Multiphase flow and Encapsulation simulations using

the moment of fluid method 1

Guibo Lia, Yongsheng Lianb, Yisen Guoc, Matthew Jemisond, MarkSussmane, Trevor Helmsf, Marco Arientig

aMechanical Engineering, University of Louisville, Louisville, KY, U.S.AbMechanical Engineering, University of Louisville, Louisville, KY, U.S.AcMechanical Engineering, University of Louisville, Louisville, KY, U.S.A

dMathematics, Florida State University, Tallahassee, FL, U.S.AeMathematics, Florida State University, Tallahassee, FL, U.S.AfMathematics, Florida State University, Tallahassee, FL, U.S.A

gSandia National Labs, Livermore, CA, U.S.A

Abstract

A moment of fluid method is presented to study incompressible flows in-volving more than two materials. This work is an extension of Jemison et al.(2013b) in which they presented a moment of fluid method for simulatingtwo-phase flows. For the present method the interfaces between differentphases are captured using the moment of fluid method, a directionally splitcell integrated semi-Lagrangian method is used to calculate interface andmomentum advection, a projection method is used to calculate pressure, anda block structured adaptive mesh refinement method is used to locally in-crease the resolution in the regions of interest. Various multiphase problems,including problems illustrating contact line dynamics, triple junctions, andencapsulation are studied using the new method in order to demonstrateits capabilities. Examples are given in 2D, 3D axisymmetric (R-Z), and 3D(X-Y-Z) coordinate systems.

1M. Sussman and M. Jemison acknowledge the support by the National Science Foun-dation under contract DMS 1016381. M. Arienti acknowledges the support by SandiaNational Laboratories via the Laboratory Directed Research and Development program.Sandia National Laboratories is a multi-program laboratory managed and operated bySandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for theU. S. Department of Energy’s National Nuclear Security Administration under contractDE-AC04-94AL85000.

Preprint submitted to International Journal for Numerical Methods in FluidsJuly 8, 2014

Keywords: multi-phase flow, Moment of fluid method, Interfacereconstruction, Interface advection, contact line dynamics, triple junctions,gas-liquid jet flow, droplet impact, droplet collision

1. Introduction

Multiphase flow plays an important role in many technical applicationsincluding ink-jet printing, spray cooling, icing, combustion and agriculturalirrigation. The instability of the interface, mass and heat transfer acrossthe interface and phase change make multiphase flow problems challenging.Theoretical studies of multiphase flows are mainly based on linear theories(Rayleigh, 1878; Taylor, 1963) while most phenomena in multiphase flows arenonlinear. Experimental studies employ high-speed cameras, pulsed shadow-graph or holograph techniques (Sallam et al., 1999, 2004; Palacios et al.,2010) to understand these complex processes, but it is still very difficult tocapture the detailed flow fields.

Computational fluid dynamics (CFD) is a powerful tool in the study ofmultiphase flows. Three major challenges exist for the accurate and effi-cient computation of multiphase flows. First, the density and viscosity ratiosbetween different phases can be high and it is difficult to accurately cal-culate the flux during the momentum advection (Raessi, 2008; Raessi andPitsch, 2009). Second, surface tension takes effect only at the interface be-tween different phases and this singularity may cause problems when solvingthe Navier-Stokes equations (Brackbill et al., 1992; Desjardins et al., 2008).Third, the drastic change of interface topology and disparity in length scalesmake interface capturing and interface advection challenging (Menard et al.,2007; Li et al., 2010).

Among the different interface capturing methods, the volume-of-fluid(VOF) method (Hirt and Nichols, 1981; Brackbill et al., 1992; Pilliod Jrand Puckett, 2004) , the level set method (Osher and Sethian, 1988; Suss-man et al., 1994, 1998, 1999; Osher and Fedkiw, 2001) and their derivativesare widely used.

The VOF method tracks the volume fraction function within each com-putational grid cell. The interface is reconstructed and advected based onthe volume fraction function information. The benefit of VOF methods isthat there have been devised either directionally split advection algorithmsRudman (1998); Scardovelli and Zaleski (2003); Jemison et al. (2013a); Wey-mouth and Yue (2010) or unsplit advection algorithms (Liovic et al., 2006;

2

Chenadec and Pitsch, 2013; Jemison et al., 2013c) which have excellent vol-ume preservation properties.

The level set method uses a smooth distance function to implicitly rep-resent the interface. On the one hand, the level set method is amenable tocomputing flows with surface tension effects since the distance function issmooth across the interface, on the other hand, volume conservation for thelevel set method is not guaranteed, even if the level set advection equation isdiscretized in conservation form. The coupled level set and volume-of-fluid(CLSVOF) method (Sussman and Puckett, 2000; Son and Hur, 2002; Suss-man, 2003; Menard et al., 2007; Wang et al., 2009) has also been developed.Even though the specific implementations of CLSVOF methods vary, theoverriding theme of CLSVOF methods is that they maintain the advantagesof both the level set method and the VOF method in simultaneously con-serving volume and accurately capturing the interface (Sussman and Puckett,2000; Wang et al., 2009).

Recently, the moment of fluid (MOF) method (Dyadechko and Shashkov,2005; Ahn and Shashkov, 2007; Dyadechko and Shashkov, 2008; Ahn andShashkov, 2009; Ahn et al., 2009; Jemison et al., 2013a,b) has been devel-oped. In addition to the volume fraction function used in the VOF method,the MOF method considers the material centroid information in the inter-face reconstruction process and can be considered a generalization of the VOFmethod. The MOF interface reconstruction is completely local because of theintroduction of centroid information. This feature enables the MOF methodto capture corners and sheets significantly better than the VOF method forpassively advected flows. Results of benchmark tests comparing MOF toVOF and CLSVOF are reported in (Kucharik et al., 2010; Wang et al., 2012;Jemison et al., 2013a). A further advantage of the moment of fluid represen-tation is that the added centroid information enables the MOF method tostraightforwardly reconstruct triple junctions (> 2 materials) in a given cell,conserving volume of each material (Ahn and Shashkov (2007); Jemison et al.(2013c)). We remark that there have been developed VOF triple junctionreconstruction algorithms. We refer the reader to the “onion skin” approachin which the materials do not intersect (Sijoy and Chaturvedi (2010)), thepower law diagram approach for 2D reconstruction of triple points (Schofieldet al. (2008)), an optimization approach for 2D passive transport problems(Caboussat et al. (2008)), and an approach that uses Voronoi diagrams as anaid in the optimization procedure (Schofield et al. (2009)). We have takenthe “nested dissection” MOF approach; if a computational cell contains P

3

materials, then there will be P − 1 linear cuts in the cell, some cuts po-tentially intersecting previous cuts. The MOF algorithm can reconstructinterface configurations where 3 materials meet at a single point, albeit oneof the angles separating 2 of the materials must be 90 degrees.

In this paper, we describe a novel method for 3D multiphase flow, to in-clude contact line dynamics and triple junctions. The interfaces separatingmaterials are represented using the moment of fluid method. A temporarydistance function, the signed distance from the moment-of-fluid reconstructedinterface, is created in order to approximate the surface tension force. Forthe purposes of constructing the temporary signed distance function, we havewritten a “non-tessellating” moment-of-fluid reconstruction algorithm, whichenables one to find the signed distance function for each material at a triplejunction (or any number of materials meeting in a cell) in a systematic fash-ion. The temporary signed distance functions are independent of the materialorder of the “non-tessellating” reconstruction.

As in our recent work (Jemison et al., 2013b) for compressible flow, thedirectionally split cell integrated semi-Lagrangian method is used to updatethe velocity in a manner that conserves momentum for each material. Alsoas in (Jemison et al., 2013b), we maintain and update the velocity at bothcell centers and face centers. The velocity is interpolated from cell centers toface centers using mass weighting where the MOF reconstruction determinesthe weights. Pressure is interpolated from cell centers to face centers usingthe condition of constant contact, where again the weights depend on theMOF reconstructed interface.

In what follows we describe our multi-phase MOF method, and we de-scribe the algorithm that we have developed for the simulation of contact linedynamics and triple point dynamics. Example simulations are given with val-idations through grid refinement studies and comparisons with experimentsand finally conclusions are drawn.

2. Numerical methods

2.1. Governing equations

The incompressible Navier-Stokes equations for immiscible multiphaseflows and the continuity equation in vector form are written as follows:

ρ(H)

(∂u

∂t+ u · ∇u

)= −∇p+∇ · τ + F tension + ρ(H)g (1)

4

∇ · u = 0 (2)

where H is denoted as the material indicator vector governed by

∂H

∂t+∇ · (uH) = 0 (3)

For a problem with M materials, vector H contains M material indicatorfunctions Hm,

H ≡ (H1, . . . , HM), (4)

and the material indicator function Hm(x, t) is defined as follows:

Hm(x, t) =

1 x ∈ material m0 otherwise

(5)

For two-phase flows, M = 2 where m = 1 corresponds to liquid and m = 2corresponds to gas. Vector u = (u, v, w) is the velocity vector, t is the time,p is the pressure, g is the gravitational acceleration vector, τ is the shearstress tensor,

τ = µ(H)(∇u + (∇u)T

),

ρ(H) ≡ ∑m ρmHm is the density, and µ(H) ≡ ∑m µmHm is the viscosity.At material interfaces, if there is no mass transfer, the velocity for all

materials are the same. At a material interface, the stress will have thefollowing jump condition due to the effect of the surface tension force:

[(−pI + τ) · n] = σκn (6)

where σ is the surface tension coefficient, κ is the curvature of the interfaceand n is the unit normal vector of the interface.

If M = 2, the surface tension force term, F tension, is written as,

F tension = −σ1,2(∇ · n1)∇H1

where

n1 =∇H1

|∇H1|.

If M = 3, and x is at a triple junction, then the surface tension force is (Kim(2007)):

F tension = −M∑m=1

γm(∇ · nm)∇Hm

where σm1,m2 = γm1 + γm2.

5

2.2. Moment of Fluid interface representation

In our numerical implementation, we represent deforming material bound-aries using the zeroeth and first order moments of the material indicator vec-tor H (Dyadechko and Shashkov, 2005; Ahn and Shashkov, 2007). In otherwords, we discretize the computational domain into rectangular cells, Ωi,j,k,and for each rectangular cell we represent the distribution of each materialwith the volume fraction (zeroeth order moment),

Fm =1

|Ωi,j,k|

∫Ωi,j,k

Hm(x)dx, (7)

and the centroid of the material (the first order moment),

xm =

∫Ωi,j,k

Hm(x)xdx∫Ωi,j,k

Hm(x)dx. (8)

The motivation for the moment of fluid method is as follows: let v(x, t)be any function (scalar or vector) satisfying,

Dv

Dt≡ ∂v

∂t+ u · ∇v = 0. (9)

For incompressible flow, it is straightforward to show using Eq. (3) and Eq.(9) that

(Hv)t +∇ · (Huv) = 0 (10)

Now, for a given computational cell, Ωn+1 ≡ Ωi,j,k, let T−1Ωn+1 = Ωn

be the advective preimage of Ωn+1. In other words, we define T to be themapping induced by the flow field that maps a parcel of fluid from Ωn att = tn to Ωn+1 at t = tn+1. In the discussion that follows we assume that Tis a linear mapping.

Integrating Eq. (10) over space (Ω(t)) and time (tn ≤ t ≤ tn+1) and usingGauss’s theorem and the Reynolds’ transport theorem, we have the followingrelationship:∫

Ωn+1H(x, tn+1)v(x, tn+1)dx =

∫Ωn

H(x, tn)v(x, tn)dx (11)

Equation (11) indicates that property Hv remains constant when it movesin an incompressible flow field. For material m if we let v(x, t) ≡ 1 (zeroeth

6

order case) and use the incompressibility condition of |Ωn+1| = |Ωn|, then wehave

F n+1m =

1

|Ωn+1|

∫Ωn+1

Hm(x, tn+1)dx =1

|Ωn|

∫ΩnHm(x, tn)dx (12)

In other words, the new volume fraction of material m in the cell Ωn+1 is thevolume fraction of the same material in its advective preimage, Ωn.

Numerically, Hm(x, tn) is approximated with Hreconstructm (x, tn) which is

derived from the moment of fluid reconstruction. Therefore, to compute areasonably accurate volume fraction function F n+1

m it is essential to recon-struct an interface to preserve the following condition∫

ΩnHreconstructm (x, tn)dx =

∫ΩnHm(x, tn)dx. (13)

For material m if we let v(x, t) satisfy v(x, tn+1) = x, v(x, tn) = Tx,Dv/Dt = 0 (first order case), and use the incompressibility condition of|Ωn+1| = |Ωn|, then we have (using (11)),∫

Ωn Hm(x, tn)Txdx∫Ωn Hm(x, tn)dx

=

∫Ωn+1 Hm(x, tn+1)xdx∫Ωn+1 Hm(x, tn+1)dx

. (14)

We note that if the advective velocity u corresponds to purely translationalmotion, then one has,

Tx = x + u(tn+1 − tn) (15)

v(x, t) = x− u(t− tn+1). (16)

If we also continue to assume that T is a linear mapping, then the centroidof material m in cell (i, j, k) is,

xn+1m,i,j,k =

∫Ωn+1 Hm(x, tn+1)xdx∫Ωn+1 Hm(x, tn+1)dx

= (17)∫Ωn Hm(x, tn)(Tx)dx∫

Ωn Hm(x, tn)dx= (18)

T

(∫Ωn Hm(x, tn)xdx∫Ωn Hm(x, tn)dx

). (19)

Note that Eq. (18) is derived from Eq. (17) using Eq. (11): letting v(x, tn+1) =x in the numerator and v = 1 in the denominator. In other words, the first

7

order moment of material m in a computational cell, Ωn+1, is the first ordermoment in the departure region advected to the target cell (See Eq.( 19)).Equations (17) through (19) illustrate that numerically in order to accu-rately calculate the centroid at t = tn+1, it is essential that the reconstructedinterface at the time level t = tn preserves as close as possible,∫

Ωi,j,k

Hreconstructm (x, tn)xdx =

∫Ωi,j,k

Hm(x, tn)xdx. (20)

We remark that T may not be a linear mapping, so in the moment offluid algorithm, the moment at t = tn+1 is computed as follows:

xn+1i,j,k =

∫Ωn+1 Hreconstruct(T−1x, tn)xdx∫Ωn+1 Hreconstruct(T−1x, tn)dx

(21)

The conditions that (13) be exactly satisfied and (20) be satisfied as closeas possible define the moment of fluid reconstruction algorithm.

2.3. MOF interface reconstruction

We first discuss the interface reconstruction when there are two materials(M = 2)in a computational cell. For problems with M > 2 we will discuss inSection 2.3.1. When there are two materials in a cell, the material interfacecan be represented by a plane in a three-dimensional (3D) problem or a linein a two-dimensional (2D) problem. This interface representation is calledthe piecewise linear interface calculation (PLIC). Take a 2D case for example,an interface Γ in cell Ωi,j can be represented by a straight line as shown inFig. 1 using the following vector form equation:

Γ = Ωi,j ∩ x|n · (x− xi,j) + b = 0 (22)

where n is the interface unit normal vector, xi,j is the cell center of Ωi,j

and b is the distance from xi,j to the interface. Thus, the interface can beconstructed when the normal vector n and distance b are known.

The MOF interface reconstruction method was first introduced by Dyadechkoand Shashkov (Dyadechko and Shashkov, 2005). Later on the multi-materialMOF method (Ahn and Shashkov, 2007; Dyadechko and Shashkov, 2008),the adaptive MOF method (Ahn and Shashkov, 2009) and the coupled LevelSet and MOF (Jemison et al., 2013a) methods were developed. In additionto the volume fraction function which is considered in the VOF method,the MOF method also considers the centroid of a given reference phase,

8

n

xi,j

0

y

x

Figure 1: The gas-liquid interface can be represented by a straight line ina 2D problem. The square represents a computational cell and xi,j is thecoordinate of the cell center.

9

n

xcact x

cref

n

(a) (b)

Figure 2: MOF interface reconstruction. The solid curved line on the leftrepresents the real interface and the dashed straight line on the right is thereconstructed interface.

xc = xc(n, b), within each cell. During the interface reconstruction process,a reference volume fraction, Fref , and a reference centroid, xcref , both corre-sponding to the real interface (not necessarily a straight line), are given. Theactual volume fraction function, Fact = Fact(n, b), and the actual centroid,xcact = xcact(n, b), corresponding to the reconstructed interface, are computedbased on the given information. The real interface and the reconstructedinterface are illustrated in Fig. 2. The curved solid line in the left figurerepresents the true interface and the dashed line in the right one is the re-constructed interface. In Fig. 2 xcref is the centroid of the reference phasewhose volume fraction Fref is the blue area under the solid curved line, andxcact is the computed centroid of the reference phase whose volume fractionFact is the blue area under the dashed straight line.

The MOF interface reconstruction method requires that the actual vol-ume fraction, Fact, be equal to the reference value, Fref , to satisfy massconservation requirement:

|Fref − Fact(n, b)| = 0 (23)

Besides, the MOF interface reconstruction requires that the actual centroid,xcact(n, b), be as close to the reference centroid, xcref , as possible. This can

10

be achieved by minimizing the following functional with a constraint givenby Eq. 23.

EMOF = ‖xcref − xcact(n, b)‖2 (24)

Since the normal vector n can be parameterized using the following equa-tion

n =

sin(Φ)cos(Θ)sin(Φ)sin(Θ)

cos(Φ)

(25)

EMOF becomes a function of angles Φ and Θ. Therefore, we need to find(Φ∗,Θ∗) such that

EMOF (Φ∗,Θ∗) = ‖f(Φ∗,Θ∗)‖2 = min‖f(Φ,Θ)‖2 (26)

wheref(Φ,Θ) = xcref − xcact(Φ,Θ)

The problem becomes a non-linear least square problem for (Φ,Θ). Eq. 26is solved numerically by the Gauss-Newton algorithm and the detailed step-by-step procedure is as follows:

0. choose initial angles (Φ0,Θ0) and set tolerance tol = 10−8∆x with ∆xthe grid size.

while not converged set k = 1

1. find bk(Φk,Θk) such that Eq. 23 holds

2. find centroid xck(bk,Φk,Θk)

3. find Jacobian matrix Jk of f evaluated at (Φk,Θk) and fk = f(Φk,Θk)

4. stop if one of the three conditions is fulfilled:

• ‖JTk · fk‖ < tol · 10−2∆x

• ‖fk‖ < tol

• k = 11

5. solve the linear least square problem: find sk such that

‖Jksk + fk‖2 = min‖Jks+ fk‖2

using the normal equations: (JTk Jksk = JTk fk)

6. update the angles: (Φk+1,Θk+1) = (Φk,Θk) + sk

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7. k := k + 1 and go back to step 1.

The detailed process for the minimization of Eq. 24 can be found in(Jemison et al., 2013a). Unlike the VOF method, the MOF interface recon-struction method only uses information from the computational cell underconsideration. This property makes the MOF method more suitable for de-forming boundary problems with sharp corners, with slender filaments, orcontaining greater than two materials. Also, the MOF reconstruction algo-rithm makes itself more suitable for block structured dynamic adaptive meshrefinement since conditions at coarse/fine grid interface can be interpolatedfrom the coarse grid using a stencil that does not depend circularly on theneighboring fine grid.

2.3.1. MOF interface reconstruction when M > 2 materials in a cell.

For the scenario when the number of materials M in cell Ωi,j,k is greaterthan 2, the following MOF reconstruction procedure determines M polygonalregions, Ωm, that tessellate Ωi,j,k (see Figure 3):

1. Let p = 0 and initialize the uncaptured space in a cell,

Ωpu = Ωi,j,k, (27)

where Ωi,j,k represents cell (i, j, k). The centroid of Ωpu is denoted as

xpu. Tag all materials as “not defined.”

2. In the cell identify the material whose centroid (xmi,j,k) is furthest to theuncaptured centroid xpu among all “not defined” materials, i.e.,

mp = argmaxm,Fm

i,j,k>0,mnot defined|x

mi,j,k − xpu|. (28)

3. Calculate the slope n and intercept b that minimizes the centroid errorwith the constraint that |Fmp

act (n, b)−Fmp

ref | = 0 and construct the MOFinterface. Ωmp is defined as:

Ωmp = Ωpu ∩ x|n · (x− xi,j,k) + b ≥ 0 (29)

The Gauss-Newton method as written above is used to solve the opti-mization problem. Tag material m as “defined.”

4. Update Ωp+1u

Ωp+1u = Ωp

u ∩ ΩComplementmp (30)

5. Let p = p+ 1 and go back to step 2.

12

Figure 3: TOP: Illustration of tessellating MOF reconstruction with threematerials; it is important that the MOF reconstruction tessellates a cell whenit is used for advection (section 2.4). The uncaptured region is initialized asthe whole cell, Ω0

u = Ωi,j,k, and then progressively reduced as each new

material m fills the cell: Ωp+1u = Ωp

u ∩ΩComplementmp . BOTTOM: Illustration of

non-tessellating MOF reconstruction. The non-tessellating reconstruction isused for finding the temporary level set functions, φm, which are then usedto find the surface tension force (section 2.5.3). The uncaptured region foreach new material is always the whole cell: Ωu = Ωi,j,k. The solid circles arethe reference first order moments for each material.

2.3.2. Determining volumes and moments of polygonal regions.

In contrast to earlier implementations of the Moment-of-Fluid InterfaceReconstruction method that used a Gauss-Green discretization to computevolumes and moments of polygonal regions (Ahn and Shashkov, 2007), ourimplementation makes use of triangulation (tetrahedralization in 3D) to com-pute reference volumes and reference centroids. Rectangular cells are sub-divided into triangles. The intersection of two triangular (tetrahedral) re-gions is expressed as the union of multiple triangular (tetrahedral) regions.A lookup table is utilized to efficiently cut a triangle (tetrahedron) with alinear (planar) interface and triangulate (tetrahedralize) the cut region. Vol-ume of the cut region is then computed as the sum of the volumes of the

13

constituent triangles (tetrahedra). Since all polygonal (polyhedral) regionsare decomposed into triangles (tetrahedra), it is easy to compute moments.For a given triangle (tetrahedra) T , the centroid xT is the average of itsvertices xj shown as follows:

xT =

∫T x · dx∫T dx

=1

3

3∑j=1

xj (31)

Any polygonal region P that we consider can be written as the union of Ntriangles Ti,

P =N⋃i=1

Ti. (32)

If each triangle has volume Vi and centroid xTi, then the volume VP of the

region P is equal to the sum of the volumes of Ti and the centroid xP of Pis the volume-weighted sum of the centroids:

VP =N∑i=1

Vi (33)

xP =

∫P x · dx∫P dx

=

N∑i=1

VixTi

VP(34)

2.4. MOF interface advection

2.4.1. Directionally split method

The directionally split method (Strang, 1968; Jemison et al., 2013a) in-tegrates the interface position sequentially, and a backward or forward pro-jection of characteristics may be carried out for each direction. Here weillustrate the backward and forward projection only in the x direction.

We introduce the “Cell Integrated Semi-Lagrangian” (CISL) mappingfunction,

TCISLi : Ωdeparti → Ωtarget

i . (35)

For backwards tracing directionally split advection, we define Ωtargeti as,

Ωtargeti =

[xi−1/2, xi+1/2

]. (36)

14

i-1 i i+1

, 1mV −

1nT +

nT1,n

iV−0,niV

11,n

iV +−

10,niV +

( )a

( )bDeparture region depart

iΩ

arg

Target region t et

i iΩ = Ω

1/2ix − 1/2ix +

1/2 1/2i ix u t− −− ∆ 1/2 1/2i ix u t+ +− ∆

1,n

ix− 0,n

ix

11,

nix +

−1

0,n

ix +

Figure 4: Backward projection for the directionally split method. The dashedrectangle in (b) represents the departure region (Ωdepart

i ). We denote thedeparture region of material m as Ωdepart

m,i = V n−1,i ∪ V n

0,i which is shaded in(b). V n

−1,i and V n0,i are the intersection of the departure region with material

m in cell i−1 and i, respectively, i.e., V n−1,i ≡ Ωn

m,i−1∩Ωdeparti and V n

0,i ≡ Ωnm,i∩

Ωdeparti . xn−1,i and xn0,i are the centroid of regions V n

−1,i and V n0,i, respectively.

The target region of material m is denoted as Ωtargetm,i = V n+1

−1,i ∪ V n+10,i which

is shaded in (a). The overall target region in (a) is cell i and we denote thisas Ωtarget

i

15

Given face velocities ui−1/2 and ui+1/2, we define the departure region as,

Ωdeparti =

[xi−1/2 −∆tui−1/2, xi+1/2 −∆tui+1/2

]. (37)

If |Ωdeparti | > |Ωtarget

i |, then the material in the departure region undergoescompression as the material is translated from the departure region to thetarget region Ωi = Ωtarget

i . Alternatively, if |Ωdeparti | < |Ωtarget

i |, then thematerial in the departure region undergoes expansion.

We will use Fig. 4 to illustrate the backward projection process. Thoughthe MOF method works for general multiphase flow simulations (M > 2materials) we use a two phase example to illustrate the key ideas. We usethe shaded region to represent material m. To determine the new volumefractions and moments in cell i, we first determine the departure region(Ωdepart

i ) which is the dashed rectangle in Fig. 4(b). The departure region attime tn will move into cell i at tn+1.

As shown in Figure 4(b), V n−1,i and V n

0,i are the intersection of the de-parture region with material m in cell i − 1 and i, respectively, i.e., V n

−1,i ≡Ωnm,i−1 ∩ Ωdepart

i and V n0,i ≡ Ωn

m,i ∩ Ωdeparti . Here Ωn

m,i−1 and Ωnm,i represent

the material m region in cell i − 1 and i at time instant of tn, respectively.After convection, the volume fraction of material m in cell Ωi at tn+1 (shadedregion in Figure 4(a)) is

F n+1m,i =

|V n+1−1,i |+ |V n+1

0,i ||Ωtarget

i |=|TCISLi (V n

−1,i)|+ |TCISLi (V n0,i)|

|Ωtargeti |

(38)

=|TCISLi (Ωn

m,i−1 ∩ Ωdeparti )|+ |TCISLi (Ωn

m,i ∩ Ωdeparti )|

|Ωtargeti |

(39)

and the centroid of material m in Ωi is

xn+1m,i =

∫V n+1−1,i

xdx +∫V n+10,i

xdx

F n+1m,i |Ω

targeti |

= (40)∫TCISL

i (Ωnm,i−1∩Ωdepart

i ) xdx +∫TCISL

i (Ωnm,i∩Ωdepart

i ) xdx

F n+1m,i |Ω

targeti |

= (41)

xn+1−1,i|TCISLi (Ωn

m,i−1 ∩ Ωdeparti )|+ xn+1

0,i |TCISLi (Ωnm,i ∩ Ωdepart

i )|F n+1m,i |Ω

targeti |

(42)

Here bmxn+1−1,i and xn+1

0,i are the centroid of material m in regions V n+1−1,i

and V n+10,i , respectively (See Figure 4(a)).

16

We remark that if TCISLi is a linear mapping (which will be the case in2D or 3D split advection, but not 3D axisymmetric split advection) then thenew centroid can be written as (referring to Figure 4(b)),

xn+1m = TCISLi

xn−1,i|Ωnm,i−1 ∩ Ωdepart

i |+ xn0,i|Ωnm,i ∩ Ωdepart

i ||Ωn

m,i−1 ∩ Ωdeparti |+ |Ωn

m,i ∩ Ωdeparti |

. (43)

In practice for the backwards tracing of characteristics CISL method, theupdated volume fraction and centroid for material m are calculated as

F n+1m,i =

∑1i′=−1 |TCISLi (Ωn

m,i+i′ ∩ Ωdeparti )|

|Ωtargeti |

(44)

xn+1m,i =

∑1i′=−1

∫TCISL

i (Ωnm,i+i′∩Ωdepart

i ) xdx

F n+1m |Ωtarget

i |(45)

For the forward tracing of characteristics CISL method, the forward pro-jection mapping function is defined as follows:

TCISLi : Ωdeparti → Ωtarget

i . (46)

This function maps the departure region, cell i (Ωdeparti = Ωi), to the target

region:Ωtargeti = [xi−1/2 + ∆tui−1/2, xi+1/2 + ∆tui+1/2] (47)

Fig. 5 illustrates the forward projection process. Cells i − 1 and i inFig. 5(a) are the departure regions ( Ωi−1 ≡ Ωdepart

i−1 and Ωi ≡ Ωdeparti respec-

tively ) and the dashed rectangles in Fig. 5(b) represents their target regions(Ωtarget

i−1 and Ωtargeti respectively). The material m regions in cells i− 1 and i

at time tn are denoted as Ωnm,i−1 and Ωn

m,i respectively. At t = tn+1 the depar-ture regions Ωn

m,i−1 and Ωnm,i are advected to the target regions TCISLi−1 Ωn

m,i−1

and TCISLi Ωnm,i which overlay cell i. We denote the overlapping region with

cell i as

Vm,−1 ∪ Vm,0 ≡ (TCISLi−1 Ωnm,i−1 ∪ TCISLi Ωn

m,i) ∩ Ωi.

See Fig. 5(b).In practice for forward tracking the volume and centroid for material m

at tn+1 in cell i are affected by neighboring cells i + i′, i′ = −1, 0, 1 and arecalculated as

F n+1m,i =

∑1i′=−1 |Vm,i′ ||Ωi|

=

∑1i′=−1 |TCISLi+i′ (Ωn

m,i+i′) ∩ Ωi||Ωi|

(48)

17

i-1 i i+1

, 1mV −

, 1nm i−Ω ,

nm iΩ

1 , 1CISL n

i m iT − −Ω,0mV

,CISL n

i m iT Ω

Departure region departiΩ1Departure region depart

i−Ω

arg1Target region t et

i−Ω argTarget region t etiΩ

(a)

(b)

1nT +

nT

1/2ix − 1/2ix +

1/2 1/2i ix u t+ ++ ∆1/2 1/2i ix u t− −+ ∆

Figure 5: Forward projection for the directionally split method. The dashedrectangles in (b) represent the target regions (Ωtarget

i−1 and Ωtargeti ). The de-

parture regions are cells i− 1 and i in (a). The departure regions of materialm in cells i− 1 and i are Ωn

m,i−1 and Ωnm,i, respectively. The target regions of

material m are TCISLi−1 Ωnm,i−1 and TCISLi Ωn

m,i. Vm,−1 and Vm,0 are the overlap-ping regions of the target regions with cell i, i.e., Vm,−1 ≡ TCISLi−1 (Ωn

m,i−1)∩Ωi,and Vm,0 ≡ TCISLi (Ωn

m,i) ∩ Ωi

18

xn+1m,i =

∑1i′=−1

∫TCISL

i+i′ (Ωm,i+i′ )∩Ωixdx

F n+1m,i |Ωi|

(49)

We have implemented two different directional splitting advection algo-rithms. In either algorithm, we use “Strang splitting” (Strang, 1968; Sussmanand Puckett, 2000) which advects the interface successively in the X, Y andZ directions, then the order is reversed to be Z, Y , and X directions.

The first directionally split algorithm that we implemented is based onthe alternating Eulerian-Implicit Lagrangian Explicit (EI-LE) algorithm de-scribed in Scardovelli and Zaleski (2003) and Aulisa et al. (2007). In two-dimensions, we advect in the X direction first using Eulerian-Implicit timediscretization (backwards tracing) and then we advect in the Y directionusing the Lagrangian-Explicit (forwards tracing) time discretization. In 3D,the advection ordering is X (EI), Y (LE), Z (EI), Z (LE), Y (EI), X (LE).

The second directionally split algorithm that we implemented follows thealgorithm described in Weymouth and Yue (2010). The approach advects inthe X and Y (X-Y -Z in 3D) directions using the following modified EulerianImplicit (backwards tracing) scheme (X direction shown as example, afterthe dth direction sweep):

Fn+1,(d+1)m,i,j,k =

∑1

i′=−1|Ωdepart

i ∩Ωnm,i+i′ |

|Ωi,j,k|F nm,i,j,k ≤ 1/2

1−∑1

i′=−1|Ωdepart

i ∩Ωni+i′,j,k

|−|Ωdeparti ∩Ωn

m,i+i′ ||Ωi,j,k|

F nm,i,j,k > 1/2

(50)

Fn+1,(d+1)m,i,j,k is the volume fraction after the (d+1)th advection sweep. Ωn

m,i+i′ isderived from the MOF reconstructed interface after the dth advection sweep.F n is the volume fraction prior to advection in any direction.

2.5. Fluid solver

2.5.1. Cell Integrated Semi-Lagrangian Multiphase Advection

We have implemented the same directionally split, semi-Lagrangian methodfor advecting the velocity field as the method used in section 2.4 for advectingthe volume fraction and moment for each material. Without loss of general-ity, we present the backwards tracing of characteristics algorithm. The CellIntegrated Semi-Lagrangian advection of material interfaces and velocity isdescribed as follows:

1. Advect and update volume fractions and moments for each material,m

19

F n+1m,i =

∑1i′=−1 |TCISLi (Ωn

m,i+i′ ∩ Ωdeparti )|

|Ωtargeti |

(51)

xn+1m,i =

∑1i′=−1

∫TCISL

i (Ωnm,i+i′∩Ωdepart

i ) xdx

F n+1m |Ωtarget

i |(52)

TCISL is the mapping defined in Equations (35)-(37).2. Advect momentum

(a) Find the MINMOD piecewise linear reconstruction of the momen-tum for each material:

Unm,i ≡ ρmun

i (53)

Unm,i(x) = (U ′)nm,i(x− xnm,i) + Un

m,i (54)

U ′i =

0 (D+U i)(D−U i) ≤ 0

SGN ·min(|D+U i

∆x|, |D−U i

∆x|) otherwise

(55)

D+U i ≡ U i+1 −U i D−U i ≡ U i −U i−1 SGN ≡ D+U i

|D+U i|(56)

(b) Update the overall velocity using the individually reconstructedmomentum from (54):

uai =

∑i′

M∑m=1

∫Ωn

m,i+i′∩Ωdeparti

Unm,i′(x)dΩ

∑i′

(M∑m=1|Ωn

m,i+i′ ∩ Ωdeparti |ρm

) (57)

2.5.2. Multiphase, Semi-Implicit Pressure Correction

The action of advection can be separated from the pressure force termsin the Euler equations as in Kwatra et al. (2009). Once the advected cell-averaged velocity, ua, is found, velocity is updated in the typical way fromincompressible flows, as in Chorin (1968):

un+1 = ua −∆t∇pn+1

ρn+1(58)

Take the divergence of (58) to obtain

∇ · un+1 = ∇ · ua −∆t∇ ·(∇pn+1

ρn+1

)(59)

20

Strictly enforcing incompressibility equates to setting ∇ · un+1 = 0, resultingin an implicit equation to solve for pressure:

1

∆t∇ · ua = ∇ ·

(∇pn+1

ρn+1

). (60)

The cell-centered advective velocity, uai , is taken as the cell-integratedaverage of velocity over the cell Ωi. To discretize ∇ · ua, the velocity, ua,must be defined at cell faces (65). Each cell is separated into left and rightcontrol volumes, Ωi,L and Ωi,R. The projection of cell centered velocity tocell faces must respect conservation of momentum, so it is required that theintegral over all face centered control volumes (61) be equal to the integralover all cells.

We define the face control volume Ωi+1/2 as

Ωi+1/2 = Ωi,R ∪ Ωi+1,L (61)

where Ωi,R is the right half control volume of cell i and Ωi+1,L is the left halfcontrol volume of cell i+ 1 as illustrated in 6.

The density over each half control volume is:

ρi,R =1

|Ωi,R|

M∑m=1

|Ωm,i ∩ Ωi,R|ρm (62)

ρi+1,L =1

|Ωi+1,L|

M∑m=1

|Ωm,i+1 ∩ Ωi+1,L|ρm (63)

Then, the face centered density ρi+1/2 is defined as the mass in the half-cell regions Ωi,R and Ωi+1,L divided by the volume of the face-centered controlvolume Ωi+1/2, as in (64).

ρi+1/2 =ρi,R|Ωi,R|+ ρi+1,L|Ωi+1,L|

|Ωi+1/2|(64)

The face centered advective velocity is now defined as a mass-weightedinterpolation of the cell centered advective velocity,

uai+1/2 =uai ρi,R|Ωi,R|+ uai+1ρi+1,L|Ωi+1,L|

ρi+1/2|Ωi+1/2|(65)

21

1/2i+Ω

iΩ 1i+Ω

,i RΩ 1,i L+Ω

Figure 6: Left and right control volumes Ωi,R, Ωi+1,L, with face-centeredcontrol volume Ωi+1/2. Cell centers are shown as dots, and the boundary ofthe face-centered control volume is shown as dashes.

A finite volume-style discretization of (66) is used. Divergence terms areintegrated over the control volume Ω and divided by the magnitude of thecontrol volume. For smooth solutions, this will converge to the divergenceoperator as the magnitude of the control volume goes to zero.

∇ ·(∇pρ

)≈

∫Ω∇ ·

(∇pρ

)dx∫

Ωdx

=

∫∂Ω

(∂p∂n/ρ

)dx∫

Ωdx

(66)

The pressure gradient at the cell face is discretized using centered differenc-ing. The ∇ · ua term is handled similarly as the ∇ · ∇p

ρterm:

∇ · ua ≈

∫∂Ωu · ndx∫Ωdx

(67)

In 1D, scheme (66) can then be written in fully discretized form (68):

22

1/2i+Ω

iΩ 1i+Ω

,m iρ, 1m iρ +

, ,m i i RΩ Ω , 1 1,m i i R+ +Ω Ω

,i Rρ 1,i Lρ +

Figure 7: To compute the half cell densities ρi,R and ρi+1,L in cut cells, theMOF reconstructed interface is used to determine the half cell volume frac-tions Fm, i, R and Fm, i+ 1, L, so that one can derive ρi,R =

∑m Fm,i,Rρm,i

and ρi+1,L =∑m Fm,i+1,Lρm,i+1.

(pn+1

i+1 −pn+1i

ρn+1i+1/2

∆x

)−(pn+1

i −pn+1i−1

ρn+1i−1/2

∆x

)∆x

=1

∆t

uai+1/2 − uai−1/2

∆x(68)

Once the linear system, (68), is inverted in order to determine a newpressure pn+1, the pressure correction terms can be computed in order toupdate the velocity.

The face velocity, un+1i+1/2, used to define characteristic tracing (37), is

updated as follows:

un+1i+1/2 − uai+1/2

∆t= − 1

ρn+1i+1/2

pn+1i+1 − pn+1

i

∆x(69)

The cell-averaged momentum is updated in a conservative fashion. Itis required to interpolate the cell centered pressure to the cell faces. As in

23

Kwatra et al. (2009), we define the momentum equation in each half cellregion, Ωi,R and Ωi+1,L:

Dui,RDt

=un+1i,R − uai,R

∆t= −

pn+1i+1/2 − p

n+1i

ρn+1i,R ∆x/2

(70)

andDui+1,L

Dt=un+1i+1,L − uai+1,L

∆t= −

pn+1i+1 − pn+1

i+1/2

ρn+1i+1,L∆x/2

. (71)

We apply the constraint that the interface between cells must remain incontact (Kwatra et al. (2009)); i.e.

Dui,R

Dt=

Dui+1,L

Dt. Using this constraint

and Equations (70 - 71), the pressure at the cell face pi+1/2 (72) is found:

pi+1/2 =ρi,Rpi+1 + ρi+1,Lpiρi,R + ρi+1,L

(72)

The derivations of the half cell densities, ρi,R and ρi+1,L, are given by (62)and (63).

With pressure defined at cell faces, we can conservatively update thecell-averaged momentum (73):

ρn+1i (un+1

i − uai )∆t

= −pn+1i+1/2 − p

n+1i−1/2

∆x(73)

2.5.3. Viscosity and surface tension

Our multiphase pressure correction algorithm requires the following mod-ifications if the stress tensor τ or surface tension forces are not zero:

1. Do the CISL advection step as outlined in (51-57).

2. Compute the intermediate velocity, u∗

ρn+1(

u∗ − ua

∆t

)= ∇ · τ(ua) (74)

The viscosity coefficient at cell faces is

µi+1/2 =|Ωi+1/2|∑M

m=11µm

(|Ωmi ∩ Ωi,R|+ |Ωm

i+1 ∩ Ωi+1,L|)

If the diffusion time step ∆tdiffuse associated with (74) is smaller thanthe advection time step ∆t, then a sub-cycling procedure is imple-mented for (74).

24

3. Calculate the conservative pressure correction for momentum exactlyas in (73) except for the following changes:

(a) replace ua with u∗ in section 2.5.2.(b) Construct temporary signed distance functions φm which are de-

rived from the piecewise linear moment-of-fluid reconstructed in-terface; see section 2.5.4. The temporary distance functions areused to approximate the curvature of interface(s).

(c) Calculate the surface tension force similarly as in Jemison et al.(2013a) except generalized for the scenario when 3 materials meet.The ghost fluid method is used; see section 2.5.5.

2.5.4. Signed distance function(s)

In order to approximate the curvature of the interface(s) (see section2.5.5), temporary signed distance functions φm,i,j are constructed. φm,i,j isthe signed distance from the cell (i, j) center xi,j to the piecewise linear(planar in 3D) “non-tessellating” moment of fluid reconstructed material minterface (See Figure 8). The sign is positive if xi,j ∈ Ωm,i,j and negativeotherwise.

The algorithm for finding the exact signed distance is explained in Suss-man and Puckett (2000). The “non-tessellating” moment of fluid recon-structed interface matches the moment of fluid reconstruction, described insection 2.3, in cells containing less than three fluid materials. For cells con-taining three fluid materials or more, and for purposes of finding the tempo-rary level set function, it is not necessary, nor is it recommended, to enforcethe condition that the reconstructed materials tessellate a cell. The tessellat-ing condition makes it logistically difficult to find the exact signed distance tothe reconstructed interface(s). Also the tessellating reconstruction is depen-dent on the order in which materials are reconstructed. Instead, we relax thetessellating condition and replace the algorithm in section 2.3.1 with the fol-lowing simpler reconstruction when three or more fluid materials are presentin a cell (i, j):

For each material m (m = 1, . . . ,M) if Fm > 0 then we find the slope n andintercept b that minimizes the centroid error with the constraint that|Fmact(n, b)− Fm

ref | = 0.

The Gauss-Newton method as written in Section 2.3 is used to solve theoptimization problem. The minimization problem is carried out assuming,for each m = 1, . . . ,M , that the uncaptured space Ωu (27) is the whole

25

cell: Ωu = Ωi,j,k. In Figure 3, we illustrate the difference between the MOFreconstruction that tessellates a cell and the MOF reconstruction that is notrequired to tessellate a cell.

We remark that our algorithm for constructing the signed distance func-tions near rigid boundaries differs slightly from the procedure just described.As in Arienti and Sussman (2014), the signed distance functions φm repre-senting fluid materials are extrapolated into rigid boundaries. In otherwords,when creating the signed distance to material m, where m represents a fluid,we ignore interfaces that separate m from rigid boundaries. The extrapolatedvalue of φm,i,j, where xi,j is inside of a rigid body material, is taken as thevalue of φm,i′,j′ where xi′,j′ is the closest cell outside of the rigid body to xi,j.See Figure 10.

2.5.5. Ghost fluid method for surface tension; two or three materials

The surface tension force is discretized similarly as in Jemison et al.(2013a) in which the Ghost Fluid Method is used. What distinguishes thiswork from Jemison et al. (2013a) is that our algorithm has been generalizedto handle contact line dynamics (2 fluids meet at rigid boundary) and triplepoints (3 fluids meet). We describe our surface tension algorithm in the xdirection, which has an analogous description in the y or z directions:

1. Suppose that material m1 dominates cell (i, j) and material m2 dom-inates cell (i + 1, j). Then the interface separating materials m1 andm2 will be the zero level set of,

φ ≡ φm1 − φm2

2, (75)

and φi,jφi+1,j ≤ 0.2. For the equation for pn+1

i in (68), the equation for pi±1/2 in (72) whenupdating un+1

i in (73), and when updating un+1i+1/2 in (69), we replace

pn+1i±1 with,

pn+1i±1 + σκi±1/2

H(φi±1)−H(φi)

∆x. (76)

σ is the surface tension coefficient, κi±1/2 is the interfacial curvature,

and H(φ) is the Heaviside function,

H(φ) =

1 φ ≥ 0

0 φ < 0(77)

26

3. The level set height function method Arienti and Sussman (2014); Suss-man and Ohta (2009) is used to approximate the curvature away fromtriple points and contact lines. Referring to Figure 8, κi±1/2 in (76) isapproximated as follows:

κi±1/2 =

κi |φi| < |φi±1|κi±1 otherwise

(78)

h′′ ≈ hi+1 − 2hi + hi−1

∆x2h′ ≈ hi+1 − hi−1

2∆x(79)

κi =−h′′

(1 + (h′)2)3/2(80)

4. In the vicinity of triple points, we approximate the curvature usingcentral differencing techniques. The level set functions, φm, are firstlinearly interpolated to the 3x3 stencil centered at the point (i + θ, j)where,

θ =|φi,j|

|φi,j|+ |φi+1,j|. (81)

Referring to Figure 9, if the 3x3 stencil of cells about cell (i + θ, j)contains a 3rd fluid material m3 (i.e. φm3,i+θ+i′,j+j′,m3 > 0 for somei′ = −1, 0, 1, j′ = −1, 0, 1), then we discretize the curvature usingcentral differences:

κi+1/2,j = γ1∇ ·∇φm1

|∇φm1 |− γ2∇ ·

∇φm2

|∇φm2 |

γ1 =σm1m2 − σm2m3 + σm1m3

2σm1m2

γ2 =σm1m2 + σm2m3 − σm1m3

2σm1m2

∇φ is discretized at the nodes surrounding cell (i+ θ, j); e.g.

(φx)i+θ+1/2,j+1/2 ≈φi+θ+1,j + φi+θ+1,j+1 − φi+θ,j − φi+θ,j+1

2∆x(82)

27

5. Referring to Figure 11, in the vicinity of contact lines we approximatethe curvature using central differencing techniques. κi±1/2 in (76) isapproximated as follows:

κi±1/2 =

κi |φi| < |φi±1|κi±1 otherwise

(83)

κi = ∇ · (H(φm3)nghost + (1−H(φm3))n) (84)

H(φm3)i+1/2,j+1/2 =

0 φm3,i+i′,j+j′ ≤ 0, all i′, j′ = 0, 11 otherwise

(85)

nghost is defined as follows (see Arienti and Sussman (2014)):

nm3 = − ∇φm3

|∇φm3|n =

∇φ|∇φ|

t1 = nm3 × n t2 = nm3 × t1

nghost = sign(nm3 · t2) sin(θ)t2

|t2|− cos(θ)n nghost =

nghost

|nghost|

Figure 8: Left: A temporary level set function φ is the exact signed distanceto the piecewise linear Moment-of-Fluid reconstructed interface. Right: Thelevel set height function method is used to approximate the curvature of theinterface. The square symbols are located at the zero crossings of φ; e.g.hi = (1− θi,j+1/2)yj + θi,j+1/2yj+1 where θi,j+1/2 ≡ |φi,j |

|φi,j |+|φi,j+1| .

28

Figure 9: Illustration of the 3x3 stencil used to calculate the curvature κi+1/2,j

when the stencil contains a 3rd fluid material (material 3 in this diagram).(i + θ, j) is the Γ1,2 interface crossing between cells (i, j) and (i + 1, j) (θ =|φi|/(|φi|+|φi+1|), φ = (1/2)(φ1−φ2)). Given a 4x3 stencil of level set values,φm,i+i′,j+j′ , i

′ = −1, . . . , 2, j′ = −1, 0, 1 (open circles), the level set functionsare first interpolated to a 3x3 stencil (filled circles) centered at (i + θ, j).Then given the 3x3 stencil of values for the level set functions φm,i+θ+i′,j+j′ ,i′ = −1, 0, 1, j′ = −1, 0, 1, the central difference curvature discretizationtechnique is used (see (82)).

29

Figure 10: When constructing the temporary distance functions φ1 and φ2

outside of the rigid boundary (φ3 < 0), the rigid boundary interface is ig-nored. The thin dashed lines represent the closest distance to the interfaceseparating materials 1 and 2. The values of φ1 and φ2 where φ3 ≥ 0 areextrapolated from the nearest cell in which φ3 < 0 (thick dashed lines). Theextrapolated fluid interface is the thick solid line.

3. Results and discussion

We simulate the following cases using the proposed method to illustrateits robustness and accuracy. These test problems are: (1) relaxation to staticshape for 2D droplet on a slope, (2) the liquid lens triple point problem, (3)downward liquid jets, (4) upward liquid jets, (5) Binary collision of two waterdrops (6) Binary collisions of a water drop and a diesel drop (7) dropletimpingement onto a thin liquid film, and (8) droplet impingement onto asmooth solid wall.

30

Figure 11: Illustration of the 3x3 stencil used to calculate the curvatureκi when the stencil contains a rigid material (material 3). Given the 3x3stencil of values for the level set function φi+i′,j+j′ , i

′ = −1, 0, 1, j′ = −1, 0, 1(φ ≡ (1/2)(φ1−φ2)), the central difference curvature discretization techniqueis used (see (84)). φ3 and φ are defined at the closed circles and the normalsnghost and n are defined at the open circles.

31

3.1. 2D droplet on a slope

As in (Arienti and Sussman, 2014; Noel et al., 2012), we tested our MOFalgorithm by performing a convergence study for the relaxation of a two-dimensional water droplet in gas on an 180 inclined solid plane. See Figure12. There is no gravity. The inclined plane is defined as,

Ωplane = (x, y)| tan(180)(x− 3) + 1− y > 0. (86)

The initial droplet is prescribed so that the initial contact angle is 900. Theinitial droplet shape is defined as,

Ωdrop = (x, y)|1− ((x− 3)2 + (y − 1)2) > 0 ∩ ΩCplane (87)

The dimensionless parameters are: ρ1 = 1, ρ2 = 0.0013, µ1 = 0.16, µ2 =0.0025, σ12 = 1118.0, σ13 = 559.0, and σ23 = 1118.0. These parameters arenon-dimensionalized by the initial droplet radius r0 = 0.06348cm (see Figure12) and a characteristic velocity U = 1cm/s. Material “1” corresponds towater (red), “2” corresponds to air (white), and “3” corresponds to solid(blue). The values of σij correspond to a contact angle of 600.

The expected drop height is,

e0 = (1 + cos(π − θ))Rθ, (88)

and the expected base length is

L0 = 2 sin(π − θ)Rθ, (89)

where Rθ is the final radius,

Rθ =

√π

2θ − sin(2θ)R0. (90)

In table 1, we give the percent error for e0 and L0 when computed onsuccessively refined grids.

3.2. Liquid Lens test problem

We repeat the Liquid Lens test problem as in Kim (2007) (section 4.4of Kim (2007)). Initially there are 3 materials with material 2 occupying acircle of diameter 0.3 in the center of the domain, material 1 occupying theremaining top half of the computational domain, and material 3 occupying

32

Table 1: Percent error for the relaxation of a water drop on a sloped inclinefrom a 900 contact angle to a 600 contact angle. The computational domain is6x3 in dimensionless units. The computational grid is a hierarchy of adaptive,dynamic, rectangular grids with a base coarse grid of 96x48 grid cells. Theerror is checked for 3 different grid resolutions.

Levels 100|e0−eexact

0 |eexact0

100|L0−Lexact

0 |Lexact

0

1 3.4 4.02 0.9 1.43 0.4 0.3

the remaining bottom half of the computational domain. All 3 materialshave the same unit density and share the same viscosity of 1/60. The surfacetension coefficient between materials one and two and between materials twoand three was set to σ12 = 2/45 and σ23 = 2/45 respectively. We tested ouralgorithm for four different values of the surface tension between materialsone and three (σ13): σ13 = 3/90, 4/90, 5/90, and 6/90. The expected steadystate solution has material 2 being stretched into a lens shape with majoraxis length of Lexact0 = 0.381, 0.416, 0.460, and 0.522 for the four cases ofσ13. In Figure 13 we illustrate the initial and steady state interfaces that wecompute using our multimaterial MOF method for the case when σ13 = 5/90.In Table 2, we give the percent error in major axis length for the four differentcases of σ13. In all of our test cases, the volume fluctuated by less than 0.001percent throughout the simulations.

3.3. Downward liquid jets

We study a downward liquid jet from a 3D round nozzle. Fig. 14 illus-trates the geometry of the nozzle and the computational conditions. Thegeometry of the nozzle can be defined by the nozzle diameter D and nozzleheight Hn. Uniform flow velocity is assigned at the nozzle inlet. Nonslipboundary condition is applied at the nozzle walls and top surface. Outflowboundary condition is applied on the rest boundaries.

The following fluid properties are used: water density ρl = 1.0×103 kg/m3

, air density ρg = 1.225 kg/m3 , water viscosity µl = 1.3× 10−3 kg/m.s , airviscosity µg = 2.0× 10−5 kg/m.s, surface tension σ = 7.28× 10−2 kg/s2 andthe gravitational acceleration g = 9.8 m/s2.

In the simulation the nozzle diameter D is 0.4 cm and the nozzle height

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Figure 12: Relaxation of droplet (red) on a slope (blue) from 900 contactangle to 600 contact angle. Base coarse grid is a rectangular 96x48 grid. Ef-fective fine grid resolution is 384x192. Left: initial conditions, Right: dropletat final static shape t = 0.8.

Figure 13: Stretching of a liquid lens. Red region indicates material 1, whiteregion is material 2 and blue region is material 3. All materials have thesame constant viscosity µ = 1/60. σ12 = 2/45, σ13 = 1/18, and σ23 = 2/45.The computational domain dimensions are 1x1. The base coarse grid is arectangular 64x64 grid. Effective fine grid resolution is 256x256. Left: initialconditions, Right: liquid lens at t = 6.

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Inflow

D Hn

Outflow

Nonslip

Figure 14: Illustration of nozzle geometry and computational domain for adownward jet. D is the nozzle diameter and Hn is the nozzle height.

35

Table 2: Percent error for the stretching of a liquid lens at static shape.Initially material 2 is a circle centered at (1/2, 1/2), material 1 occupies theremaining top half, and material 3 occupies the remaining bottom half. Thedensity and viscosity is constant for all three materials: ρ = 1, µ = 1/60.The surface tension coefficients are σ12 = 2/45, σ23 = 2/45, and σ13 = 3/90,4/90, 5/90, and 6/90. The computational domain dimensions are 1x1. Thecomputational grid is a hierarchy of adaptive, dynamic, rectangular gridswith a base coarse grid of 64x64 grid cells and two levels of refinement.

σ13 L0 Lexact0 100|L0−Lexact

0 |Lexact

0

3/90 0.374 0.381 1.8%4/90 0.407 0.416 2.2%5/90 0.463 0.460 0.7%6/90 0.531 0.522 1.7%

Hn is 0.6 cm. Two levels of adaptive mesh refinement are used, resulting ina mesh which has 12 grid points per nozzle diameter.

When the flow rate is lower, jet behavior is dominated by the gravita-tional force and the surface tension force. Dripping regime will occur if thegravitational force is larger than the surface tension force. Fig. 15 showsthe time evolution of the dripping regime at a low nozzle inlet velocity ofv0 = 6.0 cm/s; When the flow rate is higher, the jet behavior is dominatedby the inertial force and surface tension force, which leads to the jetttingregime. Fig. 16 shows the time evolution of the jetting regime at a highnozzle inlet velocity of v0 = 20.0 cm/s.

For purpose of generality, we introduce three non-dimensional parameters:the Reynolds number, the Weber number and the Ohnesorge number. TheReynolds number, Re = ρv0D/µ, gives a measure of the ratio of inertialforces to viscous forces. The Weber number, We = ρv2

0D/σ, represents therelative importance of inertia and surface tension. The Ohnesorge number,Oh =

√We/Re = µ/

√ρσD, relates viscous forces to the inertial and surface

tension forces.For the studied high inlet velocity case, the Reynolds number is 307,

the Weber number is 1.1 and the Ohnesorge number is 0.0034. The smallOhnesorge number means that the viscous forces are less important than theinertial force and surface tension. The breakup length, which is defined asthe distance between the nozzle tip and the point just before the first drop of

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Figure 15: Snapshots of jet interface with a low velocity of v0 = 6.0 cm/s.

37

Figure 16: Snapshots of jet interface with a high velocity of v0 = 20.0 cm/s.

38

the liquid jet, can be computed using the following formula (Ashgriz, 2011):

L

D= 1.04C

√We (91)

where L is the jet breakup length, and C is an empirical parameter. (Grantand Middleman, 1966) suggested the value of C = 13 based on experimentaldata for Rayleigh breakup of low viscosity jets.

The computed breakup length, as shown in Fig. 16, is about 6 cm and itis very close to the value of 5.7 cm using Eq. 91.

3.4. Upward liquid jets

We simulate upward liquid jet based on a geometry shown in Fig. 17. InFig. 17 SLOW and SUP represent the lower and upper section of the nozzleoutlet respectively. Two different nozzle configurations are used in our sim-ulations. In the first case, both upper and lower surfaces are circles with thesame diameter of 3.2 mm. In the second case, the lower surface is a circle butthe upper surface is an ellipse which has the same cross sectional area as thelower surface and the ratio of its major axis and minor axis is 3. The nozzlehas an inlet of diameter (D) of 17.4 mm, a wall thickness H1 of 3 mm, anda height H2 of 9.4 mm. Three levels of adaptive mesh refinement are usedfor these cases and the effective grid size is about D1/10. i.e., 10 grid pointsper nozzle outlet diameter. The physical properties of the liquid and air arethe same as that in the downward jet example.

In the first case, a uniform velocity v0 = 4.0 cm is applied at the nozzleinlet. Fig. 18 shows the jet interface evolution with time. The maximum jetheight is 14 cm. The jet height first increase, reach the maximum height andthen decrease with water accumulating at the jet tip because of the gravityeffect.

In the second case the same nozzle inlet velocity of 4.0 cm/s is applied.Fig. 19 shows the jet interface evolution. It is noticed that after leaving thenozzle, the major axis and minor axis of the jet switch. This phenomenonwas discussed by (Lin, 2003) who argued that this switch is due to the effectof surface tension, which causes the jet section vibration about a equilibriumcircle shape.

3.5. Binary collisions of two water drops

Binary collisions of liquid drops have been experimentally studied byAshgriz and Poo (1990) and Orme (1997) and numerically studied by Tanguy

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D

H1

H2

SLOW

SUP

Figure 17: Illustration of nozzle geometry for upward jet. SUP is the uppersection of nozzle outlet, SLOW is the lower section of nozzle outlet, D is thediameter of nozzle inlet, H1 is the nozzle wall thickness and H2 is the nozzleheight.

Figure 18: Snapshots of upward liquid jet. A round nozzle is used.

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Figure 19: Snapshots of upward liquid jet. The nozzle has a round lowsurface and an elliptical top surface

and Berlemont (2005), Chen and Yang (2014), and Nikolopoulos et al. (2009).For equal sized head-on collisions the outcomes depend on the Reynoldsnumber, Weber number which are defined as follows:

Re =ρdu

µWe =

ρdu2

σ

where ρ and µ are the liquid density and viscosity, respectively, d is thedrop diameter, u is the relative velocity of the two drops, and σ is the liquidsurface tension. With the Reynolds number in the range of 500 and 4000,Ashgriz and Poo found that the Reynolds number did not play a significantrole on the outcome (Ashgriz and Poo, 1990). In Figure 20, we show resultsof the simulation of the head-on collision of two equal sized water drops atthe Weber number of 25, 40 and 96. In the experiments, it was found thatreflexive separation occurs for a Weber number greater than nineteen andcoalescence or bouncing occurs for a Weber number less than twenty. Ournumerical results (Figure 20) corroborate these results.

3.6. Binary collisions of one water drop and one diesel drop

In Figure 21, we show results of the collision of a diesel oil drop with awater drop. These results can be compared with the experimental results

41

reported by Chen and Chen (2006). As with the case for the collision of twowater drops, again we have agreement between simulation and experimentfor capturing the Weber number cutoff separating the coalescence regimefrom the reflexive separation regime.

3.7. Impingement of droplet on a thin liquid film

The complex phenomenon of droplet impingement on a thin film phe-nomenon is characterized by the Reynolds number Re, the Weber numberWe, the Ohnesorge number Oh and the nondimensional film thickness Hdefined as follows:

Re = ρv0D/µWe = ρv2

0D/σOh = µ/

√ρσD

H = h/D

(92)

where v0 is the impact velocity, D is the droplet diameter and h is the filmthickness. Also, a nondimensional time T = tv0/D is introduced and it is setto zero at the first contact between the droplet and the film.

At high We numbers, a crown structure is formed on the film surface andliquid jets or secondary droplets are splashed from the crown rim. This iscalled the splashing regime. At low We numbers, the droplet may depositon the film surface without secondary droplets. This is called the deposi-tion regime. The critical We number separating the splashing regime andthe deposition regime is studied by many researchers (Cossali et al., 1997;Okawa et al., 2006). Two Weber numbers are simulated: We = 100 whichcorresponds to the deposition regime and We = 600 which corresponds tothe splashing regime.

Fig. 22 shows the time evolution of the deposition process and Fig. 23shows the splashing process. The effective grid size is about D/70 for thedeposition case and D/135 for the splashing case. Different Rieber (Rieberand Frohn, 1999) who introduced random disturbance in his simulation, norandom disturbance was introduced in our simulation. The initial distancefrom the droplet to the film is 0.13D. For the deposition regime, as shownin Fig. 22, a crown-like structure is formed at the first stage of the impact.This crown-like structure travels outwards with a maximum crown height of0.5D without any secondary droplet splashing. For the splashing regime, asshown in Fig. 23, fingers are formed at the top of crown rim. These fingersare then breakup into secondary droplets due to Rayleigh instablility.

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3.8. Impingement of droplet onto a solid wall

The impingement of a droplet on a smooth solid wall can result in differentregimes such as spreading, splashing, receding and rebounding (Yarin, 2006).The Weber number, Ohnesorge number as well as the contact angle play animportant role in the impact dynamics. We choose the droplet diameterD = 3.6 mm, density ρ = 1.0× 103 kg/m3, viscosity µ = 8.67× 10−4 kg/m.s,surface tension σ = 7.17 × 10−2 kg/s2 and impact velocity v0 = 0.77 m/s,which correspond to We = 30 and Oh = 0.0017. A dynamic contact anglemodel from (Jiang et al., 1979) is used, and the equilibrium contact angle isset to 87. Two levels of grid adaptation are used and the effective grid sizeis D/72.

Figure 24 shows the evolution of the droplet shape. The computed dropletshape is also compared with experimental data from Kim and Chun(Kim andChun, 2001). As can be seen from Fig. 24, the experimental results and thenumerical results are in good agreement.

4. Concluding remarks

A new method was presented to study incompressible flows involvingmore than two materials. In this method, the MOF interface reconstructionmethod was used to capture the material interfaces. The directionally splitcell integrated semi-Lagrangian method was used to advect interfaces andmomentum. The block structured adaptive mesh refinement was used toincrease the resolution near material interfaces. Various 2D, 3D axisymmetricand 3D problems were simulated. Comparisons were made with analyticalor experimental results. Our simulation showed that the proposed methodis able to simulate multiphase problems involving more than two materials.The method is also able to capture the contact line dynamics and triplejunctions.

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Figure 20: Numerical simulation of head-on collision of two water drops.From top to bottom Weber Number is 15, 25, 40, and 96, respectively. Thecomputational grid is a block structured mesh with 16 cell per initial dropdiameter. One grid refinement is used(effective fine grid resolution is 32cells per initial drop diameter). Results are in agreement with experimentsAshgriz and Poo (1990). We capture the correct transition point for reflexiveseparation.

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Figure 21: Numerical simulation of head-on collision of a diesel oil drop(cyan) with a water drop (gold) and resulting encapsulation. Weber Numberequals 9.6, 45.3, and 58.9 for the top, middle and bottom rows respectively.Dimensionless drop diameter is 1 and computational domain size is 0 <r < 1.3, −3.9 < z < 3.9. The computational grid is a block structureddynamic adaptive mesh with 48x288 coarse grid cells and 1 additional level ofadaptivity (effective fine grid resolution is 96x576). Results are in agreementwith experiments Chen and Chen (2006). We capture the correct transitionpoint for reflexive separation. Simulations are done in 3d axisymmetric (RZ)coordinate system.

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Figure 22: Snapshot of droplet-liquid film impingement, We = 100. Fromleft to right, top to bottom: T = 0.0, T = 1.0, T = 2.0, and T = 3.0

Figure 23: Snapshot of droplet-liquid film impingement, We = 600.

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Figure 24: Droplet impinging on a solid wall. Left, experimental results from(Kim and Chun, 2001). Right, present numerical results.

53