The Surface Tension Challenge in Air-Water Interfaces using the Volume-of-Fluid Method

Transcription

1 The Surface Tension Challenge in Air-Water Interfaces using the Volume-of-Fluid Method Konstantinos Politis a,d, Patrick Queutey b,d, Michel Visonneau c,d a. b. c. d. LHEEA/CNRS, Centrale Nantes, France Introduction The steadily increasing computing power, challenges modern CFD codes to take into account phenomena that are important in smaller scales. Specifically in naval engineering, modeling of spray/foam resistance related phenomena or modeling the air-water interface behavior in airassisted methods for drag reduction (air-cavity method, air-lubrication method (Allenstrom & Leer- Andersen, 2010)), could prove of major importance in the pursuit of a more efficient ship. To this end, surface tension(st) is an important phenomenon to be included in CFD simulations. In the literature, one comes across a large variety of different ST modeling approaches that can be directly coupled with a volume-of-fluid(vof) code (see, for example,tryggvason,2011). Although the performance of ST methods is traditionally demonstrated in bubble test cases with exotic fluids, their numerical behavior is altered when an air-water interface is considered. This is due to large differences in the fluid density and viscosity. The purpose of this work is to briefly exhibit the problems encountered in certain ST methods, discuss their strong and weak points in steady and unsteady simulations and propose new alternatives. Two classic ST methods and two methods which enforce the natural definition of surface tension will be considered. The classic algorithms are the continuous surface force (CSF) method and the continuous surface stress (CSS) method. The first alternative algorithm (DCM) is based on the discontinuous field reconstructions proposed by Queutey and Visonneau (Queutey & Visonneau, 2007). Finally, we propose a new algorithm (isoft) which is based on geometrically reconstructing the air-water interface. The above algorithms are part of ISIS-CFD, an unstructured URANS-VOF flow solver, distributed commercially by NUMECA as the flow solver of the package FINE /Marine. Each method is tested in the following test cases: (a) the static bubble test case, for which theoretical to calculated pressure jumps are compared and (b) the rising bubble test case, for which calculated terminal velocities for rising bubbles with radii spanning from R=0.5 mm to R=1 cm are compared to experimental/numerical results found in previous studies. Each method s implementation, advantages/disadvantages and performance for the above cases are discussed in detail. The results demonstrate that, if pressure loses its discontinuous nature,

2 the dynamics of the interface are greatly affected and may result in unrealistic behavior. The two proposed methods retain the natural definition of ST on the interface (two dimensional manifold) and give good predictions in a range of length scales relevant to ST. Future work will be orientated towards evaluating the effects of surface tension in flows encountered in naval engineering, using automatic grid refinement and multiphase macroscale modeling to address small length scales. Classical Surface Tension Methods for VOF Surface tension(st) naturally occurs in the vicinity of an air-water interface. Thus, in order to properly take into account ST when simulating multi-fluid flows, ST must be introduced as a dynamic boundary condition for the interface. In multi-fluid flows (no mass transfer across the interface), ST enforces a jump in the normal component of the fluid stresses at the interface (Panton, 2005): (1) Where, the stress tensor, the ST coefficient(considered constant), the interface s curvature and the interface s unit normal pointing from fluid 1 to 2. From a numerical point of view, an exact application of ST as a boundary condition has been introduced by Tukovic (Tukovic & Jasak, 2012) for a moving mesh method. In VOF, however, the interface is treated as an immersed boundary rather than an actual boundary. Therefore (1) cannot be treated directly as boundary condition. To that end, the following source term is added to the discretized momentum equations of cells containing a part of the interface s area, (Tryggvason, et al., 2011): (2) where is the ST coefficient and is the cell s volume. In the context of VOF methods, that use compressive reconstruction schemes for discretizing the volume fraction evolution equation without any interface reconstruction (as is the case in ISIS-CFD), only a region of cells that the interface is present is known rather that its actual location. Therefore, is not known. An appropriate approximation of (2) is proposed by classic ST methods. The most common approaches used are the continuous surface force method (Brackbill, et al., 1992) and the continuous surface stress method (Lafaurie, et al., 1994): Both methods use directly the volume fraction the geometric quantities required near the interface: [ ( ) ] (3) (or a smoothed volume fraction) to approximate (4) Moreover, both are very easy to implement, however, the natural definition of ST as a vector field defined on the interface s surface is lost. The problems are mainly associated with the poor

3 convergence properties of relation set (4) (Cummings, et al., 2005) and it has been shown (Francois, et al., 2006), that a bad approximation of the curvature results in spurious velocity currents and an inaccurate calculation of the pressure jumps related to surface tension. When the normal vector and curvature are calculated directly by the volume fraction, without kernel smoothing or the height function method, they do not converge as the grid size becomes smaller. Both kernel smoothing and the height function method define an intermediate field whose derivatives give converging approximations of the normal and curvature. To clarify, kernel smoothing generates a smooth volume fraction field, that replaces in relation set (4), using a kernel function. Field is generated by approximating using the midpoint rule the integral: ( ) ( ) ( ) ( ) ( ) (5) Where is the cell s center, is the smoothing length and ( ) is the domain of integration, centered in and defined by the smoothing length. For example if ( ) is a spherically symmetric kernel (e.g. ( ) ), then ( ) is an ε-ball centered at. For details about Kernel smoothing in structured grids we refer to Cummings (2005) and Tryggvason (2011). For unstructured grids, proper smoothing implies the need to conduct neighborhood searches that might result in unacceptable simulation times. Another difficulty occurs near the boundaries. Approaching the boundary, the smoothing length naturally has to be smaller and thus, smoothing becomes more and more absent. The height function method, as the name implies, uses a height field. To generate, a cell stencil is defined locally to the point where the curvature needs to be evaluated. The stencil contains more cells towards the direction that varies more rapidly. For example, if this is the minus y-axis, the stencil contains 3x4x3 grid cells and the height function is (see also figure 1): ( ) (6) There are cases where the definition of (6) is ambiguous, especially when the volume fraction changes rapidly inside the stencil. Cases as such are extensively described by Bornia(2010) and Popinet(2011). j=4 j=3 : Interface : Column i=1,k=2 Δy j= i=1 i=2 i=3 j=2 j=1 (i 3 k 2) : Cell where k is evaluated Figure 1. The stencil of the height function method. Only k=2 is shown

4 The calculation of the normal vector and curvature follows from the local implicit representation of the interface s surface ( ( ) ). Similar definitions can be given for different stencil orientations. The height function method is used mainly with structured grids (Cummings, et al., 2005), (Tryggvason, et al., 2011) or octree-grids generated by structured grids (Popinet, 2009). For general unstructured grids, the generalization of the height function method is not obvious since the alignment of the cells in a structured grid plays an important role. For CSS an explicit curvature calculation is not required and therefore it is not prone to problems related to curvature. Furthermore, since it is formulated as a conservative term, it is not prone to momentum conservation errors as explained by Popinet (2009). Discontinuity Preserving Surface Tension Methods for VOF First of all, we should emphasize that a ST method that uses a surface tension force defined at the interface s manifold, is difficult to be devised using compressive discretization schemes without introducing an interface reconstruction. In the methods discussed in this section, the interface is located by constructing the isosurface, which we consider as a surface close enough to the true interface and plays a key role to both methods described here. The implemented isosurface algorithm generates a surface patch in a cell-wise manner and provides a set of points representing the cell s edges-isosurface intersections. The algorithm is similar to the marching cube algorithm. We shall skip the details and the interested reader is directed to Agoston (2005), where the general marching-cube algorithm and several variations are extensively discussed. The first method is similar to the ghost fluid method(gfm), introduced by Fedwik(2000) to calculate derivatives of discontinuous field for solving PDEs. Quetey and Visonneau (2007), specialized GFM in the context of VOF with compressive discretization schemes. The method introduces a special reconstruction scheme used when calculating the derivatives of discontinuous fields, as is the pressure in our case. To clarify, the normal to the interface components of relation (1) relates both pressure and the strain rate tensor to ST as: 2 (7) Together with the hydrostatic discontinuity (Queutey & Visonneau, 2007): (8) we arrive to two relations that will be taken into account with the discontinuous reconstructions. The formulation will be given for two neighboring cells, in one of which the interface is present, as shown in figure 2. The method proposes to enforce the discontinuities (7),(8) at the faces of cells where the fluid will begin changing for water to air, instead of approximating the ST source term (2). Therefore, (2) in DCM is not used. Although, the pressure discontinuity is not exactly enforced on the interface,

5 this natural approximation allows producing very simple interpolation formulas to obtain the pressure at the face. Since the discontinuity is enforced at certain faces, we refer to this method as the Discrete Contour Method (DCM). Moving from the left cell to the left side of the face ( ) and from the right cell to the right side of the face ( ), the reconstruction formulas read: (9) (10) where, and the density of left and right cell respectively (see figure 2). Both relations are taken into account when calculating the pressure gradient of the right and left cell respectively. Even the slightest smearing of the volume fraction makes it difficult to predict the position of the interface in order to snap the discontinuity properly to the cell s faces. To solve this problem we used the isosurface to generate a sharp background volume fraction,. is defined by., the volume of water in the cell, is found by separating the interface cell into two regions whose common boundary is the cell-local isosurface patch and calculating the volume of the water region. Finally, the curvature is estimated using the same approach followed by the last method, presented next. Water Cell Interface Cell Water Cell Interface Cell L f n f R L f R L, R : left, right cell centers f, n f : face center and unit normal Isosurface local patch Face where p is enforced Figure 2. The discontinuity is snapped to faces with and (or and ) The basic idea behind the last method is to obtain a better approximation of (2) using the isosurface. In the sense that the isosurface represents the interface the method can be best viewed as a VOF method coupled with front tracking method by tracking the isosurface (isoft). The differences between isoft and the classic front tracking (FT) are: 1. In FT the interface is defined by a set of points, in isoft the interface is defined by the volume fraction and the set of points are obtained by constructing the isosurface. 2. In FT the points are advected with the local flow velocity, in isoft the volume fraction is advected with the local flow velocity.

6 3. In FT the volume fraction is constructed by the interface for each time step, in isoft the volume fraction can be periodically reconstructed using the isosurface. Using isoft we are not facing the difficulties that arise in FT when interface merging/separating takes place. At the same time, the good approximation of the normal vector and curvature, achieved by FT is maintained. Of course, the question of how appropriate is the isosurface to represent the interface still lies, especially in the unfortunate case of volume fraction smearing caused by numerical diffusion. However, having a way to keep track of the smearing, replacing by will effectively remove the smeared regions. isoft follows a similar calculation of the ST force, (2), as FT. Approximating (2) by the midpoint rule we get: ( ) ( ) ( ) (11) is the centroid of the cell local isosurface patch. The normal vector and area are easily evaluated by the patch. The curvature is evaluated by a least square fit approach. The least square fit has been referenced widely as a successful approach to evaluate the curvature (Renardy & Renardy, 2002), (Gois, et al., 2008), (Popinet, 2009). The differences lie on the sample used and the type of fit. First of all, we generate a sample of points for each cell by gathering the isosurface s patch points found in and every nearby cell to. To ensure that the fit is well defined, the point set are transfered into a local Cartesian coordinate system constructed with, the unit normal of axis, and arbitrary chosen but mutually orthogonal,. The fit function is ( ) ( ), are the coefficients to be determined and ( ) are the 2D polynomial basis functions truncated to 3 rd order, i.e. the set of functions. If there are not enough points to use 3 rd order polynomials, the basis is truncated even further. The coefficients,, are determined by minimizing the functional: ( ) [ ( )] (12) With the coefficients known the curvature is (indices denote partial derivatives): ( ) ( )( ( ) ) ( )( ( ) ) 2 ( ) ( ) ( ) ( ( ) ( ) ) (13) Since is known at the interface we need to transfer the information to the cell centers. The information was transferred to cells centers for isoft using the same approach as Shin(2005). For DCM, the curvature was used as calculated. We shall skip several other interesting details regarding the parallel programming of the method (i.e. the neighborhood information management strategy for multiblock computations) and continue by presenting our main results.

7 Results and Discussion: Static Bubble We tested the performance of each method in the static(steady) case of a three dimensional bubble of radius 2 on a Cartesian grid generated in a unit cube with 2 3 number of cells per cube s edge. The bubble contained air ( 2, ), the fluid surrounding the bubble was water (, ) and ST coefficient was constant 2. Initial pressure and velocity fields were everywhere zero. The volume fraction was initialized by setting and was generated by the sphere s level set function. The exact solution proposes that in both fluids the velocity should be everywhere zero and the pressure jump is 2. The performance of each method was evaluated by calculating the mean pressure of air and water, the pressure jump relative error defined by ( ), the maximum velocity magnitude (as measure of spurious currents) and an indicator of over(under)shoot ( ) ( ( ) ). Each reflects a quantification of well-known bad arithmetical behavior of ST methods. The results are summarized in tables 1 and 2, for the classic methods and the proposed methods respectively. CSF N e e e e e e e e e e e e e e e e-4 Table 1. Static bubble - Results using CSF and CSS DCM N CSS isoft e e e e e e e e e e e e e e e e e e e e-4 Table 2. Static Bubble - Results using DCM and isoft As expected, CSF produces a divergent pressure jump error. This is due to the divergent curvature calculation. CSS performs better giving constant error. In both cases, an increase in the undershoot is observed as the grid becomes finer, in addition to an increase in the spurious currents. For the discontinuity preserving methods the results are improved. In particular, judging from the pressure jump errors, isoft seems to converge. Although, the spurious currents are a bit smaller than in CSS, we observe the highest undershoots for isoft. The similar trends of the results observed by isoft/csf/css can be related to their similar ST modeling approach. DCM on the other hand outperforms the rest. The pressure jump errors are very small, the spurious currents are two orders of magnitude less than the other methods and undershoots are negligible. This supports the major

8 hypothesis of DCM, i.e. enforcing the ST discontinuity on properly chosen cell s faces will have a similar effect as enforcing it to the interface. Conclusively, the results indicate the improved performance of discontinuity preserving methods and that the ST modeling approach suggested by DCM should be the main reason behind the method s success. Results and Discussion: Rising Bubble Simulations The purpose of this section is to evaluate the numerical behavior of each method in relation to the interface evolution in unsteady simulations and their ability to predict the terminal velocity of a bubble. The simulations were performed on grids of about 200k up to 400k cells using 4 processors, each covering a rectangular region whose edges lengths were up to 2, up to. Inside each region the grid was initially refined inside a rectangular region centered on the initial region, with up to, up to 2. The grids were generated using Hexpress, the grid generator of FINE /Marine. The bubble s center was initially located at 3. The volume fraction was initialized as in the static cases. In every simulation we tried to maintain a small maximum Courant number about 0.2 (based on expected terminal experimental velocities given by Clift, 2005), to ensure that the compressive scheme used did not introduce numerical errors. In most cases, during the initial stages of the simulation, the Courant number reached a maximum of As in the steady cases, no symmetry assumption for the bubble s shape was made. The terminal velocity was calculated by numerically evaluating the temporal derivatives of the bubble s centroid. Results obtained by each method, experimental values for purified water and values computed by other researchers are summarized in table 3. Note that Clift s book is a 2005 reprint of his original work originally published in A short discussion of table 3, follows. ISIS-CFD Experimental Computational R CSF CSS DCM isoft Clift (2005) Duineveld (1995) Tomiyama (2002) Talia (2007) Tukovic (2012) Hua (2007) Lorstad (2004) N/A N/A N/A N/A N/A N/A N/A R= N/A N/A N/A 0.31 N/A 0.32 N/A Table 3. Terminal velocities in m/s (multiple sources) for the radii studied (R in mm) Tukovic (2012), used a moving mesh method (which is quite different from our approach) and calculated the terminal velocity based on the velocity on the bubble s centroid (personal communication), while Lorstad (2004), used compressive discretization schemes with an interface reconstruction similar to PLIC, CSF for surface tension and calculated the terminal velocity as the mean air velocity. Both Tukovic and Lorstad, compared their results with experimental values obtained by Duineveld(1995) and found very good agreement with computed values. Hua(2007),

9 calculated the terminal velocity as the mean velocity of the bubbles centroid over a period of time and used a classic FT approach. Hua compared his result with experimental values by Tomiyama (2002). Experimental results vary widely, mainly, due to the way each researcher generated the bubble and the contamination of water by surfacants (Clift, et al., 2005), (Wu & Gharib, 2002), (Tomiyama, et al., 2002), (Talaia, 2007). Tomiyama (2002), studied the effect of the bubble generation method giving a range of calculated terminal velocities which are based on the initial bubble deformations that the generation method produced. Smaller velocity values, given in table 3 for Tomiyama s results, refer to terminal velocities of bubbles produced with smaller initial deformation and the higher values with large deformations. Wu (2002), presented result with similar lower limits as Tomiyama but upper limits reaching the values of Duineveld. We chose the bubble radii to fall inside the spherical regime ( ), the ellipsoidal regime ( and ) and the spherical cap regime ( ). Each regime is named after the interface s shape. In general, for smaller bubbles we expect that surface tension dominates the interface evolution and in larger bubbles buoyancy dominates the interface evolution (Clift, et al., 2005). Our analysis begins with some general remarks about the calculations. The terminal velocities calculated by ISIS-CFD, using each method, follow specific trends based on the method used. With CSF we obtained a constantly decreasing terminal velocity until where the simulation gave an unphysical break-up of the interface. Using CSS, smearing was excessively present and did not allow a proper computation for spherical and ellipsoidal regime cases. Similar to CSF, DCM gave a decreasing terminal velocity as the bubble s radius increased. Finally, isoft gave the smallest values of the rest of the methods. When compared to experimental results, CSF, DCM and isoft, all predict terminal velocities within the Tomiyama ranges for our sample of radii in the ellipsoidal regime. CSF predicted values closer to values obtained by Clift. For the spherical regime, CSF and DCM predictions fell closer to Clifts s values and isoft to Talaia s values. To obtain a more precise estimation of the computations quality, the results were compared with an analytical relation for the terminal velocity given by Tomiyama(2002). The difference between the values obtained and the experimental values can be explained by the initial deformations of the bubble s interface that each method produced. Since the curvature estimations and the resulting pressure fields inside air are different for each method (as the static case supports), the initial evolution of the interface changes. In turn the final result is affected. The interface s shape differences and calculated pressures differences were pronounced for the smallest radius. In figure 3, the volume fraction fields are plotted on a yz-slice of the computational domain, during the initial stages of the interface evolution for. It is evident that DCM and CSF allow larger initial bubble deformations. The different initial deformations result into different bubble shapes in later time steps (figure 4). The effect of the curvature calculation is reflected on the

10 pressure jumps. In figure 3, the pressure fields are also plotted (zero pressure level defined to be at a xy-plane passing through the bubble s centroid). The expected value inside the bubble is 2. CSF underestimates the pressure jump as in the static case while for the other methods the results are closer to the expected value. Figure 3. Volume fraction and pressure slices at t= s, R=0.6mm. Left to right: DCM, CSS, isoft, CSF Figure 4. Bird s eye view of isosurfaces obtained for DCM and isoft (t=0.023, R=0.6mm) Tomiyama proposed the following analytical relation for calculating the terminal velocity inside the spherical/ellipsoidal regime using the bubble s aspect ratio (defined as the ratio of the maximum distance of trailing to leading interface part, projected on the axis of symmetry, to the maximum diameter normal to the axis) and initial (or equivalent)diameter : ( ) 2 Assuming that the area defined by half the total projected area of the isosurface patches on the xy- plane corresponds to the area of a disk, the denominator of (14) is easily calculated. Assuming the same

11 for the distance of the numerator of (using both xz and yz planes), results to the formula: 2 Like terminal velocities, each method provided different aspect ratios (table 4, column (15) ) for the same bubble radius. isoft produced bubble interfaces which are closer to spherical, CSF disk-like interfaces and DCM elliptical interfaces. With the aspect ratio known, the terminal velocity can be calculated by relation 14. The results are given in table 4 (column ). The terminal velocity relative errors (based on computed terminal velocities) are also given in table 4. The results show that, in comparison with relation (14), every method provided a terminal velocity approximation close to the Tomiyama terminal velocity for the given. However, note that the experimental aspect ratios (also given in table 4) suggest that CSF results might be unrealistic, since in every case the calculated aspect ratio is well below the experimental values. R CSF DCM isoft Experimental E Clift (2005) Duiniveld (1995) N/A N/A Table 4. Aspect ratios, terminal velocities (14), terminal velocity errors per method and experimental values Comparison of the bubble velocity and mean air velocities over time, for each method, also provides some insight on the results obtained. In figure 5, we compare the time evolution of the mean air velocity and bubble s centroid velocity for CSF, DCM, isoft and bubble s centroid velocity obtained by Tukovich for. Comparing the centroid velocity curves (Vt) of DCM and isoft with Tukovic s curve, we notice a certain agreement up to. Afterwards the curves begin to split with DCM and isoft curves converging quickly. This behavior can be attributed to the grid size. Probably using a finer grid the response of the proposed methods could be improved. Another important observation has to do with the centroid velocities and mean air velocities that, as expected, they are not the same. However, the mean air velocity is very close to the terminal velocity of the bubble. This observation is also supported by Lorstad, whose computation of mean air velocity is in agreement with our estimation. Although we do not have any evidence supporting that the mean air velocity is the terminal velocity, we believe it is a matter of chance. Turning our attention to the spherical cup regime, for CSF the interface collapsed as the approaching its spherical cap shape giving a toroidal bubble. We couldn t find any experimental result producing a toroidal bubble in air-water systems of so we consider the result nonrealistic. CSS gave excessive smearing near the cap s edges producing an elongated toroidal part

12 4,00E-01 3,50E-01 V (m/s) 3,00E-01 CSF - Vair_mean 2,50E-01 DCM - Vair_mean 2,00E-01 isoft - Vair_mean 1,50E-01 CSF - Vt 1,00E-01 DCM - Vt isoft - Vt 5,00E-02 Tukovic - Vt 0,00E+00 0,00E+00 2,00E-02 4,00E-02 6,00E-02 8,00E-02 1,00E-01 1,20E-01 t (s) Figure 5. Comparison of terminal velocities and mean air velocity for R=1 mm of the interface. This structure is known as skirt. Skirted interfaces normally are not maintained in air-water systems but they form smaller bubbles as they collapse (Landel, et al., 2008), so we consider the result nonrealistic. DCM and isoft successfully provided a spherical cap. To compare the interface s shapes, in figure 6 a yz-slice of and is plotted for DCM, isoft and CSS respectively. Relative to the bubble velocity fields are also plotted along plots. The bubble skirts are clearly noticeable in CSS and are not removed by. For isoft, the smearing on the interface edge s is contained in certain areas in the flow and it is removed by. Thus, we do not consider it a skirted interface. Velocities of isoft also provide nice representation of the expected vortices, i.e. a thin elongated vortex near the trailing surface and a larger vortex near the leading surface. For DCM an interesting phenomenon occurred during the simulation. An attached to the bubble toroidal structure of air formed behind the bubble that resembled four lobes (figure 7). The lobes broke up, each forming a bubble (satellite bubbles). The same mechanism of satellite bubble formation is described by Landel (2008). A comparison to Landel s spherical cap forming smaller bubbles and the result we obtained is given in figure 8. The photograph corresponds to 2. Qualitatively, the result is in accordance with experimental observations, for the capturing capabilities of the grid used. To sum up, we presented preliminary results of four unsteady rising bubble computations using classical ST and methods that use a discontinuity preserving approach. Each method gave different results, each following a certain pattern. The differences obtained with the discontinuity preserving methods are difficult to be attributed to a specific reason without a systematic analysis of the same problem resolved with finer grids and smaller time steps. A space-time convergence analysis for the same test cases, would provide insight on each method behavior and the cause of the differences obtained. From the other hand, both experimental results and computational results presented by other researchers vary widely. One could say, that he could always find a set of

13 experimental results to match his computational results. In order to understand the computational trends and the prediction capability of each method we should perform computations in a wider range of bubble radii to draw a sound conclusion. Figure 6. and comparison for the spherical cup case. Left to right: DCM, isoft, CSS Figure 7. Bubble lobes formation. Note the very small connections between the lobes and the main bubble Figure 8. The bubble breaks up to form smaller satellites bubbles. Photograph by Landel(2008)

14 Concluding Remarks Surface tension is a challenging phenomenon for many reasons. First of all, from a modeling perspective, surface tension is defined on a Lagrangian manifold whose evolution must be taken into account through Eulerian modeling. From a practical perspective, this leads to the need of small time steps. Moreover, surface tension couples interface dynamics with the momentum equation through the interface s curvature resulting in a pressure discontinuity that naturally affects the surrounding flow. If we don t take into account the discontinuous nature pressure we might not be able to produce realistic results in some surface tension dominated simulations. A more accurate approach requires a surface representation of the interface. Even though not computationally tome consuming, this new entity introduced is not intrinsic to the Navier-Stokes discretization with finite volumes. In our implementation, the entity introduced is the volume fraction isosurface. The proposed methods have certain similarities with previously presented methods, yet, they are different in many respects. Besides their promising results for the test cases studied, their most interesting feature is that they provide results in the whole range of geometric regimes of rising bubbles in air-water interfaces. The ongoing work is directed on improving and extending the results for more test cases, with our main goal being to improve the interface dynamics. The proposed methods will also be used to understand the effects of surface tension in ship generated water waves, on air-water cavities underneath the ship s hull and ventilation. Specifically for ventilation, bubble dynamics is considered of major importance. Even though surface tension is not dominating the flow in larger scales our capability of addressing small scales through automatic grid refinement and by adding a macroscale multiphase flow model might provide new insight on the underlying dynamic phenomena.

15 Bibliography Agoston, M., Computer Graphics and Geometrical Modeling. London: Springer. Allenstrom, B. & Leer-Andersen, M., Model Test with Air-Lubrication. Instanbul, SMOOTH-Ships. Aulisa, E., Manservisi, S., Scardovelli, R. & Zaleski, S., Interface reconstruction with least-squares fit and split advection in three-dimensional Cartesian geometry. Journal of Computational Physics, 225(2), pp Bornia, G. et al., On the properties and limitations of height function method in two-dimensional Cartesian geometries. Journal of Computational Physics, 230(4), pp Brackbill, J., Kothe, B. & Zemach, C., A continuum Method for Surface Tension. Journal of Computational Physics, 100(2), pp Clift, R., J, G. & Weber, M., Bubbles, Drops and Particles. (reprint 1978) 1st ed. New York: Dover. Cummings, S., Francois, M. & Kothe, D., Estimating curvature from volume fractions. Computers and Structures, 83(6), pp Duineveld, P., The rise velocity and shape of bubbles in pure water at high Reynolds number. Journal of Fluid Mechanics, Issue 32, pp Fedwik, A boundary condition capturing method for multiphase incompressible flow. Journal of Scientific Computing, Issue 15, pp Francois, M. et al., A balanced force algorithm for continuous and sharp interfacial surface tension models withing a volume tracking framework. Journal of Computational Physics, 213(1), pp Gois, J., Nakano, A., Notano, L. & Buscaglia, G., Front tracking with moving least squares. Journal of Computational Psysics, Volume 227, pp Hua, J., Stene, J. & Lin, P., Numerical Simulation of 3D bubbles rising in viscous liquids using a front tracking method. Journal of Computational Physics, Issue 227, pp Kenwal, P., Generalized Functions: Theory and Technique. 1st ed. New York: Academic Press. Lafaurie, B. et al., Modelling Merging and Fragmentation in Multiphase Flows with SURFER. Journal of Computational Physics, 113(1), pp Landel, J., Cossu, C. & Caulfield, C., Spherical cup bubbles with a toroidal bubbly wake. Physics of Fluids, Issue 20. Panton, R. L., Incompressible Flow. New Jersey: Wiley. Popinet, S., An accurate adaptive solver for surface-tension-driven interfacial flows. Journal of Computational Physics, 228(16), pp Prosperetti, A. & Tryggvason, G. eds., Computational Methods for Multiphase Flows. s.l.:cambridge University Press. Queutey, P. & Visonneau, M., An interface capturing method for free-surface hydrodynamic flows. Computers and Fluids, 36(9), pp Renardy, Y. & Renardy, M., PROST: A Parabolic Reconstruction of Surface Tension for the Volume-Of-fluid Method. Journal of Computational Physics, 183(1), pp Scardovelli, R. & Zaleski, S., Direct Numerical Simulation of Free-Surface and Interfacial Flow. Annual Review of Fluid Mechanics, 31(1), pp Shin, S., Computation of the curvature field in numerical simulation of multiphase flow. Journal of Computational Physics, 222(2), pp Talaia, M., Terminal Velocity of a Bubble Rise in Liquid Column. World Academy of Science, Engineering and Technology, Issue 28, pp Tomiyama, A., Celata, G., Hosokawa, S. & Yoshida, S., Terminal Velocities of Single Bubbles in Surface Tension Force Dominant Regime. International Journal of Multiphase Flow, Issue 28, pp Tryggvason, G., Scardovelli, R. & Zaleki, S., Direct Numerical Simulations of Gas-Liquid Multiphase flows. s.l.:cambridge University Press. Tukovic, Z. & Jasak, H., A moving mesh finite volume interface tracking method for surface tension dominated interfacial fluid flows. Computers and Fluids, Volume 55, pp Unverdi, S. & Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows. Journal of Computational Physics, 100(1), pp Wu, M. & Gharib, M., Experimental Studies on the shape and path of small air bubbles rising in clean water. Physics of Fluids, 14(7).