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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After Reinit.
(c)
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After Reinit.
(f)
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.
0
0
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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
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.
0
0
0.5
0.5
1
1
0 0
0.5 0.5
1 1Translation
0
0
0.5
0.5
1
1
0 0
0.5 0.5
1 1Solid Body Rotation
0
0
0.5
0.5
1
1
0 0
0.5 0.5
1 1Single Vortex
0
0
0.5
0.5
1
1
0 0
0.5 0.5
1 1Deformation Field
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.
0.25
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1 1
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(d)
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(e)
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0.5 0.5
0.75 0.75
1 1
64x64
Exact32x32
128x128
(f)
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
0.25
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0.5
0.5
0.75
0.75
0.5 0.5
0.75 0.75
1 1
128x128
Exact32x3264x64
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128x128
(e)
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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1 1VOF-DA-128x128
(b)
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1 1CLSVOF-Neuman Bc-128x128
(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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