processors that the whole calculation takes place. In turn due to the relative movement of the surface grid to the volume grid, the set of processors might dynamically change. From the implementation process of the above to ISIS-CFD resulted the ISIS-CFD extensions framework.

ISIS-CFD Extensions Framework and its Interactions with ISIS-CFD

The ISIS-CFD extensions framework is an independent add-on to ISIS-CFD. It has been written using modular and object-oriented concepts of Modern Fortran (Fortran 2010, "Modern" is a characterization of Fortran 2010 adapted from the book of Metcalf, 2011, [START_REF] Metcalf | Modern Fortran Explained[END_REF]) and MPI. The ISIS-CFD extensions framework seeks to incorporate into its design the following:

1. unified descriptions of the grid entities (of the surface or the volume grid) and adjacency relations are required 2. the implementation complexity of numerical schemes and algorithms are directly related to the implementation complexity the grid data-structure, thus a simple grid data-structure is imperative for implementing complex grid manipulation procedures or numerical schemes 3. changes to the grid data-structure must not affect the numerical schemes used 4. mathematical relations should be straightforward to interpret from the computer code 5. the use of parallel communications and vectorization should be made easy

The concentrated (and ongoing) effort towards incorporating the above "design specification", has resulted in the ISIS-CFD extensions framework. The purpose of the ISIS-CFD extension framework is to provide to the developer the tools to efficiently perform the early and late development stages of a scientific code. Firstly, it provides generalized and adaptable grid data-structures and, secondly, it tightly relates them to numerical methods. Finally, it assists to their parallelization and vectorization for modern hardware.

In practice, the ISIS-CFD framework has been used to revisit several aspects, for example, the grid representations used: what connectivities have to be stored, how can we construct connectivities that are not already defined primarily in parallel. A basic design feature, which has been used to our advantage, is that grid representations used (i.e. the set of connectivities that define the grids) can be utilized for both volume and surface grids, including the features provided by the Distributed Grid Manager (introduced briefly next). These allow introducing surface grids in ISIS-CFD, a finite volume code whose formulation is based on volume grids, with the ultimate purpose of introducing discretization schemes based on the surface grids. Both volume and surface grid used are general unstructured grids.

Moreover, since surface grids become an important component, all the required differencing schemes, interpolations schemes, coupling procedures and visualization procedure had to be developed. One of the main features of the management of the grid and data fields (required to implement numerical operations) for both serial and, most importantly, parallel execution is that they are transparent to the user and are performed by a very rich data structure, the Distributed Grid Manager.

Besides the flexibility obtained for implementing numerical schemes, the ISIS-CFD extensions framework seeks to cover another need in future developments. Advancements in numerical methods used in Computational Fluid Dynamics, follow the progress of computer hardware. The recent trends show that memory size tends to increase (128 GB per desktop machine), processor cores are numerous (ten for a processor of a desktop computer), and expansion cards of computing processors are marketed by leading technology companies, namely, Nvidia (Tesla P100, 2048 threads) and Intel (Xeon Phi

Coprocessor, 68 cores/272 threads).

At the same time, the ratio price to resources tends to become smaller. Advances in hardware are also expressed in new concepts introduced to programming languages. For example, the Fortran 2008 "do, concurrent" loop indicates that the loop can be vectorized. More specifically, these loops are taken into account by the Intel Fortran compiler that performs the vectorized operation automatically to a coprocessor (of course if a system is equipped with one). Therefore, ISIS-CFD extensions seek through its flexible design to profit from the steadily (if not exponentially) increasing advances in computer hardware and modern computer languages.

The ISIS-CFD extensions are not meant to be used in a stand-alone manner. Instead, the framework First of all, the user prepares the input files. One of input files is dedicated to the multifluid configuration. The user can choose from a list of specified configurations, which are described in the multifluid configuration subroutine (MCS). Alternatively and due to a large number of possible initial multifluid settings, the MCS can be overridden by an MCS developed by the user. The development of the MCS is performed using a set of tools that are provided by the framework. The user compiles his MCS to a shared object and overrides the default MCS shared object employed by the ISIS-CFD extensions framework.

During the first steps of the code's execution, the input is read, and basic initialization takes place.

Afterward, execution falls to the MCS (either the default or the one provided by the user). The code enters the temporal loop. For each time step, ISIS-CFD initializes the fields used, and execution passes to the library that performs the initialization of the volume fraction based on the multifluid configuration as described by the user. The code enters to the nonlinear iterations loop.

After execution of the grid manipulation procedures (moving grids, grid refinement) of ISIS-CFD and the (re)initialization of the fields, the grid data-structure is updated. The volume fraction and transport equations related to turbulence are solved next. Subsequently, the interface is reconstructed by the volume fraction (probably after certain manipulations) and the MPA. The source term is calculated and passed to the discretization procedures of the momentum equations. These steps are repeated till the nonlinear convection term has converged, or enough trials have been performed.

An Overview of Subsequent Chapters

Due to the extent of the material discussed, which in turn reflects different parts of the implementations, we have decided not to describe the programming techniques used. Even though these are relevant to the applicability of the proposed methods (and without them, the methods could not be implemented in practice), the clarifying content would make the presentation rather extensive. Therefore we will not discuss the underlying grid representation we have adopted, the generation of connectivities in parallel for the classic multiblock approach, the required extensions to obtain a generalized multiblock approach and the construction of cells neighborhoods (presented by the author to the 3rd ECOMASS Young Investigator Conference, YIC 2015, [START_REF] Politis | Serial and parallel cell neighborhood construction algorithms in unstructured grids for implementing high order finite volume methods[END_REF]). The material encountered in the remainder of this work is organized as follows.

In the second chapter, we present a derivation of the one-fluid formulation with surface tension. We try to formulate formally the one-fluid equations beginning from the classic formulation of the Navier-Stokes under the hypothesis that the air-water interface does not break-up and in the absence of air-water sources. From the derived relations we can directly conclude that the geometric boundary representing the air-water interface in ISIS-CFD is replaced with by the volume fraction. We give a concise definition of the volume fraction through which approximate relations, used to relate volume fraction and geometric quantities of the interface, follow naturally. Also, the source terms are formulated as surface integrals over the region of the interface and, consequently we argue that an explicit interface representation is required. Finally, we compare analytical solutions of the one-fluid formulation with the analytical solutions that are derived for simple problems and conclude that the one-fluid formulation can profit from an explicit interface representation of the air-water interface. In fact, we should note that this is a well-recognized observation specifically for cases where the interface velocity is required (see for example [START_REF] Senocak | Interfacial dynamics-based modelling of turbulent cavitating flows, part-2: Model development and steady-state computations[END_REF]).

Start

In the third chapter, we discuss the relevant applications of the Marching Polyhedra Algorithm (MPA). The MPA performs two basic actions related to the volume fraction, the volume fraction initialization, and the interface reconstruction. The volume fraction initialization problem can be quite complicated in the context of general unstructured grids. The treatment of multiple interfaces adds up to the difficulty, but it is a necessary preliminary work for the study of air cavity hulls. Through the presented methods, ISIS-CFD can be initialized for every possible problem with multiple interfaces.

The interface reconstruction is performed by introducing an interpolation scheme from nodes to faces and appropriate correction near boundaries. We also test the consistency of the described methods by comparing reconstructed interfaces to analytical ones. The material covered in this chapter was presented at the 6th European Conference on Computational Fluid Dynamics, ECFD 2014, [START_REF] Politis | A surface tension method for vof using a marchingcube isosurface construction algorithm[END_REF].

In the fourth chapter, we present the implemented surface tension methods and the results obtained by the verification and validation procedures. We compare the results of four different surface tension methods, two classic methods and two novel methods. The novel methods propose the coupling of surface tension methods classically adopted in the front tracking framework, with the compressive volume fraction transport schemes of ISIS-CFD, a front capturing framework. For the former the interface is explicitly/parametrically represented by a surface grid, for the latter, the interface is not represented at all as a surface grid but solely by the volume fraction. The author presented the validation tests at the 16th Numerical Towing Tank Symposium, NuTTS 2013, [START_REF] Politis | The Surface Tension Challenge in Air-Water Interfaces using the Volume-of-Fluid Method[END_REF], and the 4th International Conference on Computational Methods in Marine Engineering, Marine 2015, [START_REF] Politis | Application of a Surface Tension Method for VOF using a Marching-Cube Isosurface Construction Algorithm to Complex Hydrodynamic Flows[END_REF]. The thesis concludes with Chapter 5, where we discuss the future research axis that stems from this work.

Summary

In conclusion, we have built the required tools for ISIS-CFD that allows a user to set up simulations for drag reduction with air injection and take into account effects of surface tension in the context of VOF with compressive discretization schemes, and incorporated the necessary modifications of ISIS-CFD into the ISIS-CFD extensions framework. The required changes so that the code performs simulations with a multiphase approach were partially studied. To that end, we present a critical overview of the multifluid problem in terms of a multifluid approach in Chapter 2. In Chapter 3, we present the essential components of the framework, which include the flexible volume fraction initialization approach, the interface reconstruction method for the VOF approach of ISIS-CFD and the normal vector and curvature calculations. Finally, we present several applications of the previous methods as applied to problems of academic and engineering interest.

Chapter 2

Theoretical Formulations of the Multifluid

Navier-Stokes Problem

The Classic and the One-Fluid Formulation

Introduction

The term multifluid describes the treatment of distinct flow regions separated by interfaces that are allowed to break up and reconnect. However, these processes do not generate interfaces of smaller length scales than the length scale treated by the solution procedure (the grid size). Thus, multifluid problems describe only mechanisms through which the interface's topology changes, but we are always able to capture them numerically. In this chapter, we examine the classic formulation of the multifluid problem and the one-fluid formulation (in the volume averaged sense) after a short presentation of the equations solved by ISIS-CFD in the second section.

In the third section, we shortly revise the classic formulation of the multifluid Navier-Stokes problem.

Since the surface that separates two fluids (the interface) is a boundary whose movement is obtained by solving the Navier-Stokes equations, the fields of different flow regions are related to "compatibility conditions" at the surface. The term "compatibility condition" is used interchangeably with the term "interface boundary condition". However, we prefer the first term which clearly distinguishes between the boundary conditions of the computational domain and the "interface boundary conditions." We might say that the former are appropriately chosen, while the latter is enforced by the physical modeling of an observable entity, i.e. the surface separating two flow regions. We will formulate the terms found in the compatibility conditions for the stresses using solely the velocity field at the interfaces and its derivatives.

In the fourth section, we provide a theoretical formulation of the so-called "one-fluid" Navier-Stokes equations. More specifically in our analysis, we will formulate the one-fluid equations in the volume averaged sense, and we will derive the conditions under which the volume averaged equations obtain a form similar to the equations treated by ISIS-CFD. To that end, we shortly develop the necessary relations of volume averaging that constitute the mathematical tool used in this work to formulate the one-fluid Navier-Stokes equations. The volume fraction is directly derived, and the compatibility conditions are implicitly taken into account, by the source terms introduced by the volume averaging formulation. One of these source terms is surface tension. Other terms must be dropped to obtain the one-fluid formulation used by ISIS-CFD. The drawback of this process is that we make certain assumptions for the averaged flow fields.

25

The chapter concludes with two simple test cases that are used to compare the analytical solutions obtained by the classic Multifluid Navier-Stokes equations, to the analytical solution obtained by the one-fluid Navier-Stokes equations. Specifically, we note that the volume averaging formulation is also discussed by other researchers, e.g. by Drew [START_REF] Drew | Theory of Multicomponent Flow[END_REF] and Prosperetti and Tryggvason (Eds), 2009, [85, p. 237-281]. However, in contrast to other works, we do not seek to obtain a Navier-Stokes formulation for multiphase modeling but for multifluid modeling, which means that in this work, an interface always represents a well-defined surface on a length scale that is comparable to the grid length scale, where we perform a computation. The resulting formulation introduces terms that are not found in the "onefluid" formulation suggested in other works and discuss similar matters, such as, the extensive work of Tryggvason, Scardovelli and Zaleski (2011), [105, p.41-42]. We should recognize that since here we present the volume averaged one-fluid equations for a multifluid problem, the one-fluid formulation frequently encountered using the Dirac-δ function for the interface S, δ S , will be recovered as the averaging length tends to zero. The terms derived as by-products of the volume averaging procedure cannot be simply neglected since, as we will demonstrate in the third section, the errors obtained (when these terms are ignored even for a simple case), might not be negligible. At the same time, these terms cannot be taken into account in a simple manner and need to be eventually reformulated or probably modeled. This chapter motivates the need of an explicit interface representation in a solution framework which is usually absent or implied.

Governing Equations of ISIS-CFD Introduction

In the domain of Naval Hydrodynamics numerous scientific software, such as ISIS-CFD (Queutey and Visonneau 2007), REFRESCO [START_REF] Vaz | Free-Surface Viscous Flowcomputations. Validation of Urans Code Fresco[END_REF] [START_REF] Vaz | Free-Surface Viscous Flowcomputations. Validation of Urans Code Fresco[END_REF], CFDShipIOWA [START_REF] Carrica | An unsteady single-phase level set method for viscous free surface flows[END_REF] [13], OpenFOAM [START_REF] Jasak | OpenFOAM: Open source CFD in research and industry[END_REF] [START_REF] Jasak | OpenFOAM: Open source CFD in research and industry[END_REF] and GERRIS [START_REF] Popinet | Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries[END_REF] [START_REF] Popinet | Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries[END_REF], propose solution methods to the Reynolds Averaged Navier-Stokes equations (RANS) following Finite Volume practices. Furthermore, each code is equipped with different solution techniques to enhance the fidelity of the predictions, for instance sliding or moving grids, grid adaptation, multigrid acceleration and so on. The previous are related either to the phenomena we seek to capture and specific details of the adopted numerical solution approach. The purpose of this section is to state the formulation adopted by ISIS-CFD clearly.

Even though the material shortly revised here are covered in detail by [START_REF] Queutey | An interface capturing method for free-surface hydrodynamic flows[END_REF], we provide a short description of the equations for the sake of completeness.

Governing Equations Suppose that a given domain D is occupied by two fluids I and J . The domain is covered by a grid (or mesh) that discretizes D. The grid consists of volume regions V c , the cells, such that D = ∪ c∈C V c , where C is the set of cells. The cells must conform to certain topological conditions as that the cells should not overlap, the intersection of the boundary of the cells, ∂V c , of two neighboring cells defines a face and so, as described by [START_REF] Beall | A general topology-based mesh data structure[END_REF] [START_REF] Beall | A general topology-based mesh data structure[END_REF]. Moreover, suppose that the domain D is moving with a velocity v i . Then the space conservation law (see for example Ferzinger and Peric [28, p.321-327]) dictates that the volume of the domain must be conserved during the movement of the boundary:

∂ t V c dV - ∂V c v i n i d S = 0 (2.2.1)
where n i is the normal vector (exterior to the cell V c ) of the boundary of the cell V c . The numerical treatment of the last equation is described by Leroyer [55,.

Each fluid, either I or J , is distinguished in the domain D locally, by the volume fraction field, C I and C J respectively. We define the volume fraction as a local field which obtains the values 1 or 0, based on whether the point lies in the region that the volume fraction tracks or not, and values between 0 and 1 at points that are close to the interface. The tracking of the regions is taken into account by solving the volume fraction transport equation:

∂ t V c C I dV + ∂V c C I (u i -v i )n i d S = 0 (2.2.2)
In practice, due to the solution of the above equation, for a point that rests close to the interface the volume fraction obtains a smooth variation of its value from 0 to 1 in the range of 2 to 3 cells.

The mixture model provides the following constitutive relations for the density and the dynamic viscosity:

ρ = ρ I C I + ρ J C J and µ = µ I C I + µ J C J (2.2.3)
Given any point of D that is located far from the interface, each fluid maintains its original density and CHAPTER 2. THE MULTIFLUID NAVIER-STOKES PROBLEM dynamic viscosity and when the point is near the interface the modeling assumption specifies that the fluid resembles a miscible mixture. For the fluid defined as above, ISIS-CFD solves the conservation of mass:

∂ t V c ρdV + ∂V c ρ(u i -v i )n i d S = 0 (2.2.4)
in the sense described next, and the conservation of momentum:

∂ t V c ρu i dV + ∂V c ρu i (u j -v j )n j d S = ∂V c -pn i d S + ∂V c τ i j n j d S + V c ρg i dV (2.2.5)
where

ρ = ρ I C I +ρ J C J , τ i j = µ(Z i k ∇ k u j + Z j k ∇ k u i
) and µ = µ I C I +µ J C J given appropriate boundary conditions at the boundary of the domain. Moreover, the numerical solution procedure captures the following compatibility conditions at the interface:

u i = 0 , p = 0 , ∇ i p ρ = 0 (2.2.6)
From the above only the third has to be explicitly accounted for in the numerical procedure since the term appears when the pressure equation is constructed. To clarify we shortly discuss the approach developed by Queutey and Visonneau (2007) [86] without providing exhaustive details.

The pressure equation First of all, it is straight forward to show that from (2.2.2),(2.2.4) and the density relation of (2.2.3) we obtain the classic continuity equation:

∂V c u i n i d S = 0 or f ∈c f u f n f S f = 0 (2.2.7)
The second relation is derived directly by writing the first integral as the sum over the faces of the boundary of V c (the discrete counterparts of ∂V c , symbolically written as the set of faces c f ). From this point on, the procedure to produce the pressure equation is (relatively) straightforward.

First, the momentum equation is written in its discretized form by proceeding with the discretization of every term except the term that contains the pressure gradient. Following the classic discretization practices of finite volume method we obtain from (2.2.5):

K c u c + ûc = - ∇p ρ c + g (2.2.8)
where K c contains terms derived by the temporal and spatial discretizations (the schemes used by ISIS-CFD are described in detail in [86]):

K c = T c + A c ρ c V c (2.2.9)
and the pseudo-velocity field ûc is:

ûc = 1 ρ c V c S c + κ∈c t f A κ u κ (2.2.10)
where S c is a source term generated by the explicit parts of the temporal and spatial discretizations, c t f denotes the cells that belong to the face neighborhood of cell c (every face-adjacent cell κ to the cell c) and A κ is derived by the spatial discretizations. We note that the term ρ c V c , which appears in the denominators of the RHS in relations (2.2.9), (2.2.10), is obtained from the discretization of the time derivative (specifically the term refers to the current time step where the solution is sought) and not an equation for the local acceleration derived by the momentum equation (divided by the density).

At this point we perform a Rhie and Chow interpolation procedure:

K → f u f + û→f = - ∇p ρ f + g (2.2.11)
where → f denotes an interpolation procedure or reconstruction procedure from the (discrete) cellcentered fields K c and ûc to obtain the fields evaluated at the faces. The velocity u f is used to reconstruct the fluxes at the faces. Thus from (2.2.7) we obtain (we drop the notation → f since from this point on no confusion should arise regarding how the field K f and û f were calculated):

f ∈c f 1 K f ∇p ρ f n f S f = f ∈c f - 1 K f û f n f S f + 1 K f g n f S f (2.2.12)
The equation we obtain is a (variable-coefficient) Poisson equation for pressure which is solved to obtain the pressure field.

Because the density will be rapidly changing wherever the LHS is discretized near the interface, Queutey and Visonneau (2007) [86] developed a particular set of modified reconstruction relations for fields with prescribed discontinuities. The relations provide approximations for the field values or the direction derivative of the field in the sense of the face's normal. We describe these relations in greater detail in the last chapter of this work. For the moment it suffices to note that ISIS-CFD solves the system resulting from (2.2.12) by enforcing the compatibility conditions:

p = 0 ∇ i p ρ = 0 (2.2.13)
at the faces of the cells where the interface is close. might provide erroneous results. Nevertheless, the above condition is not derived as a compatibility condition at the interface by a formal mathematical derivation. It is rather introduced for numerical reasons and previous experience shows that the solutions are improved.

In the following sections, we present formulations which are slightly different from the standard formulation used in Naval Hydrodynamics. Specifically, in the next section, we present the "classic formulation" where we do not introduce a constitutive equation model for the fluid properties. Instead, we define different fields around the interface for which the Navier-Stokes equation hold and appropriate boundary conditions (compatibility conditions) that we should take into account when solving the resulting system of equations. Afterward, we work towards obtaining the one-fluid Navier-Stokes formulation from the classic Navier-Stokes formulation. To that end, we introduce the appropriate mathematical tools and along with them notations which make the equations appear as if they were different from the equation found in this section. However, the equations are not different but instead modified, so that we can distinguish the local fields of the classic formulation, introduced in section 2.3, from the local fields of the one-fluid formulation (in the context of this work), i.e. averaged fields of the fields that appear in the classic formulation, introduced in section 2.4. Several source terms stem from this process. The source terms appear as surface integrals of the interface, and one of these terms is the surface tension source term. The analysis gives rise to the conditions under which we may obtain the equations governing the fluid flow with the constitutive model (2.2.3) from the classic formulation.

The Classic Multifluid Navier-Stokes Formulation

In this section, we shortly formulate the basic multifluid Navier-Stokes equations. We begin by the standard Navier-Stokes equations of a single fluid and introduce the essential mathematical tools used for the problem's formulation when more than one fluid is defined. Several other works, Ishii (2011) [START_REF] Ishii | Thermo-Fluid Dynamics of Two-Phase Flow[END_REF],

Drew(1999) [START_REF] Drew | Theory of Multicomponent Flow[END_REF], refer to this formulation as the local instant formulation of the Navier-Stokes. We use general tensor notations which are imperative for appropriately stating the flow compatibility conditions on the interface. However, we should note that the general tensors are not required in practice for the numerical solution of the Navier-Stokes (as performed by ISIS-CFD) since the ambient space is Cartesian. The material discussed in this section can be also found using slightly different formulations in [START_REF] Tryggvason | Direct Numerical Simulations of Gas-Liquid Multiphase Flows[END_REF] [105, p.32-41].

The Navier-Stokes Equations The flow of a single fluid of density ρ moving in the domain D with a velocity field u under a pressure field p, is governed by the Navier-Stokes system of partial differential equations (found in many classical textbooks, see for example Panton [START_REF] Panton | Incompressible Flow[END_REF]) and appropriate boundary conditions imposed on the domain's boundaries ∂D. The system consists of the continuity equation:

∂ t ρ + ∇ j (ρu j ) = 0 (2.3.1)
and the momentum equation:

∂ t (ρu i ) + ∇ j (ρu i u j ) = ∇ j T i j + ρg i (2.3.2)
T i j is the stress tensor that can be expressed under the common assumption that the fluid is Newtonian with dynamic viscosity µ as:

T i j = -(p + 2 3 µ∇ k u k )Z i j + 2µS i j (2.3.3)
and S i j is the strain rate tensor:

S i j = 1 2 Z i k ∇ k u j + Z j k ∇ k u i (2.3.4)
Where Z i j is the contravariant metric tensor and the operator ∇ i represent covariant differentiation. If the fluid is incompressible then the continuity equation (2.3.1) simplifies to:

∇ j u j = 0 (2.3.5)
and the stress tensor is:

T i j = -p Z i j + µ(Z i k ∇ k u j + Z j k ∇ k u i ) (2.3.6)
In this work we study incompressible flows of Newtonian fluids.

Description of a Domain Occupied by Different Fluids

When I . For a time-dependent problem all subdomains and their boundaries can be time dependent, for example, D I (t ), ∂D I (t ). Furthermore, all boundaries are assumed to have the necessary smoothness properties so that the normal vector (and the curvature) is defined as an almost everywhere continuous (vector) function of position on the boundary. The unit normal vector of a fluid interface S J I points from fluid I to fluid J , n S J I and, thus n S J I =n S I J . When no confusion arises, we use the notation:

n = n I = n S J I = -n J = -n S I J (2.3.7)
We note that, later on, when tensor notations is used, capital indices indicate fluid regions and not tensor components. We give an example of a multifluid domain in the appendix, see p.233.

The region of each subdomain D I (t ) is identified by an indicator function I I ( r , t ). The range of an indicator function is either 0 or 1 based on whether the point r rests outside or inside the region:

I I ( r , t ) = 1 , r ∈ D I (t ) 0 , otherwise (2.3.8)
which closely resembles a Heaviside function. The indicator functions of any region can be constructed in a simple manner using classic Heaviside functions:

H (x) = 0 , x < 0 1 , x >= 0
Consider that each interface in the given domain and at a given instant (for now we drop the time variable in our notations), can be implicitly defined by a function f J I ( r ), through the condition:

r ∈ S J I if f J I ( r ) = 0.
The regions in either side of the interface can be then identified by f J I ( r ) < 0 or f J I ( r ) > 0. In that case, we will say that the region I is inside the interface S J I and that the region J is outside the interface. Subsequently, the indicator function of each subdomain can be written as a single Heaviside function, or a sum of Heaviside functions (either directly or beginning by a product and sums of Heaviside functions, as in the example given in the appendix p.233) for more complex regions.

In more a general setup, the surfaces may be too complicated to be represented implicitly by a function f ( r ) = 0. Consider that an explicit (or parametric) representation of the closed surface containing the region D I is given by ∂D I = r : r = x(u, v) then an indicator function for that region can be constructed by:

I I ( r , t ) = 1 4π ∂D I (t ) x -r | x -r | 3 n( x)d S( x) (2.3.9)
The notation d S( x) is used to emphasize that the integration is over the surface position vector x.

The last integral obtains the desired values (as long as the surface is smooth), see also [START_REF] Unverdi | A front-tracking method for viscous, incompressible, multifluid flows[END_REF] [START_REF] Unverdi | A front-tracking method for viscous, incompressible, multifluid flows[END_REF].

If an explicit representation of the surface of an interface S J I is given by r = x S (u, v, t ), where u, v denote surface coordinates and t time, then a point (u, v) of the interface S J I moves with velocity:

w S J I = ∂ x S ∂t (2.3.10)
Since a single fluid must be present at any point of the geometric space in the flow domain, the interfaces cannot be displaced so that the regions they represent will be overlapping, or that a point will be left without being occupied by a fluid. Therefore, we obtain the following compatibility condition for the velocities of the interfaces of the adjacent fluid regions I and J :

w S J I = w S I J = w (2.3.11)
When no confusion arises, we will use w for the interface velocity without referring to a particular interface.

Consider that the interface can be located for any instant by an implicit function, f ( r , t ) and the condition r ∈ S J I if f ( r , t ) = 0. Therefore for each instant we should have f ( x S (u, v, t ), t ) = 0 and thus the total differential of the last expressing must be zero:

d f d t x S = 0 ⇒ ∂ f ∂t x S + ∇ i f x S w i = 0 (2.3.12)
The notation | x S signifies that the values are evaluated at the surface. The last expression simply states that the values where f ( r , t ) are zero (or in general constant), are transfered with the points of the surface.

Description of Fluid Fields and Modeling of Interfaces

Up to this point, we have provided a basic description of how we can identify all the geometric points where particles of a single fluid can be found at a certain instant, using the indicator functions. In each fluid subdomain D I , a field f I ( r , t ) is defined for each fluid property that enters the Navier-Stokes equation. We consider that inside each of the subdomains, the Navier-Stokes equations hold up to the boundary of the interface. Therefore, for any point inside the subdomain D I we obtain by the continuity equation (note that only lowercase indices denote components):

∇ j u j I = 0 (2.3.13)
and the momentum equation:

ρ I ∂ t u i I + ρ I ∇ j (u i I u j I ) = -Z i j ∇ j p I + µ I Z j k ∇ j ∇ k u i I + ρ I g i (2.3.14)
Note that a capital letter index, such as I in the above, is a fluid index and not a tensor component. The above equations are valid as long as the derivatives are defined in each fluid subdomain.

One of the main problems encountered in the mathematical formulation of the multifluid problem is that the classical argument that the fields which enter the equations are everywhere continuous do not to hold. For example, the density is defined at the interface S J I as we approach it from either the side of fluid I or the side of fluid J . The last argument also holds for the viscosity, the velocity gradient, the pressure and their derivatives. However, the values of these fields are not a priori known but are derived as solutions of the Navier-Stokes equations.

The field values on each side of the interface are related by the flow compatibility conditions that are obtained from the following physical arguments:

1. Fluid particles adjacent to the interface follow its movement.

2. There is no distributed mass at the interface.

Intrinsic tractions to the surface act isotropically.

We may summarize the above by stating that each interface is a massless impermeable membrane.

Many authors discuss the mechanics of interfaces and their mathematical formulation beginning from conservation theorems for entities distributed to the interface. Specifically, the above case is derived for the simplest case where the interface is massless. The interested reader may consult Jaric and Kuzmanovic (2006) [99, Reality and Compatibility of Physical and Mathematical Formalisms, p. 273-274] and [START_REF] Dziubek | Equations for two-phase flows: A primer[END_REF] [START_REF] Dziubek | Equations for two-phase flows: A primer[END_REF] for more details.

The first item in the above list enforces the continuity of the velocity fields u I and u J at the interface:

u I = u J = w (2.3.15)
or using the bracket operator, which defines the jump of a field at the interface S J I as • = (•) J -(•) I (note that the order of the indices in S J I is important):

u i -w i = 0 or u i = 0 (2.3.16)
As a result the normal and tangential components of the velocity fields to the interface are continuous.

The second condition does not allow either mass or momentum to be stored in the interface. The continuity equation for the interface simplifies to :

ρ I ( u I -w) n = ρ J ( u J -w) n or ρ(u i -w i ) n i = 0 (2.3.17)
which states that, the mass flux of fluid I entering the fluid region J by passing through the interface, ρ I ( u Iw) n, must be equal to the mass flux of fluid J exiting the fluid region I by passing through the interface, ρ J ( u Jw) n. In our case, since no mixing of the two fluids is allowed, both sides of (2.3.17) are identically zero by (2.3.15).

The third condition allows us to write the surface stress tensor τ αβ as: [START_REF] Moeckel | Thermodynamics of an interface[END_REF] [START_REF] Moeckel | Thermodynamics of an interface[END_REF]. [START_REF] Kralchevsky | Theory of curved interfaces and membranes: Mechanical and thermodynamical approaches[END_REF] [START_REF] Kralchevsky | Theory of curved interfaces and membranes: Mechanical and thermodynamical approaches[END_REF] give several examples where the surface stress tensor is not symmetric. The surface tension coefficient depends only on the material of the ambient fluids that generate the interface and varies with temperature or the presence of surfactants. In this work, we will consider the surface tension coefficient as constant, but we give the general relations where surface tension is allowed to vary throughout the surface. Relation (2.3.18) introduces surface tension as a mechanical property of the interface. In general, the surface stress tensor defines the force exerted in a "fiber" (a differential line element) of the surface.

τ αβ =
Suppose that the unit tangent vector that determines the fiber is t (a vector that belongs to the tangent space of the surface, t i = Z i α t a , where Z i α is the shift operator that transfers components from the tangent space of the surface to components of the ambient space). If the unit normal vector to the fiber is m i = i j k t j n k (also an vector that belongs to the tangent space of the surface m i = Z i β m β ), then the (distributed) force on the fiber due to surface tension is Z i α τ αβ m β . If S is a closed patch of the interface and its boundary is the fiber ∂S, then the total force exerted by the fiber to the surface patch is:

∂S Z i α τ αβ m β d l (2.3.19)
We continue by using the integral theorem of Gauss for a surface (see for example Grinfeld, 2013 [38, p. 241]):

S ∇ α u α d S = ∂S u α m α d S (2.3.20)
where ∇ α denotes covariant differentiation with respect to the surface coordinates, to obtain:

∂S Z i α τ αβ m β d l = S ∇ β Z i α τ αβ d S = S n i B αβ τ αβ + Z i α ∇ β τ αβ d S (2.3.21)
where B αβ is the (extrinsic) curvature tensor of the surface, B αβ = -Z i α ∇ β n i . By (2.3.18) we obtain:

∂S σm i d l = S n i σB α α + Z i α ∇ α σ d S (2.3.22)
The last relation is found in classic textbooks such as [START_REF] Panton | Incompressible Flow[END_REF] [76, p.633-634] and we emphasize that it provides the resultant surface tension force acting from the fiber to the surface patch (or equivalently, from the surface to its surrounding). We refer to the trace of the curvature tensor, B α α , as the curvature. When σ is constant the second term in the integrand is zero, and the surface tension force acts only in the direction of the normal vector.

The momentum balance equation at the interface simplifies to a force balance relation between the forces exerted from the fluids to the surface and the surface tension forces since the interface does not store any mass and it is impermeable. The balance of forces on the interface gives:

S T i j n j d S + ∂S σm i d l = 0 (2.3.22) ⇒ S T i j n j d S = S -n i σB α α + Z i α ∇ α σ d S (2.3.23)
and since the last relation holds for an arbitrary surface:

T i j n j = -n i σB α α -Z i α ∇ α σ (2.
Z i α µ (w β B αβ + ∇ α w) + µdu α + n i -p -2 µ (∇ α w α -wB α α ) = -Z i α ∇ α σ -n i σB α α (2.3.25)
Where w α is the contravariant component of the tangential velocity to the surface, w is the normal component of the velocity to the surface, i.e.:

w i = Z i α w α + n i w (2.3.26)
and d denotes the directional derivative normal to the surface (or simply normal derivative), d(•) = n l ∇ l (•). Note that the term du is expressed by the ambient velocity and not the interface velocity, to emphasize that there will be a jump in the normal derivative of the tangential velocity component. The derivation of relation (2.3.25) is rather lengthy, and the details can be found in the appendix p. 217.

From relation (2.3.25) we arrive to the balance of the normal forces in the interface which prescribes the pressure jump as:

p = σB α α + 2 µ du or p = σB α α -2 µ (∇ α w α -wB α α ) (2.3.27)
and the balance of tangential forces which prescribes the jump of the normal derivative of the tangential velocity component:

µd(Z i γ u γ ) = -Z i α ∇ α σ -µ n j Z i α ∇ α u j or µdu γ = -Z γα ∇ α σ -Z γα µ (w β B αβ + ∇ α w) (2.3.28)
If the velocity of the interface is known, we can calculate the RHS of relations (2.3.28) and (2.3.27) since both are expressed using the velocity field and its derivatives of the interface. Note that for the case where σ is constant (or zero), equation (2.3.28) predicts a jump in the tangential shear traction at the surface due to the fact that two different fluids are present and in order to obtain equal shear forces at the surface there must be a jump in the normal derivative of the tangential velocity component. This result is independent of surface tension. In figure 2.1 we depict schematically the normal derivatives du and d(Z i γ u γ ). Finally, we note that relation (2.3.27) was derived by supposing that (i) the normal, n, points from fluid I to fluid J , (which also defines the bracket operator • = (•) J -(•) I ) and (ii) the surface intrinsic normal also points from I to fluid J . In any other case, if the intrinsic definition of the surface's normal is denoted by n S (recall that only lowercase indices denote components), i.e. n S is obtained as the unit vector of the cross product of the tangent vectors:

n i S = 1 2 i j k αβ Z α j Z β k (2.3.29)
then relation (2.3.27) becomes:

p = n i S n i σB α α -2n i S n i µ (∇ α w α + wB α α ) (2.3.30)
The inner product n i S n i corrects the sign of the expression to conform to the definition of the bracket operator. Details of the derivation of relations (2.3.27) and (2.3.28) can be found in the appendix p.229.

I J S J I u α I = u I s α u α J = u J s α n s α (a)
∇ j u j = 0 ρ∂ t (u i ) + ρ∇ j (u i u j ) = -Z i j ∇ j p + µZ j k ∇ j ∇ k u i + ρg i (2.3.31)
Kinematic Boundary Condition for the Interface The velocity field is continuous near the interface:

u = 0 (2.3.32)
and the interface is transported according to:

d f d t x S = 0 ⇒ ∂ f ∂t x S + ∇ i f x S u i = 0 (2.3.33)

Dynamic Boundary Condition for the Interface

The fluid forces must balance the surface tension forces, as a result, the pressure jump must be:

p = σB α α + 2 µ du = σB α α -2 µ (∇ α w α -wB α α ) (2.3.34)
and the jump of the tangential tractions must be:

µdu γ = -Z γα ∇ α σ -Z γα µ (w β B αβ + ∇ α w) where f ( x S , t ) = 0 (2.3.35)
"External" Boundary Condition Appropriate boundary conditions must be also specified in the boundary of the whole fluid flow domain.

A direct numerical solution of the above problem poses several difficulties. First of all, since we are interested in the solution using the finite volume approach, the grid in each flow subdomain has to be dynamically generated during each time step. The Navier-Stokes equations have to be solved for each fluid subdomain, and their solutions must be matched to the interface boundary conditions. Both procedures pose several implementation difficulties, while complex topological interface manipulations due to merging and separation of the interface can be quite difficult to be treated. To that end, most of the methods developed for the solution of multifluid problems use the so-called "one-fluid" formulation. We present a formulation of the one-fluid equations of ISIS-CFD in the next section.

The One-Fluid Multifluid Navier-Stokes Formulation

Introduction In the one-fluid formulation, a single Navier-Stokes equation is written for the whole fluid flow domain and the indicator functions are used to distinguish between different fluids and their properties. The compatibility conditions are either absent or taken into account in an appropriate way that is based on the numerical method used for the problem's solution. In locations where the differential operators need to be discretized and involve discontinuous field variables, such as the density or the viscosity, one introduces appropriate interpolation schemes. Finally, a characteristic function for the interface location is transferred by solving the interface kinematic equation (2.3.33).

A solution method for the one-fluid formulation can be briefly specified by the following. Firstly, how the Navier-Stokes equations are solved. Specifically, the grid used (e.g. either structured or unstructured) and the variable placement (staggered or collocated) significantly affects the details of the solution method. Secondly, how the interface is represented, which also defines how the indicator functions are represented and how the interface is transferred to its new locations to determine the fluid flow subdomains in subsequent time steps. We can distinguish three methods for these parts, methods that use exclusively explicit interface representations, methods that use exclusively implicit interface representations and methods that use both explicit and implicit representations in different sections of the algorithm. Thirdly, how the interpolation schemes are defined for the (otherwise discontinuous) variables at discretization stencils that the interface is present. In this section, we seek to derive formally the one-fluid formulation used by ISIS-CFD.

Through the above elements, we can briefly describe the one-fluid formulation of ISIS-CFD, following [START_REF] Queutey | An interface capturing method for free-surface hydrodynamic flows[END_REF] in [86]. ISIS-CFD is a finite volume solver using collocated variables.

The volume fraction implicitly distinguishes each fluid subdomain and its interfaces. Given a grid covering the computational domain, the volume fraction, C I , is calculated at the initialization step for each cell center. It results as the quotient of the volume of fluid I occupied in the cell c (symbolically "V c in I "), over the volume of the cell, V c :

C I c = V c in I V c (2.4.1)
The above relation provides one of the possible definitions for the volume fraction.

If we suppose that the volume fraction is a smooth representation of the indicator function, then we claim that the volume fraction replaces the indicator function in the interface kinematic equation 2.3.33. Therefore we obtain the following evolution equation for the volume fraction:

∂ t C I + u i ∇ i C I = 0 (2.4.2)
or equivalently the volume fraction transport equation:

∂ t C I + ∇ i u i C I = 0 (2.4.3)
Given the velocity field, the last equation defines the new volume fraction values for subsequent time steps and thus the new locations of the I -th flow subdomain. Since every fluid is incompressible, the continuity equation remains the same in the whole computational domain and is expressed by the requirement that the velocity field must be solenoidal.

Moreover, we consider that the dynamic characteristics of the each fluid flow can be sufficiently captured by supposing that the whole flow can be approximated through a single momentum equation that has exactly the same form as the classic momentum equation of a single fluid: To that end, we use the mathematical tools developed for the treatment of multiphase flow modeling.

∂ t (ρu i ) + ∇ j (ρu i u j ) = ∇ j T i j + ρg i (2.
The mathematical background is based on generalized functions and details can be found in [START_REF]GeneraIized Functions: Theory and Technique[END_REF] [48, ch. 5, p. 105-130] and Estrada and Kanwal (1991) [START_REF] Estrada | Non-classical derivation of the transport theorems for wave fronts[END_REF]. Both in multifluid and multiphase models we are interested in the coupling of the flow dynamics at the regions where the interfaces are present. However, in the ISIS-CFD multifluid model and contrary to the purpose of multiphase models, the length scales of the interface is comparable and larger to the length scale where the equations are discretized and solved. In any case, the terms generated by a formal mathematical formulation must be added to the solution procedure as long as there are cases where these terms are important. Finally, from this procedure we formally derive an equivalent formulation of the Navier-Stokes equations with surface tension without δ function. Thus we avoid interpreting equations in the weak sense.

One-fluid Formulations with Dirac-δ Functions A different formulation to the one-fluid formulation presented here, is presented by [START_REF] Chang | A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows[END_REF] [START_REF] Chang | A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows[END_REF]. Chang describes "the equivalent weak formulation [of the original classic multifluid momentum equation]" as the level set formulation given by:

ρ ∂ t u i + ∇ j (u i u j ) = ∇ j T i j + ρg i + σB α α ∇ i φδ(φ) (2.4.5)
where φ is the level set function (φ = 0 represents the interface) and δ is a one dimensional Dirac-δ function (and ρ is the discontinuous density field). The last term is the surface tension source term.

The above formulation of the momentum equation and its equivalent form (δ S is a surface Dirac-δ function):

ρ ∂ t u i + ∇ j (u i u j ) = ∇ j T i j + ρg i + σB α α n i δ S (2.4.6)
is frequently adopted in the literature, to cite a few,

∂ t (ρu i ) + ∇ j (ρu i u j ) = ∇ j T i j + ρg i + σB α α n i δ S (2.4.8)
that better suits the Navier-Stokes one-fluid formulation developed next in this section. However, as long as the delta function appears at the RHS, the equal sign has to be conceived in the weak sense.

We note that the continuity equation still holds and we may formally pass from (2.4.8) to (2.4.4). The continuity equation follows directly from the conservation of mass equation and the fact that any indicator function evolves in a Lagrangian manner:

DI I D t = 0 (2.4.9)
In the formulation described here the density and the dynamic viscosity fields are defined by the volume fraction and not the indicator function. Since the volume fraction is not an indicator function, its evolution as a Lagrangian field is an assumption rather than a derived result. The analysis that follows seeks to take these differences into account and to introduce a strong formulation of the onefluid multifluid problem with surface tension. To that end, we propose a volume averaged formulation of the one-fluid problem. The shortcoming is that the fields found in the Navier-Stokes equations should be replaced by their averaged counterparts to obtain definitions consistent to the definitions of the density and the viscosity using the volume fraction. From the discussion that follows, we will have to agree that the the velocity field that enters the Navier-Stokes equations is the center of mass velocity field.

Therefore, subtle differences between one-fluid formulations are encountered in the literature and in this section, where we present a volume averaged one-fluid formulation. We shall attempt to formulate the equations from a viewpoint of methods used in multiphase flow modeling but for immiscible fluids. This approach has been chosen in order to better understand what are the mathematical details that enter the one-fluid formulation and how ISIS-CFD can be extended to obtain multiphase models eventually. Nevertheless, in this section, we discuss only the multifluid case. Finally, we note that in the approach adopted here the generalized functions are regularized by averaging kernels. The following discussion seeks to motivate the use of averaging kernels.

The Volume Fraction As in any one-fluid formulation, we must define how the different fluids subdomains are distinguished in the computational domain. In the one-fluid approach of ISIS-CFD, the computational domain is not partitioned in distinct grids, and there is no explicit representation of the interfaces (besides when the initialization takes place). Instead, in ISIS-CFD we follow the Volumeof-Fluid (VOF) method (with compressive discretization schemes). Given a general unstructured grid covering the computational D domain and oriented surfaces ∂S I , such that the initial flow subdomain regions are specified, we define the volume fraction of the I -th fluid, C I , as the discrete cell-centered field:

C I c = 1 V c V c I I dV (2.4.10)
Note that the index c refers to a certain cell. The initialization of the volume fraction field is not a straightforward task for general unstructured grids especially when multiple interfaces are given, and we will return to this problem in the next chapter.

For a cell that rests inside the subdomain D I the indicator function is 1 and therefore the above integral is V c and the volume fraction is equal to 1, C I c = 1. For a cell that rests outside the subdomain D I the indicator function is 0 and therefore the above integral is 0 and the volume fraction is equal to 0,

C I c = 0.
For every other case where the interface rests inside the cell, the integral is the volume of the region defined by the intersection of the cell with the subdomain D I and therefore, 0 < C I c < 1. Since every cell must be completely occupied by fluids:

I =1,n f l C I c = 1 (2.4.11)
Recognizing that definition (2.4.10) is a mean value integral of the indicator function, the volume fraction can be considered as a smoothed indicator function, since, if we let the cell size (a length scale) depend on a parameter then:

lim →0 C I c ( ) = I I ( x c ) (2.4.12)
The volume fraction is better viewed as a regularization of the indicator function and through similar regularizations, we can treat any discontinuous function. From a physical point of view, such regularizations allows us to specify a length that defines the transition between the two fluids. Furthermore, if the averaged fields are dense (and sufficiently smooth) in space (and time) we can formally define their derivatives.

To that end, we define the volume fraction in any point of space as:

C I ( x, t ) = I I ( y, t )K ( y; x)dV ( y) (2.4.13)
where the integration is defined over the whole space, K ( y; x) is an averaging (or smoothing) kernel based on the averaging length and centered at the evaluation point, x, with either compact support or rapidly decaying support (in this work the "support" is the region where the kernel obtains non zero values or sufficiently larger than zero values) as xy >> . The kernel is also normalized to 1:

K ( y; x)dV ( y) = 1 (2.4.14)
In order to simplify the notation, the time variable for I I is not explicitly stated but it is implied. Note that the kernel does not depend on time. As before, as tends to zero, we recover the indicator function.

The discrete definition can be recovered by choosing the top-hat function as the averaging kernel:

K ( y; x) = 0 , otherwise 1 V c , y ∈ V c (2.4.15)
and performing the evaluation of (2.4.13) at x = x c :

C I c (t ) = C I ( x c , t ) = I I ( y, t )K ( y; x c )dV ( y) = 1 V c V c I I ( y, t )dV ( y) = V c in I (t ) V c (2.4.16)
By definition (2.4.13), the volume fraction represents the fraction of the volume occupied by fluid I in the region of the kernel's support. So we obtain the continuous counterpart of (2.4.11):

I =1,n f l C I ( x) = I =1,n f l I I ( y)K ( y; x)dV ( y) = K ( y; x) I =1,n f l I I ( y)dV ( y) = K ( y; x)dV ( y) = 1
(2.4.17)

We will frequently use the top-hat function as the averaging kernel in order to obtain simplified versions of the developed relations.

Preliminary Definitions and Basic Derived Relations of Volume Averaging

Basic Definitions of Volume Averaging The argument that a multifluid problem is formulated as a one-fluid problem is contradictory since a multifluid problem is defined as a problem where more than one fluid is present. However, the one-fluid formulation seeks to handle the different Navier-Stokes equations for each fluid, using a single Navier-Stokes equation. The problems appear early in the formulation since we must deal with the presence of fields that are discontinuous, i.e. the density, the viscosity, the pressure and the velocity gradient. To begin our analysis, firstly we recognize that a field f ( x, t ) for the whole flow domain can be written using the fields f I ( x, t ), which are defined for each flow subdomain, and the indicator functions as:

f ( x, t ) = I =1,n f l f I ( x, t )I I ( x, t ) (2.4.18)
Since this is a discontinuous field, we cannot formally define its derivatives in the classic sense. In the generalized sense, the derivatives can be defined using the derivatives of the indicator function.

An alternative to obtain a strong formulation is to introduce averaging operators which take the local discontinuous field values in a region near the ambient space around the interface and assign an averaged value for the field. The averaged fields will be smooth.

We introduce two groups of averaging operators, the bar operators, which define the average field value and the hat operators which define mass weighted average for the point x. These operators will be used to transform the discontinuous fields into continuous fields and formally (from a mathematical point of view) differentiable fields. We will apply the bar operator to the classic Navier-Stokes equations.

To pass the bar operator under the differentiation, we will derive relation to recast the averages of derivatives as derivatives of averages. From this process every surface Dirac-δ function will be directly expressed as a surface integral. The fields that appear in the Navier-Stokes equations will be averaged fields and we will state the equations using the center of mass velocity field.

For a field f , the bar-I operator, (.) I , defines the average of the field over the region where the support of the kernel rests inside the I -th subdomain:

f I ( x, t ) def = f I ( y, t )I I ( y, t )K ( y; x)dV ( y) (2.4.19)
Note that the domain of integration is not explicitly stated which implies that it covers the whole space.

However, the kernel has a compact support and therefore, it is also implied that only field values near the point x will be used to provide the integral's result. Moreover, since the indicator function multiplies these, the values of the field that contribute to the above integration, refer to points located inside the kernel's support and the fluid subdomain I . A simple example of a field generated by the bar-I operator is the volume fraction, C I ( x, t ) = 1 I . As another example, consider the case in which the averaging kernel is the top-hat function. Then the bar-I operator provides the mean value (note that we should recognize that the values of f I are not defined inside the domain D J ): Thus, given as input a field f , the bar-I operator assigns to the point x the value f I using only the values of the field f from the I subdomain. The integration region is depicted in figure 2.2a

f I ( x, t ) = 1 V c V c ( x) f I ( y, t )I I ( y, t )dV ( y) = 1 V c V c in I ( x,t ) f I ( y, t )dV ( y) (2.
The bar operator (without an index), (.) , defines the average of the field f for the whole support of the kernel:

f ( x, t ) def = f ( y, t )K ( y; x)dV ( y) (2.4.18) = I f I ( x, t ) (2.4.21)
Given as input the field f , the bar operator assigns to a point x the value f . This value is calculated by performing the above integration in the support of the kernel as depicted in figure 2.2b. The volume fraction can be defined by the bar operator as C I = I I .

For a field f , the hat-I operator, (.) I , defines the "center of mass" values of the field over the region where the support of the kernel rests inside the I -th subdomain:

f I ( x, t ) def = ρ f I ( x, t ) ρ I ( x, t ) = f I ( x, t ) C I ( x, t ) (2.4.22)
and the hat operator, (.) (without fluid subdomain index), defines the "center of mass" value of the field for the whole support of the kernel:

f ( x, t ) def = ρ f ( x, t ) ρ( x, t ) = I ρ I ( x, t ) f I ( x, t ) I ρ I ( x, t ) (2.4.23)
We refer to the hat operators as the center of mass operators due to their similarity with the definition of the center of mass. In the literature the hat operator is also named Favre average or the "mass weighted" average. The RHS of the defining relations (2.4.22), (2.4.23) are derived by taking into account that the densities ρ I are constant. Through the above definitions we observe that the field f can be written as:

f ( x, t ) = I f I ( x, t )C I ( x, t ) (2.4.24)
By comparing the above with (2.4.18) and noting that the volume fraction tends to the indicator function as tends to zero, the field f I must also tend to the field f I , as tends to zero. Therefore, the field f ( x, t ) can be considered as a regularization of the discontinuous field f . When the averaging kernel is the top-hat function, then the hat-I operator is the mean value of the field f at the I subdomain:

fI ( x, t ) = 1 C I ( x, t )V c V c ( x) f I ( y, t )I I ( y, t )dV ( y) = 1 V c in I ( x, t ) V c in I ( x,t ) f I ( y, t )dV ( y) (2.4.25)
Using the above averages for a field that is constant in each fluid subdomain, such as the density or the viscosity fields, we obtain: In order to write the differential equations involving only averaged field, it is essential to specify how the average of products of fields, such as ρu j f , are given as the products of the averaged fields. This is a well known problem when an simplification of averages of products are sought, and from this step, the derived terms (such as the Reynolds stresses) need to be modeled. For further details we refer the interested reader to Ishii (2011) [42, p.78,86-92]. A general enough procedure can be obtained if we define the following "error" operator:

ρ I ( x, t ) = ρ I C I ( x,
E ( f , h) = ρ f h -ρ f h (2.4.27)
and thus, we can write:

ρ f h = ρ f h + E ( f , h) (2.4.28)
Note that by applying the operator to a continuous constant field, f = c and the velocity field we obtain E (c, u i ) = 0, as a consequence of the hat operator definition, (2.4.23), for the velocity field. We will use the hat operator to decouple terms as ρu i and the error operator for products of the type ρ f u i , and express both using the velocity field u i .

Relations betweem Averages of Derivatives and Derivatives of Averages

To develop the equations for the averaged fields, we need to relate the averaged gradient field with the gradient of the averaged field and, similarly, for the time derivatives. The averaged gradient of a field, ∇ i f I , is (general chain rule for definite/indefinite integrals):

∇ i f I ( x, t ) = ∇ i f I ( x, t ) + J (I ) S J I f I ( y, t )n i ( y, t )K ( y; x)d S( y) (2.4.29)
and the time derivative is (Leibniz theorem):

∂ t f I ( x, t ) = ∂ t f I ( x, t ) - J (I ) S J I f I ( y, t ) n( y, t ) u( y, t )K ( y; x)d S( y) (2.4.30)
The sum in the above relations is defined over the subdomain indices J that are adjacent to the subdomain I (where the averaging operator is defined) and are symbolically denoted by J (I ). Moreover, the integrals in both relations are surface integrals over the interfaces that separate the subdomain I from the subdomains J . To clarify, we provide an example in the appendix, see p.234.

From this point on, we drop the function arguments to simplify the notation. Through the above relations, we obtain the following expressions for the gradient of the field f :

∇ i f = I ∇ i f I = ∇ i f - I J S J I f n i K d S (2.4.31)
and its time derivative:

∂ t f = I ∂ t f I = ∂ t f + I J S J I f n j u j K d S (2.4.32)
The sum in the RHS of these relations implies a summation over the different interfaces as viewed from the fluid subdomain I . The symbolic notation of the sum's index I J with the fluid's subdomain index I leading J , acts as a reminder that it uniquely enumerates and identifies the different interfaces, S J I , as viewed from the subdomain I . Therefore, the normal of the interface obtains an orientation and at the same time the jump operator is appropriately defined only once for the fluid subdomains that it connects (we give an example in the appendix, see p.234).

Relations (2.4.29), (2.4.30), (2.4.31) and (2.4.32) are derived in the appendix (see p.221) and similar relations can be also found along with their proofs in several references discussing multiphase modeling, see for example, [START_REF] Drew | Theory of Multicomponent Flow[END_REF] [21, p. 116-120], Prosperetti and Tryggvason (Eds) (2009), [85, p. 237-241]) and are also completely analogous to their counterparts when the averaging is defined over time (Ishii, 2011,[42, p. 78-82]). The second term in the RHS of the above derivative relations cannot be formally computed without an explicit interface representation. However, we can simplify them by introducing the characteristic interface value, as described next.

The Fields of Characteristic Interface Values By evaluating the derivatives of the volume fraction we arrive to interesting conclusions concerning the connections between fields that are intrinsic to the interface and the averaged fields. Since the volume fraction is derived as C I = 1 I , we obtain from relation (2.4.29):

∇ i 1 I = ∇ i 1 I + J (I ) S J I n i K d S ∇ i 1=0 ⇒ ∇ i C I = - J (I ) S J I n i K d S (2.4.33)
The last relation provides the connection between an entity that can be approximated if there existed an implicit interface representation, i.e. the sum of the integrals at the interfaces (the RHS), and the gradient of the volume fraction that can approximated algebraically, i.e. given the values of the volume fraction in a grid (the LHS). We note that relations as (2.4.33), and every relation that stems from the relations that connect the averages of derivatives to derivatives of averages (as (2.4.31)), can be viewed as derived by the definition of the surface Dirac-δ function and the sense of the Dirac-δ is recovered when → 0. This is due to the fact that we have already taken into account the definition of the surface Dirac-δ during the derivation of relations (2.4.31) and (2.4.32) which is described in the appendix (see p.221).

To simplify even further the relation, consider the case in which the support of the averaging kernel is small enough so that a single part of interface S J I is located inside the kernel's support. Denote this part as "S J I inV " (as in the case shown in figure C.3a for the subdomains 1 and 2, and in figure C.3c for the subdomains 1 and 3). Then the gradient of the volume fraction is:

∇ i C I = - S J I inV n i K d S (2.4.34)
which is immediately related to an integral that could be evaluated only if an explicit interface representation is available.

Furthermore, consider that the averaging is performed in the cells of a grid. In that case, the averaging kernel is the top-hat function, and the last relation becomes:

∇ i C I = - 1 V c S J I inV c n i d S (2.4.35)
which states that the gradient of the volume fraction is the vector area of the interface's region inside the support of the kernel divided by the volume of the cell. As a result, the norm of the volume fraction's gradient can be viewed as an indicator of the norm of the vector area of the interface S J I that is inside the averaging region. If the norm of volume fraction's gradient is small, it indicates that the vector area of the interface inside the averaging region is also small. Moreover, it will be zero only when the interface does not intersect the averaging region. Also from (2.4.35), we derive a frequently encountered formula for approximating the unit normal at the interface by the volume fraction:

n I i = - ∇ i C I |∇ i C I | (2.4.36)
These observations are true given that the following basic hypothesis holds: "The support of the averaging kernel is small enough so that a single part of interface S J I is located inside the kernel's support." In the following, we refer to the above hypothesis as the "basic hypothesis".

The case where the basic hypothesis holds is of rather particular importance because we arrive in elegant formulas for the interface integrals found in the RHS of the derivative relations. In order to simplify even further the derived relations we introduce the field of characteristic interface values, a field that "masks" surface integrals. Given a field f I at the interface S J I , we define the field of characteristic interface values at subdomain I by the tilde-I operator, f I ( x, t ), as the value at the point x that the following relation holds:

f I ( x, t )∇ i C I | ( x,t ) def = - S J I f I ( y, t )n i ( y, t )K ( y; x)d S( y) (2.4.37)
Given as an input the field f , the tilde-I operator assigns to a point x in the ambient space around the interface a value fI that is defined from the values of f at the interface as we approach it from the side of the subdomain I . Since we may write (we drop the arguments):

f I = - ∇C I | ∇C I | 2 S J I f I nK d S (2.4.38)
we recognize that the tilde-I value is an average interface value obtained from the values of the field f I that lies lies on the surface and inside the support of the kernel, shown in figure 2.3a. Whenever the characteristic value of the interface, f I , is defined then above relation removes the hustle of performing the integration of the RHS which requires an explicit interface representation. Note that for a continuous field we have f I = f J , whenever the support of the kernel is occupied by two fluids. The tilde-I operator is analogous to the hat-I operator but acts on interface values (recall that C I f I = f I ).

Consider now the general case, where the basic hypothesis does not hold. In that case, the norm of the volume fraction gradient is an indicator of the norm of the sum of the vector area of the interfaces that are inside the averaging region, as (2.4.33) suggests. We can also define in a similar manner the characteristic value of the interface by using a definition equivalent to (2.4.38):

f I ( x, t )∇ i C I | ( x,t ) def = - J (I ) S J I f I ( y, t )n i ( y, t )K ( y; x)d S( y) (2.4.39)
Through the above definition, the characteristic value might not be properly defined. Consider the case for example where the terms of the sum in the RHS of relation cancel out. Then, even when the interfaces intersect the averaging regions (so the integral term in the RHS is not zero), the gradient of the volume fraction can be zero and the characteristic value is left undefined. Also, in the general case, the definition does not imply that for a continuous field:

f I = f J .
By introducing the characteristic interface value the gradient relation (2.4.29) becomes:

∇ i f I = ∇ i f I -f I ∇ i C I (2.4.40)
To transform the last relation into a relation that contains the hat-I operator, replace the average field value f I using the definition of the hat-I operator, (2.4.22), f I = C I f I , to arrive to:

∇ i f I = ∇ i (C I f I ) -f I ∇ i C I ⇒ ∇ i f I = C I ∇ i f I + f I -f I ∇ i C I ⇒ ∇ i f I = ∇ i f I + f I -f I ∇ i C I C I (2.4.41)
The last relation shows that the center of mass value (in the subdomain of fluid I ) of a field's gradient will be almost everywhere the same as the gradient of the center of mass value (in the subdomain of fluid I ), except near the interface. In the proximity of the interface the last term cannot be considered to be zero in general.

When we defined the averaging operators at the ambient space, we first defined the subdomainspecific operators (bar-I , hat-I ) and then we defined the global averaging operators (bar, hat without an index). In the same manner, the tilde-I operator (the characteristic field value operator) represents

an averaging operator with respect to interface. The tilde-I operator assigns a value, fI , in the ambient space around the interface using the values of the field f we approach the side I of an interface S J I . Next we define the equivalent global operator which assigns a value, f (without an index), using the values of the field f we approach both sides of an interface S J I .

First, observe that the gradient of the average density is given by (2.4.31):

∇ i ρ = I J S J I ρ n i K d S (2.4.42)
and, thus we define the characteristic interface value as the value at the point x that the following relation holds:

f ( x, t )∇ i ρ| ( x,t ) def = I J S J I ρ f ( y, t ) n i ( y, t )K ( y; x)d S( y) (2.4.43)
Therefore, the tilde operator can be viewed as the equivalent definition of the hat operator for fields at the interface. As we have previously noted for the tilde I operator, its equivalent domain-independent definition, the tilde operator, assigns a value f to a point x around the interface. It acts on a given field f , but in contrast to the subdomain specific operator f I , it uses both field values on the interface to perform the calculation and weights them by the density. The integration region is depicted in figure 2.3b. For the case where the basic hypothesis holds and the field f is continuous, we obtain

f = f I = f J .
Working in the same way as before we arrive to a relation, equivalent to (2.4.41), for the Favre average of a field's gradient:

∇ i f = ∇ i f + f -f ∇ i ρ ρ (2.4.44)
The manipulations are given in the appendix.

The last terms in the RHS of relations (2.4.41) and (C. 1.11) are terms that complicate a purely "algebraic approach". In that case, an explicit representation of the interface's surface is not available and the gradients of the volume fraction, the density and viscosity are approximated by their values at the cell centers in a grid. However, the exact relations, (2.4.33) or (2.4.42) indicate that to evaluate exactly the values of these gradients (and similarly for the viscosity) an explicit representation of the interface is required. The smoothing procedure was introduced to avoid using the interface directly. However, the problem of generating a grid for each of fluid subdomain can be viewed as shifted to accurately evaluating terms as the above. In the remainder of this section, we develop the basic equations using the developed derivative formulas and thus, unavoidably, terms as such will be obtained. The arrow acts as a reminder that the result is assigned to the point x.

Deriving the Equations of the One-Fluid Formulation

The Volume Fraction Conservation Equation A relation that describes the evolution of the volume fraction is identically derived by evaluating the time derivative of the volume fraction, and the gradient of the averaged velocity field, as we descibe next. Beginning with the time derivative of the volume fraction we obtain by relation (2.4.30):

∂ t C I = J (I ) S J I n i u i K d S (2.4.45)
The last relation is a generalization of space conservation equation and is purely of geometric nature.

To demonstrate the last argument, replace the volume fraction by its definition (2.4.1) and consider the case where the averaging kernel is the top hat function and the basic hypothesis holds. Then we can write the last relation in a similar manner to (2.4.34):

1

V c ∂ t V c in I = 1 V c S J I inV c u i n i d S or ∂ t V c in I = S J I inV c u i n i d S (2.4.46)
This is an equation of geometrical nature that prescribes the rate of change of the volume of the subdomain I inside the support to the averaging kernel. We observe that the RHS of (2.4.45) is also

given by the divergence of the center of mass velocity field of the subdomain I , by applying (2.4.29):

∇ i u i I = ∇ i u i I + J (I ) S J I u i n i K d S ∇ i u i =0 ⇒ ∇ i u i I = - J (I ) S J I u i n i K d S (2.4.47)
By adding relations (2.4.45) and (2.4.47) we obtain:

∂ t C I + ∇ i u i I = 0 (2.4.22) ⇒ ∂ t C I + ∇ i ( u i I C I ) = 0 (2.4.48)
which is a space conservation equation for the part of the subdomain I contained inside the support of the averaging kernel. The last equation also relates the volume fraction to the divergence of the center of mass velocity field of the I -th fluid subdomain by:

∇ i u i I = - 1 C I ∂ t C I + u i I ∇ i C I (2.4.49)
The term inside the parenthesis in the RHS of the equation is not evident that is zero. The last equation expresses the volume dilation rate of the part of the I -th fluid subdomain that is inside the support of the averaging kernel, which is due to the interface movement.

To directly relate the interface velocity (as the characteristic interface velocity) to a transport equation for C I we apply (2.4.41) to the divergence of the hat-I velocity field, to obtain:

∇ i u i I = ∇ i u i I + u i I -u i I ∇ i C I C I ∇ i u i =0 ⇒ ∇ i u i I = -u i I -u i I ∇ i C I C I (2.4.50)
If we substitute the last relation to the LHS of (2.4.49) then we obtain:

-u i I -u i I ∇ i C I C I = - 1 C I ∂ t C I + u i I ∇ i C I ⇒ ∂ t C I + u i I ∇ i C I = 0 (2.4.51)
(the above can also be obtained directly by (2.4.45) and the definition of the tilde operator). The RHS can be recognized as the material derivative ( D I /D t ) based on the characteristic velocity of the interface as viewed from the I -th subdomain. If the basic hypothesis holds and only two fluid subdomains are present inside the averaging kernel, then the characteristic interface velocity of the I -th subdomain, u i I , is the same as the characteristic interface velocity of the J -th subdomain, u i J . Moreover, in that case, we immediately derive from the gradient of the averaged density, (2.4.42), and the definition of the characteristic interface value, (2.4.43), that

u i = u i I = u i J .
In conclusion, the volume fraction of either the I -th subdomain or the J -th subdomain at a given point always follows the movement of the interface, which is well defined by the characteristic interface velocity when the support of the kernel is small enough so that a single interface is present inside. The last observation deems the basic hypothesis as essential to ensure that as the volume fractions are transferred inside the whole computational domain, they retain their meaning as indicators of the presence of a fluid subdomain.

In other words, when the basic hypothesis holds and two fluid subdomains are inside the averaging kernel then equation (2.4.51) can be written in the simple form:

∂ t C I + u i ∇ i C I = 0 (2.4.52)
The last equation can be immediately related to the mass conservation equation for the one-fluid formulation. Since we study only incompressible flows, the total volume of each fluid subdomain must be conserved. As we will show next, the volume fraction transport equation and the mass conservation equation for the one-fluid formulation are not independent equations.

To develop the one-fluid equation, we begin by the classic mass and momentum equation and perform the averaging operators. From this procedure, we wish to produce equations that have the same formula as the classic mass and momentum equation but use the averaged variables instead the local field variables. The derivatives are treated using the formulas derived above. Thus we obtain terms that require an explicit representation of the interface's surface. For the LHS of both mass and momentum equation, these terms cancel (massless interface, p.33). However, from the treatment of the gradient of the stress tensor, these terms do not cancel, and surface tension naturally enters the formulation of the problem.

The Mass Conservation Equation of the One-Fluid Formulation

By averaging the continuity equation, we obtain the one-fluid mass conservation equation:

∂ t ρ + ∇ j (ρu j ) = 0 ⇒ ∂ t ρ + ∇ j (ρu j ) = 0 ⇒ ∂ t ρ + ∇ j (ρ u j ) + I J S J I ρ u j -ρu j n j K d S = 0 (2.4.53)
The velocity is continuous at the interface and therefore the third term is identically zero. The resulting equation has exactly the same form as the original equation,

∂ t ρ + ∇ j (ρ u j ) = 0 (2.
ρ I ∂ t C I + ∇ i ( u i I C I ) = 0 ⇒ ∂ t ρ I C I + ∇ i (ρ I C I u i I ) = 0 ⇒ ∂ t ρ I + ∇ i (ρ I u i I ) = 0 (2.4.55)
Adding the adove equations for each fluid subdomain, we obtain the one-fluid mass conservation equation (2.4.54):

I ∂ t ρ I + I ∇ i (ρ I u i I ) = 0 ⇒ ∂ t I ρ I + ∇ i ( I ρ I u i I ) = 0 ⇒ ∂ t ρ + ∇ i (ρ u j ) = 0 (2.4.56)
where to obtain the last equation we replaced I ρ I = ρ and I ρ I u i I = ρ u j (by definition of the hat operator).

By equation (2.4.54), the divergence of the center of mass velocity is given by:

∇ j u j = - 1 ρ ∂ t ρ + u j ∇ j ρ (2.4.57)
and it is not evident that the term inside the parenthesis in the RHS of the last equation is zero. If we recognize the term inside the parenthesis as the material derivative D/D t , based on the center of mass velocity field, then the divergence of the center of mass velocity field is related to the material derivative of the averaged density field by:

∇ j u j = - 1 ρ Dρ D t (2.4.58)
which resembles the classic equation relating the volume dilation rate to the density change of a particle in compressible flow:

∇ j u j = - 1 ρ Dρ D t (2.
∇ i u i = 0, since ∇ i u i = 0): ∇ i u i = -u i -u i ∇ i ρ ρ (2.4.60)
By replacing the last relation to (2.4.58) we arrive to:

u i -u i ∇ i ρ ρ = - 1 ρ Dρ D t ⇒ ∂ t ρ + u j ∇ j ρ = 0 (2.4.61)
If we recognize the term on the LHS as the material derivative D/D t based on the characteristic velocity of the interface, then the averaged density field is transported by the characteristic interface value of the velocity field:

Dρ D t = 0 (2.4.62)
When absurd changes in the interface topology are not expected then it reasonable to suppose that the relative velocity, u iu i , will be zero and thus, the center of mass velocity near the interface is the same as the characteristic interface velocity. The last assumption can also be considered as an equivalent way of enforcing the no-entry boundary condition between the ambient flow and the interface, without directly involving the interface. In that case:

Dρ D t = Dρ D t = 0 (2.

4.63)

There are two important points stemming from the above discussion. Firstly, if we consider that the assumption of a zero relative velocity, u iu i = 0, is valid, then the whole flow defined by u i velocity field behaves as a single incompressible fluid:

∇ i u i = 0 (2.4.64)
Secondly, if the support of the averaging kernel in small enough so that the basic hypothesis holds and only a single interface is inside the averaging kernel then the u i velocity field can replace the characteristic interface velocity u i in the volume fraction evolution equation (2.4.52):

∂ t C I + u i ∇ i C I = 0 (2.4.65)
and since ∇ i u i = 0 we obtain the volume fraction transport equation:

∂ t C I + ∇ i ( u i C I ) = 0 (2.4.66)
which is exactly the differential equation for the volume fraction solved by ISIS-CFD.

We note however that the assumption u iu i = 0 is a stronger condition that is related with the discussion of the momentum equation following next. As long as by the numerical solution algorithm we enforce ∇ i u i = 0, then from (2.4.60) we can see that the relative velocity vector u iu i is tangent to ∇ i ρ. If the basic hypothesis holds, then the vector ∇ i ρ is parallel to ∇ i C I . Since the vector ∇ i C I is directly related to the normal of the surface, then a crude interpretation of the condition "the relative u iu i is tangent to the ∇ i ρ" is that the relative velocity field u iu i is tangent to the normal of the surface. This argument stems from the theoretical formulation and is also confirmed by velocity fields obtained by the rising bubbles computations that are presented in chapter 4 (see p.179). Thus ( u iu i )∇ i C I = 0 suffices to obtain from (2.4.65) the volume fraction transport equation (2.4.66). This condition is also numerically enforced since the velocity field constructed to be solenoidal.

The Momentum Conservation Equation of the One-Fluid Formulation By applying the averaging operator to the momentum equation we derive the momentum equation for the averaged variables:

∂ t (ρu i ) + ∇ j (ρu i u j ) = ∇ j T i j + ρg i ⇒ ∂ t (ρ u i ) + ∇ j (ρu i u j ) = ∇ j T i j + ρg i - I J S J I T i j n j K d S (2.4.67)
The integral terms that are generated in the LHS by passing the averaging operator under the derivatives cancel, because the velocity field is continuous (as in the mass equation) and were directly omitted.

As we have already observed in the previous section, the traction jump T i j n j gives the force acting from the fluids to the interface and thus -T i j n j gives the force acting from the interface to the fluids that occupy the ambient space. Therefore, the last term in (2.4.67) can be recognized as the surface tension force acting on the ambient space from the interface (see also p.234). Moreover, due relation (2.3.24), repeated here:

T i j n j = -n i σB α α -Z i α ∇ α σ
we can see that the term depends solely on the intrinsic geometry of the interface or interfaces and the surface tension coefficient. We refer to this term as the surface tension source term Σ whose components are:

Σ i = I J S J I -T i j n j K d S (2.4.68)
Next, we express all terms using the Favre averaged velocity field u i . To write the convective term as a product of averaged fields we use "error" operator, (2.4.27):

∇ j (ρu i u j ) = ∇ j (ρ u i u j ) + ∇ j E (u i , u j ) (2.4.69)
Suppose that we are working in an affine coordinate system in order to simplify the derivation, then the averaged stress tensor can be written as:

T i j = -p Z i j + 2µS i j (2.4.70)
In order to rewrite the average of product µS i j as a product of averages using the error operator, we express the dynamic viscosity by the kinematic viscosity to obtain:

µS i j = ρνS i j = ρ ν S i j + E (ν, S i j ) (2.4.71)
Note that by the definition of the hat operator,(2.4.23), the center of mass kinematic viscosity, ν, is derived by the same fraction as the kinematic viscosity with the averaged variables replacing the local variables:

ν = I ρ I ν I ρ = I ρ I ν I C I ρ = I µ I C I ρ = µ ρ (2.4.72)
Therefore we obtain:

µS i j = µ S i j + E (ν, S i j ) (2.4.73)
And the averaged stress tensor is:

T i j = -p Z i j + 2µ S i j + 2E (ν, S i j ) (2.4.74)
Moving on to the term S i j , by the definition of the stress tensor, we obtain (for an affine coordinate system):

2 S i j = Z i k ∇ k u j + Z j k ∇ k u i (2.4.75)
and by relation (C.1.11) we obtain for the hat velocity gradient:

∇ k u j = ∇ k u j + u j -u j ∇ k ρ ρ (2.4.76)
Substituting the above in (2.4.75), and the resulting expression to the averaged stress tensor results to the final expressing for the stress tensor:

T i j = -p Z i j + µ Z i k ∇ k u j + Z j k ∇ k u i + ν Z i k u j -u j + Z j k u i -u i ∇ k ρ + 2E (ν, S i j ) (2.4.77)
The first term is the average pressure that "replaces" the local pressure for the one-fluid formulation, the second term is the viscous stress tensor of the bulk flow for the one-fluid formulation, τ i j :

τ i j = µ Z i k ∇ k u j + Z j k ∇ k u i (2.4.78)
and the third term is zero if the zero relative velocity assumption holds. Taking the above into account, we arrive to the following form for the one-fluid momentum equation:

∂ t (ρ u i ) + ∇ j (ρ u i u j ) = -Z i j ∇ j p + ∇ j τ i j + ρg i - I J S J I T i j n j K d S + ∇ j E i j (2.4.79)
where we have gathered the approximation errors derived by expressing averages of products as products of averages in a single term E i j = -E (u i , u j ) + 2E (ν, S i j ), whose gradient is considered to be negligible. Approximate Differential Equations Under the assumption that the length of the averaging kernel is small enough and that the relative velocity u iu i is negligible in the proximity of the interface, the incompressible Navier-Stokes volume averaged equation must hold in each fluid flow region:

Summary and Concluding Remarks

∇ i u i = 0 ∂ t (ρ u i ) + ∇ j (ρ u i u j ) = -Z i j ∇ j p + ∇ j τ i j + ρg i - I J S J I T i j n j K d S (2.4.80)
where

ρ( x, t ) = I ρ I C I ( x, t ) , µ( x, t ) = I µ I C I ( x, t ) and τ i j = µ Z i k ∇ k u j + Z j k ∇ k u i

Approximate Kinematic Boundary Condition for the Interface

The zero jump condition of the local velocity field is implicitly taken into account by the vanishing interface integral in the LHS of the averaged mass and momentum equation. Each interface is transported in the compuational domain by:

∂ t C I + ∇ i C I u i = 0 (2.4.81)
The above relation is valid as long as from u iu i ∇ i ρ = 0 we can deduce that u iu i ∇ i C I = 0, i.e. when the characteristic length scale of the interface is small enough so that a single interface is found inside the support of the kernel.

Dynamic Boundary Condition for the Interface

The jump of the stress tensor is implicitly taken into account by the source term: An important aspect of the volume fraction as employed in this work is that we track the surface of the interface via the volume fraction. At least formally, the surface is not allowed to breakup into smaller parts whose length scales are smaller thanthe averaging parameter . The following transport equation is proposed:

Σ i = - I J S J I T i j n j K d S = I J S J I Z i α ∇ α σ + n i σB α α K d S (2.
∂ t C I + ∇ i u i C I = 0
The support of the averaging kernel must be small enough to ensure that a single interface rests inside its support. This is the basic hypothesis that differentiates the formulation from a multiphase modeling formulation. The above relation implies that mass is conserved during the displacement of the subdomains as long as ∇ i u i = 0, a condition that enforces the relative velocity field u iu i to be tangent to the interface. We also require that the volume fraction remains sharp enough to provide an indicator of the interface location and sufficiently smooth to allow the calculation of its derivatives.

In both multifluid and multiphase formulations, the equations are written using the averaged fields.

The difference is that for a multiphase modeling formulation, the length scales of the interfaces are smaller than the grid size, so the interactions of the sub-scale phenomena with the main flow need to be modeled. In such cases, the volume fraction is not related to a single surface of the interface. The purpose of multiphase flow models is that the flow conditions are such that one of the fluids can be considered as dispersed inside another fluid. If all the interfaces were to be separately tracked then, the solution could not be produced in reasonable time by today's computers. In that case, the details are discussed in length by Ishii [START_REF] Ishii | Thermo-Fluid Dynamics of Two-Phase Flow[END_REF] and Drew [START_REF] Drew | Theory of Multicomponent Flow[END_REF]. In ISIS-CFD, we are interested in resolving free-surface flows, i.e. water-air flows by capturing as accurately as possible free-surface dynamics and we suppose that the topological changes of the free-surface would never produce interfaces that cannot be captured by the current grid length scale of the computational grid. Besides the hypothesis that, throughout the whole simulation, the averaging is performed in small enough regions, there are two other assumptions that produce the last system of approximate equations.

The first assumption is that the error term derived by expressing averages of products as products of averages E i j = E (u i , u j ) -2E (ν, S i j ) is considered to be zero. In other formulations (as given by Ishii [42, p.103-104]), the term E (ν, S i j ) is not present. Instead, the averaged viscous stress term is written as:

2µS i j = I µ I 2 S i j I = I µ I Z i k ∇ k u j I + Z j k ∇ k u i I (2.4.83)
and using relation (2.4.41) we obtain:

2µS i j = I µ I Z i k ∇ k u j I + Z j k ∇ k u i I + .
..other terms containing the characteristic interface velocities.

(2.4.84)

In that case the viscous stress term is not the same as the modified viscous stress term. However, in the one-fluid formulation we do not separately solve for the two velocity fields u j I and u j J so they are not available.

The second assumption is that relative velocity u iu i is negligible. As we will show in the following section, this assumption can be considered as valid for air-water flows for the water side. In this case, the significant differences in the density make the averaged momentum equation "favor" water. Indeed in that case and almost everywhere in the proximity of the interface, we have ρ ≈ ρ I , µ ≈ µ I and u ≈ u I .

"Almost everywhere" because in locations where the air volume fraction obtains large values the above approximations do not hold. Therefore, the averaged momentum equation without the surface tension term is almost every everywhere the same as the classic momentum equation for water. The effects of the above simplifying assumptions, on the solutions obtained for a simple multifluid problem by the one-fluid equations, are studied in the next section.

Practical Considerations

In practice, if we consider that the relative velocity u iu i might not be negligible, then we have to invent an approach to obtain the velocity field u i . This task is not straightforward since the velocity field u i is evaluated from the local velocity field of the interface, which is not available. A possible workaround is to include a model of a fictitious jump to the solution. As we have noted, by:

∇ j u j = 0 (2.4.85)
we enforce a no-entry condition to the interface. Thus we may argue that the velocity u iu i is tangential to the interface:

u i -u i = Z i γ v γ (2.4.86)
where Z i γ v γ is the slip velocity. This slip velocity should be acting to the air side of the interface where the equations favor water due to its significantly larger density. As a result, we will enforce a continuity condition to the interface in the following sense. First of all, we note that when we are outside the interface's influence, the solution from the air subdomain should be the same as the solution for the classic Navier-Stokes formulation. In turn, we expect that (by choosing a kernel with a small support) the interface velocity should be obtained by two manners. Firstly, the interface velocity should result if we extend the solution from the air subdomain towards the interface (an extrapolation-like step).

Secondly, it should result from the solution obtained from the side of the water subdomain (including the interface). Finally, we argue that the two approaches should provide identical results. The last condition enforces matching solutions for the velocity of the water side of the interface (as obtained by solving the one-fluid momentum equations) and the extrapolated solution from the air side to the interface (using the values of the velocity field near the interface but where C J , the volume fraction of air, is one). In the next section, we will show that by enforcing this condition we may improve the results obtained by the analytical solution of the one-fluid model for the plane Couette flow.

It is important to emphasize that the derived equations can be considered as equations for the local fields only in the sense of a small kernel support. In ISIS-CFD, the sense of a kernel is not directly introduced. Instead, the volume fraction varies sharply within two or three grid length scales around the interface. The variation of the volume fraction is governed by two different elements of the solution procedure. The first is the numerical schemes used to solve the transport equation of the volume fraction, which "compress" the volume fraction around the interface. The second is the addition of automatic grid refinement near the interface (or automatic grid adaptation based on the interface).

After the adaptation technique, the volume fraction is redefined so that it conforms to the requirement that it should vary from two to three grid length scales. Therefore, using the terms of this section, there is an "implied" kernel (since otherwise, in the context of this work, the volume fraction can not be defined) and this kernel is also manipulated either numerically or by AGR. Both elements tend to make the support of this implied kernel as small as possible for the whole course of the simulation. The sharp variation of the volume fraction poses difficulties when surface tension has to be included in the calculation.

Specifically for surface tension, several numerical schemes found in the literature, are based on calculating (second order) derivatives of the volume fraction, such as methods derived by the approach of [START_REF] Brackbill | A continuum method for modeling surface tension[END_REF] [10]. In this case, large variations of the volume fraction do not enable us to produce accurate approximations for either the normal vector or the curvature of the interface. In this work, actual kernels (in contrast to implied kernels) are used to build the source term obtained by the surface tension force as calculated by the reconstructed interface (we introduce a reconstruction procedure of the interface by the volume fraction in the following chapter) and distribute it to the surrounding space around the interface (via the tilde operator as we will demonstrate in the last chapter).

Comparison of Analytical Solutions of the Classic and the

One-fluid Formulation

Introduction In this section, we compare the solutions obtained by solving two simple multifluid problems, using the local field equations and the one-fluid formulation. The first problem is that of a static bubble. In this problem we demonstrate the effect of the surface tension source term, given by (2.4.82):

Σ i = - I J S J I T i j n j K d S = I J S J I Z i α ∇ α σ + n i σB α α K d S
in the resulting equation for the pressure. For the second problem, we recast the standard plane Couette flow as a multifluid problem. We will show that the zero relative velocity assumption is invalid in this case and propose a simple correction for the problem. Before developing each problem, we start with some basic observations concerning the volume fraction fields for some cases that can be analytically evaluated. These volume fraction fields will be used to generate the analytical solution of the test cases studied. We provide details related to the derivations presented here in the appendix (p.237).

Analytical Volume Fraction Fields

The volume fraction is an important intermediate field that defines the points that are near the interface.

It can be viewed as a smooth indicator function, even though it is not an indicator function since the presence of the interface is specified by a range of values (the points where the volume fraction is between [0, 1]). We should note that the volume fraction depends, by definition, on the averaging kernel we choose. In the following examples we will use the analytical expression obtained by the definition of the volume fraction, (2.4.13):

C I ( x, t ) = I I ( y, t )K ( y; x)dV ( y)
in the proximity of a planar and a spherical interface. For the planar interface, we use the kernels:

K ( y; x) = 1 8 3 , y -x ∞ < 0 , otherwise (2.5.1)
where |•| ∞ is the maximum norm and we refer to this kernel as the "top-hat in-cube kernel" since its support is a cube whose side is 2 , and:

K ( y; x) = 3 4 π 3 , y -x < 0 , otherwise (2.5.2)
and we refer to this kernel as the "top-hat in-sphere" since its support is a sphere whose diameter is 2e. For the spherical interface, whose radius is R, we provide the result obtained for the top-hat insphere kernel. The relations are produced by evaluating simple volume regions that can be calculated analytically and we do not give the proofs.

Table 2.4 summarizes the volume fractions of the J subdomain, C J , and C I is obtained by

C I = 1-C J .
The function's argument is the dimensionless (signed) distance of the centroid of the kernel's support say x to the closest point x p at the interface l = d / = ( xx p ) n J I / , l is positive when x is inside the region J and negative when it is inside the region I . Note that the volume fraction obtained for the spherical interface,(2.5.5), also depends on the sphere's dimensionless radius r = R/ , a parameter of the problem. Specifically, when r → ∞, relation (2.5.5) is the same as (2.5.4). spherical support. Since the top-hat in-sphere kernel is symmetric, the result does not depend on the orientation of the interface. However, for the top-hat in-cube kernel, the volume fraction depends on the relative orientation of the interface and the support. The general case is treated in [START_REF] Tryggvason | Direct Numerical Simulations of Gas-Liquid Multiphase Flows[END_REF] (see [105, p.284-287], first published by Scardovelli and Zaleski [START_REF] Scardovelli | Analytical Relations Connecting Linear Interfaces and Volume Fractions in Rectangular Grids[END_REF]) and we will return to it in the next chapter.

C J (l ) =        0 , l -1 1 2 + l 2 , - 1
1 , 1 l (2.5.3) where l = d / d 2 J I C J (l ) =        0 , l -1 1 2 + 3 4 l - 1 4 l 3 = 1 4 (l + 1) 2 (2 -l ) , -1 l 1 1 , 1 l (2.5.4) where l = d / d 2 J I C J (l ; r ) =        0 , l -1 1 4 (l + 1) 2 (2 -l ) + 3 16 (l + 1) 2 (l -1) 2 l + r , -1 l 1 1 , 1 l (2.
The same calculation was performed for a spherical interface using the top-hat in-sphere kernel and the results obtained are given in figures 2.5e and 2.5f.

Secondly, the value of the volume fraction at the interface (the volume fraction values obtained for l = 0) is, in general, not 0.5, as shown figure 2.5e. However, for the smaller value of , figure 2.5f, this difference is smaller. As a result, a reasonable approximation is that the volume fraction obtains the value 0.5 at the interface. In the next chapter, we use this approximation as the basis to reconstruct the surface of the interface using a marching cube algorithm. gives the basic nomenclature for each case. We plot the volume fractions in figures 2.5b, 2.5c for two values of the averaging length . The x-axis corresponds to the volume fraction values, C I and C J and the y-axis to the distance from the interface d . Figure 2.5d schematically represent the support of a spherical kernel near a spherical interface. We plot the volume fraction in figures 2.5e,2.5f for a sphere of radius, R = 1. Dashed lines were produced using the top-hat in-sphere kernel.

Static Bubble

Classic Formulation The static bubble problem is a very simple application that demonstrates the effects of surface tension at pressure. The interface separating fluid I and fluid J , is a sphere of radius R (fluid I is inside the bubble). We consider that the transient terms drop, the velocity field is zero everywhere and gravity is considered zero. The classic multifluid problem or the local formulation (see Differential Equations and Boundary Conditions for the Multifluid Problem, p.38) is:

I subdomain : Z i j ∇ j p I = 0 J subdomain : Z i j ∇ j p J = 0 Interface : p = σB α α External : p J | ∂D = 0 (2.5.6)
Suppose that we are working on a spherical coordinate system. Then the pressure obtains constant solutions for either subdomain, p I (θ, φ, ρ) = c I , p J (θ, φ, ρ) = c J (in the following we drop the arguments of the functions). Due to the external boundary condition c J is zero and c I is obtained by the interface boundary condition:

p = σB α α ⇒ p J (ρ = R) -p I (ρ = R) = σB α α ⇒ c J -c I = σB α α ⇒ c I = -σB α α (2.5.7)
The mixed curvature tensor of the sphere is (the intrinsic normal points outwards with respect to the region enclosed by the sphere):

B β α = - 1 R δ β α
and the curvature is B α α = -2 R (note that when we refer to curvature we actually refer to "twice the mean curvature"). Therefore, the pressure in the I subdomain is:

p I = 2σ R (2.5.8)
The result can be elegantly represented using the classic Heaviside function as the indicator function, by (2.4.18):

p(r ) = 2σ R H (R -ρ) (2.5.9)
By solving the classic multifluid problem we obtain local fields, as opposed to the fields we will obtain next by using the one-fluid formulation through which we obtain averaged fields. Since the pressure is constant for both fluid subdomains the averaged pressure is given by a formula similar to the formulas of averaged density and the averaged viscosity (which are also constant, see (2.4.26)):

p = I p I C I (2.5.9) ⇒ p = 2σ R C I (2.5.10)
In the following we will show that the above solution is obtained by the solving the one-fluid Navier-Stokes equations.

One-Fluid Formulation

The one-fluid formulation replaces the two equations of the problem (2.5.6)

by the problem (see Differential Equations and Boundary Conditions for the One-Fluid Formulation, p.57):

Any point in D : Z i j ∇ j p = -

S J I T i j n j K d S External : p| ∂D = 0 (2.5.11)
The two differential equations are replaced by a single differential equation. The source term in the momentum equation accounts for the surface tension effects. For a point in either fluid subdomain far from the interface, where the volume fraction is either zero or one, the source term is zero since the support of the kernel does not meet the interface. For every other point where C I ∈ (0, 1) the source term is not zero. Suppose that ρ is the distance from the sphere's center, then in view of the analytical relations provided for the volume fraction we can write C I ∈ (0, 1) : ρ ∈ (0, R + ) and

C J ∈ (0, 1) : ρ ∈ (R -, ρ| ∂D )
where ρ| ∂D is the distance of a point in the external boundary to the center of the sphere. The source term can be written as:

Σ i = - S J I T i j n j K d S = S J I σB α α n i K d S (2.5.12)
Since the surface tension coefficient and the curvature are constants we obtain:

Σ i = σB α α S J I n i K d S (2.5.13)
The final result, expressed using the dimensionless lengths r = R/ and l = d / = (z -R)/ (where z is the distance from the center of the bubble), is:

Σ 1 = 0 , Σ 2 = 0 , Σ 3 (l ) = - 3σ 8 R 1 -l 2 (l + 2r -1)(l + 2r + 1) (l + r ) 2
(2.5.14)

The source term in the RHS is plotted in figure 2.6(left) for different values of . As the averaging parameter becomes smaller (the support of the source term becomes smaller), the maximum value of the source term becomes larger. The above procedure, actually constructs a Dirac-δ sequence of functions (a "sequence" with respect to the parameter ) concentrated at the surface of the interface. In the limit of → 0 the source term becomes a surface Dirac-δ function for the interface.

So we can solve the equation for pressure to obtain:

p =            2σ R , z ∈ (0, R -) 2σ R (l -1) 2 l 2 + 2l + 4 r l -3 + 8r 16(l + r ) , l = z -R , z ∈ [R -, R + ] 0 , z ∈ (R + , ρ| ∂D ) (2.5.15)
The averaged pressure field is plotted in figure 2.6 (right) for different values of . As the averaging parameter becomes smaller (the region where the averaged pressure field varies becomes smaller) and the value of the averaged pressure in either water or air region remains the same. The above procedure actually constructs a Heaviside sequence of functions (a sequence for the parameter ) concentrated at the surface of the interface. In the limit → 0 the term: Comments For this simple case, the source term can be written in terms of the volume fraction gradient. Observe that in relation (2.5.13) (derived during the evaluation of the source term):

(l -1) 2 l 2 + 2l + 4 r l -3 + 8r 16(l + r ) ( 2
Σ i = σB α α S J I n i K d S
the integral in the RHS is exactly the gradient of the volume fraction marking the I region. This is clear by relation (2.4.34) (derived by applying (2.4.33) when the basic hypothesis holds) repeated here for convenience:

∇ i C I = - S J I inV n i K d S
Therefore in this case the source term can be written as Σ i = -σB α α Z i j ∇ j C I and the the pressure gradient as:

Z i j ∇ j p = -σB α α Z i j ∇ j C I ⇒ ∇ j p = 2σ R ∇ j C I (2.5.18)
The last relation can be integrated to get relation (2.5.17). We can directly check in practice the validity of the expression of the source term Σ i = -σB α α Z i j ∇ j C I , used to obtain (2.5.18). Since C I = 1 -C J we have ∇ i C I = -∇ i C J , and by the volume fraction relation given by (2.5.5) we obtain:

∇ 1 C I = 0 , ∇ 2 C I = 0 , ∇ 3 C I = - 3 16 1 -l 2 (l + 2r -1)(l + 2r + 1) (l + r ) 2 (2.5.19)
To form the gradient of the averaged pressure we multiply the result by 2σ/R to arrive to the same expression as (2.5.14) that was derived by directly evaluating the surface tension integral (2.5.13).

The static bubble case nicely demonstrates some simple concepts developed in the previous paragraph. To summarize, first of all, the analytical solution of the classic two-fluid Navier-Stokes formulation coincides to the one-fluid formulation in the classic sense far from the interface, and in the generalized sense (as the averaging length tends to zero) near the interface. Secondly, the analysis demonstrated that the gradient of the volume fraction and the normal of the interface are indeed related by:

∇ i C I = - S J I inV n i K d S
This is an interesting observation since the methods used to evaluate surface tension numerically, are discretizing similar relations. We will return to the discretization problem in a subsequent chapter and for now, we move to the second problem we study in this section. x-axis:

Plane Couette Flow with Two Fluids

µ I d 2 u I d y 2 = 0 J subdomain y-axis: d p J d y = -ρ J g x-axis: µ J d 2 u J d y 2 = 0 Interface : u = 0 : p = 0 : µ d u d y = 0 External -Walls : u(y = h I ) = 0 u(y = h J ) = V (2.5.20)
From the y-axis momentum equations we obtain the local pressure field:

p I (y) = -ρ I g y p J (y) = -ρ J g y (2.5.21)
Integrating the x-axis equations we obtain the following solutions for the velocity:

u I (y) = V µ J h J µ I -h I µ J y + V int u J (y) = V µ I h J µ I -h I µ J y + V int (2.5.22)
where V int is the velocity of the interface:

V int = - V h I µ J h J µ I -h I µ J (2.5.23)
The fields obtained by the solution of the classic multifluid problem are the local fields, in the sense that their values refer to the value that one would find if we were able to perform a measurement in a specific point in the flow. We plot the resulting fields in figures 2.8a and 2.8b (black lines).

Averaged Fields Obtained by the Local Fields

Using the local pressure and the local velocity field obtained we can calculate directly the averaged fields. By applying the bar operator (2.4.21) (which produces an averaged field) we obtain for a two fluid case:

f = f I + f J (2.5.24)
Since the fields we are dealing with in this example are all linear, i. Thus the difference in the fluid densities is expressed by the u field as a fictitious gradient discontinuity, and the field u is not the same or even close to the local velocity value of air. We should also note that the necessary assumption that the relative velocity u iu i is zero, approximately holds in the proximity of the interface and close to the subdomain of the heavier fluid.

Evaluation of the Basic Assumptions of the One-Fluid Formulation In this case, the characteristic interface velocity is the velocity of the interface, u = V int (derived by direct calculation using the definition of the characteristic interface velocity, (2.4.43)). As long as we are close enough to the interface, the center of mass velocity will be close to the local velocity field. Since the density of water (fluid I ) is much larger than the density of air (fluid J ) the center of mass velocity is primarily derived from the velocity field u I :

u = ρ I u I + ρ J u J ρ I C I + ρ J C J ρ I >>ρ J ≈ ρ I u I ρ I C I = u I (2.5.25)
that can be evaluated using relations, (C.2.14) and (C.2.16):

u ≈ u I = V int + a I (l -1)(3 + l ) 4(2 + l ) (2.5.26)
From the last relation we obtain the following expression for the relative velocity: And since:

u -u = u -V int ≈ a I (l -1)(3 + l ) 4(2 + l ) (2.
a I = V µ J h J µ I -h I µ J = - V int h I (2.5.28)
we have:

u -u V int = u -u u ≈ h I (1 -l )(3 + l ) 4(2 + l ) (2.5.29)
However, the second fraction in the last expression is a decreasing function of l ∈ [-1, 1] whose image is bounded in the interval, [0, 1] and thus we can readily obtain the bounds of the LHS (recall that h I < 0):

h I u -u u 0 (2.5.30)
In conclusion, the term that we considered as negligible indeed obtains small values in the proximity of the interface as long as the C J values are small enough to obtain u ≈ u I . Moreover, the smaller the value of the parameter , the smaller is the error introduced by neglecting the relative velocity term in the one-fluid formulation. The last result would be found valid, if we had calculated the term in the LHS of (2.5.29) without introducing an approximation as ρ I >> ρ J . For that case we obtain the values ( uu)/ u given in figure (2.9), plotted against the volume fraction C I .

As we move away from the interface and inside the subdomain J (air, C I < 0.5) the hypothesis that the relative velocity uu is small eventually fails. Compare for example, the values where the volume fraction is close to one and the values where the volume fraction is zero. For the former, the value of the term ( uu)/ u approaches its previously derived lower limit /h I . For the latter, as indicated in the same figure, the term ( uu)/ u becomes much larger than /h I and the hypothesis that the relative velocity uu is negligible, fails. In other words, it is in this region that the one-fluid equations would produce the most erroneous result. However, in any case, as the averaging parameter becomes The effects of the relative velocity term are of great importance, since it enters the averaged momentum conservation. Another term that was neglected was the gradient of the error term, introduced by writing the average of the viscous stresses as a product of the average viscosity and the Favre averaged strain tensor. For the case studied, this term is 2E (ν, S 12 ). This term is also non negligible in comparison to the actual averaged viscous stress term, 2µ S 12 , which is constant for the plane Couette flow:

2µ S 12 = 2 µ I S 12 I + µ J S 12 J = V µ I µ J h J µ I -h I µ J (2.5.31)
This is demonstrated in figure 2.10, where we plot the ratio of the error over the exact averaged viscous stresses:

E (ν, S 12 ) µ S 12
(2.5.32) obtained using the local velocity fields.

Therefore, the error term becomes quite important specifically for the air subdomain where it is more than 50% of the exact averaged viscous stresses. By completely neglecting this term we expect errors to be formed eventually everywhere but more importantly to the values obtained for the air subdomain. In the next paragraph, we will obtain the solution of the plane Couette via the one-fluid formulation, and we will compare the derived fields with the field arising from the classic formulation relations. As we discuss next, the larger errors due to the neglected terms will be introduced in the air subdomain, and we will propose a simple heuristic workaround for the solution of the one-fluid equations.

u(y = h I ) = 0 u(y = h J ) = V (2.5.33)
The averaged density properties are:

ρ(y) = ρ I C I (y) + ρ J C J (y) and µ(y) = µ I C I (y) + µ J C J (y)
Beginning with the pressure, we can easily find the resulting averaged pressure field by solving the y-axis momentum equation:

p(l ) =        -ρ I g l , l -1 -g 16 (ρ I -ρ J )(l 4 -6l 2 -3) + l ρ I + ρ J 2 , -1 l 1 -ρ J g l , 1 l (2.5.34)
which is same relation that we would have obtained by evaluating the averaged pressure by p = p I + p J with the fields p I and p J resulting from relations, (C. We obtain the Favre averaged velocity by integrating the x-axis momentum equation:

u(y) =          a I y + b I , y - a y 1 µ(y ) d y + b , - y a J y + b J , y (2.5.35)
The relative relations for the top-hat in-sphere kernel become quite complicated due to the fact that a cubic polynomial appears in the denominator of the above integral (the second part of the u function).

To simplify the resulting relations we provide the results using the top-hat in-cube kernel instead the kernel with spherical support. Changing the kernel does not affect the derived conclusions, but the resulting relations become tractable. In this case, the averaged viscosity is:

µ(y) = µ I + µ J 2 + µ J -µ I 2 y (2.5.36)
and the required integral is:

y 1 µ(y ) d y = 2 µ J -µ I ln µ (2.5.37)
The constants of integration can be determined by the boundary conditions and enforcing continuity for the different parts of the Favre averaged velocity field and, as a first approach, the values of the function's first derivatives. Even though it is evident from figure 2.8b that the first derivatives of the u parts do not match, it is a straightforward approach to determine the constants of integration. We refer to this solution as the C 1 continuous solution. In that case the constants of integration are:

a I = V µ J (µ J -µ I ) D , b I = V h I µ J (µ I -µ J ) D a = V µ I µ J (µ J -µ I ) D , b = V µ J [h I (µ I -µ J ) -(µ J -µ I + 2 µ I ln µ I )] D a J = V µ I (µ J -µ I ) D , b J = V h I µ J (µ I -µ J ) + µ 2 I -µ 2 J -2 µ I µ J log µ I µ J D
where:

D = (h J µ I -h I µ J )(µ J -µ I ) + µ 2 I -µ 2 J -2 µ I µ J ln µ I µ J
The constants, a I , b I , a J and b J are exactly the same as the constants of integration obtained for the local velocity fields (2.5.22), when = 0. This is an expected result since as we have already discussed the term uu tends to zero as tends to zero and the local velocity fields are recovered by solving the one-fluid equations. This is also depicted in figure 2.11 where the u is plotted for different values of . The smaller the value of , the closer the solution is to the local velocity field. However, the local velocity fields in either fluid subdomain will never be the same as the u field far from the interface and the larger errors are found in the subdomain of air. The reason behind the large difference between the local velocity field and the Favre averaged velocity field for air, can be explained by the following two facts. Firstly, the dominant terms in the one-fluid equations for an air-water flow are the terms of water.

Their effects to the solution is shown in figure 2.11. Even though the solution obtained by the one-fluid equation will never be the same to the local solution for water, it is much closer to the local velocity of water in comparison to the local velocity field of air. Secondly, the solution was derived by enforcing C 1 smoothness. Returning to figure 2.8b, we may visually confirm that the u field is not C 1 smooth.

If we perform the derivative calculations we would obtain the same result for both fluid subdomains.

However, by enforcing C 1 smoothness we implicitly enforce the zero stress jump condition at the interface:

µ d u d y = 0 (2.5.38)
as:

µ J d u d y y= + -µ I d u d y y=-- = 0 (2.5.39)
(the notation + emphasizes that the evaluation is performed as we approach from larger values) which holds since for any point we directly obtain from the x-axis momentum equation of the one-fluid problem:

µ d u d y = a (2.5.40)
Therefore, the C 1 smoothness can be considered as a reasonable requirement for the solution which is physically meaningful. Indeed, the source of error lies in the term, uu, that has been considered as small and neglected. On the other hand, to add the term that depends on the relative velocity uu to the equations, we must know beforehand the interface velocity (in order to evaluate the term u), which is not known. Thus, a two step procedure would be necessary, one step to obtain an approximation of the interface velocity and a second step to solve the one-fluid equation with the term that contains uu. A simpler approach is to acknowledge that the effect of the relative velocity term is to enforce the validity of the zero velocity jump condition at the interface. Simultaneously, we should take into account that it is due to the large density difference between the two fluids that the u field tends very fast to the local field value near the interface from the side of the lighter fluid (subdomain J ).

A Heuristic Correction to the One-Fluid Formulation

u(y = + ) - d u d y y= + = V int
Here the interface velocity V i nt is not known but it can be approximated by the value of the Favre averaged velocity field at the interface, and the above relation becomes:

u(y = + ) - d u d y y= + = u(y = 0) (2.5.41)
Relation (2.5.41) replaces the condition that enforces continuity of the parts of the function u at y = + .

In that case the constants of integration are determined by the following conditions:

External boundary conditions: Prescribed velocity at external boundaries:

u(y = h I ) = 0 ⇒ a I h I + b I = 0 u(y = h J ) = V ⇒ a J h J + b J = 0
Partial C 0 continuity: Continuous solution at the water subdomain:

u(y = --) = u(y = -+ ) ⇒ -a I + b I = a 2 µ J -µ I ln µ I + b
Matching solutions Interface-Air: Continuous velocity at the interface by matching the solution of the one-fluid equation and the extrapolated value from the air subdomain,(2.5.41). This condition replaces the condition enforcing a continuous solution in the air subdomain, relation (2.5.41):

u(y = + ) - d u d y y= + = u(y = 0) ⇒ b J = a 2 µ J -µ I ln µ I + µ J 2 + b
C 1 continuity: Continuous first derivatives at water and air subdomains:

d u d y y=-+ = d u d y y=-- ⇒ a I = a 1 µ I d u d y y= + = d u d y y= - ⇒ a J = a 1 µ J
The solution of the linear system for the integration constants gives:

a I = V µ J (µ J -µ I ) D , b I = V h I µ J (µ I -µ J ) D a = V µ I µ J (µ J -µ I ) D , b = V µ J [h I (µ I -µ J ) -(µ J -µ I + 2 µ I ln µ I )] D a J = V µ I (µ J -µ I ) D , b J = V µ J h I (µ I -µ J ) + µ I -µ J + 2 µ I ln 1 2 + µ J 2 µ I D
where:

D = (h J µ I -h I µ J )(µ J -µ I ) + µ I µ J -µ 2 J + 2 µ I µ J ln 1 2 + µ J 2 µ I
In figure 2.12, we compare the two solutions. Observe that the second method produces a discontinuous solution. The weaker continuity properties imposed to the solution enabled us to obtain an improved solution both fluid subdomains through the following two physical assumptions:

• The velocity of the interface is approximately obtained by the u evaluated at the interface from the side of water

• The local velocity field of air match the interface velocity when the solution is extended from the air subdomain to the interface, thus enforcing in a certain sense u = 0 at the interface.

. Observe that even though the solution is not continuous the derived u field in both the water and air subdomain is much closer to the local velocity field, black line in figure 2.12, in comparison to the C 1 continuous solution. For both solutions, the relative errors of the local velocity fields tend to zero as tends to zero, and in both solutions the larger errors are observed in the region near the interface, where some terms have been neglected on purpose to obtain the one-fluid equations. These terms also seem to affect not only the solutions near the interface, but also the solutions at the water and air subdomain, where the Favre averaged velocity fields should coincide with the local fields. As shown in figure 2.12b, the local velocity field in the water subdomain will never be the same as the solution provided by the one-fluid formulation (an effect that is more pronounced to the solution that does not include the correction for uu). Therefore, the solution derived by the one-fluid formulation will always be an approximate solution not only near the interface but everywhere. To that end, it is important to be able to obtain an approximate a solution that provides more accurate results specifically in the locations far from the interface where the u value must approach the local velocity, as is the case of the one-fluid formulation solution with correction for uu (or, equivalently, discontinuous solution or the solution that enforces u = 0). To further support the last observation we turn our attention to the solution errors.

We can construct relations for the relative errors for the two cases by the relations we provide for the exact averaged fields and those derived by solving the one-fluid equations, as:

| u sol -u exact | | u exact | (2.5.42)
We plot directly the results of the above relation for each solution in figures 2.13. In each figure, the lines corresponds to the error of a certain solution that we derived for a certain characteristic length of the kernel's support, . The "discontinuous solution approach" provides more accurate results than the C 1 solution. Even though, both the continuous and the discontinuous solutions are approximations of the local velocity fields, in both cases, the larger errors are found near the interface and become everywhere smaller as the parameter becomes smaller (the errors are smaller for = 0.001 m, 2.13b, in comparison to the errors obtained for = 0.002 m, 2.13a). The most interesting observation, however, is that the errors far from the interface for the discontinuous solution are much smaller in comparison to the continuous solution. In conclusion, the discontinuous solution provides results that are closer to the local velocity fields in both air and water subdomains.

[105, p.61-64] discuss another type of correction that stems from arguments which concern the discretization of the viscous stresses. The approach they describe suggests a modification of the manner that the normal derivative of the tangential velocity is approximated near the interface of a shear parallel two-fluid fluid (the same configuration to the plane Couette). From their discussion it follows that one way to improve the discretization of the normal derivative of the tangential velocity is to introduce a harmonic average of the dynamic viscosity around the interface. Here, we provide the analytical solution derived by introducing a harmonic average variation of the dynamic viscosity:

1 µ = C I µ I + C J µ J (2.5.45)
The solution of the one-fluid model follows exactly the same steps as before, and thus we begin directly from relation (2.5.35). We evaluate the required integral for (2.5.45):

1 µ(y) d y = 1 µ J y + 1 µ I - 1 µ J C I (y)d y (2.5.46)
We plug the above to (2.5.35) and write the resulting system of equation for the integration constants.

As we eliminate the constants to solve for a, we arrive to:

a 1 µ I - 1 µ J - - C I d y + b J -b I = 0 (2.5.47)
But for any C I , we have:

-

C I d y = (2.5.48)
And thus we show that for any C I the system we solve simplifies to the system solved to obtain the classic solution (2.5.22). Specifically, the classic solution will be recovered at y ∈ (h J , -] ∪ [ , h I ), i.e.

everywhere besides the region where the C I varies. In conclusion, this approach can effectively correct the results everywhere. For y ∈ (-, ), we can show after calculating the integration constants that u = u. Therefore, the harmonic average for the dynamic viscosity recovers the averaged velocity from the equation we solve.

We note that it is interesting to derive an equivalent formulation of the Navier-Stokes which is based on the averaged velocity field. Indeed, this is feasible with the material discussed in this chapter. The averaged velocity is a more plausible interpretation of the velocity we calculate since it is not affected by the large density differences. However, this formulation should begin by the momentum equation divided by the density (an equation for acceleration) which is not the equation from which the velocity field is derived by ISIS-CFD.

Conclusions

In this section, we have studied two multifluid problems where we can find analytical solutions for both the classic formulation and the one-fluid formulation. We began with the static bubble test case, where we have shown the effects of the surface tension source term, present in the one-fluid formulation. We note that, for this problem, the averaged momentum equation is exact because the velocity terms drop.

p(l ) = -σ B α α C I (l ) (2.5.49)
Despite the fact that there is no need to enforce a boundary condition for pressure for the one-fluid formulation, for the classic formulation at the interface, in both cases, we have to evaluate the curvature,

B α α .
Therefore, an explicit surface representation of the interface is required in both cases. For the numerical solution of the classic formulation, this explicit surface representation would be used to generate a grid fitted to the interface and therefore would be readily available. On the other hand, for the one-fluid approach, such an explicit surface representation is required for the curvature calculation and the evaluation of the surface integrals that appear as source terms in the one-fluid equations.

For the second problem, we have chosen the plane Couette flow. This simple problem has two different interesting aspects. Firstly, it demonstrates the effects of the modified source term due to gravity, i.e. the averaged gravity source term ρ g . Secondly, it shows the velocity solutions obtained by the approximate one-fluid momentum equation.

As we have demonstrated, by solving the y-axis averaged momentum equation, we obtain exactly the averaged pressure field that would have been obtained if the local pressure fields were used along with the definition of the averaged pressure field. An important characteristic is that the jump in the pressure gradient normal to the interface is implicitly taken into account by the averaged gravity source term. However, the last observation does not deem unnecessary the condition:

∇ i p ρ = 0 (2.5.50)
On the contrary, the analytical solutions were derived by exact integration, while the above relation has to be taken into account explicitly by the numerical solution due to the fact that it will not be possible to obtain accurate and sharp pressure fields with very steep variations of the volume fraction. Similar problems arise for the calculation of the surface tension term. Moreover, we should note that it is crucial to set the pressure level near the interface to the same value that the local pressure field has, in order to obtain the correct solution.

For the x-axis momentum equation the velocity terms do not drop, and since the plane Couette flow is dominated by viscous effects, we can obtain a clear understanding in an introductory level about the simplifying assumption introduced for the viscous stress tensor. To clarify, we have emphasized that in order to simplify the resulting averaged momentum equation, the major difficulties have been encountered to recast the convection term and the viscous stress term. Both terms contain products of local fields that were written as products of averaged fields. Specifically the viscous stress tensor in our formulation introduced the modified viscous stress tensor:

τ i j = µ Z i k ∇ k u j + Z j k ∇ k u i + ν Z i k u j -u j + Z j k u i -u i ∇ k ρ -2 E (ν, S i j ) (2.5.51)
The second term and third terms of the above sum was considered to be negligible. Through the resulting exact local velocity fields we have shown that neither the second term (the relative velocity term) nor the third term (obtained by expressing the average of a product as product of averages) can be considered as negligible. Moreover, we have shown that the terms become important as we approach the interface from the side of the lighter fluid. The effects of neglecting the terms are apparent from the solution produced by requiring C 1 continuity for the different parts of the function. Even though the solutions eventually converges to the local velocity fields as tends to zero; the solution contains significant errors even far from the interface.

Since in these locations, the standard Navier-Stokes equations are recovered, the solution of the onefluid formulation should coincide to the local velocity field. To that end, by some simple observations, we proposed a heuristic workaround. The second approach produced a solution that is not continuous, instead it is based on enforcing more strictly the compatibility condition u = 0 at the interface. The derived solution suggests that the Favre averaged velocity fields obtained by the one-fluid formulation are better approximations of the local velocity fields. In conclusion, a similar treatment in the numerical solution might considered as unavoidable and necessary to improve results.

Closing

In this chapter, we have presented two commonly encountered formulations of the Navier-Stokes problem, namely, the classic and the one-fluid formulation, and compared the analytical solutions of two simple problems. In the classic formulation, surface tension enters through the interface compatibility conditions by the requirement that the force exerted from the fluids to the interface is equal to the force exerted from surface tension:

p = σB α α -2 µ (∇ α u α -uB α α ) (2.6.1)
For the one-fluid formulation that we have constructed without Dirac-δ function (or by replacing the Dirac-δ function by their equivalent sequences), surface tension appears as the following source term in the momentum equation:

Σ i = S J I Z i α ∇ α σ + n i σB α α K d S
This source term is naturally generated when we formulate the one-fluid Navier-Stokes equations using volume averaging. Moreover, from this procedure, we have derived two underlying assumptions through which the one-fluid Navier-Stokes equations obtain the same formulas as the standard Navier-Stokes equations.

The first assumption is that the characteristic velocity of the interface must be equal to the center of mass or Favre averaged velocity. This assumption is required in addition to the assumption that the Favre averaged velocity field is solenoidal. Specifically, we have shown that:

∇ i u i = -u i -u i ∇ i ρ ρ (2.6.2)
where u i is the Favre averaged velocity and u i is the characteristic interface velocity. Therefore, when we numerically enforce ∇ i u i = 0 then two possible solutions can be obtained, either the relative velocity u iu i will be zero or it will be normal to ∇ i ρ i.e. tangent to the interface. The second assumption is that the error obtained by expressing the average of a product as the product of averages is small. The primary conclusion is that, even though the interface is not explicitly represented in the onefluid formulation, the one-fluid formulation can benefit from an explicit representation of the interface.

Firstly, though an explicit interface representation, the source term of surface tension can be calculated without introducing intermediate approximations. Secondly, using an explicit interface representation we can add corrections to the one-fluid formulation which improve the derived solution, in the sense that they approximate better the averaged fields obtained by the field produced by the solution of the classic formulation (a formulation that is free of simplifying assumptions). Finally, the argument is further backed up in the literature specifically regarding the introduction of multiphase models by modifying the transport equations. In the next chapter, we present an interface reconstruction method for the VOF formulation of ISIS-CFD, which generates a set of surface grids that represent the air-water interface or interfaces.

Chapter 3

Applications of the Marching Polyhedra

Algorithm for the Volume of Fluid Method

Volume fraction Initialization and Interface Reconstruction

Introduction

In this chapter, we describe the adopted solution procedures for solving the volume fraction initial- To solve both problems, we have introduced to ISIS-CFD a Marching Polyhedra Algorithm (MPA).

The MPA is an isosurface construction algorithm, given as an input a certain field, it produces a surface grid which corresponds to an isosurface of the field. Specifically our implementation generates an unstructured surface grid that consists of the iso-nodes (points), iso-edges (linear segments) and isopatches (surface patches), their connectivities and connectivities between the surface and the volume grid (embedment connectivities). Even though we will not discuss the algorithm in details, its most essential parts are summarized in the first section of this chapter.

After a general introduction to the algorithm, we formally state the volume fraction initialization and the interface reconstruction problem. In the second section, the volume fraction initialization problem is solved by introducing post-processing steps to the procedures that locate the iso-edges and the iso-patches. In the third section, we recast the interface reconstruction problem to an isosurface construction problem. As a result, both problems are solved in a straightforward manner by the MPA.

Both sections conclude with some tests that demonstrate the consistency of the presented methods.

Finally, we describe the normal vector and curvature calculation procedures using a local least square differencing approach for the surface grid.

The Marching Polyhedra Algorithm

Introduction In this section, we provide a general description of the Marching Polyhedra Algorithm.

The algorithm is based on the Marching Cube Algorithm (MCA), classically encountered in the domain of computational geometry and geometric modeling. After a brief description of the MCA, we present the main components of the proposed approach.

The family of Marching Simplices Algorithms (MSA) generates an explicit representation of an isosurface by constructing polygonal surface cells embedded in a given volume grid. By embedded we mean that the surface cells, or iso-patches, rest inside the cells of the volume grid. Moreover, the surface nodes, or iso-nodes of the iso-patches, always lie on the cell's faces. Banks et al. ( 2004) [START_REF] Banks | Counting cases in substitope algorithms[END_REF] describe MSA methods as "Substitope Algorithms". The distinctive procedure of MSA methods is that one constructs "by hand" a set of cases that illustrate how a patch of the isosurface will be defined inside the cell of the volume grid (the simplex substituted), given that the isosurface is somewhere inside the given cell.

Lorensen and Cline (1987) [START_REF] Lorensen | Marching Cubes: A High-Resolution 3{D} Surface Reconstruction Algorithm[END_REF] developed the original method, the Marching Cubes Algorithm, also In the original method, Lorensen and Cline (1987) count 14 cases that describe the patches of the isosurface generated inside a cube. A "case" is defined topologically. First we identify the edges of the cube that a function g ( r ) (the function whose isosurface is constructed), obtains the desired value, After identifying at which case the cell belongs to, the actual intersection points of the isosurface and the cell's edges are found by solving the equation for t :

attributed
g ( r ) = a.
g p i + ( p j -p i ) t -a = 0, t ∈ (0, 1) (3.2.1)
Here p i and p j are the starting and ending point of an edge, such that g ( p i ) • g ( p j ) < 0. We name points obtained by solving the above equations as iso-nodes. The iso-nodes are then appropriately connected, to define the "in-cell" triangular patches of the surface.

Since the publication of the original MCA, the relevant literature has had rapidly grown. The original algorithm uses cubes. Other methods use mixed cell types, as Takahashi (2003) [START_REF] Takahashi | A method of equi-valued surface construction from volume data on a face-centered cubic lattice[END_REF], or octahedra Strand (2007). In general, the use of a predefined cell shape is necessary to obtain a tractable set of cases, which specify the isosurface patches inside the cells. However, for certain combinations of nodal values, it is not possible to link a single patch configuration inside a cell, as shown in figure 3.2.

Van Gelder (1990) [START_REF] Van Gelder | Topological considerations in isosurface generation extended abstract[END_REF] characterized these cases as ambiguous, a term frequently encountered in the relevant literature. Several "disambiguation approaches" are discussed by van Gelder (1990) [START_REF] Van Gelder | Topological considerations in isosurface generation extended abstract[END_REF] and [START_REF] Newman | A survey of the marching cubes algorithm[END_REF] [START_REF] Newman | A survey of the marching cubes algorithm[END_REF]. Disambiguation criteria take into account values of the function at certain points, other than the nodes, to specify a single iso-patch configuration. A straightforward approach is to use refined grids near the interface. For example, Manson (2010) [START_REF] Manson | Isosurfaces Over Simplicial Partitions of Multiresolution Grids[END_REF] presents a MCA that uses a dynamically refined background grid. His approach results to isosurfaces that contain more nodes only wherever the isosurface construction would be inconsistent.

An in-depth discussion of the MCA-like approaches is outside the scope of the current work. The interested reader may refer to the review of [START_REF] Newman | A survey of the marching cubes algorithm[END_REF] [START_REF] Newman | A survey of the marching cubes algorithm[END_REF] for further details on the subject.

Elvins (1992) [START_REF] Elvins | A survey of algorithms for volume visualization[END_REF] discusses other methods that generate isosurfaces and their relation to computer graphics. For more modern approaches see Martin (2012) [START_REF] Martin | Direct isosurface visualization of hex-based high-order geometry and attribute representations[END_REF]. However, every MCA-like approach consists of, (i) a method to recover the topological connections of iso-nodes to generate iso-patches, and (ii) disambiguation criteria, we describe the former next.

Basic Procedures

The proposed MCA-like method has similar features as these described above. However, it must also account for the following. In ISIS-CFD there is no predefined cell shape (for example, the grid might consist of cubic or tetrahedra cells or both). Moreover, due to the fact that calculations might be performed with a deforming grid, the faces of the cells might not be planar. Thus the algorithm should produce iso-patches for any grid generalized in the above manner (a grid that we characterize as a general unstructured grid). Finally, C 0 continuous iso-patches must be produced in parallel boundaries of the grid. We will not discuss the parallelization of the numerical method, which is related to the parallelization of the interpolation scheme presented in section 3.4. As a result, we work in a low-level manner using a minimum amount of simplifying topological assumptions regarding the faces and cells.

The only requirement for both the faces and cells of the grid are that their regions are simply connected. The algorithm has to make decisions proactively, to generate the iso-edges and iso-patches from the iso-node configuration and the cell's shape. To emphasize this characteristic feature of the algorithm, we named it Marching Polyhedra Algorithm (MPA). The MPA consists of four steps:

1. Characterize nodes as "in","out,"at"

2. Construct the iso-edges for each face 3. Construct the iso-patches for each cell

We briefly describe each step next.

The first step begins by calculating the values of the function g ( r ) at the nodes of the background grid, g ( p n ). From these values, the MPA characterizes each node as "in" or "out", g ( p n )a < 0 or g ( p n )a > 0 respectively, and "at", g ( p n )a = 0. We emphasize that we have to take into account "at" nodes, if we wish to ensure that we generate proper manifold surfaces (where the normal vector is uniquely defined for each point). The node characterizations are used to find the iso-nodes.

Since an edge (of the background grid) begins and ends at two nodes, we can identify for each face, the edges where iso-nodes belong. We assume that the background grid is fine enough so that for each edge corresponds a single iso-node (except "at"-"at" iso-edges that we will not describe here).

Thus, for each edge that rests between an "out" and an "in" node, we solve equation (3.2.1), using one of the following approaches. Firstly, if there exists an analytical solution, then the equation is solved analytically. Secondly, if the function's formula is known but a nonlinear equation must be solved, then the equation is solved numerically, by a simple bisection method. Thirdly, if the function g ( r ) is given solely by the discrete sample g ( p n ), then the equation is solved using a local linear approximation of the equation g ( r (t )). The iso-nodes located at the edges of a face are subsequently used to define the iso-edges.

To construct the iso-edges, we scan the edges of each face, following their order so that they form a closed loop (this closed loop identifies an oriented boundary for each face). For each face, the algorithm joins the iso-nodes with linear segments, which represent the iso-edges. When more than two iso-nodes are found, the construction of the iso-edges is ambiguous (see for example the top face of the cube in figure 3.2). For ambiguous iso-edge constructions, a disambiguation criterion is introduced. To construct a unique configuration of the iso-edges, the disambiguation criterion uses the characterization "in","out","at" for the centroid of the face. Particular care must be taken for "at" nodes because they mark iso-nodes that can be considered to belong to, either simultaneously two edges of the face or none (e.g. when the surface osculates the polygon but does not intersect it).

When the iso-edge construction is complete, the algorithm constructs the iso-patches for each cell.

For the construction of an iso-patch, the algorithm gathers the iso-edges found at the cell's faces and orders them to form closed loops. Each loop defines an iso-patch. Ambiguous cases arise when the iso-patch loops cannot be uniquely defined. These cases are related to iso-edges with "at" nodes. The disambiguation criterion for the iso-patches uses the characterization "in","out","at" for the centroid of the cell, to decide which iso-patch configuration is the appropriate one.

Finally, it is important to observe the MPA does not directly propose a method that generates a surface grid. The algorithm provides only the patches approximating the isosurface (the iso-patches).

To obtain a surface grid (in this work, meshes are named "grids" to emphasize that they are used to state numerical schemes), a post-processing step is required. As in any grid generation procedure, the surface grid results by:

1. enumerating the iso-grid entities (iso-nodes,iso-edges,iso-patches)

enriching the iso-grid entities with connectivities

The MPA also takes into account that the surface grid must be generated. Thus, it takes specific actions to ensure that the above procedures should not prohibitively waste computational resources.

Concluding Remarks

The Marching Polyhedra Algorithm (MPA) can be viewed as a generalized version of the Marching Cube Algorithm (MCA). In contrast to MCA, the MPA can be used for general unstructured grids where the cells do not have a predefined shape. The MPA makes decisions to obtain the final configuration of the iso-patches inside a cell at two distinct levels. Firstly, for each face, it locates the iso-nodes belonging to the face's edges and connects them to generate iso-edges. Secondly, for each cell, it uses the iso-edges belonging to the cell's faces to generate the iso-patches. As a post-processing step, a surface grid generation procedure follows, from which we obtain a surface grid that represents the isosurface.

The procedures described above use low-level (topological) concepts to perform the construction.

Even though the procedures are considered as intuitive, it is precisely due to the same reason that they are difficult to be stated as simple algorithms. To that end, it would be necessary to go into details.

Firstly, concerning how ISIS-CFD represents grid entities and how the regions of the grid entities are specified. Secondly, discuss the grid data-structure used and present the decision-making approaches of the MPA. Finally, describe the surface grid generation procedure and provide the particularities of the parallel execution of the method. Therefore a detailed description is outside the scope of this work.

Nevertheless, the applications of MPA-related to VOF are directly relevant, and the remainder of this chapter is devoted to such applications.

The Volume Fraction Initialization Problem

Introduction The general volume fraction initialization problem can be stated as:

The Volume Fraction Initialization Problem

Given a general unstructured grid covering the computational domain D and a set of surfaces S I = ∂D I , find the values of the cell-centered field C I c :

C I c = 1 V c V c I I dV (3.3.1)
where I I is the indicator function of the subdomain D I and V c is the volume of the cell.

As we have already noted in the previous chapter, this is an application of the general relation that defines the volume fraction (2.4.13):

C I ( x, t ) = I I ( y, t )K ( y; x)dV ( y)
evaluated at the centroid of the cell, x = x c and for t = 0, when K is the top-hat function kernel for a cell, (2.4.15):

K ( y; x) = 0 , otherwise 1 V c , y ∈ V c
The problem is intrinsically grid dependent and analytical solutions are known for very simple configurations. We provided in paragraph Analytical Volume Fraction Fields(p.61) some simple cases

where the problem has an analytical solution. The solution depends on of the configurations of the interfaces. For example, the solution becomes much more complicated when a fluid subdomain is not simply connected, and eventually, the user/solving code interactions become necessary. Moreover, the problem depends on the representation (either explicit or implicit) of the surfaces in the code.

All the volume fraction initializations performed for the study of practical multifluid problems were carried out by providing the interfaces by an implicit surface. The volume fraction initialization problem of an interface given by an explicit surface is rather useful for the so-called immersed boundary method.

The principle behind immersed boundary methods is to represent a solid by using its indicator function (see for example [START_REF] Mittal | Immersed Boundary Methods[END_REF]). The difference between the Immersed boundary methods and surface tension in VOF is only conceptual, and the volume fraction indicates the presence of a solid instead of a fluid. The ISIS-CFD extensions framework also solves the initialization problem given an explicit surface, but we do not describe it here. Specifically, in this section, we restrict the discussion to interfaces represented by implicit surfaces.

Volume Fraction Initialization for Interfaces Represented by Implicit

Surfaces

Preliminaries Given a set of implicit surfaces (i.e. given a real function g I , the set of points p S , such that g I ( p S ) = 0, defines the surface S I ), we define the set of points that are "outside" the region bounded by the surface by the property g I ( p) > 0, and the set of points that are "inside" the region bounded by surface by the property g I ( p) < 0. The volume fraction integral can be equivalently written using the classic Heaviside function as:

C I c = 1 V c V c H -g I ( p) dV (3.3.2)
Since the values of H -g I ( p) are 1 whenever a point is inside the subdomain D I , we can write the above integral by replacing the integration region, which coincides with a cell in this case (the support of the kernel), with the region of the cell that rests inside the subdomain D I . This generates a truncated region, that is symbolically denoted as V c in I and thus the volume fraction is:

C I c = 1 V c V c in I dV = V c in I V c (3.3.3)
Note that both the region and its volume are denoted by V c in I . The purpose is to find the volume V c in I .

An analytical solution does not exist for general settings. For simple configurations, such as a plane and Cartesian grid, an analytical solution was published by Scardovelli and Zaleski [START_REF] Scardovelli | Analytical Relations Connecting Linear Interfaces and Volume Fractions in Rectangular Grids[END_REF] for the solution of the general volume fraction initialization problem for cubic cells. Their approach performs the integration using a Gauss quadrature and allows accurate approximations for cubic cells since explicit actions are performed to calculate the integration limits for complicated configurations.

Even though the approach of Lopez et al. ( 2009) has similarities with the approach followed here, in ISIS-CFD, we cannot always ensure that the faces of the cell will be planar and further complicates a method that could be based on quadrature formulas.

In our approach, we formulate a method for the most general case where a curved interface might intersect the cell in disjoint regions for a general unstructured grid. The resulting method directly calculates the metrics (a "metric" is a calculation related to certain volume or the area integral that involves only geometrical entities of either a polyhedral region or a surface region) of the truncated faces, the cell can be obtained by the MPA, more than one iso-patch can be defined inside each cell. The regions S f in I are found when the iso-edges are generated. Because the MPA explicitly constructs ambiguous iso-edges, we directly obtain compound regions, S f in I , whose boundary consist of parts of the face's edges and more than one interface segment (such as the top face in 3.3b). Approximations of the volume of such regions with simple quadrature formulas (to evaluate the volume integral) might be difficult to be achieved, especially if we take into account that the our cell might not be hexahedral. The interested reader may consult [START_REF] Bna | Numerical integration of implicit functions for the initialization of the VOF function[END_REF] for details and we note that the methods presented there could be used for general unstructured grids. The method we present here uses a simpler and general approach.

C I c = 1 V c V c in I dV = 1 3V c m=1,k S I c(m) p nd S + f ∈c f c [ f ] S f in I p nd S (3.3.4)
where V c is the cell's volume, n is a normal vector, c f denotes the set of boundary faces of the cell c and c [ f ] is a orientation correction coefficient, which ensures that the normals of the cell's faces point outwards with respect to the cell's region. We evaluate the above relation for each cell that is intersected by the interface.

Approximation of the Interface Surface Integral

For demonstration purposes, we work with the embedment connectivities. The embedment connectivities are connectivities that define the relations of the surface grid generated by the MPA with the background volume grid. For example, if c s is an iso-patch, then the embedment connectivity isopatch-cell c s c gives the cell of the volume grid that contains the iso-patch c s (we note that an upper index s, as in c s , denotes surface grid entities). For a cell c that is intersected by the interface, the MPA has already constructed at least one iso-patch. The iso-patch is identified by the cell-isopatch connectivity, c c s .

An iso-patch, c s , is a surface patch that is represented by a cyclic set of points. The cyclic set identifies a sample of points (iso-nodes) that belong to the boundary of the iso-patch. We denote an iso-node as n s and the cyclic set that identifies which iso-nodes lies on the boundary of the iso-patch, c s n s (the MPA provides the isopatch-isonodes connectivity). The order of the iso-nodes as defined by c s n s is such that (i) two subsequent iso-nodes define an iso-edge (ii) the normal vector of the iso-patch points away the subdomain I . Each iso-patch is related to its iso-nodes by the MPA algorithm.

We obtain the iso-nodes from the solution of the intersection problems of linear segments and the given implicit function. If a linear segment is given by p(t ) = a + bt then the iso-node is defined by the solution of the (non-linear, in general) equation:

g I ( p(t )) = 0 (3.3.5)
The linear segments that provide iso-nodes are the edges of a face and linear segments between a node of the face and the face's centroid. From the solution of these problems, a set of iso-nodes results as shown in figure 3.4a. The figure schematically depicts the iso-node construction of an iso-patch for the case where the interface's surface g I ( r ) = 0 would be that shown in figure 3.3a.

Therefore, each isopatch is an approximation of a part of the interface inside the cell. Only the isonodes of the isopatch (symbolically denoted as members of the set c s n s ) will lie exactly on the surface S I , by definition. Specifically, we define the region occupied by an iso-patch, S c s as (the symbol ∆ is interpreted as "triangle"):

S c s = n s ∈ c s n s c s ∆ n s where c s ∆ n s = p(u, v) = p n s u + p n s +1 v + p c s (1 -u -v), ∀u, v ∈ [0, 1] : 0 1 -u -v 1 , n s ∈ c s n s (3.3.6)
Each isopatch approximates the part of the actual surface that rests inside the cell, by the sequence of triangles, c s ∆ n s . The isopatch's representative point p c s (either a point of the surface or the centroid) and the iso-nodes taken in groups of two (following the order encountered in the set c s n s ), define the triangles c s ∆ n s , figure 3.4b. The geometrical space representation of an iso-patch (the manner we defined the iso-patch's region) is equivalent to the geometrical space representation of a face of the general unstructured volume grid. .4: Iso-nodes and geometrical space representation of an iso-patch. We obtain the iso-nodes (black crosses) by finding the intersection point of an edge and the interface. We obtain the iso-nodes (red crosses) by finding the intersection point of the line connecting the centroid of each face and a boundary node of the face. The isopatch-isonodes connectivities, c s n s , specify the nodes that belong to the boundary of the iso-patch. The triangles c s ∆ n s define the region of the iso-patch and are shown using dashed lines.

As a result, the isopatch metrics are calculated in exactly the same fashion as the grid's face metrics, i.e. the face's area and centroid. The only difference is that the face-nodes connectivities are replaced by the isocell-isonodes connectivities. For example, the surface vector of a face is:

S f = 1 2 n∈ f n p n × p n+1 (3.3.7)
where p n is a node from the face-nodes connectivity, f n (the cyclic set of nodes that lie on the face's f boundary), and p n+1 is the node found directly afterwards in f n . Therefore, the surface vector of an isopatch is:

S c s = 1 2 n s ∈ c s n s p n s × p n s +1 (3.3.8)
The centroid of the iso-patch, p c s , is calculated in a similar manner as the centroid of the faces. The centroid of the iso-patch will not be lying on the surface. However, this choice does not significantly hamper the accuracy of the method. As the background grid tends to be smaller, the patches tend towards the tangent plane of the interface and therefore, the centroid also tends closer to the tangent plane.

Since an iso-patch consists of planar triangles, the first integral of the sum inside the brackets of relation (3.3.4) can be easily obtained. Suppose that the m-th isopatch of the cell c, c c s (m), approximates the m-th part of the interface inside the cell c, S c c s (m) ≈ S I c(m) , then the first term inside the brackets of (3.3.4) can be written as:

m=1,k S I c(m) p nd S = c s ∈c c s p c s S c s (3.3.9)
where p c s is a point inside the isopatch, S c s is the surface vector of the isopatch and c c s is the cellisopatch embedment connectivity (that specifies for a given cell the isopatches that were constructed inside by the MPA). Each term of the sum can be directly calculated for any iso-patch.

Approximation of the Surface Integrals on the Faces

The second integrals of the sum inside the brackets of relation (3.3.4) refer to each face that is intersected by the interface. For each of these faces we evaluate the term:

S f in I p nd S
The above integral is written as a sum over the regions of the triangles that rest inside the subdomain

D I , denoted as f ∆ n in I : S f in I p nd S = n∈ f n inI f ∆ n in I p nd S (3.3.10)
In the last relation, the notation n ∈ f n inI denotes the nodes of the face that rest inside the subdomain D I . Since the regions f ∆ n in I are planar, the product p n is constant and we obtain:

S f in I p nd S = p f n∈ f n inI n f ∆ n A f ∆ n in I (3.3.11)
where p f is the centroid of the face, A f ∆ n in I is the area of the region f ∆ n in I , n f ∆ n is the normal vector of a triangle f ∆ n of the face (the triangle formed by the centroid of the face, the node f n and its subsequent node that, along with f n , define an edge of the face). In the general case, each of these regions is defined by a boundary consisting of a curved segment, defined by the interface's trace on the triangle. We replace these curved segments by linear segments and calculate the areas of the resulting quadrilaterals. For the special case where f ∆ n = f ∆ n in I , the triangle is not truncated (the interface does not intersect the triangle) since it is entirely immersed to the inside region.

First of all, we notice that each of the face's triangles that is intersected by the interface, has either a single "in" node (a node n where g I ( p n ) < 0) or a single "out" node (a node n where g I ( p n ) > 0).

Consider first the case where a single "in" point is found at the triangle ABC of the face. Suppose that point A is "in" and is either the centroid of the face or a node of the face. Then two isonodes would be located at the two edges of the triangle (one for each edge), that stem from point A and end to points B and C (B,C are either two nodes of the face, or one of them the centroid of the face and the other a node of the face), i.e.:

Edge

A → B : p A→B (u) = p A + ( p B -p A )u , u ∈ (0, 1) Edge A → C : p A→C (v) = p A + ( p C -p A )v , v ∈ (0, 1)
Therefore, for these edges we have to calculate the values u, v that intersect the interface, or, since the interface is given by g I ( r ) = 0 the solutions of g I ( p A→B (u)) and g I ( p A→C (v)). The points identified by these procedures are exactly the isonodes. However, for each of these edges, the MPA algorithm has already calculated the location of the iso-nodes and, thus the values u = u s and v = v s , are known. The required area easily follows:

A f ∆ n in I = 1 2 ( p A→B (u s ) -p A ) × ( p A→C (v s ) -p A ) = 1 2 u s v s ( p B -p A ) × ( p C -p A ) = u s v s A f ∆ n (3.3.12)
Similarly for the case, where the face's triangle ABC consists of a single "out" node, say that node A would be an "out" node, the edges are similarly defined as, B → A and C → A, and the required area becomes:

A f ∆ n in I = (u s + v s -u s v s )A f ∆ n (3.3.13)
To unify the calculation so that it refers to any triangle, independently of whether a part of it is immersed to D I or not, we define the area fraction of the face's triangle as:

C I f ∆ n = A f ∆ n in I A f ∆ n (3.3.14)
If the g ( f ∆ n ) is the set of characterizations for the nodes of the triangle f ∆ n , i.e. the set g

( f ∆ n ) = g ( p f ), g ( p n ), g ( p n+1
) for a node n ∈ f n , and the number of "in" nodes, "out" nodes and "at" nodes of the triangles are given symbolically as |g

( f ∆ n ) < 0|, |g ( f ∆ n ) > 0|, |g ( f ∆ n ) = 0|
respectively, then the area fraction is:

C I f ∆ n =                  0 , |g ( f ∆ n ) > 0| + |g ( f ∆ n ) = 0| = 3 and |g ( f ∆ n ) = 0| = 3 u s + v s -u s v s , |g ( f ∆ n ) > 0| = 1 and |g ( f ∆ n ) < 0| = 2 u s v s , |g ( f ∆ n ) < 0| = 1 and |g ( f ∆ n ) > 0| = 2 u s , |g ( f ∆ n ) = 0| = 1 and |g ( f ∆ n ) < 0| = 1 and |g ( f ∆ n ) > 0| = 1 1 , |g ( f ∆ n ) < 0| + |g ( f ∆ n ) = 0| = 3 (3.3.15)
The last relation can be easily implemented alongside the MPA, during the step where the iso-edges are constructed. Note that the 10 cases summarized in the above relation, define a "marching triangle" solution for the area fraction of the triangle f ∆ n .

In conclusion, we can write the RHS of (3.3.11) as:

S f in I p nd S = p f n∈ f n in I n f ∆ n A f ∆ n in I = p f n∈ f n n f ∆ n C I f ∆ n A f ∆ n (3.3.16)
Note that the summation over the nodes of the specific triangles of the faces that are immersed to D I , was replaced by a summation over all the triangles using the area fraction of the face's triangles. We also define the area fraction vector of a face as:

C I f = 1 | S f | n∈ f n n f ∆ n C I f ∆ n A f ∆ n (3.3.17)
Using the above definition, we write (3.3.16):

S f in I p nd S = p f C I f | S f | (3.3.18)
which is the final formula for the second summation term of (3.3.4) inside the brackets.

Summary: The Volume Fraction Calculation

We begin the calculation by rewriting the volume integral for the region of a cell c partially immersed to the subdomain D I :

C I c = 1 V c V c I I dV
as a surface integral. For each cell, the surface integral is separated to integrations over the parts of the isosurface inside the cell and parts of face regions that are immersed to D I . The area of the isosurface inside the cell is approximated by the iso-patches and the curved segments of the face regions immersed to D I , by linear segments. Through these approximations, we arrive to the (approximate) relations (3.3.9) and (3.3.18) for the evaluation of the integrals. When these relations are substituted to (3.3.4), we obtain the following (approximate) formula for the volume fraction calculation:

C I c = 1 3V c   c s ∈c c s p c s S c s + f ∈c f c [ f ] p f | S f | C I f   (3.3.19)
When the cell is outside the subdomain D I , the interface does not rest inside the cell. In that case the first term is not present, |c c s | = 0 (no isopatches are found inside that cell), each area fraction of the triangles of the face is equal to 0, C I f ∆ n = 0 , ∀n ∈ f n , ∀ f ∈ c f , and the area fraction vector of the face is C I f = 0 , ∀ f ∈ c f . Therefore the volume fraction is zero. When the cell is completely immersed in the subdomain D I , the interface does not cross the cell. In that case, the first term is not present, |c c s | = 0, each area fraction of the triangles of the face is equal to 1,

C I f ∆ n = 1 , ∀n ∈ f n , ∀ f ∈ c f ,
and the area fraction vector of the face is

C I f = n f , ∀ f ∈ c f . Therefore (3.3.19) becomes: C I c = 1 3V c f ∈c f c [ f ] p f S f = V c V c = 1 (3.3.20)
In conclusion, given an interface S I (that separates two fluids) as an implicit function g I ( r ) = 0, the volume fraction for each cell c of the grid can be expressed using the characterizations of the nodes of the cell. If the set of characterizations for the node of a cell is

g I (c n ) = g ( p n ),
C I c = 1 V c V c H -g I ( p) dV (3.3.21)
is given by the following relation:

C I c =                        0 , |g I (c n ) > 0| + |g I (c n ) = 0| = |c n | and |g I (c n ) = 0| = |c n | 1 3V c   c s ∈c c s p c s S c s + f ∈c f c [ f ] p f | S f | C I f   , |g I (c n ) > 0| + |g I (c n ) < 0| 2 1 , |g I (c n ) < 0| + |g I (c n ) = 0| = |c n | (3.3.22)
The above calculation refers to a single fluid subdomain D I whose surface S I can be identified a single function g I through the indicator I I = H -g I ( p) . When the volume fraction of more complex configurations is sought, such as when multiple interfaces with different fluids are specified, the indicator function will not be the same. However, almost every volume fraction initialization problem can be solved by the method presented above through the MPA and simple modifications.

Solution of the Volume Fraction Initialization Problem for Multiple

Interfaces

In the previous chapter (see Description of a Domain Occupied by Different Fluids p.32), we have identified cases where multiple interfaces might be required to describe the initial configuration. For these cases, multiple indicator functions might be necessary to describe a given configuration and specifically we have demonstrated that the indicator function can be written as a sum of Heaviside functions, say for example:

I I = i λ i H -g i ( p) (3.3.23)
where i is a counter of the functions g i required to identify the subdomain I and λ i is an integer that is either -1, 0, 1. In that case the framework provides the result of:

1

V c V c H -g i ( p) dV by (3.3.22): 1 V c V c H -g i ( p) dV =                        0 , |g I (c n ) > 0| + |g I (c n ) = 0| = |c n | and |g I (c n ) = 0| = |c n | 1 3V c   c s ∈c c s p c s S c s + f ∈c f c [ f ] p f | S f | C I f   , |g I (c n ) > 0| + |g I (c n ) < 0| 2 1 , |g I (c n ) < 0| + |g I (c n ) = 0| = |c n | (3.3.24)
and therefore the volume fraction for the subdomain I is: The framework calculates each term inside the brackets of the above sum, and it is up to the user to define the resulting volume fraction field by adding or subtracting them appropriately. When multiple subdomains are given, the above calculations can be repeated for each subdomain.

C I c * = 1 V c V c I I dV (3.3.23) = 1 V c V c i =1,n f l λ i H -g i ( p) dV = i =1,n f l λ i 1 V c V c H -g i ( p)

Verification Exercises and Examples

We begin by solving simple initialization problems where a single interface is defined and next, we demonstrate the solution of volume fraction initialization problems for complex configurations.

Initialization of Basic Two-Fluid Problems

The simplest case, that can be treated by the proposed solution method, is the initialization of two-fluid problems separated by a single interface, given implicitly as g I ( r ) = 0. To begin, we consider the initialization of a plane, that might represents a free surface, g I ( r ) = n J I ( p cp o ). Reformulating the Scardovelli-Zaleski analytical relation for the calculation of a volume fraction of a planar interface in a single cell, as the solution of the relative volume fraction initialization problem, we arrive to the following solution:

C I (l ; κ 1 , κ 2 , κ 3 ) =        1 , l -1 C S Z I (l c ; κ 1 , κ 2 , κ 3 ) , -1 l 1 0 , 1 l (3.3.26)
where l is the signed dimensionless distance of a cell's centroid to the interface (a plane with unit As an example, consider the case of a background Cartesian grid that consist of ν 3 cubic elements and discretizes the standard cube (centered at O = (0, 0, 0), side's length 2). We solve the volume fraction initialization problem for the interface g I (x, y, z) = x + 1.5y + 2z = 0 and calculate the absolute errors of the calculated solution using the proposed method and the analytical solution provided by applying the Scardovelli-Zaleski relation. The results are given in figure 3.5. As we can observe, the absolute errors of the volume fraction field are below machine precision, indicating that the method provides the exact results. The result is expected since the MPA represents exactly planar interfaces and the results furnished by the volume fraction initialization procedure coincide with the results of the analytical relations. The volume fraction initialization problem for planar interfaces in a background Cartesian grid is the simplest volume fraction initialization problem. However, its practical uses are rather limited.

normal n J I ), l = d / = n J I ( p c -p o )/ ,
Even for simple domains, a grid generator might not produce cells that are rectangular parallelepipeds. An example is given in figure 3.6. The MPA is used to generate a surface grid of a planar interface around a rectangular body, placed inside the computational domain. The computational domain has an exterior boundary surface (standard cube) and an interior boundary surface (a rectangular parallelepiped, 1 × 1 × 0.5 centered at p O = (0, 0, -0.25)), as shown in figure 3.6a. The grid (generated by Hexpress) does not consist only of rectangular parallelepipeds. The unstructured surface grid generated by the MPA is shown in figure 3.6b. As we can observe the surface grid does not consist of rectangles, and since the grid of the surface is defined by the intersections of the surface and the background grid, the background grid does not consist of rectangular parallelepipeds. Therefore, we cannot ensure that the volume fraction of cells intersected by the interface will obtain the same value everywhere.

Instead, the volume fraction varies because cells represent different regions in space. For example, we can observe that the volume fraction values visualized as intersections with the cut planes in figures 3.6c and 3.6d, change as we move closer to the interior boundary of the domain.

Another commonly encountered volume fraction initialization problem is that of a spherical interface. This case will be used later, to set up test cases with bubbles. Even though an analytical solution for the volume fraction calculation could be devised for simple cell configurations (up to now there is no such published relation for Cartesian grids), we can derive an approximate solution by the proposed method. Consider that we are working with a grid which discretizing a unit cube centered at the origin.

The solution of the volume fraction initialization problem for the sphere represented by the implicit

function g ( r ) = | r | 2 -R 2 = 0, where R = 0.5 in a uniform Cartesian grid (ν = 20), is shown in figure 3.7.
The volume fraction is calculated for the subset of cells, whose region is intersected by the interface, i.e.

the cells, where the cells-isopatches connectivities are defined. In figures 3.8 we visualize the cells (gray edges) that are intersected by the interface, the iso-patches (blue edges) defined on these cells and the volume fraction values for the cells intersected by the plane x = 0.01, x = 0.21 and x = 0.41 (figures 3.7a-3.7c).

The validity of the volume fraction approximations is directly related to the calculation of the centroids of the iso-patches. By calculating the distance errors of each iso-patch from the actual interface for different values of ν, we can demonstrate that the centroids of the iso-patches tend to the actual interface. The distance errors for ν = 20, 40, 80 are given in figures 3.8a, 3.8c, 3.8e respectively.

Observe that the errors are larger in iso-patches that represent larger regions of the spherical interface.

The max and mean errors are plotted in figure 3.9a over the dimensionless length scale δ/(2R), and are also summarized in the table 3.1. The results indicate, that the centroid of the iso-patches converge to the actual points of the spherical surface as the grid length scale tends to become smaller and that the convergence is second order. In conclusion, we can expect that the volume fraction approximations also converge.

RE (V ) = V sph - c∈C C I c V c V sph
, where

V sph = 4πR 3 3 (3.3.27)
as an indicator of the validity of the volume fraction calculation. Following Cummins et al., we have calculated the total volume error (also given in table 3.1) and plotted results in figure 3.9b. The results indicate that the proposed calculation is second order accurate. The fact that, both the distance from the actual interface and the total volume are converging, is a good indicator that the proposed volume fraction calculation provide correct results.

Initialization of Two-Fluid Problems with Multiple Interfaces

The proposed method can also be used to initialize volume fraction fields when the configuration consists of different interfaces. Therefore, it allows the investigation of more complicated settings. We consider two examples. First, the initialization of a problem involving a bubble and a free surface. For this problem, the two solutions previously demonstrated are used simultaneously. Each problem is separately solved, and the resulting volume fraction is obtained by appropriately applying relation (3.3.25). A slice of the resulting volume fraction field (C I c = 1 indicates the presence of water) is shown in figure 3.10a. We note that it is very important to properly define the normal vectors of the interface with respect to the volume fraction field, which should be defined in order to conform to relation (2.4.34):

∇ i C I = - S J I inV n i K d S ( 3

Summary

In this section, we have described the solution of the volume fraction initialization method adopted in the ISIS-CFD extensions. The volume fraction initialization method uses the MPA to locate the isopatches of each of the interfaces given by their implicit representations. This process naturally generates truncated cells, whose volume must be calculated to obtain the volume fraction. The truncated cells are implicitly defined, and there is no need to construct their connectivities to define them (besides the connectivities defined by the MPA). The proposed method can deal with cells containing multiple parts of the interface, and its implementation is straightforward, given the surface grid as generated by the MPA. Moreover, the method is second order accurate, in the sense that it provides the exact volume fraction values for planar interfaces in grids with planar faces. For other surfaces, even though we do not directly compare with analytical results, we can show that the centroid of the resulting surface converge to the analytic surface as the characteristic length of the grid becomes smaller.Thus,the initialization procedure will produce a converging result. The proposed method can deal with a large variety of initialization problems, from simple problems where a single interface separates two fluids, to cases where two or more interfaces are defined, cases with interfaces bounded by specific regions and cases where different fluids are considered. Last but not least, we can dynamically generate grids as performed by the cut-cell method, by introducing appropriate connectivities for the truncated faces and cells by the interface. We present an example in the last section of this chapter. In the next section, we describe the solution of the inverse problem, the interface reconstruction problem.

ISIS-CFD, can be found in Gopala and van Wachem (2008) [START_REF] Gopala | Volume of fluid methods for immiscible-fluid and free-surface flows[END_REF]. However, the relevant research on PLIC schemes and its various improvements is quite pronounced in the literature. Some examples include, Aulisa et al. ( 2007) that used better methods to obtain approximations of the interface's normal vector and thus a better PLIC representation [START_REF] Aulisa | Interface reconstruction with least-squares fit and split advection in threedimensional Cartesian geometry[END_REF]. [START_REF] Mencinger | A PLIC-VOF method suited for adaptive moving grids[END_REF] proposed modifications for moving grids [START_REF] Mencinger | A PLIC-VOF method suited for adaptive moving grids[END_REF]. [START_REF] Wang | A new volume-of-fluid method with a constructed distance function on general structured grids[END_REF] performed PLIC reconstructions in the independent variable domain and mapped those to the physical domain allowing its efficient use in mapped structured grids [START_REF] Wang | A new volume-of-fluid method with a constructed distance function on general structured grids[END_REF]. Besides PLIC methods, other methods have stemmed which use other higher order local representations, such as the PROST method of Renardy and Renardy (2002) [START_REF] Renardy | PROST: A Parabolic Reconstruction of Surface Tension for the Volume-of-Fluid Method[END_REF] using parabolas for two-dimensional problems or the method of Gois et al. ( 2008) [START_REF] Gois | Front tracking with moving-least-squares surfaces[END_REF] using moving least square. Due to the extensive use of PLIC in the literature, during the initial stages of this work, several PLIC methods have been developed and tested with ISIS-CFD.

In figures 3.12, we demonstrate four PLIC reconstructions (generated by the framework) that use two different normal vector calculations, two different background grids and the initial volume fraction of a sphere. The original purpose was to use compressive discretization schemes for advecting the volume fraction along with PLIC for the reconstruction of the interface. However, such practice was finally abandoned since even the slightest numerical diffusion of the volume fraction due to the compressive discretization schemes make the PLIC reconstruction produces fields without physical meaning. The MPA reconstruction replaced the PLIC method that we have initially developed.

Interface Reconstruction with the MPA

The Marching Polyhedra Algorithm is the basis of the method used in this work to dynamically reconstruct the interface. Previously, we have arrived at the conclusion that since, by definition, the volume specified, then we must introduce appropriate "interpolation correction terms" whose purpose is to provide a better volume fraction value for nodes at the computational boundaries. In the next paragraph, we describe the interpolation method and the interpolation correction terms.

Cells to Node Interpolation

We use cell-centered field C I c to approximate of the nodal field C I n by the method we describe here. Subsequently, the MPA constructs an approximation of the interface using the field C I n . We use a simple inverse distance weighted interpolation, known also as Shepard's interpolation (see [START_REF] Franke | Smooth interpolation of large sets of scattered data[END_REF]): where n c is the set of cells that are adjacent to the node n. The sense of adjacency is topologically defined. The set n c is the set of cells that share the node n on their boundary. The above relation is used to interpolate data for every node that does not reside to a physical boundary. For nodes that lie to physical boundaries, an error will be introduced by the above relation due to the missing boundary data.

C I n =
For example, in a Cartesian grid (where the cell's region represents a rectangular parallelepiped) any node that does not belong to a physical boundary, will obtain its data by the above interpolation using |n c | = 8 adjacent cells. However, when a node belongs to the physical boundary, the number of adjacent cells depends on the relative placement of the n next to the boundary. For each physical boundary face the node n belongs to, at least one cell less will be encountered to the set n c . For a Cartesian topology, if the node belongs to two boundary faces of the same cell then at least three cells less will be encountered to n c and if the node belongs to three boundary faces of the same cell then at least seven cells less will be encountered to n c . Therefore, specific actions must be taken so that interpolation does not lose its symmetric features for boundary nodes.

To that end, for physical boundary nodes, we also use the volume fraction field values found at "ghost" cells adjacent to these nodes (the physical boundary nodes where the interpolation is performed). For each ghost cell, we will define points along with their volume fraction field values that contribute to the interpolation corrections. As a result, the above relation is modified for the physical

NC or r 1( f ) = q f v w p f v ; p n DC or r 1( f ) = w p f v ; p n (3.4.5)
and NC or r (c f * ) = NC or r 1( f ), NC or r (c f * ) = NC or r 1( f ). Note that the relations are actually applied for the volume fraction, q = C I , but instead we use the general field q to simplify the notations. The index f v indicates that the field is evaluated to the ghost cell connected to the face f v . The point p f v represents the "centroid" of the ghost cell, defined as p f v = 2 p fp c . The field's value at the boundary, q f v , is obtained either by the boundary condition or, it is extrapolated from the field's value of the adjacent boundary cell and its gradient (a procedure implemented directly to the gradient calculation of field whose gradient is not known, see "Gradient Calculations at Boundary Cells", p.241 for further details). For a cell whose region corresponds to a rectangular parallelepiped, the points used to calculate the above correction term are shown in figure 3.13a.

If the cell has two boundary faces that contain the node n, i.e. c f * = f 1, f 2 , then the term:

NC or r 2( f 1, f 2) = q c + ∇q | c p f 1 v + p f 2 v -p c w p f 1 v + p f 2 v -p c ; p n DC or r 2( f 1, f 2) = w p f 1 v + p f 2 v -p c ; p n (3.4.6)
is added as boundary face corrections in order to take into account the "corner" of the two boundary faces. This term acts like generating a ghost point at boundary corners located at

p 12 = p f 1 v + p f 2 v -p c .
The value of the field is evaluated by linear extrapolation from the value of the cell-center. The correction terms are:

NC or r (c f * ) = NC or r 2( f 1, f 2) + f ∈c f * NC or r 1( f ) DC or r (c f * ) = DC or r 2( f 1, f 2) + f ∈c f * DC or r 1( f ) (3.4.7)
the first term compensates for the absence of data in the corner of the boundary and the second term for the boundary faces. For a cell whose region corresponds to a rectangular parallelepiped, the points used to calculate the above correction terms are shown in figure 3.13b.

Finally, if the cell has three boundary faces that contain the node n, i.e. c f * = f 1, f 2, f 3 , then the term:

NC or r 3( f 1, f 2, f 3) = q c + ∇q | c p f 1 v + p f 2 v + p f 3 v -2 p c w p f 1 v + p f 2 v + p f 3 v -2 p c ; p n DC or r 3( f 1, f 2, f 3) = w p f 1 v + p f 2 v + p f 3 v -2 p c ; p n (3.4.8)
is added as boundary face correction, in order to take into account the "corner" of the three boundary faces. This term acts like generating a ghost point at corner of the three boundary faces located at

p 123 = p f 1 v + p f 2 v -p c
. The field's value at this point is evaluated by linear extrapolation from the field's value at the cell-center. The correction terms are:

NC or r (c f * ) = NC or r 3( f 1, f 2, f 3) + i =1,3 N cor r 2 c f * \ f i + f ∈c f * NC or r 1( f ) DC or r (c f * ) = DC or r 3( f 1, f 2, f 3) + i =1,3 Dcor r 2 c f * \ f i + f ∈c f * DC or r 1( f ) (3.4.9)
The first term compensates for the absence of data in the corner of the boundary of the three boundary faces. The second term compensates for the data of the corners edges (defined by two boundary faces)

and the third term for the corner of the three boundaries.

Summary

The following steps define the interface reconstruction method adopted in this work:

1. Given the cell-centered volume fraction field C I c , approximate the nodal volume fraction field C I n using Shepard's interpolation with boundary corrections.

2. With the nodal volume fraction values known, construct the surface given by the implicit function g I ( r ) = 0.5 -C I ( r ) = 0 given C I n using the MPA. The resulting surface is considered as the reconstructed isosurface.

Note that the appropriate choice of the implicit function g I ( r ) is g I ( r ) = 0.5 -C I ( r ) and not g I ( r ) = C I ( r )-0.5. Even though for both functions the level-0 isosurface is the same, only the g I ( r ) = 0.5-C I ( r ) will generate an appropriately oriented isosurface. To clarify, consider a point in the region inside the interface where C I = 1. The function g I ( r ) = 0.5 -C I ( r ) is negative for these points and thus the MPA algorithm will correctly identify these points as points of the "inside" region. If the function g I ( r ) = C I ( r ) -0.5 had been chosen, then for a point where C I = 1 the function g I ( r ) would obtain a positive value and the point would erroneously be identified as an "outside" point. In the next following we consider some test cases used to evaluate the accuracy of the proposed reconstruction method.

Verification Exercises and Examples

The proposed interface reconstruction method will not, in general, reconstruct exactly the interface. Specifically, there are three sources of error:

1. The values of the volume fraction at the interface will not be 0.5, i.e. if r ∈ S I then C I ( r ) -0.5 = 0. In order to evaluate the accuracy of the interface reconstruction procedure we solve the following exercise for two interfaces.

Initially, we consider that the interface is given implicitly by g I ( r ) = 0. We perform the volume fraction initialization to obtain the cell-centered volume fraction field, C I c , as demonstrated in the previous section. From the field C I c , we calculate the nodal volume fraction field C I n , using the interpolation method demonstrated in the previous paragraph. Finally, an unstructured surface grid is obtained by the MPA, which completes the interface reconstruction. To evaluate the method's accuracy, we calculate the distance of the iso-nodes of the reconstructed surface to the initial surface (denoted here as S I ):

d ( p n s ; S I ) = min p∈S I p n s -p (3.4.11)
The distance d ( p n s ; S I ) is a natural measure of the overall reconstruction error. By the following examples, we will show that the distance of the reconstructed surface to the original surface tends towards zero, as the characteristic grid length scale of the background grid tends towards zero.

Interface Reconstruction Far from Boundaries For the first example, we reconstruct a spherical interface that does not intersect the boundary of the computational domain. In this case, the interpolation corrections for the boundaries do not affect the reconstruction of the interface. We perform the calculations for the standard cube with ν = 20, 40, 60, 80, 120 discretization intervals per edge of the computational domain and an initial spherical interface represented by g I ( r

) = | r | 2 -R 2 , where R = 0.5. The distance d ( p n s ; S I ) is: d ( p n s ; S I ) = p n s -R (3.4.12)
Note that for the same interface, we have defined in the previous section the relative distance error as:

RE (d ) = p c s -R R (3.4.13)
Here, instead of using the iso-nodes, we use the centroids of the iso-patches.

We perform this modification because in the previous section the MPA was used to generate a surface grid of the actual (or exact) surface. Thus, it generated iso-nodes that belonged to the surface by definition, and the centroids of the iso-patches were approximations of actual points of the surface.

In this section we use the dimensionless distance d ( p n s ; S I )/R as a measure of the error, since the iso-nodes of the reconstructed surface are approximations of points that belong to the actual surface S I . In table 3

: L 1 (d n s ) = 1 N s n s ∈ N s d ( p n s ; S I ) R (3.4.14)
where N s is the set of iso-nodes. We present the error diagrams in figure 3.14.

In the last column of table 3.2 we provide the local slope of the L max error curve, i.e. the slope calculated by the value of the error indicated at the same row as and the value of the previous row.

In comparison to the second order accurate results provided by the isosurface of the actual interface (given in the previous section for the centroids of the iso-patches, see figure 3.9), there is an accuracy deterioration (less than the second order accuracy for L max ). We believe that the interpolation scheme contributes most to the accuracy deterioration observed since it will not provide exact results for the volume fraction values at the background grid's nodes. For the L1 errors, the method is second order accurate.

ν To obtain a qualitative notion of the observed convergence, we plot the initial and reconstructed isosurfaces for two different values of ν. Figure 3.15 depicts the two isosurface grids by different colors of the iso-edges. The surface grid shown as a black wireframe (iso-edges), represents the initial isosurface grid. The same initial surface grids were generated in the previous section to solve the volume fraction initialization problem of a spherical interface. The reconstructed grid, obtained after performing the "cells to node" interpolation has gray shaded patches and a blue wireframe.

First of all, observe that the "exact" iso-nodes and the "reconstructed" iso-nodes will not be the same. Specifically, each "reconstructed" iso-node is found in the "inside" region of the initial isosurface. This is clearly the case for the figure 3.15a, where the coarsest background grid (ν = 20) is used. For the finer background grid (ν = 40), the two surfaces are much closer, but we can still distinguish the two grids. The iso-nodes of the reconstructed grid are closer to the exact iso-nodes for cells that contain iso-patches whose normal vectors are aligned to the normal of one of the cell's faces. We also confirmed the last observation by the local distance errors shown in figure 3.16.

From the same figure, we deduce that the reconstructed interface will not be spherical. The iso- nodes are "oscillating" around a mean radius that is smaller than the radius of the initial sphere. These "oscillations" tend to be smaller as the background grid discretization tends to be finer. For example, observe in figure 3.16d that the location where the distance error changes more rapidly are concentrated near the intersection of the lines (x, y, z) : z = x, y = 0 , (x, y, z) : z = y, x = 0 and (x, y, z) : x = y, z = 0 with the surface grid. The same pattern also arises for the relative distance errors of the centroid's for the initial sphere, presented in the previous section, figure 3.9, p.106. Since the volume fraction calculation depends on the centroid of the iso-patches and to reconstruct the interface the volume fraction is required, an erroneous computation of the centroid leads to an inaccurate volume fraction calculation, and thus the two errors are expected to be closely related. As a consequence, the fact that the geometric space representation of an iso-patch is exact only for planar surface regions can account for the observed accumulation of errors.

Interface Reconstruction Near Boundaries In the previous example, the interpolation corrections near the boundaries did not affect the result since the interface did not intersect the boundaries of the computational domain. In the following example, we construct an interface that intersects the boundaries of the computation domain. For comparison purposes we use the same grids as before. In this example we reconstruct the interface of an oblique sinusoidal wave whose initial surface is given by: As in the previous case, we have calculated the distance of the actual isosurface to the approximate isosurface. However, in this case, the distance d ( p n s ; S I ) is approximately evaluated by:

d ( p n s ; S I ) = min r ∈ r S p n s -r ≈ min r ∈ r n s p n s -r (3.4.16)
where p n s is the finite set of points of the approximate surface generated by the MPA algorithm using g I ( r ) = 0.5-C I ( r ), and r n s is the finite set of points of the exact surface generated by the MPA algorithm using g I ( r ) = 0 (given from (3.4.15)). The relative distance error RE (d ), L max and L 1 are calculated as before, using the wave's amplitude A to obtain non-dimensional results, e.g. the relative distance error is defined by:

RE (d ) = d ( p n s ; S I ) A (3.4.17)
As in the previous case we discuss the results obtained for ν = 20, 30, 40, 60, 80, 120. The relative distance errors for ν = 20 and ν = 40 are plotted in figure 3.18. As we can observe, significant errors are located near regions of large curvature and the physical boundaries. The result is reasonable since regions of high curvature indicate regions where the volume fraction values will be rapidly changing and, to obtain better results for the nodal volume fraction field, refinement is necessary near these regions. The last result is also nicely depicted in figure 3.19 where we plot the intersections of the reconstructed interfaces obtained for ν = 20, 40, 80 with the planes x = 0 and x = 1, figures 3.19a, 3.19b respectively. The figures clearly demonstrate that the distance between the reconstructed interface (colored red, blue and green for ν = 20, 40 and 80 respectively) and the actual interface will always be larger to the regions of larger curvature. Note that by "exact" in the figure we mean that the MPA uses the implicit interface function to calculate the iso-nodes (and so construct the exact isosurface). The intersection with x = 1, figures 3.19b and 3.19d consists entirely of iso-patches generated at cells of the physical boundary. In comparison to the iso-patches at section x = 0 there is a small displacement between the iso-patches and the exact interface, even in locations where the curvature is small. We may consider the above as an effect of performing the reconstruction near the boundaries and a consistent observation concerning the distances calculated in figures 3.18.

Finally, we observe that the maximum error is obtained at the furthest right iso-patch in figures 3.19b and 3.19d. The cell containing this iso-patch is a corner cell with two boundary faces (this case is shown in figure 3.13b). Therefore, the nodal field is reconstructed with even less available information.

INTERFACE RECONSTRUCTION 123

The effect of the boundary correction is that, even though the reconstruction is less accurate, we obtain a reconstructed interface that converges to the actual interface. If we have dropped the correction terms, then the reconstruction would provide a constant relative error. For finer discretizations of the background grid, the distance of the exact to the reconstructed interface will become smaller and when ν = 80 the actual, and the reconstructed interfaces can hardly be distinguished both for boundary iso-patches and non-boundary iso-patches.

Concluding Remarks

In this section, we have described the solution of the interface reconstruction problem. The main idea behind the proposed reconstruction is to take advantage of the fact that the set of points for which C I ( r ) = 0.5 holds, are approximations of points that belong to the actual interface's surface. The connectivities between the points are constructed after the MPA to obtain an approximation of the explicit (or parametric) representation of the interface. The method begins by interpolating a cellcentered volume fraction field to nodal point to obtain an approximation of a nodal volume fraction field. The MPA uses the nodal volume fraction field to construct the isosurface g I ( r ) = 0.5 -C I ( r ) = 0.

In comparison to the commonly used interface reconstruction method PLIC, the proposed interface reconstruction produces C 0 -continuous patches of the interface while PLIC constructs discontinuous patches. Conversely, for the PLIC reconstruction the volume of fluid contained in a cell truncated by the reconstructed patch, which represents the interface, will always be the same as the volume of fluid defined by the (cell-centered) volume fraction field provided as input. For the MPA reconstruction, the above property is only retained in the limit of a vanishing grid length scale. Therefore, if the volume fraction were to be recalculated by the reconstructed surface (in the same manner presented in the first section), we would not obtain the initial volume fraction field.

Even though the above observation might be understood as a drawback, it can also be considered as a simplification which allows us to perform a sequence of more complicated steps. Such simplifications are reasonable since the results show that the derived isosurfaces converge to the surface of the "exact" interface with at least linear rates for iso-patches near physical boundaries and second order rates for iso-patches far from boundaries, as the grid is refined.

Finally, we should note that the proposed interface reconstruction method performs the construction of an unstructured surface grid in reasonable execution times. In the last column of the table 3.3 we present the total execution time for the sinusoidal wave test case. The algorithm's execution time is linear with respect to the initial load (the initial number of cells). For the highest loading, the total execution time is around 10 seconds in serial (around 170000 per second which results to around 15000 surface patches). Thus we can use the reconstruction procedure whenever the volume fraction is updated during the solution of the Navier-Stokes equations. The reconstructed interface will be employed along with the normal vector and curvature approximation method describe in the next section, eventually to develop surface tension method in the final chapter.

Normal Vector and Curvature Calculation Using Iso-patch

Neighborhoods

Introduction During different stages of this work, we have tested several methods to approximate the normal and the curvature. During the initial stage, our approximations were based solely on the volume grid and the volume fraction (either directly the volume fraction or a smoothed volume fraction). Having tested several combinations of smoothing methods and methods for approximating the derivative, we could not find an approach that results to converging curvature estimations for the whole range of discretization used. Subsequently, we have tested methods using the surface grid and approximations solely using compact stencils. These approaches provide second order accurate converging approximations with the exact surface grids (exact in the sense that the MPA generates a surface grid using the actual surface's implicit function). However, we have found that they are very sensitive to errors of the normal vectors and prone to the poor grid quality of the generated surface grid.

Finally, we have developed a different set of approaches based on a local least square reconstruction of the isosurface (LSqR). In contrast to the previous methods which use compact stencils of the surface grids, LSqR uses iso-patch neighborhoods. In this section, we present the proposed approach.

Least square approximations are well-established techniques in Computer Graphics, see for example [START_REF] Pighin | Practical Least-Squares for Computer Graphics[END_REF]. In CFD for multifluid flows, the use of local quadratic surface fits or quadratic least square fits to derive approximations of the normal vector or curvature or both,is a common practice followed by front tracking methods (using surface grids) and front capturing approaches (used together with local reconstruction schemes as PLIC). To name a few, [START_REF] Tryggvason | A front-tracking method for the computations of multiphase flow[END_REF] [START_REF] Tryggvason | A front-tracking method for the computations of multiphase flow[END_REF], and Tukovic and Jasak 2004) [START_REF] Sousa | A front-tracking/front-capturing method for the simulation of 3D multi-fluid flows with free surfaces[END_REF] and [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial flows[END_REF] [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial flows[END_REF], implemented a least square quadratic fit to evaluate the curvature of the interface.

The differences between the above methods are subtle and depend on the framework used to represent the interface. A frequently encountered practice for methods using explicit representations of the surface grid, as in [START_REF] Tryggvason | A front-tracking method for the computations of multiphase flow[END_REF] [START_REF] Tryggvason | A front-tracking method for the computations of multiphase flow[END_REF] and Tukovic and [START_REF] Tuković | A moving mesh finite volume interface tracking method for surface tension dominated interfacial fluid flow[END_REF] [106], is to use a quadratic fit method and the sample of nodal points of the patch or adjacent patches. Our approach uses stencils that are generated by nodes of iso-patch neighborhoods. The neighborhood used is a control parameter of the method. A second control parameter is the order of the polynomial basis function. This polynomial approximates the iso-patch neighborhood locally, in the classic sense of a Monge patch.

Monge Patches Consider that we wish to calculate the normal vector and the curvature for a point P , which belongs to a surface S J I (a surface whose normal points from region I to J ). For this point, we construct a Monge patch that explicitly represents the surface as R = R(s 1 , s 2 , z(s 1 , s 2 )), where (s 1 , s 2 ) (denoted afterwards as u, v i.e. u = s 1 , v = s 2 ) are the surface coordinates, mapping in a one-to-one manner nearby points of P on the surface, and z(u, v) is a continuous and differentiable function (as many times as required). By this explicit representation we can obtain all the required differential objects of the surface (as described in the appendix p.225). The normal vector in the ambient space, whose covariant basis vectors are Z i , is:

n = n i Z i = n 1 Z 1 + n 2 Z 2 + n 3 Z 3 (3.5.1)
where the contravariant components are:

n i = 1 S - ∂z ∂s 1 - ∂z ∂s 2 1
and the area element is

S = 1 + ∂z ∂s 1 2 + ∂z ∂s 2 2 (3.5.2)
The curvature of the interface is:

B α α = 1 S 3/2 1 + z 2 v z uu + 1 + z 2 u z v v -2 z u z v z uv (3.5.3)
Therefore, in order to calculate the normal vector and the curvature we have to calculate the first order derivative and the second order derivative of the function z(s 1 , s 2 ) or equivalently z(u, v). Hereafter, the method is straightforwardly formulated by a least squares procedure, which we refer to as the local Least Squares Surface Reconstruction (LSqR).

Local Least Squares Surface Reconstruction

Construction of Point Samples The centroid of a given iso-patch c s is the point, P , where the Monge patch is defined. For the iso-patch, we construct an iso-patch neighborhood c s t s k . The previous notation for the neighborhood implies that for the iso-patch c s we gather its neighboring patches and their neighboring iso-patches and repeat the same action k-times in total. The neighborhood of iso-patches is a set of iso-patches that are topologically close. We define the sense of topological proximity, quantified by the integer k, in the following two manners.

The members of the set of patches that share a node on their boundary with the patch c s are topologically near the patch c s and k = 1. These are the adjacent neighbors to the patch's nodes and we say that they belong to the level 1 neighborhood. The set of iso-patches that share an iso-node on their boundary with the iso-patches of the level 1 neighborhood belongs to the level 2 (k = 2) neighborhood of the iso-patch c s and so forth. We can similarly define different notions of topological proximity by specifying the level 1 neighborhood as the set of iso-patches that share an iso-face on their boundary with the patch c s , and define the other levels as before. To distinguish between the two different notions (underlined above) of topological proximity, we specify whether a neighbohood is based on the iso-nodes by writing c s t s k n s , or the iso-edges (faces for the surface grid) by writing c s t s k f s . Therefore, to specify a neighborhood, we have to specify the sense of topological proximity and the level of the neighborhood. We can describe the set of iso-nodes that the least square procedure uses as a sample, through the iso-patch neighborhoods by: In the figure, we generate the level 1 and level 2 neighborhoods based on the iso-nodes for the blue triangle, the iso-patch c s in the middle of the figures, i.e. the neighborhoods c s t s k n s and c s t s 2 n s shown respectively in figures 3.20a and 3.20b. The iso-patches of the level 1 neighborhood are symmetrically placed around the iso-patch c s . However, the level 2 iso-patch neighborhoods are not symmetrically placed around the iso-patch. Consequently, for such iso-patch neighborhoods the corresponding isonode sample might not be symmetrically distributed around the iso-patch where the approximation using least squares is performed.

c s t s k n s = κ s ∈ c s t s k
Instead, the iso-patch neighborhood is constrained by the level k cell neighborhood of the cell c s c , i.e. the cell of the volume grid that generated the iso-patch. Using this approach the sample of iso-nodes will be symmetrically distributed around a given iso-patch. These neighborhoods are shown in figure 3.21a and 3.21b. Since we still use iso-patch neighborhoods, the reconstructed Monge patch

X Y Z (a) Unconstrained level 1 iso-patch neighborhood X Y Z (b) Unconstrained level 2 iso-patch neighborhood
Figure 3.20: Unconstrained iso-patch neighborhoods based on iso-nodes. The blue iso-patch in the center of the image denotes the iso-patch where we construct the neighborhood. Red iso-patches are members of the neighborhood. The unconstrained iso-patch neighborhoods might result to asymmetric sample of points as in 3.20b.

will refer to the surface local grid, even when interfaces approach each other.

Consider for example the configuration demonstrated in figure 3.22. Suppose that we perform the approximation at the iso-patch κ s contained inside the cell of the volume grid c. Moreover, assume that we use the sample derived by the level 1 topos neighborhood c t n (which contains the cells adjacent to the nodes of cell c), depicted as the shaded 3 × 3 stencil in figure 3.22a. In the same figure, circled crosses mark the iso-nodes that correspond to the sample passed to the least square procedure. As we can observe, some iso-nodes belong to the surface where that iso-patch κ s approximates and some iso-nodes belong to the nearby surface. If we use every node for the least square approximation of the Monge patch, then the derived patch would correspond to a "mean" surface around both actual surfaces rather than a local approximation for the surface of interest. This problem has been implicitly solved since we use iso-patch neighborhoods.

Since we define the iso-patch neighborhood as the set of iso-patches around κ s and constrain them within the node topos neighborhood of cell c (note that κ s c = c, we remind the reader that κ s c is an iso-patch-cell embedment connectivity), the resulting neighborhood is depicted by the shaded region on the surface shown in figure 3.22b. The neighborhood results from the iso-patches in the topos neighborhood c t n in the absence of the upper surface. Therefore, iso-patches that might belong to a different interface do not contaminate the generation of the resulting sample, even when the interfaces are close.

Construction of the Local Orthonormal System

We construct an orthonormal local coordinate system located at the centroid of the iso-patch, by specifying its basis vectors Z 1 , Z 2 , Z 3 . For the simplest case, where the normal vector of the iso-patches does not vary fast, we may choose the basis vector Z 3 so that it coincides with n I | c s , the unit normal vector approximated by the iso-patch c s . The basis X Y Z (a) Iso-patch neighborhood constrained by the level 1 cell neighborhood of its generating cell X Y Z (b) Iso-patch neighborhood constrained by the level 2 cell neighborhood of its generating cell Figure 3.21: Constrained iso-patch neighborhoods based on iso-nodes. Unconstrained iso-patch neighborhoods based on iso-nodes. The blue iso-patch in the center of the image denotes the iso-patch where we construct the neighborhood. Red iso-patches are members of the neighborhood. Constrained iso-patch neighborhoods to cell neighborhoods, c t k n , are symmetrically distributed around the isopatch.

vector Z 1 should be orthogonal to Z 3 . We define the vector Z 1 as:

e n s = a n s -( n a n s ) n a n s -( n a n s ) n where a n s = p n s -p c s (3.5.5)
Each member from the set of iso-nodes in the neighborhood c s t s k may serve as a possible choice to obtain Z 1 . In order to avoid making a specific choice we set:

Z 1 = e | e|
where e =

ν s ∈ c s n s e ν s (3.5.6)
If | e| is close to zero, then we set e as defined by (3.5.5) for the first iso-node of the iso-patch. Subsequently, we set

Z 2 = Z 3 × Z 2 .
Through this local coordinate system we define a mapping S : p → (u, v) from a point on the isosurface p to the surface coordinate (u, v) and a mapping z( p) as the projections of pp c s to the basis vectors:

S( p) = p -p c s Z 1 , p -p c s Z 2 and z( p) = p -p c s Z 3 (3.5.7)
When the normal varies fast in neighboring isopatches, the above definition of (u,v,z) might not provide a one-to-one mapping between (u,v) and z. We should revisit the above construction for these cases, however, we do not provide the details. The basis functions b M are constructed by choosing two sets of one-dimensional basis functions and finding their tensor product. In every calculation we used the set of polynomial basis functions of order n, 1, x, x 2 , ..., x n , that derives m = (n + 1)(n + 2)/2 two-dimensional basis functions. Quadratic polynomials are the most frequently encountered choice [START_REF] Tryggvason | A front-tracking method for the computations of multiphase flow[END_REF], [START_REF] Sousa | A front-tracking/front-capturing method for the simulation of 3D multi-fluid flows with free surfaces[END_REF], [START_REF] Aulisa | Interface reconstruction with least-squares fit and split advection in threedimensional Cartesian geometry[END_REF], [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial flows[END_REF], [106]. In our approach the input provided to LSqR is the maximum allowable order. The coefficients a M are determined by minimizing the functional: there are not enough points in the sample, the implementation drops the order of the basis function used so that the problem can be solved, we will refer to specific cases near boundaries later on.

L( z) = ν s ∈ c s t s k w( p ν s ; p c s ) z( p ν s ) -z S( p ν s ) 2 where w( p ν s ; p c s ) = 1 | p ν s -p c s | 3
The first and second order derivatives of the function z(u, v) evaluated at the centroid of the iso-patch are approximated by the function z(u, v), e.g.:

∂z ∂u c s = ∂ z ∂u c s = M =1,m a M ∂b M ∂u (0,0) (3.5.10)
Substituting to (3.5.3) we obtain an approximation for the curvature at the iso-patch and from (3.5.1), we obtain a new approximation of the normal vector at the centroid of the iso-patch:

n I c s = - 1 S c s ∂ z ∂u c s e | e| - 1
S c s ∂ z ∂v c s n| c s × e | e| + 1 S c s n| c s (3.5.11)
Note that in the last relation, n I c s is the new approximation of the normal vector and n| c s is its previous approximation defined by the iso-patch. Therefore, the above relation proposes a correction for the normal vector. Similar corrections can be found for the centroid of the iso-patch and its area. However, the implementation changes neither the centroid nor the area of the iso-patch, and for the time being, we have not studied the effects of these corrections.

To summarize, the proposed least square method acts on an iso-patch c s and:

1. Provides approximations for both the normal vector and curvature of the iso-patch 2. Uses the sample of points generated by gathering the iso-nodes of the iso-patch neighborhoods constrained to the level k neighborhood of the cell where the iso-patch lies 3. Expects as input the maximum allowable polynomial order, to define the basis functions

In the next part of this section, we present the results obtained for a spherical interface.

Application to a Spherical Interface

We apply the LSqR to a spherical interface in the background Cartesian grid used in the previous sections (for spherical interfaces, R = 0.5) and calculate the error fields for the cells that are intersected by the interface. To define the errors, we use the most frequently encountered definitions in the literature, see for example [START_REF] Cummins | Estimating curvature from volume fractions[END_REF], [START_REF] Ivey | Accurate interface normal and curvature estimates on three-dimensional unstructured non-convex polyhedral meshes[END_REF]. The error of the normal vector is:

RE ( n I c ) = n I c -N I c , if |c c s | = 0 0 , if |c c s | = 0
where

N c = p c s | p c s | (3.5.12)
where n I c is the approximation obtained for the normal vector of the interface at the cell c and c c s is the set of isopatches generated for the cell c (which always contains at most one member for the spherical interface). Similarly, the error field of the curvature is given by:

RE (κ c ) = |κ c -K c | K c , if |c c s | = 0 0 , if |c c s | = 0 where K c = - 2 R (3.5.13)
where κ c is the approximation obtained for the curvature of the interface at the cell c. To obtain a notion of the performance of each method, we calculated the maximum and mean values of the RE fields denoted as L max and L 1 , respectively. The mean values refer to the total number of volume grid cells that contain iso-patches.

In the following, we begin by the case where the MPA reconstructs isosurface using the implicit function. Thus the isosurface presents the actual or exact interface. We show that we can achieve high-order of accuracy approximations. After establishing the theoretical validity of the method, we continue with practical cases where the MPA reconstructs isosurface using the volume fraction or equivalent smoothed fields. We show that we can obtain second order accurate approximations for both the normal vector and the curvature regarding the L max norm for the whole range of discretizations used, given that the MPA reconstruction uses a smooth enough C I field.

Results for Isosurfaces Derived by the Implicit Function

We use the LSqR along with different isopatch neighborhoods and polynomial orders to approximate the normal vector and curvature. To obtain an initial approximation of the normal vector, we use the geometrical space representation of the iso-patch. We provide the calculated error alongside the normal vector errors of LSqR. For LSqR approximations, the indicated levels refer to the levels of the cell neighborhood that contains the iso-patch neighborhood, and not the levels of the iso-patch neighborhood. We repeated the calculation for each level using different orders of the polynomial basis functions. Note that we used only even orders. If we express the function z(u, v) = (R 2u 2v 2 ) -R derived for any point of the sphere in the local coordinate system ( Z 1 , Z 2 , n) in a Maclaurin power series, only even orders remain. Therefore, we expect that the LSqR based on basis functions of an even order n and the odd order n + 1 will ideally provide the same coefficients.

We present the results in figure 3.23, for both the normal vector (figures on the left) and the curvature (figures on the right). We always obtain an improved approximation for the normal vector in comparison to the normal vector estimations by the iso-patches. The estimates of the normal vector are always third order accurate, and we did not achieve a significant improvement using higher order polynomials, after fourth order. The results of the curvature calculation indicate that we can attain higher order approximations, up to sixth order accurate.

We have systematically used samples that contain more points and at the same time higher order basis functions. For example, by using fourth order basis function and level 1 neighborhoods, we obtain 2R δ 2R δ

L max ( n) Isopatch Lvl1 -Ord2 Lvl1 -Ord4 Lvl2 -Ord4 Lvl2 -Ord6 Lvl3 -Ord6 O(x -1 ) O(x -3 )
2R δ L max (κ) Lvl1 -Ord2 Lvl1 -Ord4 Lvl2 -Ord4 Lvl2 -Ord6 Lvl3 -Ord6 O(x -2 ) O(x -4 ) O(x -6 )
L 1 ( n) Isopatch Lvl1 -Ord2 Lvl1 -Ord4 Lvl2 -Ord4 Lvl2 -Ord6 Lvl3 -Ord6 O(x -1 ) O(x -3 )
Lvl1 -Ord2 Lvl1 -Ord4 Lvl2 -Ord4 Lvl2 -Ord6 Lvl3 -Ord6 O(x -2 ) O(x -4 ) O(x -6 )
Figure 3.23: Normal vector and curvature L max and L 1 error diagrams using least square differencing with different neighborhoods and polynomial basis for the exact isosurface of a spherical isosurface (R = 0.5) and Cartesian background grids.

better approximations than second order basis functions and level 1 neighborhoods, for both L max and L 1 norms. If we continue to increase the order, then we would eventually observe a saturation to the results since some point samples might not contain enough points (in that case the LSqR automatically decides to drop the polynomial order so that approximations can be derived). Increasing the level of the neighborhood to two, and keeping the same order, we observe a slight improvement for the L max norm. However, this improvement does not induce any improvement in the L 1 norm, green markers on figures curvature diagrams of 3.23. When we increase the order to sixth for level 2 neighborhoods, we improve the curvature approximation, and we achieve faster convergence. Similarly to the fourth order polynomials, if we increase the level using sixth order polynomials we obtain slightly less accurate approximation but now for both L max and L 1 norms. By repeating this exercise, we will establish a notion of good parameter choices for the combination (level, order) used as input parameters to LSqR.

We note that the above results were obtained using the weight function given in (3.5.9). Weights systematically improve the approximations for the above cases. However, the same improvement was not observed for the cases we describe next. Therefore, the results presented next were derived by LSqR without using weights.

Application for Reconstructed Interfaces

The results of the previous paragraph established the performance of the method for the ideal isosurface, constructed by iso-nodes belonging the spherical interface. In this subsection, the MPA constructs the isosurface by the volume fraction. As we have already mentioned in the previous section, the iso-nodes contain reconstruction errors which tend to become smaller when we refine the background grid. Consequently, in this paragraph, we present results derived by the LSqR method and samples of iso-nodes that approximate the actual points of the interface.

Since the construction introduces some errors, the results obtained by the quadratic polynomials (second order) and cubic polynomials (third order) will not be the same. Nevertheless, the difference for larger neighborhoods tends to be small, as we will observe next. To that end, we used polynomials of second, third and fourth order, and constrained the iso-patch neighborhoods by the level 1, level 2 and level 3 cell neighborhoods of the cells containing the iso-patches. The L max error norms are plotted in figure 3.24 and the L 1 error norms in figure 3.25. Shapes correspond to neighborhood levels, circles: level 1, squares:2, diamonds:3, and the colors to polynomial orders, blue: second, green: third, red:

fourth.

As we can observe, from figure 3.24, fourth order polynomials tend to provide less accurate (for level 1) or at least as accurate results (for level 2 and 3) as third order polynomials for both the normal and the curvature. More precisely, the maximum errors of the normal vectors obtained using fourth order polynomial tend to coincide to the errors of cubic polynomials, when larger discretizations and larger samples are used (higher levels k). The maximum errors of the curvature for third order polynomials tend to coincide for coarser discretizations to the errors of the of quadratic polynomials. Thus, the contributions of third order terms have a significant effect on first-order terms, while fourth order terms have a considerable impact on the calculation of second-order terms. The same observation holds 2R δ 2R δ for the L 1 errors, see figure 3.25. In short, higher order polynomials are prone to errors that result by including to the least square procedure coefficients for third order terms and fourth order terms. These contribute significantly to altering the lower order coefficients. Using larger neighborhoods results

L max ( n) Lvl2 -Ord2 Lvl2 -Ord3 Lvl2 -Ord4 10 1 10 -2 10 -1 10 0 2R δ L max (κ) Lvl2 -Ord2 Lvl2 -Ord3 Lvl2 -Ord4
L max ( n) Lvl3 -Ord2 Lvl3 -Ord3 Lvl3 -Ord4
to richer samples and, as we observe from the figures, the richer neighborhoods consistently provide better approximations for both normal vectors and curvature.

The approximations for the normal vector have converging tendencies for smaller discretization and eventually begin diverging. The onset of divergence seems to be affected by both the level of the neighborhood and the polynomial order used. Specifically, for larger neighborhoods and higher order polynomials the convergence is retained till finer discretizations. Similar observations hold for the results of the curvature where the convergence tendencies are clearly less pronounced.

Turning our attention to the less strict L 1 norms, figure 3.25, we can observe that the convergence degradation (as described before for the L max norm) is expressed by a systematic change of the asymptotic convergence from second-order to almost constant errors. For both the normal vector and the curvature, we obtain second order converging approximation in the sense of the L 1 norm for level 3 neighborhoods and quadratic polynomials. The LSqR method using level 3 neighborhoods and cubic polynomials also provides estimates that converge. 1. The LSqR method using quadratic polynomials provides the best approximations in the sense of the L max norm, for both the normal vector and the curvature 2. The LSqR method using large samples (samples based on higher level k neighborhoods) improves the approximations for finer discretizations in the L max sense 3. The LSqR method using polynomials of higher order improve the estimates for larger neighborhoods and coarser discretizations in the L 1 sense

2R δ L 1 ( n) Isopatch Lvl1 -Ord2 Lvl1 -Ord3 Lvl1 -Ord4 Lvl2 -Ord2 Lvl2 -Ord3 Lvl2 -Ord4 Lvl3 -Ord2 Lvl3 -Ord3 Lvl3 -Ord4 O(x -2 )
Lvl1 -Ord2 Lvl1 -Ord3 Lvl1 -Ord4 Lvl2 -Ord2 Lvl2 -Ord3 Lvl2 -Ord4 Lvl3 -Ord2 Lvl3 -Ord3 Lvl3 -Ord4 O(x -2 )
We have already established that when the volume fraction is the input to the MPA, the (output)

isosurface is a second-order accurate approximation of the surface of the actual interface (see section 3.4, p.110). The errors of the curvature indicate that there are regions where the reconstruction errors introduced to the iso-nodes are interpreted by the LSqR as local curvatures. When the LSqR method uses higher order polynomials, the curvature errors tend to be more sensitive to the reconstruction errors of the MPA. Moreover, when the LSqR method uses larger neighborhoods, the effects of the MPA errors to the curvature errors are less pronounced, indicating that the MPA errors are smoothed out by the least squares procedure.

Interfaces Reconstructed by the K k-smoothed Volume Fraction Fields K k smoothing is a simple kernel smoothing method. The chosen kernel is a top-hat function whose support coincides to the level k cell neighborhood based on node-adjacent cells. By approximating the smoothing integral by the midpoint rule we obtain:

K k (C I )| c = 1 V (c t k n ) κ∈c t k n C I κ V κ (3.5.14)
The lowercase k in both LHS and RHS obtains integer values and defines the level of the neighborhood used. For example, one might generate the field K 1(C I ) c or K 2(C I ) c and so on. The K k smoothing proposes a simple smoothing technique that is applicable both on Cartesian and grids without specific topological features. An interesting feature of this approach is that it does not require to extend the neighborhood multiple times (as when we perform a neighborhood search based on a fixed kernel radius), and thus it is very fast. However, the most interesting feature is that K k smoothing always

2R δ L max ( n) Lvl2 -Ord2 Lvl2 -Ord3 Lvl2 -Ord4
2R δ L max ( n) Lvl3 -Ord2 Lvl3 -Ord3 Lvl3 -Ord4
Isosurface reconstructed by K 2(C I ) 2R δ 2R δ

L max ( n) Lvl2 -Ord2 Lvl2 -Ord3 Lvl2 -Ord4
L max ( n) Lvl3 -Ord2 Lvl3 -Ord3 Lvl3 -Ord4
L max (κ) Lvl2 -Ord2 Lvl2 -Ord3 Lvl2 -Ord4
L max (κ) Lvl3 -Ord2 Lvl3 -Ord3 Lvl3 -Ord4
Isosurface reconstructed by K 2(C I )

L max (κ) Lvl2 -Ord2 Lvl2 -Ord3 Lvl2 -Ord4
L max (κ) Lvl3 -Ord2 Lvl3 -Ord3 Lvl3 -Ord4 O(x -2 )
Figure 3.27: Curvature L max error diagrams using least square differencing with level 2 and 3 neighborhoods for the reconstructed interface by K 1(C I ) and K 2(C I ) of a spherical isosurface (R = 0.5) and Cartesian background grids polynomials. We also observe that for the coarse discretizations, the errors for the K 1(C I ) isosurface are smaller, in comparison to the errors for the K 2(C I ) isosurface. However, the latter estimations begin diverging, while the former are second-order accurate. In conclusion, even thought the LSqR method using samples of the K 2(C I ) isosurface does not provide more precise results for coarse discretization, the result for finer discretization converge, and eventually we obtain better approximations for the finer discretizations.

Summary and Conclusions

In this section, we have presented the Least Squares Reconstruction method (LSqR) method. The LSqR method generates a local parametric equation (a Monge patch) that approximates a part of the surface. We approximate the normal vector and the curvature of the interface from the explicit local representation of the surface patch. The method has two distinguishing features related to the construction of the Monge patch:

1. The part of the reconstructed surface is locally defined by point samples, which are generated using both the iso-patches and cell neighborhoods. Each point sample consists of the iso-nodes belonging to the iso-patch neighborhoods, restricted to a level k cell neighborhood. The above approach allows to obtain point samples that cover the surrounding space, as uniformly as allowed by the isosurface grid generated by the MPA.

2. The basis functions used can be arbitrary chosen. Therefore, we can test a variety of parametric equation and define different basis function near boundaries. In the previous test cases, we have used this approach to test polynomial basis functions of different orders.

We have tested the proposed method using sample constructed by approximate isosurfaces of a spherical interface. For each case, we have examined different sets of the method's parameters, namely, (i) the level of the cell neighborhood where the iso-patches neighborhoods are constrained to and, (ii) the polynomial order used. We have demonstrated that quadratic and cubic polynomials for iso-patch neighborhood restricted to level 2 and level 3 neighborhood, provide the best results.

Moreover, the LSqR method provides second-order accurate results when the interface reconstruction uses the smooth volume fraction field K 2(C I ). Finally, the results indicate that the method is sensitive to the approximation errors of iso-nodes.

Spherical interfaces are ideal as test cases for the presented methods. Firstly, they allow a relatively easy analysis of the results from a practical perspective, because their curvature is constant and there are no iso-patches generated near boundaries. Secondly, for spherical interfaces the MPA provides surface grids with different topological features, namely, iso-patches with small and large aspect ratios, iso-patches whose neighbors of considerably different areas, and iso-patches defined by a different number of iso-nodes and iso-edges. Finally, some classic test cases of surface tension methods use spherical interfaces. Therefore, understanding the accuracy of the normal and curvature estimations permits us to draw conclusions for the performance of a surface tension method as well, which is the subject of the next chapter.

Conclusions

In this chapter, we have presented two fundamental procedures related to the multifluid problem, the volume fraction initialization procedure, and the interface reconstruction procedures. The solution of both problems directly used the Marching Polyhedra Algorithm, developed for this work as a part of the ISIS-CFD extensions. The method's generality offers two significant features. Firstly, the volume fraction initialization procedure can handle complicated configurations, as when more than one interface is defined. Secondly, the interface reconstruction procedure can generate approximations of either a single interface or multiple disjoint interfaces and results to a general unstructured surface grid. As a result, the method handles automatically merging and separation of the interfaces, because the MPA uses the volume fraction provided by the solution of the volume fraction transport equation. Another indirect application of the MPA, but directly related to the unstructured surface grid that represents the air-water interface, is the calculation of the normal vector and the curvature of the interface.

The proposed method, named LSqR, locally generates Monge patches that approximate the isosurface. For the cases we have tested and under the condition that the volume fraction field is smooth enough, we have demonstrated that the discretizations of LSqR are second order accurate. Specifically, to obtain smooth volume fraction fields, we have proposed the K k smoothing, probably the simplest kernel smoothing method.

Up to now, we have repeatedly used several concepts related to grid entities that are not classically encountered in the finite volume method, using collocated variables and compressive discretization schemes. First of all, the VOF method using compressive discretization schemes does not require for its formulation a surface grid. Moreover, the collocated finite volume method uses for its discretizations the cell-faces c f connectivity. In the previous sections, we have defined other connectivities, such as the node-cells n c connectivity and the notion of ghost cells, which are used to perform the cells to node interpolation. Finally, we have also repeatedly used the concepts of neighborhoods for both cells and iso-patches and defined their level, which quantifies their topological proximity. The idea of grid entity neighborhoods and subsequently their implementation is crucial for both LSqR and K k smoothing.

All of the above describe different stencils that are used by the proposed numerical schemes. These stencils must be derived using simpler stencils that are given by a grid generation software.

Since we must implement any numerical scheme in a computer code, two problems arise. Firstly, how can we formulate the stencil generation problem to write an algorithm used to prepare this stencil (and thus subsequently be able to use the stencil)? Secondly, how can we ensure that the stencils will be correctly generated in parallel and, of course, that all the required data will be available to implement correctly the numerical schemes related to the stencil? It is clear that both problems should be treated in a unified manner and not by a case-dependent approach, as specific methods (procedures) for each connectivity or numerical scheme introduced.

The solution of these problems resulted to a rich data-structure implemented in the ISIS-CFD extensions framework. The data-structure consists of the required components to generate new stencils (represented by connectivities). At the same time data transfers take place dynamically, between both CHAPTER 3. APPLICATIONS OF THE MPA adjacent and non-adjacent processes, to ensure that all the data will be available in each process.

Therefore, we implemented an approach that generalizes the classic multiblock (parallel) computations.

To clarify, the data transfers of classic multiblock approaches ensure that the foreign data correspond to the data required by numerical schemes that use the f c connectivities. In the dynamic (generalized) multiblock approach, the data transfers ensure that foreign data correspond to the data required by numerical schemes that use any connectivity that specifies a cell stencil. The relevant details are interesting, since they propose simple and concrete formulations using notions of the set theory to describe numerical schemes (as the formulations presented in this section), and concepts encountered in parallel processing. However, the implementation in parallel involves a significant number of details (which include, for example, how the classic multiblock approach is formulated, and what are the necessary extensions to obtain the generalized multiblock approach) that are outside the scope of this work. The reader should note, that for the realization of the methods described above, we have performed an extensive revision of the parallel concepts encountered in the classic finite volume method.

Besides the directly relevant application of the MPA to VOF, the parts of the MPA can be used in a variety of ways. For example, the MPA can be refactored to a volume grid generation method, figure 3.28, or a method for the initialization of the volume fraction of triangulated surfaces and a method to dynamically construct matching faces of sliding grids. Therefore, the MPA can be used in a variety of ways to introduce to ISIS-CFD other numerical schemes, as, immersed boundary methods, see Peskin (2002) [START_REF] Peskin | The immersed boundary method[END_REF] and Mittal and Iaccarino (2005) [START_REF] Mittal | Immersed Boundary Methods[END_REF]. Finally, every method that we have introduced in this chapter is directly related to the new surface tension methods proposed for the context of this work to ISIS-CFD, and the realization of their computer simulations, both described in the following chapter. suggested by experiments and in excellent comparison to Tomiyama's [START_REF] Tomiyama | Terminal velocity of single bubbles in surface tension force dominant regime[END_REF] analytical relation, which links the bubble's aspect ratio to its terminal velocity. Moreover, through the developed surface tension methods we can capture complex physical phenomena where merging and separation of the interfaces take place. Therefore, ISIS-CFD could be used to perform direct numerical simulations of multifluid flows with surface tension, which is a crucial capability to assist the development of other more complicated numerical models for multiphase flows. Finally, we demonstrate that the MPA can be used in test cases of engineering interest to separate small scales from larger scales, which subsequently proposes the further study of both surface tension and multiphase models in hydrodynamic flows.

Normal Vector Approximation Errors

L max ( n) C DS -K 2(C I ) C A -K 2(C I ) LSqR -K 2(C I ) O(x -2 ) 10 1 10 2 10 -3 10 -2 10 -1 2R δ
L 1 ( n) C DS -K 2(C I ) C A -K 2(C I ) LSqR -K 2(C I ) O(x -2 )
Curvature Approximation Errors

L max (κ) C DS -K 2(C I ) C A -K 2(C I ) LSqR -K 2(C I ) O(x -2 ) 10 1 10 2 10 -3 10 -2 10 -1 10 0 2R δ
L 1 (κ) C DS -K 2(C I ) C A -K 2(C I ) LSqR -K 2(C I ) O(x -2 )
Figure 4.1: Comparison of methods for normal and curvature approximations with discretizations based on the volume grid, the surface grid with compact stencils and the surface grid with iso-patch neighborhoods. Blue: Volume grid method with compact stencils, green: Surface grid method with compact stencils, red: Surface grid method with extended stencils

Smoothed Surface Tension Methods

Smoothed surface tension methods can be easily implemented in any volume of fluid code. Provided that, a generic estimate of the normal vector, N , and the curvature of the interface, K , are available for each cell, we add the following source term for the momentum equations written for the cell c:

Σ c = σK c N c (4.2.1)
where σ is the surface tension coefficient, a known constant. We can write indirect, direct or hybrid methods, using the above formulation. For each method, both the generic scalar cell-centered field, K c , and the generic vector cell-centered field N c , have to be specified.

Continuous Surface Force -CSF Methods following the CSF approach of Brackbill et al. (1992) [10] are formulated directly using:

N = - ∇C I ∇C I (4.2.2)
which provides the cell-centered field:

N c = - ∇C I | c ∇C I | c (4.2.3)
The cell-centered field m c = ∇C I | c is approximated by gradient discretization methods based the on volume grid. The curvature is obtained by either evaluating directly:

K c = -∇ N | c (4.2.4)
and discretizing the divergence operator of the above relation, or for the case where a local least square approximation of the volume fraction is available, K c is obtained by the extended form of the above relation as:

K c = Z i j ∇ i m j | c ∇ k C I | c - Z i k Z j l ∇ i C I | c ∇ j C I | c ∇ k ∇ l C I | c ∇ k C I | c 3 (4.2.5)
Finally, following [START_REF] Brackbill | A continuum method for modeling surface tension[END_REF] [10], the source term that appears in the momentum equation is scaled using a density factor:

Σ c = σK c N c ρ| c 〈ρ〉 (4.2.6)
where ρ is the smooth density field, ρ = ρ I C I + ρ J C J for the two fluid case, and 〈ρ〉 is the mean density value for the (two) fluids separated by the interface, i.e. 〈ρ〉 = (ρ

I + ρ J )/2.
Continuous Surface Stress -CSS Our implementation of CSS method of Lafaurie et al. (1994) [51] is formulated using the cell-centered vector field m = ∇C I . The source term is:

Σ i | c = σ ∇ i m k c -∇ j n i m j c (4.2.7)
The above term is calculated after the construction of the cell centered field m| c . The implementation of the CSS method uses classic Gauss approximations for the gradients. The method can be indeed implemented using least squares approximation. However, for the time being, Gauss approximation for the gradients of CSS is the only available option.

The discretization of the gradient operators are based to Gauss theorem for both m and the source term Σ. Specifically for the first term of the sum in (4.2.7) we have:

∇ m c = 1 V c f ∈c f m f s f (4.2.8)
and for the second term:

∇ j n i m j c = 1 V c f ∈c f n i | f m j | f s f j where n| f = m| f m| f (4.2.9)
We note that s f j are the components of the surface area vector of the face f oriented with respect to the cell c. Moreover, the indices c, f ,n are reserved to indicate a cell, face or node respectively and not tensor components. The variables that are denoted by (•)| f indicate that these are evaluated using the face reconstruction scheme at the centroids of the faces.

Volume fraction Manipulations Both methods described above provide approximations for the source term using the volume fraction directly. During a numerical simulation, the initial volume fraction field will lose its strict sharp characteristics, i.e. it might not vary from 0 to 1 over almost one characteristic grid length scale in the direction of the interface's normal. Since the schemes require the discretization of second-order derivatives for the volume fraction for every cell, the calculation is sensitive to volume fraction smearing. To avoid the accumulation of these errors, we have implemented two different trimming methods. Even though the modification to the volume fraction might also remove some values which are correctly defined, trimming prevents the accumulation of curvatures that result by numerical errors.

For the first trimming method, we choose two cut-off values for the volume fraction. Each value corresponds to the closest allowable volume fraction value to either zero or one. Volume fraction values below the small cut-off value are forced to be zero, and volume fraction values above the large cut-off value are forced to be one. This simple modification does not allow small and large values of the volume fraction to contribute the surface tension calculation. We choose the cut-off values 0.05 and 0.995, so volume fraction values which are smaller than 0.05 are set to 0 and volume fraction values which are higher than 0.995 are set to 1. On the other hand, this trimming method does not enforce a sharp volume fraction field.

The second trimming method is derived directly by the MPA. First, the volume fraction is used to build the C I = 0.5 isosurface. Subsequently, the volume fraction is replaced by the volume fraction that corresponds to the C I = 0.5 isosurface. This approach can be viewed as a reinitialization approach of the volume fraction and furthermore, enforces a volume fraction that is sharp.

Subsequently, the volume fraction can be smoothed to provide the field that will be used by either the CSF or the CSS methods. For the second trimming method, smoothing the volume fraction is required to ensure that the generated source term will envelop the initial volume fraction field (the volume fraction before trimming). However, using the MPA to trim the volume fraction, smooth and calculate the normal and the curvature and finally to obtain the source term, does not have any advantage over the following method which formulates of surface tension as a hybrid method. Instead, it adds a time penalty to the whole procedure, which indirect formulations of surface tension avoid by introducing approximations of the normal vector and the curvature using only the volume fraction.

Isosurface Front Tracking -IsoFT Both CSF and CSS belong to the category of indirect surface tension methods. An explicit representation (a surface grid) of the interface, is not required for their implementation. The isosurface front tracking or IsoFT discretizes the surface tension surface integral as obtained in the one-fluid formulation:

Σ = S J I σB α α nK d S (4.2.10)
where S J I is the surface of the interface separating the fluids I , J , n is the oriented normal vector of the surface S J I (pointing from the region of fluid I to the region of fluid J ), and B α α is the curvature of the interface (as defined by the normal vector with the previously described orientation). Note that we use B α α for the curvature of the interface defined on the surface and K for the curvature field as defined in the ambient space. The IsoFT method in ISIS-CFD is formulated either as a direct surface tension method (every term of the integral is obtained using data of the surface grid) or as a hybrid surface tension method (surface grid approximations are employed for some terms and volume grid approximations for others).

Beginning by the volume fraction obtained by the solution of the advection equation, the MPA builds the isosurface C I = 0.5. The least square reconstruction (LSqR) method approximates the isopatch-centered fields of the the normal vector and the curvature, n c s and B α α | c s respectively. Recall that the LSqR method approximates the normal vector and curvature by constructing locally to an iso-patch, a high order polynomial representation of the surface. We briefly described the details covering the calculations performed, the construction of the neighborhoods, and the choice of the parameters that enter the method, in the previous chapter (Section 3.5).

Next, we choose a certain kernel K ( x; x c ) centered at each cell center x c . The kernel's support defines the region V ( x c ) = V c . In turn, this region defines a cell neighborhood that we denote c(V c ). This cell neighborhood (i) might contain cells where the MPA constructed iso-patches (ii) or consists entirely of cells that are far from the interface and thus the MPA did not construct iso-patches. For a cell placed as described in the second case, the surface tension source term is zero. For a cell c placed described in first case case, if we denote the set of isopatches that belong to the its cell neighborhood c(V c ) as c s (V c ), then the integral (4.2.10) can be directly discretized using the midpoint rule for each isopatch in c s (V c ):

Σ c = κ s ∈ c s (V c ) σB α α κ s S κ s K ( x κ s ; x c ) (4.2.11)
where S κ s is the surface area vector of the isopatch κ s , i.e. its normal vector multiplied by the area of the isopatch.

Consider the case where the kernel is a top-hat function whose support coincides to the region of the grid's cells, i.e. K ( x; x c ) = 1/V c for every point inside the cell c. Then we obtain the simplest implementation of the method. The support of the kernel coincides with the region of the cell, V c = V c , and thus the cell neighborhood defined by this region contains a single cell, i.e. the cell c, c(V c ) = {c}.

As a result, the set of isopatches that belong to this cell (the cell neighborhood for this degenerate case) coincides to the cell-isopatches embedment connectivities, c s (V c) = c c s , and the source term obtained from (4.2.11) becomes:

Σ c = 1 V c κ s ∈c c s σB α α κ s S κ s (4.2.12)
For the most frequently encountered case, where a single isopatch is generated inside the cell c, we have:

Σ c = 1 V c σB α α c κ s S c κ s ( 4.2.13)
Therefore, in this case the source term would be concentrated at the cells that are intersected by the interface. Since the source term will be numerically integrated to obtain the (averaged) pressure field, a single value of the source term (in the sence of the normal) is not enough to obtain the correct values of pressure. As a result, a larger kernel has to be used.

The kernel controls the "spreading" of the source term around the isosurface. In principle, the larger the support of the kernel, the more cells the neighborhood c(V c ) contains. Thus, for a cell c intersected by the interface, the iso-patch neighborhood c s (V c ) contains more iso-patches and the source term of the cell c corresponds to larger regions of the surface. Moreover, the source term obtains values for wider regions around the ambient space of the interface. In theory, the above observations were expressed by the characteristic length scale of the kernel (see 66, figure 2.6), instead the cell neighborhoods.

When we derived the one-fluid equations, we performed the averaging by using a single kernel for all equations. The kernel defined the mean fields, the source terms, and the volume fraction. As the volume fraction is transferred inside the computation domain, the kernel used for the construction of the original volume fraction might lose its original significance, and it is only implicitly defined.

Therefore, a formulation using a kernel as described above could be utilized in an ideal numerical framework that explicitly retains the kernel's support. In practice, isoFT is formulated as a hybrid surface tension method, as presented next, rather than a direct surface tension method (as presented above).

Consider that the surface tension source term is derived from a kernel K . We rename the surface tension source term that we get from the direct formulation, relation (4.2.11), as:

Σ c = κ s ∈ c s (V c ) σB α α κ s S κ s K ( x κ s ; x c ) (4.2.14)
so that we can clearly distinguish it from the source term that appears in the momentum equations.

The source term Σ corresponds to the distributed surface tension force to the support of the kernel K .

We seek the distributed surface tension force to the support of the kernel K , where < , the kernel where the advected volume fraction C I is defined. We emphasize the following particularities:

1. The kernel K is different than the kernel K used for the definition of the volume fraction C I .

2. The kernel K is not explicitly present in any other part of the formulation but is implied through its volume fraction C I i.e. the volume fraction obtained directly by ISIS-CFD.

Suppose that we redefine a volume fraction C I defined by the kernel K . The gradient of this volume fraction can be obtained by (see also p.47):

∇C I = - S J I nK d S (4.2.15)
which can be explicitly approximated by the midpoint rule:

∇C I | c = - κ s ∈ c s (V c ) S κ s K ( x κ s ; x c ) (4.2.16)
The discretization of the gradient of the volume fraction C I , ∇C I , obtained by (4.2.16) will not be the same as the discretization of the gradient of the volume fraction C I . Specificially, the discretization of the gradient of the volume fraction C I depends on the background volume grid. It is also evident by comparing (4.2.14) and (4.2.16), that the above term (4.2.16) coincides to the source term of surface tension for a unit surface tension coefficient and unit curvature everywhere. As a result, the distributed surface tension "stress" due to the curvature in the support of the kernel K is:

(σB α α ) = - Σ ∇C I | ∇C I | 2 (4.2.17)
We have dropped the "approximation at cell c operator (•)| c for now, but it is indeed implied for each quantity on both sides. Recall that the tilde operator is the characteristic interface value operator (introduced in p.50) and the above relation contains entities that are derived by the isosurface.

In view of relation (4.2.15) applied for the volume fraction C I (defined by the implied kernel K ) we obtain:

∇C I = - S J I nK d S (4.2.18)
Since the kernel K is implied we cannot evaluate the right hand side. However, by approximating the gradient of C I using the background volume grid, we obtain the above distributed quantity that corresponds to the implied kernel, i.e. all the geometrical information required to set up the distributed surface tension force for the implied kernel K .

As a result, to obtain the distributed surface tension source term that corresponds to the support of the kernel K , with < , from (4.2.14) and (4.2.16) we evaluate (4.2.17), and from the gradient of C I approximated by the background volume grid, we obtain the surface tension source term which enters the momentum equation by:

Σ = -(σB α α ) ∇C I (4.2.19)
Since the surface tension coefficient is constant, we finally obtain the following source term:

Σ = -σ B α α ∇C I or Σ| c = -σ B α α | c ∇C I | c (4.2.20)
The hybrid method described above is similar to the method proposed by Shin et al. ( 2005) [START_REF] Shin | Accurate representation of surface tension using the level contour reconstruction method[END_REF]. The difference is that since the approach of Shin is formulated in a front tracking framework, the volume fraction is replaced by the smoothed indicator function. 1. In front tracking methods the interface is represented by a set of points (and their connectivities), in IsoFT the interface is defined by the volume fraction and the set of points is obtained by constructing the C I = 0.5 isosurface using the MPA.

Closing

2. In front tracking methods, the points that represent the interface are advected with the local fluid velocity (in a Lagrangian manner), while IsoFT remains a front capturing method where a transport equation is solved. Specifically, ISIS-CFD uses implicit integration schemes (second order backward) in time.

3. In front tracking methods, the indicator function is constructed by the interface for each time step, in IsoFT the volume fraction can be periodically reconstructed by the interface. However, for the time being, even though intermediate sharp volume fraction fields may be defined, they do not interact with the solution of the volume fraction transport equation.

Specifically, the third element in the above list is an important component of the sharp surface tension method we implemented in ISIS-CFD. Sharp methods treat the problem differently and a source term is absent from the formulation.

The subscript F can be considered as an operator which provides the above value. Similarly we define its "dual", F by:

q F = a q L + (1 -a) q R (4.2.26)
The reconstruction relations applied for (4.2.21), provide the pressure values at the left side of the face and the right side of the face:

p f R = (1 -κ) p L + κ p R -ρ t f ∇p ρ * f + (1 -κ) p p f L = (1 -κ) p L + κ p R -ρ t f ∇p ρ * f -κ p (4.2.27)
where an asterisk marks terms that are approximated and:

κ = a ρ f L ρ F and ρ = ρ f L ρ f R ρ F (4.2.28)
The third term in the RHS of the reconstruction relations can be explicitly approximated by the known (from a previous iteration) values of the term ∇p/ρ at the cells, following the same approach as in the classic reconstruction relations, i.e. by setting:

t f ∇p ρ * f = t f ∇p ρ F ( 4.2.29)
The term n f ( ∇p/ρ) f is required for the construction of the pressure equation. The reconstruction relation for the pressure gradient is:

∇p ρ f n f = 1 ρ F p R -p L h - t f ρ F h ∇p * - s ρ F h a ∇p * f L + (1 -a) ∇p * f R - 1 ρ F h p (4.2.30)
The second and third terms are explicitly treated by one-sided approximations from the left and right cells respectively:

∇p * f L = ∇p L and ∇p * f R = ∇p R (4.2.31)
More details about the construction of the pressure equation can be found in [86].

The reconstruction relations enforce the compatibility conditions as:

p f R -p f L = p (4.2.32)
We directly derived the last relation by forming the difference of relations (4.2.27). However, the bracket operator, as defined in this work, gives the jump of the pressure (and any discontinuous field) for the orientation chosen for the normal vector of the interface. To elucidate, if the normal vector of the interface points from the fluid subdomain I to the fluid subdomain J , then the bracket operator is:

p = p J -p I (4.2.33)
The normal vector of the face is defined independently of the fluid subdomains that occupy the cells on the right and left of the face. Therefore, we must appropriately specify the jump p so that it complies with p f Rp f L , as is specified by the normal vector of the face.

An important issue that arises in the above formulation, in the context of VOF using compressive discretization schemes, is that the faces where the compatibility conditions are to be applied to are not directly available. A simple way to identify these faces is to use the volume fraction. To that end, two options are available. A face f belongs to the set of faces where we enforce the compatibility conditions, The above procedure may directly use the volume fraction, as obtained by the volume fraction advection equation, to identify the faces where the compatibility conditions are to be applied. However, due to numerical diffusion, we might enforce the compatibility conditions to faces that are far from the actual interface. Furthermore, for these locations the approximated curvature will not be an accurate approximation of the curvature of the interface. MPA resolves these difficulties. As a preliminary step, we construct the C I = 0.5 isosurface. The isosurface is used to obtain the curvature and to define a sharp volume fraction field, C # I . The sharp volume fraction is used to identify the set of faces where the compatibility conditions are enforced. Since the sharp volume fraction does not contain the smeared regions of the original field, it allows identifying faces that are close to the interface. We consider an example to demonstrate the different sets of faces near the interface as found by using the volume fraction and the sharp volume fraction.

In figure 4.3, we compare the set of faces where the compatibility condition would be enforced, as identified by either the volume fraction field or the sharp volume fraction field for a three-dimensional rising bubble simulation. Only half of the configuration is shown in figures 4.3a and 4.3c, . One of the adjacent cells of the exposed faces is immersed in the water region (the region outside the bubble, where the volume fraction is 1) while the other intersected by the interface. The volume fraction corresponds to the (trimmed) volume fraction C I and the sharp volume fraction C # I respectively. We have plotted the configuration obtained in the y z plane, figures 4.3b and 4.3d, to clearly visualize the resulting set of faces,. The identified faces are shown as the black line contour near the locations where the volume fraction changes from 0 to 1. As we can observe, the sharp volume fraction identifies the faces near the interface, while the original volume fraction field locates faces that are relatively far from the interface.

As we have previously noted, the reconstruction relations enforce the pressure jump through the pressure values of the left and right cells and not the values of the adjacent subdomains. By snapping the interface to the faces of a single subdomain, we can distinguish the fluids that occupy the left and right cells of each face. Subsequently, the sign of the pressure jump is corrected as required. To clarify, consider the case where the interface is snapped near the region I . We have the following two cases: Similarly, we have the following two cases when the interface is snapped near the region J :

1. 0 < C I f L < 1, C I f R = 0 thus p f R -p f L = p J -p I = p 2. 0 < C I f R < 1, C I f L = 0 thus p f R -p f L = p I -p J = -p
In conclusion, when we choose the set of faces that the compatibility conditions are enforced, we also appropriately set the sign of pressure jump that will be used by the reconstruction relations.

At this point, we should note that the pressure jump:

p = σB α α -2 µ (∇ α w α -wB α α ) (4.2.34)
contains contributions from a term related to surface tension and a term related to viscous stress.

The second term must be continuous and should be calculated by the velocity field of the interface.

This procedure requires to interpolate the velocity field to the isosurface and to calculate the surface divergence subsequently. Up to now, these procedures have not been implemented. However, we can work with its equivalent term which (as demonstrated in the Appendix, see p.229) is the normal component of the normal viscous stresses, and we have:

p = σB α α -2 µ n i n j ∇ j u i (4.2.35)
where the required velocity gradient is estimated from the last known velocity field through the volume grid data.

The above method enforces the pressure jump at the face instead of the actual locations of the interface. Nevertheless, this is a reasonable simplifying hypothesis. Due to this peculiarity of the method, the members of the set of faces where the compatibility conditions are enforced, form a Volume fraction field obtained by advection equation

Validation: the Static Bubble Test Case

Introduction A static bubble refers to the steady configuration of two fluid subdomains where the interface is spherical in the absence of gravity. Suppose that fluid I occupies the region inside the sphere and fluid J the surrounding space, then the pressure for a static bubble, is given by:

p = 2σ R I I (4.3.1)
where σ is the surface tension coefficient, R is the radius of the sphere and I I is the indicator function of the sphere. The above relation follows directly from the pressure jump at the interface, by setting the pressure of the external region, the fluid subdomain J , to zero. The averaged pressure of the one-fluid equations follows from the above relation by applying the bar (averaging) operator:

p = 2σ R C I (4.3.2)
In chapter 2 (see "Static Bubble", p.64) we have presented the relevant details, and we have demonstrated that the above is an analytical solution of the one-fluid equations. In this section, we use the static bubble test case to arrive at conclusions regarding the performance of the implemented methods.

We present and compare the solutions obtained by the surface tension methods, introduced in the previous section. Afterward, we demonstrate the effects that automatic grid refinement has on the generation of isosurfaces, and their implications on surface tension methods.

Basic Tests without Automatic Grid Refinement

Test Configurations In the computational domain (unit cube) we set up a Cartesian grid. 1. CSF using the volume fraction field, derivatives calculated using CDS and density scaling.

2. CSS using the K 1(C I ) volume fraction field, derivatives calculated using CDS without density scaling.

3. IsoFT using level 1 neighborhoods for the distribution of the surface tension force of the interface.

Curvature calculated using iso-patch neighborhoods constrained in level 2 neighborhoods and cubic polynomials for the interface reconstructed by the K 1(C I ) field. Density scaling is used.

4. DCM using the same isosurface and curvature calculation as IsoFT. Density scaling is not used.

Quantification of Numerical Errors Numerical errors are expressed as overshoots and undershoots

of the pressure near the interface at locations where the pressure must obtain constant values (zero for the region outside the interface and 2σ/R = 0.5824N /m 2 inside the interface), and spurious currents.

To quantify these errors we have calculated the following. Suppose that the set of cells that occupy the air subdomain is a(C ), i.e. it consists of cells whose volume fraction C J (where J refer to the water region) is zero, then we define the mean pressure of air, m(p; a), as:

m(p; a) = 1 |a(C )| c∈a(C ) p c (4.3.3)
where |a(C )| signifies the number of cells in the air subdomain. We define similarly m(p; w) for the water subdomain, the relative error of the pressure jump from the mean values as:

E ( p ) = m(p; w) -m(p; a) -p p (4.3.4)
As a characteristic value for overshoots for pressure, we calculate:

o(p) = max(p; a) -m(p; a) p (4.3.5)
and for undershoots:

u(p) = min(p; w) -m(p; w) p (4.3.6)
The maximum and minimum values of pressure are defined for the water and air subdomain respec- Finally, the norm of the maximum velocity, | u| max is used to quantify the spurious currents. We note that the flow is steady and the velocity field should be everywhere zero. The results obtained for CSF and CSS are summarized in table 4.1 and the results for IsoFT and DCM in table 4.2.

Results and Discussion CSF does not provide accurate predictions for the range of discretizations used. We can relate the above observation to the erroneous curvature calculations, since we used directly the volume fraction (without smoothing) to approximate the normal and subsequently the curvature. For the CSS method, we used the gradients of the smoother volume fraction field K 1(C I ). As we can observe the mean pressure of air is more accurately calculated. The undershoots of pressure are still present, and indeed these are affected by the smoothness of the volume fraction field. Notice that for CSS, where we evaluate the derivatives by the smoother volume fraction field, the undershoots are smaller while the overshoots are larger in comparison to CSF (for the smaller discretizations). The third method that treats surface tension using a source term, IsoFT, also provided large undershoots.

To construct the smoothed surface tension source term derived by the IsoFT method, we approximate of the normal vector and curvature from the isosurface rather than approximations using gradients of the volume fraction. For this case, we obtained consistently improved estimates for the pressure of the air subdomain and spurious currents in the same order of magnitude with the other methods.

Similarly to the previous methods, we obtained considerable pressure undershoots. The undershoots of pressure are probably related to the density scaling which forces the smoothed surface tension source term to have a more pronounced effect on the water region, where the pressure should normally be zero. Finally, DCM, the sharp surface tension method outperformed the rest. In comparison to the other methods the spurious currents are smaller by two orders of magnitude, the overshoots and undershoots of pressure are less pronounced and excellent estimations for the mean pressure of are obtained.

The pressure fields obtained from the computations using the IsoFT method and the DCM for ν = 40 at the plane x = 0 are shown in figure 4.4. As we have already mentioned in the previous chapter, the DCM enforces the pressure discontinuity at the faces near the interface. These locations are clearly visible by the changes of pressure from zero to the value obtained for the interior region of the sphere.

The pressure inside the bubble is very close to the expected value, 2σ/R, a distinguishing characteristic of the method which is due to the fact that pressure jump enters the discretization at certain faces (where also ∇/ρ is discretized). For the DCM, the overshoots and undershoots of pressure are not visible while the same errors are clearly visible for IsoFT (and the other methods).

To compare the results for the spurious current, we have plotted the velocity fields obtained from the computations using each method and ν = 40 at the plane x = 0. The results are plotted in figure

Effects of Automatic Grid Refinement for Methods Using the Isosurfaces

Generated by the MPA

ISIS-CFD has many different features which can be used along the ISIS-CFD extensions in a straightforward manner. In this paragraph, we are particularly interested in automatic grid refinement (AGR).

AGR allows a user of ISIS-CFD to select from a list of implemented criteria that determine a subset of cells that are refined. Thus, the grid is adapted so that it conforms to the chosen criterion. One of the available options offers the capability of refining the grid in locations where the volume fraction obtains certain values within the range zero and one, i.e. values where the interface is present. AGR improves the results of surface tension without extensive memory requirements. The interested reader may refer to Wackers et al. ( 2014) [START_REF] Wackers | Combined refinement criteria for anisotropic grid refinement in free-surface flow simulation[END_REF] for more details about general aspects of automatic grid refinement.

For the developed methods whose curvature approximations are converging, improved results can be obtained by automatic grid refinement.

Specifically, the methods based on curvature calculations using the isosurface of the MPA can directly profit by AGR. When AGR refines the cells of the volume grid near the interface, the generated isosurface is also refined in a direct manner. Since the MPA performs the isosurface construction in a refined background grid, it constructs a refined isosurface. Nevertheless, the requirements that specify a volume grid of high quality and a surface grid of high quality in the context of this work are different.

The cells of the volume grid discretize the three-dimensional space. Therefore, when AGR tessellates a cell to smaller cells, a finer discretization of the volume region represented by the original (parent cell) is directly derived. When AGR performs this action for the subset of cells whose members comply with a physical constraint (that guides refinement), the result is a finer discretization of particular regions of the computation domain. For this case, a "good" arrangement of cells should comply with classic "grid quality" criteria, for example, misalignments should not be introduced by AGR. To summarize, AGR acts on the volume grid to a subset of cells determined by physical constraints and geometric considerations suffice to ensure that AGR produces a grid of acceptable quality.

On the other hand, the surface grid discretizes a two-dimensional region that represents a physical boundary and more precisely the interface of two fluid subdomains. Consequently, the quality of the surface grid is related to the capability of the MPA to generate a better approximation of the interface (the physical boundary) through AGR. Recall that the MPA constructs the isosurface given as input the volume fraction and the background volume grid. It follows that to achieve an improved approximation, we have to ensure that the volume fraction field will also be refined along with the cells of the background volume grid. At this point, the grid dependent characteristics of the volume fraction field become important. When the volume grid changes, the volume fraction should also be redefined so that it conforms to the new grid.

As a result, "automatic grid refinement" must also be performed along with an "automatic field refinement" for the volume fraction. To that end, the volume fraction field that corresponds to the adapted cells must be, (i) ideally, a better approximation of the volume fraction field that corresponds to the new cells, or (ii) practically, an estimate of the volume fraction field that corresponds to the new cells, probably based on the volume fraction field that corresponds to the old cells. Considering the above as two post-AGR tasks that the implementation should perform, the first requires the implicit function of the interface and the second some manipulations of the old volume fraction field. In the following, we give an example that demonstrates the effect of AGR by performing the first task.

As an example, consider the static bubble test case, where AGR acts on the ν = 20 grid for the subset of cells that the interface intersects, and their neighborhoods. the interface intersection of the grid with the plane x = 0 (after the adaptation procedure).

After AGR generates the adapted grid, the volume fraction is reinitialized by the MPA using the implicit function of the sphere and the adapted grid. The section with the plane x = 0 of the resulting volume fraction field is shown in figure 4.6b. The same figure depicts the isosurfaces generated from the initial and the refined volume fraction fields colored by the calculated (iso-patch local) curvature.

We achieve better estimates for the curvature of the isosurface generated by volume fraction field of the refined grid. As we can observe the volume fraction maintains its sharp characteristics.

Indeed, if we consider that the MPA generates the isosurface in a background grid that corresponds to the ν = 40 grid, it is reasonable to expect that the curvature estimation will be more accurate (recall that we have already shown that for the discretization used the curvature calculation converges). Figure 4.7a illustrates the pressure field provided by the computations with the IsoFT method. In this case, the mean pressure of air is m(p; a) = 0.556 and the error is E ( p ) = -0.052, which are comparable to the findings for the ν = 40 grid without AGR. In conclusion, an improvement to the solution is obtained by AGR.

As long as the volume fraction field is refined together with the background grid, the MPA will generate a more accurate approximation (in comparison to the isosurface without AGR). In turn, we achieve more precise curvature estimations and thus, a more accurate calculation of the pressure field.

For the case studied, the reinitialization of the volume fraction field is allowable because there are no transient terms. For unsteady cases, we have to resort to a different way to specify the volume fraction.

A straightforward and general way to redefine the values of the volume fraction field for cells obtained after refinement is to suppose that they are equal to the volume fraction values of their parent cells (i.e. the tessellated cell from which the refined cells stem). We perform the same calculation with AGR and IsoFT without reinitialization of the volume fraction field but with the above approach, performed automatically and by default from AGR. The resulting pressure field is shown in figure 4.7b. To obtain a better approximation of the volume fraction, we could use the iso-patches generated before grid refinement, to reinitialize the volume fraction for the adapted cells of the background grid. This procedure is straightforward since the MPA ensures that iso-patches are generated inside cells the background grid. Thus, for the set of refined cells generated by a single cell corresponds at least an iso-patch from which we may reinitialize the volume fraction. Moreover, the volume fraction produced in this manner retains its sharp characteristics. However, for the time being, the only option available is the default option used by AGR. We note that even though the proposed method will produce better approximation of the volume fraction at the refined grid, when multiple refinement steps are undertaken a more accurate approximation of the surface, probably based to LSqR, will be needed. This procedure should reinitialize the volume fraction of interface in a more accurate manner to avoid having a reinitialization that gets caught in the locally linear iso-patch approximations of the isosurface.

Comparison of Results for the Rising Bubble Test Cases

Introduction In the previous section, we have shown the solution obtained for static bubbles, a steady problem. In this section, we present the results derived from computations using the four surface tension methods for unsteady cases, specifically, rising bubbles and compare with experimental and numerical simulation results obtained by other researchers. Transient effects provide another perspective to the methods studied, specifically when significant deformations of the interface occur.

For these cases, methods whose curvature calculations use the reconstructed interface, provide superior results in comparison to methods that solely use the volume fraction. The feature that enables the isosurface methods to provide better results is that they respect the local geometrical characteristic of the interface.

The section consists of two parts. In the first part, we study rising bubbles in the ellipsoidal regime, in the second part, we study a rising bubble in the spherical cap regime. After a short presentation of the simulation setup and the post processing of the results, we demonstrate and discuss the results obtained by the simulations.

Preliminaries Similarly to the static bubble test case, no symmetry assumptions for the bubble's shape were made. The grid used for the simulations, generated by Hexpress, discretized the interior of a rectangular parallelepiped centered at O = (0, 0, 0). The length of the computational domain's edges parallel to the x,y-axis was L x = L y = 10R -12R and its height L z = 15R -18R, where R is the radius of the bubble. The initial Cartesian grid resulted by tessellating the edges to n x = n y = 20, n z = 30 uniform partitions, and the final grid resulted after performing two levels isotropic refinement inside a box.

Figure 4.9 depicts a grid that resulted from the above procedure. The grids consisted of about 200k cells, and the simulations were performed using four processors. The bubble's center was originally located at (0, 0, -3R).

In every simulation, we tried to maintain a small maximum Courant number of about 0.2 to ensure that the compressive scheme used did not introduce numerical errors. In most cases the Courant number reached a maximum of 0.35. The time step used was ∆t = 5 • 10 -5 and we note that for this time step the analysis of [START_REF] Kang | A boundary condition capturing method for multiphase incompressible flow[END_REF] [START_REF] Kang | A boundary condition capturing method for multiphase incompressible flow[END_REF] predicts stable solutions with respect to surface tension.

Simulations were performed for the following bubble radii, R = 0.6 mm, R = 0.8 mm, R = 1 mm and R = 10 mm.

To establish a baseline regarding the effects of surface tension to the deformations of the interface, we performed a computation without surface tension for R = 0.6 mm. The isosurfaces obtained by the MPA, for three instances during the simulation are given in figure 4.10, as bird's and fish's eye views (from top to bottom and bottom to top views respectively). Since surface tension is absent, the bubble is driven by its buoyancy. The initially spherical configuration rapidly deforms and forms a tongue. The tongue eventually touches and pierces the upper part of the surface and the interface which finally resembles a toroidal bubble.

Experimental research studies have not reported such interface configurations for air-water bubbles of that size. On the contrary, research studies [START_REF] Clift | Bubbles, Drops and Particles[END_REF][116] [START_REF] Tomiyama | Terminal velocity of single bubbles in surface tension force dominant regime[END_REF] suggest that small bubbles either retain their spherical shapes or become ellipsoidal, based on the water's "purity" and the bubble's generation method. Even if this behavior is expected due to the lack of surface tension, the above calculation provides an insight into the effects of the interface's static pressure distribution to its deformation (without surface tension).

Results Derived by Post-Processing

The purpose is to compare our numerical predictions with measured experimental results. Experimental studies, for instance [START_REF] Clift | Bubbles, Drops and Particles[END_REF][102] [116][16], measure the bubble's terminal velocity and the bubble's aspect ratio by treating subsequent images of the bubbles.

We calculated the terminal velocity and the aspect ratio of the interface using consecutive instances of the isosurface, which resulted from the simulations. An approximation of the aspect ratio of the interface can be calculated by the mean value of two characteristic lengths obtained by the projected area of the iso-patches to the xz-plane, A xz and the y z-plane, A y z divided by a characteristic length of the iso-patches projected area to thez y plane, A x y :

E = A xz + A y z 2 A x y (4.4.1)
An example of the variation of the calculated aspect ratio over time for R = 0.8 mm is given in figure 4.11.

The velocity of the bubble is defined as the velocity of the centroid of the isosurface. The centroid of the bubble b was evaluated using the isopatches by:

b = 1 2V b c s ∈ C s | p c s | 2 n I c s S I c s where V b = c∈C V c C # I c (4.4.2)
where C # I c is the sharp volume fraction, i.e. the volume fraction reconstructed by the isosurface. The time derivative of the centroid was found using first order finite differences between two subsequent time steps i + 1 and i :

V b i +1 = b i +1 -b i t i +1 -t i (4.4.3)
An example of the calculated centroid's components and its velocity is given in figure 4.12 for R = 0.8 mm. Notice that we also obtained some small velocities for the x-axis and the y-axis, a side effect related to the sensitivity of the calculation using the isosurfaces provided by MPA. Finally, we note that, the aspect ratio reached its terminal value very fast while the velocity required twice more time steps to converge towards its terminal value. 4.13a, each method resulted in different distortions of the interface during the initial stages of the bubble's evolution. These are more pronounced for the calculation using the DCM and less pronounced for the calculations using the IsoFT method. In general, the observed differences in the interface's shape and terminal velocities can be attributed to two factors from an experimental point of view.

Either contaminated water by surfactants or small initial deformations of the interface, see for example [START_REF] Tomiyama | Terminal velocity of single bubbles in surface tension force dominant regime[END_REF] [START_REF] Tomiyama | Terminal velocity of single bubbles in surface tension force dominant regime[END_REF] and Wu and Gharid (2002) [START_REF] Wu | Experimental studies on the shape and path of small air bubbles rising in clean water[END_REF].

Tomiyama et al. [START_REF] Tomiyama | Terminal velocity of single bubbles in surface tension force dominant regime[END_REF] commented on the computational difficulties that came across in rising bubble simulations and related the under-predictions of terminal velocities by CFD software to the "perfect" spherical initial geometrical configuration. Such ideal configurations for bubbles are without The computations we performed indicate that the prediction of the terminal velocity is directly related to the initial distortions of the interface. These initial distortions result due to numerical, rather than physical, reasons and seem to govern the initial stages of the problem's evolution. Nevertheless, a solution is eventually obtained and, as we will show next, these solutions can be considered as reasonable from a physical point of view. [START_REF] Tomiyama | Terminal velocity of single bubbles in surface tension force dominant regime[END_REF] and Wu and Gharid (2002) [START_REF] Wu | Experimental studies on the shape and path of small air bubbles rising in clean water[END_REF] suggest an alternative explanation.

Comparisons with

Both studies independently confirmed that the initial deformations of a bubble affect the evolution of its shape even in cases where the water is purified. More precisely, the initial deformations result from the experimental technique used to form the bubble (the bubble formation or generation method).

Bubble formation methods that produce small initial deformations provide smaller measured terminal velocity than bubble generation methods that cause large initial deformations of the bubble.

Previous computational research studies compare results with the experimental results of Tomiyama et al. ( 2002) [START_REF] Tomiyama | Terminal velocity of single bubbles in surface tension force dominant regime[END_REF] or with the experimental results of [START_REF] Duineveld | The rise velocity and shape of bubbles in pure water at high Reynolds number[END_REF] [START_REF] Duineveld | The rise velocity and shape of bubbles in pure water at high Reynolds number[END_REF]. Specifically, Tukovic and [START_REF] Hua | Numerical simulation of 3D bubbles rising in viscous liquids using a front tracking method[END_REF]. The IsoFT method resembles the surface tension method of Hua et al.

(2007), that used a front tracking approach and a surface tension method based entirely on the tracked interface, while isoFT describes a surface tension method that performs similar approximations but for a purely front capturing framework. At the same time, both DCM and isoFT are surface tension methods which resemble methods used in the context of a front tracking framework. Therefore, a reasonable question is why they provide a different approximation of the terminal velocity.

The answer is probably related to the deformations of the interface, produced during the initial time steps of the simulation. To validate this observation, we compared the results obtained with DCM and IsoFT with the experimental results of [START_REF] Wu | Experimental studies on the shape and path of small air bubbles rising in clean water[END_REF]. [START_REF] Wu | Experimental studies on the shape and path of small air bubbles rising in clean water[END_REF] used different nozzle diameters to generate the bubbles in uncontaminated water and compared the differences in terminal velocities and aspect ratios. Nozzles of smaller diameters generate bubbles with large initial deformations and nozzles with a larger diameter generate bubbles with small initial deformations.

As we can observe from figure 4.15, the evolution of a bubble's shape with large initial deformations compares well with the evolution of interface shapes obtained from DCM, and the bubble shapes with small initial deformations compare well to the resulting interface shapes derived by isoFT. The last observation indicates that there are indeed small differences during the initial interface deformations that affect the evolution of the shape of the bubble and the terminal velocity obtained. However, a systematic treatment is required to reach a definite conclusion.

The above observations confirm Tomiyama's remark [START_REF] Tomiyama | Terminal velocity of single bubbles in surface tension force dominant regime[END_REF]:

CFD researchers have faced with a trouble in the prediction of V T [terminal velocity]. The trouble is that in spite of using "clean" continuity and Navier-Stokes equations, interface tracking simulation is apt to yield V T close to that of a bubble in a contaminated system. Now this paradox is easy to answer. In interface tracking simulation, a spherical shape is usually assumed as an initial bubble shape. This initial condition corresponds to a bubble with small initial shape deformation so that the predicted V T is likely to be close to the lower bound of V T [..] Tomiyama et al. (2002) [START_REF] Tomiyama | Terminal velocity of single bubbles in surface tension force dominant regime[END_REF] suggested the following correlation between the aspect ratio of a bubble and its terminal velocity:

V t = sin -1 1 -E 2 -E 1 -E 2 1 -E 2 8σ ρ w d e E 4/3 + (ρ w -ρ a )g d e 2ρ w E 2 1 -E 2 (4.4.4)
The above relation suggests that for a given equivalent bubble diameter the terminal velocity depends on the bubble's aspect ratio. to the interface as predicted by the proposed theoretical formulation). The smearing that we have previously observed for the volume fraction field is due to the vortical structure of the flow. Large recirculation regions were obtained behind the bubble as expected as in any bluff body flow. For the CSS method, the sharp volume fraction field C # I indicates that the smeared regions do not disappear. Therefore, we may interpret these formations as bubble skirts of the interface's shape. Persistent bubble skirts have not been reported experimental studies for bubbles of air in water. From the C # I field obtained for the results of the DCM and isoFT the smeared regions captured in the bubble's wake seem to disappear. However, the detached parts entrain the water region and are caught in the bubble's wake.

The result is that small satellite bubbles are formed behind the leading spherical cap, as we will show next.

Details Provided by the MPA Further examination of the solutions through the generated isosurfaces revealed interesting details. Figure 4.18 shows the interface's intersections with the x = 0 plane for the initial stages of the isosurface's evolution. The plotted intersections expose the leading and trailing parts of the surface. The results were obtained using the IsoFT method. Initially, the bubble's shape is governed by buoyancy. The trailing surface of the interface formed a tongue, shown in figures 4.18a (in the same figure, the blue dashed line marks the interface's initial position on the plane). The tongue moved towards the leading surface, 4.18b and when the curvature became large enough at the tip of the tongue, the tongue decelerated.

In figure 4.19 we plot the calculated curvature field over the isosurface for the initial stages of the bubble's evolution for two particular time steps. The first figure, 4.19a, corresponds to the time step when the tongue's tip was well formed, and the second figure, 4.19b, when the skirt of the bubble was well formed. The view exposes the trailing part of the surface. The curvature reached its maximum value at the tip of the tongue. When the skirts were formed, the curvature of the tongue's tip was smaller than in the previous time steps, which implies the flattening of the tongue's tip. The curvature was correctly calculated as positive at the tip of the tongue and negative at the extremity of the skirt. Recall that the normal vector of the interface points from the water region to the air region (from the reader to the When lateral parts of the surface skirts touched, small structures are formed that resemble stalactites hanging from the bubble, figure 4.22c. When these detached from the bubble's main body, they were captured as minuscule pendant like subscale bubbles. Some time steps later, small bubbles were not captured by the MPA and the same process takes place for the skirt's parts that were aligned to the x-axis and y-axis, as shown in the figures 4.22d and 4.22e. Finally, ISIS-CFD predicts that the leading bubble establishes the spherical cap shape shown in figure 4.22f. The wrinkled and wavy trailing surface of the bubble captured by ISIS-CFD is also experimentally observed, see for example [START_REF] Landel | Spherical cap bubbles with a toroidal bubbly wake[END_REF] [START_REF] Landel | Spherical cap bubbles with a toroidal bubbly wake[END_REF]. The result indicates that isoFT diffuses parts of the volume fraction when the discretization is not fine enough to be captured by the MPA. This process occurred in a non-isotropic manner. As we have observed previously, for the parts of the isosurface whose normal vector is parallel to the x-axis, y-axis, and the z-axis, the diffusive character is less pronounced. This observation is probably related to the solution of the volume fraction transport equation. To clarify, if we correlate the volume fraction transport with the transport of an iso-patch of the surface (and small Courant numbers) for a given cell, the volume fraction should change at adjacent cells that share the same nodes, i.e. cells in the node topos neighborhood. Compressive discretization schemes can only generate fluxes of the volume fraction at cells in the face topos neighborhood, which is presumably related to the anisotropic behavior observed. We observed a similar non-isotropic behavior by the isosurfaces derived by the solution using DCM. However, the diffusive character was less pronounced.

The generated isosurfaces for DCM revealed similar solutions to IsoFT, up to the point the interactions of the lateral parts of the surface begun. From that point on the interface obtained different characteristic shapes. Figures 4.23 summarize the evolution of the isosurface as recorded in the simulation using the DCM (from the point where the detachment of the skirt begun). The skirts detached in the same manner for isoFT, but they did not break down to subscale structures. Instead, loops were formed that remained attached to the bubble. Subsequently, the loops detached from the main bubble and created small bubbles, figure 4.23d. We observe the same anisotropic features that cause the skirt to become detached in parts of the surface that are not aligned to the x,y-axis. The bubbles followed the leading bubble till the end of the simulation. The configuration obtained in the simulation resembles experimental results where satellite bubbles follow the main spherical cap bubble, as demonstrated in Landel et al. ( 2008) [START_REF] Landel | Spherical cap bubbles with a toroidal bubbly wake[END_REF].

To verify the previous observations regarding the generation of smaller satellite bubbles behind the main bubble, we have repeated the calculation using a finer grid with isoFT. approximated interface obtained, for several time instances. As for the two previous cases, a tongue is formed that approaches the leading part of the surface. When we compare the approximation of the curvature for the fine and the coarse grids, figures 4.24b and 4.22a respectively, we observe that the largest curvature differences are located at the skirt's edges. The curvature for the fine grid was estimated around -1600 m -1 , while for the coarse grid was around -800 m -1 . It appears that the coarse grid is not adequate to capture the very sharp edge. Before reaching this point, similar curvature values have been achieved for the same time instances, notably for the lateral surfaces around the edge of the skirt and the tongue, as we can observe by comparing figures 4.19 and 4.20. After the detachment of the skirt, a toroidal bubble was formed. The toroidal bubble was captured in the wake of the leading bubble and broke down to form smaller bubbles that follow the spherical cap.

The results agree with the results obtained by the DCM and indicate that the skirt interactions have significant effects on the evolution of the bubble. Moreover, they further enforce the argument that the IsoFT method introduced a numerical diffusion to the volume fraction which acts non-isotropically and selectively. The affected locations appear to be those where the normal vector of the isosurface is not aligned to the grid's faces normal vector since the bubbles that diffuse are those for which the above statement is true. Finally, from a physical point of view, we can capture the details of the flow occurring behind large bubbles. This kind of bubble formations was described by Landel et

Conclusions

In this chapter, we have presented the results obtained by ISIS-CFD and the different surface tension methods developed in the context of this work. We have compared the results for bubbles on the ellipsoidal regime and a bubble in the spherical cap regime, derived by the different surface tension methods that we have implemented in ISIS-CFD. Moreover, for the bubble on the ellipsoidal regime we have also compared our results with results of experimental studies and computational studies.

More specifically, the terminal velocity results are in excellent agreement with the terminal velocities provided by [START_REF] Tomiyama | Terminal velocity of single bubbles in surface tension force dominant regime[END_REF] relation [START_REF] Tomiyama | Terminal velocity of single bubbles in surface tension force dominant regime[END_REF], which relates the aspect ratio and the terminal velocity. Therefore, ISIS-CFD predicts excellently the dynamic evolution of the bubbles. By comparing the dynamic evolution of approximated interface (using the volume fraction and the isosurfaces), with experimentally observed bubbles from [START_REF] Wu | Experimental studies on the shape and path of small air bubbles rising in clean water[END_REF] [START_REF] Wu | Experimental studies on the shape and path of small air bubbles rising in clean water[END_REF], we have confirmed that physically realistic answers are derived regarding the observed interface shapes. Furthermore, we have detected several robustness issues that are related to the choice of the surface tension method.

The surface tension method is mainly responsible for the accurate prediction of the bubble's shape which eventually affects the bubble's terminal velocity. The coupling with classic methods found in the literature such as the CSF method and the CSS method derived results whose physical significance is questionable and, only for specific ranges of the bubble diameters. For example, the CSF method led to the formation a toroidal bubble for the bubble at the beginning of the spherical cap regime, when either an elliptic bubble or a spherical cap bubble would be expected. The CSS method produced regions where the volume fraction of water diffuses inside air. Both the DCM and the isoFT method provided of reasonable physical significance both smaller bubbles (in the ellipsoidal regime) and larger bubbles (in the limit of the ellipsoidal and spherical regime).

Finally, the results demonstrate that complex physical interactions can be reproduced and are in good agreement with experimental observations. For example, the formation mechanism of small

Other Simulations and Preliminary Results

Introduction The purpose of this section is to demonstrate some features of the ISIS-CFD extensions framework and their practical use for setting up simulations and interpreting results of other phenomena. The simulation presented here differ from the simulations in the previous section regarding their complexity. In the preceding section, we presented a sample of computations modeling the dynamic evolution of rising bubbles. In such simple initial configurations the volume fraction initialization problem is simple and its solution straightforward. In simulations of engineering interest, we are interested in more complicated configurations, where multiple interfaces are defined in a single domain.

Moreover, the rising bubble simulations produced sophisticated enough isosurfaces, but in principle much simpler from the interactions of interfaces observed in cases of practical engineering interest.

In this section, we present two preliminary simulations. The purpose is to demonstrate the volume fraction initialization capabilities of the code the applicability of the MPA to cases of engineering interest. We present an analysis of the results and briefly discuss some of the problems encountered.

Interactions of a Rising Bubble with a Free Surface

The rising bubble/free surface problem or equivalently the falling droplet/free surface interaction problem is a classically encountered in the literature, see for example [START_REF] Sousa | A front-tracking/front-capturing method for the simulation of 3D multi-fluid flows with free surfaces[END_REF] [START_REF] Sousa | A front-tracking/front-capturing method for the simulation of 3D multi-fluid flows with free surfaces[END_REF] (for the falling droplet test case). The problem illustrates the applicability of a surface tension method for a more general configuration. Specifically, we demonstrate the capabilities of the proposed methods considering, first the initialization of a more complicated problem and the capturing the physical mechanisms of the interfaces' merging.

For this problem, one of the main difficulties encountered by front tracking methods, besides the management of connectivities when interface merging takes place, is the management of the surface grid near the boundaries of the computational domain. Since parts of the interface might exit or enter the computational domain, grid entities of the surfaces grid should be added or removed as appropriate. ISIS-CFD treats automatically such cases because the solution of the volume fraction transport equation implicitly handles the problem.

Issues Regarding the Curvature As mentioned in the previous chapter (see p.109), the MPA requires the addition of explicit corrections for the node-cells interpolation scheme to provide first order converging results. Consequently, an accurate curvature calculation near boundaries is not possible using the current state of the curvature calculation method. Therefore, we define the curvature of the isopatches located near boundaries as zero to ensure that the surface tension source terms do not interfere with the calculations. Finally, the curvature calculation is sensitivity to small disturbances. Since the method uses third order polynomials (to construct the local Monge patches), small disturbances in the free surface can locally generate large curvature values especially for interfaces that are initially flat.

To that end, we introduced a cutoff planarity criterion. When the criterion detects an interface that is "almost planar" it sets the curvature equal to zero.

Configuration The computational domain spans a box whose lengths are L x = L y = 0.01 m, L z = 0.02 m. The boundary conditions were, for the top and bottom plane, updated hydrostatic pressure, and for the side planes zero imposed velocity (as for the rising bubble computations). The grid was similar to the grid used for the rising bubble computation. We also performed a two-level refinement around the planar interface (the free surface). The sphere's radius was R = 0.8 mm and its center was located at (0, 0, -0.0044). The observed burst of the film and the formation of the jet can be both explained as surface tension effects. The pressure of the air inside the bubble is higher than the pressure of ambient air, due to surface tension. During the collision, the interface movement is driven by the large pressure difference from the bubbles cavity to its exterior. This pressure difference causes the burst of the film and drives the formation of the jet. Since the curvature at the pike and the troughs of the free surface are large and of opposite sign, surface tension acts to stabilize the interface by making it flat. These observations are also confirmed by the pressure fields obtained, plotted in figure 4.27.

The MPA does not capture the burst of the film and the jet emanating from the pike. It is reasonable that a smaller grid size is required to examine these details and to check if the results are consistent. In figure 4.29, we compare the solution obtained by ISIS-CFD with the experimental results of Sangeeth and Puthenveettil (2015) [START_REF] Sangeeth | Dynamics of Collapse of Free Surface Bubbles[END_REF]. We should note that the results of [START_REF] Sangeeth | Dynamics of Collapse of Free Surface Bubbles[END_REF] are probably for air bubbles in water (since it is not clearly stated in the article) and correspond to a different equivalent bubble diameter. However, the results are in good agreement judging by the simulated geometrical patterns of the interface. The jet formation process, based on the previous experimental results takes about 4 ms to complete. ISIS-CFD predicts around 3.3 ms. This faster evolution of the interface can be explained by the fact that the simulated bubble's diameter was smaller than the bubble's diameter of the experiment. Thus, the surface tension effects are more pronounced and cause the jet to be formed faster.

Turning our attention to the predicted shapes of the interface, we observe similarities to the formations observed experimentally. Firstly, the numerical simulation predicts the formation of a thin film of the free surface and bubble which is also experimentally observed. Secondly, in both the experimental and the simulation results, a filleted region between the two surfaces is generated when the interface and the bubble touch, see the first MPA figure in 4.29 and figure f of Sangeeth and Puthenveettil, also in 4.29. From that point on the ISIS-CFD results slightly resemble the experimental results. The shape of the interface observed by the experimental results is conical. ISIS-CFD predicts that this region will be hemispherical. Both the experimental results and the simulation results indicate that when a small critical formation is reached, the lower part of the free surface begins moving upwards.

Concluding Remarks Through the rising bubble/free surface interaction test case, we have demonstrated two important aspects. The first is that the ISIS-CFD extensions framework can be used to analyze more complicated problems that involve multiple interfaces whose dynamics involve merging between different interfaces. In the previous section, we demonstrated that separation of interfaces could be modeled and captured by the MPA for a single interface. The modeling of merging and separation of the interface is an expected feature since merging/separation are taken implicitly into account by the solution of the volume fraction evolution equation. The MPA acts by reconstructing the interface which is used to perform the necessary calculations for surface tension and visualize details of the flow. At the same time, as the results demonstrate, the highly localized surface tension effects are crucial for the accurate capturing of problem's evolution. Therefore, they encourage the further study the related mechanisms that take place in this kind of cases. Finally, even though the considered phenomena take place in smaller scales, the problem can be used as a benchmark case for the most general spraying modeling. The same can be said for another group of phenomena arising when an air jet intrudes the free surface and initializes air entrainment. For both cases, the results obtained using the proposed methods can be used to build a database of direct numerical simulation whose results could be subsequently used to tune air-to-water and water-to-air entrainment models. However, the results mut be thoroughly reexamined and compared to other experimental studies such as Ghabache et al. (2014) [START_REF] Ghabache | On the physics of fizziness: How bubble bursting controls droplets ejection[END_REF] and numerical studies, also discussed by

Simulation and Flow Visualization of a Vertical Sharp Edged Plate in Drift

In this part, we present an application directly relevant to naval engineering, related to the study of complex hydrodynamic flows. The purpose it to visualize the interactions of free surface with vorticity generated by a partially immersed body to the flow. The experimental results presented by Broglia et al.

(2012) [START_REF] Broglia | Measurements of the Velocity Field Around the DELFT 372 Catamaran in Steady Drift[END_REF] have motivated the current study. The configuration is shown in figure 4.32b.

The free surface coincided to the z = 0 plane and the distance from the plate's submerged (bottom) edge to the free surface was 0.17 m. The figures 4.32c and 4.32d present the intersection of the volume fraction field with a yz-plane (also cutting the plate) and the unstructured surface grid that represented the free surface (constructed for the volume fraction initialization procedure). We can directly visualize the different refinement levels of the background grid (generated by Hexpress), from the initial surface grid of the free surface. The grid was refined in the vicinity of the free surface, both the leading and trailing edges of the plate, and the near plate's submerged edge. The final grid consisted of about 3 • 10 6 cells and the computations were performed using 16 processors.

During the first time steps of the simulation, the plate constantly accelerated and reached its An overview of the solution as obtained by the evolution of the interface, from the initial time steps, up to the end of the simulation is shown in figure 4.34. First of all, we should emphasize that the missing patches of the isosurface imply the presence of the plate which is not shown in the figures. In figure 4.34a, we observe the formation of the waves near the leading and the trailing surface of the plate (recall that the plate moves in the direction of the negative x-axis). Both waves are clearly visible in figure 4.34b. When the wave of the high-pressure side (starting from leading edge and in contact with the leading surface), was better developed, it formed the diverging wave crest shown in 4.34c. Figure 4.34d provides a wider field of view for t = 1.707 s near the end of the simulation, from which both the wake of the body and the large wave formed are visible. The most interesting details of the flow are provided by the evolution of the free surface behind the trailing surface of the plate that takes place during the first 0.8 s. In figure 4.35, the contour of the isosurface denotes the "relative depth" of each iso-patch: An example of a breaking wave's after-effects behind the plate, as captured by the MPA, is shown in figure 4.36. In figure 4.36a the wave begins its formation. The whole air-water interface above the plane zd = 0.3 is removed to allow a better visualization, and so only a part of the interface is visualized. We see the interface as if we were standing in front of the leading surface of the plate and looking towards the trailing surface (from the large wave crest to the region where ventilation begins, see also figure 4.33). The wave is formed from the left side of the cavity close to the plate. When the wave hits the free surface, its after-effects are captured by the MPA as small structures containing air that resemble elongated bubbles. These are shown in figure 4.36b-4.36d. The structures were visible for very short periods of time of around, 0.5 s. Each time a wave appeared close to the plate's leading surface and hit the free surface, the cavity became wider. After a second wave had hit the free surface behind the plate, the free surface near the leading edge of the plate had already reached the bottom edge of the plate. also demonstrate the applicability of the curvature calculation method using the isosurface to cases of practical interest and propose the further examination of this case with surface tension.

zd c s = z c s + 0.

Conclusions

In this section, we have described two possible applications of the ISIS-CFD extensions framework with practical interest. We presented two test cases as candidates for the further numerical investigation of surface tension effects to the physical mechanisms that govern the generation of vortex generation and the entrainment of water to air or water to air. From a Naval Engineering point of view, the creation of large vortices is a mechanism of energy loss and an increase of ship's resistance. The accurate capturing of the bow ventilated vortex in CFD simulations, could result in proposals for alternative bow geometries, which suppresses the creation of such vortices. To that end, it is interesting to investigate further the effects that surface tension could have in such flows and in what extend it affects such cases.

From a physical modeling point of view, systematic direct numerical simulation of air/water interface interaction mechanisms, as demonstrated by the first test case, can be used to provide a database of results. This database will eventually be used to compare with the results obtained by an air-entrainment or spray model. Therefore, they provide a mean of tuning such sub-scale models using the results produced by ISIS-CFD, which will be eventually used for practical cases.

In conclusion, the ISIS-CFD extensions provide the required tools for initializing different simulations and studying the details of flows of practical interest both from a naval engineering and a numerical modeling point of view. Among others, these include, dynamics of multiple interfaces and their interactions, probably improving shallow water waves dynamics and the improvement of hull resistance by taking into account complex physical mechanisms, an emerging task in the development of naval technology.

Conclusions

In this chapter, we have presented the four methods implemented to ISIS-CFD for surface tension modeling along with results for several test cases. The first two approaches, frequently cited in the literature, is the CSF method of Brackbill et al. (1992) [10] and the CSS method of Lafaurie et al. (1994) [51]. Both propose the addition of a surface tension source term that is expressed using gradients of the volume fraction. Their implementation is straightforward. However, converging calculations of the normal vector and curvature are hard to be obtained using derivative approximations of the volume fraction solely based on the volume grid. To that end, two new methods were developed and tested alongside the CSF and CSS methods.

The first method we presented was the Isosurface Front Tracking method, IsoFT. The IsoFT method directly discretizes the surface integral of the surface tension source term, similar to discretization approaches used in front tracking approaches. The second method, the Discrete Contour Method, DCM, directly enforces the surface tension pressure jump on faces near the interface. A source term for surface tension is absent, and DCM uses a similar approach to the Ghost Fluid Method instead, developed for unstructured grids by Queutey and Visonneau (2007) [86]. Overall, the novel methods perform a direct coupling of surface tension with the one-fluid formulation and compressive discretization schemes through the Marching Polyhedra Algorithm (MPA), a core component used in the formulation of the methods. Moreover, we can ensure that accurate and converging approximations of the normal vector and curvature are achieved due to the LSqR.

For each of the above methods, we have compared the results obtained for the static bubble test case, where a simple analytical solution is available, and rising bubble test cases, where a large number of results from experiment and numerical simulations are available. The results show that both methods that use the MPA as a basis for the surface tension coupling provide better results in comparison to the other methods, both from a numerical point of view and when compared to experimental results.

Specifically, DCM provided the best results for the static bubble test case, where we compared the obtained pressure field of the four different methods with the analytical solution. The result proposes that the treatment of the pressure jump as a captured boundary condition is preferable to the addition of a source term. Considering the results obtained from the rising bubble test cases, we have found excellent agreement between the terminal velocities proposed by Tomiyama's relation [START_REF] Tomiyama | Terminal velocity of single bubbles in surface tension force dominant regime[END_REF] and the calculated terminal velocities of ISIS-CFD. CSF and CSS provided results that are not always consistent with experimental results for the whole range of computations performed. For example, the aspect ratios calculated by CSF were small while CSS caused water to be diffused into the region occupied by air. DCM and IsoFT gave physically reasonable results both for bubbles in the ellipsoidal regime and the spherical cap regime. More specifically, for the spherical cap bubble, both DCM and IsoFT can predict the generation of a toroidal bubble behind a leading bubble and the production of smaller satellite bubbles. The results indicate that the proposed methods along with the compressive discretization schemes used by ISIS-CFD can accurately capture complex mechanisms that govern the interface's topological changes.

Therefore, we can conclude that the two surface tension methods based on the MPA, propose accurate alternatives to classic methods and can be used with compressive discretization schemes and general unstructured grids. Since we reached the above conclusions by studying cases, which may be considered trivial, we have also demonstrated the applicability of the proposed methods to two more demanding simulations.

Through the first simulation, we shortly examined the interactions of a bubble and a free surface.

The solution is in good agreement to free surface/bubble interactions observed in experiments. Several physical mechanisms were captured, such as the formation of a jet that begins from the collapse of the bubble to the free surface. The jet appeared as a sub-scale structure that separated from the air-water interface. Therefore the interface reconstruction scheme accomplishes one of its primary purposes, to distinguish in a simulation using compressive discretization schemes the large scales (modeled by the one-fluid formulation) from small scales (where explicit modeling of the dispersed state is required).

The second test case demonstrates that the proposed isosurface generation method can also be used for the reconstruction of interfaces that exhibit significant deformations as in cases of practical interest, relevant to naval engineering. Besides visualizing the interface, we have also shown that the approximated curvatures are consistent with the geometrical structures observed. As a result, the proposed curvature calculation method can be used for the study of problems of naval engineering with surface tension.

In conclusion, the results encourage the further development of the proposed methods and the extensions of the results. Firstly, the methods can be optimized on the different numerical discretization schemes used. For example, the spurious currents can be eliminated by introducing the appropriate modifications to obtain a force balanced approach as proposed by Francois et al. ( 2006) [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF]. Furthermore, the implementation of a boundary condition for curvature, Luo (2016) [START_REF] Luo | Curvature boundary condition for a moving contant line[END_REF], can aid the development of an accurate curvature calculations near boundaries, and it is essential for the prediction of wetting. Secondly, since we can ensure that the curvature computations will be second order accurate, it is interesting to study further the details provided by the simulation in this chapter by a systematic grid refinement study. Thirdly, a volume fraction reinitialization scheme has to be devised alongside automatic grid refinement to obtain improved approximations for the reconstructed interface and converging curvature estimates. Finally, the surface tension methods should be used for the study of phenomena which are directly related to the study of air lubrication methods in hydrodynamic flows.

We believe that using the proposed tools; interesting conclusions can be derived which can help us further understand the mechanics governing air entrainment from ventilated cavities (an important issue to predict the required amount of air to maintain the cavity). These are directly related to the design of air lubricated hulls. These perspective are summarized in the concluding chapter of this work that follows.

Chapter 5

Summary and Conclusions

In this work we have presented all the components required to introduce to ISIS-CFD, a modern parallel unstructured finite volume code for multifluid flows, surface tension methods that use high order curvature approximations and representation of the interface by isosurfaces of the volume fraction. To that end, we have studied the different aspects of the problem that motivated this work, namely, the study of the physical mechanisms that govern air lubrication methods for drag reduction in hydrodynamic flows and subsequently, the implementation to ISIS-CFD.

In the first chapter, we have summarized some recent experimental studies of air lubrication methods in hydrodynamics flows. We have emphasized the importance of the accurate capturing of the cavity closure and the modelling of mechanisms through which air escapes the cavity. These kind of phenomena can be described by the simultaneous action of different physical mechanisms with the air-water interface. Therefore, two different scales must be recognised, a large scale that concerns the air-water interface and a small scale, where multiple smaller air-water interfaces, i.e. mainly bubbles, are observed. Since these kind of phenomena are strongly related to the capturing of details near the interface, such as merging and separation of the interface, where surface tension can locally be the driving mechanism. Working towards that end, we have implemented a number of different tools, numerical techniques and finally, surface tension methods, which we developed in three chapters.

In the second chapter, we revisited the one-fluid formulation. We emphasized that in the volume averaged one-fluid formulation appear terms that have to be expressed as surface integrals of the air-water interface. Therefore, the volume grid that discretizes the ambient space and the surface grid that discretizes the air-water interface are naturally related by the one-fluid formulations. Consequently both representations are required simultaneously.

In the third chapter we described different applications of the Marching Polyhedra Algorithm (MPA).

Specifically, the algorithm is used to perform two necessary tasks. The first is the volume fraction initialization and the second is the construction of the surface grid through the volume fraction, that represents the air-water interface. We have also discussed other possible applications which can be used in the context of future work.

Finally, in the last chapter we have introduced the surface tension methods that have been implemented and tested in ISIS-CFD. The new surface tension methods introduced in ISIS-CFD have the novel characteristic that they use to their formulation the surface grid that represents air-water interface and is dynamically generated by the MPA. The results demonstrate that, besides the good agreement of measured and calculated terminal velocities for rising bubbles, the code can also model cases where interface merging and separation takes place. Furthermore we have shown that the methods can be applied to a large set of problems.

To that end, the next step is the use of the developed method for the study of air-injection method 209 for drag reduction in hydrodynamic flows. First of all, the effects of surface tension modeling can be introduced to problems such as the sharp edged plate discussed in the previous chapter and one can compare the solutions with and without different surface tension models. Secondly, the capabilities of ISIS-CFD and the proposed extensions can be applied to study both the partial cavity and the air cavity methods for axisymmetric bodies in a systematic manner with the aid of the versatile volume fraction initialization methods introduced. Since for these cases air must be artificially ventilated at the cavity, a boundary condition of air injection is required that has to be specified at certain regions of the solid boundary. Working towards that we have already proceeded in the preliminary design of a cavity underneath the hull of a KVLCC2 tanker, figure 5 We consider two examples. In the context of cavitation modelling, the following transport equation is proposed by Senocak and Shyy (2004) [START_REF] Senocak | Interfacial dynamics based modelling of turbulent cavitating flows, Part 1: Model development and steady state computations[END_REF][94], for the volume fraction of water :

∂ t C L + ∇ i u i C L = ρ L min( p -p V , 0)C L ρ V [( u i V -u i )n I i ] 2 (ρ L -ρ V ) t ∞ + max( p -p V , 0) (1 -C L ) [( u i V -u i ) n I i ] 2 (ρ L -ρ V ) t ∞
where ρ V , ρ L are the densities of vapor and water (liquid), p V is the vaporization pressure, t ∞ = l ∞ /U ∞ is a characteristic time scale, and C L is volume fraction of the liquid. This is clearly a modified version of the volume fraction transport equation. In this relation the RHS terms describe the replacement in space of water by vapor, with a significant change in the density. Note that when p < p V then the first term is "active" and when p > p V the second term is "active" thus we can recognize the first term as a "destruction"("production") term for the liquid (vapor) and the second term as a "production"

("destruction") term for the liquid (vapor). Senocak comments [START_REF] Senocak | Interfacial dynamics based modelling of turbulent cavitating flows, Part 1: Model development and steady state computations[END_REF]:

The model requires that an interface be constructed in order to compute the interface velocity V I ,n , as well as the normal velocity of the vapour phase. However, in steady-flow computation, the interface velocity

( u i V -u i ) n I i = -1.9ρ L /ρ V 1 -ρ L /ρ V
For cavitation, the transport equation takes into account that the changes between the two phases occur locally and around the interface. When mixing takes places the phenomenon is different. For air entrainment there is a concentrated source or air at the interface which is given by Ma et al (2011) [START_REF] Ma | Two-fluid modeling of bubbly flows around surface ships using a phenomenological subgrid air entrainment model[END_REF] as :

q = C 2 d ( u i n I i )| I k g (5.0.1)
where q is the volume of air entrained, C is a parameter of the problem, d ( u i n I i ) is the normal derivative of the normal velocity at the interface and k is the turbulent kinetic energy. Again this term can be naturally defined since an explicit interface representation can be reconstructed. In order to calculate the term d ( u i n I i ) we can greatly profit from a representation of the interface using a surface grid, As we have previously mentioned, we have to distinguish between two scales of the flow, a large scale that describes the evolution of the interface and a small scale that describes the evolution of a dispersed state of either air to water or water to air. Moreover, we have demonstrated that the interface reconstruction technique proposes a method to perform this distinction. Therefore all the locations where air is dispersed to water and water is dispersed to air can be dynamically found and thus these locations represent the locations where appropriate terms should be added for the modelling of the dispersed state. Recently, Han et al. (2015) proposed a similar approach where this distinction is made using the level set function. Therefore, a possible application in the context to future work is to formulate the appropriate source terms to handle the interactions of the dispersed phase and the continuous phase in the context of an Eulerian framework and the one-fluid formulation. Moreover, probably some modification are also required to ensure that the mass of the dispersed phase air to water will be conserved so that smaller structures air structures do not diffuse when found inside air.

More recently, an interface reconstruction approach based on an algorithm similar to the MPA has also been implemented in openFOAM by Roenby, Bredmose and Jasak (2016) [START_REF] Roenby | A Computational Method for Sharp Interface Advection[END_REF]. The purpose is to improve the capturing of the interface using the concept of an isoAdvector, a patch of the isosurface that is allowed to move inside a cell. From this patch the fluxes of the volume fractions can be dynamically obtained and used for the volume fraction transport equations directly. Their results demonstrate that isoAdvectors retain better the overall interface's shape for Courant numbers up to 0.5.

The above indicate that a reasonable conclusion is that the methods introduced in this work might significanlty contribute to the improvement of CFD software. Last but not least, the extensions provided to ISIS-CFD have also interesting academic applications for educational purposes. The developed framework offers students that attend CFD courses the capability to directly implement in three dimensions the numerical schemes taught without having to spend time on grid generation (which for the time being only inputs from GMSH and Hexpress are acceptable) or the visualization of the results (for the time being only Matlab output and Tecplot binary files are generated) which can be a quite bothersome process, especially for unstructured grids. These capabilities are offered by the highly modular and object oriented design which hides from the framework's user most of the programming details. Moreover, if the user desires he can easily navigate to locate some detail that he is interested in. Through this procedure besides shortening the "slow beginning" time period of the learning curve,

gives the opportunity to work with modern object oriented aspects of Fortran, classic visualization tools and, for the enthusiastic student, to learn the basic concepts of parallel processing with MPI. Due to the general manner the framework is written, this does not apply only to finite volume method, but also the finite element method and, since the framework treats in the same manner and in parallel surface grids, even boundary element methods.

In conclusion, a variety of extensions has been proposed to the ISIS-CFD software. These provide to the code a large set of capabilities to perform complex operations regarding surface tension modelling and the separation of interface scales to two group. One group, where the equations of the one-fluid formulation govern the dynamics of the flow and a second group, where multiphase modelling must be introduced. The representation of the "large" scale air-water interface by a surface grid is performed by the Marching Polyhedra Algorithm. Through the Marching Polyhedra Algorithm the development of two new surface tension models was possible. The methods were tested and compared to surface tension method proposed by other researchers. An important tool developed in the context of this work is the Distributed Grid Manager, through which the classic multiblock approach can be extended to a generalized multiblock approach and allows the implementation of higher order differencing schemes used to obtain robust normal vector and curvature estimation. Even though, the developed tools have not yet reached a saturation limit regarding their possible evolution, we have demonstrated that they can be used in a variety of cases and the results support their further development oriented towards the study of the whole range of phenomena involved in air cavities as air lubrication methods in hydrodynamic flows to achieve the demanding goal of successfully designing air-lubricated ships.

A.3 Curvature Tensor Calculation of a Monge Patch

In this part of the appendix, we present the calculation of the curvature tensor of a surface that can be locally represented by a Monge patch. Suppose that we work in the three-dimensional Euclidean space parameterized using a global Cartesian coordinate system. We refer to its coordinates using Z i , i = 1, 2, 3 for the x, y, z-axis respectively. In this space, there exists a surface for which we wish to calculate its curvature tensor. We shall suppose that at a point of the surface P , where its coordinates are Z i (P ), we can parameterize the surface using a Monge patch, i.e. at this point of the surface we can define its the local parameterization as:

Z 1 = Z 1 (P ) + s 1 Z 2 = Z 2 (P ) + s 2 Z 3 = Z 3 (P ) + z(s 1 , s 2 ) (A.3.1)
Thus if the position vector is R(Z 1 , Z 2 , Z 3 ) then the above defines the surface as R(S; P ) = R(s 1 , s 2 , z(s 1 , s 2 )).

The above representation of the surface has to be constructed in a local manner, around the point P , where the curvature tensor is to be calculated. The notation R(S; P ) (where S denotes the surface coordinates s 1 and s 2 ) reminds us that the patch is locally defined. The calculation of the mean and Gaussian curvatures of the Monge patch are well-documented (at least for the classic case above, see for example Oprea, 2004 [75, p.177]) and the calculations are presented for the sake of clarity, so that no confusion arises for the notation used.

We begin by calculating the basic geometric objects of the surface. Then we calculate the normal vector and finally the curvature. The shift tensor is: where the column's position refers to α and the row's position to i . Matrices are only used for presentation purposes, to avoid explicitly stating the components and so we do not imply any connection between matrices and tensors. We note again that the notation (S; P ) denotes that the derived geometric objects of the surface are defined locally. The components of the covariant basis vectors of the tangent space coincide to the column-α of the shift tensor. If s α are the covariant basis vectors then they can be expressed in the ambient coordinate system as: From this point on we will use the notation (S; P ) only for comments and in most relations it will be implied.

The covariant metric tensor is: Here the covariant metric tensor of the ambient space is Z i j (S; P ) = δ i j . If the ambient space is not Cartesian then the metric must be evaluated appropriately at the point marked by the surface coordinates S. Thus, we also require a mean of expressing the metric of the ambient space with respect to the point P where the Monge patch is constructed. The determinant of the covariant metric tensor (or the square of the surface element) is:

s αβ = s α s β = Z i j Z i α Z j β =        1 +
S = 1 + ∂z ∂s 1 2 + ∂z ∂s 2 2 (A.3.6)
The contavariant metric tensor is: The normal vector is given by:

s αβ = 1 S        1 + ∂z ∂s
n i = 1 2 i j k αβ Z j α Z kβ = 1 S i j k Z j 1 Z k2 (A.3.8)
The component of the shift operator Z j α are the same as the components of Z j α and also the covariant and contravariant components of the normal vector coincide, since the background coordinate system is Cartesian, so we obtain:

n i = 1 S - ∂z ∂s 1 - ∂z ∂s 2 1 (A.3.9)
The above notation acts as a reminder that the proper definition of the normal vector written as n uses the covariant basis vectors of the ambient space, so: With the normal vector known we can calculate the covariant components of the curvature tensor:

n = n i Z i = n 1 Z 1 + n 2 Z 2 + n 3 Z 3 = - 1 S ∂z ∂s 1 Z 1 -
B αβ = -Z i α ∇ β n i = -Z i α ∂n i ∂s β (A.3.12)
where the last equation holds since the Christoffel symbols of the ambient space vanish. Equivalently we may write the curvature tensor using the frequently encountered expression:

B αβ = -s α ∇ β n = -s α ∂ n ∂s β (A.3.13)
where in the last relation the derivatives of the normal with respect to the surface coordinates (the surface derivatives), naturally gives rise to the Christoffel symbols of the ambient space (in our case these are zero). The last equality in relation (A.3.13) holds for any parameterization of the ambient space, in contrast to the last equality in relation (A.3.12) that holds only when the ambient space is parameterized using Cartesian coordinates. Here we will calculate using the above, the contravariant components of the curvature tensor and derive the components of the mixed tensor as B β α = B αγ s γβ . This two step approach simplifies the calculation specifically when they are performed by hand.

Alternatively we could have used one step by evaluating:

B β α = -Z β i ∇ α n i (A.3.14)
However, the shift tensor Z α i has not been calculated. Even though no difficulty is posed during its evaluation, the shift operator Z α i = s α Z i , which provides the components of the contravariant basis vectors of the surface's tangent space with respect to the ambient space, (1) the shift operator Z α i must be calculated while the shift operator Z i α , required by (A.3.12), is already available since it coincides to the shift operator Z i α , (2) the shift operator Z i α is sparser than the shift operator Z α i : and, (3) the covariant curvature tensor is symmetric so we can calculate only three out of its four components, while the mixed curvature tensor is not symmetric. Eventually, the same result will be obtained but in a slightly more painful manner. Note that (i) the elements of the column-α o f Z α i coincide to the components of the contravariant basis vector s α , since the ambient space is Cartesian, and (ii) that Z i α Z where the symbol c i α is used for presentation purposes and is equal to :

Z α i = Z i β s βα = 1 S              1
c i α = - ∂( Sn i ) ∂s α =             ∂ 2 z ∂s 1 ∂s 1 ∂ 2 z ∂s 1 ∂s 2 ∂ 2 z ∂s 2 ∂s 1 ∂ 2 z ∂s 2 ∂s 2 0 0             =          z uu z uv z uv z v v 0 0          (A.3.17)
As you can observe from the last relation, we rename the surface coordinates s 1 and s 2 as u, v and the partial derivatives, z u , z v , z uu , z uv , z v v to obtain a more elegant presentation. Finally, by substituting (A.3.16) to (A.3.12), the curvature tensor is given by: The mixed curvature tensor is:

B αβ = 1 S Z i α c i β (A.
B β α = B αγ s γβ = 1 S 3/2     1 + z 2 v z uu -z u z v z uv 1 + z 2 u z uv -z u z v z uu 1 + z 2 v z uv -z u z v z v v 1 + z 2 u z v v -z u z v z uv     (A.3.20)
The trace of the curvature tensor (the curvature) is: We will show that if the ambient velocity field at a fluid interface is continuous, u = 0, the velocity field from both fluid regions I and J , is equal to the interface velocity w = u I = u J , and the flows are incompressible ∇ i u i I = 0 and ∇ i u i J = 0, then:

B α α = 1
S 3/2 1 + z 2 v z uu + 1 + z 2 u z v v -2 z u z v z uv (A.
T i j n j = Z i α µ (w β B αβ + ∇ α w) + µdu α + n i -p -2 µ (∇ α w α + wB α α ) (B.1.2)
We refer to the LHS of the above equation, as the traction jump. First, the traction will be decomposed to its normal and tangential components with respect to the interface. The traction jump follows from this decomposition. Finally, we simplify the formulas obtained.

First of all, the stress tensor is:

T i j = -p Z i j + 2µS i j (B. 1.3) where S i j is the rate of strain tensor:

2S i j = Z i k ∇ k u j + Z j k ∇ k u i (B.1.4)
The above tensors are defined at the interface, only as we approach either the side of fluid I or the side of fluid J . In the above relations, we have skipped the index denoting the fluid (either I or J ) which is considered as implied to simplify the notations. The traction jump is:

T i j n j =p n i + 2 µS i j n j (B.1.5)

The jump of viscous stresses The strain tensor can be decomposed to its normal and tangential components, from either side of the interface, using relation (A. The traction jump The traction jump is:

T i j n j = -p + µ n j du j n i + Z i α µ u β B β α + ∇ α u + µdu i (B.1.18)
Finally, in order to emphasize that the above velocity field refers to the surface, we replace the components of the velocity field u with the velocity field w defined on the surface:

T i j n j = -p + µ n j dw j n i + Z i α µ w β B β α + ∇ α w + µdu i (B. 1.19) For the last term, the velocity field was not replaced by the field w in order to point that there will be a discontinuity in the normal derivative of the velocity components, since there is no relation that stems from physical considerations and specifies that the last term is continuous. The evaluation of the derivatives is meaningful only as we approach the interface from fluid I and fluid J the last term acts as a reminder of the last observation.

The normal components of the jump are obtained by multiplying the last relation by n i and replacing n j du j by (B.1.13) with ∇ j u j = 0, so we have:

n i T i j n j = -p + µ n i du i + µn i du i = -p + 2 µ n i du i = -p -2 µ ∇ α w α -wB α α (B.1.20)
The tangential components of the jump are obtain by multiplying by the shift operator Z where we emphasize that the normal vector in the above relation is n J that points from the region J to the region I (in this work capital letters never refer to components), and (ii) the force acting from fluid J to the interface: Therefore the integral S T i j n j d S is the force acting from the ambient space to the surface and thus -S T i j n j d S is the force acting from the surface to the fluids occupying the ambient space. (or ∇ i f 2 ), the integral would be evaluated over the part of the interface S 2 1 (or S 1 2 ) that rests inside the support of the kernel. Specifically, for the case of C.3b, the integral would have to be broken into to two integrals, one for each part of the interface that lies inside the support of the kernel. In figure C.3c, two different interfaces are present inside the support of the kernel. When relation (C.1.5) is evaluated for either fluid subdomains 1 (or 3), thus we evaluate ∇ i f 1 (or ∇ i f 3 respectively), the the sum contains a single term, i.e. for the fluid subdomain 1, the integral over the interface S 2 1 (for fluid 3, the integral over the interface S 2

3 ). When the relation is evaluated for the fluid subdomain 2, ∇ i f 2 the sum contains two terms one for the interface S 1 2 and one for the interface S Note that symbolic notation of the sum's index I J with the fluid's subdomain index I leading J is different than the symbolic notation J (I ). The notation I J acts as a reminder that it uniquely enumerates and identifies the different interfaces, S J I , as viewed from the subdomain I (the first index). Simultaneously, the normal of the interface obtains an orientation along with the jump operator which is appropriately defined only once for the fluid subdomains that it connects. For example, for the case shown in figure C.3a (also C.3b), the sum contains a single term that is evaluated by either the interface S 2 1 when I = 1 and J = 2, or S 1 2 when I = 2 and J = 1 but not both. Either way, we obtain the same result since the orientation of the normal vector changes along with the definition of the jump, e.g. if the integral is evaluated by S 2 1 then the unit normal vector points from region 1 to region 2 and the jump is defined by • = (•) 2 -(•) 1 . For the case shown in figure C.3c, the sum contains two terms, one the interface S 2 1 (or S 1 2 ) and one for S 3 2 (or S 2 3 ).

Relations between center of mass derivatives to derivatives of the center of mass

We provide the manipulation to derive relation (C. 1.11). We arrive to a relation equivalent to (2.4.41), for the Favre average of a field's gradient:

∇ i f = ρ∇ i f ρ = I ρ∇ i f I ρ = I ∇ i (ρ f ) I ρ = ∇ i (ρ f ) ρ (2.
∇ i f = ∇ i f + f -f ∇ i ρ ρ (C.1.11)

C.2 Notes on Analytical Solutions

Static Bubble: Surface Tension Source term To facilitate the source term evaluation we work in a local spherical coordinate system and a local background Cartesian coordinate system, both defined with respect to the point where the source term is evaluated. For any point on the sphere p S , set a background Cartesian coordinate system whose origin coincides with the sphere's center p O and whose z-axis has the same orientation as the the unit exterior normal vector of the sphere at this point, i.e. the unit vector n J I ( p S ) pointing from the subdomain I (inside the region bounded by the sphere) to the subdomain J (outside the region bounded by the sphere). We then define for any point in space p the spherical coordinates as:

ρ( p) = p -p O cos φ( p) = p -p O ρ( p) n J I ( p S ) = p -p O p S -p O p -p O R
The angle θ can be defined by choosing any x-axis. Due to the spherical symmetry of the problem the final result will be the same. The point where the averaging is performed is somewhere on the z axis (or, equivalently the φ = 0 axis).

As mentioned earlier the integrand is different from zero only when the interface is located inside the support of the kernel (the interface intersects the support of the kernel). Since we have chosen a kernel with finite support, the region of integration is the part of the surface that rests on the support of the kernel. The integration region is shown as a thick gray line in in figure C.4 and it can be defined, in the spherical system introduced as "S J I inV " = (ρ, θ, φ) : ρ = R, θ ∈ [0, 2π), φ ∈ [0, φ 1 ] . The angle φ 1 is defined in the same figure. Note that z is also the distance of the kernel's center to the sphere's center. We also provide the contravariant components of the surface normal vector in the background Cartesian system where we perform the integration. Substituting the curvature, the kernel function and writing the surface tension integral (2.5.13) in

R z J I φ 1
The angle φ 1 is such that:

cos(φ 1 ) = z 2 + R 2 -2 2 R z (C.2.1)
The contravariant components n i are: The final solution is:

u I (y) = V µ J h J µ I -h I µ J y + V int u J (y) = V µ I h J µ I -h I µ J y + V int (C.2.12)
where V int is the velocity of the interface:

V int = - V h I µ J h J µ I -h I µ J (C.
∇q κ = q κ V κ f ∈κ f a f c →κ s f + 1 V κ f ∈κ f a f c →κ q f c →κ s f + 1 V κ f ∈κ f ∇q κ v f c →κ s f + 1 V κ f ∈κ f ∇q f c →κ v f c →κ s f (D.1.6)
Observe that the sum of the first two terms is the gradient that would be calculated if the vector coefficients where considered to be zero. To simplify even further the above, we introduce the term:

∇q • κ = q κ V κ f ∈κ f a f c →κ s f + 1 V κ f ∈κ f a f c →κ q f c →κ s f (D.1.7)
which can be recognized as the gradient that would be calculated if the vector coefficients where considered to be zero. If we consider that all the vectors above can be viewed as 3 × 1 columns, then we may write (D.1.6) in matrix notation as:

∇q κ = ∇q • κ + 1 V κ f ∈κ f s f v T f c →κ ∇q κ + 1 V κ f ∈κ f s f v T f c →κ ∇q f c →κ (D.1.8)
Note that the order of multiplications must be respected and the result of a b T is: With this notation the gradients are multiplied by some matrix coefficients.

a b T =    
The gradient at cell κ was factored out of the summation in the second term of (D.1.8) and the result of the sum is a matrix that multiplies the gradient at κ. Similarly, since the third sum contains the gradient evaluated at the cell adjacent to the face's of the cell κ, it can be reformulated as a sum of these cells. By definition, the cell adjacent to the face's of the cell κ is the topos (or level 1 cell) neighbors (based on the faces) of cell κ, κ t f . Therefore, as final step we introduce the matrix coefficients:

Āκ = 1 V κ f ∈κ f s f v T f c →κ and Bκλ = 1 V κ s f v T f c

1. 1

 1 Different drag reduction methods concepts using air injection . . . . . . . . . . . . . . . 1.2 Artificial cavity designs in different hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Air cavity characteristics in design and off-design conditions . . . . . . . . . . . . . . . . 1.4 Model scale experiments with an air cavity by Matveev et al. (2009) . . . . . . . . . . . . 1.5 Model scale experiments with an air cavity by Arndt et al. (2009) . . . . . . . . . . . . . . 1.6 Natural cavities and artificial cavities in axisymmetric bodies . . . . . . . . . . . . . . . . 1.7 Artificial cavities generated by disk cavitators . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Artificial cavities generated by a disk cavitator and two different cavitation numbers . . 1.9 The two types of cavity closure zones for ventilated cavities of axisymmetric bodies . . 1.10 The ISIS-CFD Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 A schematic representation of the normal derivatives of the normal velocity component and the tangential velocity components at the interface . . . . . . . . . . . . . . . . . . . 2.2 A schematic representation of the bar operators . . . . . . . . . . . . . . . . . . . . . . . . 2.3 A schematic representation of the tilde operators . . . . . . . . . . . . . . . . . . . . . . . 2.4 Analytical relations of volume fraction fields . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Examples of analytical volume fraction fields . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Source terms and averaged pressures field for the static bubble test case . . . . . . . . . 2.7 Domain description for the two fluid plane Couette flow . . . . . . . . . . . . . . . . . . . 2.8 Local and averaged fields for the water-air plane Couette flow . . . . . . . . . . . . . . . 2.9 The ratio ( uu)/ u for the plane Couette flow . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 The ratio E (ν, S 12 )/µ S 12 ) for the plane Couette flow . . . . . . . . . . . . . . . . . . . . . . 2.11 The Favre averaged velocity field derived by solving the one-fluid equations and enforcing C 1 smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Favre averaged velocity solutions of the one-fluid equation by enforcing different conditions near the interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Relative errors of Favre averaged velocity solutions of the one-fluid equation by enforcing different conditions near the interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The 14 isosurface patch construction cases for the classic Marching Cubes Algorithm . 3.2 An ambiguous case of patch construction for Marching Cubes . . . . . . . . . . . . . . . 3.3 Two different configurations of an interface inside a cell . . . . . . . . . . . . . . . . . . . 3.4 Isonodes and geometrical space representation of an isopatch . . . . . . . . . . . . . . . 3.5 Volume fraction calculation and absolute errors for a planar interface . . . . . . . . . . . 3.6 Representation of a planar interface in a background grid generated by Hexpress . . . . 3.7 Volume fractions for a spherical interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii viii LIST OF FIGURES 3.8 Local relative distance errors (based on the centroids of the isopatches) and volume fractions for a spherical interface and a background Cartesian grids . . . . . . . . . . . . 105 3.9 Error diagrams for spherical isosurfaces generated in Cartesian background grids . . . . 106 3.10 Volume fraction initialization of a free surface and a submerged bubble . . . . . . . . . . 107 3.11 Volume fraction initialization of a free surface and a planar interface inside a cavity of a body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.12 PLIC reconstructions of a spherical interface in a uniform Cartesian grid and a tetrahedral grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.13 Locations of points and data for the interpolation corrections . . . . . . . . . . . . . . . . 114 3.14 Reconstruction error diagrams for a spherical isosurfaces generated in Cartesian background grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.15 Comparison between the initial and the reconstructed isosurface grids for a spherical interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.16 Local relative distance errors (based on iso-nodes) for the reconstructed surface of a spherical interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.17 Reconstruction error diagrams for a sinusoidal isosurface generated in a Cartesian background grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.18 Local relative distance errors (based on iso-nodes) for the reconstructed surface of a sinusoidal interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.19 Comparison of reconstructed surface grids for the sinusoidal wave case . . . . . . . . . 122 3.20 Unconstrained iso-patch neighborhoods based on iso-nodes . . . . . . . . . . . . . . . . 127 3.21 Constrained iso-patch neighborhoods based on iso-nodes . . . . . . . . . . . . . . . . . 128 3.22 Iso-node samples generated by volume grid neighborhoods and surface grid neighborhoods at nearby surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.23 Normal vector and curvature L max and L 1 error diagrams using least square differencing for the exact isosurface of a spherical isosurface (R = 0.5) and Cartesian background grids.132 3.24 Normal and curvature L max error diagrams using least square differencing for the reconstructed interface by the volume fraction of a spherical isosurface (R = 0.5) and Cartesian background grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.25 Normal and curvature L 1 error diagrams using least square differencing for the reconstructed interface by the volume fraction for a spherical isosurface (R = 0.5) and Cartesian background grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.26 Normal L max error diagrams using least square differencing with level 2 and 3 neighborhoods for the reconstructed interface by K 1(C I ) and K 2(C I ) of a spherical isosurface (R = 0.5) and Cartesian background grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.27 Curvature L max error diagrams using least square differencing with level 2 and 3 neighborhoods for the reconstructed interface by K 1(C I ) and K 2(C I ) of a spherical isosurface (R = 0.5) and Cartesian background grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.28 Cut-Cell volume grid generation using the MPA . . . . . . . . . . . . . . . . . . . . . . . . 142

. 1 .

 1 From the alternatives shown in the figure, the PCDR (Partial Cavity Drag Reduction) and the multi-wave PCDR concepts propose a straightforward lubrication mechanism.

figure 1 . 1 .

 11 figure 1.1. The characteristic formation of the air-water interface, for the air chamber's design point (freestream velocity V d , air influx Q d ) is shown in the upper photograph. Due to technical reasons the researchers could not achieve a higher air influx than Q d . The characteristic formation of the air-water interface, for the air chamber's off-design point (freestream velocity V , air influx Q d ) is shown in the lower photograph.

Figure 1 . 3 :

 13 Figure 1.3: Air cavity characteristics in design and off-design conditions. Measured forces and photographs of artificial cavities generated. Experiments performed in a cavitation tunnel. Results adapted by Allenström and Leer-Andersen (2010) [3].

  photographs on the right show the actual configurations. The air chamber has a rectangular outline, beginning with a step upstream and extends downstream covering the whole length of the ship, similar to the air chamber of the hull in figure1.2a.

Figure 1 . 4 :

 14 Figure 1.4: Model scale experiments with an air cavity. Centerline length 0.558m, beam 0.3 m. Corresponding Froude numbers are 0.16, 0.26, 0.37. On the left, schematic representations of the cavity and the interface, on the right, photographs of the experiments. Adapted from Matveev et al. [67].

(

  transparent) flat air-water interface to an opaque (perturbed) one. Some photographs adapted from the experiments of Arndt et al. (2009) are presented in figure 1.5. The height of the air chamber varies as shown in figure 1.1 for PCDR. The freestream flow is from left to right. The step of the air chamber is visible in the left corner of each figure. The "beach", the surface that smoothly recovers from the step's depth to the model's stern, is visible at the top right of the figures. The corresponding Froude numbers are 0.2, 0.29, 0.41, 0.446, 0.504 and 2.935 (from bottom to top).

Figure 1 . 5 :

 15 Figure 1.5: Model scale experiments with an air cavity. Centerline length 0.5m. Corresponding Froude numbers are 0.2, 0.29, 0.41, 0.446, 0.504, 2.935. Adapted from Arndt et al. (2009) [4].

Figure 1 . 6 :

 16 Figure 1.6: Natural cavities and artificial cavities in axisymmetric bodies. Photographs adapted from Brennen (1970) [11]. Incoming flow is from right to left.

  (a) Cavity near disk. Incoming flow from right to left. V ∞ = 6.1 m/s, Re = 4.6 • 10 5 . (b) Disk cavitator, its cavity and a schematic representation of its cross sections. Incoming flow from left to right. Side view with gravity acceleration pointing downwards.

Figure 1 . 7 :

 17 Figure 1.7: Artificial cavities generated by disk cavitators. A steady ventilated cavity as observed at Brennen's experiments (1970) [11], left figure, and modern experiments of Kawakami and Arndt (2011) [49], right figure.Figure 1.7b also demonstrates the disk cavitator used and an artistic representation of the cross sections of the cavity's interface.

Figure 1 .

 1 Figure 1.7: Artificial cavities generated by disk cavitators. A steady ventilated cavity as observed at Brennen's experiments (1970) [11], left figure, and modern experiments of Kawakami and Arndt (2011) [49], right figure.Figure 1.7b also demonstrates the disk cavitator used and an artistic representation of the cross sections of the cavity's interface.

Figure 1 . 8 :

 18 Figure 1.8: Artificial cavities generated by a disk cavitator and two different cavitation numbers for same F r = 24.5. Upper figure σ c = 0.0334, lower figure σ c = 0.0644. The first cavity is significantly larger. Flow from left to right, adapted from Semenenko (2001)[START_REF] Semenenko | Defense Technical Information Center Compilation Part Notice[END_REF]

Figure 1 . 9 :

 19 Figure 1.9: The two types of cavity closure zones for ventilated cavities of axisymmetric bodies. In the first two figures, foam is rejected from the cavity. In the third and fourth figure, showing a closeup of the cavity closure, a vortex pair is formed through which gas escapes and mixes with the exterior fluid. Details of the vortex pair are demonstrated in the second and third figures. The figures were adapted from Kawakami et al. (2011) [49].

  ISIS-CFD is a modern parallel multiphase flow solver used by the commercial software FINE TM /Marine oriented towards Naval Hydrodynamics. It is developed by the Research Laboratory in Hydrodynamics, Energetics, and Atmospheric Environment (Laboratoire de la Recherche en Hydrodynamique, Énergétique et Environnement Atmosphérique) situated at Ecole Centrale de Nantes.

4 .

 4 ISIS-CFD is fully parallelizable via a multiblock approach written in MPI, and GPU implementations are currently under study.ISIS-CFD uses purely algebraic VOF schemes for the solution of the volume fraction transport equation. For the case of algebraic VOF approaches, there is no implemented interface reconstruction procedure coupled with the volume fraction transport equation. Moreover, they allow transporting the volume fraction with Courant numbers which might locally be larger that 0.3, usually a safe threshold to ensure stability. The ability to use large Courant numbers is an essential aspect for engineering applications since local grid scales might otherwise not allow practical computations which will complete in reasonable execution times.[START_REF] Queutey | An interface capturing method for free-surface hydrodynamic flows[END_REF] [86] discuss certain implemented compressive schemes in ISIS-CFD, and the interested reader may refer to this publication for more details.One of the capabilities of Automatic grid refinement (AGR) is the "tracking" of the interface (a geometric boundary represented by the volume fraction -a scalar field) by periodic modification of the volume grid. Thus, when we visualize the grid we can observe the subsequent location of the free surface. An interesting discussion along with application examples can be found in Wackers et al.[112][START_REF] Wackers | Combined refinement criteria for anisotropic grid refinement in free-surface flow simulation[END_REF].Finally, multiphase modeling in the context the one-fluid formulations, as cavitation modeling, is one of the latest developments (along with sliding grids). The implemented method is based on the solution of a transport equation for the volume fraction of vapor which includes a source term that depends on the local pressure. Some of the implemented methods are described bySenocak and Shyy (2004) [START_REF] Senocak | Interfacial dynamics based modelling of turbulent cavitating flows, Part 1: Model development and steady state computations[END_REF][START_REF] Senocak | Interfacial dynamics-based modelling of turbulent cavitating flows, part-2: Model development and steady-state computations[END_REF]. More recently, such methods have successfully been used to predict results for artificial supercavities of axisymmetric bodies by[START_REF] Ji | Numerical investigation of the ventilated cavitating flow around an under-water vehicle based on a three-component cavitation model[END_REF] [START_REF] Ji | Numerical investigation of the ventilated cavitating flow around an under-water vehicle based on a three-component cavitation model[END_REF]. All of the above are directly relevant to this work. The main contributions to ISIS-CFD are:1. General initialization methods for the volume fraction for single or multiple interfaces 2. An interface reconstruction algorithm that generates unstructured surface grids embedded to volume grids (for either a single or multiple interfaces) 3. Surface tension modeling 4. High-order differencing schemes both for volume and surface unstructured grids 5. Extensions of the classic multiblock strategy for the implementation of the aforementioned items in parallel

  interacts with the ISIS-CFD main code during different steps of its execution, either acquiring data or providing data. The flow diagram of the ISIS-CFD main code is shown in figure 1.10. The diagram represents a simplified version of the actual flow diagram and emphasizes the steps where interactions with the ISIS-CFD extensions framework take place. Gray-shaded shapes denote the procedures related to the ISIS-CFD extensions framework.

Figure 1 . 10 :

 110 Figure 1.10: The ISIS-CFD flowchart, highlighted processes denote part of the ISIS-CFD extensions

  we are interested in the flow of more than one fluid, then the problem is characterized as a multifluid problem. In this case, the (open) flow domain, D, is separated to a number of mutually exclusive non-overlapping subdomains D I , I = 1, n f l where n f l denotes the number of the subdomains covering D. Formally, D = ∪ I =1,n f l D I and D I ∩D J = , ∀I , J = 1, n f l , I = J . Each open subdomain, D I is bounded by a closed boundary ∂D I and may have a common boundary to more than one neighboring subdomain. We denote by S J I the common boundary between the subdomain I and the subdomain J (equivalently, S I J is the common boundary of D J with the neighboring D I ). The boundaries S J I are parts of the entire boundary of D I , so if we introduce for the subdomain D I the list N I containing the indices scanning the neighboring subdomains then ∂D I = ∪ J ∈N I S J

3 . 24 )

 324 A relation also given[START_REF] Tryggvason | Direct Numerical Simulations of Gas-Liquid Multiphase Flows[END_REF] [105, p.39]. The the jump on the RHS of the last relation is the resultant fluid stresses (or simply the fluid stresses) acting on the interface (see appendix, p.234). By expressing the fluid stresses to their tangential and normal components at the interface we arrive to:

  The derivative of the normal velocity component du is continuous as we move in the direction The derivative of the tangential velocity component d(Z i γu γ ) is discontinuous as we move in the direction normal to interface

Figure 2 . 1 :

 21 Figure 2.1: A schematic representation of the normal derivatives of the normal velocity component and the tangential velocity components at the interface. The values of each field are given in different colors to distinguish the subdomain where they are defined. A thick line marks the interface. The red vectors represent its normal and tangent vectors.

2. 3 . 3

 33 Summary Differential Equations and Boundary Conditions for the Multifluid Problem To summarize, a multifluid problem describes the simultaneous evolution of more than one fluid flow. The fluids are not allowed to mix and each flow region can be distinguished by a single surface separating the fluids. The multifluid problem is mathematically formulated as: Differential Equations The incompressible Navier-Stokes must hold in each fluid flow region:

4 . 4 )

 44 and represents the flow of a "fluid ensemble" whose density and viscosity obey the relations: ρ = I ρ I C I and µ = I µ I C I We should recognize that the presence of the interface is indicated by a thin layer whose length scale is indirectly defined by the volume fraction. At first hand, this derivation allows us to obtain a simplified model which we refer to as the one-fluid model. The purpose of the rest of this section is to mathematically formulate the above model by explicitly taking into account into our formulation that the length where the fluid properties are distributed around the interface is introduced by the volume fraction.

Figure 2 . 2 :

 22 Figure 2.2: A schematic representation of the bar operators. A thick line marks the interface. The bar operators perform the integrations of a field f in the gray regions to assign a field value f I or f to the point x.

Figure 2 . 3 :

 23 Figure 2.3: A schematic representation of the tilde operators. A thick line marks the interface. The tilde operators perform the integrations of a field f in colored regions of the interface to assign a field value f I or f to the point x. The arrow acts as a reminder that the result is assigned to the point x.

4 . 54 )

 454 However, the variables are the average density and the center of mass (or Favre) averaged velocity instead of the "local or particle" density and the "local or particle" velocity. Because each fluid flow is incompressible, the volume of each fluid subdomain must be conserved and the volume fraction conservation equation and the mass conservation equation are not independent equations. If we multiply (2.4.48) by ρ I , we obtain the mass conservation equation for the I -th fluid:

4 . 59 )

 459 However,(2.4.58), does not imply that the volume dilation rate of the Favre averaged velocity field is related to incompressibility effects. Compressibility effects are absent since pressure is not related by an equation of state with the density and relation (2.4.58) expresses the volume dilation rate of the fluid subdomains that are inside the support of an averaging kernel and are constantly being replaced by other fluid subdomains. Using relation (C.1.11), to obtain the divergence of the Favre averaged velocity field we obtain (note that

  Equations and Boundary Conditions for the One-Fluid Formulation The classic Navier-Stokes equation and the boundary (or compatibility) conditions for the interface that need to enforced to obtain the local field values (the term "local" for a field signifies that these fields are different from the averaged fields) were replaced by the volume averaged one-fluid Navier-Stokes equations. The equations provide the Favre averaged velocity and the average pressure. The multifluid problem is mathematically formulated in the one-fluid approach by:

4 . 82 )

 482 introduced in the RHS of the averaged momentum equation. The term represents the sum of the averaged surface tension forces of the interfaces that are near the point were the equation is written. "External" Boundary Conditions Appropriate boundary conditions must be also specified in the boundary of the whole fluid flow domain. Concluding Remarks In this section we presented a mathematical formulation of the one-fluid Navier-Stokes equations based on volume averaging. To that end, we introduced the basic definitions, derived fields and relation used by the volume averaged technique that is well-developed in the framework of multiphase flow modeling. The basic characteristic of the one-fluid formulation is that the boundary conditions that need to be enforced in the free surface (a physical boundary) are implicitly taken into account, along with the movement of the interface with respect to the stationary kernels. These are consequences of reformulating the equations for the averaged local fields instead of the classic local fields. The averaged values are not the same as the local field values and they depend on the averaging kernel K . As the parameter tends to zero the classic Navier-Stokes equation are recovered. Therefore we have to agree on the following points that are included in the formulation: 1. The fluids are imminsible 2. The interface is massless and surface tension acts from the interface to the ambient fluids 3. The volume fraction is an averaged indicator function 4. We solve for the volume fraction, the averaged pressure and the Favre averaged velocity field

When d (or 1 2 J

 12 y) the kernel's support does not intersect the interface and lies entirely in the subdomain J , so C J = 1. When d -(or l -1) the kernel's support does not intersect the interface and lies entirely in the subdomain I , so C J = 0. Examples demonstrating the volume fraction fields obtained by the above relations are given in figure 2.5. The volume fractions that correspond to a planar interface are plotted in figures 2.5b and 2.5c for = 0.3 and = 0.1 respectively. The volume fractions that correspond to a spherical interface are plotted in figures 2.5e and 2.5f for = 0.3 and = 0.1 respectively. The y-axis of the figures is the distance d from the local tangent plane (see also 2.5a and 2.5d) and the x-axis is the volume fraction values. Firstly, observe that for smaller supports of the kernel (when a smaller averaging length is used), the volume fraction tends to the indicator function. For the two kernels, the volume fraction obtained are different. The dashed lines in figures 2.5b and 2.5c are the volume fraction values for the kernel with Interface and Kernel's support Volume Fraction -Domain J d I

5 . 5 )Figure 2 . 4 :

 5524 Figure 2.4: Analytical relations of volume fraction fields. Each relation provided correspond to the case shown in the figure on the left. From top to bottom: planar interface with kernel (2.5.1), planar interface with kernel (2.5.2), spherical interface with kernel (2.5.2).

1 Figure 2 . 5 :

 125 Figure 2.5: Figure2.5a schematically represent the support of two kernels near a planar interface and gives the basic nomenclature for each case. We plot the volume fractions in figures 2.5b, 2.5c for two values of the averaging length . The x-axis corresponds to the volume fraction values, C I and C J and the y-axis to the distance from the interface d . Figure2.5d schematically represent the support of a spherical kernel near a spherical interface. We plot the volume fraction in figures 2.5e,2.5f for a sphere of radius, R = 1. Dashed lines were produced using the top-hat in-sphere kernel.

  .5.16) in the solution of the averaged pressure, (2.5.15), becomes a Heaviside function and the averaged pressure field coincides with the local pressure field. More specifically, the term (2.5.16) is the volume fraction of the I regions, C I (as we can verify directly from C I = 1-C J and the relation that defines C J , (2.5.5)). Hence, relation (2.5.15) can be written as: p = 2σ R C I or p = -σB α α C I (2.5.17) which is the desired result. In conclusion, the analysis we followed here is in agreement with the theoretical construction of the one-field Navier-Stokes equations. Tryggvason et al. (2011) [105] arrive at the same relation through the discretized equations. The above analysis shows that the result is the analytical solution of the static bubble for the one-fluid formulation of the Navier-Stokes.

Figure 2 . 6 :

 26 Figure 2.6: Source term (left) and averaged pressure field (right) for the static bubble test case for different values of . The plotted results are based on the relation (2.5.14) and (2.5.15) and the y-axis is the signed distance from the tangent plane d (d > 0:air region, d < 0:water region, d = 0 interface). The parameter values are R = 1 m, σ = 0.073 N m -1 and the units of the source term are N m -3 , pressure in P a and in m.

AFigure 2 . 7 :

 27 Figure 2.7: Domain description for the plane Couette flow with two fluids

  e. f I (y) = a I y +b I and f J (y) = a J y +b J , we can directly evaluate the averaged fields, f I . The relations we use are given in the appendix (see p.239). Using relations (C.2.14) and (C.2.15) we can evaluate the averaged pressure p and the averaged velocity u, by substituting the appropriate values for the constant a I , a J , b I and b J . The center of mass or Favre averaged velocity u easily follows. Figures 2.8a and 2.8b demonstrate the derived fields,for the case where the fluid I is water and the fluid J is air and = 0.002 m. For both averaged fields (pressure and velocity) the gradient discontinuity (due to the difference in the fluid density and viscosity respectively) is replaced by a smooth variation from the local field of the subdomain I to the local field of the subdomain J (plotted as red lines in the figures). In the one-fluid formulation, we are solving for the averaged pressure field, and the Favre averaged or center of mass velocity field (blue line in figure2.8). We can observe that the Favre averaged velocity will always deviate from the local velocity field of the lighter fluid, in this case, air (fluid J , the black line above y = 0). Specifically, the Favre averaged velocity tends to be almost equal to the local velocity of the heavier fluid, in this case, water (fluid I , the black line above y = 0). As we approach y = , the Favre averaged velocity field quickly recovers from the side of the air subdomain to the local velocity value of air.The u field behaves in this manner due to the significant difference in the density of the two fluids and tends to be almost equal to the local velocity field of the heavier fluid, in this case of water (fluid I ).

Figure 2 . 8 :

 28 Figure 2.8: Local and averaged fields for the water-air plane Couette flow. The local pressure field (black line) in figure 2.8a is derived by the analytical solution of the classic problem, relation (2.5.21). The averaged pressure field (red line) is directly calculated by the definition of the bar operator. Similarly for the velocity, the local field (black line) was generated by (2.5.22), and the averaged velocity, u, and the Favre or center of mass average velocity, u, directly by the definition of the related operator. Application for, fluid I :water, fluid J :air, h I = -0.005 m, h J = 0.005 m, V = 0.005 m/s, = 0.002 m.

2 Figure 2 . 9 :

 229 Figure 2.9: The ratio uu u for the plane Couette flow. Note that the y-axis is the volume fraction of the subdomain I , C I .

Figure 2 . 10 :

 210 Figure 2.10: The ratio E (ν,S 12 ) µ S 12 for the plane Couette flow. Note that the y-axis is the volume fraction of the subdomain I , C I .

  2.14) and (C.2.15), for a I = -ρ I g , b I = 0 and p J and a J = -ρ J g , b J = 0 respectively.

Figure 2 . 11 :

 211 Figure 2.11: The Favre averaged velocity field derived by solving the one-fluid equations. Each line in the figure is the solution obtained for a different value of . The black line depicts the local velocity field.

  A heuristic solution to this problem can be obtained by allowing the u to have a jump discontinuity, where the one-fluid momentum equation switches entirely to the classic Navier-Stokes momentum equation for air. The jump discontinuity is implicitly determined by the requirement that: the analytic continuation of the u field as we approach the interface from the side of the lighter fluid's J subdomain to the interface (obtained by a Taylor expansion of u around y = from the side of the fluid subdomain J ), must be equal to the interface velocity:

  Detail near discontinuity at the water subdomain

Figure 2 . 12 :

 212 Figure 2.12: Favre averaged velocity solution of the one-fluid equation by enforcing different conditions near the interface. The black line depicts the local velocity field.

Figure 2 . 13 :

 213 Figure 2.13: Relative errors of Favre averaged velocity solutions of the one-fluid equation by enforcing different conditions near the interface for two values of .

  The pressure obtained by the classic formulation and the one-fluid formulation are firmly related. The classic formulation gives the local pressure field. The local pressure field obtains a jump discontinuity at the interface p(l ) = -σ B α α H (Ry). The one-fluid formulation gives the averaged pressure field. The averaged pressure varies smoothly between the local pressure values of the ambient fluid. Through the solution of the averaged momentum equation, the Heaviside function is replaced by a smooth Heaviside function H . We have demonstrated that H is equal to the volume fraction field C I :

  To compare the solutions obtained by the one-fluid formulation with the solutions obtained by the classic formulation and test the validity of the one-fluid formulation, we have studied two simple cases where analytical solutions can be derived, the static bubble and the plane Couette flow. The analytical solution for the static bubble test case, demonstrated in practice, that the surface tension source term introduces a mollified Dirac delta function around the interface. When the source term is integrated, a mollified Heaviside function is obtained for the averaged pressure. Moreover, we have shown the averaged pressure field obtained by the one-fluid formulation corresponds to the averaged local pressure field (obtained by the average of the local pressure field). These are expected results and in accordance with the fundamental mathematical concepts of volume averaging.The analytical solution for the plane Couette flow demonstrated that the one-fluid formulation does not always produce velocity fields that compare well to the averaged local velocity field (obtained by finding the averages of the fields derived from the solution of the classic formulation). By comparing the derived fields, we have demonstrated evidence which supports that the last observation is related to the underlying assumptions of the one-fluid formulation. To that end, we have proposed a simple heuristic correction which appears to improve the solution obtained by the one-fluid formulation (in the sense that they compare better to the averaged fields obtained by finding the averages of the fields derived from the solution of the classic formulation).

  ization problem and the interface reconstruction problem. Both are directly related to the one-fluid formulation of ISIS-CFD and ISIS-CFD uses them in various ways during the solution procedure. Previously, we have presented the volume fraction initialization problem from a theoretical point of view and provided some simple analytical solutions. Here we describe the discrete counterpart of the problem and the inverse problem, the interface reconstruction. The solution of the interface reconstruction problem is an approximate explicit representation of the interface's surface (given the volume fraction as in input). Since the solution of interface reconstruction problem replaces the volume fraction by an explicit representation of the surface, integrals that appear in the one-fluid formulation of the Navier-Stokes (see section Deriving the Equations of the One-Fluid Formulation, p.51), can be directly evaluated. Specifically, we use this approach to calculate the surface tension source term in the next chapter.

  to Wyvill et al.(1986), see for example Agoston (2005) [1, pp.365].

  An ordered set of eight nodes defines a cube. Consider that we know the values of the function at the nodes, g ( p n ), n ∈ c n (where c n is the set of nodes belonging to the boundary of the cell c). Then we obtain fourteen different combinations (actually, there are 2 8 = 256 possible combinations but owing to symmetry only 14 need to be considered). Finally, we map to each combination a topological pattern linking iso-nodes to iso-patches, as shown in figure 3.1.

Figure 3 . 1 :

 31 Figure 3.1: The 14 cases for constructing the isosurface patches with the Marching Cubes Algorithm. The function's values at the nodes identify the case. Nodes marked with black dots represent negative function values. Otherwise, the function is positive at the node.Figure adapted from Newman(2006) [74] .

2 Figure 3 . 2 :

 232 Figure 3.2: An ambiguous case of patch construction for Marching Cubes. For the combination of nodal values in figure3.2a (negative values are indicated by circled node and iso-nodes by crosses), two possible iso-patch configurations are acceptable (iso-edges depicted by red lines and iso-patches by red).

  , and also provided byTryggvason et al. [105, using the name Cube-chopping algorithm. As the name "cube-chopping" algorithm suggests, a procedure is described to define the truncated cell that represents the part of the interface inside the cell, whose volume is calculated to obtain the volume fraction. The cube-chopping algorithm is used along with PLIC (Piecewise Linear Interface Computation or Construction) approaches to solve the volume fraction transport equation. Therefore the volume fraction initialization problem of a planar interface is indirectly discussed in the context of PLIC methods. To give a few example, Rider and Kothe (1998)[START_REF] Rider | Reconstructing Volume Tracking[END_REF] and Lopez et al.(2008) [START_REF] López | Analytical and geometrical tools for 3D volume of fluid methods in general grids[END_REF] review the relevant procedures for general polyhedral grids.[START_REF] Fan | Piecewise linear volume tracking in spherical coordinates[END_REF] [START_REF] Fan | Piecewise linear volume tracking in spherical coordinates[END_REF] present similar calculations for grids in spherical coordinates, and Ahn and Shaskov (2007)[START_REF] Ahn | Multi-material interface reconstruction on generalized polyhedral meshes[END_REF] for general polyhedral cells for representing linear patches of interfaces using multiple materials.[START_REF] Cummins | Estimating curvature from volume fractions[END_REF] [START_REF] Cummins | Estimating curvature from volume fractions[END_REF] approximate the interface by linear segments after performing four levels of refinement in the original Cartesian grid. Lopez et al. (2009)[START_REF] López | An improved height function technique for computing interface curvature from volume fractions[END_REF] shortly describes the same volume fraction initialization of an interface given by an implicit function for the case where a single patch of the surface truncates a typical polyhedral cell (consisting of planar faces). However, we cannot always ensure that a single interface is located inside a cell. Recently,[START_REF] Bna | Numerical integration of implicit functions for the initialization of the VOF function[END_REF] [START_REF] Bna | Numerical integration of implicit functions for the initialization of the VOF function[END_REF] [9] present a method

S

  f in I , and the iso-patches S I c(1), S I c(2), ... in order to calculate the required volume. Two examples of regions truncated by an interface's surface, V c in I , are given in figure 3.3.The boundary of the regions consists of one or more parts of the interface surface, S I c , that rest inside the cell, and parts of the cell's faces that rest in the region of the interface, S f in I . Clearly a method that does not explicitly takes into account the second case cannot always provide a correct result. The method proposed in this section uses the MPA to obtain an explicit representation of the isosurface g I ( p S ) = 0 which represents the interface. The result of the MPA is an approximation of the surface patches S I c as iso-patches. Since the correct representation of the disjoint interface regions intersecting

Figure 3 . 3 :

 33 Figure 3.3: Two different configurations of an interface inside a cell. In the first case, a single part of the interface is inside the cell. In the second case, two parts of the interface are inside the cell. Thick lines depict the intersections of the interface(s) with the faces. The light gray shade (only visible in figure1) distinguishes the part of the interface that lies inside the cell, S I c , and a dark gray shade the parts of the faces that are in the fluid domain, S f in I .

  Figure 3.4: Iso-nodes and geometrical space representation of an iso-patch. We obtain the iso-nodes (black crosses) by finding the intersection point of an edge and the interface. We obtain the iso-nodes (red crosses) by finding the intersection point of the line connecting the centroid of each face and a boundary node of the face. The isopatch-isonodes connectivities, c

  ∀n ∈ c n where c n are the cell-nodes connectivities and the number of "in" nodes, "out" nodes and "at" nodes of the cell c are given symbolically by |g I (c n ) < 0|, |g I (c n ) > 0|, |g I (c n ) = 0|, then the volume fraction field:

  dV *: by definition (3.3.1) (3.3.25)

  is the maximum distance of the plane from the centroid of the cell, C S Z I (l c ; κ 1 , κ 2 , κ 3 ) is the Scardovelli-Zaleski modified analytical formula for the volume fraction and the κ 1 , κ 2 , κ 3 are problem parameters related to the relative position of the cell with the interface (see appendix, p.246). The condition -1 < l < 1 indicates that the region where the planar interface intersects the cell, is specified by the points that satisfy -< n J I ( p cp o ) < i.e. the regions between planes, above (in the sense of n J I ) n J I [ p -( p on J I )] = 0, and below the plane n J I [ p -( p o + n J I )] = 0. In this case, the volume fraction initialization procedure derives the same results as the above analytical relation.

55 Figure 3 . 5 : 51 Figure 3 . 6 : 41 Figure 3 . 7 :

 553551364137 Figure 3.5: Volume fraction calculation and absolute errors for g I (x, y, z) = x + 1.5y + 2z = 0. The interface's surface grid (iso-edges represented by thick black lines) is shown over the computational domain (standard cube, 20 cells per edge, boundary edges represented by thin black lines). Each of the sets of slices depicts the volume fraction field, figures 3.5a,3.5b, and the absolute errors of the volume fraction field, figures 3.5c,3.5d. The absolute error values are below machine precision.

80 Figure 3 . 8 :

 8038 Figure 3.8: Local relative distance errors (based on the centroids of the iso-patches) and volume fractions for a spherical interface and a background Cartesian grids. The figures show the isosurfaces of the spherical interface colored by the local distance error and slices of calculated volume fraction fields for the sphere g ( r ) = | r | 2 -R 2 = 0, R = 0.5 in a Cartesian grid discretizing the standard cube using different number of discretizations.

LFigure 3 . 9 :

 39 Figure 3.9: Error diagrams for spherical isosurfaces generated in Cartesian background grids, g ( r ) = | r | 2 -R 2 = 0, R = 0.5 .

Figure 3 . 10 :

 310 Figure 3.10: Volume fraction initialization of a free surface and a submerged bubble.Figure 3.10a shows a slice of the resulting volume fraction field. The grid used consists of rectangular parallelepipeds.Figure 3.10b demonstrates the normal vectors of the interfaces with the orientations derived by the MPA. The orientation of the normals is appropriately for both interfaces (surface grids colored by their local distances from the z=0 plane)

Figure 3 .

 3 Figure 3.10: Volume fraction initialization of a free surface and a submerged bubble.Figure 3.10a shows a slice of the resulting volume fraction field. The grid used consists of rectangular parallelepipeds.Figure 3.10b demonstrates the normal vectors of the interfaces with the orientations derived by the MPA. The orientation of the normals is appropriately for both interfaces (surface grids colored by their local distances from the z=0 plane)

Figure 3 . 12 :

 312 Figure 3.12: PLIC reconstructions of a spherical interface in a uniform Cartesian grid, figures 3.12a and 3.12b, and a tetrahedral grid, figures 3.12c and 3.12d. The initialization problem of a sphere provides the volume fraction values used for the reconstruction. The normal vector is approximated by two different methods. The PLIC generates surface patches that are piecewise continuous.

; p n where w p c ; p n = 1 p c -p n 2 (

 12

  boundary nodes n ∈ N b (N b is the set of physical boundary nodes), by adding corrections that depend on the data of the physical boundary faces that contain this node. Since this kind of connectivity (nodes to boundary faces) is not defined, we work in an implicit manner. Suppose that the set c f * (defined for one of the cells c ∈ n c used by relation (3.4.2)) contains the physical boundary faces f ∈ F b of the cell c and share the node n: c f * = f : f ∈ c f and f ∈ F b and n ∈ f n (3.4.3) then the relation (3.4.2) for the node n (that is a node that belongs to a physical boundary) is modified as: C I n = c∈n c C I c w p c ; p n + NC or r (c f * ) c∈n c w p c ; p n + Dcor r (c f * ) (3.4.4) The definition of the correction terms for the numerator and denominator, NC or r and DC or r respectively, depends on the number of physical boundary faces of the cell. If the cell has one boundary face f that contains the node n, i.e. c f * = f , then the corrections are:

Figure 3 .Figure 3 . 13 :

 3313 Figure 3.13: Locations of points and data for the interpolation corrections. A node n that lies on the boundary of a different number of physical boundary faces (highlighted by darker blue shade) obtains different correction terms. Each figure shows the points used for the interpolation corrections. The locations marked with different shapes to indicate different correction terms, NC or r 1: circled dot, NC or r 2: cube, NC or r 3: rhombus. Other points are indicated by dots.

  However, as the characteristic length of the kernel tends to smaller values the volume fraction tends to the indicator function and thus for a point r ∈ S I (and strictly speaking not belonging to the boundary of the computational domain): lim →0 C I ( r ) -0.5 = I I ( r ) -0.5 = 0 (3.4.10) 2. The nodal values are obtained by an interpolation scheme that does not provide exact values. 3. The iso-nodes are obtained by solving the linearized version of the equation that defines them (see p.88).

LFigure 3 . 14 :

 314 Figure 3.14: Reconstruction error diagrams for a spherical isosurfaces generated in Cartesian background grids, g ( r ) = | r | 2 -R 2 = 0, R = 0.5 .

40 Figure 3 . 15 :

 40315 Figure 3.15: Comparison between the initial and the reconstructed isosurface grids for a spherical interface. Both grids shown in each of the above figures were generated by the MPA algorithm for a different of discretization, ν. Black (iso)edges represents the initial isosurface grid, g I ( r ) = | r | 2 -R 2 = 0 and blue (iso)edges the reconstructed isosurface, g I ( r ) = 0.5 -C I ( r ). The gray shade represents reconstructed surface's patches.

Figure 3 . 16 :

 316 Figure 3.16: Local relative distance errors (based on iso-nodes) for the reconstructed surface of a spherical interface. The figures show the isosurface grids of the reconstructed interface colored by the local relative distance error as obtained for different discretizations of the background Cartesian grid. The volume fraction of the original surface g I ( r ) = | r | 2 -R 2 = 0 is used for the reconstruction.

Figure 3 . 17 : 40 Figure 3 . 18 :

 31740318 Figure 3.17: Reconstruction error diagrams for a sinusoidal isosurface generated in a Cartesian background grids

1 Figure 3 . 19 :

 1319 Figure 3.19: Comparison of reconstructed surface grids for the sinusoidal wave case. Figures 3.19a, 3.19b show the intersections of the generated isosurface with the x = 0 plane and the x = 1 plane respectively, for ν = 20, ν = 40 and ν = 80 using different colors for each isosurface. In figures 3.19c, 3.19d, only the parts of surface grid slices that correspond to the window [0.5, 1] × [-0.5, 0.5] are shown.

(

  2012) [106] use quadratic fits to estimate the normal vector of the interface.[START_REF] Renardy | PROST: A Parabolic Reconstruction of Surface Tension for the Volume-of-Fluid Method[END_REF] [START_REF] Renardy | PROST: A Parabolic Reconstruction of Surface Tension for the Volume-of-Fluid Method[END_REF] proposed the interface reconstruction method PROST (Parabolic Reconstruction of Surface Tension), which performs a constrained parabolic fit using the volume fraction and subsequently evaluate the curvature.[START_REF] Gois | Front tracking with moving-least-squares surfaces[END_REF] [START_REF] Gois | Front tracking with moving-least-squares surfaces[END_REF] used a moving least squares reconstruction of the interface based on second order polynomial basis functions, from which both the normal vector and curvature of the interface are obtained.[START_REF] Aulisa | Interface reconstruction with least-squares fit and split advection in threedimensional Cartesian geometry[END_REF] [START_REF] Aulisa | Interface reconstruction with least-squares fit and split advection in threedimensional Cartesian geometry[END_REF] proposed an improved PLIC reconstruction which uses quadratic least square approximations to achieve better estimates for the normal vector and improve the PLIC reconstruction.Sousa et al. (

s.

  is the set of adjacent nodes to the iso-patch κ s . As we have already mentioned, the samples of[START_REF] Tryggvason | A front-tracking method for the computations of multiphase flow[END_REF] [START_REF] Tryggvason | A front-tracking method for the computations of multiphase flow[END_REF] are generated using the nodes of the level 1 patch neighborhoods based on faces(edges) of the patch, samples of Tukovic and[START_REF] Tuković | A moving mesh finite volume interface tracking method for surface tension dominated interfacial fluid flow[END_REF] [106] the nodes of the level 1 patch neighborhoods based on nodes of the patch, c s n s Both approaches are specific cases of the general method used here. The sample generation approach must also take into account the peculiarities of the grid generated by the MPA. Specifically, the sample of points c symmetrically distributed around the point where we derive approximations. This is demonstrated in figure3.20.

  Iso-nodes of both surfaces are contained in the sample. The least square procedure will be affected by iso-nodes that do not belong to the surface whose normal and curvature are sought. Iso-nodes of the required surface are contained in the sample. The least square procedure provides local approximations to the iso-patch κ s and is not affected by the presence of the other surface.

Figure 3 . 22 :|

 322 Figure 3.22: Volume grid neighborhoods and surface grid neighborhoods generate different iso-node samples. The sample construction method might affect the consistency of LSqR. Crosses denote isonodes of the surfaces. Each figure demonstrates different sample ensembles (points marked by circled crosses) that can be used to perform LSqR to the iso-patch κ s . The sample are constructed by are the topos neighborhood c t n , or the iso-patch neighborhood constrained by c t n , each marked as a gray shaded region in figures 3.22a and 3.22b respectively.

( 3 .|

 3 5.9) with respect to the coefficients a M included in z and some weights. The problem reduces to a linear system whose coefficient matrix is positive definite and symmetric. Therefore, it can be solved by a Choleski decomposition which allows an efficient implementation. For the case m = | c the coefficients a M are obtained by fitting (3.5.8) to the sample S( p ν s ) , z( p ν s ) : ∀ ν s

Figure 3 . 24 :

 324 Figure 3.24: Normal and curvature L max error diagrams for a spherical isosurface (R = 0.5) and Cartesian background grids obtained by least square differencing using different neighborhoods and polynomial basis for the reconstructed interface by the volume fraction

Figure 3 . 25 :

 325 Figure 3.25: Normal and curvature L 1 error diagrams obtained by least square differencing using different neighborhoods and polynomial basis for the reconstructed interface by the volume fraction of a spherical isosurface (R = 0.5) and Cartesian background grids

  figures 3.24 to 3.26). In conclusion, the normal vector approximations are not second-order convergent for the whole range of discretization using samples of the K 1(C I ) isosurface. When we use samples

Figure 3 . 26 :

 326 Figure 3.26: Normal L max error diagrams using least square differencing with level 2 and 3 neighborhoods for the reconstructed interface by K 1(C I ) and K 2(C I ) of a spherical isosurface (R = 0.5) and Cartesian background grids

Figure 3 . 28 :

 328 Figure 3.28: Cut-Cell volume grid generation using the MPA. A grid is tessellated dynamically by a spherical interface. The figures depict half of the resulting unstructured volume grids, the "out" grid (left) and the "in" grid (right). Cells colored by their volumes.

  The methods presented up to this point use the volume fraction to generate an approximation of the surface tension source term that appears in the one-fluid formulation. The Continuous Surface Force (CSF) and the Continuous Surface Stress (CSS) method have the advantage that they do not require an explicit interface representation. The Isosurface Front Tracking (IsoFT) method, resembles methods used in front tracking, where an explicit representation of the interface is always available.The differences of the IsoFT method and other front tracking methods encountered in the literature, see for instance[START_REF] Tryggvason | Direct Numerical Simulations of Gas-Liquid Multiphase Flows[END_REF] [105, p.133-134,150-155], are:

either if: 1 .

 1 fluid I occupies entirely one of the adjacent cells of the face, and the interface intersect the other cell, i.e. 0 < C I f L < 1 and C I f R = 1, or C I f L = 1 and 0 < C I f R < 1 2. fluid J occupies entirely one of the adjacent cells of the face and the interface intersect the other cell, i.e. 0 < C I f L < 1 and C I f R = 0, or C I f L = 0 and 0 < C I f R < 1 The first option "snaps" the discontinuity to the face near the regions of the I fluid, while the second near the regions of the J fluid. An example is shown in figure 4.2.

Figure 4 . 2 :

 42 Figure 4.2: Snapping to faces where the compatibility conditions are enforced. Specifically for this figure, we refer to the subdomains by the fluid which occupies them, i.e. air and water.

1 .

 1 the interface occupies the left cell of the face, 0 < C I f L < 1, and fluid I occupies the right cell of the face, C I f R = 1 and therefore p f Rp f L = p Ip J =p 2. the interface occupies the right cell of the face, 0 < C I f R < 1, and fluid I occupies the left cell of the face, C I f L = 1 and therefore p f Rp f L = p Jp I = p

Figure 4 . 3 :

 43 Figure 4.3: Identifying faces where compatibility conditions are enforced. Each set of figures corresponds to the relevant set of faces derived from a different volume fraction field.

  For the demonstration we have used ν = 20, 30, 40, 60 uniform discretizations for each of the cube's edges. The spherical interface of radius R = 0.25 contains air (ρ a = 1.2kg /m 3 ,µ a = 1.85•10 -5 kg /ms) and its exterior region water (ρ w = 998.4kg /m 3 ,µ w = 10 -3 kg /ms). The surface tension coefficient is σ = 0.0728N /m. The volume fraction is obtained by the volume fraction initialization procedure presented in chapter 3 (see The Volume Fraction Initialization Problem, p. 91). The methods used for the calculations along with their options are the following:

4. 5 .Figure 4 . 4 :Figure 4 . 5 :

 54445 Figure 4.4: Pressure fields obtained for the static bubble test case using different surface tension methods. The plotted results correspond to the calculated velocities at the x = 0 plane for ν = 40. First column, left: CSF, right: CSS. Second column, left: DCM, right: IsoFT

  Firstly, the volume fraction field C I is initialized for the original grid. AGR adapts the cells targeted by this volume fraction field. Specifically, the adapted subset of cells contains cells where C I is between zero and one and their neighborhoods. AGR refines the cells isotropically. The section of the initial volume fraction field C I with the plane x = 0, is shown in figure 4.6a. After AGR, the generated subset of cells corresponds to some cells that occupy the same regions in space for the ν = 40 grid.Figure 4.6 visualizes the refined cell region around (a) Initial volume and surface grid (b) Volume and surface grid after AGR

Figure 4 . 6 :

 46 Figure 4.6: Volume fraction initialization and isosurface generation with AGR. The volume fraction field is reinitialized after AGR and maintains its sharp characteristics.

Figure 4 . 7 :

 47 Figure 4.7: Pressure fields obtained using AGR with and without volume fraction reinitialization

Figure 4 . 8 :

 48 Figure 4.8: Isosurfaces and local curvature fields using AGR with and without volume fraction reinitialization

Figure 4 . 9 :

 49 Figure 4.9: An example of a grid for the rising bubble test case. The slice exposes the different refinement levels used for the rising bubbles simulations.

Figure 4 .

 4 Figure 4.10: A rising bubble computation without surface tension. Bird's and fish's eye views for three instances. The contours of the surface correspond to the z-axis coordinates of the iso-nodes.

Figure 4 . 11 :

 411 Figure 4.11: Typical aspect ratio diagram as a function of time R = 0.8 mm, results obtained by isoFT

Figure 4 . 12 : 4 . 4 . 1

 412441 Figure 4.12: Typical centroid and velocity diagrams as functions of time R = 0.8 mm, results obtained by isoFT

"

  any initial interface deformation." In other words, air bubbles that are initially spherical and both fluids are initially immobile, are not observed in nature. To obtain valid comparisons between the experimental results and results from rising bubble simulations, the model has to take into account the bubble formation method used in the experiments.(a) R = 0.6 mm, t = 0.0078 s (b) R = 0.8 mm, t = 0.0253 s (c) R = 1 mm, t = 0.0253 s

Figure 4 . 13 :

 413 Figure 4.13: Volume fraction fields obtained with different surface tension methods for bubbles in the ellipsoidal regime. From left to right, results from simulations using CSF, CSS, IsoFT and DCM

Figure 4 . 14 :

 414 Figure 4.14: Estimations of terminal Velocities by ISIS-CFD, experimental results and computational results plotted on the terminal velocity-equivalent radius diagram. Marks without lines denote experimental results, red circles: Duineveld (1995), black circles: Wu et al. (2002) large initial interface deformations, black crosses: Wu et al. (2002) small initial interface deformations, blue: Clift et al. (1978). Lines denote computation results of previous studies, blue line: Tukovic and Jasak (2012), red line: Hua et al. (2008). Square marks with dashed lines denote results obtained by ISIS-CFD, blue: CSF, red: DCM, black: IsoFT.

Figure 4 . 15 :

 415 Figure 4.15: Comparison of interface shapes obtained by experiments, DCM and IsoFT by ISIS-CFD. The photographs were taken from Wu et al. (2002) [116], Bubble radii are 0.975 mm and 0.94 mm for 4.15a and 4.15b respectively. The initial bubble radius for computation is R = 1 mm for both figures. DCM results are given in figure 4.15a and IsoFT results are given in figure 4.15b.

Figure 4 . 17 :

 417 Figure 4.17: Sharp volume fraction fields obtained by different surface tension methods for bubbles in the spherical cap regime. From left to right: DCM, IsoFT, CSS

  t = 0 s (blue line) and t = 0.0325 s (black line) t = 0.1205 s

Figure 4 . 18 :

 418 Figure 4.18: Isosurface intersections with the x = 0 plane, obtained during the initial stages of a spherical cap bubble's development, R = 10 mm. Results obtained by isoFT. The isosurface contours represent the calculated curvature field.

Figure 4

 4 t = 0.0635 s

Figure 4 . 19 :

 419 Figure 4.19: Curvature during the initial stages of a spherical cap bubble (coarse discretization). The three-dimensional view exposes the trailing part of the surface.

Figure 4 . 20 :

 420 Figure 4.20: Curvature during the initial stages of a spherical cap bubble (fine discretization). The three-dimensional view exposes the trailing part of the surface.

Figure 4 . 21 :

 421 Figure 4.21: Comparison of multi-bubble system obtained by ISIS-CFD and a photograph of Landel's experiments. The ISIS-CFD bubble corresponds to a bubble of equivalent radius d e = 20 mm. The bubble from Landel's experiments corresponds to a bubble of equivalent diameter d e = 49 mm

Figure 4 . 22 :

 422 Figure 4.22: Generated isosurfaces (as viewed from below) over time during the final stages of a spherical cap bubble's evolution, R = 10 mm (results obtained by IsoFT). The contour corresponds to the iso-patch curvature.

  (a) t = 0.1405 s (b) t = 0.1785 s (c) t = 0.1825 s (d) t = 0.1946 s (e) t = 0.2365 s Fish's eye view (f) t = 0.2365 s Bird's eye view

Figure 4 . 23 :

 423 Figure 4.23: Generated isosurfaces (as viewed from below) over time during the final stages of a spherical cap bubble's evolution R = 10 mm (results obtained by DCM). The iso-patches are colored by the z-axis components of their centroids (Matlab visualizations).

  t = 0.18175 s (closer view) t = 0.23275 s

Figure 4 . 24 :

 424 Figure 4.24: Generated isosurfaces (view from bottom to top) over time during different stages of a spherical cap bubble's evolution R = 10 mm (results obtained by IsoFT). The contour represents the calculated curvature field.

  The plane was placed at a distance d = 0.0081 m above the sphere. The computation used DCM for surface tension modeling. The volume fraction is calculated for the planar surface and the spherical surface separately and the result is obtained by the union of the two previous volume fraction fields. The resulting initial configuration and its volume fraction field for R = 0.8 mm are shown in figure 4.25a (volume fraction),4.25b (input surfaces for initialization).

  (a) Volume fraction -Intersection with x = 0 plane X Y Z (b) Surfaces of the air-water interface

Figure 4 . 25 :

 425 Figure 4.25: Initial configuration for the rising bubble/free surface interaction problem

Figure 4 . 26 :

 426 Figure 4.26: Basic observations before the collision

Figure 4 . 28 :

 428 Figure 4.28: Rising Bubble/Free surface interactions, volume fraction fields over time visualized as intersection with the z y plane.

Figure 4 .

 4 Figure 4.30: A naturally ventilated vortex emanating from the bow. Figure reproduced from Broglia (2012) [12].

Figure 4 .

 4 Figure 4.31: A schematic representation of the sharp-edged plate's geometry. The shaded area indicate a slope of 45 • .

  maximum and constant velocity value of 2.5 m/s after the first 25 time steps. The body moved along the negative x-axis (towards the paper and away from the reader, figures 4.32a or 4.32b). The domain moved rigidly following the movement of the body during the simulation. The Reynolds number was Re = 1.75 • 10 6 and the Froude number F r = 0.95. For turbulence modeling, we used the Nonlinear EASM-kω model. The boundary conditions used were the following, for the top and bottom planes of the domain, updated hydrostatic pressure, for the every other exterior plane of the domain, imposed velocity, and wall functions for the solid boundary. Finally, the time step was 0.003 s.

Figure 4 . 32 :Figure 4 . 33 :

 432433 Figure 4.32: Domain and initializations specifications for the marching plate test

Figure 4 . 34 :

 434 Figure 4.34: Interface evolution. The contours of the isosurface indicate the interface's height from the xy-plane. Bird's eye view.

Figure 4 . 35 :

 435 Figure 4.35: The evolution of the interface behind the trailing surface of the plate. The contours of the isosurface indicate the interface's height from the xy-plane.

Figure 4 . 36 :

 436 Figure 4.36: Structures captured after a breaking wave hits the free surface. Only a part of the cavity is shown. The plate moves to the right. Top to bottom views.

Figure 4 . 38 :

 438 Figure 4.38: The roll-up of the ventilated vortex, initial stages, and the final configuration. Views from top to bottom.

Figure 4 . 39 :

 439 Figure 4.39: Different views from the final configuration (t = 1.707 s) of the interface with and without the unstructured surface grid.

Figure 4 . 40 :

 440 Figure 4.40: Curvature values at different locations for the final configuration of the interface and intersections of the ventilated vortex tube by different x = const planes

  .1 with the assistance of Prof. Zarafonitis of the Ship Design Laboratory (headed by Prof. Papanikolaou) of the National Technical University of Athens. Another interesting application is the study of sloshing in tankers, which especially for oil tankers, still posses a safety hazard with significant environmental consequences. The problem requires the simultaneous solution of a set of at least two one-fluid multifluid Navier-Stokes problems,and their coupling. The management of the two problems which would require at least two different grids can be performed with the developed grid manager. Subsequently two Navier-Stokes problems must be solved which implies that the solver should be recast to a single subroutine which accepts simultaneous different inputs. Such an approach could also aid the further development towards introducing multiphase models.

Figure 5 . 1 :

 51 Figure 5.1: Preliminary design of the principal air cavity of the KVLCC2 tanker

  V I ,n is zero and the normal direction of the vaporous cavity can be computed by taking the gradient of the phase volume fraction [...]. Owing to the single fluid formulation, ũ [Favre averaged velocity] represents the liquid velocity in the liquid phase and the vapour velocity in the vapour phase. and for unsteady computations the model is supplied with the following relation:

  where we would not have to express the term using intermediate approximations. A proposed model for mixing and the generation of bubbly flows have not yet been proposed in the context of the one fluid formulation. Air-entrainment formulations such as these proposed by Ma et al (2011)[START_REF] Ma | Two-fluid modeling of bubbly flows around surface ships using a phenomenological subgrid air entrainment model[END_REF] and Carrica et al.[START_REF] Carrica | Towards an air entrainment model[END_REF], were developed in the context of an ensemble average formulation, where multiple Navier-Stokes equations are solved and are coupled through the addition of several source terms.

  s α (S; P ) = Z i α (S; P ) Z i (S; P ) (A.3.3)and thus:s α=1 = Z 1 + Z 3 ∂z ∂s 1 and s α=2 = Z 2 + Z 3

  d(•) = n l ∇ l (•) of an invariant φ (a field that remains constant under coordinate transformations) becomes: dφ = n l ∇ l φ = -

βi

  = s α s β = δ β α .Moving on with the calculations, the surface derivatives of the components of the normal vector are:

1

 1 Decomposition of the stress traction jump T i j n j to its normal and tangential componentsIn this section, we provide the details for deriving the LHS of equation (2.3.25) repeated here for convenience:Z i α µ (w β B αβ + ∇ α w) + µdu α + n ip -2 µ (∇ α w α + wB α α ) = -Z i α ∇ α σn i σB α α (B.1.1)

  1.25) (see appendix A.1,p.220) :2S i j = Z i α Z j β + Z j α Z i β ∇ α u β -uB β α + Z i α n j + Z j α n i u β B β α + ∇ α u + n i du j + n j du i (B.1.6)229of the normal derivative of the normal velocity component will be zero:du = d(u i n i ) = n i du i + u i dn i = 0 (B.1.14)since the first term in the resulting expression is zero, as we have already demonstrated, and the second since the velocity field is continuous. Therefore we have:du j = d Z j α u α + un j = d Z j α u α = Z j α du α (B.1.15)As a result the third term in (B.1.8) becomes:n i µn j du j = µ n i n j du j (B.1.16)Substituting the above relations to (B.1.8) we obtain:2 µS i j n j = Z i α µ u β B β α + µ ∇ α u + µ n i n j du j + µdu i (B.1.17)

1 )

 1 figures C.1a,C.1b and C.1c) are given by:

  that the normal vector in the above relation is n I i.e. the normal pointing from the region J to the region I . The resultant is the force exerted from the fluids to the surface and it is written using the normal vector n = n I =n J as:

f 1 = 0 f 1 < 0 f 1 2 < 0 f 2 > 0 f 2 3 > 0 f 3 = 0 f 3 Figure C. 2 :

 010120202303032 Figure C.2: Range of values of the functions f 1 , f 2 , f 3 for the domain of figure C.1. Each function defines implicitly an interface where f I ( r ) = 0, an interior region where f I ( r ) < 0 and an exterior region where f I ( r ) > 0. Using these function we can write the indicator functions of the fluid subdomains using Heaviside functions.

3 2 . 2 1( 2 3(Figure C. 3 :

 2223 Figure C.3: Configurations of interfaces inside the support of an averaging kernel.

  i K d S ⇒ ∇ i f = ∇ i f + 1 ρ f ∇ i ρ -I J S J I ρ f n i K d S (C.1.9)We introduce the field of characteristic interface values by the tilde operator, f (without a fluid subdomain index), defined in p.50 as:f ( x, t )∇ i ρ| ( x,t ) def = I J S J I ρ f ( y, t ) n i ( y, t )K ( y; x)d S( y) (C.1.10)Through the last definition, relation (C.1.9) becomes:

n 1 = 2 )Figure C. 4 :

 124 Figure C.4: Integration region and nomenclature for the evaluation of (2.5.13)

2 . 13 )

 213 Averaged Field for Local Linear Fields When the local fields are linear we can obtain very simple relations for the averaged fields. Specifically, for the top-hat in-sphere kernel these are: l = y/ . Both averaged fields are zero when the support of the kernel rests outside the field's domain of definition, for example, the field f I is zero for 1 l . Moreover, for the special case f I = 1 and f J = 1 the above relation give the volume fraction fields, C I and C J respectively since by definition Substituting to (D.1.3) we obtain:

a 1 b 1 a 1 b 2 a 1 b 3 a 2 b 1 a 2 b 2 a 2 b 3 a 3 b 1 a 3 b 2 a 3 b 3 

 333

E. 4

 4 →λ where λ ∈ κ t f (D.1.10) And (D.1.8) can be written as: (I -Āκ ) ∇q κ -λ∈κ t f Bκλ ∇q λ = ∇q • κ (D.1.11)The above relations constitute a system of n C × 3 equations for the gradient of every cell (n C × 3 unknowns). The system can be solved using a simple Jacobi-like iterative procedure beginning by Par exemple, un utilisateur final du code doit disposer d'une façon simple et unifiée de résoudre les problèmes d'initialisation multi-interfaces pour la fraction volumique, nécessaire à l'étude des cavités partielles. De plus, un développeur doit proposer des solutions en regard des différentes méthodes de tensions superficielles qui demandent des stencils[START_REF] Tryggvason | Direct Numerical Simulations of Gas-Liquid Multiphase Flows[END_REF] différents et plus larges que ceux des schémas de discrétisation de second ordres classiques. Pour cela, des modifications importantes, particulièrement pour l'exécution parallèle du code, ont dû être effectuées comme point de départ pour ce travail, tout en maintenant simultanément l'interaction minimum avec le code source principal d'ISIS-CFD. Pour les raisons exposées ci-dessus, nous avons développé un cadre de développement pour ISIS-CFD, le "ISIS-CFD Extensions Framework", pour permettre une plus grande flexibilité sur le développement du code, avec une approche de pointe pour la technologie Fortran. Dans la thèse, nous ne discuterons pas des détails de la programmation. Au lieu de cela, nous nous concentrerons sur les aspects théoriques de la formulation multi-fluides et des interventions d'ISIS-CFD Extensions sur ISIS-CFD pour la modélisation numérique de la tension superficielle. Nous présenterons une courte description du contenu de la thèse dans la section suivante. E.3 Formulation théorique des équations Navier-Stokes multi-fluides L'écoulement d'air à l'intérieur de la cavité est important car il indique la quantité d'air qui s'échappe de la cavité au niveau de sa clôture, la stabilité de l'interface et de ce fait, la performance de la cavité. Pour cela, l'écoulement d'air devrait être résolu finement, globalement et à proximité de l'interface. La tension superficielle pourrait jouer un rôle majeur, du fait que, sur la clôture de la cavité, des variations importantes des courbures sont observées. Le but est de fournir une formulation formelle mono-fluide du Navier-Stokes dans le contexte du VOF et du moyennage de volume, qui vise à (i) dériver l'équation constitutive du modèle multi-fluides à proximité de l'interface pour la densité (c.-à-d. ρ = ρ w C w + ρ a C a où C a est la fraction du volume d'air) et de façon similaire pour la viscosité dynamique (deux approximations pour les codes de calculs d'ingénierie navale), (ii) introduire les termes de tension superficielle et (iii) remplacer les fonctions généralisés comme Dirac-δ par leurs equivalantes lisées. Nous présentons une formulation qui suit les concepts de moyennes du volume, qui est habituellement introduite dans le contexte de modélisation mutliphasique tel que Drew et Passman (1999) [21]. Dans notre cas, la technique de moyennage de volume est présentée dans le contexte de modélisation multi-fluides (les fluides sont immiscibles et les mélanges ne sont pas permis). A partir de cette procédure, nous dérivons une définition formelle du terme source de la tension superficielle (en tant que terme qui distribue la tension superficielle à partir de l'interface de l'espace environnant) et d'autres termes sources que nous dérivons du besoin d'introduire les relations constitutives pour la viscosité dynamique. La difficulté principale pour calculer ces termes sources vient du fait qu'il n'y a pas de représen-tation explicite de l'interface disponible dans la méthode VOF de ISIS-CFD, alors que pour évaluer les termes sources visés, nous devons évaluer les intégrales de surface de l'interface. A la suite, nous comparons les solutions analytiques simples pour les mono-fluides vers les formulations classiques Navier-Stokes. Nous arrivons à la conclusion que, du fait des importantes variations de densité, les équations de quantité de mouvements s'adaptent mieux au côté aqueux proche de l'interface. Les erreurs apparaissent dans les solutions analytiques d'un écoulement Couette avec interface. Nous présentons une approche pour les corriger. Après cette discussion, nous présenterons les applications pratiques d'une formulation basée sur les discrétisations rendues par le maillage volumique et le maillage surfacique construit par le code pour la représentation d'interface. Utilisation de l'algorithme "Marching Polyhedra Algorithm" pour la méthode "Volume of Fluid" Le "Marching Polyhedra Algorithm" (MPA) est une des principales techniques implémentées dans ISIS-CFD. Nous avons développé le MPA dans le cadre de ce travail, suivant les principes du "Marching Cube Algorithm", un algorithme de construction d'isosurface fréquemment utilisé dans la Modélisation Géométrique (Newman et al, 2006) [74]. Le MPA est utilisé avec la formulation de maillage non structuré d'ISIS-CFD pour résoudre les problèmes d'initialisation de fraction volumique et à l'inverse, pour le problème de reconstruction d'interface. Après une présentation de l'algorithme, nous discutons les solutions et les problèmes, démontrons plusieurs exemples pour des cas simplifiés et montrons que le MPA peut être utilisé comme méthode de reconstruction, du fait qu'il produit des approximations convergentes de la surface réelle de l'interface. Notre approche présente un point de vue différent pour les méthodes VOF, avec des schémas de discrétisation compressive (les schémas décrits par Queutey et Visonneau (2007) [86] qui n'utilisent pas de reconstruction d'interface telle que le PLIC). En plus de la fraction volumique qui capture l'interface, la notion d'interface en tant que surface est récupérée (et, dans ce sens, l'interface est repérée simultanément) par le MPA. Le but est d'utiliser le maillage de surface pour, soit calculer les termes sources requis, soit renforcer les conditions de frontières sur les interfaces. Ensuite, nous présentons les méthodes adoptées pour évaluer le vecteur normal et la courbure de l'interface. La méthode est basée sur la discrétisation effectuée sur le maillage de surface. A fin que le MPA génère des maillages surfaciques non-structurés qui peuvent être de faible qualité, la méthode s'appuie sur une procédure de reconstruction locale des moindres carrés. Au cours de ce travail, nous avons testé différentes fonctions de bases polynomiales et différentes contructions de voisinages sur les cellules surfaciques. Nous démontrons que la méthode de calcul de courbure qui en résulte est d'une précision de second ordre, dans le sens de la norme L max , étant donné que le champ de fraction volumique est representé par une définition suffisante sur le maillage volumique. Les calculs précis et convergents de la courbure et du vecteur normal sont essentiels pour les applications dans lesquelles la tension superficielle est importante et pour la réalisation des calculs avec raffinement adaptatif du maillage. avoir présenté les techniques utiles implémentées dans ISIS-CFD via le cadre ISIS-CFD extensions, nous présentons les méthodes de tension superficielle implémentées et les résultats obtenus pour plusieurs cas tests. Les matières discutées dans ce travail forment divers axes de futures recherches qui pourraient être envisagées pour ISIS-CFD et les études dans l'hydrodynamique navale. Premièrement, nous formons une hypothèse sur les améliorations possibles du modèle au point de vue académique. Des applications pratiques de la théorie peuvent être employées aux solutions numériques. Si nécessaire, nous pourrons par la suite procéder à l'amélioration du modèle. Deuxièmement, le MPA peut être utilisé pour renforcer les conditions de pression et de vitesse sur l'interface plutôt que sur les faces voisines comme le propose la DCM. Troisièmement, les procédures d'initialisation de fractions volumiques implémentées peuvent être utilisées pour l'initialisation d'objets solides immergés dans l'écoulement. Finalement, une méthode peut être conçue pour utiliser l'interface reconstruite et le champ de vélocité avoisinant, pour évaluer les termes volumétriques sources d'un écoulement de bulles autour de l'interface (suivant Ma et al. 2011 [61]) pour introduire un modèle d'entrainement d'air dans ISIS-CFD. En conclusion, ISIS-CFD pourrait se munir des outils de modélisation nécessaires pour la conception de cavités partielles pour les coques de bateaux, en permettant une analyse détaillée des mécanismes physiques qui gouvernent les écoulements d'air et d'eau pour un vaisseau naval lubrifié.

  The following table summarizes the results obtained for the maximum relative error of the I -th subdomain solution, RE max I :Comparing the error values for the two solutions, it is evident that the discontinuous solution provides more accurate results than the continuous solution for both subdomains.

			RE max I ( u) = max {y:CI (y)=1}	u sol (y) -u exact (y)			(2.5.43)
	and the mean L 1 norm of the I -th subdomain solution, L 1I ,
			L 1I ( u) =	{y:CI (y)=1} u sol (y ) -u exact (y ) d y {y:CI (y)=1} d y		(2.5.44)
			RE max errors				L 1 errors
		C 1 Continuous Solution	Discontinuous Solution	C 1 Continuous Solution	Discontinuous Solution
	(m)	water	air	water	air	water	air	water	air
	2 • 10 -3

5.173 • 10 -1 7.416 • 10 -1 2.718 • 10 -3 3.896 • 10 -3 1.409 • 10 -5 7.618 • 10 -4 7.405 • 10 -8 4.003 • 10 -6

Table 3 .

 3 1: Summary of L max and L 1 distance errors and the total volume error for a spherical interface.

	ν	s	C	L max d c s	L 1 d c s	RE (V )
	20	416	1.3973e-02 8.4432e-03 2.0415e-02
	30	1064	5.2421e-03 3.2940e-03 8.4518e-03
	40	1832	3.4411e-03 1.9246e-03 5.0420e-03
	60	4064	1.5246e-03 8.6054e-04 2.1923e-03
	80	7472	8.3102e-04 4.6463e-04 1.2150e-03
	120 16736 3.7218e-04 2.0999e-04 5.4432e-04

  .2 we summarize the maximum distance error, symbolically denoted by L max d n

	s	, and
	the mean value distance error, symbolically denoted by L 1 d n s

Table 3 .

 3 2: Summary of L max and L 1 distance errors for the iso-nodes of a spherical interface and local slopes of L max and L 1 .

		s	N	L max d n s	L 1 d n s	Slope L max Slope L 1
	20	414	3.0133e-02 2.4582e-02	-	-
	30	1062	1.2683e-02 1.0414e-02	2.134	2.118
	40	1830	8.1282e-03 6.0784e-03	1.547	1.872
	60	4182	3.8696e-03 2.6905e-03	1.831	2.010
	80	7470	2.4606e-03 1.4797e-03	1.574	2.078
	120 16854 1.3124e-03 6.4995e-04	1.550	2.029

Table 3 .

 3 3: Summary of L max , L 1 distance errors and execution times.

	ν	s	N	L max d n s	L 1 d n s	T s (s)
	20	1449	1.1232e-01 2.0210e-02 0.064
	30	3561	8.7087e-02 1.2075e-02 0.208
	40	5969	5.2000e-02 5.2262e-03 0.444
	60	13565 4.8894e-02 2.5314e-03 1.352
	80	24221 2.7246e-02 1.5599e-03 3.060
	120 54745 2.0167e-02 8.0408e-04 9.861

The calculated maximum distance error, L max d n s , and the mean error norm, L 1 d n s are summarized in table 3.3 and plotted in figure

3

.17. From the results, we arrive at the conclusion that when the interface reconstruction involves nodes of physical boundaries the method is first order accurate.

In this case, the local slopes of the errors are not indicative of the method's accuracy. Using linear regression the order of the error has been found to be 0.93 for L max and 1.73 for the L 1 .

Table 4 .

 4 1: Characteristic values of performance for the static bubble test case using CSF and CSS

	ν	m(p; a) E ( p )	CSF o(p)	u(p) max(| u|)	m(p; a) E ( p )	CSS o(p)	u(p) max(| u|)
	20	0.358	-0.383 -6.62e-4 0.432	1.62e-4	0.517	-0.114 -1.31e-2 0.398	8.18e-5
	30	0.276	-0.526 -7.43e-3 0.545	1.28e-4	0.500	-0.145 -3.41e-3 0.389	1.34e-4
	40	0.252	-0.566 -3.31e-2 0.902	3.60e-4	0.520	-0.111	2.24e-2	0.557	3.75e-4
	60	0.138	-0.763 -2.42e-1 1.105	4.47e-4	0.523	-0.105	5.71e-2	0.642	4.49e-4
	ν	m(p; a) E ( p )	IsoFT o(p)	u(p) max(| u|)	m(p; a) E ( p )	DCM o(p)	u(p)	max(| u|)
	20	0.712	0.231	-6.07e-2 1.126	1.50e-4	0.603	0.035	1.73e-2	2.51e-2	4.39e-6
	30	0.639	0.099	-1.57e-3 0.509	1.71e-4	0.590	0.014	-2.17e-2 2.28e-2	4.04e-6
	40	0.610	0.052	-1.92e-2 1.312	1.97e-4	0.589	0.011	-2.67e-2 2.21e-2	3.38e-6
	60	0.568	-0.023 -3.27e-2 1.019	2.48e-4	0.588	0.009	-2.55e-2 3.62e-2	2.23e-6

Table 4 .

 4 2: Characteristic values of performance for the static bubble test case using IsoFT and DCM

Experimental and Numerical Results from Previous Studies

  Based on experimental observations, two cases are distinguished in the literature for a bubble in the ellipsoidal regime.First, cases where the bubble's aspect ratio is considerably different from one and the bubble's terminal velocity acquires its maximum value. Second, cases where the bubble's aspect ratio is almost one, and the bubble's terminal velocity is considerably smaller than the first case. Experimental studies suggest two explanations of this phenomenon. A wide range of studies, seeClift et al. (1978) [START_REF] Clift | Bubbles, Drops and Particles[END_REF], propose that in principle surfactants contaminate the air-water interface. The contamination results to a modified surface tension coefficient which in turn modifies the observed air-water interface. More recent studies of[START_REF] Tomiyama | Terminal velocity of single bubbles in surface tension force dominant regime[END_REF]

Table 4 .

 4 3: Terminal velocity estimations obtained by the implemented surface tension methods in ISIS-CFD, characteristic terminal velocity values by experimental results and previous CFD studies tension and predict the evolution of the interface. The experimental results of Clift et al. (1975), Wu et al. (2002) and Duineveld (1995), and the computational results from Tukovic and Jasak (2012) [106], and Hua et al. (2008) [41] are plotted alongside the results obtained by ISIS-CFD in figure 4.14. The x-axis of the diagram corresponds to the bubble's equivalent radius and the y-axis to the terminal velocity. The calculations with CSF provided the largest estimations of the terminal velocity. The DCM gave smaller terminal velocities than CSF but higher terminal velocities than IsoFT. From the calculation by IsoFT, we obtained the smallest estimations of terminal velocities. For the smallest radius tested, CSF and DCM did not provide terminal velocities that fall inside the limits suggested by the experimental study of Tomiyama et al. (given in table4.3). Nevertheless, the experimental results from[START_REF] Duineveld | The rise velocity and shape of bubbles in pure water at high Reynolds number[END_REF] and other researchers such as[START_REF] Wu | Experimental studies on the shape and path of small air bubbles rising in clean water[END_REF], suggest that bubbles with small equivalent radii can obtain large terminal velocities and the result obtained by DCM is much more reasonable. These observations are also supported in the context of the calculated aspect ratios as we will discuss next.The DCM and IsoFT methods provide terminal velocity values that fall inside the Tomiyama limits and the upper limit suggested by theClift et al. (1978) [START_REF] Clift | Bubbles, Drops and Particles[END_REF] for larger bubbles. The results of IsoFT provide good comparisons with the experimental results of[START_REF] Wu | Experimental studies on the shape and path of small air bubbles rising in clean water[END_REF] [START_REF] Wu | Experimental studies on the shape and path of small air bubbles rising in clean water[END_REF] and the CFD results from[START_REF] Hua | Numerical simulation of 3D bubbles rising in viscous liquids using a front tracking method[END_REF]

	Jasak (2012) [106] compare their results to Duineveld's results and Hua et al. (2008) [41] with Tomiyama's
	results. In table 4.3 we summarize the terminal velocities computed by ISIS-CFD using CSF, DCM and
	IsoFT and characteristic values from terminal velocities measured experimentally by Clift (1978) [19],
	Duineveld (1995) [22] and Tomiyama et al. (2002) [102], and computational results obtained by Hua et
	al. (2008) [41] and Tukovic and Jasak (2012) [106]. We emphasize that by "characteristic values," we
	refer to either values that dominantly appear in experimental results or interpolated values for CFD
	results of previous studies (when these are not explicitly provided). Each method provides different
	results, which is reasonable considering that each method uses a different approach to model surface
		ISIS-CFD V t (m/s)	Experimental V t (m/s)	CFD V t (m/s)
	R (mm)	CSF	DCM isoFT	Clift (1978)	Duineveld Tomiyama (1995) (2002)	Tukovic (2012)	Hua (2007)
	0.6	0.345 0.290	0.18	0.24	0.32	0.14-0.17	N/A	0.22
	0.8	0.300 0.266 0.209	0.33	0.36	0.15-0.33	0.359	0.24
	1	0.278 0.257 0.222	0.31	N/A	0.15-0.31	0.357	0.256

Table 4 .

 4 Through the calculated aspect ratio we compared for each case the terminal velocities obtained by Tomiyama's relations and the results we obtained by the calculated terminal velocity. The results are summarized in table 4.4. The table summarizes the aspect ratio 4: Aspect ratio estimations of isosurfaces produced using different surface tension methods of ISIS-CFD, terminal velocity obtained Tomiyama relation and aspect ratios by other researchers.

	R (mm)	E	ISIS-CFD: CSF V t (m/s)	V er r	ISIS-CFD: DCM E V t V er r	ISIS-CFD: IsoFT E V t V er r	E (Other Works) Clift Duineveld Tukovic
	0.6	0.393	0.361	0.045	0.710 0.320 0.093	0.933 0.175 0.031	0.86	0.78	N/A
	0.8	0.438	0.319	0.059	0.705 0.281 0.076	0.859 0.213 0.017	0.78	0.61	0.595
	1	0.399	0.287	0.030	0.641 0.271 0.051	0.793 0.225 0.015	0.73	N/A	0.522

  al. (2008)[START_REF] Landel | Spherical cap bubbles with a toroidal bubbly wake[END_REF]that performed experiments on spherical cap bubbles, as crowned cap bubbles.[START_REF] Landel | Spherical cap bubbles with a toroidal bubbly wake[END_REF], described the formation mechanisms observed for the crowned cap bubble in their experiments as "crowned bubble systems were also formed when the jet was not sufficiently strong to penetrate the upper surface of the original bubble. In such circumstances, the crown of satellite bubbles appeared to be formed

	by the strong toroidal vorticity in the liquid which is shed from the trailing edge of the accelerating initial
	bubble."
	X	X
	Z	Z
	Y	Y
	k	k
	800	800
	700	700
	600	600
	500	500
	400	400
	300	300
	200	200
	100	100
	0	0
	-100	-100
	-200	-200
	-300	-300
	-400	-400
	-500	-500
	-600	-600
	-700	-700
	-800	-800
	(a) t = 0.0395 s	(b) t = 0.0635 s

  The bubble from Landel's experiments corresponds to a bubble of equivalent diameter d e = 49 mm

	Z		Z
		X		X
	Y	k	Y	k
	800	800
	700	700
	600	600
	500	500
	400	400
	300	300
	200	200
	100	100
	0		0
	-100	-100
	-200	-200
	-300	-300
	-400	-400
	-500	-500
	-600	-600
	-700	-700
	-800	-800
	(a) t = 0.1205 s	(b) t = 0.1385 s
	Z		Z
		X		X
	Y	k	Y	k
	800	800
	700	700
	600	600
	500	500
	400	400
	300	300
	200	200
	100	100
	0		0
	-100	-100
	-200	-200
	-300	-300
	-400	-400
	-500	-500
	-600	-600
	-700	-700
	-800	-800
	(c) t = 0.1435 s	(d) t = 0.1475 s
	Z		Z
		X		X
	Y	k	Y	k
	800	800
	700	700
	600	600
	500	500
	400	400
	300	300
	200	200
	100	100
	0		0
	-100	-100
	-200	-200
	-300	-300
	-400	-400
	-500	-500
	-600	-600
	-700	-700
	-800	-800
	(e) t = 0.1595 s	(f) t = 0.2385 s

  Tryggvason et al. [105, p.232-235].

  [START_REF] Broglia | Measurements of the Velocity Field Around the DELFT 372 Catamaran in Steady Drift[END_REF] performed stereo PIV measurements for a high-speed catamaran. Among other details, naturally ventilated vortices were observed for high drift angles and Froude number, figure 4.30. Previous experience with ISIS-CFD results has demonstrated the excellent interface capturing capabilities of the code for this case. The purpose of this subsection is to demonstrate that the MPA can be applied to a test case of naval engineering interest, where similar phenomena occur.

  zd = 0 indicates that the interface has reached the plate's sharp edge that is submerged in water. The isosurface is depicted as viewed by an observer standing behind the plate (the plate moving from the reader to the paper). The crest of the large wave, which was formed at the leading surface of the plate (high-pressure side), reached a maximum of zd = 2. When zd = 0, as in figure4.35b and 4.35d, the formation of the ventilated vortex begun. Other structures that contained air have been already formed several times due to breaking waves.

	0.17	17	(4.5.1)
	Z	Z
	Y		Y
	X	X
	Zsdr	Zsdr
	1.8	1.8
	1.7	1.7
	1.6	1.6
	1.5	1.5
	1.4	1.4
	1.3	1.3
	1.2	1.2
	1.1	1.1
	1	1
	0.9	0.9
	0.8	0.8
	0.7	0.7
	0.6	0.6
	0.5	0.5
	0.4	0.4
	0.3	0.3
	0.2	0.2
	0.1	0.1
	0	0
	(a) t = 0.375 s -View from top to bottom	(b) t = 0.741 s -View from top to bottom
	Z	Z
	X	X
	Y		Y
	Zsdr	Zsdr
	1.8	1.8
	1.7	1.7
	1.6	1.6
	1.5	1.5
	1.4	1.4
	1.3	1.3
	1.2	1.2
	1.1	1.1
	1	1
	0.9	0.9
	0.8	0.8
	0.7	0.7
	0.6	0.6
	0.5	0.5
	0.4	0.4
	0.3	0.3
	0.2	0.2
	0.1	0.1
	0	0
	(c) t = 0.371 s -View from bottom to top	(d) t = 0.741 s -View from bottom to top

1.3. AIR/VAPOR CAVITIES FOR DRAG REDUCTION OF UNDERWATER VEHICLES

Interface Reconstruction Introduction

The interface reconstruction problem is the inverse of the volume fraction initialization problem and it is stated as:

The Interface Reconstruction Problem

Given a general unstructured grid covering the computational domain D and the values of a cell-centered field g I c whose continuous counterpart defines a surface S I = ∂D I by the implicit function g I ( r ) = a, construct an unstructured surface grid that represents the surface S I .

The interface reconstruction problem can be considered as a discretized isosurface construction problem, since instead of having the continuous function g I ( r ) we have its discrete counterpart g I c , i.e. the values of the function evaluated at the centers of the cells. Specifically, we are interested in the case where the field g is the volume fraction field, g I c = C I c . As we have previously pointed out, a reasonable approximation of the interface can be obtained by the implicit surface C I ( r ) = 0.5.

Therefore, the isosurface of the implicit function C I ( r ) = 0.5 proposes an approximation of the interface. To construct the isosurface using the MPA we have first to approximate the volume fraction wherever required, using cell-centered values of the volume fraction, C I c . In this section, the purpose is to obtain an approximation of the interface using the volume fraction field solely as given by the solution of volume fraction transport equation by ISIS-CFD. In the literature, the interface reconstruction procedure is central to many numerical methods treating multifluid problems and usually follow a different approach.

The standard practice to obtain an approximation of the interfaces is the so-called Piecewise Linear Interface Calculation (or Construction), abbreviated as PLIC. Specific numerical schemes use PLIC-like schemes to solve the volume fraction transport equation. Usually "PLIC" refers to both the interface reconstruction scheme and the volume fraction convection scheme. As the name implies, the interface is represented locally by a plane that is constructed through the volume fraction of the cell C I c . To clarify, consider that the plane is given by g I c ( r ) = ( rr Oc ) n c , the normal vector, n c , is obtained by finding an approximation of the interface's normal at the cell and the point r Oc is found by enforcing:

where is a small number, i.e. the volume fraction of the local plane should be the same as the volume fraction of the cell. Further details about the PLIC scheme can be found in [START_REF] Tryggvason | Direct Numerical Simulations of Gas-Liquid Multiphase Flows[END_REF]. The Scardovelli-Zaleski analytical volume fraction relation [START_REF] Scardovelli | Analytical Relations Connecting Linear Interfaces and Volume Fractions in Rectangular Grids[END_REF] was developed to solve both the direct problem (the volume fraction initialization) and the PLIC reconstruction in cells that are rectangular parallelepipeds. Later, Lopez et al. (2008) [START_REF] López | Analytical and geometrical tools for 3D volume of fluid methods in general grids[END_REF] proposed a solution for polyhedral cells with planar faces. The difficulties encountered when cells are not rectangular parallelepipeds, and specifically, the Courant number condition limitation for PLIC schemes might limit the method's applicability for the problems dealt by ISIS-CFD. A comparison between PLIC schemes and compressive schemes, similar to those used by Chapter 4 The Development of Surface Tension Methods

under ISIS-CFD

Verification and Validation

Introduction

In the previous chapters, we have introduced the necessary tools to develop surface tension modeling schemes to ISIS-CFD and setup numerical simulations for the study of multifluid problems with multiple interfaces. Surface tension plays a significant role in the dynamic evolution of phenomena of practical interest. These flows include phenomena with various length/time scales, from the smaller, which are usually out of engineering interest, to larger such as the detailed capturing of cavity closure regions in air lubrication cavities. The evolution of these phenomena is characterized by very complex interface topological changes, including merging and separation, in which the modeling of surface tension plays a crucial role.

In this chapter, we present the surface tension methods that were implemented in ISIS-CFD.

The ISIS-CFD extensions framework becomes an important component which initializes the volume fractions for the setup of the simulations, calculates the normal vector and curvature of the interfaces and finally, performs the coupling of the surface tension with ISIS-CFD. We present results obtained through different stages of this work, which implies that some of the resulting isosurfaces are depicted as wireframes or patches generated by Matlab. During the initial stage, the surface grid generation method was only partially complete, and we could not achieve better visualizations. The chapter is organized as follows.

In the first section, we present in detail the classic and modern surface tension methods implemented in the ISIS-CFD fluid solver. In the second section, we compare the results obtained by each method for the static bubble test case and verify the results with the analytical solution. In the third section, we present the results obtained for the rising bubble test cases by coupling ISIS-CFD with different surface tension methods. We performed computations for various flow regimes, where the interface undergoes either no changes at all or very severe topological changes. For each case, we compare the results of ISIS-CFD to the experimental and numerical results obtained by other researchers.

The chapter concludes with two more complicated test cases of practical interest where the interface undergoes fast topological changes of large scale.

The results demonstrate that ISIS-CFD captures the dynamics of rising bubbles accurately. Specifically, the terminal velocities obtained by ISIS-CFD are in good comparison with terminal velocities

Surface Tension Methods with Compressive Discretization

Schemes

Introduction In this section, we present the surface tension methods, implemented to cooperate properly with ISIS-CFD. Surface tension models for compressive discretization schemes (that do not include an interface reconstruction procedure) and general unstructured grids have not been thoroughly studied. We believe that this is due to the practical difficulties for the accurate calculation of the interface's normal vector and curvature.

Indeed, accurate normal and curvature calculations cannot be easily achieved using the volume fraction and differencing schemes that complete in reasonable execution times for general unstructured volume grids. In figure 4.1 we compare the best approximation errors obtained by implemented evaluation methods for the normal vector and curvature of a spherical interface (again we emphasize that these methods complete in practically reasonable execution time and can be used with unstructured grids). The x-axis represents the number of cells in the diameter of the interface and the y-axis error norms. Specifically, the following methods were used:

1. approximations using the classic Central Difference Scheme (CDS with misalignment corrections)

and Kernel smoothing using a top-hat function whose support covers the level 2 neighborhoods of the cells, denoted K 2(C I ) 2. approximations using a modified CDS scheme for the surface grid generated by passing as input to the MPA the field K 2(C I ) 3. approximations using the LSqR method generated by passing as input to the MPA the field K 2(C I )

The first two methods derive approximations that do not converge. The LSqR achieved the best approximation for both the normal vector and the curvature. Therefore, converging normal vector and curvature approximations were obtained using the unstructured surface grid reconstructed by the MPA and local least squares approximations. Specifically, this approach gives second-order approximations for both the normal vector and the curvature.

In this section, we present surface tension methods that are used with compressive discretization schemes and general unstructured grids. We present two classic and two novel methods. The novel methods, developed specifically for ISIS-CFD, use the reconstructed interface obtained by the MPA, as a core component of their formulation. Firstly, we present the smoothed surface tension methods and afterward a sharp surface tension method approach using a method that resembles the Ghost Fluid Method, introduced to ISIS-CFD by Queutey and Visonneau (2007) [86].

The Discrete Contour Method

In this subsection, we present the implementation of the Ghost Fluid Method used in ISIS-CFD. [START_REF] Queutey | An interface capturing method for free-surface hydrodynamic flows[END_REF] in [86] have introduced the reconstruction relations used to obtain the gradient approximations of a discontinuous field for a general unstructured grid. We present the required modifications to take into account surface tension which formulates the Discrete Contour Method (DCM).

Sharp methods use a straightforward procedure to take into account surface tension in the calculations, as long as we establish the compatibility conditions for the interface. We will consider the case where the boundary conditions are given by:

Both relations hold for points of the interface. The fundamental assumption of the proposed method is that there exist a face f in the vicinity of the interface where we assume that the pressure discontinuity p is enforced. Therefore, for this face, the pressure will be defined as we approach the face from its two sides, say left and right, with the face normal pointing from the left to the right. The adjacent cells to the face are called left and right. By convention, the normal vector of the face is exterior to the region of the left cell and interior to the region the right cell. Then the reconstruction relations will provide the pressure values at the face f , as we approach the face from its left and right cell respectively. Before providing the reconstruction relations, we shortly revise the geometrical quantities required.

We need the following geometrical entities and definitions to state the reconstruction relations. If the centroids of the left and right cells to the face (see also figure D.1, p.242) are r L and r R , the centroid of the face r f and its normal vector n f , then we define the length ratio a of the face, as:

The intersection of the linear segment LR with the face f is the point r F :

We define the following tangent vector, t f , to the face f as:

For the linear segment LR, we consider that a field q varies linearly over LR, and we obtain the value of the field at F :

discrete contour that surrounds the actual interface. The name Discrete Contour Method is used to emphasize this characteristic and to distinguish the method from the other "smoothed" surface tension methods that use a source term.

Summary

In this section, we have presented the methods that we have implemented in ISIS-CFD for the modeling of surface tension. We have described four methods, CSF, CSF, and IsoFT belong to the category of smoothed surface tension methods, and DCM to sharp surface tension methods.

IsoFT and DCM use the marching polyhedra algorithm to perform specific actions which make them resemble methods used in front tracking framework, while ISIS-CFD uses a front capturing framework.

A common characteristic of both methods is that the curvature results from the isosurface which provides better results in comparison to methods using only the volume fraction. In the following section, we demonstrate the results obtained by each method for three-dimensional static bubbles and compare them to the analytic solution.

Conclusions

In this section, we have presented the results obtained using the different surface tension methods implemented in ISIS-CFD for the static bubble test case. The smooth surface tension methods provide pressure fields that follow similar patterns. The pressure near the interface oscillates until it reaches the constant values outside the bubble and inside the bubble. For the cases studied, the CSS method performs better than CSF. However, we attributed the last observation to the smoothed volume fraction field that we used for the CSS computations. Experience has shown that using the CSF method produces comparable results to the CSS method only for coarser discretization. For finer discretizations, the curvature obtained directly by the volume fraction diverges.

Both the CSF method and the CSS method, use solely the volume fraction to obtain approximations for the surface tension source term. In comparison to the IsoFT method that uses the isosurface to obtain curvature approximations, the computations with CSF and CSS provide less accurate approximations for the mean pressure field inside the bubble. The DCM provides better results as compared to the previous methods we tested. Specifically, the pressure oscillations practically vanish, the predicted pressure values are close to the theoretical values, and the spurious velocities are smaller than these obtained by the other methods.

All the methods presented here can directly profit from Automatic Grid Refinement. An important aspect of AGR, when used together with a method based on interface reconstruction, is the volume fraction reinitialization procedure. After refining the background volume grid, the volume fraction field must be refined as well, because it represents the presence of the interface (a physical boundary) for a given cell. The reinitialization procedure should ensure that the MPA will generate a more accurate approximation of the interface, to subsequently ameliorate the curvature approximations.

Finally, due to the resulting spurious currents we can conclude that the implementation is not "force balanced," in the sense described by [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF] [START_REF] Francois | A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework[END_REF]. A force balanced approach does not allow the surface tension source term to interact with the reconstruction of volumetric fluxes, by interpolating the source terms from cell centers to the faces. Instead, the source term is treated in the same manner as the pressure gradient that, during the interpolation step to obtain the velocity field values at faces, it is directly discretized at the faces. Another important feature of a "force balanced" method, is that the gradients of the source term are discretized in the same manner as the pressure gradient at the faces. For the DCM, the curvature is obtained using LSqR on the interface while the pressure gradient is discretized at the grid's faces (using the left and right cell of the face), is a possible cause of spurious currents. calculated for each method as described before, the terminal velocity that corresponds to Tomiyama's relation and the relative error of the terminal velocity V er r .

The terminal velocities obtained by ISIS-CFD and Tomiyama's relations are in excellent agreement.

Therefore the results indicate that ISIS-CFD correctly predicts the motion of the bubble. The prediction of the interface's deformation is directly related to the surface tension method used. This observation is further supported by the fact that CSF provides for every case tested much smaller aspect ratios than aspect ratios measured by experiments and, at the same time, the calculated terminal velocities compare well to terminal velocities Tomiyama's relation. The DCM and IsoFT provide consistent results comparable to both experimental and computational studies.

A Bubble test case in the Spherical Cap Regime

General Remarks Spherical cap bubbles demonstrate fascinating interactions, which are difficult to be accurately captured due to the highly localized surface tension effects. We demonstrate the results obtained using the same methods as before for a spherical cap bubble whose initial radius is r = 10 mm.

We note that the simulated bubble stands at the limit between the spherical cap (for larger bubble radii) and the ellipsoidal regime (for smaller bubble radii).

In figure 4.16, we compare the volume fraction field obtained at the final stages of the bubble's evolution. Due to the lack of CSF to take into account the local curvature effects, the bubble took a toroidal form, similar to that observed when the surface tension coefficient is zero. The other methods produced results which are closer to the anticipated spherical cap shape. The computations coupled with the CSS method produced a highly skirted interface, and the skirts were not detached. The volume fraction obtained using DCM show the volume fraction was smeared near the lateral parts of the surface, making the effective region of the bubble (occupied by air) smaller. The volume fraction field obtained by using the IsoFT method indicates that the interface's shape is closer to the anticipated results for a single spherical cap bubble.

The intersection of the relative velocity V -V b field and the sharp volume fraction field C # I (the calculation of the sharp volume fraction field is based on the generated isosurface) are shown in figure 4.17 for DCM, IsoFT, and CSS (also note that we can visually confirm that relative velocity is tangent [START_REF] Sangeeth | Dynamics of Collapse of Free Surface Bubbles[END_REF] [START_REF] Sangeeth | Dynamics of Collapse of Free Surface Bubbles[END_REF]. The isosurfaces colors correspond to their (signed) distance from the z = 0 plane.

When the interface reached the edge of the plate, air displaced water and mounted the sharp edge. This appears as the cavity shown in figures 4.37a and 4.37b. In figure 4.37a a section of the plate with the zy-plane is added for demonstration purposes. We look at the interface from behind the plate, and the gray area of the figure denotes the thickness of the plate. Air partially covered the sharp edge that was entirely immersed underwater at the initial setup. The same instance as viewed from the top is shown in figure 4.37b. The grid is transparent, and the parallel cross-sections (black lines) visualize the interface. The cavity is visible by the cross-sections that form almost the same angle, located at the bottom right. A breaking wave is also visible near the plate, in the upper left, where the cross-section lines begin folding. Shortly afterward air passes inside the cavity, and the cavity widens, figures 4.37c and 4.37d (during the same time instance the wave on the verge of breaking). The entrainment or air inside this cavity marks the beginning of the generation of the ventilated vortex.

Appendices

Chapter A

Diverse Notes in Mathematics

A.1 Decomposition of the gradient of a vector field in symmetric and antisymmetric parts using its tangential and normal components on a surface

We express the components of the gradient of a vector field using only the tangential and normal components of the vector field on a surface. The surface crosses the same point where the gradient of the vector field is evaluated. The relations for the divergence, curl and the symmetric and antisymmetric parts of the gradient easily follow.

Nomenclature -Basic Differential Objects

Consider a representation of R 3 by a coordinate system whose covariant basis vectors are Z k , k = 1, 2, 3 (tangent vectors to coordinate lines), the covariant metric tensor is Z i j = Z i • Z j , the contravariant base vectors are Z k , k = 1, 2, 3 (normal vectors to coordinate lines taken two at a time) and the contravariant metric tensor is

consider that there is an embedded surface in R 3 whose points are given by Z = s(η, ξ). For each point on the surface we define the coordinate system whose covariant base vectors are s 1 , s 2 , n (the first two vectors are tangent vectors to ξ and η coordinate lines respectively and the third is their unit normal)

and the covariant metric tensor of the surface is S αβ = s α • s β . The contravariant base vectors are s 1 , s 2 , n (each of the first two vectors are normal vectors to the other coordinate line and the unit normal) and the surface contravariant metric tensor is S αβ = s α • s β . Latin letters refer to coordinates representing points in the ambient space, while Greek letters refer to the coordinates representing points on the surface. In these notes, we use the same nomenclature as used in Grinfeld's book [38, Part II: Tensors on surfaces]. Every relation used in these notes without proof can be found there. Finally, note that we rarely distinguish between surface and space metrics using the symbols S αβ and Z i j but instead by the alphabet used for the indices, and for both we use the letter Z . So Z αβ (with Greek letters) is the covariant surface metric and Z i j (with Latin letters) the covariant metric of the ambient space.

For each point in space, which is also a point of the surface we define the components of the shift operator as the tensor

relates the components of vector u in the ambient space to the components of a vector v in the tangent space of the surface v α = Z α k u k . Similarly, the components of the shift operator as the tensor Z k α = s α • Z k which relates the components a vector v in the tangent space of the surface to its components in the ambient space v k = Z k α v α . The former expresses for each k = 1, 2, 3 the covariant surface components of the covariant basis of the ambient space. The contravariant components of the unit normal are ( i j k is the Levi-Civita tensor):

where S is the area element (S is the determinant of S αβ ).

A vector can be in decomposed as:

where u = n i u i , or using only components:

Thus the components of the gradient of a vector field can be expressed as:

In the above the indexed nabla symbol refers to covariant differentiation, either with respect to the ambient space (Latin index) or the surface (Greek index). Our purpose is to write the components of the gradient in the ambient space using only the normal derivatives, the vector field values, and derivatives defined on the surface.

The second term in the RHS of (A.1.4) is as far as we can get regarding the normal component of the gradient to the surface and we denote it symbolically as d(•) = n l ∇ l (•):

If we consider that in vicinity of the evaluation point at the interface the components of Z i α and the normal n i do not vary as we move along the normal direction then:

and (A.1.4) would then become:

The components of the tangential derivative To continue our derivation we focus on the covariant derivative ∇ α u j . By decomposing the velocity component in the ambient space u j by (A.1.3):

The surface covariant of the shift tensor and the normal vectors are related to the curvature tensor B αβ .

The relations for

and for ∇ α n j :

Both relations are given in [38, pp .193-195] and are commonly known as Gauss and Weingarten formula respectively. Substituting (A.1.9) and (A.1.10) to (A.1.8) we obtain:

By grouping tangential and normal contravariant components we arrive to:

This expression of the surface covariant derivative of the ambient contravariant component of the vector field contains only derivatives and components of the vector field on the surface. Substituting (A.1.12) to (A.1.5) results to:

The first two terms of the above sum, can be calculated by the values of the vector field on the surface and the metrics.

Divergence From the last relation we may immediately calculate the divergence of the vector field, by contracting the indices k, j :

For the first term of the sum Z α j Z j β = δ α β , the second term vanishes because the components of the normal vector in the tangent space vanish, Z α j n j = 0:

Curl Similarly the curl of the vector field follows through relation (A. 1.13):

Each of these terms can be simplified as follows. To begin with the first term, note that Z 1 j Z 1 k and Z 2 j Z 2 k are symmetric over k, j so i j k Z 1 j Z 1 k = 0 and i j k Z 2 j Z 2 k = 0 and we obtain

In general, the following identity holds:

that is obtained by performing the operations in the RHS. To continue with the first term, since the curvature tensor is symmetric, we have

For the other terms we will use the identity:

So the second term is:

and the third term is:

As before we may suppose that in the vicinity of the evaluation point and along the normal direction, the components of Z k δ do not change, so that Z

Symmetric and Antisymmetric Parts The symmetric part, S i j , of the tensor Z i k ∇ k u j is:

Substituting from (A.1.5) we obtain:

and the antisymmetric part A i j is:

A.2 Relations between the average of the derivative of a field and the derivative of the average values of a field

Preliminaries The purpose is to provide proofs of the equation relating the average of the derivative of a field and the derivative of the average values of a field for the case where the field is allowed to be discontinuous in the vicinity of a surface. Suppose that a given domain D, is tessellated into n non-overlapping subdomains D I , I = 1, 2, ..., n, which are separated by a certain number of surfaces S J I (called the interfaces). Near the proximity of an interface, say S J I , a field f ( x, t ) is defined from either region of the interface, in a certain manner that depends on the subdomain. We use the notation, f I ( x, t ) and f J ( x, t ) to indicate the region that the field refers to. In general the field f might be discontinuous at the interface, this if y is a point of the surface S J I , then f I ( y, t ) = f J ( y, t ). To discern the set of points that belong to a given subdomain D I from the points of other subdomains, we build the indicator function

The domain of definition of an indicator function, I I ( x, t ), is defined as the whole domain D, but the domain of definition of a field f I ( x, t ) is the subdomain D I . Using the indicator function we can write the field f ( x, t ) for the whole domain D using the fields f I ( x, t ):

and the field f I is allowed to obtain any value in a point that rests outside the domain D I . We define the average value of the field f for the I -th subdomain by:

The domain of integration is the whole space and K ( y; x) is an averaging kernel centered at x, i.e. a function that:

• the set of points in space y where K ( y; x) is not zero (or sufficiently close to zero for the case of exponentially decaying kernels) is called the support of the kernel and it is specified by the length parameter (also named averaging length).

• is normalized to one, in the following sense:

• in the limit where the support vanishes the kernel converges to the Dirac-δ function:

in the sense that:

which is one of the basic derived properties of the Dirac-δ, the shifting property, usually stated as:

A function with the above properties is a Dirac-δ sequence and for further details and examples the interested reader may refer to Kanwal's book which covers a wide discussion about generalized function and the related theory [48, pp. 60-64]. In our notation K ( y; x) implies that the two variables y and x in the formula implementing K act as a remainder of the difference yx. Moreover, an equivalent notation would be

3 ). However, we prefer the notation K ( y; x) which emphasizes also that the second point, x, is the center of the kernel's support, while the first point is an integration point.

We will show that the average gradient of a field is related to the gradient of the average field by:

and the average time derivative is related to the time derivative of the average field by:

We refer to these relations as the averaging derivative relations. We note that the first relation can be viewed as a general chain rule for definite/indefinite integrals and the second as the theorem of Leibniz.

Proofs

The proofs are directly derived by several manipulation of the term ∇ i f I . By definition of the bar-I operator, we have:

The gradient acts over the free variable x, as pointed out by introducing the notation ∇ ( x) i . Therefore it can be passed inside the integral:

i K ( yx), the RHS of the last relation can be written as:

and by using the product rule for the gradient found at the integrand, we obtain:

By the Gauss theorem, the first integral can be transformed to a surface integral defined at points far enough from the support of the kernel. Therefore the first term is zero. The second term can be written by applying the product rule as:

The first term of the RHS can be recognized to be ∇ i f I ( x, t ). For the second term, we should note that the gradient of the indicator function cannot be formally defined everywhere in the classic sense, because the indicator function is discontinuous at the interface (and it obtains the value one for a point inside the domain D I and zero in any other point). However, in the generalized sense the gradient of the indicator function generated by a moving surface, is given by (see Kanwal [48,):

where n i are the components of the normal vector of the surface pointing at the exterior of domain D I (the outward normal ofthe surface ∂D I ) and δ ∂D I is the Dirac-δ defined on the surface ∂D I . The Dirac-δ function on the surface has the property:

15)

Using (A.2.14) and (A.2.15) to relation (A.2.13) we arrive to:

The boundary of D I is defined by the interfaces S J I , so we write the last term as a sum over the interfaces for all the adjacent regions of the subdomain D I , which we denote by J = int(I ). Finally, by rearranging the relation and dropping the superscript denoting the differentiation variable, we arrive to the desired equation:

The relation for the time derivative is derived in a similar manner. By definition of the bar-I operator, the term ∂ t f I is:

Since the integration is over the whole space, the time derivative can be passed inside the integration:

The first term in the RHS can be recognized as the average of the time derivative of the field f inside the domain D I , ∂ t f I . The second term is the time derivative of the indicator function generated by a moving surface, which is given by (see Kanwal [48,):

Substituting to relation (A.2.19), using the property of the Dirac-δ on a surface and rearranging the terms, we obtain the desired relation:

Multiplying by µn j we obtain the normal and tangential components of the viscous stress from either side of the interface:

and the jump is:

Note that the normal component to the surface of the velocity u is denoted without an index, u, while the components at the tangent components are denoted as u α (using a Greek index, here α = 1, 2)

and the components in the ambient coordinate system are denoted as u i (using a Latin index, here i = 1, 2, 3):

Every term on the LHS of relation (B.1.8), besides the last, can be simplified as follows. For the first term, since the velocity field is continuous, we obtain:

The second term involves the jump of the tangential derivative of the normal velocity component. By

Hadamard's lemma, the jump of the tangential derivative equals the tangential derivative of the jump (see for example, Truesdell [103, pp. 250]). Therefore, we obtain for the tangential velocity derivative:

However, the tangential derivative of the velocity jump is zero since u is constant (and equal to zero) everywhere at the surface. So the tangential derivative of the velocity is continuous. Working in the same manner, we obtain that the tangential derivative of the tangential and normal velocity components are continuous. As a result, it follows that the second term can be written as:

Finally, we will show that the normal derivative n j du j , the third term of (B.1.8), is also continuous. The normal derivative of the normal velocity component can be related to the divergence of the velocity field by equation (A.1.15) as:

Since we are working with incompressible fluid flows, ∇ j u j = 0 (from either side of the interface) and thus the jump of n j du j is also continuous because the other terms in (B.1.13) are continuous so n j du j = n j du j = 0. Of course the last relation does not imply that du j = 0 but instead that the vector whose components are du j will be confined in the tangent plane of the interface's surface.

The last observation implies that du j = Z j α du α which we verifu next. First, we verify that the jump Closing Suppose that we construct extensions from the a point of the surface to the ambient space such that du i = Z α i du a + n i du, then the relations can be simplified. For the normal direction we obtain:

and for the tangential:

The first term on the LHS was obtained by the components of n∇ β u and it can be related to time derivative of the tangent vector to the surface and the time derivative of the normal:

where d (•)/d t is the material derivative. This observation is also verified by Lundgren and Koumoutsakos (1999) [START_REF] Lundgren | On the generation of vorticity at a free surface[END_REF].

A simple application of the above expressions is due to Hirt and Shannon (1968) [START_REF] Hirt | Free-surface stress conditions for incompressible-flow calculations[END_REF]. Hirt and Shannon, used as boundary condition for the two dimensional free surface of a single fluid the relations:

where u n is the normal velocity component , u m is the tangential velocity component and the partial derivatives denoted by n and m denote differentiations with respect to the normal and tangential directions respectively. They state that this relation is approximately true when the curvature of the interface is small. Both relations are directly derived by applying the derived relations for the case of a vanishing curvature tensor. We note that we must use the physical components of the vectors in practice, so the tangential relation must be multiplied once more with a shift operator, say Z k γ (which forms the projection operator computable form we have:

The final result, expressed using the dimensionless lengths r = R/ and l = d / = (z -R)/ (where z is the distance from the center of the bubble), is:

So the equation for pressure in the local Cartesian coordinate system is:

The solution is:

where c I , c, c J are integration constants. By the external boundary condition we obtain c J = 0. Since the averaged pressure field is everywhere continuous, the constants c and c I are determined by enforcing a continuous solution at z = Rand z = R + . The final result is:

Plane Couette: Classic Solution From the y-axis momentum equations, we obtain:

The constant of integration for the J subdomain equation is found to be zero, c J = 0, by enforcing zero pressure at the interface, p J (y = 0) = 0. Since in this case pressure is continuous at the interface, we also obtain c I = 0. Therefore the solutions of the y-axis momentum equations are:

9)

C I = 1 I and C J = 1 J . For example, we obtain:

which is the same result obtained by evaluating C I by the identity C I = 1 -C J and using for C J , relation (2.5.4)).

Two Fluids Plane Couette Flow: Averaged Pressure By integrating the y-axis one-fluid momentum equation (see p.73) we obtain:

The constants of integration, c I , c J and c are obtained by setting a pressure level and enforcing continuity of the function's parts at the starting and ending locations of the transition regions, l = -1 and l = 1. In the classic formulation, the pressure was defined as zero at the interface from the side of water.

In view of the fact that the averaged pressure field coincides with the local pressure field far from the interface, we set the averaged pressure equal to the pressure where we enter the averaging region. Thus, where the C I values start changing from one to less that one, i.e. l = -1 or y = -, the averaged pressure field will be the same as the local pressure field (C.2.9), p(l = -1) = p J (y = -). The solution finally reads:

Chapter D

Diverse Notes in Methods of Computational

Fluid Mechanics

D.1 Gradient Calculations at Boundary Cells

When we approximate the gradient at a boundary cell, the symmetry of the two-sided reconstruction formula for its boundary faces is lost. If the field's values at these faces are not a priori known, errors will be inevitably introduced to the gradient computations. For these cases, the method presented here improves the gradient calculation by modifying both the reconstruction formula used to obtain a field's value at boundary faces and the gradient calculation.

Nomenclature

In general, to compute the gradient of a field at the center of a cell, the first step is to obtain an approximation of the field's values at the centroid of the cell's faces. Such approximations are achieved by interpolating the field values found at the adjacent cells of the face. More specifically, suppose that a face f , shown in figure D.1, is represented as a plane whose normal is n f and the face's centroid is r f . Even though this will not always be an accurate representation of the geometrical space defined by the face, we may consider that, for the purpose of generating reconstruction formulas, each face is represented as a "mean" plane defined by the points, ( rr f ) n f = 0.

We also suppose that the face's normal vector points from the cell f c (1) to cell f c [START_REF] Ahn | Multi-material interface reconstruction on generalized polyhedral meshes[END_REF]. The last statement is equivalent to saying that the face will always be pointing to the exterior of the cell f c [START_REF] Agoston | Computer Graphics and Geometric Modeling: Implementation and Algorithms[END_REF]. In order to simplify the notation we will say that cell f c (1) is "on the left" of the face and cell f c (2) is "on the right" of the face and we name these cells as "cell L" and "cell R". In order to simplify the notation, every field, say q( r ), evaluated at the centroids of these cells will be distinguished by the indices L, R instead f c (1) and f c (2) respectively. So for example q L = q( r L ) where r L = r f c [START_REF] Agoston | Computer Graphics and Geometric Modeling: Implementation and Algorithms[END_REF] is the the centroid of the cell on the left of the face L. Points f , L and R do not belong to the same line.

Instead there is a point on the face's plane that belongs to the line LR, point F such that:

Then the value of the field at the face's centroid is evaluated by the second order accurate relation:

From now on we refer to field value at the face's centroid as the value at f . The third term in the above expression is called the misalignment term, since it vanishes whenever r f = r F , i.e. when the line LR 241 passes from the centroid of the face. For more details, the interested reader may consult Queutey and Visonneau (2007) [86].

Gradient Calculation

The gradient at a given cell κ is obtained by:

where s f is the surface vector of the face κ f multiplied by the orientation correction for the face κ [ f ] ,

V κ is the volume of the cell κ and q f is the value obtained by (D.1.2). Since for a given face f ∈ κ f , either the left or the right adjacent cell will be the cell c, the value at f depends on the gradient of the field at c which is the gradient we are about to calculate. As a result, the above gradient calculation cannot be performed in an explicit manner.

To clarify, we recast relation (D.1.2) for a face f ∈ κ f to:

where the notation f c → κ denotes the adjacent cell with respect to face f (the left or the right cell) that is the same cell as κ, and thus the coefficient (either a or v) or the field's value is the appropriate one for the cell κ. Respectively the notation f c → κ refers to a cell adjacent to f and different than κ. For example, if the left cell of the face is cell κ then a f c →κ = 1a, q f c →κ = q L = q κ , ∇q f c →κ = ∇q L = ∇q κ , and then the the right cell of the face is a cell different that κ, a f c →κ = a, q f c →κ = q R , ∇q f c →κ = ∇q R .

To simplify even further (D.1.4), we write directly q f c →κ = q κ and ∇q f c →κ = ∇q κ :

the initial guess ∇q i =0 κ = ∇q • κ and using the block diagonal matrix C (κ, κ) = (I -Āκ ), κ = 1, n C as a preconditioner to obtain the solution improvements for each subsequent iteration i as:

Notice that when no misalignments are present then the elements of matrices Āκ , Bκλ are zero and

For boundary cells where the value of the field at the boundary faces is given, no modifications are required. In this case, there is no need to calculate the field value at the faces (since it is given). When the field value is not given another approach is required. The application in mind is the gradient calculations for the volume fraction.

Gradient Calculations at Boundary cells with a Prescribed Linear Variation

The characteristic of this problem is that there is no apparent boundary condition to be applied if the volume fraction is not prescribed at the boundary. It is reasonable to assume that the gradient is specified at the boundary, and it is zero. This simplification leads to the reconstruction relation for a boundary face that is adjacent to the cell κ:

However, the above does not provide an approximation for the gradient at the cell when we use for every other (non-boundary) face of the cell the classic reconstruction relation. To clarify, suppose that we work in a uniform Cartesian grid. The boundary cell, where the gradient is approximated, is named 0 and in the sense of the x-axis it is next to cell 1 whose distance is l . The cells are separated by the face b and the boundary face is named a. Then the gradient component in the x-axis would be calculated by:

The above relation does not approximate the gradient at of the cell.

To obtain a gradient approximation, we modify the reconstruction by interpolating the field's value at the boundary faces from the value at cell κ:

This is a relation similar to (D.1.5) with a f c →κ = 1, v f c →κ = ( r fr κ ) and the other coefficient zero. Using this coefficients for boundary faces solve (D.1.12) to obtain the gradient at boundary cells. We note that even when misalignments are absent we obtain:

And actually when misalignment are present we use the above value as the initial guess for the gradient of the boundary cells. To demonstrate the effect of the above consider the one-dimensional case. The gradient would then be:

which is the first order forward finite difference approximation for the derivative. It is interesting to note that this one-iteration improvement for the accuracy of the gradient is systematically observed even in non-boundary cells when using higher-order reconstruction relations. For example, the third-order accurate reconstruction relation (disregarding misalignments):

leads after the one iterative improvement of the type (D. 1.16) to the fourth order central finite difference scheme for the calculation of the gradient for a uniform Cartesian grid. The most interesting feature of this approach is that it does not require a computational stencil larger than the directly adjacent cells of the cell where we use to approximate the gradient. Instead, the stencil is "built" by the iterative improvement and therefore it is not a priori required for the calculation.

D.2 Modifications of the Scardovelli-Zaleski Analytical Volume Fraction Relation

The

Pure 3D case

If N i > 0, ∀i , i ∈ [START_REF] Agoston | Computer Graphics and Geometric Modeling: Implementation and Algorithms[END_REF][START_REF] Allenstrom | Model tests with air lubrication[END_REF]:

3D case simplifies to 2D case

3D case simplifies to 1D case

and the function F is the truncated power function defined as:

We note that the above calculation implies that for a given cell, the plane is represented in an appropriately constructed coordinate system local to the cell. For this coordinate system, the point A is the point that, when the plane is parallelly translated till meeting the point, the volume of the inside region (i.e. the region indicated by the opposite sense of the normal vector) will be zero. Since the plane is usually better described in a global coordinate system rather than a local coordinate system, we present the required modifications to recast the above relation to a form which suits the way we initialize the volume fraction, i.e. using directly the global coordinate system.

Suppose that the plane is given by g ( p) = n J I ( pp o ) where p o is a point on the plane. The plane separates the cell in an inside region and an outside region defined by the orientation of the unit vector n J I (expressed in a global coordinate system not shown in the figure) which always points from the inside region (region I ) to the outside region (region J ). For the case of shown in the figure, the region K N M LHGB A is "in" and the region K N M LDC F E is "out" (the normal vector is not shown in the figure). Point A is the origin of the local coordinate system where (D.2) are written. Point A is chosen as the point that if the plane were parallelly translated so that it passes from that point, the volume of the inside region would be zero (and the cell would completely be laying in the outside region). Point F is chosen as the point that if the plane were to pass from that point the inside region would occupy the whole volume of the cell (and the cell would completely be laying in the inside region). The unit vectors of the coordinate system are chosen as the normal vectors of the faces that share point F (directed outwards with respect to the cell), say f 1, f 2, f 3 ∈ c f . Therefore, if p A , p F are the representations of the points A, F in a global coordinate system then:

where

) n f i are the lengths of the edges. Recall that c [ f ] are the orientation correction coefficients which make the normal vectors of a cell's face f ∈ c f point outwards the region of the cell.

For the local coordinate system the plane can be expressed by its implicit representation as, G( P ) = N Pa where a = n J I ( p op A ). Note the capital letters for points or vectors, e.g. P , N , indicate the representation of the vector (or point) in the local coordinate system of the cell. The distance a max is defined as before:

In the last relation, the values N f k = c [ f k] n J I n f k are the coordinates of the normal vector in the "local to cell" coordinate system, N , and by construction N f k 0, k = 1, 2, 3. The distance a can be written as:

recall that p c is the centroid of the cell. We can identify the (signed) distance d = n J I ( p cp o ), which was previously defined as the distance of the centroid of the kernel's support, the cell's center in this case, to its closest point of the interface (see also p. 61). Moreover, p Fp A = 2( p cp A ) and thus we arrive to:

With this modifications the Scardovelli-Zaleski volume fraction formula can be directly used to solve the volume fraction calculation problem as stated in p.42 and at the same time implemented in a general unstructured grid framework. We present the resulting relation and procedures next.

Suppose that we are given a cell c that represents the kernel of the calculation of (2.4.13) and a planar interface that defines two fluid subdomains I and J . Moreover, the unit normal vector of the plane (the interface) in the global coordinate system is n J I and the position vector of a point in the plane p o . Then the following procedure is used to calculate the volume fraction of region I in any point in space. First, calculate the sequence of products c [ f ] δ f n J I n f , ∀ f ∈ c f defined in the same order as the faces are found in c f and gather the positive elements of this set to a set M . The resulting set M contains the terms M i = N i δ i , i = 1, |M | that will be used by the Scardovelli-Zaleski formula. The sum of the elements of M is a max and their product is the result of the product in the denominator of the volume fraction relations. The volume fraction relation can be directly evaluated for each cell that is intersected by the interface, i.e.:

-

by the Scardovelli-Zaleski formula written as a function of the dimensionless signed distance l = d / = n J I ( p cp o )/ where = a max 2 (the characteristic length of the kernel's support with respect to the given plane) and the problem parameters κ i = 1 -M i / (note that κ i obtains values in -1 κ i < 1):

Using the the function (D.2.10) we can write for any cell:

Notice that (D.2.10) for |M | = 1 is the same expression that we would obtain for C I from (2.5.3),62. Also for any |M | and for l = 0 we obtain C I = 0.5, which is the expected result and can be directly obtained using simple factorization rules and that i κ i = 1.

An Analytical Relation for the Area of a Plane Contained in a Rectangular Parallelepiped Exactly the same procedure used by Scardovelli and Zaleski for the calculation of the volume inside the cell can be used for the calculation of the area of the interface inside the cell. The proof is based in simple area calculations of triangles (for full 3D cases) and rectangles (for 3D cases that degenerate to 2D cases) and the proof is omitted. The final result is:

where V c is the volume of the cell. Note that for |M | = 1, the interface is parallel to the face of the cell.

Suppose that this is the face f 1, then 2 = a max = δ f 1 is the length of the grid's edge in the direction of n J I and therefore the result of the above relation is,

where f 2 and f 3 are the faces of cell whose normal vectors are orthogonal to the normal vector of the face f 1. It is interesting to note that (D.2.12) is directly derived by the derivative of (D.2.10) as expected by norm of the volume fraction gradient.

Chapter E

Résumé Entendu

E.1 Introduction

La lubrification par injection d'air est un secteur actif de recherche en hydrodynamique. La recherche expérimentale est orientée vers la compréhension des phénoménes physiques en lien avec les effets lubrifiants d'un fluide moins visqueux que l'eau sur des corps immergés dans des petites échelles [START_REF] Vakarelski | Drag reduction by Leidenfrost vapor layers[END_REF] [START_REF] Vakarelski | Drag reduction by Leidenfrost vapor layers[END_REF], comme l'effet Laidenfrost, jusqu'au dans des échelles plus grandes comme l'evolution temporelle des supercavités entourant des véhicules sous-marins [START_REF] Kawakami | Investigation of the Behavior of Ventilated Supercavities[END_REF] [49] et les cavités qui entourent partiellement les coques des bateaux lubrifiées par injection d'air (Makiharju et al., 2013) [START_REF] Mäkiharju | On the scaling of air entrainment from a ventilated partial cavity[END_REF] [START_REF] Zverkhovskyi | Ship drag reduction by air cavities[END_REF] [START_REF] Zverkhovskyi | Ship drag reduction by air cavities[END_REF] par la méthode de la Cavité Partielle. Le concept de cette méthode de réduction de la traînée sur les navires propose la conception d'une ou plusieurs creux sous la coque d'un bateau où l'air est injecté en continu. Le but ultérieur de cette étude est de développer dans le code de calcul ISIS-CFD [86], qui utilise la méthode de volumes finis et une formulation multi-fluides, les outils nécessaires pour modéliser finement les phénomènes dynamiques d'écoulements avec interfaces sur les coques lubrifiées par la méthode de la Cavité Partielle. A cette fin, une gamme plus large de modèles numériques est considérée comme nécessaire.

En principe, un paramètre fondamental de la conception de la cavité partielle est la quantité d'air qui s'échappe de la cavité, puisque cela affecte directment sa stabilité [4] et ainsi sa capacité à lubrifier la coque. L'estimation précise de d'influx d'air est directement reliée à l'estimation de la quantité d'air qui s'échappe de la cavité. Donc la prédiction rigoureuse de la région de la clôture de la cavité joue un rôle décisif. Nous considérons que les effets de tension superficielle pourraient être importants [START_REF] Mäkiharju | On the scaling of air entrainment from a ventilated partial cavity[END_REF], en particulier au niveau de la région de la clôture de la cavité où les courbures changent notablement.

Un autre mécanisme associé avec l'échappement d'air est l'entraînement d'air, particulièrement actif quand les effets de turbulences sont importants. Dans ce travail, nous présentons les efforts de la recherche orientés sur l'introduction dans ISIS-CFD de méthodes de modélisation numérique de la tension superficielle et nous étudions certains aspects de la formulation multiphasique d'un point de vue multi-fluides, pour but final d'introduire la modélisation d'entraînement d'air dans des études de recherches qui pourraient succéder à celle-ci.

E.2 Le cadre de développement "ISIS-CFD extensions"

Ce travail est caracterisé par deux niveux distincts et interdépendants. Premièrement, au niveau de la programmation, nous avons adopté une stratégie pour développer des codes qui suivent les pratiques de la programmation modulaire et orientée objet du Fortran Moderne (un terme que nous avons adoptés par Metcalf, 2011 [69]). Une telle approche était forcée par les considérations contingentes.

Development of Surface Tension Models and Numerical Techniques for Air-Water Interface Dynamics

Résumé

Les méthodes de lubrification par injection d'air sont considérées par la communauté scientifique comme la principale percée technologique à venir pour la réduction de la trainée des navires de commerce. A cette fin, il est impératif de modéliser finement les phénomènes physiques en jeu dans la lubrification par injection d'air, qui intègrent la représentation précise de la tension superficielle, des instabilités d'interface et des écoulements avec entraînement d'air. Au cours de ce travail, nous nous sommes attachés au développement d'outils de programmation, de schémas de reconstruction d'interface et de modélisations de tension superficielle dans le code de calcul ISIS-CFD.

Deux nouvelles méthodes de calcul de la tension superficielle sont présentées. Elles utilisent un schéma de reconstruction globale de l'interface et sont couplées avec les schémas de discrétisation compressifs de la fraction volumique utilisés dans la formulation volumes finis nonstructurés sur laquelle est basé le code ISIS-CFD. Les résultats démontrent que des interactions dynamiques complexes, d'une ou plusieurs interfaces, peuvent être modélisées de façon précise. Ce travail permet d'envisager la réalisation ultérieure de calcul d'écoulements sur des navires lubrifiés par injection d'air, d'améliorer la compréhension physique et de contribuer à la modélisation macroscopique de modèles d'entraînement.

Mots clés hydrodynamique, tension superficielle, simulation numérique, modélisation géométrique, VOF, interface air-eau

Abstract

Air Lubrication methods are regarded by the scientific community as the next major technological breakthrough in Naval Engineering to achieve the reduction of drag in commercial vessels. The accurate modeling of the physical phenomena governing the drag reduction mechanisms of Air Lubrication methods, namely, the dynamics of surface tension, the instabilities of the air-water interfaces and air entrainment, are imperative for the design of air-lubricated hulls. To that end, we have implemented to ISIS-CFD several programming tools, interface reconstruction schemes and surface tension modeling.

Two new surface tension methods were developed. Both use a global interface reconstruction scheme and are coupled with the compressive discretization volume fraction schemes for the unstructured finite volume formulation that the flow solver ISIS-CFD is based on. The results demonstrate that complicated dynamic interactions of either a single or multiple interfaces can be accurately captured. In the context of a future research study, the proposed approaches could lead to the further enhancement of the modeling capabilities of ISIS-CFD by introducing a macroscopic air entrainment model and eventually the assessment of different physical effects encountered in lubricated naval vessels using ISIS-CFD.