1 Introduction 1.1 Two Fluid Flows

A flow situation wherein two immiscible, non reacting fluids flow together is known as

two-fluid flow. Some instances where they occur in engineering systems are flow of molten

metal during casting, flow in pipe carrying steam and water, liquid sloshing in tankers, liquid jet

issuing into gaseous environments, bubble formation, breakup and propagation in the

surrounding fluid.

Analysis of such flows occurring in engineering systems is required to be done. One thing

that separates the analysis of two-fluid flows from single fluid flow is existence of interface

(boundary which separates the two fluids) across which step change in properties exists. At the

interface the surface tension comes into effect and in some flow situations phase change process

at interface also occurs.

1.2 Numerical Methods for Two-Fluid Flows Physical experiments in two-fluid flow situations are difficult and expensive to perform,

numerical simulation methodologies on the other hand can be used for variety of problems and

prove as a relative cheaper tool for analysis.

Numerical solution of single fluid problem results in velocity, pressure and temperature (if

it is non-isothermal flow) at discrete points in the flow/computational domain. In addition to that,

in two-fluid flow simulation methodology, representation of interface in the computational

domain, its advancement with respect to time and application of the boundary conditions at the

interface are main issues.

There exist many two-fluid flow simulation methodologies such as Marker and Cell

(MAC) method [Hyman (1984)], Volume of Fluid method [Hirt and Nichols (1981)] and Level

set method [Sussman et al. (1994)]. In all the methods a flag or some maker particles are used to

represent region of a fluid or the interface. Governing equation of that flag or any marker

particle is derived from physical principles and is a pure convection equation.

In Marker and Cell (MAC) method, mass-less lagrangian particles are spread in one of the

fluid. Navier-Stokes equations are solved in the region occupied by that fluid. Once the

velocities in the respective cells are known their values are interpolated to marker particles and

then those marker particles are moved in the grid with interpolated velocities. Interface is defined

1

as the “boundary” between regions with and without marker particles. This method has no

applicability restriction but it requires careful implementation and also the overall cost of

simulation is enormous.

In Volume of Fluid method, interface motion is not followed but volume fraction of the

cell (ratio of volume of one fluid to volume of the cell) is used to store information of a region of

a fluid. Volume of Fluid governing equations is derived naturally from the mass conservation

law and its solution provides updated information of the region occupied by that fluid. Interface

information is derived from the volume of fluid function field based on the approximation of the

interface shape. Mass conservation characteristic of volume of fluid methods are good but

because of discontinuous nature of volume of fluid function, surface tension force is calculation

is inaccurate.

Level set methods find a very large range of applications like two-fluid flows, flame

propagation, crystal growth, image processing and many more. In level set method used fro two-

fluid flows the interface is modeled as the zero level set of a smooth function known as level set

function, defined over whole computational domain. The level set function is advected with the

background flow field. As the level set function is a continuous function calculation of interface

normal and curvature are easy and accurate but level set methods are known to violate mass

conservation.

1.3 Literature Survey

In this section topic based literature survey for Volume of Fluid (VOF), Level Set Method

(LSM) and Combined Level Set-Volume of Fluid (CLSVOF) method is presented. One of the

earliest and most referred works on Volume of Fluid (VOF) and Level Set (LS) methods are

discussed in detail in subsection 1.3.1 and 1.3.3, respectively. Further developments in these

methods are reviewed briefly in subsection 1.3.2 for VOF method and 1.3.4 for LS method.

Thereafter, work on Combined Level Set Volume of Fluid (CLSVOF) method is mentioned in

subsection 1.3.5. Finally, the literature review is summarized in subsection 1.3.6.

1.3.1 VOF Method proposed by Hirt and Nichols (1981) It was mentioned in last section that the main issues in any two-fluid simulation

methodology are,

(a) Representation of interface in the computational domain.

(b) Evolution of interface in space and time.

(c) Application of the boundary conditions at the interface.

2

In this section, solutions to the above mentioned issues by VOF method of Hirt and

Nichols (1981) are presented, which ultimately sum up to VOF two-fluid solution methodology.

1.3.1.1 Representation of Interface in VOF method In VOF method the interface data is converted into discrete volume data (volume occupied

by a particular fluid in a cell). The volume data is retained in from of fractional volume of the

cell (C) (ratio of volume of one fluid in a cell to volume of the cell). Fig. 1.1(a) shows the actual

interface position in a computational domain and Fig. 1.1(b) shows the values of volume

fractions or VOF function (C) in all computational cells. Such representation of interface avoids

the logical problems involved in explicit interface tracking; also one variable per cell is required

to store the information of the cell.

(a)

(b)

Figure 1.1: Two-fluid flow situation: (a) Actual Interface Position (b) Discretized domain with VOF function values.

When the interface information is required out of the volume fraction field, it is done so

geometrically. In a discrete space the interface has to be represented by a curve in a cell. The

collection of curves in domain represents the interface. Hirt and Nichols (1981) approximated

the interface is as a line segment in the cell which can be aligned to any of the co-ordinate axis

i.e. either horizontal or vertical in Cartesian co-ordinates. This type of interface reconstruction is

known as Simple Line Interface Calculation (SLIC).

The SLIC interface reconstruction method involves (a) determination of the alignment of

the interface, (b) determination of the position of one fluid relative to other across the interface

and (c) locating the interface in cell.

Choice of the alignment of the interface is made by calculating the gradient of VOF

function along x and y direction. The interface lies normal to the direction in which the absolute

value of gradient is more i.e. if value of C changes more along x-direction than along y-direction,

then the interface is vertical and vice versa.

3

Hirt and Nichols (1981) presented expressions of gradient of C for non-uniform Cartesian

grids; the simplified expressions for 2D square Cartesian grids shown in Fig. 1.3(b) are derived

as

( ) ( )( ) ( )

* * * *

* *

* *

,2 2

,,

,

E W N S

W E

S N

C C C CC Cx x y y

WhereC C C C y C C C C yNW W SW NE E SEC C C C x C C C C xSW S SE NW N NE

− −∂ ∂= =

∂ Δ ∂ Δ

= + + Δ = + + Δ

= + + Δ = + + Δ

(1.1)

It is seen from Eq. 1.1 that the expressions of the gradient are merely a measure of change

in value of C in x and y direction. The relative position of the fluids across the interface is

determined by examining the sign the gradient of C along the normal direction to the interface.

Consider a case when the interface is found to be vertical and the sign of gradient of C along x is

negative, this means that the value of *WC is more than *

EC and so the fluid (on which the VOF

function is defined) lies on the west side to the interface. The possible cases of interface

orientation and relative position of fluid is shown in Fig. 1.2(a).

(a) (b)

(c)

Figure 1.2: Simple Line Interface Calculation (SLIC) (a) Possible cases of interface orientation and relative position of fluid (b) Circular Fluid Body (c) SLIC reconstruction of circular fluid body.

After determining the orientation of interface and position of fluid, co-ordinates of a line

are determined such that the reconstructed interface divides the cell in to two parts with correct

amount of fluid volume lying on the fluid side. The SLIC reconstruction of a circular fluid body

(Fig. 1.2(b)) is shown in Fig. 1.2(c), it is seen that SLIC is very inaccurate but it is the simplest

method of interface reconstruction.

1.3.1.2 Evolution of VOF function in Space and Time The VOF method of Hirt and Nichols (1981) is used for free surface flows, the dynamics

of lighter fluid in comparison to heavier fluid is neglected. Navier-Stokes equations are solved

4

only in the region occupied by heavier fluid; the interface is treated as computational boundary.

The governing equations are,

Continuity Equation:

0u∇⋅ = (1.2)

Momentum Equation:

( ) 2 ˆu uu p u gjt

ρ μ ρ∂⎡ ⎤+∇ ⋅ = −∇ + ∇ −⎢ ⎥∂⎣ ⎦ (1.3)

For accurate calculation of interface position with time and space the volume fraction field

must be evolved accurately. The governing equation of VOF function as given by Hirt and

Nichols (1981) is,

0C u Ct

∂+ ⋅∇ =

∂ (1.4)

The reasoning given by Hirt and Nichols (1981) for the use of Eq. 1.4 is, the VOF function

C simply moves with the fluid and in a lagrangian mesh the value of VOF function will remain

constant with time. This statement seems to be inspired by the MAC method, wherein the marker

particles are simply convected with the flow. Conservative form of VOF governing equation (Eq.

1.4) is obtained by using continuity equation (Eq. 1.2) as

( ) 0C Cut

∂+∇⋅ =

∂ (1.5)

Eq. 1.5 is solved explicitly after Eq. 1.2 and Eq. 1.3 are solved in domain. Eq. 1.5 is a pure

convection equation and volume conservation equation. When Eq. 1.5 is integrated over a cell,

flux of C at the cell face is needs to be evaluated. A convection scheme is used to interpolate the

value of the C at the cell face, but use of a convection scheme in case of VOF method will smear

the VOF function in partially filled cells because of interpolation. Due to smearing, the interface

will loose its definition.

To overcome this problem, Hirt and Nichols (1981) used the donor-acceptor method.

Donor-acceptor method is combination of upwind and downwind difference schemes to find the

flux of fluid at cell faces in partially filled cells. The time advanced value of VOF function (C) is

found as

( )1

, , ,

1 n nf

f e w n sP

C C Volume of Fluid Fluxed OutV

+

=

= −Δ ∑ (1.6)

The expression for the amount of fluid fluxed out of a cell face in one time step is

( ) ( ) { }( ) ,f AD f D D ffVolume of Fluid Fluxed Out sign u MIN C u t AF C x S= Δ + Δ Δ (1.7)

Where, Additional Fluid, ( ) ( ){ }1 1 ,0.0AD f D DAF MAX C u t C x= − Δ − − Δ ,

5

Sign of velocity at face, ( ) ( )f f fsign u u abs u=

In Eq. 1.7, fSΔ is the cell face area vector. Subscript A means acceptor/downwind cell, D

means donor/upwind cell and AD means either donor or acceptor cell, the choice of the donor or

acceptor cell is made in following way,

, if interface moves normal to itself , Otherwise

AAD

D

CC

C⎧

= ⎨⎩

(1.8)

However, if the acceptor cell is empty or the cell upstream of the donor cell is empty, then

CAD = CA regardless of the orientation of the interface, this is done to ensure that a donor cell

must fill before any fluid can enter a downstream empty cell.

For the cells which are completely filled or empty, donor cell approximation (CAD = CD) is

used. Even if the acceptor cell is used (CAD = CA) to calculate amount of fluid fluxed, the amount

of fluid fluxed is always subtracted from or added to the donor cell fluid. The MIN feature in Eq.

1.7 prevents fluxing more fluid than a cell can donate. Additional fluid, AF is the amount of fluid

in the donor cell that is needed to be fluxed in case the acceptor cell (CAD = CA) is used to

calculate flux to avoid incorrect flux calculation.

Donor-Acceptor method is basically a 1D method and it is extended in 2D by operator

splitting i.e. intermediate values of volume fractions are calculated by solving 1D form of Eq. 1.5

in one of the directions and then based on those intermediate values, final volume fractions are

obtained by solving the other 1D form of Eq. 1.5 in the remaining direction.

1.3.1.3 Application of Boundary Condition at Interface As already mentioned, Hirt and Nichols (1981) treated the interface as computational

boundary. Surface tension on the interface is neglected and pressure is assumed to be constant on

the interface. To implement this boundary condition, pressure in the partially filled cell is set

equal to pressure obtained by linear interpolation (or extrapolation) between the desired interface

pressure and pressure of a fully filled nearest neighbor cell. Navier-Stokes equations are not

solved in the partially filled cells.

1.3.1.4 VOF algorithm of Hirt and Nichols (1981)

Hirt and Nichols (1981) used a MAC type numerical algorithm to solve Navier-Stokes

equations (Eq. 1.2 and Eq. 1.3). Their VOF algorithm can be summarized as

(1) Initialize velocities, pressure and volume fractions in all cells.

(2) All the partially filled cells are marked as boundary cells.

(3) Using old time level pressure, velocities are predicted.

6

(4) Predicted velocities in general do not satisfy the continuity equation. Pressure in the

interior cells is changed and the velocity change due to the pressure change is added to

predicted velocities. Pressure in the partially filled cells is calculated as explained in

sect. 1.3.1.3. Iterations are done till the continuity equation is satisfied.

(5) The VOF equation is solved explicitly using the newly calculated velocity field.

(6) Go to Step 2 until the end time or steady state.

Many times the values of volume fractions calculated using donor acceptor method exceed

1 or become negative, in such case the volume fractions are simply reset to 1 or 0 respectively.

This resetting introduces mass error.

1.3.2 Developments in VOF method

There has been a considerable work in VOF method since the original algorithm. There

have been developments in nearly all aspects of VOF method like representation of interface,

evolution of interface with time, application of boundary conditions and overall VOF algorithm.

Major developments in all the aspects of VOF method are reviewed and presented in this section.

1.3.2.1 Development in Representation of Interface The SLIC method of Hirt and Nichols (1981) is not able to reconstruct a curved interface

accurately. There have been many different approximations about the shape of the interface

inside a cell.

(a)

(b) Figure 1.3: Piecewise Linear Interface Calculation (PLIC) Method (a) Interface

representation (b) Stencil required for determination of normal in Young’s method.

One step ahead of SLIC is to assume that the interface inside a cell can be variably aligned,

as shown in Fig. 1.3 (a). Interface inside a cell is represented by,

x yn x n y d+ = (1.9)

nx ny

d

7

,x yn n are components of normal vector (pointing in to the fluid) of interface and d is the

line constant in Eq. 1.9. Analogous to SLIC the main steps in PLIC method are (a) determination

of interface normal, (b) determination of fluid position relative to interface and (c) locating the

interface in the cell.

Different PLIC methods are characterized by different interface normal finding techniques.

Youngs’ (1982) gave simplest of all normal finding methods. In Youngs’ (1982) method the

gradients of the VOF function are taken as the components of normal vector. Generally the

Youngs’ method is used for square cartesian grids but extension of Young’s method to non-

uniform cartesian grids is presented by Rider and Kothe (1998). A stencil of 3x3 cells is used to

determine the interface normal as seen from Fig. 1.3(b). Components of normal vector,

and x yn n for square cartesian grids are given as

( )e wx

C Cn

x−

=Δ

(1.10)

n sy

C Cny−

=Δ

(1.11)

With ( )0.25ne P E NE NC C C C C= + + + , ( )0.25se P E SE SC C C C C= + + +

( )0.25nw P W NW NC C C C C= + + + , ( )0.25sw P W SW SC C C C C= + + +

( )0.5w nw swC C C= + , ( )0.5s sw seC C C= + , ( )0.5e se neC C C= + , ( )0.5n ne nwC C C= +

Owing to the PLIC assumption, the interface segment can be inclined in the cell arbitrarily

leading to existence of various cases of interface inclinations and relative fluid positions. The

relative position of fluid with the interface by a procedure similar to SLIC method, as an

example if both nx and ny are negative then the fluid is likely to be located in the south-west

corner of cell as shown in Fig. 1.3(a).

(a)

(b)

Figure 1.4: Locating interface in a cell in PLIC method (a) Iterations performed by positioning interface without changing alignment between maximum and minimum possible limits. (b) Reconstructed fluid polygon at the end of the iterations.

Iter 1

Iter 2

Fluid Polygon Min

Max

8

Locating the interface segment in the cell is equivalent of finding the line constant, d . As

such there is no fixed working method to perform this task, Rider and Kothe (1998) used an

iterative procedure. In their method the interface segment of known alignment and fluid position

is placed in a cell corresponding to maximum and minimum volume fractions possible for that

configuration of interface. Then the interface is moved iteratively in the cell while keeping its

alignment constant till the volume enclosed by the reconstructed interface segment represents the

actual fluid volume in that cell as shown in Fig. 1.4(a). The fluid region representing the volume

fraction in a cell is known as fluid polygon, shown in Fig. 1.4(b).

Rudman (1997) has given implementation of Youngs’ method wherein algebraic

expressions are used and geometrical reconstruction is completely avoided. Expressions given by

Rudman (1997) are derived from the same geometrical principles of Youngs’ (1982) and provide

very easy implementation of Youngs’ method.

(a)

(b)

Figure 1.5: Least Squares Volume-of-fluid Interface Reconstruction Algorithm (LVIRA) (a) Calculation of volume fractions in neighboring cells by extending interface. (b) Changing the orientation of interface to minimize error in neighboring cells, the volume fraction in the center cell has to be same as actual volume at all times.

Pilliod and Puckett (2004) gave a criterion to determine order of any PLIC scheme; any

method is of second order if it is able to exactly reconstruct a line that is not horizontal or

vertical. Youngs’ method [ Youngs’ (1981)] according to Pilliod and Puckett (2004) is having

order between 1 and 2. Pilliod and Puckett (2004) proposed a second order interface normal

calculation procedure named as Least Squares Volume-of-fluid Interface Reconstruction

Algorithm (LVIRA).

In LVIRA, firstly interface is reconstructed in a cell by Youngs’ method. The

reconstructed interface is extended to its neighboring cells in a 3x3 stencil. Volume fractions are

determined in the all the neighboring cells due to intersection by extending the reconstructed

interface as seen from Fig. 1.5(a). The difference of actual volume fractions of the cells and the

calculated volume fractions in the neighboring cells is defined as error. The purpose of LVIRA is

to minimize root mean squared value of error (rms error) for all neighboring cells. The error is

9

minimized by changing the orientation of interface under the constraint that this reconstructed

interface segment exactly reproduces the volume fraction in the center cell as shown in Fig.

1.5(b) i.e. volume fraction calculated in center cell is always same as actual volume fraction.

In summary, LVIRA minimizes the error due to PLIC approximation in neighborhood of

3x3 cells. The discontinuity between successive interface segments in LVIRA is found to be less

than for Youngs’ method. Improvement over LVIRA is also given by Pilliod and Puckett (2004),

namely the Efficient Least squares VOF Interface Reconstruction Algorithm (ELVIRA). In

ELVIRA the basic methodology is same but the error norm used for minimization is different.

Figure 1.6: Illustration of Circle Fit technique: A circle is fitted through midpoints of interface segments of two selected neighbors. New direction of interface normal is the along the line joining the midpoint of interface segment of center cell and center of circle.

Circle fit technique proposed by Mosso et al. (1996) was successfully implemented by

Garrioch and Baliga (2006). In this technique, initially the interface segments are constructed

using Youngs’ method. With reference to an interface segment in cell, a circle is drawn through

its midpoint and the midpoints of two its neighbors as shown in Fig. 1.6. The new interface

normal is taken to be in the direction joining the midpoint of reference line segment and

midpoint of circle. As the change of normal in one cell affects the circle calculation of

neighboring cell, iterations are needed to be performed. Iterations are stopped when the interface

normal calculated during successive iterations is same.

Generally, the interface is approximated to be located inside a cell but in Flux Line-

segment model for Advection and Interface Reconstruction (FLAIR) of Ashgriz and Poo (1991),

the interface segments are fitted at the boundary of every two neighboring cells.

In Parabolic Reconstruction of Surface Tension Force (PROST) method of Renardy and

Renardy (2002) the shape of the interface in a cell is approximated as a parabola. The parameters

of the curve are found such that the difference between the known volume fractions and those

based on the parabolic reconstruction is minimized in the 3x3 stencil of cells.

ˆ old n

ˆ new n

10

eu tΔ

Many researchers [Rudman (1997), Ubbink and Issa (1999)] have represented the interface

by volume fraction contour of level 0.5 while plotting the interface. By representing the interface

by a contour plot, smooth interface is obtained which is otherwise difficult to get from a PLIC

method.

1.3.2.2 Development in Volume Fraction Advection The donor-acceptor method used for volume advection preserves the step discontinuity of

volume fraction field and generally maintains global mass conservation. But it is found in

different studies [ Rudman (1997), Rider at al. (1995)] that donor-acceptor method gives

inaccurate results in vortical flows as it introduces fluid breakup due to over all 1D nature of

algorithm.

To overcome the limitations of donor-acceptor method, various methods are used for

solution of VOF advection equation such as the geometric advection method, flux corrected

transport method and high resolution convection schemes are discussed in this section.

Rider and Kothe (1998) presented the geometrical interpretation of donor-acceptor method.

Consider a case as shown in Fig. 1.7(a), flux calculation across a face can be calculated by

donor-acceptor method in following way. In donor-acceptor method (Eq. 1.7) the quantity

f fu t sΔ Δ represents the net fluid fluxed across a cell face. If the volume fraction of the

donor/upwind cell is multiplied to the net fluid flux then the resulting quantity ( )D f fC u t sΔ Δ is

the amount of fluid of interest fluxed across east face of donor cell.

Figure 1.7: Geometric interpretation of advection (a) Advection using donor-acceptor

method. (b) Geometric advection.

The same information can be obtained geometrically as shown in Fig. 1.7(b). It is seen that

the area of polygon ‘f1f2f3f4’ represents the quantity f fu t sΔ Δ , polygon ‘f1f2f3f4’ is known as the

flux polygon. Polygon ‘e1e2e3e4’ represents the fluid of interest inside the cell; polygon ‘e1e2e3e4’

(fluid polygon) is constructed by interface reconstruction algorithms. The common area to both

the polygons, represented by polygon ‘c1c2c3c4’ is the amount of fluid of interest is fluxed across

ue

f1, c1 f2, e2, c2

f4 f3

e1

e3, c3 e4 c4 ˆ

es yiΔ = Δ

Donor Cell Acceptor Cell

CAD = CD

ue

11

a cell. Area of the polygon has the value of U f fC u t sΔ Δ in this case. The advantage of this

geometric interpretation is that better interface reconstruction algorithm can be used in the

advection procedure i.e. PLIC methods combined with geometric advection method to solve

VOF advection equation.

After the fluid polygon is constructed using interface reconstruction procedure, flux

polygons at all faces are geometrically constructed. In Fig. 1.7(b) flux polygon (‘f1f2f3f4’) is

constructed at the east face of a cell, vertices 2 and 3 of the flux polygon are at cell vertex se and

ne respectively, so for

( ) ( )( ) ( )

2 2,

3 3,

, ,

, ,se seeast flux polygon

ne neeast flux polygon

f x y x y

f x y x y

→ =

→ = (1.12)

As the area enclosed by the flux polygon should be equal to f fu t sΔ Δ , the co-ordinates of

remaining two vertices of the flux polygon are,

( ) ( )( ) ( )

1 1,

4 4,

, ,

, ,se e seeast flux polygon

ne e neeast flux polygon

f x y x u t y

f x y x u t y

→ = − Δ

→ = − Δ

The area of intersection of fluid polygon and flux polygon represents the amount of fluid

of interest fluxed across the cell face. As only the normal velocity at the cell face was

considered, the fluxes are calculated in operator split way, which means the advection algorithm

is first order. In fact, SLIC method coupled with operator split geometric advection gives exact

result as donor-acceptor method.

Figure 1.8: Determination of co-ordinates of multi dimensional flux polygon.

Rider and Kothe (1998) in their geometric advection method incorporated the effect of

transverse velocity at a face to avoid the operator splitting and enable solution of VOF equation

in single step. They constructed multi-dimensional flux polygon. Calculation of co-ordinates of

the vertices of multi-dimensional flux polygon is demonstrated using Fig. 1.8. In Fig. 1.8 the two

vertices of the east flux polygon, 2 and 3 are same as that for operator split fluid polygon (Eq.

1.12). Since the flux polygon represents multi-dimensional flux passing across a face, the co-

f1

f3

f4

*sev tΔ

*nev tΔ

u tΔ

f2

12

ordinates of other two vertices of the multidimensional flux use transverse velocity at cell

vertices, the other two co-ordinates of multi-dimensional flux polygon are determined as

( ) ( )*1 1,

, ,se e se seeast flux polygonf x y x u t y v t→ = − Δ − Δ

( ) ( )*4 4,

, ,ne e ne neeast flux polygonf x y x u t y v t→ = − Δ − Δ

Different multi-dimensional geometric methods are characterized by different procedures

to get vertex velocities. Rider and Kothe (1998) used upwinding to get vertex velocity, for east

face in case of v velocity,

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

* *, , i,j, ,

* *1, 1, i,j, ,

if 0

if 0

ne n se s ei j i ji j i j

ne n se s ei j i ji j i j

v v and v v u

v v and v v u+ +

= = ≥

= = <

Garrioch and Baliga (2006) calculated a common velocity for both vertices as

( ) ( )( ) ( )

* *i,j

* *1 i,j

0.5 if 0

0.5 if 0ne se n s ei

ne se n s ei

v v v v u

v v v v u+

= = + ≥

= = + < (1.13)

Similarly, co-ordinates of vertices of flux polygons for other faces are determined. Once

fluid polygon and all the flux polygons are in order, the common area to both fluid polygon and

each of flux polygons is determined.

The common area shown in Fig. 1.9, represents amount of a particular fluid passing across

east cell face in one time step. The total common area for all the faces is the total amount of a

particular fluid fluxed out of the cell. The new volume fraction is determined by solving Eq. 1.6.

The geometric advection procedure is only applied in the cells which have interface, for all other

cells first order upwind gives accurate results.

Figure 1.9: Common area of Fluid polygon and Flux polygon at east face: Shaded area is the

common to Fluid polygon and Flux polygon, representing the amount of fluid fluxed out from east face in one time step.

13

Donor-acceptor method used by Hirt and Nichols (1981), discussed in subsection 1.3.1.2

is a variation of Flux Corrected Transport (FCT) method but it is 1D algorithm. Zalesak (1979)

had presented multidimensional algorithm of flux corrected transport method but did not use it to

solve VOF governing equation. The basic methodology of FCT method as given by Zalesak

(1979) is,

(a) Fluid flux across a cell face is calculated using a lower order scheme (e.g. FOU). Use

of lower order flux introduces the numerical diffusion in the solution but does not

produce overshoot (C > 1) or undershoot (C < 0).

(b) Secondly, fluid flux across that cell face is calculated using any higher order scheme,

Use of lower order flux introduces minimum numerical diffusion in the solution but may

produce overshoot or undershoot.

(c) The difference of the higher order flux and lower order flux is known as anti-

diffusive flux.

(d) The maximum possible flux from the anti-diffusive flux is added to the lower order

flux without producing overshoots or undershoots. The maximum possible anti-diffusive

flux is calculated by procedure called flux limiting.

(e) The end result is that a flux across a face is obtained having minimum numerical

diffusion without producing overshoots and undershoots.

Although Rudman (1997) used FCT method of Zalesak (1979) to solve VOF governing

equations, the results were not comparable to geometric advection method. On the other hand,

implementation of geometric advection method takes lot of effort and it is not easily extendible

to unstructured grids and 3D whereas FCT method is easy to implement and extend.

Alternative to geometric method and FCT is use of high resolution convection schemes.

When the VOF governing equation is discretized, value of the volume fraction is required to be

calculated at cell face center. As values of volume fractions are available only at cell centers, a

method is used to interpolate or extrapolate the value of volume fraction using the neighboring

cell center values. The volume fractions distribution exhibits a step discontinuity and simple

interpolation in zone of discontinuity is an inaccurate approximation. Due to higher order

interpolation like in central difference scheme and QUICK, undershoots and overshoots known

as spurious oscillations occur at zone on discontinuity [Wang and Hutter (2001)].

High-resolution methods are nonlinear methods that use a limiting mechanism to control

the spurious oscillations at zone on discontinuity. In the limiting mechanism, the coefficients of

interpolating polynomial are modified by multiplying them with a slope limiter. A slope limiter

is a function of local slopes of the volume fraction. Many different interpolation schemes and

14

slope limiter functions exist [Wang and Hutter (2001)] and can be used in any combination

giving rise to many different schemes.

Rider et al. (1995) used a Piecewise Parabolic Method (PPM) [Colella and Woodward

(1984)] and Ubbink and Issa (1999) presented the Compressive Interface Capturing Scheme for

Arbitrary Meshes (CICSAM). CICSAM can be easily implemented on 2D and 3D unstructured

grids; it has also been implemented in latest version of one of the commercial software.

1.3.2.3 Development in Application of Boundary Condition at Interface Hirt and Nichols (1981) neglected lighter fluid of the two fluids and treated the interface

as a computational boundary. Application of pressure boundary condition in PLIC method is

more difficult. In PLIC method the interface may be inclined in a cell and interpolation of

pressure across an inclined interface will require extra effort during implementation. Also, in

some important problems like rise of gas bubble in a liquid, the gas phase (lighter fluid)

dynamics cannot be neglected.

To overcome the problem of implementation of boundary conditions and solution of

Navier-Stokes equations in complete domain, a single fluid model with mean properties is used

in modern VOF method [Rider et al. (1995), Rudman (1998), Welch and Wilson (2000)

Ginzburg and Wittum (2001), Garrioch and Baliga (2006)]. The mean properties are calculated

in all the cells by simple volume weighed average,

( )2 1 2m Cρ ρ ρ ρ= + − (1.14)

( )2 1 2m Cμ μ μ μ= + − (1.15)

Here, 1ρ and 2ρ are densities of fluid 1 and fluid 2 respectively, 1μ and 2μ are dynamic

viscosities of fluid 1 and fluid 2 respectively and C is the volume fraction of the cell, defined as

ratio volume of fluid 1 to volume of cell. For a case of incompressible, isothermal, immiscible

two-fluid flow situation the governing equations are:

Continuity Equation:

( ) 0mmu

tρ ρ∂

+∇⋅ =∂

(1.16)

Conservation of Momentum:

( ) ( ) ( )( )( ) ˆTmm m m ST

uuu p u u gj F

tρ

ρ μ ρ∂

+∇⋅ = −∇ +∇⋅ ∇ + ∇ − +∂

(1.17)

The expression for average density (Eq. 1.14) can be obtained from equation of

conservation of mass but the justification of using average viscosity (Eq. 1.15) is not explicitly

15

explained in literature. Ginzburg and Wittum (2001) have mentioned that, for immiscible phases,

density and viscosity are constant along a particle path and so the following relation holds,

0mmu

tρ ρ∂

+ ⋅∇ =∂

(1.18)

0mmu

tμ μ∂

+ ⋅∇ =∂

(1.19)

Using Eq. 1.14 and Eq. 1.15 in Eq. 1.18 and Eq. 1.19 respectively, VOF governing

equation (Eq. 1.4) is obtained. Rider et al. (1995) have given a similar justification. Validity of

averaging viscosity based on some physical principle and its effect on solution accuracy is not

mentioned in the literature. However, Welch and Wilson (2000) have simulated film boiling

using average properties and presented accurate results.

It is concluded that averaging viscosity is a pragmatically made numerical approximation

to keep the model simple. Because of the averaging of properties, the interface no longer remains

sharp as it is diffused within one cell width.

1.3.2.3.1 Surface Tension Model

Surface tension is modeled by most of the researchers using the Continuum Surface Force

(CSF) method of Brackbill et al. (1992). In CSF approach, surface tension is modeled as a body

force applied in a thin transition region near the interface. The body force approaches the true

surface force when the transition region becomes very small i.e. on very fine grid. In CSF

method there is no need of interface reconstruction, only the volume fraction field must be

known. The surface tension per unit volume by CSF model is,

ˆST sF nσκ δ= (1.20)

In Eq. 1.20, σ is the surface tension coefficient, κ and n̂ are curvature and the unit

normal vector of the interface, sδ is the Dirac delta function, which is zero everywhere except at

the interface.

In VOF methods, interface normal is calculated as

n C= ∇ and ˆ CnC

∇=∇

(1.21)

Curvature of the interface is calculated from the interface normal as

ˆ CnC

κ ∇⎛ ⎞= −∇ ⋅ = −∇ ⋅⎜ ⎟∇⎝ ⎠ (1.22)

Eq. 1.21 and Eq. 1.22 are discretized using central difference scheme. The Dirac delta

function in case of VOF method is given by Brackbill et al. (1992) as

s C nδ = ∇ = (1.23)

16

In VOF method, calculating interface normal and curvature using Eq. 1.21 and Eq. 1.22

involves calculating first order and second order derivatives of volume fractions. As the volume

fraction field has step discontinuity, use of central difference scheme is invalid and leads to

incorrect normal and curvature calculation.

Brackbill et al. (1992) suggested smoothing of volume fractions by convolving volume

fraction field by a radially symmetric kernel function. The smoothened volume fraction ( )C

field can be used to calculate interface normal and curvature. The smoothened volume fraction

from the raw volume fraction field is obtained as

, ,,

,

, ,

,

ni j k l

k lk l

ni j k l

k l

r rC K x y

hCr r

K x yh

−⎛ ⎞Δ Δ⎜ ⎟

⎝ ⎠=

−⎛ ⎞Δ Δ⎜ ⎟

⎝ ⎠

∑

∑ (1.24)

In Eq. 1.24, 2 2 2r x y= + , 2h x= Δ is the smoothing length or the width of diffuse interface.

A kernel function used must be twice continuously differentiable. One of the most widely

used kernel function is K8 kernel function

( ) ( )421 10

v if vK vOtherwise

⎧⎪ − <= ⎨⎪⎩

(1.25)

Interface curvature can also be calculated by interface reconstruction algorithms such as

circle fit technique and PROST without the smoothing of volume fraction field. Curvature

calculation is a major advantage of circle fit technique and is simply the inverse of radius of the

circle fitted in the cell. In PROST, as the equation of the interface (parabola) is twice

differentiable, the interface normal and curvature are calculated by differentiating the equation of

interface [Renardy and Renardy (2002)].

1.3.2.4 Modern VOF algorithm and some Results Most of the researchers use Projection method [ Peyret and Taylor (1983), Bell and Marcus

(1992) ] to solve Navier-Stokes equations, a modern VOF method algorithm based on single

fluid model can be summarized as

(1) Initialize velocities, pressure and volume fractions in all cells.

(2) Calculate Properties in all cells using Eq. 1.14 and Eq. 1.15

(3) Navier-Stokes equations are solved in the complete domain.

(4) VOF equation is solved explicitly using the newly calculated velocity field.

(5) Go to Step 2 until the end time or steady state.

17

It is possible to get undershoots (C < 0) or overshoots (C > 1) in volume fraction field, in

that case the volume fractions in the cell are reset to 0 or 1 respectively.

(a) (b)

(c)

Figure 1.10: Some Results of 2D VOF method (a) Bubble bursting through the interface in a partially filled cavity by Martinez et al.(2006). (b) Melting of ‘ASME’ logo nu Kothe et al. (1999) (c) Simulation of horizontal film boiling by Welch and Wilson (2000)].

Since introduction, tremendous progress has been made in implementing the VOF based

methods. Martinez et al. (2006) solved the problem of bubble bursting through the interface in a

partially filled cavity, shown in Fig. 1.10 (a). Fig. 1.10(b) shows the melting simulation of

‘ASME’ logo done by Kothe et al. (1999). Horizontal film boiling is simulated by Welch and

Wilson (2000), shown in Fig. 1.10(c).

18

Bonito et al. (2003) have developed a 3D VOF code for simulating motion of viscoelastic

fluid, shown in Fig. 1.11 (a). Chen and Li (1998) simulated the rise and coalescence of bubbles

in 3D shown in Fig. 1.11(b). Barkhudarov (2006) simulated the 3D break up of liquid jet, shown

in Fig. 1.11 (c).

(b)

(a) (c)

Figure 1.11: Some Results of 3D VOF method (a) Formation of thin filaments by pulling top and bottom faces apart of a viscoelastic fluid (non-Newtonian) fluid body by Bonito et al. (2003) (b) Coalescence of bubbles in same axial position while rising Chen and Li (1998) (c) 3D simulation of breakup of liquid jet by Barkhudarov (2006).

1.3.3 Level Set Method

Level Set method has been successfully used in problems like kinetic crystal growth,

dendritic solidification, computer vision and image processing, medical imaging and many

others. Sussman et al. (1994) presented a methodology to use level set methods for

incompressible two-fluid flows. In this section, representation of interface, advection of

interface, application of boundary conditions in level set methods and level set solution

methodology proposed by Sussman et al. (1994) is presented.

1.3.3.1 Representation of Interface in LSM

19

Interface representation in LSM is based upon concept of implicit surfaces, wherein a

function is defined in a domain having a fixed value at the interface.

In level set method the interface is modeled by defining a level set function, φ in the

complete domain i.e. for both the fluids. The level set function is chosen in such a way that it

assumes negative values in one fluid and positive values in other fluid and thus the interface is

defined where the value of level set function is zero. Level set function is taken as a signed

normal distance function measured from the interface as it satisfies requirements of level set

interface representation. Fig. 1.12 (a) shows a two fluid flow situation with actual interface, the

value of the level set function in the domain, shown in Fig. 1.12 (b).

-0.0651851

-0.107593

-0.0227777

0.0196297

0.06203710.104445

0.146852

0.189259

0.1044450.146852

0.06

2037

1

φ>0Fluid 2

Fluid 1φ<0

Fluid 2

Interface

Fluid 1

(a)

-0.0651851

-0.107593

-0.0227777

0.0196297

0.06203710.104445

0.146852

0.189259

0.104445

0.146852

0.06

2037

1

φ>0Fluid 2

Fluid 1φ<0

Interfaceφ=0

(b)

Figure 1.12: Interface representation in LS method (a) Actual Interface Position (b) Value of Level set function in the domain with contours of the normal distance function.

1.3.3.2 Advection of Interface in LSM In level set methods, the single fluid model followed in VOF method is used [Sussman et

al. (1994), Chang et al. (1996), Osher and Fedkiw (2001)]. For a case of incompressible,

isothermal, immiscible two-fluid flow situation the governing equations of flow in level set

method are:

Continuity Equation:

0u∇⋅ = (1.26)

Conservation of Momentum:

( ) ( )( )( )1 ˆTm ST

m

u uu p u u F gjt

μρ

∂ ⎡ ⎤+∇ ⋅ = −∇ +∇⋅ ∇ + ∇ + −⎢ ⎥⎣ ⎦∂ (1.27)

In a flow field that satisfies continuity and momentum equations, the interface simply

moves with the flow. Thus, the level set function is simply convected with flow and its temporal

evolution is modeled as

20

0utφ φ∂+ ⋅∇ =

∂ (1.28)

This is a pure convection equation of a smooth scalar function,φ . Use of a lower order

convection scheme will introduce numerical diffusion which will cause direct mass error.

Sussman et al. (1994) used a second order Essentially Non Oscillatory scheme (ENO) for spatial

discretization of Eq. 1.28.

1.3.3.3 Reinitialization of Level Set Function LSM Level set function is initialized as signed normal distance function and once the level set

function is advected, it no longer remains a normal distance function and become irregular. In a

fluid flow problem it is necessary to maintain constant width of band i.e. the distance across

which the properties vary near the interface. If the width of band is not same along the interface

then the Heaviside function (Eq. 1.33) will be calculated inaccurately and hence the properties

are calculated inaccurately (Eq. 1.32).

Also Sussman et al. (1994) mentioned that if level set function is not maintained as normal

distance function then steep gradients of level set function will appear and incorrect flux can be

calculated while solving Eq. 1.28.

To maintain the constant width of the band, level set function must be ‘repaired’ to be

signed normal distance function every time after the level set advection equation is solved but

the position of the interface ( 0φ = ) obtained by solution of Eq. 1.28 should not be changed. The

interface normal is calculated in level set method as

n φ= ∇ (1.29)

The unit normal of the interface is given as

n̂ φφ

∇=∇

(1.30)

If magnitude of the normal i.e. φ∇ is made equal to one in Eq. 1.32 then the unit normal

vector of the interface will be equal to interface normal ( n̂ n= ), under that case the level set

function will be normal distance function.

Sussman et al. (1994) presented a procedure to reinitialize a function ( oφ ) to signed normal

distance function (φ ) by obtaining steady state solution of,

( )( )1 0os

Sεφ φ φτ∂

+ ∇ − =∂

(1.31)

In Eq. 1.33, sτ is pseudo-time, ( )oSε φ is the smoothened sign function,

21

( )2 2

oo

o

Sεφφ

φ ε=

+, Generally, xε = Δ

The steady state solution of Eq. 1.33 has the property that the value of level set function at

the interface remains unchanged i.e. the location of 0φ = will remain unchanged but away from

the interface φ will converge to 1φ∇ = , i.e. level set function will be reinitialized to normal

distance function. Moreover, the sign function, ( )Sε φ also ensures that the normal distance

function remains as signed normal distance function.

Sussman et al. (1994) showed Eq. 1.32 to be a hyperbolic equation and they discretized it

using second order ENO scheme.

1.3.3.4 Application of Boundary Condition at Interface In LSM, the single fluid model is used to solve Navier-Stokes equations and thus there is

no need to explicitly implement the boundary conditions on interface. Sussman et al. (1994) used

smoothened fluid properties across the interface. Properties were smoothened in order to avoid

numerical instabilities in the solution. The smoothened/mean density and viscosity are defined as

( ) ( )( ) ( )

2 1 2

2 1 2

m

m

H

Hε

ε

ρ ρ ρ ρ φ

μ μ μ μ φ

= + −

= + − (1.32)

Here, 1ρ and 2ρ are densities of fluid 1 and fluid 2 respectively, 1μ and 2μ are dynamic

viscosities of fluid 1 and fluid 2 respectively, ( )Hε φ is the smeared out Heaviside function,

Sussman et al. (1994) used the expression of heaviside function as

( )

0, < -

1 sin , 2 2

> 1,

ifH if

ifε

φ εφ ε πφφ φ εε π ε

φ ε

⎧⎪ +⎪ ⎛ ⎞= + ≤⎨ ⎜ ⎟

⎝ ⎠⎪⎪⎩

(1.33)

ε is a small parameter usually taken as 1.5 xΔ , which means the interface is smeared

across 3 cells when φ is normal distance function, the smeared interface is known as the band

with width 3 cells.

Surface tension in LSM is modeled using the Continuum Surface Force (CSF) method of

Brackbill et al.(1992) as explained in section 1.1.3.1; Eq. 1.20 is gives the expression of surface

tension per unit volume by CSF model. The validity of use of CSF model in level set framework

is given by Chang et al. (1996).

22

The unit normal vector and curvature of the interface and the Dirac delta function required

to evaluate surface tension force in Eq. 1.20 are calculated in a different way in LSM than VOF

method.

The unit normal vector of the interface ( n̂ ) is calculated using Eq. 1.32 and Interface

curvature (κ ) is given as

n̂ φκφ

∇⎛ ⎞= −∇ ⋅ = −∇⋅⎜ ⎟∇⎝ ⎠ (1.34)

Level set function is a smooth function, so Eq. 1.32 and Eq. 1.34 are discretized using

central difference scheme. The Dirac delta function [Sussman et al. (1994)] in case of LSM is

given as

1 1 cos

20

s

if

Otherwise

πφ φ εδ ε ε

⎧ ⎛ ⎞⎛ ⎞+ <⎪ ⎜ ⎟⎜ ⎟= ⎝ ⎠⎨ ⎝ ⎠⎪⎩

(1.35)

1.3.3.5 Level Set Solution Methodology Sussman et al. (1994) used a projection method to solve Navier-Stokes equations in the

domain, their level set solution methodology can be summarized as

(1) Initialize velocities, pressure and level set function values in all cells.

(2) Calculate Heaviside Function in all cells using Eq. 1.30

(3) Calculate Properties in all cells using Eq. 1.29.

(4) Navier-Stokes equations are solved in the complete domain.

(5) Level Set governing equation is solved in the domain explicitly for the next time step

using the newly calculated velocity field.

(6) Level Set function is reinitialized to normal distance function by solving Eq. 1.34 till

steady state (of pseudo time) in complete domain.

(7) Go to Step 2 until the end time or steady state.

1.3.4 Developments in LS method LSM method has not undergone drastic change as far as interface representation and

advection and solution of governing equations is concerned but it is reported [ Chang et al.

(1996)] that level set method in its original form is unable to conserve mass in two-fluid flow

simulation. Major work has been done to identify and rectify the source of mass error in the

solution procedure.

23

Although not mentioned explicitly anywhere, one can see the difference of formulation in

LSM and VOF method. In VOF method, mean density is calculated using Eq. 1.14 and volume

fraction field is conserved by solving VOF governing equation (Eq. 1.5)). Using Eq. 1.14 in Eq.

1.5, continuity equation used in VOF method (Eq. 1.16) is obtained. Thus by solution of VOF

governing equation the mass conservation is ensured.

In level set method, the form of continuity equation (Eq. 1.26) is incorrect for describing a

two-phase flow situation; the correct form of continuity equation should be Eq. 1.16. Moreover,

in level set method, the mean density is calculated by Eq. 1.29 but conservation of Heaviside

function field is not enforced, hence the continuity equation used in VOF method (Eq. 1.16) is

not satisfied. Mass conservation in LSM will be ensured when advection of level set function

and its reinitialization will ensure conservation of Heaviside function field. As in level set

method of Sussman et al. (1994) there is no such constraint, mass conservation is violated.

1.3.4.1 Developments to reduce Mass Error Chang et al. (1996) observed that the formulation of level set method as given by Sussman

et al. (1996) was correct but the reinitialization procedure needed modification. They mentioned

that theoretically Eq. 1.33 will reinitialize the level set function to signed normal distance

function but due to numerical errors the interface position ( 0φ = ) changes during

reinitialization, the error induced accumulates and results in mass loss/gain. Chang et al. (1996)

presented an ‘area-preserving’ reinitialization equation,

( )( )( )1 0os

m m tφ κ φτ∂

+ − − ∇ =∂

(1.36)

In Eq. 1.36, mo is the total mass at initial condition, m (t) is the mass at time instant t. Ni et

al. (2006) presented a variable time-step method for reinitialization, in their method the value of

pseudo time-step is changed improve mass conservation. The time-step chosen is calculated by

that constraint given by Eq. 1.37.

( ) 0t domainH φ∂ =∫ (1.37)

1.3.4.2 Developments in solution of LS advection equation Jiang and Peng (2000) have presented higher order ENO scheme and the Weighted

Essentially Non Oscillatory scheme (WENO).

When Eq. 1.28 is discretized, gradient of level set function needs to be calculated at cell

center. In ENO scheme, gradient of the level set function in the discretized equation is selected

from a two or more candidate gradients calculated on different stencils. Gradient of the level set

24

function is calculated by approximating a higher order polynomial in each of the stencil. The

gradient of the smoothest stencil is selected out of all candidate gradients, by doing so the

unphysical oscillations introduced due to use of higher order scheme are eliminated.

In WENO schemes, gradient of the level set function in the discretized equation is

calculated by taking weighted average of all the gradients calculated on different stencils. The

weights are calculated so that maximum weight is given to smoothest stencil. In WENO scheme

the logical statements involved in ENO while selecting the smoothest stencil are avoided.

(a)

(b)

(c)

Figure 1.13: Some results of 2D LS method (a) Rayleigh-Taylor instability with multiple perturbations at interface by Naourgaliev and Theofanous (2006) (b) Free falling liquid droplet in a channel near a wall by Ni et al.(2006) (c) Rise and merging of two bubbles in a channel filled with liquid by Chang et al. (1995).

1.3.4.3 Some Results of LS method As compared to VOF method, few results using level set method are available on literature.

Most of the results available are for hypothetical test problems. Fig. 1.13 (a) shows result of

25

Rayleigh-Taylor instability with multiple perturbations at interface by Naourgaliev and

Theofanous (2006). Ni et al. (2006) studied the effect of wall on falling droplet; one of their

results of near wall bubble drop is shown in Fig. 1.13(b). Chang et al. (1995) solved the problem

of rise and merging of two bubbles in a channel filled with liquid as shown in Fig. 1.13(c).

Osher and Fedkiw (2000) have presented a 3D simulation result of an invisible solid object

splashing in water pool, shown in Fig. 1.14.

Figure 1.13: Result of 3D LS method: simulation of invisible object splashing into water pool by

Osher and Fedkiw (2000).

1.3.5 Combined Level Set VOF (CLSVOF) method As the name suggests, Combined Level Set Volume of Fluid (CLSVOF) method proposed

by Sussman and Puckett (2000) is combination of level set and volume of fluid method intended

to combine advantages of LSM and VOF method.

In CLSVOF method the volume fractions (Heaviside function) are used to reconstruct the

interface and solve VOF advection equation, ensuring mass conservation. The level set function

is calculated from the reconstructed interface. The interface normal required to reconstruct

interface in cell (PLIC) and curvature are calculated from the level set function field.

CLSVOF method has advantages of VOF and level set method but it has a big

disadvantage of VOF method i.e. Interface reconstruction.

26

1.3.6 Summary of Literature Review Extensive literature is available on VOF method indicating that extensive work has been

done on VOF method. There are many methods to construct the interface by knowing the

volume fraction field in domain but none reproduces the actual surface because of PLIC

approximation and existence of finite number of cells. In VOF, generally discontinuous interface

will be reconstructed, however efforts have been made by researchers to get continuous

interface.

Fairly accurate interface can be reconstructed on a fine grid using LVIRA [ Pilliod and

Puckett (2004)], circle Fit technique [ Mosso et al. (1996), Garrioch and Baliga (2006)] or

PROST [ Renardy and Renardy (2002)] but the computational effort will increase such methods

involve iterations to locate interface in cell, a trade-off limit will have to be found.

To obtain the accurate solution of VOF governing equation, most of the researchers prefer

the geometric advection method. VOF method has been extended to 3D [Lörstad and Fuchs

(2004)]. Use of single fluid model has simplified the solution procedure of the two-phase

problems. Use of CSF model has been reported to give fairly accurate results.

Use of level set method is not wide as compared to VOF, yet some great problems have

been solved using level set method. There are various approaches available to tackle the problem

of mass loss but it was found that all the methods try to conserve the Heaviside function field.

CLSVOF methods are used to explicitly conserve to Heaviside function whereas Chang et al.

(1996) and Ni et al. (2006) have tried to indirectly conserve Heaviside function by applying

constraint in reinitialization procedure.

1.4 Objective and Motivation of Present Work It is seen from literature survey, considerable work has been done in fluid flow algorithms

using VOF and LS methods in 2D and 3D. Simulating boiling is a challenging task, some work

is available in 2D but simulation of 3D boiling is seldom attempted.

The objective of present work is development of a 3D two-phase flow code capable of

simulating boiling; the code will be used to simulate two-phase flow over nuclear rod bundles in

a cylindrical shell.

To achieve the objective in systematic way,

(a) 2D two-fluid flow codes based on VOF and LS methods will be developed for making

detailed comparisons and understanding the numerical strengths and limitations of both methods.

A method that suits the requirements of present objective will be selected for further

development.

27

(b) It is intended to rectify the shortcomings of simulation methodology selected after

comparative study by proposing a new formulation in 2D.

(c) Boiling heat transfer at interface will be modeled in the proposed/improved method in

2D.

(d) 3D extension of the two-phase flow code will be done.

1.5 Outline of Report In chapter 2, the mathematical formulation of two-fluid solution methodology is presented.

Numerical methodology and implementation details for development of two-fluid flow codes

developed in the present work is explained in Chapter 3. A new CLSVOF method is explained in

Chapter 4. All numerical tests, their results and discussions are presented in Chapter 5.

Conclusions drawn from present work and future work proposal are presented in Chapter 6.

28

2 Mathematical Formulation

In this chapter the mathematical formulation of two-fluid solution methodology is

presented. In computational multi-fluid/interfacial dynamics, two sets of transient equations are

solved: Navier-Stokes and interface evolution equations, discussed in section 2.1 and 2.2

respectively. The first set of equations are used to obtain flow properties (velocity and pressure)

and the second set is used to obtain the location of interface. Physically, both flow property and

interface change simultaneously with respect to time but here, numerically the two sets of

equations are solved in a sequence. For example, if Navier-Stokes equations are solved first, then

the interface location needed is lagged by one time step. Thereafter, the updated velocity field of

new time step is used to solve equations governing evolution of interface to obtain interface

location of new time step.

2.1 Navier-Stokes equation In the present work single fluid assumption model is used i.e. NS equations are solved in

the complete domain using volume weighted properties. Single fluid flow solution procedures

can be easily used in case of level set method (Eq. 1.26 and Eq. 1.27) but not in case of VOF

method as the mean density term is present in Eq. 1.16.

To keep things simple, it is desired to use same numerical methods of single fluid flow in

case VOF method. Eq. 1.16 is expanded as,

( ) ( ) 0mm mu u

tρ ρ ρ∂

+ ⋅∇ + ∇⋅ =∂

(2.1)

Eq. 2.1 is split as,

( ) 0m uρ ∇ ⋅ = (2.2)

and

( ) 0mmu

tρ ρ∂

+ ⋅∇ =∂

(2.3)

Substituting Eq. 1.14 in Eq. 2.3 and applying the fact that in incompressible flow the

individual fluid density does not change with time or space, VOF governing equation (Eq. 2.4) is

obtained. Similarly the momentum equation used in LS method (Eq. 1.27) is obtained from Eq.

1.17 using Eq. 2.1. The continuity equation (Eq. 1.26) is obtained from Eq. 2.2.

Hence, the form of Navier-Stokes equation used in VOF method is same as that for LS

method. It is seen from this transformation that solving the continuity equation (Eq. 1.26) and the

29

VOF governing equation (Eq. 1.4) is same as solving Eq. 1.16. It is observed that VOF method

has a built-in mass conservation property whereas in case of LSM, mass will be conserved only

if solving Eq. 1.28 ensures solution of Eq. 1.4.

2.1.1 Non-Dimensional form of Navier-Stokes equation Governing equations are written in non-dimensional form to generalize a large array of

problems and perform the parametric study easily. In the present work non-dimensional form of

governing equation are solved. The Continuity (Eq. 1.26) and Momentum equations (Eq. 1.27)

are written in non-dimensional form as,

0U∇⋅ = (2.4)

( ) ( )( ) 2

1 1 1 1 ˆˆRe

T

sU UU P U U n j

We Frηλ

μ κ δτ ρ

∂ ⎡ ⎤⎛ ⎞+∇ ⋅ = −∇ + ∇⋅ ∇ + ∇ + −⎜ ⎟⎢ ⎥⎝ ⎠∂ ⎣ ⎦ (2.5)

HereU ,τ , P are non-dimensional velocity, time and pressure respectively. The length

scale ( )*L , time scale ( )*t and the velocity scale ( )*U are defined based on the problem at

hand; λρ and ημ are non-dimensional density and viscosity defined as,

( ) ( )2 1 2 1 1 2

1 , 1

,

C C

with andλ ηρ λ λ μ η η

λ ρ ρ η μ μ ρ ρ

= + − = + −

= = > (2.6)

Re is Reynolds number defined as, * *

1

1

Re U Lρμ

=

Fr is Froude number defined as, *

*

UFrgL

=

We is Weber number defined as,

( )2* *1 U LWe ρ

σ=

As seen here, the non-dimensional governing parameters are Reynolds number, Froude

number and Weber number. Re and We are based on properties of heavier fluid.

2.2 Mathematical Representation of Evolution of Interface This representation depends on the numerical method used and is discussed in subsection

2.2.1 for VOF method and 2.2.2 for LS method.

30

2.2.1 Volume of Fluid Governing Equation As mentioned earlier, in VOF method, interface motion is not followed but volume of fluid

function (C) (ratio of volume of one fluid to volume of the cell) is used to store information of a

region of a fluid. Eq. 1.5 is the governing equation of VOF function.

VOF function is a step function and there is doubt over existence of its derivatives and thus

existence of VOF governing equation itself. Use of finite difference or finite volume technique to

discretize the VOF governing equation (Eq. 1.5) is also inappropriate. However, the volume

conservation has to be satisfied and instead of writing Eq. 1.5, VOF governing equation can be

written as Eq. 1.6.

There is no advantage in converting a conservation statement into a differential equation

and then using discretization procedures to convert the differential equation back to conservation

statement.

2.2.2 Level Set equations In case of LS method the evolution of interface is defined by two equations: Convection

equation and a Reinitialization equation.

2.2.2.1 Convection Equation The level set convection equation also known a level set governing equation, governs the

temporal evolution of level set function. Level set function is advected with the background flow

field, so the convection equation of interface is Eq. 1.28.

2.2.2.2 Reinitialization Equation Level set function is defined as a normal distance function. Once a time advanced level set

function field is calculated by solving Eq. 1.28, the level set function field no longer remains a

normal distance function field, properties and curvature calculation procedures require level set

function to be a normal distance function. Sussman et al. (1994) mentioned that solution of Eq.

1.31 after every time step reinitializes the level set function to signed normal distance function.

To solve Eq. 1.31, it is converted to a hyperbolic equation. The gradient of level set function is,

ˆ ˆx yi jφ φ φ∇ = + and thus ( ) ( ) ( )( )2 2

x xφ φ φ φ φ φ∇ ⋅ ∇ = + = ∇ ∇ . Using this Eq. 1.31 is written as,

( )( )( ) ( )oo

s

SSεε

φ φ φφ φτ φ

∇ ∇∂+ =

∂ ∇

31

( ) ( )oo

s

SSεε

φ φφ φ φτ φ

∇∂+ ⋅∇ =

∂ ∇ (2.7)

Eq. 2.7 is a hyperbolic equation with characteristic ( )oSε φ φ φ∇ ∇ , it is known as the

reinitialization equation.

32

3 Numerical Methodology

In this chapter, finite volume/difference discretization of the governing equations shown in

previous chapter and its solution methodology is discussed. The numerical methodology of

solution of solution of Navier-Stokes equations using projection method on a staggered grid is

discussed in section 3.1. Thereafter, implementation details of VOF and LS methodologies are

discussed in section 3.2 and 3.3 respectively.

3.1 Finite Volume Discretization: Navier-Stokes equation A numerical method adopted to approximate the governing equations along with the

relevant boundary conditions by a system of linear algebraic equations is known as discretization

method. An algorithm, known as solution methodology is devised to obtain solutions to the

algebraic equations obtained by discretization. The domain on which the governing equations are

solved is also discretized in to finite number of control volumes, and governing equations are

solved for those cells.

3.1.1 Domain Discretization: Staggered Grid The governing equations are discretized on a square staggered grid arrangement as shown

in Fig. 3.1. The pressure is located at cell center and velocity components are staggered and thus

located at the cell face centers.

Figure 3.1: Staggered Grid used for two-fluid flow simulation.

33

3.1.2 Discretization of Governing Equations The Navier stokes equations (Eq. 2.4 and Eq. 2.5) are discretized by performing volume

integral over a cell (Fig. 3.2). The finite volume discretized form of the momentum equation is

for kth velocity component is

( )1, ,

, ,, , ,

1n nP k P k P D C

gSTk P k P f ff w s e n

V U UF F P S F F

λτ ρ

+

=

⎡ ⎤⎣ ⎦Δ −

= − − Δ + +Δ ∑ (3.1)

Where,

Convective Flux, ( ),

, , ,, fk f

f w s e n

Ck P U U SF

=

= ⋅Δ∑

Diffusion Flux, ( )( )( ), , ,,

,1

ReT

k k ff w s e nP

Dk P U U SF η

λ

μρ =

= ∇ + ∇ Δ∑

Surface Tension force, ˆ SP

STV nWe

Fλ

κ δρΔ⎛ ⎞= ⎜ ⎟

⎝ ⎠, Gravitational force, 2

ˆPg

V jFr

F Δ= −

ˆ ˆ, ˆ ˆ,

w e

s n

S yi S yi

S xj S xjV x y

Δ = −Δ Δ = Δ

Δ = −Δ Δ = ΔΔ = Δ Δ

Figure 3.2: A finite volume cell / control volume with geometrical parameters

In Eq. 3.1, ,k fU in the convective flux term is calculated using a convection scheme. In the

present work, Three convection schemes, first order upwind (FOU), Second Order Upwind

(SOU) and Quadratic Upwind Interpolation for Interface Kinematics (QUICK) scheme is

implemented, either one can be used based on the problem at hand.

The gradient of velocity across a cell face in diffusion term of Eq. 3.1 is calculated using

central difference scheme.

3.1.3 Solution Methodology: Projection algorithm In projection method, continuity equation (Eq. 2.4) is solved implicitly whereas the

convective and diffusive terms of momentum equation (Eq. 2.5) are considered explicitly. The

proposed time levels are,

1 0nU +∇ ⋅ = (3.2)

xΔ

yΔVΔ

eSΔ

nSΔ

wSΔ

sSΔ

34

( ) ( )( )( ) ( )1 1 1 11 ˆˆ 2Re

1 nnn

n n TU U n n n n nU U P U U n jsWe Fr

ηλ

κ δτ

μρ

+ − += −∇ ⋅ + −∇ + ∇ ⋅ ∇ + ∇ + −Δ

⎡ ⎤⎢ ⎥⎣ ⎦

(3.3)

In The solution algorithm in projection method is,

1. Intermediate velocity is calculated by doing explicit time integration from time level n of

Eq. 3.3 without pressure terms,

( ) ( )( ) ( )*

2

1ˆ

1 1 ˆRe

n T nn n n n nn n sWe

U U U U U U jFrη

λ

κ δμτ ρ− ⎛ ⎞= −∇ ⋅ + ∇ ⋅ ∇ + ∇ + −⎜ ⎟Δ ⎝ ⎠

(3.4)

2. To determine the pressure field that satisfies the continuity equation, Eq. 3.4 is subtracted

from Eq. 3.3, 1 * 1n n

n

U U P

λτ ρ

+ +− ∇= −

Δ (3.5)

3. Taking divergence on both sides of Eq. 3.5 and using Eq. 3.2, the pressure Poisson

equation is obtained as, 1

*1n

n

P Uλρ τ

+⎛ ⎞∇∇ ⋅ = ∇ ⋅⎜ ⎟ Δ⎝ ⎠

(3.6)

4. Eq. 3.6 is solved for 1nP + subjected to homogeneous Neumann boundary condition, 1

0nPn

+∂=

∂.

5. Once the pressure of new time level is obtained from Eq. 3.6, Velocities at new time level

are calculated using Eq. 3.5.

In the present work, the pressure poisson equation (Eq. 3.6) is solved using Alternating

Direction Integration (ADI) method.

3.1.4 Calculation of Time Step The projection method implemented in this work is of semi-explicit nature as the

convective and diffusive terms are treated explicitly. The VOF and level set governing equations

are also solved explicitly. Thus there is a problem of numerical instability if higher time step is

taken. Time step calculated must obey the Courant-Friedrichs-Lewy (CFL) condition, the

restriction due to gravity and surface tension and the restriction due to viscous terms. The non-

dimensional time step is calculated as,

( )min , ,CFL ST VISRτ τ τ τΔ = Δ Δ Δ (3.7)

Where,

35

R is the reduction factor; its value is problem dependent.

CFLτΔ is restriction on time step due to CFL criterion. min2CFL domain

XU

τ Δ⎡ ⎤Δ = ⎢ ⎥⎣ ⎦ .

STτΔ is restriction on time step due to surface tension. ( ) 3

21min8ST domain

We Xλτπ

⎡ ⎤+Δ = Δ⎢ ⎥

⎣ ⎦ .

VISτΔ is restriction on time step due to viscous force. ( )2 2

2 2

Remin4VIS domain

X YX Y

τ⎡ ⎤Δ Δ

Δ = ⎢ ⎥Δ + Δ⎣ ⎦ .

3.2 Implementation of VOF method

In present work, VOF algorithm explained in Section 1.3.2.4 is followed. The VOF

function is located at the cell center with pressure. Implementation details of methodologies

adopted to solve VOF governing equation is explained in subsection 3.2.1, application of

boundary condition of volume fraction is discussed in subsection 3.2.2, modeling of surface

tension and procedure to calculate properties at required locations is presented section 3.2.3 and

3.2.4 respectively.

3.2.1 Solution of VOF governing equation From literature review it is seen that many approaches are present to solve VOF equation

However, results of detailed comparisons between different VOF solution methodologies on

wide variety of problems are not available. Rudman (1997) had performed comparative study of

flux corrected transport method and Youngs’ method without geometric reconstruction on

Rayleigh Taylor instability with low property ratio and without surface tension.

In this work, all three solution approaches of VOF governing equation discussed in 1.3.2.2

are implemented. Donor-acceptor method, a type of flux corrected transport method is

implemented. High resolution convection schemes in form of Piecewise Linear Method (PLM)

and Piecewise Parabolic Method (PPM) are used in finite volume discretized VOF equation. The

most popular solution approach in form of multidimensional geometric advection method using

Youngs’ PLIC method is implemented.

As all the different approaches are implemented, relative advantages and shortcomings can

be determined on same set of conditions. Implementation details of each method are explained

here. A test problem is considered to explain working of donor-acceptor and geometric advection

method.

36

3.2.1.1 Implementation of Donor-Acceptor method Donor-acceptor (DA) method of Hirt and Nichols (1981) used to solve VOF governing

equation is simplest and earliest of VOF solution approach. It is a form of flux corrected

transport method, wherein the numerical diffusion in solution is reduced without inducing

undershoots or overshoots. The basic algorithm is explained in section 1.3.1.2.

Implementation of DA method can be understood with the help of a test problem. Values

of volume fractions in a square domain (5x5) are shown in Table 3.1. Consider the central stencil

of 3x3 cells, the central cell has volume fraction 0.5. It is intended to calculate the value of

volume fraction in center cell at new time level. Geometrical details, velocity field and time step

is given in Table 3.2.

Table 3.1: Values of volume fractions in domain. Central Stencil of 3x3 cells is considered for calculation.

0 0 0 0 0 0 0 0 0 0.1667 0 0.1667 0.5 0.8334 1

0.8334 1 1 1 1 1 1 1 1 1

Table 3.2: Geometry, velocity field and time step for test problem Grid spacing 1X YΔ = Δ = Cell Face Area Vectors 1 and 1w s e nS S S SΔ = Δ = − Δ = Δ = Velocity Field U = 0.6 and V = 1 everywhere in domain Time Step 0.6τΔ =

Time advanced value of volume fraction in central cell can be found by following steps,

for other cells same steps need to be performed. As already mentioned, the DA method is

implemented using operator splitting and the final value of volume fraction is calculated by

executing x-sweep and y-sweep.

The steps required to execute x-sweep for this case are,

Step 1: The gradient of volume fractions are calculated using Eq. 1.1 as,

0.33 1.5x yC and C= = −

Step 2: In DA method only the information of orientation of interface is required, a variable for

orientation is defined as,

( ) ( ) [ ][ ]

0

1 x yif abs C abs C Horizontal Interface

ORTelse Vertical Interface

⎧ <⎪= ⎨⎪⎩

In this case for the central cell, ORT = 0.

37

Step 3: To solve VOF equation (Eq. 1.6), the amount of fluid fluxed across a cell face is

calculated using Eq. 1.7. The donor and acceptor cells across east and west cell faces are

determined based on the direction of velocity field, for the present test problem values of CD and

CA are,

Volume Fractions Face Upstream to donor cell (CDD) Donor Cell (CD) Acceptor cell (CA) East 0.1667 0.5 0.8334 West 0 0.1667 0.5

Step 4: The choice of CAD in Eq. 1.7 is made on basis of orientation of interface and flow

direction as,

(1 )AD A DC ORT C ORT C= × + − ×

For the east face, CADe = CDe = 0.5 but in case of west face, the cell upstream to donor cell

is empty so, CADw = CAw = 0.5.

Step 5: For the east face as the value of CAD is equal to CD, 0eAF = and for west face,

{ }max 0.6533,0 0wAF = − =

Step 6: Sign of face velocity at east and west face is calculated to be 1.

Step 7: Putting everything together in Eq. 2.7 the fluid fluxed out of east and west face is,

( ) { }( ) { }

0.18,0.5 0.18

0.18,0.1667 0.1667ee

ww

Fluid Fluxed Out MIN S

Fluid Fluxed Out MIN S

= Δ =

= Δ = −

It is seen that the MIN feature prevented more fluid flux than the donor cell can give

across the west face.

Step 8: Intermediate value of volume fraction ( *C ) in central cell is determined using Eq. 1.6 as,

( )* 0.5 0.18 0.1667 1 0.4867C = − − = . Intermediate value of volume fractions in all cells as

shown in Table 3.3 are found by following Step 1 to Step 8 for all cells.

Table 3.3: Values of intermediate volume fractions after x-sweep in domain. 0 0 0 0 0 0 0 0 0 0 0 0 0.486667 0.713333 0.94

0.713333 0.94 1 1 1 1 1 1 1 1

Once the intermediate volume fraction values are calculated the y-sweep is executed by

following steps,

Step 9: The orientation of the interface based on intermediate volume fractions may change so

the gradient of intermediate volume fractions are calculated again using Eq. 1.1,

38

* *0.3867 1.47x yC and C= = −

Step 10: In this case for the central cell, the orientation has not changed so, ORT = 0.

Step 11: The donor and acceptor cells across north and south cell faces are determined as,

Volume Fractions Face Upstream to donor cell (CDD) Donor Cell (CD) Acceptor cell (CA) North 1 0.4867 0 South 1 1 0.4867

Step 12: For the north face as the interface (horizontal) is moving normal to itself (in vertical

direction), CADn = CAn = 0. Similarly for south face CADs = CAs = 0.4867.

Step 13: For north face, ( ){ }max 0.6 0.5133 ,0 0.0867nAF = − = and for south face,

{ }max 0.308,0 0.308sAF = = . It is seen that in case of north and south face the value of

additional fluid is not zero, the geometrical interpretation for the calculation of additional fluid in

case in north face is shown in Fig. 3.3, it is seen that although the volume fraction value of

acceptor cell is used to calculate the fluid flux, there is still some fluid that can be fluxed across

the north face represented by the overlapping area on the fluid polygon and flux polygon. Similar

interpretation can be made for the south face.

Figure 3.3: Geometrical interpretation of calculation of additional fluid for the case of north

face of the present test problem.

Step 14: Sign of face velocity at east and west face is calculated to be 1.

Step 15: Putting everything together in Eq. 1.7 the fluid fluxed out of north and south face is,

( ) { }( ) { }

0 0.0867,0.4867 0.0867

0.292 0.308,1 0.6nn

ss

Fluid Fluxed Out MIN S

Fluid Fluxed Out MIN S

= + Δ =

= + Δ = −

Step 16: Time advanced value of volume fraction ( 1nC + ) in central cell is determined using Eq.

2.6 as, ( )1 0.4867 0.0867 0.6 1 1nC + = − − = .

Donor cell

Additional Fluid

Acceptor Cell

vn ( )1 DC y− Δ

nv tΔ( )( )1n n DAF v t C y x= Δ − − Δ Δ

Geometrically, Additional Fluid:

39

It is noted that if the donor cell approximation was used for all the faces (first order

upwind method) then the value of volume fraction in central cell would have been equal to 0.68,

signifying numerical diffusion. Also the MIN and MAX feature provided the optimum

combination of donor and acceptor flux so that the value of volume fraction did not go out of

bounds (C > 1 or C < 0). The time advanced values of volume fractions in all cells are found by

following Step 9 to Step 16 for all cells. Table 3.4 shows the time advanced values of volume

fractions in all cells, it is seen that the step change in values of volume fraction still exists.

Table 3.4: Time advanced values of volume fractions. 0 0 0 0 0 0 0 0.086667 0.313333 0.54

0.313333 0.54 1 1 1 1 1 1 1 1 1 1 1 1 1

3.2.1.2 Implementation of Geometric Advection Method Geometric advection method is the most popular method to solve the VOF governing

equation. In the present work multidimensional geometric advection method of Garrioch and

Baliga (2006) coupled with Youngs’ PLIC method is implemented to solve VOF governing

equation.

The basic methodology of geometric method is explained in section 1.3.2.2 but there exists

no fixed/standard implementation procedure. Implementation procedure of geometric method

developed in this work is explained in this section with the help of a test problem. Values of

volume fractions in a square domain (5x5) are shown in Table 3.1. Consider the central stencil of

3x3 cells, the central cell has volume fraction 0.5. It is intended to calculate the value of volume

fraction in center cell at new time level. Geometrical details, velocity field and time step is given

is Table 3.2.

Time advanced value of volume fraction in central cell is found by a two step process in

which Eq. 2.6 is solved by series of geometric tasks. Firstly, interface is located inside the cell

based on the approximation of the shape of the interface. Then amount of a particular fluid

fluxed across a cell faces is determined geometrically.

3.2.1.2.1 Interface Reconstruction by Youngs’ PLIC method

Interface reconstruction is the first step in geometric advection where interface co-

ordinates are determined in cell. Interface is approximated as a line segment in a cell (Eq. 1.9) as

shown in Fig. 1.3 (a). Steps required to locate the interface in a cell are as follows,

40

Step 1: Interface normal calculation

In Youngs’ method the interface normal is calculated using Eq. 1.10 and Eq. 1.11. For the

present test problem the components of interface normal are,

0.1667, 0.5x yn n= = −

The absolute value of slope of the interface is calculated as, 0.3334x

y

nabs n⎛ ⎞ =⎜ ⎟⎝ ⎠

Figure 3.4 : Possible cases of Interface orientation and fluid location

Step 2: Determination of fluid position relative to interface

Fig. 3.4 shows all the possibilities of the shape of the interface and the relative location of

fluid in the cell. It is found that out of all the possibilities one can reduce to 3 possibilities (for

interfaces not aligned to axis) by just knowing the values of ,x yn n , this idea was borrowed from

the donor-acceptor method [Hirt and Nichols (1981)], wherein the position of fluid relative to

interface is determined by checking the sign of the ,x yn n .

Table 3.5 is used to determine the existing case from Fig. 3.4, for the present test problem

as 0, 0 1xx y

y

nn n and abs n⎛ ⎞> < <⎜ ⎟⎝ ⎠

the fluid lies in south west side of the interface and the

existing case is A2.

41

Table 3.5: Determination of fluid position relative to interface

xn yn Value of ( )x yabs n n

Possible Case of interface and fluid in

cell from Fig. 3.4 > 0 < 0 > 1 A1 > 0 < 0 <= 1 A2 > 0 0 -NA- V1 > 0 > 0 > 1 B1 > 0 > 0 <= 1 B2 0 > 0 -NA- H2

< 0 > 0 > 1 C1 < 0 > 0 <= 1 C2 < 0 0 -NA- V2 < 0 < 0 > 1 D1 < 0 < 0 <= 1 D2 0 < 0 -NA- H1

Step 3: Determination of fluid shape from 3 possible shapes

From step 2 it is found that the existing case for the test problem is A2 but still the shape of

the fluid polygon can be triangle or a trapezoid or a pentagon as shown in Fig. 3.5.

(a)

(b)

(c)

Figure 3.5: Three possible shapes of fluid polygon (a) Triangle (b) Trapezoid (c) Pentagon

It is seen that no matter what the shape of the fluid polygon is, there would always be one

vertex of cell in fluid, named as fixed corner. In the present test problem, the south-east corner is

the fixed corner.

(a)

(b)

Figure 3.6: Determination of Fluid Shape (a) Largest triangular fluid polygon for present test problem (b) Largest trapezoidal fluid polygon for present test problem

42

It is seen from Fig. 3.6 (a) that for present test problem when the interface is located at

south-west vertex for the given normal and extend till the east face, the fluid polygon formed so

is the largest possible triangular fluid polygon. South-west vertex in this case is known as the

triangle forming vertex. Triangle forming vertex for other cases can be determined in similar

fashion. Similarly, for the present test problem north-east vertex is the trapezoid forming vertex

as shown in Fig. 3.6 (b).

Once the fixed vertex, triangle forming vertex and trapezoid forming vertex are known,

the steps to determine the shape of fluid polygon out of three possible shapes are,

(1) Using data of normal of interface, biggest possible fluid triangle is constructed and its area

is determined. The volume fraction of the largest possible triangular fluid polygon is calculated

and compared with the actual value of volume fraction.

For the present test problem the co-ordinates of fixed vertex are (x, y) fxd = (3, 2) and of

triangle forming vertex are (x, y) trg = (2, 2). The subscript ‘fxd’ stands for ‘fixed vertex’ and

subscript ‘trg’ stands for ‘triangle forming vertex’. The line with normal, (nx, ny) = (0.16667, -

0.5) is located at the triangle forming vertex and extended to east face, the third co-ordinate of

biggest possible triangular fluid polygon as shown in Fig. 3.6(a) is found to be, (x, y) trg2 = (3,

2.3333). The subscript ‘trg2’ stands for ‘second triangle forming vertex’.

The area enclosed by an n-sided polygon in Cartesian geometry from vertices (xv, yv)

collected in anti-clockwise direction is calculated using [Rider and Kothe (1998)],

( )1 11

12

n

polygon v v v vv

Area abs x y x y+ +=

⎛ ⎞= −⎜ ⎟⎝ ⎠∑ (3.8)

For the present case the area of the largest possible triangle is found to be,

0.16667MaxTriArea = and the volume fraction of largest possible triangle is found to be,

0.16667 1 0.16667MaxTriC = = . The volume fraction in the central cell is 0.5 thus, MaxTri actualC C< .

As the volume fraction of biggest possible triangle is less than the actual volume fraction the

fluid shape is not triangular.

(2) If the volume fraction of the biggest possible triangle is found to be less than the actual

volume fraction then the fluid polygon shape will be confirmed as triangle.

The volume fraction of the biggest possible trapezoid is calculated in a similar way and it is

found to be, 0.83333 1 0.83333MaxTrpC = = . The actual volume fraction in the central cell is 0.5

and as MaxTrp actual MaxTriC C C> > . The fluid shape is confirmed to be trapezoidal.

43

(3) If the volume fraction of the biggest possible trapezoid is found to be less than the actual

volume fraction then the fluid polygon shape will be confirmed as pentagon.

Step 4: Locating interface in cell

Once the fluid shape is determined from step 3, the interface location in the cell is

determined using an iterative procedure. Determining location of the interface segment in the

cell is equivalent of finding the line constant (d). Interface segment should be positioned in the

cell in such a way that the volume fraction of fluid polygon is same as the actual volume

fraction.

Figure 3.7: Locating the interface between known limits of interface position.

In the present test problem, from step 3 it was confirmed that the fluid polygon is a

trapezoid. It is seen from Fig. 3.7 that out of co-ordinates of four vertices (vertex 1, 2, 3 and 4) of

fluid polygon, co-ordinates of two vertices (vertex 1 and 2) are known completely and x co-

ordinate of other two vertices (vertex 3 and 4) are known.

It is seen from Fig. 3.7, the y co-ordinate of vertex 3 of fluid polygon should be between

y co-ordinate of trapezoid forming vertex of cell (ytrp) and y co-ordinate of other triangle forming

point (ytrg2). The subscript ‘trp’ stands for ‘trapezoid forming vertex’.

For the present test problem following information is known,

(x, y)1 ,fdp = (x, y)trg = (x, y)sw = (2, 2)

(x, y)2, fdp = (x, y)fxd = (x, y)se = (3, 2)

(x, y)3, fdp = (xse, ? ) = (3, ?)

(x, y)4, fdp = (xsw, ? ) = (2, ?)

Value of y3, fluid poly is between 3 (ymax = ytrp) and 2.3333 (ymin = ytrg2). Steps to determine y3,

fdp and y4, fdp are as follows,

1. Initially value of y3, fdp is guessed from the maximum and minimum values, for the present

test problem y3, fdp, guess1 = 0.5(ymax + ymin) = 0.5(3+2.3333) = 2.6667. The interface of given

normal is extended to west face and y4, fdp, guess is calculated, for present test problem y4, , fdp,

guess is calculated to be 2.3333.

44

2. The volume fraction of the fluid polygon using the guessed co-ordinates of vertices is

calculated to be, 1 0.5 1 0.5Guess actualC C= = = . In the present test case the result is obtained

in single iteration. However, if the volume fraction of fluid polygon with guessed co-

ordinates is not equal to actual volume fraction then iteration is required.

3. The guess of y3, fdp is improved as,

( )( )

1 3, , 3, , min

1 3, , 3, , max

0.5

0.5Guess actual fdp new fdp old

Guess actual fdp new fdp old

if C C then y y y

if C C then y y y

> = +

< = +

4. Steps 1 to 3 are repeated in a general case till the condition, ( )1 1 12Guess actualabs C C e− < −

is satisfied.

5. Once the final values of y3, fdp and y4, fdp are obtained the line constant of interface is

obtained from Eq. 2.9. The equation of the interface for the present test case is found to be,

0.16667 0.5 0.83333 0.3333 1 1.66667x y x y− = − ⇒ − = −

The co-ordinates of the fluid polygon are found to be, (x, y)1,fdp = (2, 2), (x, y)2,fdp = (3, 2),

(x, y)3,fdp = (3, 2.6667) , (x, y)4,fdp = (2, 2.3333).

Interface in all partially filled cells is determined in the same way. Once the fluid

polygons in all the partially filled cells are constructed, next task is geometric advection.

3.2.1.2.2 Multidimensional geometric advection

Determination of the amount of fluid fluxed across cell faces geometrically by constructing

flux polygons at cell faces is known as geometric advection. Geometric advection is executed by

following steps,

Figure 3.8: Dividing flux polygon in to two for better implementation.

45

Step 1: Construction of flux polygons at cell faces

Multidimensional flux polygon constructed about east face is shown in Fig. 1.9. For

implementation purpose the multidimensional flux polygon is represented as combination of two

polygons as shown in Fig. 3.8, the flux polygon constructed in the upwind cell is named as

normal flux polygon and the flux polygon constructed in the transverse upwind cell is named as

transverse flux polygon, the notations used for vertices of both the flux polygons is also shown.

Firstly normal and transverse velocities at cell faces are determined. For the present test

problem the normal velocity at east cell face is 0.6eU = and the transverse velocity is found

using method of Garrioch and Baliga (2006) (Eq. 1.13) and is * * 1ne seV V= = . As shown in Fig. 3.8,

the co-ordinates of the normal flux polygon at east face are obtained as,

( ) ( )1, ,, ,se e senfp e

x y x U t y= − Δ

( ) ( )2, ,, ,se senfp e

x y x y=

( ) ( )3, ,, ,ne nenfp e

x y x y=

( ) ( )*4, ,

, ,ne e ne nenfp ex y x U t y V t= − Δ − Δ

The subscript ‘nfp’ stands for normal flux polygon and ‘e’ for east face. As shown in Fig.

3.8, the co-ordinates of the transverse flux polygon at east face are obtained to as,

( ) ( )*1, ,

, ,se e se setfp ex y x U t y V t= − Δ − Δ

( ) ( )2, ,, ,se setfp e

x y x y=

( ) ( )3, ,, ,se e setfp e

x y x U t y= − Δ

The subscript ‘tfp’ stands for transverse flux polygon For the present test problem the co-

ordinates of normal and transverse flux polygons are, ( ) ( )1, ,, 2.64,2

nfp ex y = , ( ) ( )2, ,

, 3, 2nfp e

x y = ,

( ) ( )3, ,, 3,3

nfp ex y = , ( ) ( )4, ,

, 2.64,2.4nfp e

x y = and ( ) ( )1, ,, 2.64,1.4

tfp ex y = , ( ) ( )2, ,

, 3, 2tfp e

x y = ,

( ) ( )3, ,, 2.64,2

tfp ex y = . Similarly co-ordinates of vertices of normal and transverse flux polygons

for other faces are determined. Once fluid polygon and all the flux polygons are in order, the

common area to both fluid polygon and each of flux polygons is determined.

Step 2: Determination of the common area between fluid polygon and flux polygon

Determination of common area requires finding the points of intersection between fluid and all

flux polygons. The points of intersection must be found in a specific order (clockwise or anti-

clockwise) to calculate the area properly.

46

As a first step, normal and line constant of the sides of fluid and flux polygons are

determined using Eq. 1.9. The notation used for denoting the sides of fluid polygon and flux

polygons about east face is shown in Fig. 3.9.

Figure 3.9: Notation used to represent polygons (a) Fluid Polygon (b) Normal Flux Polygon at

east face (c) Transverse Flux Polygon at east face.

For the present test case the values of normal and line constant for fluid and flux

polygons is given in Table 3.6.

Table 3.6: Values of components of normal and line constant for sides of fluid and east face flux polygons. xn xn d

Side Fluid Polygon 1 0 1 2 2 1 0 3 3 -0.3333 1 1.6667 4 1 0 2 Normal Flux Polygon at East face 1 0 1 2 2 1 0 3 3 -1.6667 1 -2 4 1 0 2.64 Transverse Flux Polygon at East face 1 -1.6667 1 -3 2 0 1 2 3 1 0 2.64

Once the normal and line constant for all sides of fluid and flux polygons are calculated,

the next step is determination of points of intersection in clockwise/anti-clockwise sequence.

Firstly, each vertex of flux polygon is checked if it lies in the fluid by using the using the fact

that, A vertex (xv, yv) lies in the fluid polygon bounded by interface , ,x i y i in x n y d+ = if

, , 0x i v y i v in x n y d+ − < . Once the test has been applied to all vertices of flux polygon, points of

intersection of fluid and each flux polygon is found sequentially.

47

Considering the present test problem, vertices 1, 2 and 4 of normal flux polygon (nfp) at

east face are found to be in the fluid polygon as seen from Fig. 3.9(b). The points of intersection

of fluid and normal flux polygon at east face in anti-clockwise sequence are found as,

1. Set counter for number of vertices of common polygon as zero.

2. Start at vertex 1 of normal flux polygon at east face. As vertex 1 is inside the fluid

polygon, add 1 to counter and record the co-ordinates first vertex of the common

polygon as, ( ) ( )1, , , 1, ,, ,

cmn n e nfp ex y x y= . The subscript ‘cmn’ stands for common and ‘n’

is for contribution from normal flux polygon.

3. Go to vertex 2 of normal flux polygon at east face. As vertex 2 is also inside the fluid

polygon, add 1 to counter and record the co-ordinates of the second vertex of the

common polygon as, ( ) ( )2, , , 2, ,, ,

cmn n e nfp ex y x y= .

4. Go to vertex 3 of normal flux polygon at east face. Now vertex 3 is outside the fluid

polygon, hence there exists a point lying on interface between vertex 2 and 3 on the

line joining them. The intersection of interface and the side 2 of normal flux polygon at

east face is found using Eq. 1.9 to give the third vertex of common polygon as as

( ) ( )3, , ,, 3, 2.6667

cmn n ex y = . Add 1 to counter.

5. Go to vertex 4 of normal flux polygon at east face. Now vertex 4 is inside the fluid

polygon, hence there exists a point lying on interface between vertex 3 and 4 on the

line joining them. The intersection of interface and the side 3 of normal flux polygon at

east face is found using Eq. 1.9 to give the fourth vertex of common polygon as

( ) ( )4, , ,, 2.75,2.58333

cmn n ex y = . Add 1 to counter.

6. Since vertex 4 of normal flux polygon at east face is also in the fluid, add 1 to counter

and record the co-ordinates of fifth vertex of the common polygon as,

( ) ( )5, , , 4, ,, ,

cmn n e nfp ex y x y= .

7. At the end, vertex 1 is again visited to check if there is interface intersection between

the line joining vertex 4 and 1, in the present test problem this is not the case.

8. After the normal flux polygon is traversed in anti-clockwise direction, five points are

collected which are vertices of the common area as shown in Fig. 3.10. The area of the

common polygon, denoted by Anf,e is found using Eq. 3.8. The subscript ‘nf’ stands for

‘normal flux’. For present case Anf,e = 0.210333.

48

Figure 3.10: The common polygon formed by collecting points in anti-clockwise direction,

representing normal fluid flux.

The next step is determining the area of common polygon in the transverse flux polygon.

In case of transverse flux polygon, similar traversing is followed but the fluid polygon is taken of

transverse donor cell i.e. south cell in the present problem. But in the present problem the south

cell is completely filled and there exists no fluid polygon. In cases when the cell is completely

filled or empty the common area is equal to the fluxed area times the donor cell volume fraction

i.e for present test problem, Atf,e = CS (area of transverse flux polygon at east face) = 0.108.

The fluid fluxed across east face is calculated as Ae = Anf,e + Atf,e = 0.318333. Similarly

the fluid fluxed across west, south and north face is -0.203061, -0.6 and 0.040333. Using the

value of fluid fluxed across each face, value of volume fraction in the central cell is found using

Eq. 2.6 to be 0.944394. Time advanced value of volume fraction in all cell is shown in Table 3.7

Table 3.7: Time advanced values of volume fractions using Geometric advection method.

0 0 0 0 0 0 0 0.032267 0.316672 0.646667

0.316672 0.646667 0.944394 1 1 1 1 1 1 1 1 1 1 1 1

It summary, geometric advection method is different from convectional methods in a way

that VOF function is not interpolated from cell center values to get face center values but VOF

function distribution is used to estimate the location of one of the fluids in a cell. Thereafter,

based on velocity at the cell faces, amount of that fluid fluxed across cell faces is determined.

3.2.1.3 Implementation of High Resolution Convection Schemes VOF governing equation (Eq. 1.5) is a scalar convection equation. Solution of the VOF

governing equation using a finite volume method is difficult as the VOF function is a step

49

function. Finite volume discretization of the VOF governing equation for a cell centered at P

having volume VΔ with face surface area vectors fSΔ is,

( )1

, , ,

n nP P f

f w s e nP

C C CU SVτ+

=

Δ= − ⋅Δ

Δ ∑ (3.9)

Out of all terms on right hand side of Eq. 3.9, Velocity at cell face center, Uf is obtained

from the solution of Navier-Stokes equations; Cf is cell face center value of volume fraction. As

the volume fractions are located at cell centers, an interpolation scheme is required to determine

value of volume fraction at cell face center from neighboring cell center values, which is known

as convection scheme.

First order upwind, second order upwind, QUICK etc are known as linear convection

schemes as they use a same expression at all locations in domain. If lower order convection

scheme like first order upwind is used then unacceptable numerical diffusion is introduced and

interface appears to be smeared. Normal linear convection schemes use same differencing stencil

everywhere in the solution domain. If there is a discontinuity in solution domain then higher

order methods like second order upwind and QUICK exhibit spurious oscillations near the zone

of discontinuity.

High-resolution convection schemes on the other hand are nonlinear methods that use

some kind of mechanism to control the oscillations in domain. Some high-resolution methods

attempt to totally eliminate oscillations while others simply minimize them.

A methodology is said to be high-resolution if it has following properties,

(1) The method provides at least second order accuracy in smooth areas of flow.

(2) The numerical solution produced by the method must be free from spurious oscillations.

(3) The smearing of the zone of discontinuity must be small than smearing due to use of first

order methods.

There exist many High Resolution schemes; prominent high resolution schemes are

Monotonic Upwind Schemes for Conservation Laws (MUSCL) of Van leer, Piecewise Linear

Method (PLM) and Piecewise Parabolic Method (PPM) [Colella and Woodward (1984)]. They

maintain high accuracy in smooth regions of the solution by approximating a higher order

polynomial to interpolate volume fractions but in regions of step change or steep gradients, the

higher order scheme is modified to give result equivalent to lower order scheme. By switching

between higher order and lower order schemes, high accuracy is maintained in smooth regions

and unphysical oscillations are avoided at discontinuities.

50

3.2.1.3.1 The MUSCL scheme

In MUSCL scheme, the profile of volume fraction is taken as a line in a cell. For square

cells the, expression of C in one dimensional form is

( ) ( ) ( )1, ,, ,

i j i jni j i j

C CC x C X X

X+ −

= + −Δ

(3.10)

Equation 3.10, for convenience is written as ( ) , ,

ni j i jC Cγ γω= + (3.11)

Where,

[ ], 0.5,0.5i jX XX

γ−

= ∈ −Δ

and ( ), 1, ,, i j i j i jslope C Cω += − (3.12)

When Eq. 3.12 is used to determine value of volume fraction at a cell face, there exist two

values of volume fractions at that cell face, one each from neighboring cell. Consider a case of

east face as shown in Fig. 3.11, value of volume fraction supplied by cell i,j denoted by the

subscript L (for left) and value from i+1,j denoted by the subscript R (for right) at east face of

cell i,j are found as,

1 , ,, ,2

0.5ni j i ji j L

C C ω+

= + and 1 1, 1,, ,2

0.5ni j i ji j R

C C ω+ ++

= − (3.13)

Figure 3.11: Two values of volume fractions at east cell face due to use of MUSCL scheme

The final value at cell face out of two values is determined by upwinding and it is denoted

by no additional subscript.

In the zone of discontinuity, the slope calculated using Eq. 3.12 may lead to calculation of

cell face value of volume fraction which may be unbounded from neighboring cell center values

as shown Fig. 3.12 (a). The unbounded solution is nothing but the existence of spurious

oscillation. To keep the cell face value in limits of neighboring cell center values a mechanism

known as slope limiting is used.

1,i jC+,i jC

C

, 0.5i jγ =γ

1, 0.5i jγ + = −

1 , ,2i j LC

+

1 , ,2i j RC

+

51

(a)

(b)

Figure 3.12: Slope Limiting in MUSCL scheme (a) Without slope limiting unbounded solution may be produced (b) With slope limiting bounded solution is produced.

In slope limiting, the slope of the line is modified by multiplying it with a slope limiter

function such that the value at cell face is bounded by the neighbouring cell centre values; this

treatment removes oscillations from the solution. To apply the slope limiters, the slope ( ,i jω )

instead of Eq. 3.12 is taken as,

( ), ,

Ri j i jvω ζ ω= (3.14)

Where,

( )v slope limiter functionζ =

,

,

Li j

Ri j

v ωω= , ( ) ( ), , 1, , 1, , L n n R n n

i j i j i j i j i j i jC C and C Cω ω− += − = −

There are many slope limiter functions. The choice of the best slope limiter function

depends upon the problem at hand. Some prominent slope limiter functions are,

( ) , 1vL

v vvan Leer v

vζ

+=

+

( ) ( ) ( ) , max 0,min 2, ,min 1,2SBsuperbee v v vζ = ⎡ ⎤⎣ ⎦

The value of velocity at cell face ( 1 2,e i ju u += ) is available from the solution of NS

equations and the Riemann problem for volume fraction at a cell face is solved as,

( )1 2, , 1 2,

1 2, 1 2, , 1 2, , 1 2,

1 2, , 1 2,

if 0

0.5 if 0

if 0

ni j L i j

n n ni j i j L i j R i j

ni j R i j

C U

C C C U

C U

+ +

+ + + +

+ +

⎧ >⎪⎪= + =⎨⎪ <⎪⎩

(3.15)

Most of the high resolution schemes are derived in one dimension, the 2D extension is

implemented by operator splitting i.e. series of x and y sweeps. Algorithm to solve VOF

governing equation using MUSCL scheme is summarised as,

1,i jC+

,i jC

C

γ

1 , ,2i j LC

+1 , ,2i j R

C+

1,i jC+

,i jC

C

γ

1 , ,2i j LC

+

1 , ,2i j RC

+

1 ,, ,2 u n b o u n d ed so lu tio ni ji j R

C C+

<, 1 1,, ,2

, 1 1,, ,2

<

<

i j i ji j R

i j i ji j L

C C C

C C C

++

++

<

<

52

(1) Based on the choice of the slope limiter function calculate the slope, ,i jω in all cells using

Eq. 3.14.

(2) Calculate L and R values of volume fractions at east and west face using Eq. 3.13.

(3) Calculate the single value of volume fractions at east and west cell face using Eq. 3.15.

(4) Solve the 1D form of discretized equation (Eq. 3.9) of VOF governing equation in x-

direction to get intermediate values of volume fractions.

(5) Repeat steps 1 to 4 with intermediate volume fraction field for north and south faces to

get time advanced values of volume fractions.

3.2.1.3.2 Piecewise Linear Method (PLM)

In Piecewise Linear Method (PLM), the profile of volume fraction in a cell is taken same

as in case of MUSCL i.e. Eq. 3.11. PLM scheme is different from MUSCL scheme with regard

to the expressions used to calculate volume fraction at cell face.

To understand the difference between MUSCL and PLM consider Fig. 3.13. It is seen

from Fig. 3.13 (a) that in case of MUSCL scheme the value of volume fraction at cell faces is

always an extreme value i.e. maximum or minimum in the cell.

(a)

(b) (c) Figure 3.13: Averaging C in PLM method (a) Calculation of extreme value at cell face in case

of MUSCL scheme (b) Variation of volume fractions in a small zone near cell face. (c) The averaged value equivalent of the variation in the small zone near cell face.

In PLM the value at the cell face is averaged in a small zone near the cell face. In one time

step as the total flux across a cell face is fU τΔ , the volume fraction is averaged in the region of

width fU τΔ near the cell face. Fig. 3.13 (b) shows the linear fit of volume fraction in the region

representing fU τΔ , it is seen that taking the extreme value of volume fraction at a cell face will

be inappropriate as the value of volume fraction is continuously varying. Fig. 3.13 (c) shows the

averaged value of volume fraction in vicinity of cell face. As the distance from the cell center is

normalized by divided it by cell width U τΔ is also normalized as,

1,i jC+

,i jC

C

γfU τΔ

average

1 , ,2i j LC

+

1 , ,2i j RC

+

1,i jC+

,i jC

C

γ

v a r ia t io n

fU τΔ

fU1,i jC+

,i jC

C

γ

1 , ,2i j LC

+1 , ,2i j R

C+

53

For a case of east face, shown in Fig. 3.13 (c), the average value of volume fraction at the

cell face can be obtained from the two cells sharing the east face as

( ),0.5

1 2, , ,0.5,

1 i jN

i j L i jxi j

C C dN

γ γ−

+−

= ∫ (3.16)

( )1,0.5

1 2, , 1,0.51,

1 i jN

i j R i jxi j

C C dN

γ γ+− −

+ +−+

−= ∫ (3.17)

The expression of average value of volume fraction at cell face is obtained by using Eq.

3.11 in Eq. 3.16 and 3.17 as

( )1 2, , , , ,0.5 1ni j L i j i j i jC C N ω+ = + − and ( )1 2, , 1, 1, 1,0.5 1n

i j R i j i j i jC C N ω+ + + += − + (3.18)

The solution algorithm of PLM is same as MUSCL but Eq. 3.13 is replaced with Eq. 3.18.

3.2.1.3.3 Piecewise Parabolic Method (PPM)

In this method, a parabola is the interpolating function. It is defined by a polynomial built

from the cell centre value of a variable and cell face values. The polynomial has the form,

( ) ( ) ( )( )2, , , , , , , , ,

13 12ni j i j L i j R i j R i j LC Cγ γ γ= + Δ + Δ + Δ −Δ − (3.19)

Where,

( ) ( )

( ) ( )

, , , 1, , 1,

, , 1, , 1, ,

10.5 , 610.56

n n n ni j L i j i j i j i j

n n n ni j R i j i j i j i j

C C

C C

ω ω

ω ω

− −

+ +

Δ = − + −

Δ = − − − (3.20)

Slope, ,i jω is obtained from Eq. 3.12. The solution is made oscillations free by modifying

, , , ,,i j L i j RΔ Δ as,

( )( )

, , , , , ,

, , , , , ,

max 0,min ,2 ,

max 0,min 2 ,

Mi j L i j L i j R

Mi j R i j L i j R

S S

S S

⎡ ⎤Δ = Δ Δ⎣ ⎦⎡ ⎤Δ = Δ Δ⎣ ⎦

(3.21)

Where, ( ), ,i j LS sign= Δ . The expressions of average value of volume fraction at cell face

in PPM are obtained by using Eq. 3.19 in Eq. 3.16 and 3.17 as

( )( ) ( )2, ,

1 2, , , , , , , , , , , ,10.5 1 3

3 2 6i j i jn M M M M

i j L i j i j i j L i j R i j R i j L

N NC C N+

⎛ ⎞= + − Δ + Δ + Δ −Δ − +⎜ ⎟⎜ ⎟

⎝ ⎠ (3.22)

( )( ) ( )2

1, 1,1 2, , 1, 1, 1, , 1, , 1, , 1, ,

10.5 1 33 2 6i j i jn M M M M

i j R i j i j i j L i j R i j R i j L

N NC C N + +

+ + + + + + +

⎛ ⎞= − + Δ + Δ + Δ −Δ + +⎜ ⎟⎜ ⎟

⎝ ⎠ (3.23)

The solution algorithm of PPM is same as the MUSCL except that Eq. 3.13 is replaced by Eq. 3.22 and Eq. 3.23.

( ),, Local Courant Number

ni j x

i j

UN

XτΔ=

Δ

54

3.2.2 Application of Boundary Condition for Volume Fraction As VOF equation is a hyperbolic equation, theoretically there is no need to implement

boundary condition but in numerical implementation boundary values of volume fractions are

required in all the methods to calculate gradients of volume fractions for near boundary cells.

In case of donor-acceptor and high resolution convection schemes the Neumann boundary

condition, 0Cn

∂=

∂ is used.

In case of PPM as three cells are required to calculate the value of volume fraction at a

cell, there is a problem of availability of cells for near border cells. For easy calculation of

gradients, in addition to the boundary cell, one fictitious cell outside the boundary is taken. The

values of boundary cell and the fictitious cell are taken to be same.

In case of geometric method, the volume fraction value at boundary is required for proper

calculation of interface normal. In the completely filled or empty cells Neumann boundary

condition is used, the implementation of the boundary condition in cells having interface is

explained with an example. For a case of west boundary as shown in Fig. 3.14, the interface

intersects the west cell face at a distance yi from the south west corner of the cell. The volume

fraction at the boundary cell is set as, boundary iC Y Y= Δ .

Figure 3.14: Implementation of boundary condition in geometric method, example of west

boundary.

In some problems, periodic boundary condition is to be implemented. The boundary

condition for east and west boundary is satisfied as, 0, 1,j NI jC C −= and 1NIC C= , Where, NI is the

total number of cell centers in x-direction. The value of fictitious cell in case of PPM is set as,

1, 2,j NI jC C− −= , boundary condition is satisfied in a similar way for other boundaries.

The periodic boundary needs careful implementation in case of geometric advection, for

instance the flux polygon for a west face for a west boundary cell has to be constructed at east

face of east boundary cell having same j.

55

3.2.3 Surface Tension Force Modeling in VOF method In present work the CSF method is used for modeling surface tension. The volume

fractions are smoothened using K8 kernel. Volume fraction field is smoothened by using Eq. 1.24

and Eq. 1.25 in the domain. Interface normal and curvature are calculated based on smoothened

volume fraction field using Eq. 1.22 and 1.21 respectively.

Although straight forward, the smoothing to volume fractions is computationally very

expensive process. The reason for this is that to calculate smoothened volume fraction in one

cell, complete domain is traversed. So for N number of cells, N2 number of cells will have to be

traversed per time step while smoothing. Interface normal and curvature are discretized as,

2 2ˆ , with yxx y

CCn C C CC C

⎛ ⎞= ∇ = +⎜ ⎟

∇ ∇⎝ ⎠

( )

2 2

322 2

2y xx x y xy x yy

x y

C C C C C C C

C Cκ

− += −

+

The derivatives of smoothened volume of fluid function are calculated using second order

central difference scheme.

3.2.4 Calculation of Properties in VOF method The properties at the pressure cell centers are calculated using Eq. 1.14 and Eq. 1.15.

Density is required at cell centers of u-velocity and v-velocity cell and cell face centers of

pressure cell. Viscosity is required at face centers of u-velocity and v-velocity cell. Properties at

all required locations are calculated using simple averaging.

3.3 Implementation of LS method

In present work level set algorithm explained in Section 1.3.3.5 is followed. The level set

function is located at the north east vertex of the cell; such staggering of level set function is

done to be able to use a uniform stencil everywhere in the domain.

Implementation details of methodologies adopted to solve level set convection equation is

discussed in subsection 3.3.1, solution procedure of reinitialization equation is discussed in

subsection 3.3.2, application of boundary condition of level set function, modeling of surface

tension and procedure to calculate properties at required locations are presented in subsections

3.3.3, 3.3.4 and 3.3.5 respectively.

56

3.3.1 Solution of Level Set Convection Equation Many researchers have used higher order convection schemes like Essentially Non

Oscillatory (ENO) and Weighted ENO (WENO) to discretize level set equation (Eq. 1.28), as it

is a convection equation of a smooth function. WENO and ENO schemes are based on first order

Godunov scheme. In present work Godunov scheme and 5th order WENO scheme have been

implemented. Numerical schemes have been implemented using a finite difference method.

Finite difference discretization of level set equation and implementation of Godunov and WENO

schemes are presented in this section.

3.3.1.1 Temporal Discretization of Level Set Convection Equation The level set equation can be written as,

( ) ( ) , L Where L u vt x yφ φ φφ φ∂ ∂ ∂= − = +

∂ ∂ ∂ (3.24)

The time derivative term is discretized using a third order Total Variation Diminishing

(TVD) Runge-Kutta scheme as [Jiang and Peng (2000)],

( )1 n ntLφ φ φ= + Δ

( ) ( )( )2 1 134

nt L Lφ φ φ φΔ= + − +

( ) ( ) ( )( )1 2 1 2812

n nt L L Lφ φ φ φ φ+ Δ= + − − + (3.25)

3.3.1.2 Spatial Discretization of Level Set equation

The gradient of level set function ,x yφ φ∂ ∂⎛ ⎞

⎜ ⎟∂ ∂⎝ ⎠, denoted by ( ),x yφ φ is calculated by first order

forward difference or backward difference discretization. Finite difference discretization of the

spatial term in Eq. 3.24 is given as,

( ), , , , , , ,i j i j x i j i j y i jL u vφ φ φ= + (3.26)

Where,

( ) ( )

, , , , , ,

, , , , , , , , , ,

, , , , , , , , , ,

0 0 0 and 0

0.5 0 0.5 0

x i j i j y i j i j

x i j x i j i j y i j y i j i j

x i j x i j i j y i j y i j i j

if u if vif u if v

if u if v

φ φφ φ φ φ

φ φ φ φ

− −

+ +

− + − +

⎧ ⎧> >⎪ ⎪⎪ ⎪= < = <⎨ ⎨⎪ ⎪+ = + =⎪ ⎪⎩ ⎩

In Godunov scheme, the forward difference and backward difference discretization of level

set function is,

57

1, ,i j i jx x

φ φφ ++ −

=Δ

and , 1,i j i jx x

φ φφ −− −

=Δ

(3.27)

Similarly for y-direction, the forward difference and backward difference discretization of

level set function is,

, 1 ,i j i jy y

φ φφ ++ −

=Δ

and , , 1i j i jy y

φ φφ −− −

=Δ

(3.28)

Godunov scheme is first order and induces numerical diffusion in the solution. In case of

level set methods ENO and WENO schemes are popular with researchers. In the present work 5th

Order WENO scheme of Jiang and Peng (2000) is implemented.

Expressions of forward difference and backward difference discretization in case of 5th

order WENO scheme of Jiang and Peng (2000) are modified in this work for simplifying the

implementation. The modified expressions of forward and backward difference in 5th WENO for

x-direction are,

( ) ( )2, 1, 1, 2,1 8 8 , , ,

12x i j i j i j i j WENO A B C Dx

φ φ φ φ φ ψ± ± ± ± ±− − + += − + − ±

Δ (3.29)

Where,

( )( )( )( )

1, 2, 3,

, 1, 2,

1, , 1,

2, 1, ,

2

2

2

2

i j i j i j

i j i j i j

i j i j i j

i j i j i j

A x

B x

C x

D x

φ φ φ

φ φ φ

φ φ φ

φ φ φ

±± ± ±

±± ±

±±

±

= − + Δ

= − + Δ

= − + Δ

= − + Δ

∓

∓ ∓

A, B, C and D are different gradients of level set functions on local stencils near cell i,j

used to determine weights in WENO scheme. The WENO weights are calculated as function of

gradient of level set function on local stencils such as to give more weight to the smoothest

stencil. Similarly, the modified expressions of forward and backward difference in 5th WENO for

y-direction are,

( ) ( ), 2 , 1 , 1 , 21 8 8 , , ,

12y i j i j i j i j WENO A B C Dy

φ φ φ φ φ ψ± ± ± ± ±− − + += − + − ±

Δ (3.30)

Where,

( )( )( )( )

, 1 , 2 , 3

, , 1 , 2

, 1 , , 1

, 2 , 1 ,

2

2

2

2

i j i j i j

i j i j i j

i j i j i j

i j i j i j

A y

B y

C y

D y

φ φ φ

φ φ φ

φ φ φ

φ φ φ

±± ± ±

±± ±

±±

±

= − + Δ

= − + Δ

= − + Δ

= − + Δ

∓

∓ ∓

WENOψ in Eq. 3.29 and Eq. 3.30 is given as,

58

( ) ( ) ( )0 2

1 1 1, , , 2 23 6 2WENO A B C D w A B C w B C Dψ ⎛ ⎞= − + + − − +⎜ ⎟

⎝ ⎠

The WENO weights w0 and w2 are calculated as,

( ) ( ) ( )( ) ( )

( ) ( )

( ) ( )

0 20 2

0 1 2 0 1 2

0 1 22 2 21 2

2 2

2 21

2 2

, ;

1 6 3, and

13 3 3 ,

13 3 ,

13 3 3 .

o

o

o

w w

IS IS IS

IS A B A B

IS B C B C

IS C D C D

α αα α α α α α

α α αε ε ε

= =+ + + +

= = =+ + +

= − + −

= − + +

= − + −

Implementation of Godunov scheme for advection is straight forward,

1. The forward and backward differences of in x and y-directions form the old time level

values of level set function ( )nφ are calculated using Eq. 3.27 and Eq. 3.28.

2. The value of ( )L φ is calculated using Eq. 3.26.

3. First intermediate value of level set function ( )1φ is calculated using Eq. 3.25.

4. Steps 1 to 3 are followed to get the second intermediate value ( )2φ of level set function

using Eq. 3.25, the values of ( )1L φ and ( )nL φ are used.

5. Steps 1 to 3 are followed to get the time advanced value ( )nφ of level set function using

Eq. 3.25, the values of ( ) ( ) ( )2 1, nL L and Lφ φ φ are used.

Godunov scheme is First order upwind equivalent in finite volume method. The difference

between Godunov and WENO scheme is difference in the method to calculate the gradient i.e.

Eq. 3.29 and Eq. 3.30 are used instead of Eq. 3.27 and Eq. 3.28. The rest of algorithm is same.

3.3.2 Solution Procedure of Reinitialization Equation Importance of reinitialization is already mentioned in section 1.3.3.3. Solution of Eq. 1.31

will ensure reinitialization of a function oφ to signed normal distance function. The function

oφ need not be a normal distance function initially.

Eq. 1.31 is shown to be a hyperbolic equation (Eq. 2.7) in section 2.2.2.2. In the present

work, spatial discretization of Eq. 2.7 is done using 5th order WENO scheme and temporal

discretization using first order explicit Euler scheme. Eq. 2.7 can be written as,

( ) ( ) ( )2 2 2 2s

o yo xx y o

x y x y

SSSεε

τ ε

φ φφ φφ φ φ φ

φ φ φ φ+ + =

+ + (3.31)

59

Discretized form of Eq. 3.31 is

( ) ( ) ( ), , , , , , , ,1, , , , , , , ,2 2 2 2

, , , , , , , ,

i j o x i j i j o y i jl li j i j s i j o x i j y i j

x i j y i j x i j y i j

S SS ε εε

φ φ φ φφ φ τ φ φ φ

φ φ φ φ+ ⎛ ⎞= + Δ − +⎜ ⎟

⎜ ⎟+ +⎝ ⎠ (3.32)

In Eq. 3.32 superscript ‘l’ stands for iteration number, sτΔ is the pseudo time step. The

mollified sign function is taken as [Salih and Ghosh Moulic (2005)],

( )( )22 2

oo

o o

Sx

εφφ

φ φ=

+ ∇ Δ.

Eq. 3.32 is solved in the complete domain till steady state in the band of designated width

around the interface is achieved.

As proposed by Sussman et al. (1994), the pseudo time step is taken as, 0.1s XτΔ = Δ . The

convergence criteria of steady state is taken as 30.1 XΔ and the error norm to check convergence

is,

,

1, ,

i j

l li j i j

REINITEM

φ ε

φ φ+

<

−

=∑

(3.33)

M is the total number of grid points located in the band. Eq. 3.33 is defined in such a way

that convergence will be checked only in the band of width 2ε near to the interface. As the

interface normal and curvature are calculated in the band only and the Heaviside function is

varied smoothly in band as a function of level set function, the convergence of reinitialization

equation is measured only in the band. Outside the band the sign of level set function is enough

to calculate Heaviside function.

The steps required to perform single iteration of Eq. 3.32 are,

1. Mollified sign function ( )( )oSε φ is calculated in all cells.

2. The forward and backward difference of gradient of initial function or old iteration

function ( ),x yφ φ± ± is calculated using 5th order WENO scheme.

3. The term , ,x i jφ is determined as,

a. If ( ) ( ), , , , , , , , , , , ,0 and 0 then i j o x i j i j o x i j x i j x i jS Sε εφ φ φ φ φ φ− + +≤ ≤ =

b. If ( ) ( ), , , , , , , , , , , ,0 and 0 then i j o x i j i j o x i j x i j x i jS Sε εφ φ φ φ φ φ− + −≥ ≥ =

c. If ( ) ( ), , , , , , , , , ,0 and 0 then 0i j o x i j i j o x i j x i jS Sε εφ φ φ φ φ− +≤ ≥ =

d. If ( ) ( ), , , , , , , ,0 and 0 i j o x i j i j o x i jS Sε εφ φ φ φ− +≥ ≤

i. If ( ) ( ), , , , , , , , , , , , then i j o x i j i j o x i j x i j x i jS Sε εφ φ φ φ φ φ+ − +≥ =

60

ii. If ( ) ( ), , , , , , , , , , , , then i j o x i j i j o x i j x i j x i jS Sε εφ φ φ φ φ φ+ − −≤ =

4. Similarly the term , ,y i jφ is also obtained.

5. Values of ( ), ,i j oSε φ , , ,x i jφ and , ,y i jφ are plugged in Eq. 3.32 to get a new value of level set

function.

6. The error is calculated using Eq. 3.32 and convergence is checked using the condition 30.1REINITE X≤ Δ . If the convergence criterion is not met, iterations are performed by

following steps 2 to 6.

3.3.3 Application of Boundary Condition for Level Set Function As in case of VOF equation, level set governing equation is a hyperbolic equation so there

is no need to implement boundary condition but in numerical simulation, boundary values of

level set functions are required to calculate gradients of level set functions for near boundary

cells.

When the Godunov scheme is used, Neumann boundary condition on all boundaries is

used, 0nφ∂=

∂. In case of WENO scheme, for first cell inside the cell there is need of two cells in

addition to the boundary cell to calculate the gradient and WENO weights properly.

In the present work, two fictitious cells in addition to the boundary cell are defined outside

the domain. The approach to set the boundary values is explained with the example of west

boundary as shown in Fig. 3.15; similarly for other boundaries the expressions can be derived.

Figure 3.15: Stencil required for implementing boundary condition at west boundary of

domain for WENO scheme.

The values of fictitious cells and boundary cell can be taken same as the first real cell

inside that boundary, for west boundary the expression is, 2, 1, 0, 1,j j j jφ φ φ φ− −= = = .

Another way is to calculate the values of fictitious cells and boundary cell from interior

cells by performing quadratic extrapolation. The expressions for west boundary are,

0, 1, 2, 3,3 3j j j jφ φ φ φ= − + , 1, 0, 1, 2,3 3j j j jφ φ φ φ− = − + and 2, 1, 0, 1,3 3j j j jφ φ φ φ− −= − + .

In some problems periodic boundary condition is to be implemented in that case the

boundary condition is satisfied as, 0, 1,j NI jφ φ −= and , 1,NI j jφ φ= , Where NI is the total number of

61

level set nodes in x-direction. The values of fictitious cells is set as, 1, 2,j NI jφ φ− −= and

2, 3,j NI jφ φ− −= , similarly for other boundaries.

Unlike geometric advection method, no special treatment other than specifying boundary

condition and setting values of fictitious cells is required in WENO scheme to implement

periodic boundary condition.

3.3.4 Surface Tension Force Modeling in LS method As in case of VOF code, the CSF method is used for modeling surface tension in level set

code also. Interface normal and curvature are calculated using Eq. 1.30 and 1.34 respectively.

The Dirac delta function is calculated using Eq. 1.35. The smoothing length,ε in Eq. 1.35 is

taken as1.5 XΔ . Interface normal and curvature are discretized as

2 2ˆ , with yxx yn

φφ φ φ φφ φ

⎛ ⎞= ∇ = +⎜ ⎟∇ ∇⎝ ⎠

( )

2 2

32 2 2

2y xx x y xy x yy

x y

φ φ φ φ φ φ φκ

φ φ

− += −

+

The derivatives of level set function are calculated using second order central difference

scheme [Chang et al. (1996)].

3.3.5 Calculation of Properties in LS method Heaviside function is calculated at the cell centers of level set CV. i.e the vertex of the cell.

The properties at the level set cell centers are calculated using Eq. 1.32. Density is required at

cell centers of u-velocity and v-velocity cell and cell face centers of pressure cell. Viscosity is

required at face centers of u-velocity and v-velocity cell. Properties at all required locations are

calculated using simple averaging.

62

4 A New Combined Level Set-Volume of Fluid Method

Combined Level Set-Volume Of Fluid (CLSVOF) method was proposed by Sussman and

Puckett (2000) to rectify the mass loss problem in LS method and inaccurate calculation of

surface tension force in VOF method.

In CLSVOF method the conservation equation of Heaviside function field is solved using a

geometric advection method, conservation of Heaviside function will mean conservation of

mean density as properties are calculated using Eq. 1.32 and thus mass will be conserved. The

conservation equation of Heaviside function is,

0H u Ht

∂+ ⋅∇ =

∂ (4.1)

Once the new Heaviside function field is obtained from solution of Eq. 4.1, the level set

function is calculated from the Heaviside function field with the help of interface reconstruction.

The interface curvature for surface tension force and normal used for PLIC interface

reconstruction are calculated from level set function field.

CLSVOF method of Sussman and Puckett (2000) has been found to give good results for

surface tension dominant flows with like bubble formation and boiling. In CLSVOF method the

mass is conserved as in VOF method and the surface tension force is calculated as accurately as

in LS methods.

4.1 Motivation to Develop New Method The CLSVOF method of Sussman and Puckett (2000) has inherited a big disadvantage of

VOF method i.e. PLIC interface reconstruction and geometric advection method. Due to

requirement of PLIC interface reconstruction and geometric advection, a new formulation and

implementation procedures need to be developed to extend CLSVOF method to complex

geometry.

Motivation to develop a new CLSVOF method comes from the need to have a method in

which mass conservation is obeyed, surface tension force is calculated accurately and extension

to complex geometries and 3D is easy with no requirement of interface reconstruction and

geometric advection method.

63

The geometric advection method is preferred over other methods to solve VOF governing

equation as it maintains step discontinuity in volume fraction field without inducing mass loss. It

is found from results on many problems; PPM method does not introduce mass loss in the initial

times of solution but at later time lot of mass loss is incurred. The continuous smearing of

volume fraction field is thought to be the reason for mass loss at later times.

If the smearing of volume fractions is limited within a fixed distance across the interface at

all times, the mass loss will also be limited. Level set method provides solution to have limited

smearing of volume fractions/Heaviside function inside the band.

4.2 Working Principle of New CLSVOF method Consider this, the Heaviside function field is advected using a high resolution convection

scheme and the level set function is calculated from the advected Heaviside function field. If the

advected Heaviside function field is used to solve Eq. 4.1 then over a period of time there will lot

of numerical diffusion introduced in the solution and the advantage of using CLSVOF

formulation will be lost.

In the new approach, the Heaviside function field is advected using a high resolution

convection scheme and the level set function is calculated from the advected Heaviside function

field. To solve Eq. 4.1, a new Heaviside function field is calculated using Eq. 1.33 from the

newly calculated level set function. By following this approach the numerical diffusion in not

allowed to increase over the period of time.

In the new approach as the interface reconstruction is avoided, problem of extracting

values of level set function from the Heaviside function field arises. Outside the band of width

2ε level set function is set as

0 if Hφ ε= − = and 1if Hφ ε= = (4.2)

The level set function is to be calculated inside the band. Two different approaches have been

tried to calculate value of level set function from Heaviside function; both the approaches are

based on use of Eq. 1.33,

Approach A: Inverse Curve fit to calculate level set function in the band:

The value of Heaviside function as a function of level set function using Eq. 1.33 is plotted

in Fig. 4.1(a). The inverse relation of level set function as a function of Heaviside function is

obtained as a 6th order polynomial fit using Microsoft Excel. The relation of level set function as

a function o Heaviside function obtained as,

( ) ( )6 5 4 3 2 1H aH bH cH dH eH fH gφ ε= + + + + + + (4.3)

64

The coefficients of the polynomial fit are obtained as, a = -4e-7, b = 31.725 c = -79.312, d

= 73.244, e = -30.554, f = 6.673 and g =-0.8878. The result of the curve fit and the actual inverse

relation are plotted in Fig. 4.1(b). It is seen that although the curve fit does not give exact result,

the result are good enough. The approach of dividing the band width in to small intervals and

fitting many polynomials instead of a single polynomial has not been tried yet.

Level Set Function,

HeavisideFunction,H

-1

-1

-0.75

-0.75

-0.5

-0.5

-0.25

-0.25

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

-0.1 -0.1

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1 1

1.1 1.1Relation of H as function of φ

=1ε Heaviside Function, H

LevelSetFunction

0

0

0.5

0.5

1

1

-1 -1

-0.75 -0.75

-0.5 -0.5

-0.25 -0.25

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1Actual relation of as funtion of HCurve fit of as function of H

φφ

Figure 4.1: The relation between Heaviside function and level set function (a) Direct relation

given by Eq. 1.33 (b) Inverse relation and curve fit relation (Eq. 4.3).

Approach B: Solution of Eq. 1.33 using iterative method in band:

One other approach to determine value of level set function in band is to solve Eq. 1.33

directly in the band. In the band the equation to be solved is,

( ) 1 sin 02 2

f Hφ ε πφφε π ε+ ⎛ ⎞= + − =⎜ ⎟

⎝ ⎠ (4.4)

The limits of level set function are known i.e. toε ε− . Bisection method is used to solve

Eq. 4.4.

4.3 Complete Algorithm of New CLSVOF Method The new CLSVOF is combination of procedures that are already implemented for LS and

VOF method. Complete algorithm of new CLSVOF is as follows,

1. Initialize velocities, pressure and level set function in the domain.

2. Calculate Heaviside function at level set cells using Eq. 1.33; calculate properties at level

set cell using Eq. 1.32. Use simple averaging to get properties at required locations.

3. Calculate interface normal and curvature using level set function field. Solve the NS

equations in the domain using projection method.

65

4. Using the newly calculated velocity field, solve the Heaviside advection equation

explicitly using PPM method.

5. Calculate new level set function in the band using Eq. 4.3 or solving Eq. 4.4, set values of

level set function outside the band using Eq. 4.2

6. Reinitialize the newly calculated level set function field to signed normal distance

function.

7. Go to step 2 till the end time is arrived at or steady state has reached.

4.4 Conservative Redistribution It has been found that the sum of Heaviside function calculated from the newly calculated

level set function does not remain same as the sum of advected Heaviside function calculated by

PPM method.

In short, the mass error was found in the New CLSVOF method and it was because of

incorrect way of determining level set function from Heaviside function field i.e. Eq. 4.3 or use

of Eq. 4.4. The constraint that the sum of advected Heaviside function and sum of Heaviside

function calculated as a function of newly calculated level set function should be same has not

been devised and implemented.

In this work, mass conservation is enforced by equally distributing or taking away the

amount of mass difference incurred due to use of Eq. 4.3 or use of Eq. 4.4. The validity and

effectiveness of conservative mass distribution is been studied.

66

5 Numerical Tests and Results

In the present work two-phase flow codes based on VOF, LS and a new CLSVOF method

have been developed. A two-phase flow code consists of three modules, a Navier-Stokes (NS)

solver to solve single fluid model, an interface representation and advection module i.e. solution

method of VOF and level set governing equation and surface tension module. To develop a code

in systematic way, each of the modules is implemented and tested separately. Finally all the

modules are combined to solve a two-phase flow problem. In this chapter, the results of the

benchmark tests and two-phase flow problems are presented.

NS solver is benchmarked on a single phase problem of lid driven cavity flow. The

solution methodologies implemented for solution of VOF, level set and new CLSVOF governing

equation are tested individually on four standard interface advection test problems i.e.

translation, solid body rotation, fluid body subjected to single vortex and fluid body subjected to

multiple vortices in form of deformation field. The surface tension model is benchmarked using

standard test problem of Young-Laplace Law test.

VOF, level set and new CLSVOF codes are developed by combining all the

abovementioned modules. The two-phase flow codes are benchmarked on the standard broken

dam problem. The two-phase flow codes are then tested on rather difficult, Rayleigh-Taylor

instability, Splash of water drop in water pool and rise of two gaseous bubbles surrounded by

fluid in a channel.

5.1 NS Solver Benchmark: Lid Driven Cavity Flow This is a classic single phase problem to test the NS solver and relative performance of

different convection schemes for various Reynolds number. The problem setup is a square cavity

completely filled with fluid; the fluid in the cavity is given motion by lid of the cavity moving at

a constant velocity shown in Fig. 5.1. Ghia et al. (1982) solved the same problem using stream

function vorticity formulation, their results are considered as benchmark results. The steady state

values of u velocity along the vertical mid-plane and v velocity along the horizontal mid-plane

using QUICK convection scheme for various Re are compared with results of Ghia et al. (1982)

in Fig. 5.2, an excellent agreement of present results with benchmark results is seen in figure.

Furthermore, the streamlines are also compared in Fig. 5.3 with a good agreement.

67

Figure 5.1: Domain and boundary conditions for lid driven cavity flow problem

u

y

-1 -0.5 0 0.5 10

0.25

0.5

0.75

1

-0.5

v0

11

-1-1

x

0.5

0.5 0.75Ghia et al.QUICK

Re 100 [60x60] (c)

u

y

-1 -0.5 0 0.5 10

0.25

0.5

0.75

1

-0.5

v0

11

-1-1

x

0.5

0.5 0.75Ghia et al.QUICK

Re 1000 [60x60] (d)

Figure 5.2: Comparison of present results on a grid size of 60x60 for variation of u-velocity along vertical centerline and v-velocity along horizontal center line with benchmark results of Ghia et al. (1982) on 129x129 grid (a) Re = 100 (b) Re = 1000

5.2 VOF, LS and new CLSVOF module benchmark VOF, LS and new CLSVOF modules are implemented individually and must be tested

independent of Navier-stokes solver. Mass conserving property of a two-phase solution

methodology largely depends on the accuracy of solution of interface advection methodology.

In the present work, VOF governing equation is solved by three different solution

methodologies:

1. Donor-acceptor (DA) method of Hirt and Nichols (1981).

2. High resolution convection scheme PPM.

3. Multidimensional geometric advection method coupled with Youngs’ PLIC method (Y-

PLIC).

Cavity filled With Fluid

00

0

uvpx

==∂=

∂

00

0

uvpx

==∂=

∂

0, 0, 0pu vy

∂= = =

∂

Lid moving with constant u-velocity

X

X

Y Y

, 0, 0opu u vy

∂= = =

∂

68

Re 100 (a)

Re 100 (b)

Re 1000 (c)

Re 1000 (d)

Figure 5.3: Streamlines for Re = 100 and Re = 1000 on fine grid (60x60) (a) & (c) Present results on 60x60 grid (b) & (d) Benchmark result of Ghia et al. (1982) on 129x129 grid.

3rd order TVD scheme is used to discretize the temporal terms and 5th order WENO

scheme is used to discretize the spatial terms of level set convection equation (WENO). The

spatial terms of reinitialization equation are discretized using 5th order WENO scheme.

In case of new CLSVOF method, high resolution PPM method is used to solve the

heaviside convection equation and the spatial terms of reinitialization equation are discretized

using 5th order WENO scheme.

Interface reconstruction module in case of Y-PLIC method and the reinitialization

algorithm based on WENO are tested before the complete solution methodologies are tested.

5.2.1 Interface Reconstruction Test Accuracy of Interface reconstruction has direct relation with the mass error induced in

VOF method as it is important part of geometric advection method. Interface reconstruction

based on Youngs’ PLIC method is tested by reconstructing a circle (R = 0.15) centered at (0.5,

0.75) in a square domain of size 1. Fig. 5.4 shows the results of reconstruction of circle on 8x8,

16x16 and 32x32 grid. It is seen from Fig. 5.4 (a), on a very coarse grid the interface is

69

reconstructed to be collection of disconnected line segments; this is shortcoming of any PLIC

method. However, the reconstructed interface looks more like a circle on grid refinement as seen

in Fig. 5.4(b) and Fig. 5.4(c).

(a)

(b)

(c) Figure 5.4: Interface reconstruction of a circle using Youngs’ PLIC method (a) 8x8 grid (b)

16x16 grid (c) 32x32 grid.

5.2.2 Reinitialization module test Reinitialization of level set function is done after every time step to ‘repair’ the level set

function to signed normal distance function. Level set function should be signed normal distance

function in the band near the interface to enable accurate calculation of properties and curvature

in the band. The position of interface i.e. 0φ = line should not change during reinitialization.

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1 1

0

0

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1 1

(a)

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1 1

0

0

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1 1

(b)

Figure 5.5: Effect of boundary conditions on reinitialization of a square interface on 32x32 grid (a) Neumann boundary condition (b) Periodic boundary condition

The reinitialization module developed in the present work is tested to reinitialize a

square of size 0.3 centered at (0.5, 0.75) in a square domain of size 1. Value of level set function

is set as 1 outside the square and -1 inside the square. The reinitialization is carried out on 32x32,

64x64 and 128x128 grid. The reinitialization is carried out for Neumann and periodic boundary

condition to examine the effect of boundary condition on reinitialization. Fig. 5.5 shows the

contours of reinitialized level set function on 32x32 grid for Neumann and periodic boundary

70

condition. It is seen from Fig. 5.5, as the contours of level set function are parallel, the level set

function has been reinitialized to normal distance function for both boundary conditions. Fig.

5.5(a), shows that the level set contour near all the walls becomes normal to wall, which signifies

implementation of Neumann boundary condition and near wall level set function does not remain

a normal distance function.

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1 1

Before Reinit.

After Reinit.

(a)

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1 1

Before Reinit.

After Reinit.

(b)

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

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Figure 5.6: Effect of different grid size and boundary condition on accuracy of reinitialization procedure (a) 32x32 Neumann b.c. (b) 64x64 Neumann b.c. (c) 128x128 Neumann b.c. (d) 32x32 Periodic b.c. (e) 64x64 Periodic b.c. (f) 128x128 Periodic b.c.

Periodic boundary condition, shown in Fig. 5.5(b), near the north boundary the variation

of level set function is smooth near wall, and the same level set is introduced inside the domain

through the south boundary. Use of periodic boundary condition ensures that the level set

function remains normal distance function everywhere in the domain, even near boundaries. Fig.

5.6 shows the interface position before and after reinitialization for different grid sizes and

boundary condition. It is seen from Fig. 5.6(a) and Fig. 5.6(d) for the case of 32x32 grid, the

effect of different boundary condition is found to be negligible on a same grid size. For other

grid sizes also the effect of boundary condition is negligible. Grid refinement improves the

solution as seen from Fig. 5.6(a), (b) and (c) for the case of Neumann boundary condition.

71

The error in case of 32x32 grid is maximum, in a flow problem the error due to

reinitialization accumulates to result in mass loss/gain. Similar trend for periodic boundary

conditions is seen from Fig. 5.6(d), (e) and (f).

5.2.3 VOF, LS and CLSVOF Module Tests Advection algorithms used in VOF, LSM and new CLSVOF method are tested on four

hypothetical pure advection test problems proposed by Rider and Kothe (1998). The advection

test problems are designed to measure the accuracy of only the interface advection methodology

of VOF, LSM and CLSVOF.

In advection test problems, Navier-Stokes equations are not solved but a fluid body is

placed in a domain, subjected to a predetermined velocity field, inducing a desired movement

and deformation of fluid body in the domain. During the movement and deformation of the fluid

body its volume (area in this case) must remain constant at all times.

5.2.3.1 Advection Tests: Physical Description The computational domain and initial conditions are identical for all four test problems.

The computational domain is a square of size 1x1 and in this domain a circular fluid body of

radius 0.15 is placed having center at (0.50, 0.75) as shown in Fig. 5.7.

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Figure 5.7: Domain and Initial condition for advection test problem.

5.2.3.2 Advection Tests: Velocity Fields The different tests are characterized by different velocity fields. All the velocity fields are

solenoidal and thus continuity equation is satisfied for all test cases.

72

For different test problems, the velocity field is expressed as the stream function. The velocity

in terms of stream function is defined as u yψ= −∂ ∂ and v xψ= ∂ ∂ . The four different stream

functions corresponding to different advection tests:

1. Translation test: The stream function for translation test is x yψ = − . Under this velocity field

the circular body translates at 45o angle across the mesh. In one time unit the fluid body comes

back to initial position.

2. Solid body rotation test: A constant-vorticity velocity field is imposed at the center of the

domain represented by the stream function ( ) ( )( )2 20.5 0.5x yψ = − + − . The fluid body rotates

around the center of domain but does not undergo any topology changes. The body undergoes

one complete rotation in π time units.

3. Single vortex test: A single vortex with center at the center of domain is given by the stream

function, ( ) ( )2 2sin sin 4x yψ π π= . The single vortex stretches out the fluid body and pulls it

towards the center of domain, inducing a severe topology change.

4. Deformation field test: Deformation field is a series of 16 vortices which distorts the fluid

severely, the stream function is ( )( ) ( )( )sin 4 0.5 cos 4 0.5 4x yψ π π= + + . The velocity vectors

for all test problems considered are plotted in Fig. 5.8.

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Figure 5.8: Velocity field for advection test problems.

73

5. time-reversed flow field

In case of translation and solid body rotation, the fluid body returns to its initial state after a

fixed time-period. In case of single vortex test and deformation field test, Rider and Kothe

(1998) mentioned that multiplying the velocity field by ( )cos t Tπ gives the time-reversed

velocity field, meaning that the fluid body will undergo deformations until time, t = T/2, at t =

T/2 flow direction reverses and fluid body returns to its initial state at t = T. By using time

reversed flow fields, initial and final interface shape can be compared, ideally they should be

same.

5.2.3.3 Advection Tests: Error Measurement Each method is assessed quantitatively and qualitatively on all the four test problems.

Results are compared with benchmark results wherever possible. Qualitative comparisons

between the initial and final interface plots are made to observe the inconsistency between final

and initial fluid shape in case of translation, solid body rotation and time reversed vortex fields.

In case of VOF method, volume fraction contour level of 0.5 is considered as interface. In case

of LSM and CLSVOF, interface is represented by level set function contour level of 0.

Mass error incurred during the advection is a quantitative parameter for comparison, in

case of VOF method mass error is quantified as,

, ,, ,

,,

100%

t oi j i j

i j i jVOF o

i ji j

C Cm

C

−Δ = ×

∑ ∑

∑ (5.1)

Superscript ‘o’ stands for initial value of volume fraction; mass error is calculated at every

time step to know about the trend of mass loss. In case of level set method the sum of Heaviside

function represents the area of the fluid, hence for LSM and CLSVOF the mass error is

quantified as,

( ) ( )

( ), ,

, ,,

,,

100%

t oi j i j

i j i jLSM CLSVOF o

i ji j

H Hm

H

φ φ

φ

−Δ = ×

∑ ∑

∑ (5.2)

5.2.3.4 Advection Tests: Computational Details In case of VOF method, the volume fraction field is initialized in the domain equal to one

and zero inside and outside the circle, respectively. For those cells containing the circular

interface, the volume fraction value is set to a value between zero and one, in proportion to the

74

cell volume truncated by the circle. Periodic boundary conditions are applied at all boundaries in

case of VOF method for all problems.

In case of LSM and CLSVOF, The level set function is initialized as signed normal

distance function using equation of circle. Periodic boundary condition is used for translation test

and deformation field test, Neumann boundary condition is used for solid body rotation and

single vortex test.

Table 5.1: Solution methodologies used for VOF,LSM and CLSVOF method and their short form used in present work

Method Solution methodology adopted in present work VOF Donor-Acceptor (DA), high resolution convection scheme, Piecewise

Parabolic Method (PPM) with superbee slope limiter and geometric advection method with Youngs’ PLIC method (Y-PLIC).

LSM 5th order WENO for advection and reinitialization (WENO) New CLSVOF PPM for advection of heaviside function and WENO for

reinitialization (CLSVOF)

The solution methodologies used in this work are as shown in Table 5.1. Four problems are

solved from the available velocity fields and availability of exact result. All the test problems are

solved on 32x32, 64x64 and 128x128 grid with CFL number 0.5 for translation and solid body

rotation tests and CFL number 1 for single vortex and deformation field test. The problem

conditions used to solve advection problems are shown in Table 5.2.

Table 5.2: Different advection problems solved in present work No. Problem Max

time Time Reversal

Exact result at end time

1 Translation 1 No Fluid body returns to initial position 2 Solid Body Rotation π No Fluid body returns to initial position 3 Single Vortex 3 No Available of Rider and Kothe (1998) 4 Deformation field 2 Yes (at t =

1) Available of Rider and Kothe (1998) at t = 1 and Same as initial condition at t = 2.

5.2.3.5 Advection Tests: Results and Discussion Interface plots at time, t = 1 for simple translation test for all methods on 322, 642 and 1282

are shown in Fig. 5.9. Ideally the fluid body should come to the initial position without

undergoing any shape change.

It is seen in Fig. 5.9(a), in case of VOF-DA method the shape of fluid body changes

drastically on coarser gird, the shape improves on fine grid but still it is not to circle. The final

position of interface on coarse grid is also not near the exact interface position. The circular

fluid body starts to develop corners in VOF-DA method because of its SLIC approximation and

75

1D advection algorithm. For the case of VOF-PPM as seen in Fig. 5.9(b), the interface shape

remains circular but the final position of interface is away from initial location. On fine grid the

final position and shape tend to match with exact result. The results of VOF-Y-PLIC as seen in

Fig. 5.9(c), match well with the exact result on coarse grid with further improvement on grid

refinement.

Interface shape has not changed much in case of LSM-WENO as seen from Fig. 5.9(d) but

the interface position does not match with the exact result. The results improve on grid

refinement.

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Figure 5.9: Final (t = 1) interface plots for translation test on 32x32, 64x64 and 128x128 grid using different methods (a) VOF-DA (b) VOF-PPM (c) VOF-Y-PLIC (d) LSM-WENO (e) CLSVOF-Without conservative redistribution (f) CLSVOF-With conservative redistribution

In case of new CLSVOF method as seen from Fig. 5.9(e), the interface shape changed

drastically and apparently lot of mass loss occurred for all grid sizes. When the method was

closely examined, it was found that the sum of advected heaviside function and sum of heaviside

function calculated from the level set function were not same. A step known as conservative

redistribution has been developed to overcome this problem wherein the difference in sum of

advected heaviside function and sum of heaviside function calculated from level set function is

76

distributed equally among the partially filled cells. Fig. 5.9 (f) shows the result of CLSVOF

method with conservative redistribution, it is seen that though the mass loss has been avoided the

interface shape has not improved. The reason for shape change in CLSVOF method is not known

yet. Without conservative redistribution the new CLSVOF method will incur mass loss, so for all

the problems CLSVOF is used with conservative redistribution.

Table 5.3: Mass error (%) at time, t = 1 for translation test for all solution methodologies on different grid sizes

VOF LSM New CLSVOF

Grid size

DA PPM Y-PLIC WENO Without

Conservative Redistribution

With Conservative Redistribution

32x32 7.20 4.46 1.93E-10 2.00E-02 40.81 0 64x64 1.06 2.13 1.18E-06 2.95E-04 45.20 9.77E-14

128x128 0.59 1.09 1.41E-07 1.90E-04 40.64 5.88E-14

Final mass loss incurred by all the methods on all grid sizes for translation test is shown in

Table 5.3, VOF-Y-PLIC method performs best followed by LSM-WENO. VOF-DA and VOF-

PPM incur considerable mass loss on coarse grid; reason for mass loss in VOF-DA method is

occurrence of undershoots and overshoots. In case of VOF-PPM, mass loss is incurred as the

slope limiting does not ensure that the sum of volume fractions remain constant in domain. Mass

loss for VOF-DA, VOF-PPM and LSM-WENO improve with grid refinement. Mass error in

case of VOF-Y-PLIC seems to increase with grid refinement, one possible reason could be: as

the grid is refined the dimensions of cell become very small and the convergence criteria in

bisection method used to locate interface may be met before the actual interface position is

achieved.

Mass loss incurred in CLSVOF method without conservative redistribution is for reasons

other that advection algorithm. CLSVOF method with conservative redistribution produces

practically zero mass loss, which validates the implementation of conservative redistribution

algorithm but it cannot be considered a good result as mass conservation is enforced by a

technique whose validity is yet to be proved.

Interface plots at time, t = π for solid body rotation test for all methods on 322, 642 and

1282 are shown in Fig. 5.10, in solid body rotation test also the fluid body should come to the

initial position without undergoing any shape change.

It is seen from Fig. 5.10; the interface shape in case of VOF-DA and CLSVOF has

changed. Interface shape in case of VOF-PPM and VOF-Y-PLIC are circular but the position of

interface does not match with position of exact interface. LSM-WENO performs best among all

methods even on coarse grid. Results of all methods improve on grid refinement.

77

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Figure 5.10: Final (t = π ) interface plots for solid body rotation test on 32x32, 64x64 and 128x128 grid using different methods (a) VOF-DA (b) VOF-PPM (c) VOF-Y-PLIC (d) LSM-WENO (e) CLSVOF-With conservative redistribution

Final mass loss incurred by all the methods on all grid sizes for solid body rotation test is

shown in Table 5.4, it is seen that the mass loss for VOF-Y-PLIC is higher than for VOF-PPM or

VOF-DA. High mass loss in VOF-Y-PLIC is due to use of geometric advection, when the

velocity in the domain is not same, adjacent flux polygons may overlap or leave some fluid

untouched resulting in mass loss. In case of translation test the velocity is same everywhere in

domain so mass error in VOF-Y-PLIC is negligible.

It is further seen from Table 5.4 that mass loss for LSM-WENO is very less even on coarse

grid and it reduces on grid refinement. Mass loss for CLSVOF was forced to be zero; same is

reflected in the results.

Table 5.4: Mass error (%) at time, t = π for solid body rotation test for all solution methodologies on different grid sizes

VOF LSM New CLSVOF Grid DA PPM Y-PLIC WENO

32x32 0.77 2.21E-03 0.26 0.02 3.18E-14 64x64 0.68 1.92E-03 0.12 1.20E-03 3.90E-14

128x128 0.04 1.90E-03 0.06 1.70E-04 1.38E-13

78

(a)

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(g)

Figure 5.11: Final (t =3) interface plots for single vortex test without time reversal on 128 x128 grid using different methods (a) ‘Exact’ solution using Mac method by Rider and Kothe (1998) (b) VOF-DA (c) VOF-PPM (d) VOF-Y-PLIC (e) LSM-WENO using periodic Bc (f) LSM-WENO using Neumann Bc (g) CLSVOF-With conservative redistribution

Interface plots for single vortex test without time reversal at time, t = 3 for all methods on

1282 grid are shown in Fig. 5.11. The interface plots are compared with exact result of Rider and

Kothe (1998) as shown in Fig. 5.11(a).

It is seen from Fig. 5.11 that all methods produce a crude shape similar to the exact

solution however in case of VOF-DA as seen from Fig. 5.11(b), small fluid bodies detach from

the main body, especially the tail. The detachment of the fluid body is not acceptable as it will

lead to incorrect property calculation in domain. It is also observed from Fig. 5.11(a) that near

the center of domain, the fluid body has developed corner and the interface appears wavy at all

places; in such case the interface curvature calculation will be inaccurate.

79

In case of VOF-PPM as shown in Fig. 5.11(c), no detachment of fluid particles take place

near the tail and the interface is smooth everywhere but due to smearing the tail is shortened.

VOF-Y-PLIC exhibits best result among VOF methods when compared to exact result as

shown in Fig. 5.11(d), as it produces smooth interface than VOF-DA and has no smearing like in

VOF-PPM. Only concern in VOF-Y-PLIC is the detachment of some fluid at the tail. It is

observed that when the fluid body becomes very thin, of the order of the cell size then incorrect

value of interface normal is calculated and the piecewise linear approximation of the interface

fails in that region leading to incorrect flux calculation. False void region is generated there

which leads to breakup of fluid. The phenomenon of breakup of fluid in the thin fluid regions in

PLIC methods has been termed as ‘numerical surface tension’ by Rider and Kothe (1998) and is

observed in all PLIC methods irrespective of the method of the normal calculation.

In case of LSM-WENO, two different simulations are done to study the effect of different

boundary condition of level set function while keeping all other parameters constant. Fig. 5.11(e)

shows the result of using periodic boundary condition for level set function and Fig. 5.11(f)

shows the result of using Neumann boundary condition for level set function. Clearly, the

interface obtained by using Neumann boundary condition is better for this problem, which

contradicts an earlier test wherein the effect of using different boundary conditions was found

negligible when level set function of stationary square fluid body was reinitialized (Fig. 5.6). It is

seen from Fig. 5.11(e) and Fig. 5.11(f), in a fluid flow problem the choice of boundary condition

of level set function plays major role in mass conservation. The results of LSM using either of

the boundary condition do not match with the exact result.

The interface plot of CLSVOF method as shown in Fig. 5.11(g) is best among level set

methods. The fluid body appears to have incurred less mass error than LSM-WENO but still

does not appear to match with the exact result. Mass loss in case of CLSVOF method is zero,

due to conservative redistribution the tail thickness is never allowed to decrease and thus the

fluid body appears to be uniformly wide all along the spiral, from head to tail. In the exact result

(Fig. 5.11(a)) the fluid body appears to be thin in tail region and thick near the head of the spiral.

The final mass error (t = 3) for the single vortex test is given in Table 5.5, the mass loss in

case of VOF-DA is very high in presence of vortical velocity field for all grid sizes due to the 1D

algorithm. Mass error for VOF-PPM is very high on coarse grid, it reduces on grid refinement

but the mass error using VOF-PPM method is comparable to mass error on coarse grid using

VOF-Y-PLIC. In case of VOF-PPM, due to presence of vortical velocity field the 1D slope

calculation turn out be inaccurate and due to slope limiting mass error is incurred.

The reason for mass loss in VOF-Y-PLIC is same as for solid body rotation i.e.

overlapping of flux polygons or missing some fluid during advection.

80

Table 5.5: Mass error (%) at time, t = 3 for single vortex test without time reversal for all solution methodologies on different grid sizes

VOF LSM New CLSVOF

DA PPM Y-PLIC WENO Neumann Bc

With Conservative Redistribution

32x32 14.14 18.97 4.92 92.11 0 64x64 10.00 7.93 0.46 87.34 0

128x128 14.37 4.73 0.08 48.59 0

In case of LSM the mass error is tremendous in presence of vortical flow field; mass loss

reduces on grid refinement but still is very high in comparison to VOF methods.

Interface plots for time reversed deformation field with time period, T = 2 for all methods

on 1282 grid are shown in Fig. 5.12. The interface plots are compared with exact result of Rider

and Kothe (1998) as shown in Fig. 5.12(a). To get a measure of numerical diffusion in VOF

methods, volume fraction contours of 0.05, 0.5 and 0.95 are plotted for VOF-DA, VOF-PPM and

VOF-Y-PLIC.

(a)

0.95

0.5

0.05

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

(b)

0.95

0.5

0.05

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

(c)

0.05

0.95

0.5

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

(d)

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

(e)

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

(f)

Figure 5.12: Interface plots at t = 1 for time reversed deformation field with time period (T = 2) on 128 x128 grid using different methods (a) ‘Exact’ solution using Mac method by Rider and Kothe (1998) (b) VOF-DA (c) VOF-PPM (d) VOF-Y-PLIC (e) LSM-WENO using periodic Bc (f) CLSVOF-With conservative redistribution

It is seen from Fig. 5.12, in case of VOF-DA (Fig. 5.12(b)) and VOF-Y-PLIC (Fig.

5.12(d)) very less numerical diffusion is introduced as the volume fraction contours of level 0.05,

81

0.5 and 0.95 are very close to each other. VOF-PPM (Fig. 5.12(c)) exhibits maximum smearing,

which can affect proper property calculation in real flow problems.

VOF-Y-PLIC exhibits better result among VOF method as it introduces minimum

diffusion, the break up of fluid near south wall is less and the interface is smoother than given by

VOF-DA method. Symmetry of fluid body is maintained in all the VOF methods.

Result of LSM-WENO as shown in Fig. 5.12(e) is not up to the mark, mass loss is apparent

and the fluid body is not symmetric. Result shown in Fig. 5.12(e) suggests that level set function

diffuses/smears heavily in presence of multiple vortices as compared to presence of single

vortex. Results of CLSVOF as shown in Fig. 5.12(f) are better than LSM-WENO results, the

fluid body remains fairly symmetric but still do not match with the exact result.

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

(a)

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

(b)

0.5

0.95

0.05

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

(c)

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

(d)

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

(e)

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

(f)

Figure 5.13: Final interface plots at t = 2 for time reversed deformation field with time period (T = 2) on 128 x128 grid using different methods (a) ‘Exact’ solution (b) VOF-DA (c) VOF-PPM (d) VOF-Y-PLIC (e) LSM-WENO using periodic Bc (f) CLSVOF-With conservative redistribution

Final interface plots ( t =2) for time reversed deformation field with time period, T = 2 for

all methods on 1282 grid are shown in Fig. 5.13. The fluid shape and position are expected to be

same as at initial condition.

It is seen from Fig. 5.13; out of all the methods VOF-DA (Fig. 5.13(a)) and VOF-Y-PLIC

(Fig. 5.13(c)) are able to get back original fluid shape better than any other method, further result

of VOF-Y-PLIC is better than of VOF-DA because of multidimensional advection algorithm.

82

Numerical diffusion in case of VOF-PPM is very high, signified by the gap between volume

fraction contours of level 0.05, 0.5 and 0.95.

Interface shape in case of LSM-WENO (Fig. 5.13(d)) and CLSVOF (Fig. 5.13(e)) are

unacceptable.

5.3 Dam break simulation Two-dimensional dam break simulation is a well known benchmark problem for two-fluid

flow simulation wherein a liquid column confined to a corner in a rectangular cavity is suddenly

let off. The results of simulation are compared with experimental results of Martin and Moyce

(1952). Dam break problem is a good test problem because it has simple domain. Initial and

Boundary conditions are simple and experimental results are available. Surface tension effects

are neglected to test performance of interface advection methodologies coupled with Navier-

Stokes solver.

5.3.1 Dam Break Simulation: Physical Description The computational domain is a rectangular cavity filled with air of size 4.5lx1.5l as shown

in Fig. 5.14. Water column of size lxl is confined to bottom left corner of the cavity. All the

boundaries of the cavity are solid walls, so no slip boundary condition is applied on each

boundary. The water column is suddenly let off; the motion is induced due to gravity only.

Figure 5.14: Computational domain, initial condition and boundary conditions for Dam Break

simulation.

5.3.2 Dam Break simulation: Non-dimensionalization For the purpose of non-dimensionalization, length is scaled using the width of the liquid

column, *L l= Velocity is scaled by defining velocity scale, *U gl= and thus Froude number,

Water

Air

No Slip / Wall

No Slip / Wall No Slip / Wall

No Slip / Wall

l

l

4.5l

1.5l

Distance of Leading edge

83

Fr = 1, Time is scaled by defining the time scale, * * *t L U= . Properties of air and water are

taken at 20 oC as,

Density of Water, 31kg998.1 mρ = , Dynamic Viscosity of water, 21

N s1.00 03 mEμ ⋅= −

Density of air, 32kg1.21 mρ = , Dynamic Viscosity of water, 22

N s1.81 05 mEμ ⋅= −

Reynolds number is defined based on the properties of water. Martin and Moyce (1952)

used the column width of 2.25 inch (l = 0.05715 m). Corresponding to properties, length scale

and velocity scale the Reynolds number for Dam break problem turns out to be of order of

42710.

5.3.3 Dam Break simulation: Computational Details In the present work, simulations are carried with VOF-PPM, VOF-Y-PLIC, LSM-WENO

and CLSVOF method on 60x20 and 180x60 uniform grid till two non-dimensional time units.

Initially velocities and pressure are set as zero. The time step is calculated using Eq. 3.7 with

reduction factor as 0.48. During each time step the residual of pressure poisson equation has

been brought down to the order of 10E-8. QUICK convection scheme has been used to discretize

convective terms in momentum equation.

Non Dimensional Time

LeadingEdgePosition

0

0

0.5

0.5

1

1

1.5

1.5

2

2

1 1

2 2

3 3Martin and MoyceVOF-PPMVOF-Y-PLICLSMCLSVOF

Figure 5.15: Present result of distance of the Leading edge of water with respect to non

dimensional time for all methods on 180x60 grid in comparison with experimental result of Martin and Moyce (1952).

Evolution of interface at different time levels at both the grid sizes is plotted for methods

for comparison. The distance of the leading edge of the water column with respect to time is

plotted and matched with experimental results of Martin and Moyce (1952). The trend of mass

loss in all methods on both the grids is also plotted.

5.3.4 Dam Break simulation: Results and Discussion

84

In Fig. 5.15, present results are compared with the experimental results by plotting the

distance of the leading edge of water (position of interface along south wall) with respect to non

dimensional time.

It is seen from Fig. 5.15, the trend of result matches with the benchmark result fro all

methods but exact match is not achieved. The inconsistency in present results and the

experimental results may be due to properties selected in this work or velocity boundary

condition selected at south boundary.

0

0

0.5

0.5

1

1

1.5

1.5

2

2

2.5

2.5

3

3

3.5

3.5

4

4

4.5

4.5

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

1.25 1.25

1.5 1.5

VOF-PPM-60x20

(a) 0

0

0.5

0.5

1

1

1.5

1.5

2

2

2.5

2.5

3

3

3.5

3.5

4

4

4.5

4.5

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

1.25 1.25

1.5 1.5

VOF-PPM-180x60

(b)

0

0

0.5

0.5

1

1

1.5

1.5

2

2

2.5

2.5

3

3

3.5

3.5

4

4

4.5

4.5

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

1.25 1.25

1.5 1.5

VOF-Y-PLIC-60x20

(c) 0

0

0.5

0.5

1

1

1.5

1.5

2

2

2.5

2.5

3

3

3.5

3.5

4

4

4.5

4.5

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

1.25 1.25

1.5 1.5

VOF-Y-PLIC-180x60

(d)

0

0

0.5

0.5

1

1

1.5

1.5

2

2

2.5

2.5

3

3

3.5

3.5

4

4

4.5

4.5

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

1.25 1.25

1.5 1.5

LSM-60x20

(e) 0

0

0.5

0.5

1

1

1.5

1.5

2

2

2.5

2.5

3

3

3.5

3.5

4

4

4.5

4.5

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

1.25 1.25

1.5 1.5

LSM-180x60

(f)

0

0

0.5

0.5

1

1

1.5

1.5

2

2

2.5

2.5

3

3

3.5

3.5

4

4

4.5

4.5

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

1.25 1.25

1.5 1.5

CLSVOF-60x20

(g) 0

0

0.5

0.5

1

1

1.5

1.5

2

2

2.5

2.5

3

3

3.5

3.5

4

4

4.5

4.5

0 0

0.25 0.25

0.5 0.5

0.75 0.75

1 1

1.25 1.25

1.5 1.5

CLSVOF-180x60

(h) Figure 5.16: Interface plot at non dimensional time interval of 0.5 for all methods on 60x20

and 180x60 grid.

Results of VOF-Y-PLIC and LSM are in better agreement than the results of VOF-PPM

and CLSVOF. In case of VOF-PPM a stair shaped profile is seen suggesting, the changes in the

value of volume fraction in cells near south wall are not continuous but periodic i.e. the fluid

would not have entered a cell due to slope limiting for some time steps and for the next change to

occur it would have taken some more time steps. Of all the methods, LSM exhibits best results as

its profile matches with experimental results at later time. VOF-Y-PLIC and CLSVOF have

parallel profile to experimental results.

85

Interface plots at non dimensional time interval of 0.5 for all methods on 60x20 and

180x60 grid are plotted in Fig. 5.16.

Effect of grid refinement for the case of VOF-PPM is shown in Fig. 5.16, the interface

becomes smooth on the fine grid but at later times interface becomes wavy near the wall. Results

of VOF-Y-PLIC are interesting as shown in Fig. 5.16; the no slip boundary condition at south

wall impedes free movement of interface at the wall and leads to fluid breakup on fine grid.

Interface in case of LSM and CLSVOF is smoother than in case VOF methods as seen in

Fig. 5.16. In case of the coarse grid in LSM, interface data at non dimensional time 1.5 is not

available as mass error is very high. Results on case of LSM improve on grid refinement

moreover the effect of using no slip boundary condition for velocity at south wall is not adverse.

Interface shape in case of CLSVOF is inaccurate, because of the procedure used to calculate

level set function from heaviside function using bisection method.

The variation of mass loss with non dimensional time for all the methods on 60x20 and

180x60 grid is plotted in Fig. 5.17, very less mass loss is incurred in case of VOF-PPM and

VOF-Y-PLIC on both coarse and fine grid. Mass loss in case of level set method on coarse grid

increases with time and reaches a very high value at nearly 1.5 non dimensional time units. On

grid refinement the mass loss incurred in LSM method decreases but its value is found to be

more than mass loss incurred by VOF methods on coarse grid. Mass loss in CLSVOF method

was forced to zero; the same is reflected in the result.

It is observed from the interface plots (Fig. 5.16) and the trend of mass loss (Fig. 5.17),

VOF-PLM is better method of the two VOF methods. Though LSM seems to give a better result

the mass loss even on fine grid is very high. CLSVOF does give a smooth interface and

maintains mass conservation but the interface shape is unacceptable.

+ + + + + + + +

Non Dimensional Time

MassError(%)

0

0

0.5

0.5

1

1

1.5

1.5

2

2

-2 -2

0 0

2 2

4 4

6 6

8 8

10 10PPM-60x20Y-PLIC-60x20LSM-60x20CLSVOF-60x20PPM-180x60Y-PLIC-180x60LSM-180x60CLSVOF-180x60

+

Figure 5.17: Variation of mass error with non dimensional time for all methods on 60x20 and

180x60 grid for dam break simulation.

86

5.4 Surface Tension Module Test: Young-Laplace law 2D Young-Laplace law test, also known as equilibrium rod test is a standard test problem

proposed by Brackbill et al. (1992) to benchmark surface tension modeling using CSF model. In

this problem an infinite cylindrical rod having radius, R is placed in a quiescent surrounding

fluid. Gravity and viscous forces are neglected, so surface tension force is balanced by pressure

force only. This results in pressure jump at the interface given by the Laplace equation,

p RσσκΔ = = (5.3)

The objective of the test is to determine the pressure inside the fluid rod by solving NS

stokes equations in presence of surface tension force but without gravity and viscous terms.

5.4.1 Young-Laplace law: Physical Description The square computational domain is considered of size 3R, shown in Fig. 5.18. Fluid rod,

represented as a circle of radius, R is placed at a (1.5R, 1.5R) in the domain filled with another

fluid. Free slip boundary condition for velocity in used and pressure is set to zero on all

boundaries. Gravity and viscous forces are neglected.

Figure 5.18: Computational domain, initial condition and boundary conditions for Young-Laplace law test.

5.4.2 Young-Laplace law: Governing Parameters The present test problem is solved in dimensional form by Brackbill et al. (1992), diameter

of the liquid rod is taken as , 0.02R m= , Density of the liquid of rod is taken

Fluid RodRadius = R

Centered at (1.5R, 1.5R)

Surrounding Fluid

3R

3R

Gravity and Viscous forces are neglected

Free Slip Bc, P = 0

Free Slip Bc, P = 0

Free Slip Bc, P = 0

Free Slip Bc, P = 0

87

as 31kg1000 mρ = and density of the background fluid is taken as 32

kg500 mρ = . The co-

efficient of surface tension is taken as N0.02361 mσ = .

5.4.3 Young-Laplace law: Computational Details The interface curvature is calculated from volume fraction field in case of VOF based

methods and it is calculated from level set function in case of LSM and CLSVOF. In the present

work, VOF-Y-PLIC and LSM are used to solve present problem to determine the difference in

results due to different interface curvature calculation techniques.

In case of VOF-Y-PLIC two cases for curvature calculation are considered, first is using

the raw volume fraction field i.e. without smoothing and second using the smoothened volume

fraction field.

Calculations are carried out on 15x15 and 30x30 uniform grids. Time step is taken as 10E-

05 and computations are performed till 50 time steps. Pressure in the background fluid is zero so

the actual pressure in the rod using Eq. 5.3 should be, 2N1.1085 mRODP = . The computed

pressure inside the rod is determined as

2,,

1 N mRN

i ji jR

P PN

= ∑ (5.4)

Where, NR in equation Eq. 5.4 is the number of cells having a density, 10.99ρ ρ≥ ⋅ . Error

in the computed and actual value of pressure in quantified by defining L2 error norm as

( )0.5

2,

,2 2

RN

i j RODi j

R ROD

P PL

N P

⎡ ⎤−⎢ ⎥

⎢ ⎥=⎢ ⎥⎣ ⎦

∑ (5.5)

Computed L2 error is compared with results of Brackbill et al. (1992) to validate the

implementation of CSF model. Pressure variation across the interface must be smooth. Surface

plot of pressure is plotted for both the methods to check this and finally, velocity vectors are

plotted to check existence of any generation of currents. Ideally there should be no movement in

the domain.

5.4.4 Young-Laplace law: Results and Discussion Ratio of calculated average pressure to actual pressure in rod and L2 error for all methods

is shown in Table 5.5. Present results are compared with the benchmark results of Brackbill et al.

88

(1992) on 152 and 302 grid. Brackbill et al. (1992) used a VOF method and smoothened volume

fraction field was used to calculate curvature.

It is seen from Table 5.6, for the present result of VOF method without smoothing, L2 error

is very high on coarse grid, which does not improve too much on grid refinement. The value of

average calculated pressure is way off from the actual value of pressure for coarse grids but

improves on grid refinement.

In case of VOF method, smoothing of volume fraction field is more effective on fine grid

than on coarse grid as seen from Table. 5.5. Even with smoothing, the L2 error on both the grids

is higher than benchmark result.

LSM produces better results compared to VOF methods and even the benchmark results,

and on 30x30 grid LSM produces nearly exact solution. Level set method is a very accurate

method involving surface tension as the calculation of interface curvature is accurate in LSM

than VOF methods.

Table 5.6: Comparison of present results with benchmark result of Brackbill et al. (1992) on 15x15 and 30x30 grid.

Method ROD

PP

L2 error

15x15 grid Brackbill et al. (1992) 1.034 5.56E-02 VOF-Without smoothing 0.865 1.93E-01 VOF-Smoothing using K8 0.936 1.32E-01 LSM 0.988 2.63E-02 30x30 grid Brackbill et al. (1992) 1.016 2.82E-02 VOF-Without smoothing 0.930 1.38E-01 VOF-Smoothing using K8 0.967 5.09E-02 LSM 1.004 8.96E-03

The pressure variation in the domain is shown in Fig. 5.19 with help of 3D surface plot. X

and Y axis in the plot represent the co-ordinate directions and the calculated pressure in the

domain is rescaled ( )2,10 i jNP m× and plotted along Z-axis.

Fig. 5.19 shows that on coarse grid, the pressure across the interface changes abruptly

when VOF method without smoothing of volume fraction field is used, negative pressure is

observed near the interface. When the volume fraction is smoothened, large changes in pressure

are suppressed but still some irregularity is seen. In case of LSM the transition is very smooth.

On fine grid the result of VOF method without smoothing deteriorates as seen in Fig. 5.19,

even though the pressure change is sharp, the extent of negative pressure has increased. On

smoothing the volume fraction field, variation of pressure is suppressed but not eliminated, just

89

like on coarse grid. On grid refinement the result of LSM seems improved as the transition

region becomes thin and a sharper change in pressure is seen.

-5

0

5

10

15

CaculatedPressure

0

2

4

6 0 2 4 6

Y

Z

X

VOF-without smoothing15x15

-5

0

5

10

15

CaculatedPressure

0

2

4

6 0 2 4 6

Y

Z

X

VOF-with smoothing15x15

-5

0

5

10

15

CaculatedPressure

0

2

4

6 0 2 4 6

Y

Z

X

LSM-15x15

-5

0

5

10

15

CaculatedPressure

0

2

4

6 0 2 4 6

Y

Z

X

VOF-without smoothing30x30

-5

0

5

10

15

CaculatedPressure

0

2

4

6 0 2 4 6

Y

Z

X

VOF-with smoothing30x30

-5

0

5

10

15

CaculatedPressure

0

2

4

6 0 2 4 6

Y

Z

X

LSM-30x30

Figure 5.19: Surface plot of pressure variation in domain for all methods on 15x15 and 30x30 grid.

Fig. 5.20 shows the interface position and velocity vectors at end of the simulation on

30x30 grid. In ideal conditions there should be no velocity in the domain as pressure force

balances surface tension force.

0

0

2

2

4

4

6

6

0 0

2 2

4 4

6 6VOF-with smoothing

15x15

0

0

2

2

4

4

6

6

0 0

2 2

4 4

6 6VOF-without smoothing

30x30

0

0

2

2

4

4

6

6

0 0

2 2

4 4

6 6VOF-with smoothing

30x30

0

0

2

2

4

4

6

6

0 0

2 2

4 4

6 6VOF-with smoothing

15x15

0

0

2

2

4

4

6

6

0 0

2 2

4 4

6 6LSM30x30

Figure 5.20: Final interface position and velocity vectors for Young-Laplace test on 30x30

grid

In case of VOF method without smoothing, the velocity vectors are more prominent than

for other methods, on careful examination it is seen from Fig. 5.20 that velocity vectors are

pointing in opposite directions near the interface. Existence of negative pressure near the

90

interface is the reason for this. The final interface shape is not smooth, suggesting that some

movement of interface has taken place.

For the case of VOF method with smoothened volume fraction field as seen from Fig. 5.20,

velocity vectors have not been eliminated but they point in same direction, which means the

existence of negative pressure has been eliminated. The final interface shape in this case also has

undergone some movement.

LSM produces the ideal result as there are no major velocity vectors and the interface has

not undergone any movement.

5.5 Rayleigh-Taylor Instability Consider a system of two immiscible stratified fluids under the effect of gravity with

heavier fluid being on top. If the interface is horizontal the system will remain stable as there is

no way for movement to occur. If a small perturbation is given along the interface, due to density

difference the heavier fluid will try to fall down displacing the lighter fluid, the lighter fluid will

thus try to rise in the cavity forming bubbles, this phenomenon is known as Rayleigh-Taylor

(RT) Instability.

This problem is stringent than the dam break simulation in regard that there is vortex

formation near the interface which stretches the interface severely, testing the robustness of

interface advection algorithm coupled with Navier-Stokes solver. The simulation time is much

longer than that for broken dam problem, so trend of mass error for long time can be checked.

Figure 5.21: Computational domain, initial condition and boundary conditions for Rayleigh-

Taylor instability.

Lighter Fluid

Free Slip

Heavier Fluid

Free Slip

No Slip

No Slip

Initial interface position 3l

l

1.86l

91

5.5.1 RT Instability: Physical Description of Problem The computational domain is a rectangular cavity with width, l = 1 and height, 3l as

shown in Fig. 5.21. Unperturbed height of the interface is 1.86l from the base with heavier fluid

being on top. A cosinusoidal, single wave perturbation with amplitude 0.03l is introduced at the

interface. No slip boundary condition is applied at top and bottom boundary and free slip

boundary condition is applied at side boundaries. Uniform gravity acts everywhere in the

domain, there is no other source of motion in the domain.

5.5.2 RT Instability: Non Dimensional Governing Parameters For the purpose of non-dimensionalization, length is scaled using the width of the domain,

*L l= . Velocity is scaled by defining velocity scale, *U gl= and thus Froude number, Fr = 1,

Time is scaled by defining the time scale, * * *t L U= . Properties of heavier and lighter fluid are

taken as,

Density of heavier fluid, 31kg5 mρ = , Dynamic of heavier fluid, 21

N s0.1 mμ ⋅=

Density of lighter fluid, 32kg1 mρ = , Dynamic Viscosity of water, 22

N s0.01 mμ ⋅=

Reynolds number is defined based on the properties of heavier fluid, the length of the

column is taken as l = 1 m. Corresponding to properties, length scale and velocity scale the

Reynolds number for RT instability turns out to be 155.605.

The density ratio (λ ) and the Atwood number, ( ) ( )1 1A λ λ= − + for the given densities

turn out to be 0.2 and 0.6667 respectively. Weber number is defined on the properties of heavier

fluid; parametric study is performed for different Weber number.

5.5.3 RT Instability: Computational Details In the present work, simulations are carried with VOF-PPM, VOF-Y-PLIC, LSM-WENO

and CLSVOF method on 16x48, 32x96 and 64x192 uniform grid till four non-dimensional time

units. Initially velocities and pressure are set as zero. The time step is calculated using Eq. 3.7

with reduction factor as 0.48. During each time step the residual of Pressure Poisson equation

has been brought down to the order of 10E-8. QUICK convection scheme has been used to

discretize convective terms in momentum equation.

First set of computations are carried out without surface tension to study the effect of grid

refinement on 16x48, 32x96 and 64x192 grid. In the second set of computations, the effect of

surface tension is included. Computations are carried out for different Weber number to

92

determine if the VOF, LSM and CLSVOF solution methodologies can provide the critical Weber

number as predicted by the linear analysis. When the Weber number is equal to critical Weber

number no instability occurs. From the linear analysis, the critical Weber number is given by,

2

1

k FrmWecr λ=

−

where ( )2k lm π= is the wave number of perturbation.

VOF-PPM16x48

VOF-PPM32x96

VOF-PPM64x192

VOF-Y-PLIC16x48

VOF-Y-PLIC32x96

VOF-Y-PLIC64x192

LSM16x48

LSM32x96

LSM64x192

CLSVOF16x48

CLSVOF32x96

CLSVOF64x192

Figure 5.22:Effect of grid refinement on interface shape at non dimensional time, 4τ = for all the methods without surface tension

5.5.4 RT Instability: Results and Discussion Interface shape for all methods without surface tension at non dimensional time, 4τ = on

16x48, 32x96 and 64x192 grid is shown in Fig. 5.22. Volume fraction contours of level 0.5 are

used to represent interface in VOF methods, whereas level set contour level 0 is defined as

interface in case of LSM and CLSVOF.

93

Fig. 5.22 shows that with the exception of CLSVOF, the interface shape predicted is fairly

accurate on coarse grid. Interface shape in case of CLSVOF seems to be excessively developed

and it is asymmetrical.

In case of VOF-PPM, fluid body is symmetrical for grid sizes, fluid break up seems to be

happening on finer grids but actually due to numerical diffusion the fluid body in thin fluid

regions is highly smeared. The smeared interface in case of VOF-PPM is shown in Fig. 5.23, the

interface shape obtained using VOF-PPM is close to exact result.

In case of VOF-Y-PLIC as seen in Fig. 5.22, the fluid breakup seen on coarse grid reduces

on grid refinement but it is never eliminated. Unlike VOF-PPM, in the case of VOF-Y-PLIC no

numerical diffusion is introduced in the volume fraction field and thus fluid break up occurs. To

resolve thin fluid body even finer grid will have to be taken as Y-PLIC method cannot resolve

fluid particles smaller than the grid size.

Figure 5.23: Final fluid shape using VOF-PPM, fluid shape represented by 10 contour levels

between 0 and 1. This result is closest to exact result.

In case of LSM, it is seen from Fig. 5.22 that the interface shape is symmetrical for all grid

sizes, the results improve tremendously on grid refinement giving extremely smooth interface in

finest grid. The thin fluid regions are not captured in case of LSM, even on a very fine grid. In

case of LSM, the thin fluid regions get smeared.

Fig. 5.22 shows that CLSVOF produces good result on fine grid, symmetry is maintained

and interface is developed to larger extent as compared to other methods. The variation of mass

loss with non dimensional time for all the methods on 32x96 grid is plotted in Fig. 5.24.

It is seen from Fig. 5.24, VOF-PPM incurs very less error till non dimensional time, 2τ = ,

after which the error monotonously increases and reaches maximum value its maximum value

i.e. 0.64 % at end time. Mass error in case of VOF-PPM starts to appear at later time because of

numerical diffusion and vortex field set up in the domain by that time.

94

Non Dimensional Time

MassError(%)

0

0

1

1

2

2

3

3

0 0

0.5 0.5

1 1VOF-PPMVOF-Y-PLICLSMCLSVOF

Figure 5.24: Variation of mass error with non dimensional time for all methods on 32x96grid

for RT instability with surface tension.

Fig. 5.24 shows that VOF-Y-PLIC incurs negligible mass loss at all times and mass loss in

case of LSM increases till non dimensional time, 3τ = after which it decreases to zero, the mass

error seems to increase after touching zero error because absolute mass error values are plotted.

Trend of mass error in case of CLSVOF is random, reason for which is not still known. It is

noted from Fig. 5.24 that the mass loss for all methods is less than 1 %, low property ratios can

be a reason for good result of LSM.

VOF-PPM64x192We 144

VOF-Y-PLIC64x192We 144

LSM64x192We 144

CLSVOF32x96We 144

VOF-PPM64x192We 80

VOF-Y-PLIC64x192We 80

LSM64x192We 80

CLSVOF32x96We 80

VOF-PPM64x192We 50

VOF-Y-PLIC64x192We 50

LSM64x192We 50

CLSVOF32x96We 50

VOF-PPM64x192We 40

VOF-Y-PLIC64x192We 40

LSM64x192We 40

CLSVOF32x96We 40

Figure 5.25:Effect of different Weber number of on interface shape at non dimensional time, 4τ =

for VOF-PPM, Y-PLIC , LSM on 64x192 grid and CLSVOF on 32x96 grid.

95

Fig. 5.25 shows the interface shape for all methods except CLSVOF with surface tension

included for Weber number equal to 144, 80, 50 and 40 on 64x192 grid at non dimensional

time, 4τ = , results of CLSVOF are obtained on 32x96 grid.

As a general observation, it is seen from Fig. 5.25 for all methods that with decreasing

Weber number, development of interface is restricted. In case of VOF methods and CLSVOF

method the spike formation for We = 144 and the bulb formation for We = 80 are more

prominent as compared to LSM.

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.5 1.5

2 2

2.5 2.5

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.5 1.5

2 2

2.5 2.5

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.5 1.5

2 2

2.5 2.5

32x9664x192

VOF-PPM We 5016x48

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.5 1.5

2 2

2.5 2.5

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.5 1.5

2 2

2.5 2.5

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.5 1.5

2 2

2.5 2.5

32x9664x192

VOF-Y-PLIC We 5016x48

0

0

0.5

0.5

1

1

1.5 1.5

2 2

2.5 2.5

0

0

0.5

0.5

1

1

1.5 1.5

2 2

2.5 2.5

0

0

0.5

0.5

1

1

1.5 1.5

2 2

2.5 2.5

0

0

0.5

0.5

1

1

1.5 1.5

2 2

2.5 2.5

32x96

LSM We 50

64x192

16x48

0

0

0.5

0.5

1

1

1.5 1.5

2 2

2.5 2.5

0

0

0.5

0.5

1

1

1.5 1.5

2 2

2.5 2.5

0

0

0.5

0.5

1

1

1.5 1.5

2 2

2.5 2.5

32x96

CSLVOF We 5016x48

Figure 5.26: Effect of grid refinement on interface shape at non dimensional time, 4τ = for

all the methods with surface tension at We = 50.

At We = 50 it is seen from Fig. 5.25, there is very little change in the final interface

position from the initial condition of the interface for all the methods. The value of critical

Weber number in the present problem, calculated theoretically is 49.35, thus it is seen that all the

methods predict the critical Weber number fairly accurately.

96

It is seen from Fig. 5.25, at We = 40 the interface is exhibits negative movement and is

seen above the initial perturbed position at the final time instant, the interface actually starts to

oscillate signifying that the Weber number chosen is less than the critical Weber number.

The results on fine grid are inconclusive of exposing weakness of any method in presence

of surface tension, so the effect of grid refinement for all methods with We = 50 is plotted in Fig.

5.26.

In case of VOF-PPM and VPF-Y-PLIC, it is seen from Fig. 5.26 that the interface

undergoes considerable movement on coarse grids. Ideally the interface should move very little

as the value of Weber number used for computation is near to the critical Weber number value

(49.35). In case of VOF methods, interface movement is seen because of presence of spurious

currents near the interface. Spurious currents appear due to inaccurate calculation of interface

curvature.

It is seen from Fig. 5.26 that the results of VOF-PPM are better than for VOF-Y-PLIC on a

coarse grid. In VOF-PPM, the volume fraction field is smeared to some extent due to solution

algorithm of VOF equation using PPM method, the smeared volume fraction field aids in

calculating smooth curvature. In case of VOF-Y-PLIC the volume fraction field is maintained as

step function, even smoothing using K8 kernel leads to inaccurate curvature calculation. The

interface movement for both VOF methods is restricted on grid refinement.

It is seen in Fig. 5.26 that in case of LSM and CLSVOF the interface movement is

negligible for all grid sizes; this confirms the fact that interface curvature calculated using level

set method is very accurate. For surface tension dominant flows, VOF methods will require very

fine grid in comparison to LSM or level set based methods to capture same phenomenon.

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.6 1.6

1.8 1.8

2 2

VOF-PPM32x96We 50

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.6 1.6

1.8 1.8

2 2

VOF-Y-PLIC32x96We 50

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.6 1.6

1.8 1.8

2 2

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.6 1.6

1.8 1.8

2 2

LSM32x96We 50

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.6 1.6

1.8 1.8

2 2

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.6 1.6

1.8 1.8

2 2

CLSVOF32x96We 50

Figure 5.27: Interface position and velocity vectors for all methods at non dimensional

time, for We = 50 on 32x96 grid.

97

Based on the assumption of single fluid model used to solve Navier-Stokes equations, a

continuous velocity field is expected in domain, even across the interface. To check this, final

interface position and velocity vectors for We = 50 on 32x96 grid are plotted n Fig. 5.27.

Fig. 5.27 shows velocity field in case of VOF-PPM, LSM and CLSVOF is smooth across

the interface but in case of VOF-Y-PLIC method there are abrupt changes in the velocity across

the interface. These abrupt changes in velocity, known as spurious currents [Brackbill et al.

(1992)] impart motion to interface even if all the forces are balanced. The spurious currents arise

due to inaccurate calculation of surface tension force and are inherent shortcoming of VOF-Y-

PLIC method.

To get a measure of the computational time, average iterations per time step is defined as

ratio of summation of number of iteration of pressure poisson equation per time step to total

number of time steps required. Values of average iterations per time step for all methods on

32x96 grid for all cases of Weber number are shown in Table 5.7.

Table 5.7: Average iterations of pressure poisson equation required per time step at non-dimensional time, 4τ = on 32x96 grid.

Method Weber Number ∞ 144 80 50 40 VOF-PPM 1977.81 1895.01 1408.59 230.515 83.53 VOF-Y-PLIC 1984.28 1887.82 1437.12 305.65 151.28 LSM 1591.47 1582.11 1088.50 55.83 61.41 CLSVOF 1938.98 1928.02 1659.85 92.60 209.00

It is seen from Table 5.7 that number of iterations of pressure poisson equation per time

step required in VOF methods are more than for LSM. More number of iterations are required

for solution of pressure poisson equation in case of VOF method as the properties have a step

change whereas in LSM properties are smeared across the band.

For the flows involving high surface tension (flows involving bubbles), VOF-Y-PLIC may

prove to be computationally too expensive as on an average, it takes double or more than double

iterations of pressure poisson equation per time step to converge as compared to LSM for high

surface tension flow, and on top of that finer grid will be required for accurate calculation of

surface tension force in VOF-Y-PLIC. When the computational time required by the K8 kernel

smoothing step is considered, VOF-Y-PLIC method comes out be at least four times slower than

LSM method.

In CLSVOF method, reinitialization of a step function to signed normal distance function

is done at each time step and thus CLSVOF method is to be computationally most expensive

method out of all, followed by VOF-Y-PLIC and VOF-PPM. LSM is fastest of all the methods.

98

5.6 Splash of Water drop in water pool Splash of a water drop in water pool is a good test problem as it involves fluids with high

property ratio, the simulation time is longer than any of the other methods and most importantly

ability of the solution methodology to maintain the symmetry in the solution is tested.

In this problem a water drop placed above a water pool in the domain, is suddenly let off to

splash in to the water pool under the effect of gravity inducing motion in the pool which

gradually dampens. Surface tension effects in this problem are neglected. The objective of this

test is to determine interface position at different times and the variation of interface position at

west wall with respect to time.

5.6.1 Splash of Water Drop: Physical Description The computational domain as shown in Fig. 5.28 is a square cavity of size 8R filled with

air, water pool having depth 2R is also there in this cavity. A water drop of radius R is initially

centered at (4R, 6R). Top and Bottom boundaries are solid walls, so no slip boundary condition

is applied on them. Free slip boundary condition is applied on side boundaries.

The water drop is suddenly let off, triggering a splash into the water pool below.

Figure 5.28: Computational domain, initial condition and boundary conditions for splash of

water drop in water pool.

5.6.2 Splash of water drop: Non-dimensionalization For the purpose of non-dimensionalization, length is scaled using the radius of the water

drop, *L R= Velocity is scaled by defining velocity scale, *U gR= and thus Froude number,

Air

No Slip / Wall

Free Slip 8R

Water

8R

Free Slip

No Slip / Wall

Water Drop

Radius = R Centered at (4R, 6R)

2R

99

Fr = 1, Time is scaled by defining the time scale, * * *t L U= . Properties of air and water are

taken as same in the dam break simulation problem.

The radius of the drop in present work is taken as 1.25E-03 m. Reynolds number is defined

based on the properties of water. Corresponding to properties, length scale and velocity scale the

Reynolds number for the present problem comes out to be, 138.57.

5.6.3 Splash of water drop: Computational Details Simulation is carried using all the methods on 64x64 uniform grid till 50 non-dimensional

time units. Initially velocities and pressure are set as zero; the time step is calculated using Eq.

3.7 with reduction factor as 0.48. The convergence criterion of pressure poisson equation and

other computational details are same as for dam break simulation.

5.6.4 Splash of water drop: Results and Discussion The propagation of the drop splash for all method at various instants of non dimensional

time is shown in Fig. 5.29, time instants during which water droplet touches the water pool and

induces movement are considered here.

Overall it is seen that VOF-PPM, VOF-Y-PLIC and LSM capture similar fluid position at

same non dimensional time. CLSVOF fails to treat merging of two different fluid bodies and so

its simulation was cut short.

It is seen from Fig. 5.29 that LSM maintains symmetry of the fluid body at all times. In

case of VOF-Y-PLIC, asymmetry of fluid body begins to appear by non dimensional time,

3.0τ = on the other hand VOF-PPM maintains symmetry till a bit longer time but eventually

asymmetries in fluid body is seen at 6.0τ = . In VOF-Y-PLIC at 9.0τ = , fluid breakup is seen

but the fluid reaches maximum height as compared to other methods.

Fig. 5.30 shows the variation of interface position at west wall with time for VOF-Y-PLIC

and LSM. The interface exhibits sinusoidal behavior, with large amplitudes at initial times and

due to viscosity the amplitude decreases over a period of time and interface movement stops at

steady state.

100

τ = 2.0

VOF-PPM64x64

τ = 2

VOF-Y-PLIC64x64

LSM64x64τ = 2.0

CLSVOF64x64τ = 2.0

τ = 2.6

VOF-PPM64x64

τ = 2.6

VOF-Y-PLIC64x64

LSM64x64τ = 2.6

CLSVOF64x64τ = 2.6

τ = 3.0

VOF-PPM64x64

τ = 3.0

VOF-Y-PLIC64x64

LSM64x64τ = 3.0

CLSVOF64x64τ = 3.0

τ = 3.4

VOF-PPM64x64

τ = 3.4

VOF-Y-PLIC64x64

LSM64x64τ = 3.4

τ = 6.0

VOF-PPM64x64

τ = 6.0

VOF-Y-PLIC64x64

LSM64x64τ = 6.0

τ = 9.6

VOF-PPM64x64

τ = 9.6

VOF-Y-PLIC64x64

LSM64x64τ = 9.6

Figure 5.29: Fluid shapes at different time instants for all methods on 64x64 grid for water drop splash problem

101

It is seen from Fig. 5.30, the trend of movement of interface for VOF and LSM is similar

but the amplitude of movement in case of VOF method is much higher, and does not reduce as

easily as in case of LSM.

Non Dimensional Time

HeightofInterfaceatWestwall

0

0

10

10

20

20

30

30

40

40

50

50

0 0

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

10 10LSMVOF-Y-PLIC

Figure 5.30: Variation of interface height at west wall with respect to non dimensional time for

LSM and VOF-Y-PLIC.

Variation of percentage mass error with time for all methods is plotted in Fig. 5.31, it is

seen that mass error for VOF-PPM reaches a value after the splash has taken place an then

becomes constant. Mass error in VOF-Y-PLIC is negligible at all times as expected

In case of LSM method, the variation of mass error is interesting. The mass error in LSM

method does not increase monotonously but it exhibits a periodic behavior, similar to the

interface movement.

Non Dimensional Time

MassError(%)

0

0

10

10

20

20

30

30

40

40

50

50

0 0

3 3

6 6

9 9

12 12

15 15VOF-PPMVOF-Y-PLICLSMCLSVOF

Figure 5.31: Variation of mass error with non dimensional time for all methods.

102

6 Conclusions and Future Work Proposal

In the present work, Literature survey of Volume of Fluid (VOF) method and Level Set

(LS) method is performed and summarized.

A Navier-Stokes solver based on projection is method developed and tested on lid driven

cavity flow problem. Module to solve VOF governing equation using donor-acceptor method,

multi dimensional geometric advection method combined with Youngs’ PLIC method and a high

resolution convection scheme (PPM) is developed and tested individually on four pure advection

test problems. Module to solve LS governing equation and reinitialization equation using 5th

order WENO scheme is developed and tested individually on four pure advection test problems.

Two-fluid VOF code is developed by combining Navier-Stokes solver with VOF solution

modules based on geometric advection and PPM method. Similarly, Two-fluid level set code is

developed by combining Navier-Stokes solver with level set solution modules. VOF and LS two-

fluid codes are tested on standard dam break simulation test problem. Module to include surface

tension force in VOF and LS method is developed and combined with both the two-fluid codes.

Module of surface tension force for VOF and LS is tested on Young-Laplace law test. A new

Combined Level Set Volume of Fluid Method (CLSVOF) is devised, implemented and tested.

The new CLSVOF method is intended to reduce mass error in LS methods while avoiding

interface reconstruction. Two-fluid code based on new CLSVOF method has also been

developed with surface tension effect included and is used to solve dam break simulation

problem. All the two-fluid codes with surface tension effects are used to solve Rayleigh-Taylor

instability problem and splash of water drop in water pool.

Comparison based on qualitative and quantitative results is made between different

methodologies to solve VOF governing equation in case of VOF method and LS method. It is

concluded from the comparative study that:

• It was seen from the advection tests; among the VOF methods the donor-acceptor method

preserved sharpness of volume fraction field but induced considerable change in shape of the

fluid. Moreover in DA method, break up of fluid particles took place from various locations

of fluid body. PPM method was able to maintain the shape of the fluid body without fluid

breakup but the volume fraction field got smeared due to numerical diffusion. Geometric

advection method performed best among all methods as it maintained sharpness in volume

fraction field without distorting the fluid shape. However, fluid break up in thin fluid regions

was seen in case of geometric advection method, the PLIC approximation is reason for fluid

103

breakup. Mass error for geometric advection was negligible for all test problems and that for

PPM method was under acceptable limits. However, high mass error in case of DA method

was seen on vortical flow field.

• In case of advection tests of LS method, the mass error reduced the importance of otherwise

exceptional results. It was seen from results of single vortex test, use of different boundary

conditions affected the final results considerably.

• In CLSVOF method, the numerical diffusion of volume fractions/heaviside functions was

limited in a thin region near the interface but the procedure adopted to calculate level set

function from heaviside function was inaccurate and more mass error was incurred in raw

CLSVOF, mass conservation was enforced by conservative redistribution. Validity of

conservative redistribution is not established but the results of CLSVOF, as in case of single

vortex test give a taste of the result that can be obtained using LS method when mass

conservation will be obeyed.

• In hindsight, it is seen that CLSVOF method was devised to reduce mass error but, the same

reinitialization procedure responsible for mass loss in LS method is used in CLSVOF.

• It was seen from the solution of two-fluid flows that,

1. Mass conservation was obeyed better in VOF based methods as compared to LS

method. In LS method, presently there is no constraint in formulation that prevents mass

loss. During reinitialization procedure the interface position is changed, the error

induced in this manner is accumulated to result in mass loss.

2. Surface tension force was calculated accurately in LS method and CLSVOF method.

In case of VOF methods, even the smoothing of volume fraction field is insufficient to

calculate accurate surface tension force.

3. VOF method based on PPM showed good mass conservation without fluid break up.

However the surface tension force calculation procedure is same as in VOF method

based on geometric advection. In PPM based VOF method, even though the geometric

advection and fluid breakup is avoided, calculation of surface tension force is

inaccurate.

4. Although mass conservation was best in VOF based methods, there is a severe

problem of fluid breakup on coarse grids and break up of symmetry of fluid body in

case of geometric advection and introduction of numerical diffusion for high density

flows in case of PPM. In case of LS method, symmetry in fluid body is maintained

without fluid breakup.

5. Fluid breakup and inaccurate interface curvature calculation in VOF method based of

geometric advection and numerical diffusion in case of PPM can be avoided by using a

104

fine grid but the computational time increases tremendously, as smoothing volume

fraction is computationally expensive.

6. In its present form, the new CLSVOF method cannot treat merging of two different

fluid bodies and its results are not acceptable.

Overall it’s difficult of choose a better method from the VOF and LS method, the answers

to important issues regarding the two-fluid flow simulation methodologies presented by VOF

and LS methods after implementing them are summarized as,

Issue VOF method LS method Is the interface represented in the domain using a function ?

No, Information of region occupied by a particular fluid is stored in form of volume fraction field. Interface is reconstructed from the volume fraction field, interface reconstruction is not accurate.

Yes, Level set function of value 0 defines the interface directly and its contour gives the exact interface.

What is the storage requirement of the function representing interface ?

It was intended to be one variable per cell but due to use of geometric advection, the storage requirement per cell is very high. (It takes 64 variables per cell in present implementation of geometric method in 2D)

Only one variable per cell is required, even in 3D.

Is mass conservation obeyed ?

Yes, but only when geometric advection method is used. Use of convection schemes induces mass error.

No, not in the present form due to lack of constraint in formulation.

How easy it is to calculate interface curvature required in surface tension model ?

Not so easy, Interface curvature needs double derivatives of volume fractions but volume fraction field has sharp discontinuity. Smoothing of volume fraction field is done but it is computationally expensive and it is not exact.

Very easy, exact curvature is calculated without any special treatment.

What is implementation effort required ?

Implementation effort required for geometric advection is tremendous but that for convection schemes is relatively very less.

Very less implementation effort required as compared to VOF method

105

Issue VOF method LS method What about extendibility to 2D complex geometry and 3D ?

Difficult, Unless the 2D implementation on simple geometry is generalized. Extension to 2D complex will require a new procedure to be devised and a new code to be written, as the shapes of fluid polygons, flux polygons and their combinations will be different on complex grids from those seen on simple grid. 3D extension is most difficult but it exists

Easy, as only convection schemes are needed to be extended.

Where does the method go wrong ?

In every aspect of numerical implementation, special treatment is required. The method defines the region of fluid using a step function so convection schemes cannot be used easily to solve its governing equation. Many researches have tried to ‘improve’ the solution procedure of VOF method by giving more and more complex interface reconstruction and advection algorithms. To calculate the interface curvature, smoothing of volume fraction field is used by many researchers but it is inaccurate. In VOF method, use of a step function to represent a region of fluid makes the implementation of the method very difficult.

The formulation has no constraint of mass conservation.

106

Issue VOF method LS method What is the best way to overcome present short comings ?

The definition of VOF method cannot be changed and also there is no point in devising more complex interface advection algorithms which do not improve the results to large extent. Important thing learnt from VOF method is that for mass conservation to be obeyed the volume fraction field must be conserved.

Best way forward is to include the constraint of mass conservation i.e. conservation of volume fraction/heaviside function field in the formulation. CLSVOF is a direct way of conserving heaviside function field. An improvement in reinitialization procedure is indirect way of satisfying this constraint.

Is a CLSVOF method feasible ?

If the original CLSVOF method, with interface reconstruction is to be used then the disadvantage of interface reconstruction is inherited. If the CLSVOF method devised in the present work is improved, it may still not give good results as the reinitialization procedure is flawed. A CLSVOF method may not be needed if the reinitialization procedure is improved.

Overall it is found that Level set method a better candidate method for further

developments.

In future it is intended to,

• Improve the mass conservation in level set method from present state by the CLSVOF

proposed in this work or improving reinitialization procedure by applying the constraint

of mass conservation.

• Model boiling heat transfer at interface.

107

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