A compressible single-temperature conservative two-phase model with phase transitions

Abstract

A model for multidimensional compressible two-phase flow with pressure and velocity relaxations based on the theory of thermodynamically compatible system is extended to study liquid-€“gas flows with cavitation. The model assumes for each phase its own pressure and velocity, while a common temperature is considered. The governing equations form a hyperbolic system in conservative form and are derived through the theory of a thermodynamically compatible system. The phase pressure-equalizing process and the interfacial friction are taken into account in the balance laws for the volume fractions of one phase and for the relative velocity by adding two relaxation source terms, while the phase transition is introduced into the model considering in the balance equation for the mass of one phase the relaxation of the Gibbs free energies of the two phases. A modification of the central finite-volume Kurganovâ-Noelle-Petrova method is adopted in this work to solve the homogeneous hyperbolic part, while the relaxation source terms are treated implicitly. In order to investigate the effect of the mass transfer in the solution, a 1D cavitation tube problem is presented. In addition, two 2D numerical simulations regarding cavitation problem are also studied: a cavitating Richtmyer-€“Meshkov instability and a laser-induced cavitation problem.

Full text

This is the peer reviewed version of the following article: La Spina, G., de' Michieli
Vitturi, M. and Romenski, E. (2014), A compressible single-temperature conservative
two-phase model with phase transitions, Int. J. Numer. Meth. Fluids, 76, pages 282–
311, which has been published in final form at http://dx.doi.org/10.1002/fld.3934. This
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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
Int. J. Numer. Meth. Fluids 2014; 00:2–??
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld
A compressible single temperature conservative two-phase model
with phase transitions
G. La Spina1,2∗, M. de’ Michieli Vitturi2, E. Romenski3
1University of Pisa, Dipartimento di Matematica L.Tonelli, Italy
2Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Pisa, Italy
3Russian Academy of Sciences, Sobolev Institute of Mathematics, Russia
SUMMARY
A model for multidimensional compressible two-phase flow with pressure and velocity relaxations based on
the theory of thermodynamically compatible system is extended to study liquid-gas flows with cavitation.
The model assumes for each phase its own pressure and velocity, while a common temperature is considered.
The governing equations form a hyperbolic system in conservative form and are derived through the theory
of thermodynamically compatible system. The phase pressure equalizing process and the interfacial friction
are taken into account in the balance laws for the volume fractions of one phase and for the relative velocity
by adding two relaxation source terms, while the phase transition is introduced into the model considering
in the balance equation for the mass of one phase the relaxation of the Gibbs free energies of the two phases.
A modification of the central finite-volume Kurganov-Noelle-Petrova method is adopted in this work to
solve the homogeneous hyperbolic part, while the relaxation source terms are treated implicitly. In order
to investigate the effect of the mass transfer in the solution, a 1D cavitation tube problem is presented.
In addition, two 2D numerical simulations regarding cavitation problem are also studied: a cavitating
Richtmyer–Meshkov instability and a laser-induced cavitation problem. Copyright c
2014 John Wiley &
Sons, Ltd.
Received ...
KEY WORDS: Compressible flow; Finite volume; Hyperbolic; Partial differential equations; Two-phase
flows; Phase change
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2G. LA SPINA ET AL.
1. INTRODUCTION
Multiphase flow modeling is one of the most challenging research areas in computational and
applied mathematics and intensive efforts have been done in the recent decades in the development
of advanced models and numerical methods. Common formulations of mathematical models of
multiphase flows are originated in [1,2] and are based on the averaging theory. The governing
equations of such models consist of the balance equations for density, momentum and energy of
each phase, and phase interaction is taken into account by additional algebraic and differential
source terms. The model proposed by Baer and Nunziato [2] and studied in [3] stimulated further
modifications of the governing equations and the development of new approaches in the modeling of
two-phase flows, see for example [4,5,6]. In these papers and references therein the mathematical
properties of differential equations of two-phase flow models were studied and a variety of test
problems have been solved numerically. It is proved that the differential equations of the model
are hyperbolic that guarantee a solvability of certain classes of initial-boundary-value problems.
Some disadvantage of the Baer-Nunziato-type models is that not all equations can be written in a
conservative (divergent) form. Note that the divergent form of equations is very attractive because
it allows to apply known mathematical tools and accurate numerical methods to study problems of
practical interest.
Another, phenomenological approach, based on the theory of thermodynamically compatible
system of conservation laws was proposed in [7,8] to formulate the governing equations of
multiphase compressible flow. In this method the mixture is supposed as a continuum in which a
multiphase character of a flow is taken into account by introducing new phenomenological variables
beyond the classical density, momentum and energy. The resulting governing equations form a
hyperbolic system of equations in conservative form and these equations can be rewritten as a set of
subsystems of conservation laws for each phase coupled by the interface exchange terms. In general
∗Correspondence to: Giuseppe La Spina, Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Pisa, via della
Faggiola, 32, Pisa, 56126, Italy. E-mail: giuseppe.laspina@pi.ingv.it
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 3
the two-phase flow thermodynamically compatible equations differ from the Baer-Nunziato-type
equations, but for simplest case of one-dimensional isentropic flow they coincide [7].
Multiphase flow problems are very challenging even in the case of two-phase flow. That is why
many papers are devoted to the development of simplified models which are designed introducing
additional constraints [9,10]. These constraints usually represent an assumption that the rate of
unsteady-state relaxation to the equilibrium state is very high as compared with the characteristic
time of the process under consideration. As a rule these constraints consist in the requirement that
the pressures, velocities or temperatures must be equal in the two phases. In recent years two-phase
models with gas-liquid transition have been developed in [11], where phase change is introduced
with the use of discontinuities derived from the conservation laws. This approach requires a detailed
knowledge of properties of discontinuous solution and its numerical realization is quite complicated.
Actually the phase transition in two-phase model can also be accounted as an irreversible process
with high rate of relaxation. In [12] a simplified two-phase model is proposed in which an equality
of phase Gibbs potentials is proposed in addition to the phase velocities, pressures and temperatures
equalities. In this paper the kinetic model of phase transition with finite rate is proposed by adding
a source term to the phase mass balance equations with phase Gibbs potentials relaxation to the
common value. The similar approach, for example, is used in [13] for the description of continuous
phase transition in polytetrafluoroethylene.
In the present paper phase transition is introduced in the thermodynamically compatible system
of conservation laws proposed in [14,15] for two-phase flow with single temperature, but different
pressures and velocities. Although the temperature equalizing time is larger than the pressure and
velocity relaxation times, as noted in [16], it is possible to consider a single temperature model as
an approximate model of flow if the difference of phase temperatures is not too big. Mathematical
properties of the governing equations are studied and three test problems are solved numerically.
The effect of the finite rate of pressure relaxation on the pressure wave in two-phase flow is
investigated and it is shown that the solutions are significantly different for different rates. A 1D
cavitation problem, as it is formulated in [12], is considered here with liquid-vapor evaporation. The
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4G. LA SPINA ET AL.
evaporation is modeled by the nonlinear kinetics in the phase mass balance equations. It is shown
that the flow character strongly depends on the nonlinearity of kinetic coefficient as a function
of temperature and pressure. We present also 2 multidimensional numerical tests regarding the
cavitation problem: a cavitating Richtmyer–Meshkov instability simulation [17,18] and a laser-
induced cavitation problem [19].
For solving the proposed governing equations a finite volume numerical scheme based on the
MUSCL-Hancock method with linear reconstruction [20] is adopted. The conservative form of the
equations allows the use of central methods to compute the fluxes [21,22], and in this paper the
modification of the Kurganov, Noelle and Petrova numerical fluxes in conjunction with a second-
order reconstruction presented in [15] is adopted.
This paper is organized as follows: in Section 2 we present the governing equations and we derive
an entropy balance for the system. In Section 3 the equations of state used as closure constitutive
relations are described. Section 4 is devoted to the characteristic analysis of the system of equation
and more details on the evaluation of the eigenvalues are given in Appendix A. Section 5 describes
first the modified central finite-volume scheme adopted and the details of the integration of the
source terms, and finally the numerical results are presented for three test-cases.
2. CONSERVATIVE EQUATIONS FOR TWO-PHASE COMPRESSIBLE MODEL
2.1. Single temperature governing equations
In this section we present the extension to liquid-gas flows with cavitation of the two-phase flow
model with two pressures, two velocities and a single temperature, which has been proposed in [14]
and studied in [15]. These equations are derived using the principles of extended thermodynamics
[23,24,25,26] and the resulting system of partial differential equations can be written in
conservative form and transformed into a symmetric hyperbolic system in a similar way to those
described in [14,8]. Note that symmetric hyperbolic systems form a special class of hyperbolic
systems in the sense of Friedrichs [27], for which the initial value problem is well-posed and a deep
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 5
mathematical theory is developed, see for example [28] and references therein. Here we extend
the model considering also phase transition and the new entropy balance is derived in the next
subsection.
In the two-phase model, each phase can be characterized by its own parameters of state. The
phase volume fractions in the mixture are denoted with α1and α2, and they satisfy the saturation
constraint α1+α2= 1. Phase mass densities are ρ1, ρ2, while the velocities are ~u1and ~u2.
We introduce also the notation sifor the phase specific entropies and eifor the specific internal
energy and we assume that the equations of state for each phase eiare known function of the density
ρiand entropy si:
ei=ei(ρi, si).(1)
The phase pressures piand the temperatures Tiare computed from the equations of state as
follows:
pi=ρ2
i
∂ei
∂ρi
, Ti=∂ei
∂si
.(2)
Finally, the following mixture parameters of state, connected with the individual phase
parameters, are introduced:
T=T1=T2ρ=α1ρ1+α2ρ2, c =c1=α1ρ1
ρ
~u =c1~u1+c2~u2, ~w =~u1−~u2, e =c1e1+c2e2,
E=e+c(1 −c)|~w|2
2,
(3)
where Tis the common temperature, ρis the mixture density, cis the mass fraction of the first phase,
~u is the mixture velocity, ~w is the relative velocity, eis the specific internal energy of the mixture,
c2is the mass fraction of the second phase and Eis the total specific energy of the mixture.
Using this notation, we can write the mixture mass conservation law:
∂
∂t (ρ) + ∇ · (ρ~u) = 0.(4)
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6G. LA SPINA ET AL.
The balance law for the volume fraction of the first phase is:
∂
∂t (ρα1) + ∇ · (ρα1~u) = −φ, (5)
where φis the pressure relaxation defined as a function of the pressures difference and a relaxation
rate τ(p)by
φ=1
τ(p)(p2−p1).(6)
This term represents the phase pressure equalizing through the process of pressure wave
propagation in dispersed phase and its interaction with the interface boundaries.
The mass conservation law for the first phase is:
∂
∂t (ρ1α1) + ∇ · (ρ1α1~u1) = −ρθ, (7)
where θis defined as
θ=1
τ(c)µ1−µ2+ (1 −2c)|~w|2
2.(8)
In the previous equation, τ(c)is a phase exchange relaxation rate, ~w is the relative velocity, µ1
and µ2are the phase chemical potentials (or Gibbs free energies):
µ1=e1+p1
ρ1−s1T, µ2=e2+p2
ρ2−s2T. (9)
The mixture momentum equation is:
∂
∂t (ρ~u) + ∇ · [~u1⊗(ρ1α1~u1) + ~u2⊗(ρ2α2~u2) + (α1p1+α2p2)I] = 0,(10)
where Iis the three dimensional identity matrix and ⊗is the tensor product.
The relative velocity equation in conservative form is:
∂
∂t (~w) + ∇ · ~u1⊗~u1
2−~u2⊗~u2
2+e1+p1
ρ1−e2−p2
ρ2−(s1−s2)TI=
=−1
ρ~
λ0−~u ×~ω,
(11)
where the interficial friction ~
λ0is defined as a function of the interfacial friction coefficient ζ, the
mass fraction cand the relative velocity ~w:
~
λ0=ζc(1 −c)~w. (12)
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 7
In the case of dispersed particle flow the velocity relaxation represents the Stokes drag force in
the phase momentum equations.
As done in [8], to save a conservation-law form of the system, in the equation (11) it is introduced
an artificial vector ~ω satisfying the steady state equation:
∇ × ~w =~ω. (13)
Furthermore, this variable can be treated as a source terms in the equation for the relative velocity
[14].
The conservation energy law is written as:
∂
∂t ρE+|~u|2
2+∇ · α1ρ1~u1e1+|~u1|2
2+p1
ρ1+
+α2ρ2~u2e2+|~u2|2
2+p2
ρ2−ρc(1 −c)~w(s1−s2)T= 0.
(14)
To summarize, introducing the vector of conservative variables q, defined as
q=



















α1ρ1+ (1 −α1)ρ2
α1(α1ρ1+ (1 −α1)ρ2)
α1ρ1
α1ρ1~u1+ (1 −α1)ρ2~u2
~w
ρE+|~u|2
2



















(15)
the system of governing partial differential equations can be written in the more compact form:
∂q
∂t +∇ · F(q) = Ω(q),(16)
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8G. LA SPINA ET AL.
where F(q)are the conservative fluxes
F(q) =





















ρ~u
ρα1~u
ρ1α1~u1
~u1⊗(ρ1α1~u1) + ~u2⊗(ρ2α2~u2) + (α1p1+α2p2)I
~u1⊗~u1
2−~u2⊗~u2
2+e1+p1
ρ1−e2−p2
ρ2−(s1−s2)TI
2
X
i=1 αiρi~uiei+|~ui|2
2+pi
ρi−ρc(1 −c)~w(s1−s2)T





















,(17)
and Ω(q)are the relaxation terms
Ω(q) =




















0
−1
τ(p)(p2−p1)
−1
τ(c)ρµ1−µ2+ (1 −2c)|~w|2
2
0
−ζc(1 −c)
ρ~w−~u ×~ω
0




















.(18)
Equations (1–3),(9) and (13) represent the closure of the partial differential equations system
(16–18).
Finally we observe that, after some cumbersome transformation, it is possible to derive for each
phase a momentum balance equation. For the sake of simplicity, we present here the equations for
the 1D model:
∂α1ρ1u1
∂t +∂α1ρ1u2
1+α1p1
∂x =(α2ρ2p1+α1ρ1p2)
ρ∂
∂x α1−
−α1ρ1α2ρ2
ρ(s2−s1)∂
∂x T−ρθu1+α1ρ1θ(u1−u2)−α1ρ1α2ρ2
ρ2λ0.
(19)
∂α2ρ2u2
∂t +∂α2ρ2u2
2+α2p2
∂x =(α2ρ2p1+α1ρ1p2)
ρ∂
∂x α2−
+α1ρ1α2ρ2
ρ(s2−s1)∂
∂x T+ρθu2+α2ρ2θ(u1−u2) + α1ρ1α2ρ2
ρ2λ0.
(20)
We observe that, with respect to other models based on the approach of Baer and Nunziato [2],
the definition of interfacial pressure is different. As discussed in [7], while in the Baer-Nunziato-
type models we have pI=α1p1+α2p2, here the interfacial pressure in the momentum balance
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 9
equations for the two phases is expressed as
ˆpI=α2ρ2p1+α1ρ1p2
ρ.(21)
An analogous dual relation was also proposed in [29] for the studying Richtmyer–Meshkov
instability. In [30], the definition of the interface pressure was defined in terms of impedances of
each phase. But the same duality was present.
2.2. Entropy balance
We show here that from the system of equations (4–14) it is possible to write an equation for the
balance of the total mixture entropy S, defined as
S=c1s1+c2s2.(22)
From now on, for simplicity, we only consider the monodimensional case and, in this way, the
steady-state equation (13) and the term ~u ×~ω in the relative velocity equation can be neglected,
obtaining the following system:
∂
∂t (ρ) + ∂
∂x (ρu) = 0,(23)
∂
∂t (ρα1) + ∂
∂x (ρα1u) = −φ, (24)
∂
∂t (ρ1α1) + ∂
∂x (ρ1α1u1) = −ρθ, (25)
∂
∂t (ρu) + ∂
∂x ρ1α1(u1)2+ρ2α2(u2)2+α1p1+α2p2= 0,(26)
∂
∂t (w) + ∂
∂x (u1)2
2−(u2)2
2+e1+p1
ρ1−e2−p2
ρ2−(s1−s2)T=−1
ρλ0,(27)
∂
∂t ρE+u2
2+∂
∂x α1ρ1u1e1+(u1)2
2+p1
ρ1+
+α2ρ2u2e2+(u2)2
2+p2
ρ2−ρc(1 −c)w(s1−s2)T= 0.
(28)
To derive the entropy balance equation, first we have to write an equivalent nonconservative form
of the equations (23–28).
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10 G. LA SPINA ET AL.
Using the notation for the total energy of the mixture Ereported in Eq. (3), it is possible to write
an equivalent nonconservative form of the equations (23–28) for the variables (α1, c, ρ, u, w, E):
dα1
dt =−φ
ρ,
ρdc
dt +∂
∂x (ρc(1 −c)w) = −ρθ,
dρ
dt +ρ∂u
∂x = 0,
ρdu
dt +∂p
∂x +∂
∂x (ρc(1 −c)w2) = 0,
dw
dt +w∂u
∂x +∂
∂x (1 −2c)w2
2+µ1−µ2=−1
ρλ0,
ρdE
dt +p∂u
∂x +ρc(1 −c)w2∂u
∂x
+∂
∂x ρµ1−µ2+ (1 −2c)w2
2(c(1 −c)w)= 0,
(29)
where we used the notation d/dt for the material derivative
d
dt =∂
∂t +u∂
∂x .(30)
Now, using the formula
dE =∂E
∂α1
dα1+∂E
∂ρ dρ +∂E
∂c dc +∂E
∂w dw +∂E
∂S dS,
we find for the material derivative of the total mixture entropy
dS
dt =1
ESdE
dt −Eα1
dα1
dt −Eρ
dρ
dt −Ec
dc
dt −Ew
dw
dt .(31)
The partial derivatives of E, appearing in the previous equation, can be easily evaluated from the
definition of Ereported in Eq. (3), rewriting first the specific internal energy of the mixture ein
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 11
terms of the mixture parameters (α1, ρ, c, S):
Ew=c(1 −c)w,
Ec=∂
∂c e(α1, ρ, c, S ) + (1 −2c)w2
2=µ1−µ2+ (1 −2c)w2
2,
Eα=p1−p2
ρ,
Eρ=1
ρ2(α1p1+α2p2),
ES=T.
Using (29) and the previous expressions, we can rewrite (31) as
dS
dT =1
T−p
ρ
∂u
∂x −wEw
∂u
∂x −1
ρ
∂ρEcEw
∂x +Eα
φ
ρ
+p
ρ
∂u
∂x +Ec
ρ
∂ρEw
∂x +E2
c
τ(c)+Eww∂u
∂x +Ew
∂Ec
∂x +Ew
ρλ0.
(32)
From this equation, after some cancellation, we obtain the desired entropy balance law as
dS
dt =1
TEα
φ
ρ+E2
c
τ(c)+Ew
ρλ0=
=1
T"(p1−p2)2
ρτ(p)+1
τ(c)µ1−µ2+ (1 −2c)w2
22
+c2(1 −c)2w2
ρζ#.
(33)
We observe that the right hand side in the above equation, which represents the entropy production,
is always a non-negative quantity.
3. EQUATIONS OF STATE
The equations of state we employ in the model for gas-liquid flows, obtained from a linearized form
of the Mie-Gr¨
uneisen equations [31,32,11,14], are presented here.
The expression for the specific energy is analogous to the expression presented in [14], with
the difference of the presence of two constants ¯eiand s0,i, introduced here in order to ensure the
thermodynamic equilibrium in phase-transition problems when the chemical potentials of the two
phases are equal [32], and the additional parameter p0,i denoting a reference pressure for the i−th
phase.
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12 G. LA SPINA ET AL.
With the introduction of these parameters, the specific energy for the i−th phase can be written
in the following way:
ei(ρi, si) = ¯ei+cv,iT0,i ρi
ρ0,i γi−1
exp si−s0,i
cv,i +ρ0,iC2
0,i −γip0,i
γiρi
,(34)
where ¯eiis a constant parameter representing the formation energy of the fluid, cv,i is the specific
heat capacity at constant volume, T0,i is the reference temperature, ρ0,i is the reference density, s0,i
is the entropy at the reference state (ρ0,i, T0,i ),C0,i is a reference sound of speed at temperature T0,i
and γiis the adiabatic exponent.
While for a liquid phase the reference pressure p0,i is given as an independent parameter, for a
gas phase it is defined as
p0,i =ρ0,iC2
0,i
γi
,(35)
in order to ensure that the last term of Eq. (34) cancels when dealing with gases.
Now, using the relationships given by Eq. (2), we obtain the following expressions for the pressure
and temperature of the two phases:
pi(ρi, si) = (γi−1)ρ0,icv,iT0,i ρi
ρ0,i γi
exp si−s0,i
cv,i −ρ0,iC2
0,i −γip0,i
γi
,
Ti(ρi, si) = T0,i ρi
ρ0,i γi−1
exp si−s0,i
cv,i .
Furthermore, combining the pressure equation above with Eq. (34), the pressure for both the
phases can be written as a function of the density and the specific internal energy:
pi(ρi, ei) = (γi−1)ρi(ei−¯ei)−ρ0,iC2
0,i −γip0,i.(36)
This expression has the same form of the equation of state utilized in [32] for the stiffened gas
approximation, and it highlights the contribution of ¯eito the term (γi−1)ρi(ei−¯ei)in the phase
pressure, representing a repulsive effect present in all the media (gas, liquid and solid) due to the
molecular agitation.
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 13
We also observe that it is possible to write the energy and the pressure of the two phases in a more
convenient form as functions of the densities ρiand the common temperature Tas:
ei(ρi, T ) = ¯ei+cv,iT+ρ0,i C2
0,i −γip0,i
γiρi
,(37)
pi(ρi, T ) = cv,i (γi−1)ρiT−ρ0,iC2
0,i −γip0,i
γi
,(38)
si(ρi, T ) = s0+cv,i ln "T
T0,i ρ0,i
ρiγi−1#.(39)
Introducing the notation ¯pifor the limit of the pressure of the i−th phase at absolute zero
(T= 0K):
¯pi=ρ0,iC2
0,i −γip0,i
γi
,(40)
we obtain the more compact form for the specific energy and the pressure:
ei(ρi, T ) = ¯ei+cv,iT+¯pi
ρi
,(41)
pi(ρi, T ) = cv,i (γi−1)ρiT−¯pi.(42)
The negative term −¯piin the equation for the pressure represents the effects of the molecular
attraction guaranteeing the cohesion in the liquid and solid phases, and it is null for gas phases.
Using the expressions above for the specific energy, the pressure and the specific entropy as
functions of the density and the temperature, we find the following expression for the chemical
potentials µi:
µi= ¯ei−T cv,i s0,i
γi−γi+ ln "T
T0,i ρ0,i
ρiγi−1#! (43)
or, in terms of the reference sound speed C0,i instead of the reference temperature T0,i:
µi= ¯ei−T cv,i s0,i
γi−γi−ln "T cviγi(γi−1)
C2
0,i ρ0,i
ρiγi−1#!.(44)
4. CHARACTERISTIC ANALYSIS
In order to find the eigenvalues of the hyperbolic system of equations (23–28), we find an equivalent
quasilinear form
∂v
∂t +B(v)∂v
∂x =Z(v),(45)
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14 G. LA SPINA ET AL.
where vis the vector of primitive variables
v= (α1, S, ρ1, ρ2, u1, u2)T.(46)
We remark that with this choice of the vector v, for 0< α1<1and ρi>0, it is easy to show that
there exists a bijection Φ : v→q.
Now, from the system of equations (16) and the entropy equation (33), the following non-
conservative system of equations for the primitive variables can be obtained:
∂α1
∂t +u∂α1
∂x = 0,
∂S
∂t +u∂S
∂x =1
Tτ(p)Ω2
1
ρ+τ(c)Ω2
2
ρ2+ρΩ2
5
ζ,
∂ρ1
∂t +α2ρ2ρ1
α1ρ(u1−u2)∂α1
∂x +u1
∂ρ1
∂x +ρ1
∂u1
∂x =−ρ1
α1ρΩ1+Ω2
α1
,
∂ρ2
∂t +α1ρ1ρ2
α2ρ(u1−u2)∂α1
∂x +u2
∂ρ2
∂x +ρ2
∂u2
∂x =ρ2
α2ρΩ1−Ω2
α2
,
∂u1
∂t +u1
∂u1
∂x +1
ρ1
∂p1
∂x +p1−p2
ρ
∂α1
∂x −c2(s1−s2)∂T
∂x =
α2ρ2
ρΩ5−u1−u2
ρΩ2,
∂u2
∂t +u2
∂u2
∂x +1
ρ2
∂p2
∂x +p1−p2
ρ
∂α1
∂x +c1(s1−s2)∂T
∂x =
−α1ρ1
ρΩ5−u1−u2
ρΩ2.
(47)
In order to have the system in the desired form (45), we have to express the derivatives ∂pi/∂x
and ∂T /∂x as functions of the derivatives of the primitive variables with respect to x. We observe
that because of the equal phase temperatures, one can derive phase entropies sias a function
of volume fraction α1, phase densities ρiand mixture entropy S(i.e. we can explicitely write
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 15
s1= ˜s1(α1, ρ1, ρ2, S)and s2= ˜s2(α1, ρ1, ρ2, S)) by solving the system of equations
T1=∂(e1(ρ1, s1))
∂s1
=T, T2=∂(e2(ρ2, s2))
∂s2
=T, c1s1+c2s2=S. (48)
From the first two equations, solving for s1and s2, we can find s1=s1(ρ1, T )and s2=s2(ρ2, T ).
Now, substituting in the last equation and solving for T, we can find
T=e
T(α1, ρ1, ρ2, S).(49)
Then, we can define ˜s1and ˜s2as
˜s1=s1(ρ1,e
T),˜s2=s2(ρ2,e
T),
and, from the equations of state, we can write the phase pressures and the common temperature as
˜pi(α1, ρ1, ρ2, S) = pi(ρi,˜si),
e
T(α1, ρ1, ρ2, S) = Ti(ρi,˜si),
and their derivatives with respect to xas
∂T
∂x =∂e
T
∂α1
∂α1
∂x +∂e
T
∂ρ1
∂ρ1
∂x +∂e
T
∂ρ2
∂ρ2
∂x +∂e
T
∂S
∂S
∂x ,(50)
∂pi
∂x =∂˜pi
∂α1
∂α1
∂x +∂˜pi
∂ρ1
∂ρ1
∂x +∂˜pi
∂ρ2
∂ρ2
∂x +∂˜pi
∂S
∂S
∂x .(51)
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16 G. LA SPINA ET AL.
Substituting in the quasi-linear system (47), we can finally write the matrix B(v)as
B=







































u0 0 0 0 0
0u0 0 0 0
b3,10u10ρ10
b4,10 0 u20ρ2
b5,1b5,2
1
ρ1
∂˜p1
∂ρ1−c2(s1−s2)∂e
T
∂ρ1
1
ρ1
∂˜p1
∂ρ2−c2(s1−s2)∂e
T
∂ρ2
u10
b6,1b6,2
1
ρ2
∂˜p2
∂ρ1
+c1(s1−s2)∂˜p
∂ρ1
1
ρ2
∂˜p2
∂ρ2
+c1(s1−s2)∂e
T
∂ρ1
0u2







































(52)
where
b3,1=c2
ρ1
α1
(u1−u2),(53)
b4,1=c1
ρ2
α2
(u1−u2),(54)
b5,1=p1−p2
ρ+1
ρ1
∂p1
∂α1−c2(s1−s2)∂T
∂α1
,(55)
b5,2=1
ρ1
∂p1
∂S −c2(s1−s2)∂T
∂S ,(56)
b6,1=p1−p2
ρ+1
ρ2
∂p2
∂α1
+c1(s1−s2)∂T
∂α1
,(57)
b6,2=1
ρ2
∂p2
∂S +c1(s1−s2)∂T
∂S ,(58)
Due to the structure of the matrix B, the equation for the eigenvalues does not depend on the
coefficients bi,j defined in (53-58) and the characteristic polynomial takes the form
π(λ) = (u−λ)2·det(A−λI),(59)
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 17
where the matrix Ais
A=























u10ρ10
0u20ρ2
1
ρ1
∂˜p1
∂ρ1−c2(s1−s2)∂e
T
∂ρ1
1
ρ1
∂˜p1
∂ρ2−c2(s1−s2)∂e
T
∂ρ2
u10
1
ρ2
∂˜p2
∂ρ1
+c1(s1−s2)∂e
T
∂ρ1
1
ρ2
∂˜p2
∂ρ2
+c1(s1−s2)∂e
T
∂ρ1
0u2























.
The characteristic polynomial of Ais ˜π(λ) =
4
X
i=0
aiλiwhere
a4= 1,
a3=−(2u1+ 2u2),
a2= u2
1+u2
2+ 4u1u2−∂˜p1
∂ρ1−∂˜p2
∂ρ2−c1ρ2(s1−s2)∂e
T
∂ρ2
+c2ρ1(s1−s2)∂e
T
∂ρ1!,
a1=−2u1u2
2+ 2u2
1u2−2u2
∂˜p1
∂ρ1−2u1
∂˜p2
∂ρ2
−2c1ρ2u1(s1−s2)∂e
T
∂ρ2
+ 2c2ρ1u2(s1−s2)∂e
T
∂ρ1!,
a0= u2
1u2
2−u2
2
∂˜p1
∂ρ1−u2
1
∂˜p2
∂ρ2−c1ρ2u2
1(s1−s2)∂e
T
∂ρ2
+c2ρ1u2
2(s1−s2)∂e
T
∂ρ1
+∂˜p1
∂ρ1
∂˜p2
∂ρ2−∂˜p1
∂ρ2
∂˜p2
∂ρ1
+c1ρ2(s1−s2)∂˜p1
∂ρ1
∂e
T
∂ρ2−c1ρ2(s1−s2)∂˜p1
∂ρ2
∂e
T
∂ρ1
+c2ρ1(s1−s2)∂˜p2
∂ρ1
∂e
T
∂ρ2−c2ρ1(s1−s2)∂˜p2
∂ρ2
∂e
T
∂ρ1!.
We observe that it is possible to write the characteristic polynomial ˜π(λ)in the more compact
form
˜π(λ) = π1(λ)·π2(λ)−π3(60)
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18 G. LA SPINA ET AL.
where
π1(λ) = λ2−2u1λ+u2
1−∂˜p1
∂ρ1
+c2ρ1(s1−s2)∂e
T
∂ρ1
,
π2(λ) = λ2−2u2λ+u2
2−∂˜p2
∂ρ2−c1ρ2(s1−s2)∂e
T
∂ρ2
,
π3= ∂˜p2
∂ρ1
+c1ρ2(s1−s2)∂e
T
∂ρ1!· ∂˜p1
∂ρ2−c2ρ1(s1−s2)∂e
T
∂ρ2!.
The coefficient of the characteristic polynomial can be written in terms of the sound speeds Ci
and of the partial derivatives of the temperatures, both defined in terms of the original equations of
state (1-2). Then, once the polynomial is defined, a simple numerical method can be used to find the
eigenvalues of system. More details are given in the Appendix.
4.1. Simplified models
We analyze here the eigenvalues of the Jacobian matrix in some particular case with additional
hypothesis that simplify the model.
First, we consider the case with a single velocity, i.e. u1=u2=u, corresponding to analytical
solution obtained with a relative velocity relaxation (λ0= +∞). With this assumption the
characteristic polynomial ˜π(λ)takes the following form:
y−∂˜
P1
∂ρ1−c2ρ1(s1−s2)∂˜
T
∂ρ1y−∂˜
P2
∂ρ2
+c1ρ2(s1−s2)∂˜
T
∂ρ2−π3,(61)
where
y= (λ−u)2.
Now, from the two solutions y1ey2of the second order polynomial (61), we obtain
λ1,2=u±√y1;λ3,4=u±√y2,
where the terms y1and y2do not depends on the velocities of the two phases. Furthermore, writing
the coefficients of the characteristic polynomial in terms of the sound speeds Ci(see Appendix), we
have that
y1→C2
1for α1→0,(62)
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 19
y2→C2
2for α1→1.(63)
Thus, in the case of very dilute regime, two eigenvalues approach the usual characteristic
velocities associated with the carrier phase, given by the velocity of the phase plus and minus the
phase sound speed.
5. NUMERICAL SOLUTION OF THE MODEL
For the numerical integration of the system the fractional-step method presented in [15] is applied
by first solving the homogeneous system and then the ODEs system where the flux terms are
neglected and only the source terms are considered. Using the compact form (16) of the system and
following the Strang splitting, the fractional-step method results in the solution of the two following
subproblems:
P roblem A :qt+F(q)x= 0
P roblem B :qt=Ω(s)
(64)
5.1. Integration of the homogeneous system
Due to the complexity of the equations, the development of exact or approximate Riemann solvers
for the homogeneous part of the two-phase single temperature model can be a difficult task. In
particular, the evaluation of the eigenvalues and the eigenvectors of the the Jacobian matrix of
the fluxes can be very expensive from a computational point of view. For this reason, to solve
the hyperbolic homogeneous system of the step A, a so-called Godunov-like central (or central-
upwind) scheme formulation is adopted. This family of schemes shares some of the high-resolution
properties of classical (approximate) Riemann-solver based upwind schemes, while being much
simpler to implement. In particular, the central finite volume scheme described in this work is not
based on the complete eigenstructure of the Jacobian matrix of the fluxes, and only the numerical
fluxes and the maximum and minimum eigenvalues (largest and smallest roots of the characteristic
polynomial eπ(λ)given in Eq. (59)) are requested for its implementation.
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20 G. LA SPINA ET AL.
As said, the numerical scheme adopted is based on a central–upwind formulation and can be
written in the form:
Qn+1
i=Qn
i−∆t
∆xFi+1
2−Fi−1
2(65)
where Qn
iis an approximation of the average of qover the i-th cell at the n-th time step
and Fi+1/2≈F(Qi+1/2)is the numerical flux, which is a function of inter-cell boundary value
Qi+1/2evaluated at an intermediate time step tn+1/2. The latter can be obtained numerically as a
solution of a local Riemann problem. Here, the MUSCL-Hancock method with linear reconstruction
and limited slope is used providing second-order of accuracy. Differently from [14], the linear
reconstruction is done on a set of primitive variables U= (α1, p1, p2, u, w, T )instead that on the
conservative variables qto ensure the positivity of densities, energy and pressures, which is an
essential issue in the discretization of the model. With this choice of the vector U, there exists a
bijection
Q= Γ(U) = (Γ1(U),...,Γ6(U)).
The reconstruction is obtained as
Un,L
i=Un
i−∆x
2(Un
i)′Un,R
i=Un
i+∆x
2(Un
i)′,(66)
where (Un
i)′is an approximation of the first spatial derivative of the solution at the point xiat the
time tn. In order to prevent high derivative values and thus oscillations in the numerical solution,
nonlinear slope limiter functions should be adopted. In particular, in this work the Van-Leer limiter
[33]ϕvl has been used:
(Un
i)′=ϕvl(Un
i−1,Un
i,Un
i+1) =
=








2
∆x·∆Un
i+1
2∆Un
i−1
2
∆Un
i+1
2+ ∆Un
i−1
2
,if ∆Un
i+1
2∆Un
i−1
2>0
0otherwise
(67)
where
∆Un
i+1
2=Un
i+1 −Un
i∆Un
i−1
2=Un
i−Un
i−1.
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 21
Figure 1. Schematic illustration of a linear reconstruction. The solid line corresponds to the average integral
Un
iin the i-th cell, while the dash-dot line represent the linear reconstruction inside the i-th cell.
A schematic illustration of a linear reconstruction is reported in Fig.1. Then, from the
reconstructed interface values of the primitive variables the corresponding values of the
characteristic variables Qn,L
i= Γ(Un,L
i)and Qn,R
i= Γ(Un,R
i)are obtained. These boundary
extrapolated values are evolved in time by
e
Qn+1/2,L
i=Qn,L
i−1
2
∆t
∆x(F(Qn,R
i)−F(Qn,L
i)),(68)
e
Qn+1/2,R
i=Qn,R
i−1
2
∆t
∆x(F(Qn,R
i)−F(Qn,L
i)).(69)
Finally, using as initial value e
Qn+1/2,L
iand e
Qn+1/2,R
i, we integrate the source terms to find the
values ˆ
Qn+1/2,L
iand ˆ
Qn+1/2,R
iused to evaluate the numerical fluxes Fi+1/2of the finite volume
scheme (65).
The numerical fluxes adopted here are obtained from a modification of the Kurganov, Noelle and
Petrova semidiscrete scheme proposed in [34]. These numerical fluxes, that have been studied in
[15], are defined as follows:
Fi+1
2=
a+
i+1
2F(ˆ
Qn+1/2,L
i+1 )−a−
i+1
2F(ˆ
Qn+1/2,R
i)
a+
i+1
2−a−
i+1
2
−
a+
i+1
2
a−
i+1
2
a+
i+1
2−a−
i+1
2ˆ
Qn+1/2,R
i−ˆ
Qn+1/2,L
i+1 .(70)
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22 G. LA SPINA ET AL.
The speeds a+
i+1
2and a−
i+1
2are the maximum left-going and right-going characteristic speed at
the interface, computed as
a+
i+1
2
= max(λmax
i+1
2,R, λmax
i+1
2,L,0),
a−
i+1
2
= min(λmin
i+1
2,R, λmin
i+1
2,L,0),
(71)
where λmax
i+1
2,R and λmax
i+1
2,L are the maximum eigenvalues and λmin
i+1
2,R and λmin
i+1
2,L the minimum
eigenvalues of the jacobian matrix of the fluxes evaluated respectively at ˆ
Qn+1/2,L
i+1 and ˆ
Qn+1/2,R
i.
5.2. Integration of the source terms
After the integration of the homogeneous hyperbolic system of PDEs defined by the Problem A,
the system of ordinary differential equations associated to the Problem B in the Strang splitting
is solved. Using a segragated approach, in [15], it has been shown that the pressure and velocity
relaxation terms can be integrated analytically, under the condition of constant pressure relaxation
parameter τ(p)and frictional coefficient ζ. Here, instead, we focus the attention on the numerical
integration of the phase exchange relaxation term.
For the applications presented in this paper, we consider the gas phase being exactly the vapor
phase of the liquid and we assume the characteristic times of phase pressures and velocities
equalizing being negligible compared to the characteristic time of phase transition, and thus the
pressure and the velocity relaxations instantaneous (τ(p)= 0 and ζ= +∞). With these assumptions,
the equations for the integration of the source terms of the volume fraction and the mass fraction of
the first phase and the equation of the relative velocity become:
p2−p1= 0,(72)
∂q3
∂t =−ρ
τ(c)(µ1−µ2),(73)
w= 0.(74)
According to the sign of the difference of the Gibbs free energies, it is possible to have evaporation
or condensation, and the reaction ends when the difference becomes zero. Thus the equilibrium
density of the liquid phase (or gas phase) is the density that we have when the difference of the
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 23
Gibbs energies is zero. Furthermore, we define the relaxation time τ(c)as a nonlinear function of
the temperature:
τ(c)=τ(c),0+KT
T−Teq
Teq 
n
·Kp
p−peq
peq 
m
,(75)
where τ(c),0is a reference relaxation time, Teq and peq are the critical temperature and pressure
that we have at the thermodynamical equilibrium when the condition given by Eqs. (72) and (74)
are satisfied, and KT, Kp, n and mare empirical parameters measuring the degree of nonlinearity.
These are introduced to better model the fact that the transition only occurs in the neighborhood of
the critical values of temperature and pressure, when we are close to the thermodynamic equilibrium
and the difference of the Gibbs energies is zero. The values of the four constants KT, Kp, n and m
are not obtained directly from the available experimental data, being our goal here to demonstrate
only that the kinetic approach can give reasonable results in phase transition modeling.
We show here that it is possible to derive, when the two conditions (72) and (74) hold, an
expression of the source term for the mass fraction of the first phase as a function of q1only.
First of all, we observe that from the Eqs. (37) it is possible to write the chemical potential µias
function of the density ρiand the common temperature Tonly. Now, substituting the expressions for
the specific energies in the total energy q6, we obtain the following expression for the temperature
T:
T=q6−1
2q3u2
1+ (q1−q3)u2
2−q3¯e1−α1¯p1−(q1−q3)¯e2−(1 −α1) ¯p2
q3cv,1+ (q1−q3)cv,2(76)
where q1and q6are constant during the integration of Eq. (73).
We also observe that the velocities of the two phases, appearing in the expression of the
temperature, can be expressed as functions of the conservative variables in the following way:
u1=q4+q5(q1−α1ρ1)
q1(77)
u2=q4−α1ρ1q5
q1(78)
If we assume now that the relative motion of phases can be neglected, then the conservative variable
q5can be set to zero. With this assumption also the velocities u1=u2=q4/q1are constant during
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24 G. LA SPINA ET AL.
the integration of Eq. (73), and the equation for the temperature becomes
T=q6−1
2
q2
4
q1−q3¯e1−α1¯p1−(q1−q3)¯e2−(1 −α1) ¯p2
q3cv,1+ (q1−q3)cv,2(79)
where the only variables changing during the integration of the phase relaxation term are q3and α1.
Now, using the condition on the local pressures given by Eq. (72) for the thermodynamic
equilibrium, and the expressions for the pressures from the equations of state defined in terms of the
densities and the temperature, we have:
ρ1(γ1−1)cv,1T−¯p1=ρ2(γ2−1)cv,2T−¯p2.(80)
Using the definition of the variable q3and the conservation of the mixture density q1=ρwe can
write the density of the two phases as functions of the constant variable q1and the two variables q3
and α1:
ρ1=q3
α1
, ρ2=q1−q3
1−α1(81)
and, substituting these expressions into the equation (80), we obtain
T=α1(1 −α1)(¯p1−¯p2)
(1 −α1)q3(γ1−1)cv,1−α1(q1−q3)(γ2−1)cv,2(82)
Again, as for the other expression of the mixture temperature given by Eq. (79), the only variables
changing during the integration of the phase relaxation term are q3and α1, and from the equations
(79) and (82) we get
q6−1
2
q2
4
q1−q3¯e1−α1¯p1−(q1−q3)¯e2−(1 −α1) ¯p2
q3cv,1+ (q1−q3)cv,2
=
=α1(1 −α1)(¯p1−¯p2)
(1 −α1)q3(γ1−1)cv,1−α1(q1−q3)(γ2−1)cv,2
(83)
From this one we can explicit α1as a function of q3
α1≡α1(q3)(84)
and substituting (84) in (79) or (82) and the first of Eqs.(81), we finally have the temperature and
the density as T=T(q3)and ρ1=ρ1(q3)respectively, and from the equations of state we can write
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 25
also the chemical potential µ1in terms of q3only. Furthermore, from Eq. (81) and the conservation
of the mixture density, also ρ2can be written as function of q3, and consequently µ2. Thus, under
the assumptions of negligible relative velocity and small characteristic time of pressure relaxation
compared to that of phase transition, the difference of the chemical potentials in the right hand side
of Eq. (73) can be written as function of the the conservative variable q3only.
In order to find an expression for the relaxation time τ(c), we have to find the critical temperature
and thus, being the temperature a function of q3only during the integration of the phase change
term, the equilibrium value qeq
3. This value can be evaluated imposing the condition µ2−µ1= 0,
from which we obtain an equation in the only unknown q3. The solution can be easily computed
using an iterative method like bisection or Newton method, and then, using the equations (84) and
(81), we also obtain the equilibrium volumetric fraction α1and density ρ1.
Once we have the equilibrium values all the terms of the right hand side of Eq. (73) are defined as
a function of the integration variable q3. A second-order Cranck-Nicolson method has been applied
in this work to integrate the ordinary differential equation and to update the value of q3:
q(n+1)
3=q(n)
3−ρ∆t
2µ1µ2
τ(c)(n)
+µ1µ2
τ(c)(n+1).(85)
Now, due to the assumptions of negligible relative velocity and characteristic time of pressure
relaxation, all the other conservative variables are constant during the integration except q2. But,
substituting the updated value q(n+1)
3in Eq. (84), the new value of α(n+1)
1is found and also q2can
be updated as q(n+1)
2=α(n+1)
1q1.
6. NUMERICAL TESTS
In this section we solve numerically three test problems in order to investigate the effects of the
evaporation term on the single temperature two-phase model. Each of the test problems represents
a Riemann problem for a liquid/gas mixture with instantaneous pressure and velocity relaxations.
In the first test we study the 1D cavitation tube problem presented in [12] and studied in [35,36],
analyzing the effects of the evaporation, with both instantaneous and finite rate relaxation. Then, as a
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26 G. LA SPINA ET AL.
second test, we present a multidimensional cavitation problem: the cavitating Richtmyer–Meshkov
instability test described in [17,18]. For this test evaporation is not taken into account and a small
fraction of gas is initially present in the liquid. Finally, we present another multidimensional test,
in which the evaporation is considered: the laser–induced cavitation problem [19]. For the 2D tests,
it has been developed and used a new solver for the parallel open–source CFD software package
OpenFOAM, implementing the numerical scheme described in the previous section.
For all the runs presented in this paper, the fractional–step method illustrated in section 5has
been adopted.
6.1. One-dimensional cavitation tube
In this first test we consider an evaporation problem for a tube initially at atmospheric pressure and
at a temperature of 355 K, filled with water and a small volumetric fraction of water vapor. In this
test the water can not be treated as pure and evaporation only occurs if the pressure of the liquid
phase is lower than the saturation pressure.
The initial condition is similar to the two-phase expansion tube test described in [12], where a
seven-equation model for two-phase flows is applied to model evaporation of metastable liquids.
The left part of the tube is set to motion with a velocity ul=−2m/s while the right part is set
to motion with a velocity ur= 2 m/s. In such situation, the pressure, density and internal energy
decrease across the rarefaction waves in order that the velocity reaches zero at the center of the
domain. The pressure decreases until the saturation pressure at the local temperature is reached.
Then the mass transfer appears, a part of liquid become gas, and the flow becomes a two-phase
mixture. This problem has been studied also in [37] in a simplified situation where a small fraction
of gas is initially present in the liquid (1% gas by volume) and the mass transfer in not considered.
Here we analyze the effects of different values of the phase exchange term, in order to compare our
results with both those presented in [37] and [12].
The indexes 1 and 2 for this application are referred to the parameters of state of water vapor
and water, respectively, and the constants for the equations of state are given in Table I. The
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 27
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12x 104
x (m)
Pressure (Pa)
0 0.2 0.4 0.6 0.8 1
−3
−2
−1
0
1
2
3
x (m)
Velocity (m/s)
0 0.2 0.4 0.6 0.8 1
354.975
354.98
354.985
354.99
354.995
355
355.005
x (m)
Temperature (K)
0 0.2 0.4 0.6 0.8 1
0.94
0.96
0.98
1
x (m)
Water volume fraction
Figure 2. Cavitation Tube: Mixture pressure (top left), mixture velocity (top right), common temperature
(bottom left) and liquid water volume fraction (bottom right) at t= 3.2·10−3scomputed on 2000 cells for
τ(c)= +∞(no evaporation). The CPU time is 850 s.
computational domain considered is x∈[0,1] (in meters) with the Riemann problem being defined
with the interface located at x= 0.5. The initial data are given by:
•Left: α1= 0.99,α2= 0.01,u1=u2=−2m/s, p1=p2= 105Pa, T= 355 K;
•Right: α1= 0.99,α2= 0.01,u1=u2= +2 m/s, p1=p2= 105Pa, T= 355 K.
In Figs. 2–4we present the results of several simulations done with no evaporation, instantaneous
evaporation and finite rate evaporation. We observe that for all the tests the magnitude of the
temperature variations are small compared to the absolute value of the temperature, therefore the
single temperature approximation is appropriate.
In Fig. 2the solution for a test without evaporation (τ(c)= +∞), but with instantaneous pressure
and velocity relaxations, is shown at t= 3.2×10−3s. Two symmetric rarefaction waves propagate
to the left and to the right. At the middle of the domain the mixture velocity is zero and the gas
volumetric fraction increases due to the expansion. The solution is not physically correct because
the expansion wave make the liquid thermodynamic state metastable and evaporation should occur.
These effects are taken into account in the second test, where τ(c)= 0.
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28 G. LA SPINA ET AL.
0 0.2 0.4 0.6 0.8 1
5
6
7
8
9
10
11x 104
x (m)
Pressure (Pa)
0 0.2 0.4 0.6 0.8 1
−3
−2
−1
0
1
2
3
x (m)
Velocity (m/s)
0 0.2 0.4 0.6 0.8 1
354.8
354.85
354.9
354.95
355
355.05
x (m)
Temperature (K)
0 0.2 0.4 0.6 0.8 1
0.5
0.6
0.7
0.8
0.9
1
x (m)
Water volume fraction
Figure 3. Cavitation Tube: Mixture pressure (top left), mixture velocity (top right), mixture temperature
(bottom left) and liquid water volume fraction (bottom right) at t= 3.2·10−3scomputed on 2000 cells for
τ(c)= +0 (instantaneous evaporation). The CPU time is 1046 s.
0 0.2 0.4 0.6 0.8 1
4
5
6
7
8
9
10
11x 104
x (m)
Pressure (Pa)
0 0.2 0.4 0.6 0.8 1
−3
−2
−1
0
1
2
3
x (m)
Velocity (m/s)
0 0.2 0.4 0.6 0.8 1
354.9
354.92
354.94
354.96
354.98
355
x (m)
Temperature (K)
0 0.2 0.4 0.6 0.8 1
0.7
0.8
0.9
1
x (m)
Water volume fraction
Figure 4. Cavitation Tube: Mixture pressure (top left), mixture velocity (top right), mixture temperature
(bottom left) and liquid water volume fraction (bottom right) at t= 3.2·10−3scomputed on 2000 cells for
τ(c),0= 105,KT=Kp= 20,n=m= 3 (finite rate evaporation). The CPU time is 1176 s.
The solution at t= 3.2×10−3s of a test with instantaneous pressure and velocity relaxations
and evaporation is presented in Fig. 3. The solution consists of four rarefaction waves: two waves
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 29
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12 x 104
x (m)
Pressure (Pa)
0 0.2 0.4 0.6 0.8 1
0
1
2
x 10−4
x (m)
Vapor mass fraction
0 0.2 0.4 0.6 0.8 1
−3
−2
−1
0
1
2
3
x (m)
Velocity (m/s)
0 0.2 0.4 0.6 0.8 1
0.5
0.6
0.7
0.8
0.9
1
x (m)
Water volume fraction
Figure 5. Cavitation Tube: Mixture pressure (top left), vapor mass fraction (top right), mixture velocity
(bottom left) and liquid water volume fraction (bottom right) at t= 3.2·10−3scomputed on 2000 cells for
no evaporation (dashed line), instantaneous evaporation (solid line) and finite rate evaporation (τ(c),0= 106,
KT=Kp= 20,n=m= 3, dotted line).
0 0.2 0.4 0.6 0.8 1
2
4
6
8
10
x 104
x (m)
Pressure (Pa)
0 0.2 0.4 0.6 0.8 1
0
0.005
0.01
0.015
0.02
0.025
0.03
x (m)
Vapor mass fraction
0 0.2 0.4 0.6 0.8 1
335
340
345
350
355
360
x (m)
Temperature (K)
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
x (m)
Water volume fraction
Figure 6. Cavitation Tube: Mixture pressure (top left), vapor mass fraction (top right), mixture temperature
(bottom left) and liquid water volume fraction (bottom right) at t= 1.5·10−3scomputed on 2000 cells
for instantaneous evaporation, pressure relaxation and velocity relaxation and ul=−100m/s and ur=
100m/s. The CPU time is 551 s.
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30 G. LA SPINA ET AL.
propagate into the mixture on both sides of the initial velocity discontinuity and two evaporation
fronts propagate into the metastable region. These four waves are evident in the two symmetric
drops in the temperature and water volume fraction plots.
Finally, in Fig. 4, the solution at t= 3.2×10−3s for a test with instantaneous pressure and
velocity relaxations and evaporation occurring with finite speed is shown (τ(c),0= 105,KT=
Kp= 20,n=m= 3). In Fig. 5, the solutions obtained with instantaneous evaporation (solid line)
and without evaporation (dashed line) are compared with the solution obtained with a finite rate
evaporation (τ(c),0= 106,KT=Kp= 20,n=m= 3). We observe that, in order to better highlight
the effects of the finite rate mass transfer and the differences with the instantaneous evaporation test,
a larger evaporation rate has been used with respect to the test represented in Fig. 4. In all the cases
the gas volume fraction at the center of the tube increases due to rarefaction waves and pressure
relaxation. From the plot of the liquid phase volumetric fraction we can see that, with a finite-
rate evaporation, the evaporation front becomes less sharp. When the phase change occurs, a larger
amount of water vapor is present and free to expand, decreasing the volume of the liquid phase and
increasing its density. Furthermore, for the finite rate evaporation, we see that while in the density
and volume fraction plots it is still possible to identify the transition from the expansion waves to
the evaporation waves the velocity profile is more smooth.
As a final plot, to better appreciate the validity and the performance of the method, we also
present in Fig. 6the pressure and the water volume fraction at t= 1.5·10−3s, obtained when the
value of the initial velocities are increased to -100 m/s on the left and 100 m/s on the right for a test
with instantaneous pressure and velocity relaxations and instantaneous evaporation. With this initial
conditions, as shown in [12], four expansion waves are clearly visible.
Comparing the results of this test case with those reported in [12] for a finer grid, we can conclude
that a reasonable agreement with their solutions is achieved.
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 31
(a) (b)
(c) (d)
(e) (f)
Figure 7. Richtmyer–Meshkov instability: Pseudo-color images of gas volume fraction. The results are taken
respectively at time t= 0 (a), 2.0(b), 3.1(c), 6.4(d), 8.6(e) and 11.0(f) ms. The solution has been computed
using a grid of 1200 ×400 cells.
6.2. Cavitating Richtmyer–Meshkov instability
As a second test we present the 2D gas–water cavitating Richtmyer–Meshkov instability problem
proposed in [18,17]. For this test the mass transfer is not taken into account, while instantaneous
pressure and velocity relaxations are assumed. The indexes 1 and 2 for this application are referred
to the parameters of state of water and gas, respectively. Their values are given in Table II and are
chosen in agreement with those used in [17].
For this test a rectangular domain (x, y)∈[0,3] ×[0,1] m2is considered, where the left part is
filled with nearly pure water (α2= 10−6) and the right part with nearly pure gas (α1= 10−6). The
two phases are separated by a curved interface, a portion of circle with 0.6mradius centered at x=
1.2m,y= 0.5m. Figure 7-(a) shows the computational domain and the initial gas-water interface.
Top, bottom and left boundaries are assumed as solid walls, while the right side is considered
an outflow boundary. Both liquid and gas phases have an initial velocity of ux=−200m/s.
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32 G. LA SPINA ET AL.
Furthermore, the initial pressure is set to 105P a for both phases, while the temperature profile is
chosen in the way that the density of water in the right zone is ρ1= 1000kg/m3, and the density of
the gas in the left region is ρ2= 100kg/m3.
For this test we use a grid of 1200 ×400 cells and the simulation has been done on a High
Performance Computing 48 multi core shared memory system, using 32 cores. The numerical results
are reported in Fig. 7and 8. In Fig. 7we report the gas volume fraction, while in Fig. 8the emulated
Schlieren photographies generated from the numerical results plotting |∇ρ|in a non–linear greymap
are presented. The solutions illustrated in these figures are taken respectively at time t= 0 (a), 2(b),
3.1(c), 6.4(d), 8.4(e) and 11.0(f) ms. From the plots we can observe that, when the flow impacts the
left wall, a right-going shock wave propagates through the curved gas–water interface, producing
a Richtmyer–Meshkov instability. This is characterized by expansion waves and an elongating jet.
These expansion waves result in a decrease of pressure in the upper and lower corner of the left side
of the domain, generating cavitation pockets in these regions. Our results agree well with the ones
shown in [17], by looking at the global features of the solution structure. Comparing the plot of the
gas volume fraction presented in Fig. 7with the results obtained in [18], we can see that our solution
seems to be more diffusive in the resolution of the interfaces between the phases. We have also to
remark that the parameters for the equations of state used in [18] are different from those used in
[17], thus leading to different characteristic speeds and different propagations of the shock-waves.
6.3. Laser–Induced Cavitation–Bubble Problem
The last numerical test we present in this work is a numerical simulation for a 2D laser-induced
cavitation-bubble problem, where instantaneous evaporation is considered. Many studies related to
this problem are dedicated to medical applications, especially in ophthalmology [38,39,40,41] and
biomedicine.
As a test case, we considered here the laser-induced cavitation-bubble experiments as described in
[19]. A laser pulse is focused for a short time into the liquid producing an increasing of the pressure
and temperature. Due to the high pressure gradient, a shock wave propagates, causing an expansion
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 33
(a) (b)
(c) (d)
(e) (f)
Figure 8. Richtmyer–Meshkov instability: emulated Schlieren image generated from the numerical results
plotting |∇ρ|in a nonlinear graymap. The results are taken respectively at time t= 0 (a), 2.0(b), 3.1(c), 6.4
(d), 8.6(e) and 11.0(f) ms. The solution has been computed using a grid of 1200 ×400 cells.
and a strong decrease of the pressure in the area previously irradiated by the laser and therefore the
evaporation of the liquid. Then the gas bubble generated by the evaporation starts to expand and
after a certain time, due to the high pressure of the surrounding water, the expansion ends and the
bubble begin to collapse. Finally, when the bubble collapse entirely, a new shock wave is generated.
For the numerical simulation, as initial condition, we consider a square domain [−5,5] ×
[−5,5] mm2that contains nearly pure water (α2= 10−6, where indexes 1 and 2 are referred to
water and vapor, respectively) at ambient pressure and temperature. The parameters of state for the
liquid and gas phases are given in Table III. The zone irradiated by a laser pulse of 0.2mJ is a sphere
which center is located at (x, y) = (0,0) with a radius of 5×10−2mm. Furthermore, we assume
that the pulse is so short in time that the energy is transfered to water instantaneously, resulting,
in the zone irradiated by the laser, in an increase of the pressure up to ≈5.8×108P a and of the
temperature up to ≈520K.
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34 G. LA SPINA ET AL.
U_r (m/s)
(a) (b) (c)
(d) (e) (f)
1-1 0
Figure 9. Laser–Induce Cavitation Bubble: emulated Schlieren image generated from the numerical results
plotting |∇ρ|(left side of each panel) and radial velocity (right side of each panel). The radial velocities
greater or equal to 1 m/s are represented by the same red color, while those smaller or equal to -1 m/s with
the same blue one. The results are taken respectively at time t= 0.01 (a), 1.0(b), 30.0(c), 75.0(d), 145.0
(e) and 150.0(f) µs. The solution has been computed using a grid of 1000 ×1000 cells.
The results of a numerical simulation, obtained with a 1000 ×1000 cells grid, with instantaneous
pressure and velocity relaxation, i.e. single pressure and temperature, are presented in Fig. 9. On
the left side of each panel is presented an emulated Schlieren image generated from the numerical
results plotting |∇ρ|and on the right side the radial velocity is plotted, in order to distinguish the
expanding and collapsing regions. The Fig. 9–(a) illustrates the solution a few instants after the
laser pulse (t= 0.01µs). In Fig. 9–(b), instead, we can observe the solution at time t= 1µs. Here
we can see the spherical shock wave that propagates into water and, at the center of the domain,
the bubble nucleated due to the cavitation. Furthermore, the positive radial velocity indicates that
the bubble is expanding. In Fig. 9–(c) (t= 30µs), the shock wave exited from the computational
domain, but we can see the bubble growth with respect to Fig. 9–(b). The positive radial velocity
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 35
around the surface of the bubble indicates that is still expanding. The maximum radius of the bubble
is reached at time t= 75µs, as illustrated in Fig. 9–(d). Here, we can see that, although the radial
velocity inside the bubble is still positive, the water outside the bubble, due to the high pressure, is
starting to move towards the center of the domain. Because of that, the bubble expansion ends and
from now on the bubble starts to collapse. In Fig. 9–(e) (t= 145µs) we can observe that the radius
of the bubble is decreased, and the negative radial velocity indicates that the bubble will continue
to compress up to it entirely collapses. The collapse of the bubble produces an increase in pressure
at the center of the domain, generating again a shockwave. Due to the lower intensity of the shock
with respect to the one generated by the laser pulse, in the emulated Schlieren image reported in
Fig. 9–(f) (t= 150µs), the new shock wave is not clearly visible. However, the presence of this
shockwave is highlighted in the right side of Fig. 9–(f) where the positive radial velocity indicates
that a shockwave is propagating towards the boundary of the domain.
The bubble radius and the pressure at the center of the bubble are also reported in Fig. 10.
Comparing the oscillation time and the maximum radius of the bubble with the data reported in
[19] we notice that these results are not in agreement with real observation. This can be related to
the fact that we are considering a 2D bubble, instead of a 3D one as in the laboratory experiment.
Furthermore, as in the numerical simulation presented in [42], there is no rebound of the bubble
after the collapse. In real experiments this behavior is not observed and it is thought to be due
to the presence of a percentage of noncondensable gas inside the bubble beside the water vapor
[43,42]. However, from the data collected, the model is able to properly reproduce the dynamics
of the problem, i.e. the bubble and the shock generation, the bubble expansion and collapse and the
consequent generation of the new shock.
7. CONCLUSIONS
In this paper, the thermodynamically compatible system of conservation laws proposed in [14,15]
for two-phase flow with single temperature and different pressures and velocities, is extended
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36 G. LA SPINA ET AL.
0 50 100 150
0
0.05
0.1
0.15
0.2
0.25
0.3
Time (µs)
Radius (mm)
0 50 100 150
103
104
105
106
107
108
109
Time (µs)
Pressure (Pa)
0 50 100 150
250
300
350
400
450
500
550
Time (µs)
Temperature (K)
Figure 10. Laser–Induce Cavitation Bubble: growth and collapse of the vapor bubble. On the left panel the
bubble radius is plotted versus time, while on the right panel the pressure at the center of the bubble is
plotted.
introducing phase transition. The fluid thermodynamics has been restricted to the “Stiffened Gas”
equations of state because it was sufficiently accurate for the applications.
The governing equations form a hyperbolic system and are written in conservative form, allowing
the use of central finite-volume schemes. The modification of Kurganov, Noelle and Petrova
numerical fluxes [15], in the framework of the second-order MUSCL-Hancock method, were
employed to develop a new multidimensional solver within the parallel opensource CFD software
package OpenFOAM.
Three cavitation problems with difference relaxation time scales have been studied. Comparison
of the results with previous works showed a good agreement with the results obtained with different
numerical schemes and different governing equations.
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 37
ACKNOWLEDGMENTS
The first author’s research leading to these results has received funding from the European Research
Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant
Agreement n. 279802. The second author was supported by the Marie Curie Actions of the European
Commission in the frame of the MAMMA project (FP7-PEOPLE-2009-IOF-251833). The third
author acknowledges a financial support of the Russian Foundation for Basic Research (project 13-
05-12051) and the Siberian Branch of Russian Academy of Sciences (Integration Project No 127).
We are also grateful to two anonymous reviewers and the editor for their careful review and useful
and constructive comments.
A. EIGENVALUES EVALUATION
In section 4we have shown that two characteristic speed equal the mixture velocity and, for the
reduced matrix Adefining the remaining four characteristic speeds, we have written explicitly
the characteristic polynomial coefficients, defined as functions of known quantities and of the
derivatives ∂˜pi/∂ρjand ∂e
T /∂ρj. A possibility to evaluate the roots of the polynomial is to evaluate
these terms solving the system of equations (48) for ˜s1and ˜s2, in order to write explicitly the
expressions for ˜pi(α1, ρ1, ρ2, S )and e
T(α1, ρ1, ρ2, S), but this approach can be expensive from a
numerical point of view.
We show here that it is possible to write the two derivatives only in terms of the original equation
of state (1), i.e. without the need of evaluating explicitly the terms ˜s1and ˜s2.
Using the definition of e
T, we can write:
∂e
T
∂ρj
=∂
∂ρj
Tj(ρj,˜sj) = ∂Tj
∂ρj
+∂Tj
∂sj
∂˜sj
∂ρj
(86)
Now, from the system of equations (48), we define the function Gas:
G(α1, ρ1, ρ2,˜s2, S) = c1T−1
1,ρ1(T2(ρ2,˜s2)) + c2˜s2−S= 0 (87)
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38 G. LA SPINA ET AL.
where T−1
1,ρ1(·)is the inverse function of T1(ρ1,·)with respect to the entropy s1(i.e. s1=T−1
1,ρ1(T))
and, using the implicit function theorem, we can write
∂˜s2
∂ρ2
=−Gρ2
G˜s2
=−
∂c1
∂ρ2T−1
1,ρ1(T2(ρ2,˜s2)) + c1
∂T −1
1,ρ1
∂T
∂T2
∂ρ2+∂ c2
∂ρ2s2
c1∂T −1
1
∂T
∂T2
∂s2+c2
=
=−
∂c1
∂ρ2s1+c1∂ T2
∂ρ2/∂ T1
∂s1+∂ c2
∂ρ2s2
c1∂T2
∂s2/∂ T1
∂s1+c2
=−
∂c1
∂ρ2(s1−s2) + c1∂ T2
∂ρ2/∂ T1
∂s1
c1∂T2
∂s2/∂ T1
∂s1+c2
=
=−
∂c1
∂ρ2(s1−s2)∂ T1
∂s1+c1∂ T2
∂ρ2
c1∂T2
∂s2+c2∂ T1
∂s1
(88)
In the same way, from the system of equations (48), we define the function Has:
H(α1, ρ1, ρ2,˜s1, S) = c2T−1
2,ρ2(T1(ρ1,˜s1)) + c2˜s2−S= 0 (89)
and we obtain
∂˜s1
∂ρ1
=−Hρ1
H˜s1
=−
∂c2
∂ρ1T−1
2,ρ2(T1(ρ1,˜s1)) + c2
∂T −1
2,ρ2
∂T
∂T1
∂ρ1+∂ c1
∂ρ1s1
c2∂T −1
2
∂T
∂T1
∂s1+c1
=
=−
∂c2
∂ρ1s2+c2∂ T1
∂ρ2/∂ T2
∂s2+∂ c1
∂ρ1s1
c2∂T1
∂s1/∂ T2
∂s2+c1
=−
∂c2
∂ρ1(s1−s2) + c2∂ T1
∂ρ1/∂ T2
∂s2
c2∂T1
∂s1/∂ T2
∂s2+c1
=
=−
∂c2
∂ρ1(s1−s2)∂ T2
∂s2+c2∂ T1
∂ρ1
c2∂T1
∂s1+c1∂ T2
∂s2
(90)
Now, expanding the derivatives of ˜pi, we can write
∂˜pi
∂ρj
=∂
∂ρj
pi(ρi,˜si) = δij
∂pi
∂ρj
+∂pi
∂si
∂˜si
∂ρj
=
=δij C2
j+∂
∂siρ2
i
∂ei(ρi, si)
∂ρi∂˜si
∂ρj
=
=δij C2
j+ρ2
i
∂Ti
∂ρi
∂˜si
∂ρj
(91)
where Ciis the isentropic sound speed of the i-th phase defined as
Ci=s∂pi(ρi, si)
∂ρi(92)
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 39
We observe that, to evaluate the expressions we have found for the derivatives are ∂˜pi/∂ρjand
∂e
T /∂ρj, it is requested only the knowledge of the sound speeds Ciand of the partial derivatives of
the temperatures, both defined in terms of the original equations of state (1-2). Furthermore, being
Ti=Ti(ρi, si), we have, independently from the particular choice of the equations of state,
∂Ti
∂si
=Ti
cv,i
,∂Ti
∂ρi
=TiΓi
ρi(93)
where cV,i is the specific heat capacity at constant volume of the i−th phase and Γiis the Gr¨
uneisen
coefficient, defined in [31] as a function of the thermal expansion coefficient and the isothermal
compressibility.
When all the coefficients of the characteristic polynomial of Aare known, it is easy to evaluate
numerically its roots. First of all, due to the fact that the system of equations (1–6) is hyperbolic, the
characteristic polynomial has four real roots λ1≤λ2≤λ3≤λ4. Now, being the coefficient a4>0,
we have:
eπ(λ)>0,∂2eπ
∂λ2(λ)>0f or λ < λ1or λ > λ4.(94)
The positivity of both the polynomial and its second derivative ensures that the iterative Newton’s
method
xn+1 =xn−eπ(xn)
eπ′(xn),
with a starting point x0< λ1, converges to the smallest eigenvalue λ1, and with a starting point
x0> λ4, converges to the largest eigenvalue λ4. Finally, the initial guess x0can be easily determined
using the Gershgorin circle theorem, or with a lower and upper estimates of the solutions of the
characteristic polynomial.
B. SOLVER IMPLEMENTATION
In this section we discuss the implementation of the numerical scheme reported in this work
using the OpenFOAM framework. OpenFOAM (Field Operation And Manipulation) is a free
source CFD package written in C++ which uses classes and templates to manipulate and operate
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40 G. LA SPINA ET AL.
scalar, vectorial and tensorial fields [44]. OpenFOAM is programmed using an object-oriented
programming (OOP), in which the programmer creates classes to represent conceptual objects in
the code, classes that contain the data that make up the object. One of the strengths of OpenFOAM
is that new solvers inherit from OpenFOAM framework some very useful features like the use of
unstructured mesh and parallelization. Furthermore, it is possible to use some built-in functions to
calculate differential operators, like divergence, gradient, laplacian and curl. OpenFOAM offers the
possibility of choosing several discretization for these differential operators.
We report here the key steps of our solver to calculate the solution. Given a numerical solution
Qtat the time t, the solution at the new time step is computed as follows:
1. Computation of the time step ∆t.
2. Calculation of the primitive variables Utfrom the conservative variables Qt.
3. Linear reconstruction of Utat the cells interfaces.
Example:
fvc::interpolate(p1, pos, "reconstruct(p1)")
fvc::interpolate(p1, neg, "reconstruct(p1)")
4. Calculation of the reconstructed conservative variables Qt,L and Qt,R at the cells interfaces
from the primitive variables Ut,L and Ut,R.
5. Computation of the predictor step at the cells interfaces.
6. Application of the interface relaxation.
7. Calculation of the local speeds at the cells interfaces.
8. Computation of the numerical fluxes.
9. Computation of the solution Qt+∆t∗of the hyperbolic part of PDEs.
Example:
solve( fvm::ddt(rho) + fvc::div(flux_rho) );
10. Computation of the solution Qt+∆tintegrating the source terms.
11. Correction of the solution at the boundaries.
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 41
In this way, starting from t= 0 we can compute the solution at time t= ∆t, and iterating the
procedure we can obtain the solution at the time desired.
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42 G. LA SPINA ET AL.
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COMPRESSIBLE TWO-PHASE MODEL WITH PHASE TRANSITIONS 45
p0ρ0γ C0cv¯e s0
(P a)(kg/m3)(m/s) (J/(kg ·K)) (J/kg)
Water 105958.4 2.514 1542.98 1677 -1.167e+6 1307
Vapor 1050.527 1.324 501.37 1571 2.030e+6 7742
Table I. Parameters of state for water and water vapor for the cavitation tube test (from
www.engineeringtoolbox.com).
p0ρ0γ C0cv¯e s0
(P a)(kg/m3)(m/s) (J/(kg ·K)) (J/kg)
Water 1051000.0 4.4 1624.94 951.0 0 -43497.97
Gas 105100.0 1.4 37.4165 714.0 0 -2035.43
Table II. Parameters of state for water and gas for the cavitating Richtmyer–Meshkov instability.
p0ρ0γ C0cv¯e s0
(P a)(kg/m3)(m/s) (J/(kg ·K)) (J/kg)
Water 105998.2 2.514 1374.53 1676.97 -1.200e+6 243
Vapor 1050.527 1.324 501.37 1571.00 2.030e+6 7742
Table III. Parameters of state for water and water vapor for the laser–induced cavitation problem.
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