A Volume Of Fluid Typ e Algorithm
for Compressible Two Phase Flows
Keh Ming Shyue
Abstract We present a simple approach to the computation of a simpli ed
two phase ow mo del involving gases and liquids separated by interfaces in
multiple space dimensions In contrast to the many p opular techniques which
are mainly concerned with the incompressible ow we consider a compress
ible version of the mo del equations without the e ect of surface tension and
viscosity We use the law and the Tait equation of state for approximating
the material prop erty of the gas and the liquid in a resp ective manner The
algorithm uses a volume of uid formulation of the equations together with a
sti ened gas equation of state that is derived to give an approximate mo del
for the mixture of more than one phase of the uids within grid cells A stan
dard high resolution sho ck capturing metho d based on a wave propagation
view p oint is employed to solve the prop osed mo del We show results of some
preliminary calculations that illustrate the viability of the metho d to practical
application without the o ccurrence of any spurious oscillation in the pressure
near the interfaces This includes results of a planar sho ck wave in water over
a bubble of air
Intro duction
We are interested in a two phase ow problem with interfaces that separate regions
of two di erent uid comp onents consisting of gases and liquids We consider a
two dimensional ow as an example and use the compressible Euler equations to
mo deling the motion of the gases
u v
2
C C C B B B
u u p uv
C C C B B B
2
A A A
v uv v p
t x y
E E p u E p v
Here is the density u and v are the velo cities in the x and y direction resp ec
tively p is the pressure and E is the total energy p er unit mass We assume that
This work was supp orted in part by National Science Council of Republic of China Grants NSC
M and NSC M
K M Shyue
the equation of the state for the gas satis es the law where is the ratio of
sp eci c heats Then the internal energy is
p
e
2 2
and E e u v We note that the four comp onents of Eqs express
the conservation of mass momenta in the x and y direction and energy resp ec
tively
As it is often the case we take the liquids to b e baratropic that the pressure
p is a function of the density only and in particular it ful lls the Tait equation
of state of the form
p A B
where A and B are material dep endent constants these two parameters along with
give a fundamental characterization of the uid prop erties of interest and can
b e obtained from a tting pro cedure of lab oratory data Typical set of values
are for water A atm and B atm and for human
blo o d A MPa and B MPa approximately It is
imp ortant to note that b ecause of the mere dep endency of the pressure p on the
density the energy equation of the Euler Eqs plays no active role to the
determination of this ow b ehavior and hence can b e neglected For completeness
we write down the equations of motion for the liquid as follows
u v
2
A A A
u u p uv
t x y
2
v uv v p
where the pressure p is governed by
We want to use a state of the art sho ck capturing metho d on a uniform rect
angular grid for the computation For convenience let us supp ose that for each
grid cell we have a volume fraction function Z representing the typ e of uid within
the cell For example for the liquid only cells we may take Z and therefore
for the gas only cells we may set Z In case there are some cells cut by the
interfaces where Z we then have b oth of the gas and liquid comp onents
in the cells in which the liquid and the gas are o ccupied by the volume fractions
Z and Z resp ectively
It is clear that when Z or Z there is no problem in describing
the motion of each of the gas and liquid ows individually see Eqs The
principle problem in the current application and also in the other multicomp onent
problems cf is however directed to the development of an e cient
and yet accurate way that is capable of dealing with the uid mixture case when
Z Motivated by the previous work by the author the algorithm we
employed uses a mixture mo del based on a volume fraction or called a volume
of uid formulation of the equations that is devised to mo del the mixture of the
two di erent equations of state and by the so called sti ened gas equation
of state see Section Then from the energy equation of we cho ose
Volume Of Fluid Algorithm for Two Phase Flows
conditions that should b e satis ed to ensure the pressure in equilibrium for these
cells see Section Numerical results to b e presented in Section show that this
is a viable approach for practical problems without any spurious oscillations in
the pressure near the interfaces when we approximate the mo del equations in a
consistent manner by a uctuation and signal typ e of sho ck capturing metho d
It is true that for real applications the e ect of surface tension as well as
viscosity are two imp ortant elements to the solutions of a two phase ow problem
under studied Among many of the approaches develop ed over the years to deal
with these situations cf the metho d based on the level set formulation
has shown to b e quite e ective for an incompressible version of the problem
where the uids are mostly in a low Mach numb er regime Here in contrast to
the work just mentioned we consider a class of problems where the in uence of
compressibility of the uids to the solutions is vital but not the surface tension
and viscosity Examples of this kind cover a family of sho ck wave problems with
interfaces Our goal here is to establish a basic solution strategy for the
problem cf for other approaches Realization of this approach that
couples with adaptive grid pro cedures such as front tracking and adaptive mesh
re nement is in progress cf In a future work we will take more physical
factors into account and also lo ok at metho ds that are e cient for problems with
low and high Mach numb er ow where the time step restriction is an imp ortant
issue to b e addressed
This pap er is organized as follows In Section we remark and discuss the
equations of state that are used in our mo del two phase ow problems with gases
and liquids In Section we review the derivation of the mo del system based on
a volume of uid formulation of the equations Numerical results for some sample
problems are shown in Section
Equations of State
It is known that in gases the sp eci c entropy denoted by S has great in uence
to the b ehavior of the pressure p while in liquids the in uence of its changes is
negligible to the pressure In fact b ecause of this little dep endency of the
pressure on the entropy and a consequence of the rst law of thermo dynamics we
may write the internal energy of a liquid as a separable energy of the form
(1) (2)
e e e S
where with the Tait equation of state we nd
p B
(1)
e
From this it is easy to observe strong resemblance of the equation of state of a
liquid to that of an ideal gas With this in mind it should b e sensible to use
K M Shyue
a generalized version of Eq namely the sti ened gas equation of state
p B
e
without the explicitly dep endence on the density to the pressure p to mo deling
mixtures of the comp onents of gases and liquids within a grid cell Note that
a sti ened gas reduces to a law gas when B and to a baratropic gas of
the form when the pressure p is indep endent of the entropy S Oftentimes
the equation of state has b een used in other applications to mo del materials
including compressible liquids and solids Here we nd it is suitable for the
current application when combining with a two phase ow algorithm describ ed
in the next section see results shown in Section for numerical veri cation of this
statement
Volume Of Fluid Algorithm
We now discuss the basic idea of our approach to mo deling cells that contain b oth
of the gas and liquid phases We use the Euler Eqs as a mo del system that
describ es the motion of the mixtures of conserved variables such as u v and
E in a gas liquid co existent grid cell Then with the use of the equation of state
the pressure p in the equations can b e set by
2 2
u v
B p E
when the mixture of material dep endent parameters and B are de ned and
known a priori It is imp ortant to note that in our approach the conditions for
and B are chosen so that the pressure p in retains in equilibrium for a gas
liquid mixture cell The derivation of the evolution equations for and B in the
current case follows directly from pro cedures develop ed in However to make
the pap er self contained a brief overview of this approach is undertaken b elow
We b egin by considering a mo del two phase ow problem that the pressure
p and the velo city eld u v are constant in the domain while the other vari
ables such as the density and the equation of state parameters and B are
having jumps across a gas liquid interface We write Eqs in the following
non conservative form
v u
u v
t x y x y
u p u u
u v
t x y x
p v v v
u v
t x y y
v u
e eu ev p
t x y x y
Volume Of Fluid Algorithm for Two Phase Flows
and obtain easily equations describing the motion of the gas liquid interface as
u v
t x y
e u e v e
t x y
To see how the pressure p would keep in equilibrium as it should b e for this
mo del problem we insert the equation of state into and have
p B p B p B
u v
t x y
Then with the volume fraction function Z de ned previously in Section the
total internal energy e can b e expressed as a function of the volume fraction by
(l) (l) (l) (g )
p B p p B
Z Z e
(l) (g )
where the sup erscript l and g of the states p and B represent the values
for the liquid and gas phases resp ectively When substituting to and
after some algebraic manipulation it is not di cult to show the satisfaction of
(l) (g )
the required pressure equilibrium p p p when the following evolution
equation for the volume fraction function is hold
Z Z Z
u v
t x y
Note that in this instance the mixture of and B are computed by
Z Z
a
(l) (g )
and
(l) (l)
Z Z Z B
b B
(l) (l) (g )
In summary the mo del equations we prop osed to solve two phase ow prob
lems with gases and liquids consist of the following equations
u v
t x y
2
u u p uv
t x y
2
v uv v p
t x y
E E p u E p v if Z
t x y
Z Z Z
u v
t x y
K M Shyue
where in case Z the Tait equation of state is used to compute the pressure
p for a liquid while Eq is employed when Z with and B de ned
by a and b resp ectively We note that the mo del Eqs is not written in
the full conservation form but is rather a quasi conservative system of equations
Numerical approximation based on the discretization of the mo del equations via
the wave propagation approach has shown to b e quite robust for a wide
variety of problems of practical interests see results presented in Section The
algorithmic details as well as many other numerical asp ects of this approach may
b e found in the pap ers
Numerical Results
We now present two sample calculations to demonstrate the feasibility of the pro
p osed approach describ ed in Section for practical applications cf for more
numerical results
Example To b egin we consider a radially symmetric problem that
the computed solutions can b e compared with the one dimensional results for
numerical validation We use the same setup as done by Davis that inside
a circular membrane of radius r the uid is air with density
0
pressure p and the adiabatic gas constant while outside the
circular membrane the uid is liquid with density and the Tait equation of
state of parameter values A B and Initially b oth of the air and
the liquid are in a stationary p osition but due to the pressure di erence b etween
the uids breaking of the membrane o ccurs instantaneously For this problem the
resulting solution consists of an outward going sho ck wave in liquid an inward
going rarefaction wave in air and a contact discontinuity lying in b etween that
separates the air and the liquid
In Fig we show numerical results for the density as well as the pressure
p at time t We use a grid with the high resolution version of the
wave propagation metho d cf for the computation the computational
domain is a square with dimensions From the contours of the
plots it is easy to see the wave pattern as just mentioned after the breaking of the
membrane where the dashed line in the pressure contours is the approximate lo ca
tion of the interface From the scatter plots we nd go o d agreement of the results
as compared with the true solution obtained from solving the one dimensional
multicomp onent mo del with appropriate source terms for the radial symmetry
using the high resolution metho d with mesh side h Noticeably
here the pressure near the interface b ehaves in a satisfactory manner without any
spurious oscillations and the sho ck wave and contact discontinuity app ear to b e
very well lo cated
Example Our next example concerns a planar sho ck wave in water
over a bubble of air We take an initial condition that is similar to the one used
by Grove and Meniko where anomalous re ection of a sho ck wave at a uid
Volume Of Fluid Algorithm for Two Phase Flows
a) Density Pressure
b) Density Pressure
...... 0.8 0.9 1.0...... 1.1 1.2 ...... 0.0 0.5 1.0 1.5 2.0 2.5
0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6
r r
Figure High resolution results for a radially symmetric prob
lem at time t a Contours of the density and the pres
sure p b Scatter plots of the density and the pressure p with
lo cations measured as a distance from a p oint of the solution to
the center The solid line in the scatter plot is the true
solution obtained from solving the one dimensional multicomp o
nent mo del with appropriate source terms for the radial symmetry
using the high resolution metho d The dotted p oints are the two
dimensional result The dashed line in the pressure contour is the
approximate lo cation of the interface
interface is examined closely there Here to test the prop osed metho d we consider
a downward moving Mach sho ck wave in water with data in the
u v p
presho ck
K M Shyue
and data in the p ost sho ck state by
4
u v p
p ost sho ck
The parameters we used for the equation of state of the water are A atm
B atm and In addition to this we assume that there is a stationary
bubble of air of radius lo cated just b elow the sho ck with density di
mensional unit g cc and the adiabatic gas constant for the law equation
of state We note that the ratio of acoustic imp edances c c
w ater air
is large for the problem studied here where c is the sp eed of the sound of the mate
rial of interests In fact this is a much more di cult test than those ones considered
in Example and also in
Fig shows preliminary results of this problem using the high resolution
metho d with a grid on a unit square domain In the gure reasonable
resolution of the results are obtained by the metho d where the density and the
pressure p contours in logarithmic scale at three di erent times t
3
are present cf Some works are in progress to further validate the
approach intro duced here cf This includes results of a mo del water splash
problem that due to gravity a water drop is fallen from the air to a water surface
in b elow
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Volume Of Fluid Algorithm for Two Phase Flows
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Department of Mathematics
National Taiwan University
Taip ei Taiwan Republic of China
E mail address shyue math ntu edu tw
K M Shyue
a) Density Pressure
air air
water water
b) Density Pressure
c)
Density Pressure
Figure High resolution results for a planar Mach sho ck
wave in water over a bubble of air Contours of the density and
the pressure p are shown in logarithmic scale at three di erent
times a at time t b at time t c at time t
The dashed line in the pressure contour is the approximate
lo cation of the interface