Journal of Computational Physics 231 (2012) 3887–3895

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Journal of Computational Physics

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Short note

A new contact line treatment for a conservative level set method

Yohei Sato ⇑, Bojan NicenoPaul Scherrer Institute, CH-5232, Villigen PSI, Switzerland

a r t i c l e i n f o

Article history:Received 23 March 2011Received in revised form 20 January 2012Accepted 30 January 2012Available online 10 February 2012

Keywords:Conservative level setContact lineContact angleTwo-phase flow

0021-9991/$ - see front matter � 2012 Elsevier Incdoi:10.1016/j.jcp.2012.01.034

⇑ Corresponding author. Tel.: +41 (0)56 310 26 66E-mail address: yohei.sato@psi.ch (Y. Sato).

a b s t r a c t

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

A new contact line treatment has been developed for the three-dimensional conservative level set (CLS) method [1] with-in a Computational Fluid Dynamics (CFD) code. A contact line treatment for CLS is proposed in [2], however it is limited totwo-dimensional computations and requires two additional parameters. The contact line treatment proposed in this papercan be applied to three-dimensional computations without such additional parameters. In the proposed method, the walladhesion force of the Continuum Surface Force (CSF) model [3] works properly and many contact angle problems can be sim-ulated. The proposed method is verified by the comparison with theoretical solutions pertaining to droplet on a wall, andvalidated using experimental data of free surface deformation in a rectangular-shaped container. In all cases, mass conser-vation is strictly satisfied.

Unlike the level set (LS) method developed by Sussman et al. [4], CLS uses the color function to describe the volume frac-tion instead of the signed distance function. CLS satisfies the mass conservation equation exactly provided the equation forthe color function is discretized conservatively. The equations to be solved in CLS are (i) the advection equation of the colorfunction, and (ii) an interface sharpening algorithm for the color function. The latter is introduced in order to prevent smear-ing of the color function distribution.

The advantage of CLS is its simplicity of implementation compared to the other conservative free-surface models, such asthe volume of fluid (VOF) method [5], and the Coupled LS and VOF (CLSVOF) method [6,7]. In VOF, surface reconstruction isrequired in order to prevent numerical smearing of VOF function. The reconstruction process, such as Piecewise Linear Inter-face Calculation (PLIC) [8], requires complex geometric calculations to be performed, and the implementation is not straight-forward, particularly in the three-dimensional case. CLSVOF takes advantages of the separate benefits of LS and VOF, in thatmass conservation is satisfied using VOF and the computation of geometric quantities, such as the normal vector, can be cal-culated directly from the LS function. However, implementation of the CLSVOF method inherits the complexities of LS andVOF in addition to the coupling of the two methods.

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3888 Y. Sato, B. Niceno / Journal of Computational Physics 231 (2012) 3887–3895

Owing to its conservative feature and simplicity, CLS is widely used, and, with several improvements, has been applied tothe dam breaking problem, the rising bubble problem [9], and the turbulent atomization flow problem [10,11]. The maindrawback of CLS is lack of a contact line treatment for the interface sharpening equation. Though an algorithm has been pro-posed by Zahedi et al. [2], it is limited to two-dimensional computations only, and the extension to three dimensions is notpresented. In addition to these limitations, the method requires two additional numerical parameters. In order to obtain real-istic values of the two parameters, numerical experiments are required, and unfortunately the values are test-case depen-dent [2].

On the other hand, for other conservative methods such as VOF and CLSVOF, contact line models have been intensivelydeveloped. Bussmann et al. [12,13] first installed CSF [3] into a three-dimensional VOF code employing a piecewise-linearvolume-tracking algorithm and used the model to compute drop impact phenomena. Recently, a contact line model usingthe height function has been developed for VOF by Afkhami and Bussmann [14], enabling both static and dynamic contactline phenomena to be simulated. Meanwhile, a contact angle model for CLSVOF was developed by Sussman [15,16] incorpo-rating an embedded boundary approach [17] in which a solid-body shape can be represented within a Cartesian grid. Usingthe contact angle model, contact line dynamics problems can be handled; for example, free surface deformation due to a walladhesion force in zero gravity [18], free surface flow around a solid body [19], and drop impact phenomena [20]. In [20], theframework of CLSVOF is used, but the THINC/WLIC scheme is used as a type of VOF method. THINC/WLIC is simpler than VOFin regards to the implementation, but requires an iterative calculation between the LS function and the VOF function for thecoupling.

Recently, the phase field model, which was originally developed for solidification dynamics has been applied to the con-tact line treatment for two-phase flow [21–24]. In this model, the wetting phenomena are modeled as a free interface prob-lem in which the interface between the two phases is free to change shape so as to minimize the surface energy. Jacqmin [21]has developed a phase field model which is coupled with the Navier–Stokes equations, and made two-dimensional numer-ical investigations of the moving contact line dynamics. Villanueva et al. [22] validated their phase field model by compar-ison with experiments, but the model does not conserve mass. A numerical study of a three-dimensional droplet on a solidsubstrate in shear flow has been conducted by Ding and Spelt [23] using a geometric instead of the surface energy. The meth-od conservatives mass, but no validation has yet been performed. Lee and Kim [24] develop a high-accurate and robust con-tact angle model for the phase field method, and demonstrate its use on two-dimensional problems.

As mentioned above, in terms of contact line treatment development, CLS is less advanced than either VOF or CLSVOF.However, it is still worth developing a contact line treatment for CLS due to its simplicity. In this paper, we propose sucha treatment, and apply it to several three-dimensional, two-phase flow simulations. The interface sharpening equation ismodified from the original CLS [1], but no additional parameters need be introduced. The implementation of the schemein a standard CFD code is simple and straightforward.

The outline of this paper is as follows: In Section 2, the new contact line treatment is presented, together with a briefintroduction to CLS and the Navier–Stokes solver. In Section 3, several sample calculations are given, with results comparedwith theoretical and experimental data. Finally, conclusions are drawn in Section 4.

2. Numerical method

2.1. Conservative level set method

The color function / is introduced as the volume fraction of liquid inside a control volume. The averaged density of thecontrol volume is defined as:

q ¼ /ql þ ð1� /Þqg ; ð1Þ

where q is the density and the subscripts l and g denote the liquid and gas phases, respectively. The governing equation forthe color function can be written as:

@/@tþr � ð/~uÞ ¼ 0; ð2Þ

where t is the physical time, and ~u the flow velocity vector.The first step of the CLS method [1] is the solution of Eq. (2). A high-order advection scheme should be used to reduce the

numerical error. In this paper, a second-order scheme with a flux limiter [25] is used for the advection term. The second stepin the procedure is the solution of the equation for the interface sharpening, which will keep the interface thickness constantwithout deforming its shape. The governing equation is as follows [1]:

@/@sþr � ð/ð1� /Þ~nÞ ¼ r � ðer/Þ; ð3Þ

where s is the pseudo time,~n the normal vector of the interface between liquid and gas, and e an artificial viscosity. The sec-ond term on the left side of the equation is called the compression term, which forces the color function to be compressedonto the interface along with the normal vector ~n. The term on the right side of Eq. (3) is called the artificial diffusion term,and controls the thickness of the interface via the parameter e. By solving this equation, the interface thickness can be kept

Y. Sato, B. Niceno / Journal of Computational Physics 231 (2012) 3887–3895 3889

constant. The parameter e is set at Dx/2 or Dx0.9/2 in [1], where Dx is the grid spacing. In this paper, we set e at Dx/2. Eq. (3) issolved iteratively until @/

@s ’ 0 to a specified tolerance.At this point, the main advantage of the CLS approach becomes obvious: namely, the simplicity of its numerics. With CLS,

only two equations, Eqs. (2) and (3), have to be solved.

2.2. Contact line treatment for CLS

The discretized form of Eq. (3), using the finite volume method, is written as:

V@/@sþXðð/ð1� /Þ~nÞ �~SÞ ¼

Xeðr/Þ �~S; ð4Þ

where V is the cell volume, and~S is the area vector of the cell faces. In our discretization, the color function and the normalvector are defined at the cell centers.

In order to achieve the conservative condition of Eq. (4), a special treatment is required in which the fluxes of the com-pression term and the artificial diffusion term are set to zero at a wall boundary. These conditions are respectively written as:

ð/ð1� /Þ~nÞ �~Swall ¼ 0; ðr/Þ �~Swall ¼ 0; ð5Þ

where ~Swall is the area vector of the wall.The conditions are enforced as follows. One layer of cells adjacent to the wall is separated from the others. Hereafter, this

layer is referred to as the wall-layer. A schematic of the wall-layer is depicted in Fig. 1(a). In the wall-layer, the interfacesharpening algorithm is different from that used elsewhere in that the color function is sharpened only in the direction tan-gential to the wall; i.e. it is not sharpened in the normal direction of the wall. Thus, in place of Eq. (4), the governing equationfor the wall-layer is written as:

V@/@sþXðð/ð1� /Þð~nÞkÞ �~SÞ ¼

Xððeðr/ÞkÞ �~SÞ; ð6Þ

where the subscript k denotes the tangential component. For example, the tangential component of the vector ~a is definedas:

ð~aÞk ¼~a� ð~a �~nwÞ~nw; ð7Þ

where ~nw is the normal vector of the wall.On the wall boundary, the fluxes of the artificial compression term and the artificial diffusion term in Eq. (6) are respec-

tively written as:

ð/ð1� /Þð~nÞkÞ �~Swall ¼ 0; ðeðr/ÞkÞ �~Swall ¼ 0; ð8Þ

because ð~nÞk �~Swall ¼ ðr/Þk �~Swall ¼ 0. Then it is clear that Eqs. (8) satisfy the wall boundary conditions given in Eq. (5).In the case of a Cartesian grid system, the fluxes of the artificial diffusion and the compression terms depicted as in

Fig. 1(b) are forced to be zero, resulting from Eq. (8). Since the fluxes between the wall-layer and other parts of the flow do-main are set to zero, the interface sharpening algorithm does not function between them. As a result, the color function ofthe wall-layer smears out in the normal direction of the wall. However, the range of the smearing out is limited only to theone layer of cells. Further discussion on this issue is given in Section 3.2 in the context of a sample computation.

2.3. Incompressible Navier–Stokes equations

Based on the color function, a single set of governing equations can be used for a two-phase flow consisting of the con-tinuity equation and the Navier–Stokes equations for incompressible flow, defined as:

Wall boundary

: Flux calculated normally

: Flux enforced to be zero

: Wall-layer cells

Wall boundary

: Other cells

(a) (b)

Fig. 1. (a) Schematic of the wall-layer; and (b) wall-boundary treatment for the flux calculation.

3890 Y. Sato, B. Niceno / Journal of Computational Physics 231 (2012) 3887–3895

r �~u ¼ 0;@ðq~uÞ@tþr � ðq~u~uÞ ¼ �rpþr � flðr~uþ ðr~uÞTÞ þ q~g þ~f sv ; ð9Þ

where ~u is the flow velocity vector, t the physical time, p the pressure, ~g the gravitational acceleration vector, and~f sv thesurface tension force; q and l are the density and the dynamic viscosity, respectively. The viscosity is defined using linearinterpolation in the same way as the density, as given by Eq. (1).

The CSF model proposed by Brackbill [3] is used for the surface tension and the wall adhesion force. The curvature, whichis used for the CSF model, is calculated from the signed distance function instead of the color function, since the curvaturecalculated from the color function tends to include more numerical error. The signed distance function is calculated from thecolor function by using the re-distance scheme [26].

The Navier–Stokes equations are discretized with a semi-implicit projection method in time [27]. Diffusion terms arediscretized in time using the Crank–Nicolson scheme, and the advection terms with the Adams–Bashforth scheme. Forthe spatial discretization, the orthogonal finite volume method is used in a staggered variable arrangement. The second-order-accurate, central-difference scheme is used for the diffusion term and the second-order scheme with flux limiter[25] for the advection term.

3. Simulations

Four simulations have been carried out. The first is a verification test for a three-dimensional droplet on a wall underzero-gravity conditions. The second simulation is a re-computation of the two-dimensional droplet on a flat plate case per-formed by Zahedi [2], which is used in order to compare the methods. The third simulation features a bubble detaching froma wall due to buoyancy. This case is considered to elucidate to discuss the smearing issue described in Section 2.2. Finally, thefourth simulation is a validation case of free-surface deformation in a rectangular container. Note that mass conservation isstrictly satisfied in all cases.

3.1. Three-dimensional droplet on a wall under zero-gravity conditions

The simulation is a verification test. The computed results are compared with an exact analytical solution, and a grid-refinement study is also carried out.

A semi-spherical droplet is located in the center of a flat wall as the initial condition. A zero-gravity condition is imposed,since in this case the theoretical solution can be readily obtained. The droplet starts to deform due to surface tension andwall adhesion force. The deformation continues until these forces are balanced. The equilibrium droplet shape is theoreti-cally obtained as:

R ¼ R02

ð1� cos hÞð1þ sin2 h� cos hÞ

!13

; L ¼ 2R sin h; e ¼ Rð1� cos hÞ; ð10Þ

where R0 and R are the radius of the sphere at initial and equilibrium conditions, respectively, h is the static contact angle,and L and e are the droplet shape parameters depicted in Fig. 2. The pressure difference between inside and outside of thedroplet is Dp = 2r/R.

The initial radius, R0, is set at 0.01 m. For simplicity, the same physical properties are used for liquid and gas; the densityq = 1.0 kg/m3, the viscosity l = 0.001 Pa s, and the surface tension coefficient r = 0.01 N/m. Five cases with different contactangles, ranging from 30� to 150� at every 30�, have been computed. The computational domain has dimensions 6R0

� 6R0 � 3R0 in the two lateral and the vertical directions, respectively. The computational grid is uniform, with a grid spacingof 3R0/32 in all directions. This gives a total of 64 � 64 � 32 = 131,072 cells for the problem.

The computed droplet shapes at the equilibrium condition are shown in Fig. 3. The top view of the droplet shape is de-picted in the top row, the side view in the middle, and the center plane at the bottom. The dashed line is the theoretical solu-tion of the droplet shape. In the figure of the top view, it can be noticed that the droplet shapes have remained circular.Comparing the computed results and theoretical solution, a discrepancy is observed in the case of h = 30�: the computeddiameter of the droplet is smaller than the theoretical value in the top view. The reason for this discrepancy is the cell-cen-tered variable arrangement. As shown in Fig. 4, the droplet edge, whose thickness is smaller than half of the cell size, cannot

θ

L

e

0R

R

Fig. 2. Initial semi-spherical droplet (left) and the definitions of the droplet shape parameters for an equilibrium droplet (right).

Fig. 3. Equilibrium droplet shapes for different contact angles. Top view of the droplet (top), side view (medium) and side view of the center plane (bottom).Red dashed line (top and bottom) is the theoretical solution, and black lines (bottom) are the contour lines at / = 0.1, 0.5, and 0.9.

Liquid

Gas

Analytical solution

Iso-surface of

φ =0.1

φ =0.5

Wall

Fig. 4. Iso-surface of the cell-centered variable / based on the finite-volume method.

0.9 0.5 0.1

Liquid

Gas

Error

φ=

Wall

Fig. 5. Magnified view of the center plane for contact angle h = 150� in Fig. 3.

θ (deg.)

L/R0,e/R0

ΔpR0/σ

0 30 60 90 120 150 1800

1

2

3

4

5

0

1

2

3

L/R0, comp.L/R0, theor.e/R0, comp.e/R0, theor.Δp R0/σ, comp.Δp R0/σ, theor.

Fig. 6. Comparisons of the droplet shape parameters and pressure between the computed results (comp.) and the theoretical solutions (theor.).

Y. Sato, B. Niceno / Journal of Computational Physics 231 (2012) 3887–3895 3891

be resolved by the finite volume method. Thus, the discrepancy for h = 30� looks larger than other cases for which contactangle is closer to 90�. In the case of h = 150�, the same effect occurs for the thin gas film, which can be observed in the mag-nified view of the center plane as shown in Fig. 5. The center-plane profiles given in the bottom row of Fig. 3 show that theinterface thickness has stayed constant.

Fig. 6 gives a comparison between the computed results and the theoretical solutions of the droplet shape parameters andthe pressure difference as functions of the contact angle. The pressure difference is computed from the mean pressure of the

hi/h1q c

omp./qtheor.

1 2 3 4 50.95

1

1.05

1.1

1.15

1.2

1.25

LeΔp

Fig. 7. Grid-refinement study for the droplet simulation; qcomp. is the computed value, and qtheor. the theoretical solution.

Time(s)

M/M

0

0 1 2 3 40.9999

0.99995

1

1.00005

1.0001

Fig. 8. Mass conservation of the droplet.

3892 Y. Sato, B. Niceno / Journal of Computational Physics 231 (2012) 3887–3895

droplet, computed as the average pressure over the cells with color function / P 0.99. The computed results for L, e and Dpagree well with the theoretical solutions, with some discrepancies for the smallest and largest contact angles, h = 30� and150�.

In order to (i) investigate these discrepancies, and (ii) verify the model, a grid-refinement study has been conducted forthe case with h = 150�. The grid spacings used for the study were 3R0/32, 3R0/64, 3R0/96 and 3R0/128. Results are shown inFig. 7. Here, hi is the grid spacing of grid i, with i = 1 representing the finest grid. The ratios of the computed result to thetheoretical solution, qcomp./qtheor., monotonically converge to 1.0 as the grid spacing decreases. The discrepancy between Lcomp.

and Ltheor. is larger than for other cases, as explained above for Figs. 4 and 5.Mass conservation is satisfied exactly for all the cases. Fig. 8 shows the mass ration M/M0 for the case h = 150� and

hi = 3R0/32 as a function of time. M is defined as M ¼PN

j¼1qjV j, where q is the density of the cell defined in Eq. (1), V the cellvolume, and N the number of cells in the computational domain. M0 is the value of M at the initial condition.

3.2. Two-dimensional droplet on a wall under zero-gravity conditions

The simulation of a two-dimensional droplet on a flat plate, which was presented in Zahedi’s study [2], is re-computedhere in order to compare the accuracy of the present method for estimating the wetting velocity. The computed results arecompared against those of Zahedi, predictions obtained using a phase field approach [22], and an experiment conducted byStröm et al. [28]. In the simulation, the droplet deforms due to the action of the wall adhesion force. In contrast, in Ström’s

Fig. 9. Comparison of wetting of a liquid droplet on a solid surface between the present method and that of Zahedi [2]. For the present method, threecontour lines at / = 0.1, 0.5, and 0.9 are drawn with blue, black, and red lines, respectively. (For interpretation of references to color in this figure legend, thereader is referred to the web version of this article.)

Present method (coarse grid)

Phase field

Zahedi

Experiment

Present method (medium grid)

Present method (fine grid)

Capillary number

Contactangle(deg.)

10 10-3 10 1020

40

60

80

100

120

140

160

-4 -2 -1

Fig. 10. Comparison of contact angle versus capillary number between the experiment [28], Zahedi’s model [2], the phase field model [22], and the presentmethod.

Y. Sato, B. Niceno / Journal of Computational Physics 231 (2012) 3887–3895 3893

experiments, flat plates were inserted in a liquid at constant velocity, and the contact angle measured with a camera. Thus,the conditions of the simulation and the experiment are not exactly the same. However, Ström’s experiment may be consid-ered pseudo two-dimensional, and is a closer representation of the simulation than an experiment using capillary tube, suchas that of Hoffmann [29].

The static contact angle is set at 25� in simulation, while that of the experiment using Silicone oil III and polytetrafluo-roethylene [28] was measured at 19�, with an accuracy of ±2�.

The conditions of the simulation in regards to computational domain, boundary conditions, initial conditions and physicalproperties are the same as in [2], except for the number of grid points. In the present study, the computational grid in the xand y directions is uniform, and three grid refinements are used: a coarse grid with 128 � 64 cells, a medium grid with192 � 96 cells, and a fine grid with 256 � 128 cells.

In Fig. 9, the computed droplet shape and the apparent contact angle of the fine grid at different times are compared withthose of Zahedi [2]. The values of the contact angle are similar for the two methods, however it can be noticed that the drop-let shapes are obviously different, especially at time 0.1 s. The contact angle of Zahedi at this time is apparently smaller than90�, and the value 149� printed in the figure is an apparent contact angle obtained by extrapolation of the low-curvature partof the droplet surface shape.

Computed results of the contact angle as a function of the capillary number are shown in Fig. 10. Note that the capillarynumber is equivalent to the wetting speed in this condition. The contact angle computed from the present method increasesas the number of cells increase, and the result with the fine grid agrees well with that obtained using the phase field method.Compared to Zahedi’s results, predictions here are close to the experiment.

3.3. Bubble detached from a wall

In order to further discuss the smearing issue described in Section 2.2, a simulation of a three-dimensional bubble detach-ing from a wall has been performed. The semi-spherical bubble is located at the center of the horizontal bottom wall as aninitial condition. The droplet starts to rise due to buoyancy, and pinches at its base due to the action of the surface tensionand wall adhesion forces.

The radius of the initial semi-spherical bubble is set at 2 mm, the computational domain is an 8 mm cube. A uniformCartesian grid is employed with 64 cells in each direction. All the boundaries are set as no-slip surfaces. The densities ofthe liquid and gas are 1000 kg/m3 and 1.0 kg/m3, respectively. The viscosity is 1.0 Pa s for liquid and 0.01 Pa s for gas. Thesurface tension coefficient is 0.07 N/m, the gravitational acceleration is 9.8 m/s2, and the contact angle is set to 20�.

The bubble shape, the color function contours (red lines)1, and the flow velocity vectors in the center plane at 0.0, 0.15, 0.30,0.45, 0.58, 0.61 and 0.64 s are depicted in Fig. 11. The bubble is still attached to the wall at 0.58 s, but detaches before 0.61 s. At0.61 s, the contour line of / = 0.9 remains in the wall-layer, due to the smearing effect. However, the smearing is limited to onecell in the normal direction and four cells in the lateral direction. The smeared region disappears at 0.64 s. The reason for thedisappearance is explained by the upward advection of the color function rather than due to interface sharpening.

Mass conservation is exactly satisfied in the same way as shown in Fig. 8.

3.4. Free surface deformation in a rectangular-shaped container

The numerical model is validated by comparison with experimental data from a three-dimensional, free-surface deforma-tion test [18]. Since a Cartesian grid system is used, the sides of the container lie along grid lines. The container is filled with

1 For interpretation of references to color in this text part, the reader is refered to the web version of Fig. 11.

Time (s) = 0.0, 0.15, 0.30, 0.45,

0.58, 0.61, 0.64

Fig. 11. Evolution of the flow field around for a bubble detaching from a wall taken at 0.0, 0.15, 0.30, 0.45, 0.58, 0.61, and 0.64 s. Contour lines of the colorfunction at / = 0.1, 0.5 and 0.9 in the center plane are depicted.

Time (s) Experiment Side view Perspective view Center plane

0.00

0.50

0.75

Y X

Z

Y X

Z

Y X

Z

Y X

Z

X Y

Z

Y X

Z

X Y

Z

Y X

Z

X Y

Z

Fig. 12. Deformation of the free surface in a rectangular-shaped container. The free surface shapes observed in the experiment (left) are compared with thecomputed free surfaces from different view points (second and third columns), and the contour lines of the color function in the center plane (right column).Contour lines are at / = 0.1, 0.5 and 0.9. (For interpretation of references to color in this figure legend, the reader is referred to the web version of thisarticle.)

3894 Y. Sato, B. Niceno / Journal of Computational Physics 231 (2012) 3887–3895

perfluorocarbon (C8 F18: FC-77) and air, and is placed into a vessel equipped with the camera. The vessel is then droppedfrom a drop tower. During the fall, the vessel and contents become weightless, and the free surface deforms due to the actionof surface tension and wall adhesion forces. The simulation starts at the beginning of the fall, and ends before the vessel hitsthe ground.

The computational domain has the dimensions of 150 � 145 � 150 mm in the x, y, and z directions; the z axis is vertical,and points upward. A uniform grid is used with 60 � 58 � 60 cells in the coordinate directions. Initially, half of the containeris filled with FC-77. The contact angle is set to 20� as the advancing contact angle. Note that the contact line of the free sur-face on the walls advances only in the direction of the air, and does not recede during the simulation. Therefore, a constantadvancing contact angle can be used for the simulation. The densities of FC-77 and air are 1780 kg/m3 and 1.25 kg/m3,respectively. The viscosity is 1.42 � 10�3 Pa s for FC-77 and 1.00 � 10�5 Pa s for air, and the surface tension coefficient is0.015 N/m.

Y. Sato, B. Niceno / Journal of Computational Physics 231 (2012) 3887–3895 3895

A comparison of the predicted free-surface shape and that seen in the experiment is given in Fig. 12. The computed shapesqualitatively resemble those of the experiment, with the surface at the corners rising faster than on the flat part of the wall.However, the comparison is limited to visual observation. From the center plane perspective (right column), it is evident thatthe interface thickness has stayed constant during the simulation.

4. Conclusions

In this paper, we describe a new contact line treatment for the three-dimensional conservative level set (CLS) method. Forthe cells adjacent to the wall, the color function is sharpened only in the direction tangential to the wall. Mass conservation isstrictly satisfied and numerical smearing of the color function is also prevented. The new model has been tested throughcomparison with the theoretical solutions of a three-dimensional droplet, and against experimental data of free-surfacedeformation in the rectangular-shaped container under conditions of zero gravity.

Compared to Zahedi’s model [2], in which two additional numerical parameters are introduced to the original Olssonmethod [1], and with which only two-dimensional problems can be solved, our method does not require any additionalparameters and it can treat three-dimensional computations.

Acknowledgment

The authors gratefully acknowledge the financial support of swissnuclear for the project ‘‘Multi-Scale Modeling Analysisof convective boiling (MSMA)’’.

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