A New Approach To Solve Mixture Multi-Phase Flow Model Using Time Splitting Projection Method

160 Progress in Computational Fluid Dynamics, Vol. 19, No. 3, 2019
Copyright © 2019 Inderscience Enterprises Ltd.
A new approach to solve mixture multi-phase flow
model using time splitting projection method
Farhang Behrangi*, Mohammad Ali Banihashemi and
Masoud Montazeri Namin
School of Civil Engineering,
University of Tehran,
Tehran, Iran
Email: behrangi@ut.ac.ir
Email: banihash@ut.ac.ir
Email: mnamin@ut.ac.ir
*Corresponding author
Asghar Bohluly
Institute of Geophysics,
University of Tehran,
Tehran, Iran
Email: Bohluly@ut.ac.ir
Abstract: This paper was aimed at introducing a new approach to numerical simulation of
two-phase flows by solving the mixture model equations. In this approach, an effective numerical
algorithm was developed based on the time splitting projection method to solve the mixture
model equations for an incompressible two-phase flow. In this study, a new technique was
developed to solve the mixture continuity equation based on the time splitting projection method.
One of the advantages of the mixture model is that, in addition to determining the velocities of
each phase, it solves only one set of momentum equations and calculates the velocity differences
between the phases by solving the relative velocity equation. Accordingly, an extended finite
volume vertical two-dimensional numerical model was used to determine the pressure
distribution, velocity field and volume fraction of each phase at each time step. The model has
been used in various tests to simulate unsteady flow problems. Comparison between numerical
results, analytical solutions and experimental data demonstrated a satisfactory performance.
Keywords: time splitting; projection method; mixture model; multi-phase flow; slip velocity;
computational fluid dynamics; phase interpenetrate; numerical methods; numerical modelling;
numerical solution.
Reference to this paper should be made as follows: Behrangi, F., Banihashemi, M.A.,
Namin, M.M. and Bohluly, A. (2019) ‘A new approach to solve mixture multi-phase flow model
using time splitting projection method’, Progress in Computational Fluid Dynamics, Vol. 19,
No. 3, pp.160–169.
Biographical notes: Farhang Behrangi is a last-year’s PhD candidate at the University of
Tehran, School of Civil Engineering. He received his Master’s in Hydraulic Structures from the
College of Engineering, University of Tehran in Iran. His research interests include numerical
method, CFD simulation and design of hydraulic structures.
Mohammad Ali Banihashemi is an Associate Professor in the College of Engineering at the
University of Tehran where he has been a faculty member since 1995. He teaches computational
fluid dynamics, fluid mechanics and hydraulics. He completed his PhD in Hydraulics from the
University of Queensland and his undergraduate studies at the University of Tehran. His research
is framed in numerical methods in fluids, involving the simulation of dam break flow,
sedimentation of dam reservoirs and multi-phase flow, studying area, ranging from theory to
design to implementation.
Masoud Montazeri Namin received his PhD in Hydraulics from the University of Wales, Cardiff
in 2003. Currently, he is an Assistant Professor of Civil Engineering and Head of Division of
Hydraulic Structures in the College of Engineering at the University of Tehran. His research
interests include numerical modelling, computational fluid dynamics and development of
hydrodynamic models.

A new approach to solve mixture multi-phase flow model using time splitting projection method 161
Asghar Bohluly received his PhD in the field of Hydraulics from the Sharif University of
Technology in 2010. His thesis was about two phase flow modelling and slug flow. He obtained
his BSc in Civil Engineering from the Ferdowsi University of Mashhad in 1998 and because of
his interest in hydraulics he continued in field of hydraulic structures in the Sharif University of
Technology. In 2000, he defended his thesis for numerical modelling of reshaping breakwater for
MSc degree. He has more than 15 years’ experiences in numerical modelling and development of
hydrodynamic models. Now, he is an Assistant Professor in the University of Tehran, Institute of
Geophysics and teaches numerical modelling and coastal processes.
1 Introduction
The flows that exist in nature are all combinations of three
phases of gas, liquid and solid. Air bubbles moving upward
in a liquid, flows that are combinations of two or more
liquids with different densities, or solid particles transported
by a fluid are all examples of such flows. In recent years,
interest in the applications of computational fluid dynamics
in multi-phase engineering and industrial processes has
risen, [e.g., Ge and Gutheil (2007) for particle-fluids, Yang
and Zhang (2013) for incompressible flows, Kositgittiwong
et al. (2013) for engineering applications and Sundaresan,
(2000) for industrial processes]. Polymerisation processes,
settling tanks, chemical reactors, air-lift reactors and gas
dispersion in liquid are typical examples of different
applications of multi-phase flows in industrial processes.
The flows passing around water structures such as
spillways, stilling basins, water jets and outlets are among
the phenomena representing the application of multi-phase
flows in engineering. In a multi-phase flow, entry of the air
phase into the water flow results in significant changes in
the flow’s characteristics, including density, velocity,
viscosity and depth. Therefore, using multi-phase modelling
in simulation of such flows is inevitable.
In the simplest numerical simulation, the flow
containing suspended particles was modelled by a
homogeneous single-phase system (Bernard-Champmartin
et al., 2014; Hinch, 1977). In this case, the multi-phase flow
was described as a single-phase flow while all other phases
were ignored. In another simple simulation, flows of two
incompressible and immiscible fluids were viewed as that of
a single fluid whose density and viscosity changed abruptly
at the interface (Nandi and Date, 2009). Therefore, these
models were aimed at detecting interfaces existing between
two fluids. In order to capture the interface, several methods
such as VOF method (Hirt and Nichols, 1981), flux
corrected transport (FCT)-VOF method (Rudman, 1997)
and level-set method (Osher and Sethian, 1988) were
developed.
In cases where the density difference between the two
phases is considerable, or where the second phases change
the hydrodynamic behaviour of the flow, the multi-phase
nature of flows cannot be ignored. At the same time, when
dealing with flows with a wide distribution in particle size
or density, the numerical simulation of flows based on the
multi-phase model is quite complex. Given the physics of
the problem, different approximations are often provided to
simplify the equations. Such simplifications have resulted in
different multi-phase flow models.
The accuracy of the multi-phase flow simulation
programs often depends on the accuracy of the model
employed in the program and the relevant physical
assumptions. Depending on the physical conditions and the
desired accuracy, different models have been presented for
multi-phase flow simulations (Hiltunen et al., 2009). In the
multi-fluid method, the mass and momentum equations are
solved for each individual phase, increasing the
computational cost of the model.
The mixture model is a multi-phase model used to
determine the velocities of the second phase from the
relative velocity equation that is based on the local
equilibrium assumption (Ishii, 1975). Therefore, instead of
solving momentum equations for the second phases, an
algebraic slip velocity equation needs to be solved. This
method results in a considerable reduction in the
computational workload of the program, especially when
several phases need to be considered (Hiltunen et al., 2009).
However, the superiority of the multi-phase mixture model
over the other models that consider similar velocities for
different phases is based on two facts. In other words, the
mixture method allows the phases
1 to be interpenetrating
2 to move at different velocities (Manninen et al., 1996).
Therefore, in a phenomenon where the interpenetration of
phases is important, not only does the mixture method
reduce costs, but it also provides high accuracy. The
mixture method has yielded suitable results in many
practical applications of multi-phase flows (Qian et al.,
2009; Ozturk and Aydin, 2009).
The multi-phase mixture model was first introduced by
Ishii (1975). Manninen et al. (1996) developed a theory
based on the model, rewrote the model equations based on
various parameters and proposed a system of mixture
equations for the numerical solution. In a number of models
similar to the mixture model, the local equilibrium
assumption was used to determine the velocity differences
between the two phases. In the relevant literature, these
models have been referred to as the algebraic-slip model
(Zuber and Findlay, 1965), drift-flux model (Ungarish,
2013) and diffusion model (Pericleous and Drake, 1986).
Other researchers have also attempted to develop
mixture models. For instance, Bernard-Champmartin et al.
(2014) introduced a homogeneous equilibrium mixture
model which can determine velocity disequilibrium between
fluids based on the closure law of the relative velocity
instead of the second phase momentum equation. This

162 F. Behrangi et al.
model is only valid when the drag force has enough time to
act on the fluids. Recently, Damián and Nigro (2014)
presented an extended mixture model for simultaneous
treatment of small-scale and large-scale interfaces. In this
model, the VOF and mixture methods were combined to
determine the phase interfaces. Furthermore, an approach
was also presented to reduce the grid requirement, while
maintaining the accuracy required for quicker problem-
solving.
In order to reduce computational costs, Ge and Gutheil
(2007) proposed a novel solution scheme for the
computation of particle velocity. Kerger et al. (2011)
considered an integration domain defined by a general cross
section and derived an original 1D free-surface drift-flux
model. In a similar study, Damián et al. (2012) derived a
semi-analytical solution for sedimentation cases in 1D
mixture models. They solved the mixture model as a
hyperbolic system with restrictions.
In all numerical studies, commercial solvers such as
FLUENT® and OpenFOAM® were used to solve the
numerical equations. These solvers employed the
semi-implicit method for pressure-linked equations
(SIMPLE) or the pressure-implicit split-operator (PISO)
algorithms to solve mixture equations (Jasak, 1996; Damián
et al., 2012). These algorithms are iterative procedures used
to solve velocity and pressure equations. Both algorithms
are based on the evaluation and correction of some initial
solutions.
Namin (2003) presented a time splitting, non-hydrostatic
projection for free surface flows originally based on
Chorin’s work (Chorin, 1968). It has the advantage of being
efficient, stable and non-iterative. Most recent
improvements in the projection method were aimed at
increasing its accuracy. The algorithm is based on the time
splitting method which results in a block tri-diagonal system
of equation with pressure as the unknown. This system of
equations can be solved using a direct matrix solver without
iteration.
Although researchers have used different applications of
the time splitting projection method in numerical models to
solve the flow field and the Navier-Stokes equations
(Ahmadi et al., 2007; Chegini and Namin 2012), the use of
the time splitting projection method for solving mixture
model equations is not addressed in the literature.
In this study, a new numerical algorithm for solving the
mixture model equations was developed based on the time
splitting projection method. The projection method
comprised of two main steps. In the first step, the
intermediate velocity values were determined by solving the
momentum equation except for the pressure term. These
intermediate velocities do not satisfy the continuity equation
yet. In the second step, a modification was considered to
transform these velocities in a convergence field that can
also satisfy the continuity equation. In this work, two more
steps were added in order to solve multiphase mixture
equations based on the time splitting projection method.
The multi-phase mixture model and its governing
equations are presented in Section 2. The numerical method
including the introduction of the new mixture model
solution algorithm is presented in Section 3. The
multi-phase flow tests used to evaluate the proposed model
are provided in Section 4. The results showed that the
proposed algorithm was successful in simulating the
mixture model.
2 Mixture model
The multi-phase mixture model involves the continuity and
momentum equations for the mixture, the volume fraction
equation for the secondary (particle) phases and the
algebraic expressions for the relative velocities. Manninen
et al. (1996) derived multi-phase mixture model equations
suitable for numerical solutions.
0
.
m
m m
ρ
ρ
u
t
w

w
’ (1)
.
. .
m m
m m m m
Dm Gm m
ρ
ρ p
t
τ τ
u
u u
g
ρ
w
w
’
’ ’

’
(2)
1
. .
p
p m p c
Mp p p
p
u D c u
t
’
w

w
’ ’
D
D D D (3)
4
3
.
p m m
cp cp m
p
D c
m
ρ ρ u
u u g
d
t
ρ
u u
C
w
’
w
ª º

« »
¬ ¼
(4)
where t is time, um is the velocity of the mixture mass
centre; ucp is the velocity of phase p (particle) relative to the
continuous phase (carrier) (ucp = up – uc); Dp is the volume
fraction of particle phase; cp is the mass fraction of particle
phase; pm is the pressure; ρm is the density of mixture; ρp and
ρc are the densities of particle and continuous phase,
respectively; g is the gravitational acceleration; dp is the
diameter of the particles; CD is the drag coefficient; DMp is
the diffusion constant in the continuity equation of the
dispersed phase; τGm is the generalised stress tensor in the
mixture; and τDm is the diffusion stress. In cases where only
one dispersed phase is present, the equation simplifies to
1 c
m p c
D m p p p
τ c c u
ρ u
(5)
The generalised stress obtained from equation (6):
Gm T
T
m m m m
u u
τ μ μ ’ ’
ª º

¬ ¼ (6)
where µm is the dynamic viscosity of the mixture; and µTm is
the coefficient of the turbulent eddy viscosity.
The mixture density and the mixture velocity are
defined as:
1
n
m k k
k
ρ ρ
¦D (7)
1
1
n
k
k
¦D (8)

1
1
n
m k k k
m k
u ρ u
ρ ¦D (9)
The phase pressures are often taken to be equal, i.e., pm = pk.
3 Numerical method
In this study, a new approach to solve the mixture model
was developed based on the finite volume approximation on
a staggered grid. We use two-dimensional in vertical
simulation in our works (2DV). The scalar quantities such
as pressure and volume fraction were defined at the centre
of the cells, while the velocity components were located at
the faces.
The numerical algorithm introduced for solving mixture
model equations was based on the projection method and
consisted of four steps.
3.1 Step 1: intermediate velocity
The momentum equations to describe 2DV were shown as
equations (10) and (11). Given the continuity of the mixture,
density was factored on the left side of the equation.
2
2 2
2 2
1
1
m p p cp
m m m
m m
m m
m p p cp cp m Tm m m
m m
c c u
u u u
u w
c
ρ
p
t x z ρ x
c u w
ρ x
ρ μ μ
ρ z ρ x z
u u
w
w w w w

w w w w w
w w w
ª º

« »
w w w
¬

¼

(10)
2 2
2 2
2
1 1
m m m
m m
m
m p p cp cp m p p cp
m m
m Tm m m
m
w w
u w
c c u w c
w p
t x z ρ z
ρ ρ
ρ x ρ z
μ μ
ρ x
c w
w
z
w
g
w w w w

w w w w
w w

w w
w w
ª º

« »
w
¬ ¼

w

(11)
In this step, according to the projection method, the pressure
term was eliminated from the momentum equations.
Assuming that the velocity fields in the past time n are
known, other terms were calculated as follows to yield the
intermediate velocities.
This step was subdivided into four stages. In the first
stage, the advection terms were solved using the known
velocity field at the previous time step n to obtain the
intermediate velocity field *
m
u and *
.
m
w
* n
m m
n
m m
m m
u u u u
u w
t x z
w w
§ ·

¨ ¸
' w w
©

¹
(12)
* n
m m
n
m m
m m
w w w
t
w
u w
x z
w w
§ ·

¨ ¸
' w w
© ¹

(13)
In the second stage, the diffusion terms were solved to find
the second intermediate velocity field **
m
u and **
.
m
w
** * **
2 2
2 2
m m m
m
t
t x z
u u u u
ν
§ ·
¨
w w

' w w
¸
© ¹
(14)
** * **
2 2
2 2
m m m
m
t
t x z
w w w w
ν
§ ·
¨
w w

' w w
¸
© ¹
(15)
In the third stage, the velocities were advected using the
known slip velocity field at previous time step n to find the
third intermediate velocity field ***
m
u and ***
.
m
w
2
*** **
1
1
n
m p p cp
m
m p p cp cp
m
m m
ρ
ρ x
t ρ
c c u
u u
ρ z
c c u w
ª º
w
« »
w
« »
« »
' w
« »

«
¬ ¼

»
w
(16)
*** **
2
1
1
n
m p p cp cp
m
m p p cp
m
m m
c c u w
w w
c
ρ
ρ x
t ρ c w
ρ z
ª º
w
« »
w
« »
« »
' w
« »

« w
¬ ¼

»
(17)
In the final stage, the final velocity in z-direction was
obtained from the gravity term.
**** ***
.
m m
m
ρ
t
w w
g

'
(18)
Fromm’s second-order upwind explicit scheme and the
implicit Crank-Nicholson method were used to solve the
advection and diffusion stages, respectively.
3.2 Step 2: velocity-pressure equation
In the momentum equation of the mixture, apart from the
pressure term, other terms were solved separately to
calculate the intermediate velocities. Velocity at the
subsequent time n + 1 was determined with regard to the
pressure term in the momentum equation as follows:
1
(1 ) 0
n
m
m m
n
u p p
φ φ
t x x
ρ ρ

w w w
§ · § ·

¨ ¸ ¨ ¸
w w w
© ¹ © ¹
(19)
1
( )
(1 ) 0
m
m
n
m
n
ρ
w p p
φ φ
t z z
ρ

w w w
§ · § ·

¨ ¸ ¨ ¸
w w w
© ¹ © ¹
(20)
1
1 1 2 1 0
, , 1 , , ,
, 1/2
n
n n
m k i k i k i k i k i
k i
u AU p AU p AU

(21)
1
1 1 2 1
, 1, , ,
1/2,
0
,
n
n n
m k i k i k i k i
k i
k i
w AW AW p
AW
p

(22)
where φ is the implicit weighting factor (here is 0.5) and:
***
1 2
, ,
0
, , , 1
, 1/2
, ,
,
, ,
(1 )
k i k i
n
k
n n
m k i m k i
n
m k
i
i
m k i k i
k i
ρ ρ
ρ
t φ t φ
AU AU
x x
t φ
AU u p p
x

' '

' '
'

'
(23)

1 2
***
,
*
0
,
1 2
,
1,
,
/ ,
, ,
(1 ) n
m
n n
m k i m k i
n
m k i
k i k i
k i
ρ
t φ t φ
AW AW
z z
φ t
AW w p p
z
ρ
ρ

' '

' '
'

'
(24)
3.3 Step 3: a new approach for determining pressure
poisson equations (PPEs)
According to the common approach adopted by many
researchers for writing PPEs, in the continuity equation (1),
the velocity terms were replaced with the equivalent
pressures terms obtained from equations (21) and (22). In
the models with constant concentration, density values were
omitted from the continuity equation. In models with
negligible concentration changes, the equation can also be
simplified according to the Boussinesq approximation
(Ahmadi et al., 2007; Chegini and Namin, 2012). Therefore,
the continuity equation used in the mentioned models
lacked the temporal term of density variations (wρm/wt).
Thus, in the previous works, the only unknown term in the
continuity equation was pressure.
For the 2DV mixture model, the continuity equation of
the mixture was rewritten as follows:
0
m m m m
m ρ u ρ w
ρ
t x z
w w
w

w w w
(25)
One of the advantages of the multi-phase mixture model is
its ability to model fluids with a high density difference,
(e.g., water and air). Therefore, in this model, the above
simplifying assumptions could not be used to eliminate the
first term of the continuity equation (25). In the projection
method, pressure was the only unknown in the continuity
equation. Here, in solving the mixture model using the
projection method, the problem was that the term (wρm/wt)
should be somehow treated to the formation of PPE in terms
of pressure. The time-dependent density term could not be
written in terms of pressure due to the incompressibility of
the fluid. Another approach could be to determine (wρm/wt)
explicitly from the past time. The explicit solution required
that an iteration loop be added to the program for the
convergence of the solutions, which would increase the
computational cost. Moreover, instability was observed in
the solution in cases where the difference in concentration
of the two phases was considerable.
In this study, a new technique was presented to solve
this problem. According to this technique, the continuity
equation of the mixture should be written as the sum of the
continuity equations of the phases:
1 1 2 2 1 1 2 2
0
u u w w
x z
w w

w w
D D D D
(26)
The above equation is rewritten as:
1 1 2 2
1 1 2 2
0
m m
m m η w η w
λ u λ u
x z
w
w

w w
D D
D D
(27)
where the parameters λ and η are defined as:
,
k k
k k
m m
u w
λ η
u w
(28)
In equation (27), the values for D, λ and η were calculated
based on the previous time and the values for um and wm
were written in terms of pressure [equations (21) and (22)].
Therefore, as could be seen in equation (29), the unknown
pressure was present in all the terms. This pressure term
could be solved in the Poisson equation in a way that the
instability caused by the explicitness of the known terms in
the previous methods be removed.
1 1 2 1
, 1 , , 1 , 1
1 1 2 2 , 1
0
, 1
1 1 2 1
, , 1 , ,
,
1 1 2 2
0
,
1 2 1
1, , 1, 1,
1,
1 1 2 2
0
1,
1
1
1
n n
k i k i k i k i
k i
k i
n n
k i k i k i k i
k i
k i
n n
k i k i k i k i
k i
k i
AU p AU p
λ λ
AU
x AU p AU p
λ λ
AU
Aw p Aw p
η η
Aw
z

½

ª º

° °
« »

° °
¬ ¼

® ¾
'
ª º
° °

« »
° °

¬ ¼
¯ ¿
ª º

« »

¬ ¼
'
D D
D D
D D
1 1 2 1
, 1, , ,
,
1 1 2 2
0
,
0
n n
k i k i k i k i
k i
k i
Aw p Aw p
η η
Aw

½
° °
° °
® ¾

ª º
° °

« »
° °

¬ ¼
¯ ¿
D D
(29)
Equation (29) is rewritten as:
1 1 1
, , ,
, 1 , 1 ,
1 1
, , ,
1, 1,
n n n
k i k i k i
k i k i k i
n n
k i k i k i
k i k i
a P b P c P
d P e P f

(30)
where the coefficients are:
1
,
, 1 1 2 2 ,
2
, 1
, 1 1 2 2 , 1
1
,
, 1 1 2 2 ,
2
1,
, 1 1 2 2 1,
, , , , ,
0
, 1 1 1 2 2 , 1
,
0
,
1
k i
k i k i
k i
k i k i
k i
k i k i
k i
k i k i
k i k i k i k i k i
k i k i
k i
k i
AU
a λ λ
x
AU
b λ λ
x
Aw
d η η
z
Aw
e η η
z
c a b d e
AU λ λ
f
x AU

§ ·

¨ ¸
'
© ¹
§ ·

¨ ¸
'
© ¹
§ ·

¨ ¸
© ' ¹
§ ·

¨ ¸
© ' ¹

'
D D
D D
D D
D D
D D
D1 1 2 2 ,
0
1, 1 1 2 2 1,
0
, 1 1 2 2 ,
1
k i
k i k i
k i k i
λ λ
Aw η η
z Aw η η

½
° °
® ¾

° °
¯ ¿
½

° °
® ¾
'
° °
¯ ¿
D
D D
D D
(31)
The pressure values at the next time step were determined
by solving equation (30) using the tri-diagonal block matrix
solver. The velocity field was then determined using
equations (21) and (22).
3.4 Step 4: solving phases equations
After determining the values for pressure and velocity fields
of the mixture flow in the new time, the second phase
continuity equation [equation (3)] was solved. The volume
fraction of the particle phase should be solved as a scalar
equation.
In the advection term in equation (3), the ULTIMATE
algorithm presented by Leonard (1991) was used to prevent
the unphysical oscillations in severe gradients and to
guarantee conservation of the quantities.

The term involving ucp was treated as additional
convection with slip velocity at the past time step n. Finally,
the relative velocity was solved by equation (4) to find ucp.
As Cd depended on ucp, the solution was iterative.
4 Model validation
In order to validate the proposed numerical model, a
number of tests were conducted. The first example
corresponded to a lock exchange flow and the objective was
to verify the modelling of gravity and density flows. A
series of velocities from free surface and bottom points of
flow were compared to numerical and analytical results. In
the second example, the dam break test was used to validate
the model results in the case of flows with high velocity and
high density differences. The shape of the free surface, the
velocity of water surge movement and the height of the
water column were compared to laboratory experiments.
Finally, the Rayleigh-Taylor instability was calculated using
the proposed model and the results were compared to the
VOF solutions. A qualitative comparison of the solutions
was conducted and the dynamics of the dispersed phase in
each case were compared using an integrated measure.
4.1 Lock exchange flow
The lock exchange flow is a well-known test used to verify
the modelling of gravity and density flows. This test deals
with a practical problem in engineering, where a flow of
dense water flows below the fresh water in the channel
when the gate is opened. The specifications simulated by
Jankowski (1999) were used to simulate the physics of the
problem and to compare the results.
Figure 1 Initial condition and lock exchange test results for (a) t
= 0, (b) t = 100 s, simulated by the proposed mixture
model (present work) and (c) t = 100 s, ordinary
density current simulation (see online version
for colours)
(a)
(b)
(c)
Source: Jankowski (1999)
Two incompressible fluids of slightly different densities
confined in a rectangular basin, with impermeable walls and
a free surface, were initially divided by a very thin wall,
[Figure 1(a)]. The left half of the basin was filled by a fluid
of ρ = 1,000.722 kg/m3
and the right half by another fluid of
ρ = 999.972 kg/m3
. In the initial time step, the impermeable
thin wall in the middle of the basin was removed. In the
initial condition, the velocity in the whole domain was set to
zero (u = 0). The turbulent diffusion and viscosity
coefficients were also set to zero. The channel measured 30
m in length, 3 m in width and 4 m in depth (Jankowski,
1999).
The numerical parameters selected for this test were:
∆x = ∆z = 1 m and the time step was 1 s. The initial
condition and results for t = 100 s are presented in Figure 1.
Assuming the infinite channel length and other
modelling assumptions, dense flow velocity could be
obtained using the law of conservation of energy from the
following equation (Jankowski, 1999):
2 1
2 1
0.5
c
ρ ρ
U gH
ρ ρ

(32)
where ρ1 and ρ2 are the densities of both fluids and H is the
fluid’s depth. For the geometry and the densities of the
fluids as mentioned above, equation (32) yielded a velocity
of Uc = 0.86 m/s.
Figure 2 The lock exchange flow speed time series for points at
the free surface and at the bottom located in the middle
of the domain (see online version for colours)
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0 20 40 60 80 100 120 140 160 180 200
U(m/s)
Time (s)
Free surface, Present work Bottom, Present work
Analytical Free surface, Jankowski
Bottom, Jankowski
In the phase distribution, it was observed that the phase
interfaces were clear and smooth [Figure 1(b)]. This showed
the effectiveness of the proposed model in obtaining stable
fluid interfaces. The results of the ordinary density current
simulation for a lock exchange problem (Jankowski, 1999)
are shown in Figure 1(c). As shown, ordinary solutions
caused some diffusion errors in interfaces. In Figure 2, the
time series of velocity variations in the middle of channel
for the two upper and bottom points of the proposed model
were compared with the analytical results and the results
reported by Jankowski (1999). In this case, the flow speed
of the upper point in the free surface was almost constant
after 150 seconds with velocity being about 2% lower than
theoretically expected. The flow speed of the bottom point
becomes almost constant after 120 s with velocity being
about 5% higher than expected. In short, the simulation of
fluids, the interfaces and the movement speed of the points
were well captured.

4.2 Dam break
The dam break test was used to validate the model results in
the case of flows with high velocity and high density
differences. The test deals with a problem in the two-phase
flow simulation with simple initial and boundary conditions.
This test, however, includes changes in free surface, wave
forehead, rising of water, return of water and even entry of
air. The complex dynamics of water surface is a serious
problem in two-phase models. The experimental results of
Martin and Moyce (1952) were used to control the
numerical simulation results.
In this test, the water column was a rectangle with its
vertical side being twice long the horizontal side (Figure 3).
According to the results obtained through physical tests, the
length of the small side of the rectangle was estimated
a = 0.05715 m. The length and height of the computational
domain were considered 4a. Free pressure boundary
condition was defined for the upper boundary and no-slip
boundary condition was defined for the other three
remaining sides. The physical parameters of fluids were
ρair = 1.23 kg/m3
, µair = 1.8 u10–5
pa.s and were
ρwater = 1,000 kg/m3
, µwater = 0.001 pa.s.
Figure 3 Dam break snapshots at different dimensionless time
steps (T*) (see online version for colours)
The time sequences of volume fractions are shown in
Figure 3 to demonstrate the motion of the dam break wave
and the predicted position of the interface in dimensionless
*
( / ).
T t g a After the water column fell down at
T* = 1.71, the water occupied more than half of the bottom
wall. Then, the wave front hit the wall and started climbing
the right wall. At T*= 4.91, the water column started
detaching from the wall and overturning. At T*= 6.45, the
water fell down and the wave front moved to the left wall.
Finally, at T*= 7.55, the water hit the right wall. However, a
quantitative comparison of results can be seen in Figure 4
and Figure 5. In Figure 4, the remaining water column’s
height was compared with the experimental results. The
horizontal axis represents the dimensionless time and the
vertical axis represents the dimensionless height
(H* = H/2a), where H is the water column’s height in the
left wall after the break. Surge front locations from
numerical prediction and experimental results are shown in
Figure 5. The horizontal axis represents the dimensionless
time and the vertical axis represents the dimensionless wave
front location (x* = x/2a), where x is the wave front
location.
Figure 4 The remaining water column’s height at different
dimensionless time steps (T*) (see online version
for colours)
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6
H*
T*
Experimental, Martin and
Moyce 1952
Numerical, present work
As can be seen in the figures, the numerical results are in a
good agreement with the experimental data in terms of
predicting the surge movement and the water column’s
height. Although the mixture model is not essentially used
to determine the free surface, in this test, it determined the
free surface with an acceptable accuracy. The accuracy of
the test results indicated the satisfactory performance of the
model in the two-phase flows with high density variations.
Figure 5 Surge front location at different dimensionless time
steps (T*) (see online version for colours)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2 2.5 3 3.5 4
X*
T*
Experimental, Martin and Moyce
(1952)
Numerical, present work
4.3 Rayleigh-Taylor instability
In the Rayleigh-Taylor instability problem, a heavier fluid
layer initially lies on top of a lighter fluid. The fluid
densities were 1.225 and 0.1694 kg m–3
. The fluid volumes
were the same. The fluid viscosities were 0.00313kg m–1
s–1
(Shao, 2013). A rectangular L = 1 m u 4 m = H domain was
discretised using a 64 u 256 grid. The interface’s initial
condition was a sinusoidal wave:

0
2
sin , 0
πx
δ δ x L
L
§ ·
§ ·
d d
¨ ¸
¨ ¸
© ¹
© ¹
(33)
where δ is the disturbance of interface and δ0 = 0.05 m. In
this case, the time step size was 0.001s and the gravitational
acceleration was 9.81 m s–2
. Owing to the gravity, we
expected that the fluid with higher density would move
down. Figure 6 shows the evolution of the interpenetration
of fluids at different times. The interface between the phases
was similar to the sine wave at first and to the
mushroom-like structures later. The interface evolution
compared well with the Popinet’s work (Popinet and
Zaleski, 1999).
Linear stability analyses with linearised Navier-Stockes
equations (before t = 0.9 s) were conducted to find
analytical solutions. The time development for growth of
amplitude was analytically calculated from equation (34)
(Štrubelj and Tiselj, 2011).
0
( ) τ
y τ δ eE (34)
where E is the linear amplification rate:
2
gkAt
E (35)
The wave number k = 2π/L and the Atwood number were At
= 0.5:
1 2
1 2
ρ ρ
At
ρ ρ

(36)
The initial free surface with small sine disturbance
developed to a surface with a larger sine wave and then a
mushroom-like structure. The results obtained from the
mixture model and the analytical solutions in terms of
amplitude growth are shown in Figure 7.
Figure 6 Rayleigh-Taylor instability at different time steps
(see online version for colours)
The amplitude growth at t 0. 9 s or τ 0.16 was no longer
exponential and the linear theory and analytical equation
(34) were no longer valid. The growth of amplitudes was
simulated well (τ 0.16) and it was slightly faster than that
of the analytical solution.
Figure 7 Analytical growth of amplitude and amplitude growth
in simulation of Rayleigh-Taylor instability with the
mixture model (see online version for colours)
0.01
0.1
1
10
0 0.05 0.1 0.15 0.2 0.25
d
t
analytical
numerical, presente work
5 Conclusions
Development of an effective numerical model for the
simulation of multi-phase flows is important in
computational fluid dynamics and engineering. In this
study, a solver for the multi-phase mixture model was
developed based on the finite volume method to simulate
multi-phase flows in a two-dimensional vertical plane. The
mixture model is a multi-phase model that determines the
velocity field and volume fractions of all phases. The
advantage of the multi-phase mixture model compared with
other multi-phase models that provide all phases with full
flow field is that, for the second phase, the mixture model
uses an algebraic equation for the relative velocity rather
than solving the momentum equations, thus, significantly
reducing the computational costs. Moreover, unlike the free
surface models, the mixture model allows for the
interpenetration of the two phases. Therefore, volume
fraction of each phase is calculated for each cell in the
model in the ranging from zero to one.
In this paper, a numerical solution algorithm for the
mixture model was proposed based on the time splitting
projection method. The proposed algorithm solved the
equations governing the mixture model in four steps. The
parametric continuity phasic equation technique was
developed to solve the mixture continuity equation. Using
this technique, the modified mixture continuity equation had
exactly one unknown parameter (pressure at time n + 1).
The proposed solution algorithm had a feature that enabled
it to solve the equations in just one step and with respect to
the tri-diagonal matrix. Thus, there was no need to correct
the solutions through iterative methods. This approach
significantly reduced the program run-time. Another feature
of the proposed solution algorithm was that different
numerical schemes could be used for different steps in the
process of solving the equations.
In order to validate the model, three multi-phase flow
tests were investigated. The lock-exchange flow, the dam
break and the Rayleigh-Taylor instability tests were used to
assess the hydrodynamic modelling and its ability to
determine the interface between the two fluids. The
validation results indicated good agreement between the

proposed numerical method and experimental works and the
analytical solution in the simulation of the flow field. Based
on the results discussed in this study, it can be concluded
that the proposed mixture model was a first-time
combination of the multi-phase mixture model and the time
splitting projection method to simulate multi-phase flows.
Characteristics of the mixture models and features of
their numerical solution method developed in this paper
contributed to the development of an effective numerical
model. The proposed model can be used in many
engineering applications, where, in addition to the free
surface, phase concentration is also important.
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