Submitted to INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
Int. J. Numer. Meth. Fluids 2007; 00:1–6
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Homogeneous Equilibrium Mixture Model for Simulation of
Multiphase/Multicomponent Flows
Randy S. Lagumbay
∗‡ ,
Oleg V. Vasilyev ∗ , Andreas Haselbacher
†
‡ Alden
Research Laboratory, Inc., 30 Shrewsbury St., Holden, MA 01520-1843 USA,
of Mechanical Engineering, University of Colorado, 427 UCB, Boulder, CO 80309, USA,
of Mechanical and Aerospace Engineering, University of Florida, MAE-B 222, Gainesville, FL
32611-6300, USA
∗ Department
† Department
SUMMARY
Most of the existing approaches in dealing with multiphase and multicomponent flows are limited
to either single-phase multicomponent or multiphase single-component mixtures. To remove this
limitation, a new model for multiphase and multicomponent flows with an arbitrary number of
components in each phase is developed. The proposed model is based on a homogeneous
equilibrium mixture approach. The model is hyperbolic and gives an accurate value for the mixture
speed of sound when compared to experimental data. A suitable numerical method is developed to
perform simulations which demonstrate the capabilities of the proposed model. The Harten, Lax and
van Leer scheme (HLLC) approximate Riemann solver is extended to compute the convective fluxes
for multiphase and multicomponent flows and used to capture shock waves and contact discontinuities.
Furthermore, a novel “idealized” fluid-mixture model is developed. This model allows the derivation
of an exact solution for the multiphase and multicomponent Riemann problem in one dimension. To
verify the accuracy of the proposed numerical method and to demonstrate the physical fidelity of the
proposed model, three classical benchmark problems (single-phase two-component shock tube, shock
wave propagation in a single-phase two-component fluid, and single-phase shock-bubble interaction)
and two novel benchmark problems for the “idealized” fluid-mixture model (two-phase shock-tube
c 2007 John Wiley & Sons, Ltd.
and two-phase rarefaction problems) are presented. Copyright key words: Harten, Lax and van Leer scheme; numerical method; mixture; multiphase;
multicomponent; idealized fluid; Riemann problem
∗ Correspondence
to: Department of Mechanical Engineering, University of Colorado, 427 UCB, Boulder, CO
80309, USA
Email Addresses: [email protected] (Randy S. Lagumbay), [email protected] (Oleg V.
Vasilyev), [email protected] (Andreas Haselbacher)
Contract/grant sponsor: Argonne National Laboratory; contract/grant number: 2-RP50-P-00005-00, 3B-00061,
4B-00821, 5F-00462
Contract/grant sponsor: Department of Energy; contract/grant number: B523819
c 2007 John Wiley & Sons, Ltd.
Copyright Received 5 November 2007
Revised
2
R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
1. INTRODUCTION
Multiphase and multicomponent flows are common in many engineering applications. Relevant
examples are fuel sprays in combustion processes, liquid-jet machining of materials, and
steam generation and condensation in nuclear reactors. The physical mechanisms underlying
multiphase (in the context of this paper refers to liquid and gas) and multicomponent
(several instances of the same phase) flows as well as the interplay of these mechanisms are
very complex. In multiphase and multicomponent flows the phases and/or components can
assume a large number of complicated configurations; small-scale interactions between the
phases can have a profound impact on macroscopic flow properties [1]. Multiphase and
multicomponent flows can generally be identified at the outset as disperse flows
or separated flows. Disperse flows consist of finite particles, drops or bubbles (the
disperse phase) distributed in a connected volume of the continuous phase. On the
other hand, separated flows consist of two or more continuous streams of different
fluids separated by interfaces.
The simulation of multiphase and multicomponent flows poses far greater
challenges than that of single-phase and single-component flows. These challenges
are due to interfaces between phases and large or discontinous property variations
across interfaces between phases and/or components. Two approaches are
commonly used for the simulation of multiphase and multicomponent flows. In
the first approach, each phase and/or component is considered to occupy a
distinct volume and the interfaces between the phases and/or components are
tracked explicitly, see, e.g., [2, 3, 4, 5, 6, 7, 8]. Typical applications include the
prediction of the motion of large bubbles in a liquid, the motion of liquid after
a dam break, the prediction of jet breakup, and the tracking of any liquid-gas
interface. In the second approach, the phases and/or components are spatially
averaged to lead to a homogeneous mixture and are considered to occupy the
same volume. Many dispersed flows including bubbly flow of air in water or mist
flow can be considered as homogeneous mixture. Homogeneous mixture can either
be in equilibrium (e.g., the mechanical and thermal properties are in equilibrium)
or in non-equilibrium (e.g., the mechanical and thermal properties are not in
equilibrium) conditions. The advantage of the homogenized-mixture approach
compared to the interface-tracking approach is that it solves only one set of
equations for the mass, momentum, and energy of the mixture, supplemented
by equations for the mass or volume fraction of the mixture constituents [9].
In this work, the homogeneous equilibrium mixture approach is used to model
homogeneous multiphase/multicomponent flows. The phases and/or components
are assumed to be sufficiently well mixed and the disperse particle size are
sufficiently small thereby eliminating any significant relative motion. The phases
and/or components are strongly coupled and moving at the same velocity. In
addition, the phases and/or components are assumed in close proximity to each
other so that heat transfer between the phases and/or components would occur
at small time-scale maintaining the phases and/or components in thermodynamic
equilibrium. Furthermore, the response of the disperse phase (e.g., response
of bubbles in the liquid) to the change in pressure is assumed an essentially
instantaneous change in their volume so that the disperse phase would behave
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HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
3
quasistatically and the mixture would be in constant pressure. The frequency
disturbance of the disperse phase is assumed smaller than the natural frequencies
of the disperse phase themselves in order to maintain thermodynamic equilibrium.
Multiple approaches of dealing with multiphase or multicomponent mixtures exist. However,
most of them are limited to either single phase multicomponent fluids [10, 11, 12, 13] or
multiphase single component mixtures [14, 15, 16]. One of the goals of our work is to
remove this limitation. Hence we have developed a new model for multiphase flows with an
arbitrary number of components in each phase. The proposed model is hyperbolic, allowing
the construction of upwind methods for the computation of convective fluxes. Furthermore,
the proposed model is acoustically and thermodynamically consistent, which means that the
model gives an accurate value for the mixture speed of sound.
To perform simulations which demonstrate the capabilities of the new multiphase and
multicomponent model, a suitable numerical method is developed, based on a finite-volume
framework [17, 18, 19]. The modified Harten, Lax and van Leer scheme (HLLC) is extended
to multiphase and multicomponent flows and used to capture shock waves and contact
discontinuities [20]. The numerical method is verified by applying it to a number of test
problems. The problems were chosen to highlight the flexibility and robustness of the new
approach and cover the following cases:
1.
2.
3.
4.
single-phase single-component fluid;
single-phase multicomponent fluid;
multiphase single component fluid;
multiphase multicomponent fluid.
It should be noted that a number of benchmark problems for cases 1-3 are available
[11, 14, 16, 21, 22, 23, 24]. However, up to now, there are no known problems for case 4 which
have an exact closed-form solution for arbitrary initial conditions and arbitrary numbers of
phases and components. The second goal of this work is to address this deficiency. Accordingly,
in this paper, a novel “idealized” fluid mixture model is developed which allows the derivation of
an exact solution for the multiphase and multicomponent Riemann problem in one dimension.
A number of existing benchmark problems for single-phase multicomponent flows become a
subset of this new problem.
The rest of this paper is organized as follows. Section 2 introduces the mathematical
formulation for multiphase and multicomponent flows. The governing multiphase and
multicomponent Euler equations in conservative form and the mathematical model for the
mixture variables are presented. Section 3 introduces the novel idealized fluid mixture model.
The derivation of an exact solution for the multiphase and multicomponent Riemann problem
in one dimension is presented in Section 4. The proposed numerical method is outlined in
section 5. Section 6 presents the computational results for some known benchmark problems,
including the new benchmark problem for idealized fluid mixture model. Finally, conclusions
are outlined in Section 7.
2. Mathematical Formulation
The governing equations of the homogeneous equilibrium mixture model for the
simulation of multiphase and multicomponent flows are described in the following.
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R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
The model assumes strong coupling of the phases and/or components and moving
at the same velocity components u and v. The phases and/or components are
in close proximity to each other so that heat transfer between the phases
and/or components would occur instantaneously maintaining the phases and/or
components in thermodynamic equilibrium, i.e., temperature T and pressure P are
identical for all the phases and components. Also, the frequency disturbance of
the disperse phase is assumed smaller than the natural frequencies of the disperse
phase themselves to maintain equilibrium.
In the description to follow, the mixture is assumed to consist of two phases, namely liquid
and gas, and the gas phase is assumed to consist of two components, namely a generic gas and a
vapor. These are denoted by the subscripts l, g, and v for liquid, gas, and vapor, respectively. It
should be noted, however, that the homogeneous equilibrium mixture model can be extended
in a straightforward fashion to an arbitrary number of phases and components. Variables
without subscripts are applicable to the mixture only. The subscript i is used to denote a
specific component.
2.1. Homogeneous Equilibrium Mixture Model
The homogeneous equilibrium mixture model is based on the notion that the
velocity, temperature and pressure between the phases and/or components are
equal. The quantities associated with a given phase and/or component are
averaged to give the corresponding mixture quantity. Accordingly, quantities per
unit volume are averaged by their respective volume fraction φi . For example, the
mixture density is given by
X
ρm =
ρi φi ,
(1)
i=l,g,v
where ρi is the density of the ith phase and/or components, and the volume fractions satisfy
the constraint
X
φi = 1.
(2)
i=l,g,v
Conversely, quantities per unit mass are averaged by their respective mass fractions Yi . For
example, the specific heat at constant volume of the mixture is given by
X
cvm =
cvi Yi .
(3)
i=l,g,v
where cvi is the specific heat at constant volume of the ith phase and/or components
and the mass fractions satisfy the constraint
X
Yi = 1.
(4)
i=l,g,v
The volume and mass fractions are related through
ρi φi = ρm Yi .
(5)
The equations governing the evolution of mass, momentum, energy, and composition of the mixture for compressible homogeneous multiphase/multicomponent
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HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
5
flows (presented here in two dimensions for brevity)is given by
∂Q ∂E ∂F
+
+
= 0,
∂t
∂x
∂y
(6)
where Q is the vector of the conserved variables and E and F are the flux vectors given by






ρm
ρm u
ρm v
 ρm u 




ρm u 2 + P
ρm uv






2
 ρm v 




ρ
uv
ρ
v
+
P
m
m





,
Q=
, F=
, G=
(7)



ρ
e
(ρ
e
+
P
)
u
(ρ
e
+
P
)
v
 m mT 
 m mT

 m mT

 ρm Yg 




ρm Yg u
ρm Yg v
ρm Yv
ρm Yv u
ρm Yv v
where u and v are the x- and y-components of the velocity vector of the mixture, respectively,
and the total energy emT is defined as
emT = cvm T +
1 2
u + v2 .
2
(8)
The constitutive equations of the liquid, gas, and vapor are assumed to take the form
ρi = ρi (P, T ).
(9)
The mathematical model derived in this paper is general and can be used for arbitrary forms
of the equation of state for each phase. However, in the present study the gas and vapor are
assumed to obey the ideal-gas laws
P
,
Rg T
P
ρv =
,
Rv T
ρg =
(10)
(11)
while the liquid is assumed to be a linear dependent of pressure and temperature
2
1
βl
ρl = ρo + 2 (P − Po ) −
(T − To ),
Cl
Cl
(12)
where ρo , Po and To are the reference density, pressure and temperature of the liquid,
respectively. Cl2 and βl are the isothermal speed of sound and compressibility of the liquid,
respectively. The specific heat at constant volume of the gas and vapor is given by
Rg
,
γg − 1
Rv
=
,
γv − 1
cvg =
(13)
cvv
(14)
where γg and γv are the specific heat ratio of the gas and vapor respectively. For
the liquid, the specific heat at constant volume cvl is equal to the specific heat at
constant pressure cpl .
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R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
2.2. Mixture Speed of Sound
In the following, the equation of speed of sound of the mixture is derived and the
value is compared to the experimental data of Karplus [25] that corresponds to
the isothermal theory for the bubbly air/water mixture.
Taking the acoustic differential of Eq. (9) gives
2
1
βi
dT,
(15)
dρi = 2 dP +
Ci
Ci
1/2
where Ci = (∂P/∂ρi )1/2 and βi = (∂P/∂T )i are the isothermal speed of sound and
compressibility of the ith component, respectively. Using Eqs. (1) and (15), the mixture-density
differential can be written as
!
1
1
− 2
dP,
(16)
dρm = (ρv − ρl )dφv + (ρg − ρl )dφg +
Cφ2
Cφβ
P
P
2
where 1/Cφ2 = i φi /Ci2 and 1/Cφβ
= i φi βi2 /Ci2 .
The mixture speed of sound can be obtained easily by transforming Eq. (6) from conservative
variables to primitive variables. The eigenvalues of the transformation matrix give the speed
of sound by inspection. The speed of sound of the mixture is then found to be given by
2
Cm
X φi βi 2
ρm cvm + P
ρi Ci
i
.
=
X φi 1
ρ2m cvm
ρi Ci2
i
(17)
1/2
Note that in thePcase of multicomponent
gases,
P
P Eq. (17) becomes Cm = (γm Rm T ) , where
γm = cpm /cvm = i Yi cpi / i Yi cvi , Rm = i Yi Ri , and Ri is the gas constant of the ith gas
component. Furthermore, it should be noted that due to the assumption of thermodynamic
and mechanical equilibrium, the speed of sound predicted by Eq. (17) is applicable only to
disturbances whose frequency tends to zero.
The speed of sound of a mixture of water (equation of state is given by Eq. (12)) and air
(equation of state is given by Eq. (10)) at sea-level conditions predicted by Eq. (17) is plotted
in Fig. 1 as a function of the volume fraction of air. Note that over a wide range of volume
fractions, the mixture speed of sound of the mixture is much lower than the speed of sound of
either medium [26]. Good agreement is observed between the value predicted by Eq. (17) and
the experimental data of Karplus [25]. The predicted speed of sound was also compared to
Refs. [27, 28, 29, 30] and, although not plotted, is in good agreement. For a mixture of liquid,
gas, and vapor at sea-level conditions, the speed of sound is shown in Fig. 2.
3. Idealized Fluid-Mixture Model
The governing equations for the mixture cannot be solved exactly for arbitrary mixtures. The
equation of state for each component must be carefully chosen if an exact solution is to be
found. In this section, a novel idealized fluid-mixture model is developed which allows exact
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HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
solutions of the governing equations to be derived. Expressions for the density and entropy are
derived. In addition, the Riemann invariants and eigenvectors of the idealized fluid mixture
are presented for one-dimensional flows.
3.1. Mixture Density and Equation of State
To obtain a closed-form solution of the governing equations, the mixture entropy is assumed to
be a function of pressure, temperature and mass fractions of the species, i.e., sm = sm (P, T, Yi ).
Differentiation gives
X ∂sm
∂sm
∂sm
dsm =
dP +
dT +
dYi ,
(18)
∂P
∂T
∂Yi
i=l,g,v
and because we assume dsm to be an exact differential, we have that
∂ 2 sm
∂ 2 sm
=
,
∂P ∂T
∂T ∂P
(19)
∂ 2 sm
∂ 2 sm
=
,
∂Yi ∂T
∂T ∂Yi
(20)
∂ 2 sm
∂ 2 sm
=
.
∂Yi ∂P
∂P ∂Yi
(21)
Using the T -ds equation applied to the mixture gives
P
1
1 X
dT
dsm = cvm
+ d
+
Li dYi ,
T
T
ρm
T
(22)
i=l,g,v
where Li is the latent heat of phase change and is assumed to be a function of pressure and
temperature, i.e., Li = Li (P, T ).
¿From Eq. (5) we obtain the following relation
X Yi
1
=
,
ρm
ρi
(23)
i=l,g,v
and differentiation gives
d
1
ρm
=
X dYi
dρi
− Yi 2 .
ρi
ρi
(24)
i=l,g,v
Substituting the differential form of Eq. (9) into Eq. (24) results
X dYi
1
∂
1
∂
1
d
=
+ Yi
dP + Yi
dT .
ρm
ρi
∂P ρi
∂T ρi
(25)
i=l,g,v
Subsequent substitution of Eqs. (3) and (25) into Eq. (22) yields
X cvi Yi
P
∂
1
P
∂
1
P
Li
dsm =
+ Yi
dT +
Yi
dP +
+
dYi .
T
T ∂T ρi
T ∂P ρi
T ρi
T
i=l,g,v
(26)
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R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
Applying exact differential properties (18)-(21) to Eq. (26) results in the following constraints
X Yi ∂ρi
P ∂ρi
+
= 0,
(27)
ρ2i ∂T
T ∂P
i=l,g,v
∂
∂T
∂
∂P
Li
T
=
cvi
P
+ 2 ,
T
T ρi
=−
Li
T
1
.
T ρi
(28)
(29)
In order for Eq. (27) to be satisfied for arbitrary mass fractions, the expression inside of the
brackets must be equal to zero for each phase/component. Consequently, the constitutive
equation for each phase/component must be a function of the ratio of pressure and
temperature, i.e.,
P
ρi = ρi
.
(30)
T
Thus, to obtain an analytical solution for the mixture entropy, the density of each component
must be a function of the ratio of pressure and temperature. Because the gas and vapor are
assumed to follow the ideal-gas laws, see Eqs. (10) and (11), they automatically satisfy Eq.
(30). For the liquid, we propose to use the relation
ρl = ρo + α
P
,
T
(31)
where α = To /Cl2o , To is the reference temperature, Clo is the reference speed of sound, and
ρo is the reference density of the liquid. A liquid obeying Eq. (31) is called an idealized liquid
in this work. Equation (31) can be regarded as a model for a liquid described by Eq. (9). To
see this, note that the linearized model given in Eq. (12) can be approximated by Eq. (31),
provided that the temperature variations are small.
Combining Eqs. (10), (11), (31) and (23), the mixture density can be written as
ρm =
z
z
=
,
−1
zYl /ρl + Rg Yg + Rv Yv
zYl (ρo + αz) + Rg Yg + Rv Yv
(32)
where z = P/T . The mixture defined by Eq. (32) is called an idealized fluid mixture because
it is derived from the idealized liquid defined above and an ideal gas and vapor. Note that Eq.
(32) can also be interpreted as the equation of state of the mixture.
3.2. Mixture Entropy
Integrating Eqs. (28) and (29) from some reference state (P r , T r ) the following equation for
the latent heat of phase change is obtained
Li
T
Lr
= cvi ln r + Fi (z) + ir ,
T
T
T
where
Fi (z) = −
Z
z
zr
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1
dζ.
ρi (ζ)
(33)
(34)
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HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
Integrating Eq. (26) results in the following expression for the entropy of the mixture
X T
P Yi
P
Lr
+ Fi
+ ir Yi .
(35)
sm =
cvi Yi ln r +
T
T ρi
T
T
i=l,g,v
The expression (35) can be further simplified by using Eqs. (3) and (23)
X P Lr T
P
sm = cvm ln r +
+
Fi
+ ir Yi .
T
ρm T
T
T
(36)
i=l,g,v
Substituting equations of state (10), (11), (31) for the gas-vapor-liquid mixture into Eq. (36)
yields
z Y ρ
X Lr
Yl
Yl
ρl
T
l 0
i
−
ln
− a1 ln r −
+
Yi ,
(37)
sm = cvm ln r + a1 +
r
T
α
α
ρl
z
α ρl
Tr
i=l,g,v
where a1 = Rg Yg + Rv Yv .
Evaluating Eq. (37) at two different states and subtracting one from another, one obtains
[sm ]21
=
−
2 2
2 h
z i2
T
Yl
1
ρl
+
a
+
−
Y
ln
−
a
ln
1
l
1
Tr 1
α 1 α
ρrl 1
zr 1
2
X Lr
ρ0 Yl
2
i
+
[Yi ]1 ,
α ρl 1
Tr
cvm ln
(38)
i=l,g,v
where square brackets denote the following operation [(·)]21 = (·)2 − (·)1 .
If no mass transfer between the phases is present, i.e., the mass fractions are assumed
constant, Yi1 = Yi2 = Yi , Eq. (38) reduces to
Yl ρl2
z2
ρo Yl
1
1
T2
−
ln
− a1 ln
−
−
,
(39)
sm2 − sm1 = cvm ln
T1
α
ρl1
z1
α
ρl2
ρl1
For isentropic changes of state, Eq. (39) leads to
T2
=
T1
z2
z1
ca1 vm
ρl2
ρl1
αcYl
vm
exp
ρo Yl
αcvm
1
1
−
ρl2
ρl1
.
(40)
For the case of a pure gas, Eq. (40) reduces to
T2
=
T1
P2
P1
(γg −1)/γg
,
(41)
where γg = 1 + Rg /cvg . For the case of a pure liquid Eq. (40) reduces to
T2
=
T1
ρl2
ρl1
αc1
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vm
ρo
exp
αcvm
1
1
−
ρl2
ρl1
.
(42)
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R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
3.3. Speed of Sound
The speed of sound of the idealized fluid mixture can be obtained by applying Eqs. (10)-(31)
to Eq. (17), giving
φv
φg
φl
+
+
ρm cvm + P β 2
ρl Cl2
ρv Cv2
ρg Cg2
2
,
(43)
Cm
=
φl
φv
φg
+
+
ρ2m cvm
ρl Cl2
ρv Cv2
ρg Cg2
where Cl = (T /α)1/2 , Cg = (Rg T )1/2 , and Cv = (Rv T )1/2 are the isothermal speeds of
sound in the liquid, gas, and vapor, respectively, and β = βl = βg = βv = (P/T )1/2 is
the compressibility. Note that the compressibilities are identical for the idealized mixture.
Simplification yields
2
Cm
=
2
ρm cvm Cφ/ρ
+ P β2
ρ2m cvm
,
(44)
where
1
φl
φv
φg
=
+
+
.
2
Cφ/ρ
ρl Cl2
ρv Cv2
ρg Cg2
(45)
Equations (44) and (17) give practically identical results if the sound speeds of the idealized
and non-idealized liquid are matched at the reference temperature To .
3.4. Eigenvalues, Eigenvectors, and Riemann Invariants
For one-dimensional problems, the eigenvalues are given by
λ = (u − Cm , u, u, u, u + Cm )T ,
(46)
and the corresponding set of eigenvectors are

1
2
ρm Cm
P
− 2 2
ρm Cm cvm T







φg
2
2
Π=
 − ρg ρm C 2 C 2 ρg Cg − ρm Cm +
g m


φv

2
−
ρv Cv2 − ρm Cm
+

2
 ρv ρm Cv2 Cm

1
2
ρm Cm
−
!
βg2 P
ρm cvm
βv2 P
ρm cvm
1
Cm
0
0
0
1
T
0
0
0
1
0
0
0
1
Cm
0
0
0



0 




0 
.



1 


0
(47)
The Riemann invariants are computed from the relation dΥ = Π dK, where K =
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HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
[P, u, T, φg , φv ]T ,
dP
du
−
2
ρm Cm
Cm




P dP
dT

− 2 2
+


ρm Cm cvm T
T
!

2

β
P
φ
g
g
2
2
dΥ = 
 − ρg ρm C 2 C 2 ρg Cg − ρm Cm + ρm cvm dP + dφg
g m



φv
βv2 P
 −
2
2
ρv Cv − ρm Cm +
dP + dφv

2
ρv ρm Cv2 Cm
ρm cvm


dP
du
+
2
ρm Cm
Cm









.








(48)
4. Riemann Problem for Idealized Mixture
The Riemann problem is very important in understanding the wave structure of
the governing systems of hyperbolic partial differential equations and developing
numerical algorithms for solving these equations. The solutions of the Riemann
problem are composed of elementary waves. The structure and behavior of the
wave curves depend on the properties of the equation of state. For ideal fluids
satisfying the standard assumptions, the solution of Riemann problem consists of
shock wave, rarefaction wave, and contact discontinuity. Menikoff and Plohr [31]
examined the Riemann problem for fluid flow of real materials and described the
wave structure in fluids governed by the general equations of state allowed by
thermodynamics. It was shown in [31] that for real fluids, the class of elementary
waves includes composite and split waves in addition to shock and rarefaction
waves. Obtaining an exact solution of the Riemann problem for multiphase flow
using real fluids for the constitutive equations for each phase and/or component
is very difficult and complicated.
The Riemann problem is characterized by uniform initial conditions except for a
discontinuity at x = 0 on an infinite one-dimensional domain. The lack of an intrinsic length
or time scale means that the solution to the Riemann problem is self-similar. The solution of
the Riemann problem for scalar conservation laws, linear hyperbolic systems of equations, and
the single phase Euler equations can be derived, see, e.g., [32] and [33]. For this reason, the
Riemann problem is often used to verify numerical methods. In this section, we present the
solution of the Riemann problem for the idealized mixture.
We consider the initial conditions
QL if x < 0,
Q(x, 0) =
(49)
QR if x ≥ 0,
where Q = [ρm , ρm u, ρm em T , ρm Yg , ρm Yv ]T . The solution of the Riemann problem involves
expansion waves, shock waves, and a contact discontinuity. The structure of the solution is
shown in Fig. 3. The four regions with constant solutions are separated by five wave families.
The challenge in finding the solution of the Riemann problem lies in determining the unknown
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R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
states Q∗L and Q∗R to the left and right of the contact discontinuity, see Fig. 3. These regions
are referred to as the left and right star regions, respectively. The corresponding unknown
primitive variables are
∗
∗ T
K ∗L = [PL∗ , u∗L , TL∗ , YgL
, YvL
] ,
K ∗R
=
(50)
∗
∗ T
[PR∗ , u∗R , TR∗ , YgR
, YvR
] .
(51)
There are four possible wave patterns in the solution of the Riemann problem as shown in
Fig. 4, see, e.g, Toro [33]. These wave patterns are considered in constructing the exact solution.
The eigenstructure of the mixture formulation reveals that the pressure P ∗ and velocity u∗
are constant across the contact discontinuity, while other thermodynamic variables such as ρ∗m
and T ∗ are discontinuous. The unknown variables K∗L and K∗R are connected by the condition
that the pressure P ∗ and velocity u∗ are constant across the contact discontinuity. In the
following, detailed analyses of the conditions across the left shock wave and left rarefaction
wave, denoted by a subscript L, are presented. The conditions across the right shock and right
rarefaction wave can be obtained by replacing the subscript L by R.
4.1. Conditions Across Left Shock Wave
The left wave is assumed to be a shock wave moving with speed SL , see Fig. 4(b) and (d). The
∗
pre-shock variables are PL , uL , TL , YgL , and YvL . The post-shock variables are PL∗ , u∗L , TL∗ , YgL
,
∗
and YvL . The Rankine-Hugoniot conditions are applied across the left shock, leading to
ρmL uL − ρ∗mL u∗L = SL (ρmL − ρ∗mL ),
(ρmL u2L
(ρmL emL uL +
∗
+ PL ) − (ρ∗mL u∗2
L + PL )
PL uL ) − (ρ∗mL e∗mL u∗L + PL∗ u∗L )
∗ ∗
uL
ρmL YgL uL − ρ∗mL YgL
∗
∗ ∗
ρmL YvL uL − ρmL YvL uL
=
=
=
=
(52)
SL (ρmL uL − ρ∗mL u∗L ),
SL (ρmL emL − ρ∗mL e∗mL ),
∗
),
SL (ρmL YgL − ρ∗mL YgL
∗
∗
SL (ρmL YvL − ρmL YvL ).
(53)
(54)
(55)
(56)
Equations (52)-(56) can be solved to give
u∗L = uL − fL (PL∗ , ρ∗mL , QL ),
gL (PL∗ , ρ∗mL , TL∗ , QL ) = 0,
∗
YiL
∗
cvmL
(57)
(58)
= YiL ,
= cvmL ,
(59)
(60)
where
fL (PL∗ , ρ∗mL , QL ) =
(PL∗ − PL )(ρ∗mL − ρmL )
ρmL ρ∗mL
12
,
(61)
and
gL (PL∗ , ρ∗mL , TL∗ , QL )
= cvmL TL −
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c∗vmL TL∗
1
− (PL + PL∗ )
2
1
ρ∗mL
−
1
ρmL
.
(62)
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HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
4.2. Conditions Across Right Shock Wave
The analysis is analogous to that for the left shock wave. By replacing the subscript L by R,
the following conditions are obtained,
u∗R = uR − fR (PR∗ , ρ∗mR , QR ),
gR (PR∗ , ρ∗mR , TR∗ , QR )
∗
YiR
c∗vmR
(63)
= 0,
= YiR ,
(64)
(65)
= cvmR ,
(66)
where
fR (PR∗ , ρ∗mR , QR )
(PR∗ − PR )(ρ∗mR − ρmR )
=
ρmR ρ∗mR
12
,
(67)
and
1
gR (PR∗ , ρ∗mR , TR∗ , QR ) = cvmR TR − c∗vmR TR∗ − (PR + PR∗ )
2
1
ρ∗mR
−
1
ρmR
.
(68)
4.3. Conditions Across Left Rarefaction Wave
Let us assume that the left wave is a rarefaction wave, see Fig. 4(a) and (c). Then the unknown
state K∗L is connected to the known left state QL using the isentropic relation given by Eq.
(40) and the generalized Riemann invariants for the left wave. The Riemann invariant across
the left rarefaction is given by
dΥ1 = 0 =
du
dP
−
,
2
ρm Cm
Cm
(69)
from Eq. (48). Integration across the left rarefaction yields
u∗L = uL + fL (PL∗ , ρ∗mL , QL ),
(70)
where
fL (PL∗ , ρ∗mL , QL ) =
Z
∗
L
dP
.
ρm Cm
(71)
The integral in Eq. (71) is evaluated using adaptive Simpson quadrature [34] due to the
complexity of the integrand.
Similarly, from Eq. (38) we obtain for isentropic conditions the non-linear algebraic equation
gL (PL∗ , TL∗ , QL )
=
−
∗
∗
∗
L
Yl L
1
ρl
T L
cvm ln r
+ a1 +
−
Yl ln
T L
α L
α
ρrl L
∗
h
z i ∗
X Lr h i ∗L
ρ0 Yl L
L
i
a1 ln r
−
+
Yi
= 0,
z
α ρl L
Tr
L
L
(72)
i=l,g,v
h i
where square brackets denote the following operation (·)
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∗
L
L
= (·)∗L − (·)L .
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R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
4.4. Conditions Across Right Rarefaction Wave
The analysis is analogous to that for the left rarefaction wave, except that the Riemann
invariant across the right rarefaction is given by
dΥ5 = 0 =
dP
du
+
,
2
ρm Cm
Cm
(73)
from Eq. (48). By replacing the subscript L by R, the following conditions are obtained,
u∗R = uR − fR (PR∗ , ρ∗mR , QR ),
where
fR (PR∗ , ρ∗mR , QR ) =
Z
∗
R
(74)
dP
,
ρm Cm
(75)
and
gR (PR∗ , TR∗ , QR )
=
−
h
z i ∗
R
a1 ln r
z
R
T
cvm ln r
T
∗
R
R
∗
∗
R
Yl R
1
ρl
+ a1 +
−
Yl ln
r
α R
α
ρl R
∗
X Lr h i ∗R
ρ0 Yl R
i
+
= 0,
−
Yi
α ρl R
Tr
R
(76)
i=l,g,v
h i
where square brackets denote the following operation (·)
∗
R
R
= (·)∗R − (·)R .
4.5. Complete Solution
Now the conditions for all four possible wave patterns, shown in Fig. 4, can be determined.
The unknown states K∗L and K∗R can be computed by utilizing the condition that the pressure
and velocity are constant across the contact discontinuity, i.e.,
PL∗ = PR∗ = P ∗ ,
(77)
u∗L = u∗R = u∗ .
(78)
and
By eliminating u∗ from Eqs. (57) or (70) and (63) or (74), a single non-linear algebraic equation
is obtained,
fL (P ∗ , ρ∗mL , QL ) + fR (P ∗ , ρ∗mR , QR ) + uR − uL = 0.
(79)
Note that Eq. (79) has three unknowns, namely, P ∗ , ρ∗mL , and ρ∗mR . To close the problem,
Eqs. (58) and (72) for the state to the left of the contact discontinuity and Eqs. (64) and
(76) for the state to the right of the contact discontinuity are used, in addition to Eq. (32).
Therefore, we consider the following four cases.
1. If P ∗ > PL then a shock wave is traveling to the left and the function fL is given by Eq.
(61) supplemented by Eqs. (58) and (32) for the left star region.
2. If P ∗ ≤ PL then a rarefaction wave is moving to the left and the function fL is given by
Eq. (71) supplemented by Eqs. (72) and (32) for the left star region.
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HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
15
3. If P ∗ > PR then a shock wave is traveling to the right and the function fR is given by
Eq. (67) supplemented by Eqs. (64) and (32) for the right star region.
4. If P ∗ ≤ PL then a rarefaction wave is moving to the right and the function fR is given
by Eq. (75) supplemented by Eqs. (76) and (32) for the right star region.
The speed of the contact discontinuity can be determined from
u∗ =
1
1
(uL + uR ) + [fR (P ∗ , ρ∗mR , QR ) − fL (P ∗ , ρ∗mL , QL )] .
2
2
(80)
5. Numerical Method
The spatial and temporal discretization as described in [17, 18, 19] is adopted, except that the
equations being solved represent the mixture of liquid, gas, and vapor and that two additional
conservation equations are solved for the mass fractions of the gas and vapor. An extension
of the HLLC approximate Riemann solver of [20] is used to compute the convective fluxes of
the mixture as described below. The convective fluxes of the gas and vapor components are
computed as suggested by [13] to ensure positivity. The occurrence of pressure oscillations
near the material interface can be derived by calculating the two fluxes across
the material fronts [21]. Several approaches were reviewed in [21] to eliminate
these oscillations in multimaterial flow simulations. The face states required by the
flux computation are computed from a simplified WENO scheme [18]. The classical four-stage
Runge-Kutta method in low-storage formulation is used for the temporal discretization.
In the following, we give an outline of our extension of the HLLC approximate Riemann
solver of [20] to the mixture of liquid, gas, and vapor. The description below focuses on those
aspects critical to the extension and does not describe the HLLC method in any detail. For
an in-depth description of the HLLC method, see [35], [20], and [33].
The primary difficulty in extending the HLLC method to multiphase flows is due to the
treatment of the speed of sound. The speed of sound enters the HLLC method through the
computation of the wave speeds SL and SR , determined from
SL = min[qL − CmL , q̃ − C̃m ],
(81)
SR = max[qR + CmR , q̃ + C̃m ],
(82)
where qL , qR , CmL , and CmR are the face-normal velocities and the speeds of sound of the
mixture at the left and right state, respectively, and q̃ is the Roe-averaged [36] face-normal
velocity. In the HLLC method of [20] for single-phase flow, C̃m is computed from the the
constant ratio of specific heats and the Roe-averaged total enthalpy and velocities. For the
multiphase mixtures considered in this work, this is not possible because the speed of sound
given by Eq. (17) cannot be related to the total enthalpy in a straightforward fashion. Instead,
we propose to compute C̃m from Eq. (17) as
!2
X ρ̃m Ỹi β̃i
ρ̃m c̃vm + P̃
ρ̃2i
C̃i
i
2
,
(83)
C̃m =
X ρ̃m Ỹi 1
ρ̃2m c̃vm
ρ̃2i C̃i2
i
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R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
where, using the definition,
Rρ =
p
ρmR /ρmL ,
(84)
we have the usual Roe-averages of the mixture density
ρ̃m = Rρ ρmL ,
and mass fractions,
Ỹi =
(85)
YiL + YiR Rρ
,
1 + Rρ
(86)
and we define the Roe-averaged specific heat at constant volume of the mixture as
X
c̃vm =
Ỹi cvi .
(87)
i
In addition, we define, for lack of a better approach,
P̃ =
PL + PR Rρ
,
1 + Rρ
(88)
T̃ =
TL + TR Rρ
.
1 + Rρ
(89)
and
The remaining variables appearing in Eq. (83) are computed as ρ̃i = ρi (P̃ , T̃ ), β̃i = βi (P̃ , T̃ ),
and C̃i = Ci (P̃ , ρ˜i ).
6. Computational Results
In this section, we present numerical solutions obtained with our method for some benchmark
problems. We also present two novel benchmark problems for the idealized fluid-mixture model.
The problems considered are:
1. Single-phase two-component shock-tube problem (Gas/Gas) – tests the
accuracy with which shock waves and contact discontinuities are captured.
2. Shock-wave propagation in a single-phase two-component fluid (Gas/Gas) –
tests the accuracy of computing the shock wave refraction at a component interface.
3. Two-phase single-component shock-tube problem for idealized-fluid mixture
(Liquid/Gas) – tests the accuracy with which two-phase flows are solved if all solution
variables are discontinuous.
4. Two-phase single-component rarefaction problem for idealized fluid mixture
(Liquid/Gas) – tests the accuracy for low-density flows.
5. Single-phase two-component shock-bubble interaction (Gas/Gas) – demonstrates the ability to solve the interaction of a shock-wave and a material interface in
two dimensions.
For the first four problems, the accuracy is assessed by comparing the numerical solutions to
the appropriate exact solutions. For the fifth problem, the accuracy is evaluated by comparing
our results to the experiments of Haas and Sturtevant [37] and the simulations of Quirk and
Karni [38].
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HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
17
6.1. Single-Phase Two-Component Shock-Tube Problem (Gas/Gas)
The initial conditions correspond to two different ideal gases [10, 11, 13, 23, 39],
QL if x < 0.5,
Q(x, 0) =
QR if x ≥ 0.5,
(90)
T
where Q = [ρm , u, P, γm , Y1 , Y2 ] and
QL
QR
=
=
[ 1.000, 0.0, 1.0 · 105 , 1.4, 1.0, 0.0 ]T ,
[ 0.125, 0.0, 1.0 · 104 , 1.2, 0.0, 1.0 ]T .
(91)
The primary difficulty is the capturing of the contact discontinuity without oscillations. The
computational domain 0 ≤ x ≤ 1 is discretized uniformly with 1000 cells. Excellent agreement
between the numerical and exact solutions is obtained as shown in Fig. 5. In particular, our
results do not exhibit oscillations like the results presented in [23].
6.2. Shock-Wave Propagation in a Single-Phase Two-Component Fluid (Gas/Gas)
The problem suggested by [23] is considered, in which a shock wave refracts at a gas interface,
leading to a transmitted and a reflected shock wave. The transmitted shock wave may travel
faster or slower than the incident shock wave depending on the sound speeds of the respective
gases. The reflected wave is either a shock wave or a rarefaction wave depending on the ratio
of the acoustic impedances [40, 41]. The interface is set into motion by the shock wave.
The initial conditions correspond to a weak shock wave with a Mach number Ms = 1.1952
in air propagating toward a region occupied by helium,

 QA1 if 0.0 ≤ x < 0.25,
QA2 if 0.25 ≤ x < 0.5,
Q(x, 0) =
(92)

QHe if 0.5 ≤ x, ≤ 1.0
where Q = [ρm , u, P, γm , Y1 , Y2 ]T and
QA1
QA2
QHe
=
=
=
[ 1.7017, 98.956, 1.5 · 105 , 1.40, 1.0, 0.0 ]T ,
[ 1.2763, 0.000, 1.0 · 105 , 1.40, 0.0, 1.0 ]T ,
[ 0.1760, 0.000, 1.0 · 105 , 1.67, 0.0, 1.0 ]T .
(93)
The vectors QA1 , QA2 , and QHe correspond to the post-shock variables in air, pre-shock
variables in air, and pre-shock variables in helium, respectively. For these initial conditions,
the transmitted shock wave is very weak and the reflected wave is a slender rarefaction wave.
The computational domain 0 ≤ x ≤ 1 is discretized uniformly with 1000 cells.
Figure 6 shows the comparison of the numerical and exact solution at t ≈ 864 µs. As for
the first test problem, our results exhibit no oscillations like those presented by [23]. The
transmitted shock wave travels faster than the incident shock wave since the acoustic speed in
helium is greater than the acoustic speed in air.
6.3. Two-Phase Shock-Tube Problem for Idealized Fluid Mixture (Liquid/Gas)
The two-phase shock-tube problem of [42] is considered. The driver section contains a liquid
at high pressure and the driven section contains a gas at low pressure. The initial conditions
are
QL if x < 0.7,
Q(x, 0) =
(94)
QR if x ≥ 0.7,
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R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
T
where Q = [ρm , u, P, Y1 , Y2 ] and
QL
QR
= [ 1500.0, 0.0, 1.12 · 109 , 1.0, 0.0 ]T ,
= [
50.0, 0.0, 1.00 · 105 , 0.0, 1.0 ]T .
(95)
Note that the problem is very stiff: The density and pressure differ by ratios of 30 and about
104 across the discontinuity, respectively. The computational domain 0 ≤ x ≤ 1 is discretized
uniformly with 1000 cells.
Figure 7 shows the comparison of the numerical and exact solution at 240 µs. Small
oscillations in the density and velocity between the contact discontinuity and shock wave
are visible due to the proximity of the contact discontinuity and the shock wave. This was
also observed by [42]. However, our solution shows no oscillations at the tail of the rarefaction
wave.
6.4. Two-Phase Rarefaction Problem for Idealized Fluid Mixture (Liquid/Gas)
A two-phase rarefaction problem is considered. The solution consists of two symmetric
rarefaction waves and a trivial stationary contact discontinuity. The initial conditions are
QL if x < 0.5,
Q(x, 0) =
(96)
QR if x ≥ 0.5,
T
where Q = [ρl , φl , ρg , φg , u, P ] and
QL
QR
=
=
[ 1000.0, 0.1, 1.2342, 0.9, −200.0, 9.7 · 104 ]T ,
[ 1000.0, 0.1, 1.2342, 0.9, 200.0, 9.7 · 104 ]T .
(97)
The computational domain 0 ≤ x ≤ 1 is discretized uniformly with 1000 cells. Figure 8 shows
the comparison of the numerical and exact solution at t = 540 µs. No problems are encountered
despite the low pressure and density.
6.5. Single-Phase Shock-Bubble Interaction (Gas/Gas)
Haas and Sturtevant [37] presented experiments of a weak planar shock wave with Ms = 1.22
in air interacting with a cylindrical bubble filled with helium. The shock wave is transmitted
through the bubble and sets it into motion. The results presented below focus only on the early
stages of interaction and include a comparison with the experiments of Haas and Sturtevant
and the computations of Quirk and Karni [38].
Both air and helium are assumed to be perfect gases with the properties listed in Table ??.
As indicated by Haas and Sturtevant, the helium bubble is contaminated with air about
28% by mass [37]. The initial flow field is determined from the standard shock relations
given the strength of the incident shock wave and considering the density and pressure of
the quiescent flow ahead of the shock to be 1 kg/m3 and 105 Pa. The bubble is assumed to
be in thermodynamic and mechanical equilibrium with its surroundings. Therefore, its initial
density is given by ρHe = ρAir RAir /RHe where RAir and RHe are the gas constants for air and
helium, respectively.
The computational domain is shown in Fig. 9. Only the upper half is actually computed
because the flow is symmetric about the shock-tube axis. We have employed quadrilateral grids
with uniform resolutions of h = ∆x = ∆y of 224 × 10−6 m, 112 × 10−6 m, and 56 × 10−6
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19
HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
Experiment [37]
Computation [38]
Coarse grid
Medium grid
Fine grid
VS
410
422
426
426
423
EVS
−
+3.0
+3.9
+3.9
+3.2
VT
393
377
375
380
391
EVT
−
−4.1
−4.6
+3.3
−0.5
Vui
170
178
172
168
169
EVui
−
+4.7
+1.2
−1.2
−0.6
Vdi
145
146
136
137
140
EVdi
−
+0.7
−6.2
−5.5
−3.4
Table I: Comparison of computed velocities with those measured by Haas and Sturtevant [37]
and computed by Quirk and Karni [38]. Percentage errors with respect to the measurements
by Haas and Sturtevant are also shown. The notation is defined in Fig. 12.
m. The smallest grid spacing is comparable to that used on the finest refinement level in the
adaptive-grid computations of Quirk and Karni. (Of course, we do not advocate using grids
with uniform spacing for purposes other than focussed verification and validation studies.)
Referring to Fig. 9, solid-wall and symmetry conditions are applied to BC and AD, respectively.
Along AB, inflow conditions are specified using the conditions behind the incident shock wave.
An outflow condition is applied along CD.
Figure 10 shows a sequence of numerically generated Schlieren images, illustrating the
interaction of the shock wave with the bubble as computed on the finest grid. The computation
reproduces all the features of the interaction, and is in good agreement with the experiments
by Haas and Sturtevant and the simulations by Quirk and Karni.
Figure 10(a) shows the helium bubble at t ≈ 32 µs, after it is hit by the incident shock wave.
A curved refracted shock is generated inside the bubble. Since the helium has a higher speed
of sound than the surrounding air, the refracted shock wave travels faster than the incident
shock wave. Also, a weak expansion wave is reflected outside the bubble. The refracted shock
wave eventually emerges from the bubble to become the transmitted wave, see Fig. 10(b).
The incident shock diffracts at approximately 201 µs as shown in Fig. 10(c). The bubble is
deformed into a kidney shape and spreads laterally, see Fig. 10(d). The deformation is caused
by vorticity generated at the edge of the bubble due to the passage of the shock, which induces
a jet of air along the axis of symmetry [38]. Figure 10(e) shows the formation of the bubble
into a distinct vortical structure.
The accuracy of the numerical solutions are evaluated by conducting a convergence study.
Figure 11 shows the comparison of the results on the coarse, medium, and fine grids at
approximately 674 µs. The location of the bubble is approximately identical for the three
grids. As expected, the vortical structures are more pronounced on the fine grid.
To validate the numerical solutions, the computed velocities of some prominent flow features
as defined in Fig. 12 are compared to the measurements of Haas and Sturtevant [37] the
computations of Quirk and Karni[38]. The velocities are determined from a linear least-squares
curve fit of positions obtained from digitizing a sequence of numerically generated Schlieren
images. Table I lists the velocities and the percentage errors of our computed results relative
to the experimental values. The result are in good agreement with the measured values and
thus demonstrate the accuracy of our approach.
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20
R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
7. Conclusion
A new model for multiphase/multicomponent flows with an arbitrary number of components
in each phase has been successfully developed. The model does not require an ad hoc closure
for the variation of mixture density with regards to the attendant pressure and yields a
thermodynamically accurate value for the mixture speed of sound. The proposed numerical
method has been successfully implemented in an existing finite-volume framework. The HLLC
approximate Riemann solver, originally developed for single-phase single-component fluids, was
extended to multiphase multicomponent fluids and successfully used to capture shock waves
and contact discontinuities. A novel “idealized” fluid mixture model was developed, which
allows the derivation of an exact solution for the multiphase and multicomponent Riemann
problem in one dimension. A number of existing benchmark problems for single phase and
multicomponent flows become a subset of this new model.
The accuracy of the proposed numerical method and physical model were verified
and validated by solving a number of test problems. We have presented three classical
benchmark problems (single-phase two-component shock tube, shock-wave propagation in a
single-phase two-component fluid, and single-phase shock-bubble interaction) and two novel
benchmark problems for the idealized fluid-mixture model (two-phase shock-tube and twophase rarefaction problems). For all problems allowing an exact solutions, good agreement
between our numerical and the exact solutions was observed. For the case of single-phase
shock-bubble interaction problem, our numerical results were compared with the experiments
by [37] and the simulations by [38], and exhibited good agreement.
ACKNOWLEDGEMENTS
This work was sponsored by Argonne National Laboratory under grants number 2-RP50-P-00005-00,
3B-00061, 4B-00821, 5F-00462. This support is gratefully acknowledged. The authors also thank Dr.
Jin Wang for the fruitful discussion and support of this work. The third author was supported by the
Department of Energy through the University of California under subcontract number B523819.
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21
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HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
1400
PURE LIQUID
Speed of sound (m/s)
1200
1000
800
600
400
PURE AIR
200
0
0
0.2
0.4
0.6
0.8
1
0.8
1
Volume fraction of air
(a)
100
90
Speed of sound (m/s)
80
70
60
50
40
30
20
10
0
0
0.2
0.4
0.6
Volume fraction of air
(b)
Figure 1: Speed of sound of a water-air mixture at P = 1·105 Pa and T = 298.15K as a function
of the volume fraction of air. (a) Speed of sound predicted by Eq. (17). (b) Comparison of
predicted speed of sound with experimental data of Karplus [25] for frequencies of 1 kHz
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& Sons,
Int. J. Numer.
Meth. Fluids 2007; 00:1–6
(diamonds),
0.5 John
kHz Wiley
(squares)
andLtd.
extrapolated to zero frequency
(circles).
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R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
Figure 2: Speed of sound of a liquid-gas-vapor mixture predicted by Eq. (17) at P = 1 · 105
Pa and T = 298.15K.
t
u, u, u
u − Cm
*
KL
Q
u + Cm
*
KR
Q
L
0
R
x
Figure 3: Structure of the solution of the Riemann problem for the idealized-mixture
formulation.
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HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
t
01
101011111111
00000000
00000000
1011111111
00000000
101011111111
00000000000000000
11111111111111111
0000000000000000
1111111111111111
00000000
11111111
00000000000000000
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0000000000000000
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0000000000000000000
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00000000
11111111
000000000000000000
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00000000
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000000000000000000
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00000000
11111111
0
1
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00000000000000000
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0000000000000000000
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00000000
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0
1
000000000000000000
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00000000000000000
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000000000000000000000000000000000000000
0000000000000000
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0000000000000000000
00000000000000000001111111111111111111
1111111111111111111
x
00000000
10011111111
000000000000000000
111111111111111111
00000000000000000
11111111111111111
(a)
(b)
t
0110
0000000
1111111
10
0000000
1111111
10
0000000
1111111
1010
0000000
1111111
0000000
1111111
000000000000000000
111111111111111111
000000000000000000
111111111111111111
10111111111111111111
0000000
0000000000000000001111111
111111111111111111
000000000000000000
10111111111111111111
0000000
1111111
000000000000000000
111111111111111111
000000000000000000
10111111111111111111
0000000
1111111
000000000000000000
111111111111111111
10111111111111111111
0000000
1111111
000000000000000000000000000000000000
111111111111111111
000000000000000000
10
01
1011111111
00000000
1000000000
00000000000001011111111
1111111111111
00000000
11111111
1011111111
0000000000000
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00000000
0000000000000000
1111111111111111
1011111111111111111
000000000000000
111111111111111
0000000000000
1111111111111
00000000000000000
00000000
0000000000000000
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1011111111
000000000000000
111111111111111
0000000000000
1111111111111
00000000000000000
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00000000000000000
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00000000
0000000000000000
1111111111111111
1011111111
000000000000000
111111111111111
0000000000000
1111111111111
00000000000000000
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00000000000000000
11111111111111111
00000000
11111111
0000000000000000
1111111111111111
1011111111
000000000000000
111111111111111
0000000000000
1111111111111
00000000000000000
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00000000000000000
00000000
0000000000000000
1111111111111111
1011111111111111111
000000000000000
111111111111111
0000000000000
1111111111111
00000000000000000
11111111111111111
111111111111111111111111111111111111111
000000000000000000000000000000000000000
00000000000000000
11111111111111111
x
00000000
0000000000000000
1111111111111111
100
000000000000000
111111111111111
000000000000011111111
1111111111111
00000000000000000
11111111111111111
0110
0000000
1111111
101111111
0000000
101111111
0000000
101111111
00000000000000000
11111111111111111
0000000
101111111
00000000000000000
11111111111111111
00000000000000000
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0000000000000000
1111111111111111
0000000
000000000000000000
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101111111
00000000000000000
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00000000000000000
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00000000000000000
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0000000
000000000000000000
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101111111
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0000000000000000000
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00000000000000000
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00000000000000000
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0000000
000000000000000000
111111111111111111
101111111
00000000000000000
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0000000000000000000
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00000000000000000
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000000000000000000
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00000000000000000
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000000000000000000
111111111111111111
101111111111111111111
00000000000000000
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00000000000000000
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000000000000000000
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00000000000000000
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101111111111111111111
00000000000000000
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0000000000000000000
00000000000000000
11111111111111111
00000000000000001111111
1111111111111111
0
(c)
t
t
x
0
(d)
x
Figure 4: Four possible wave patterns of the solution of the Riemann problem [33]: (a) left
rarefaction, contact, and right shock; (b) left shock, contact, and right rarefaction; (c) left
rarefaction, contact, and right rarefaction; (d) left shock, contact, and right shock.
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R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
(a) Density.
(b) Pressure.
(c) Velocity.
(d) Temperature.
(e) Mach number.
(f) Speed of sound.
Figure 5: Comparison of numerical (◦) and exact (solid line) solutions for single-phase twocomponent shock-tube problem at t ≈ 517 µs. Number of cells = 1000.
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HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
(a) Density.
(b) Pressure.
(c) Mass fraction of helium.
(d) Temperature.
(e) Mach number.
(f) Speed of sound.
27
Figure 6: Comparison of numerical (◦) and exact (solid line) solution for shock-wave
propagation in a single-phase two-component fluid at t ≈ 864 µs. Number of cells = 1000.
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R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
(a) Density.
(b) Pressure.
(c) Mass fraction of gas.
(d) Velocity.
Figure 7: Comparison of numerical (◦) and exact (solid line) solution for two-phase shock-tube
problem for an idealized fluid mixture at 240 µs. Number of cells = 1000.
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HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
(a) Density.
(b) Pressure.
(c) Volume fraction.
(d) Velocity.
Figure 8: Comparison of numerical (◦) and exact (solid line) solution for two-phase rarefaction
problem for an idealized fluid mixture at 540 µs. Number of cells = 1000.
0.0000
B
A
C
Incident Shock
y[m]
0.0445
Cylindrical
D
Bubble
0.050
−0.0445
0.000
0.050
0.300
0.085
x[m]
Figure 9: Schematic diagram of the computational domain (not drawn to scale).
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R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
(a) 32 µs.
(b) 89 µs.
(c) 201 µs.
(d) 400 µs.
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(e) 674 µs.
Figure 10: Numerically generated Schlieren images of the shock-bubble interaction
computations on the fine grid at various times.
HOMOGENEOUS EQUILIBRIUM MIXTURE MODEL
31
(a) Fine grid.
(b) Medium grid.
(c) Coarse grid.
Figure 11: Numerically generated Schlieren images of the shock-bubble interaction
computations at approximately 674 µs for different grid spacings.
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R. S. LAGUMBAY, O. V. VASILYEV, A. HASELBACHER
Figure 12: Definition of locations at which velocities are measured. VS - incident shock, VT transmitted shock, Vui - upstream edge of bubble, and Vdi - downstream edge of bubble.
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