Journal ArticleDOI

# A fractional step method to compute a class of compressible gas–liquid flows

15 Feb 2012-Computers & Fluids (Pergamon)-Vol. 55, pp 57-69

AbstractWe present in this paper some algorithms dedicated to the computation of numerical approximations of a class of two-fluid two-phase flow models. Governing equations for the statistical void fraction, partial mass, momentum, energy are presented first, and meaningful closure laws are given. Then we may give the main properties of the class of two-fluid models. The whole algorithm that relies on the fractional step method and complies with the entropy inequality is presented afterwards. Emphasis is given on the computation of pressure–velocity–temperature relaxation source terms. Conditions pertaining to the existence and uniqueness of discrete solutions of the relaxation step are given. While focusing on some one-dimensional test cases, the true rates of convergence may be obtained within the evolution step and the relaxation step. Eventually, some two-dimensional numerical simulations of a heated wall are shown and are briefly discussed. Some advantages and weaknesses of algorithms are also discussed.

Topics: , Uniqueness (51%), Computation (50%)

### 1 Introduction

• Two distinct types of models are used in order to compute liquid-gas or watervapour two-phase flows in industrial codes: the homogeneous approach and the two-fluid approach.
• The two-fluid approach is assumed to be more general, and it is also expected to predict more accurately flows for which the phasic desequilibrium plays a crucial role (see [9, 25, 26]).
• One must be aware that at least two difficulties are hidden in these sets of PDE.
• The second section gives emphasis on the approximation of velocity-pressure-temperature relaxation effects.

### 2.1 The two-fluid model

• Throughout the paper, indexes l, g refer to the liquid and gas phases; the statistical void fractions of gas and liquid are noted classically αg and αl, which should agree with: αl + αg = 1.
• The so-called conservative variable W will be defined as: W = (αl, αlρl, αlρlUl, αlEl, αgρg, αgρgUg, αgEg) Moreover, PI(W ) and VI(W ) respectively denote in this paper the interfacial pressure and velocity, and will precised afterwards.
• The term S1,l will also be introduced later on.
• External sources might be included but are not considered herein.
• Standard viscous contributions may be included, which of course comply with the entropy inequality that will be detailed in the next subsection.

### 2.2 Closure laws for interfacial transfer terms

• The same holds when tackling three-phase flows, as emphasized in [18].
• Now, the second requirement (H1) implies that the field associated with λ = VI should be linearly degenerate.
• As shown in [8, 11], few expressions guarantee this behaviour.

### 2.4 Main properties of the two-fluid model

• The authors may now recall in brief the main properties of system (2) using the previous closure laws.
• Apart from the field associated with the eigenvalue λ =.
• Otherwise the computation of shock solutions would be meaningless, since multiple shock solutions may be obtained using various -stable- solvers (see for examples [17]) .
• Obviously, an alternative formulation of jump conditions in genuinely non linear fields that is probably more convenient may be: σ = [ρφUφ]RL/[ρφ].

### 3.2 Computing the evolution step

• Many solvers have been proposed in the literature for such a purpose.
• Approximate solutions in the evolution step may also be obtained using either the non-conservative form of Rusanov scheme, or the non-conservative form of the approximate Godunov scheme VFRoe-ncv [6]; the authors refer to [11, 10] for such a description.
• Obviously, the ultimate scheme has not been proposed yet.
• The proof is classical but is briefly recalled.
• The authors detail afterwards the relaxation step, with special focus on the pressure relaxation step which is rather tricky.

### 3.3 Computing the velocity relaxation step

• It may be easily checked that internal energies remain positive through this step.
• The proof is obvious considering formula (16) and is left to the reader.

### 3.4 Computing the temperature relaxation step

• The proof provides a practical way to compute solutions of system (18), that is actually used in the code.
• The authors emphasize that other -simpler- algorithms may be exhibited ; however, practical computations seem to show that the non-linear temperature relaxation scheme described here provides better results (see [24] and section IV).
• The authors turn now to the most difficult part which corresponds to the pressure relaxation step.

### 3.5 Computing the pressure relaxation step

• The first one is a semi-implicit scheme, that is such that the existence and uniqueness of the discrete solution is ensured, whatever the equations of state would be.
• The authors focus here on the second one which is totally implicit with respect to the unknown (Pl, Pg, αl).
• The main lines of the proof are given below, also known as Proof.
• The implicit scheme (20) is exactly the same as the one introduced for dense granular gas-particle flows in [10], also known as Remark 2.

### 4 Numerical results

• In the first subsection, the authors focus on the computation of the evolution step involving all convective terms.
• The following unities are used : m for distances, kg/m3 for densities, m/s for velocities, Pa for pressures, K for temperatures, s for times and J/s/m2 for heating fluxes.

### 4.1 Verification of the evolution step

• The authors use for this test case perfect gas EOS within each phase, setting γg = 1.4 and γl = 1.1.
• The second test case is another Riemann problem taken from [13], where initial conditions are given in table 2.
• This Riemann problem also contains a contact wave associated with Ug, and a right-going gas shock wave.

### 4.2.1 Velocity relaxation substep

• Two different series of verification test cases have been considered in [24] for the velocity relaxation step.
• The first one refers to a constant time scale τ3.
• In that case, the scheme (16) is perfect, since it computes the exact value at time tn+1.
• In that case, analytic solutions allow computing the true error occuring in the velocity relaxation step.
• Several examples can be found in [24], which confirm that a first-order rate of convergence is achieved.

### 4.2.2 Temperature relaxation substep

• The authors present below some tests corresponding to constant time scales τ4, when computing approximate solutions of (17) with the scheme (18).
• Φ, unlike the non-linear scheme (18); actually, round-off errors are found for the linear scheme in that case.

### 4.2.3 Pressure relaxation substep

• Eventually, the authors provide an example of measured convergence rates in the pressure relaxation step.
• This one is crucial, and should be handled with great care.
• Otherwise, both the present model (where the relaxation time scale τ2 is non-zero) and standard two-fluid models (corresponding to τ2 = 0) may be confused, if inadequate “rough” schemes are used to provide approximate solutions of (19).
• The initial conditions of the test case are given in table 5. Figure 6 and Figure 7 show the behaviour of the scheme (20) and the comparison with another half-implicit scheme introduced in [22] and recalled in [24], focusing on void fractions and pressures within each phase at time t = 10−5.
• This enables to retrieve the fact that the initial-value problem associated with τ2 = 0 is ill-posed: spurious oscillations arise when the mesh size is sufficiently small, and this is particularly spectacular for void fraction profiles, since the algorithm guarantees bounded variations owing to properties 4 and 5 (see [21] and [17] also for a similar study).

### 4.3 Two-dimensional numerical results

• The authors consider now the two-dimensional unsteady computation of a heated wall in an almost square domain, where the wall contains a small cavity in the middle of the lower part .
• On the contrary the two cells at the exit corners of the cavity do not receive any heat flux.
• It also seems worth mentionning that almost similar results have been obtained while changing the pressure relaxation time step within the range τ2 ∈ [10−9, 10−6], keeping other relaxation time steps unchanged and using the same mesh.
• Nonetheless, one must be aware that the 3D counterpart of the present 2D mesh would contain more than 150 millions of cells, which is of course far beyond what one can afford in an industrial situation.

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A fractional step method to compute a class of
compressible gas–liquid ows
Jean-Marc Hérard, Olivier Hurisse
To cite this version:
Jean-Marc Hérard, Olivier Hurisse. A fractional step method to compute a class of compressible gas–
liquid ows. Computers and Fluids, Elsevier, 2012, 55, pp.57-69. �10.1016/j.compuid.2011.11.001�.
�hal-01265315�

A fractional step method to compute a class of
compressible gas-liquid ﬂows
Jean-Marc H´erard
EDF, R&D
Olivier Hurisse
EDF, R&D
Abstract
We present in this paper some algorithms dedicated to the compu-
tation of numerical approximations of a class of two-ﬂuid two-phase ﬂow
models. Governing equations for the statistical void fraction, partial m ass,
momentum, energy are presented ﬁrst, and meaningful closure laws are
given. Then we m ay gi ve the main properties of the class of two-ﬂuid mod-
els. The whole algorithm that relies on the fractional step method and
complies with the entropy inequality is presented afterwards. Emphasis
is given on the computation of pressure-velocity-temperature relaxation
source terms. Conditions pertaining to the existence and uniqueness of
discrete solutions of the relaxation step are given. While focusing on some
one-dimensional test cases, the true rates of convergence may be obtained
within the evolution step and the relaxation step. Eventually, some two-
dimensional numerical simulations of a heated wall are shown and are
brieﬂy discussed. Some advantages and weaknesses of algorithms are also
discussed.
Keywords:
Two-phase ﬂows / Finite Volume schemes / Two-ﬂuid model / Hyperbolic s ys-
tems / Closure laws / Entropy inequality / Relaxation eﬀects.
1 Introduction
Two distinct types of models are used in order to compute liquid-gas or water-
vapour two-phase ﬂows in industrial codes: the homogeneous approach and
the two-ﬂuid approach. Within the framework of nuclear safety codes[16] ,
the homogeneous approach is generally adopted for codes dedicated to compo-
nents such as reactor cores and steam generators (THYC, FLICA and GENEPI
codes in France), whereas the two-ﬂuid approach is prefered in system codes
Correspond ing author. EDF, R&D, Fluid Dynamics, Power Generation and Environ-
ment, 6 quai Watier, 78400, Chatou, France. Tel: (33)-1-30 87 70 37. Email: jean-
marc.herard@edf.fr
EDF, R&D, Fluid Dynamics, Power Generation and Environment, 6 quai Watier, 78400,
Chatou, France.
1

(CATHARE or RELAP codes for instance) and also in 3D commercial codes
(CFX, Star-CD, Fluent) and inhouse codes (NEPTUNE-CFD). The two-ﬂuid
approach is assume d to be more general, and it is also expected to predict more
accurately ﬂows for which the phasic desequilibrium plays a c rucial role (see
[9, 25, 26]).
Now, two distinct two-ﬂuid approaches may be considered. The ﬁrst one,
which is the most standard one, relies on an instantaneous pressure equilibrium
between both phases. The second one no longer assumes this hypothesis, which
means that the seven unknowns corresponding to the statistical fraction of liq-
uid, the two mean velocities, the two mean temperatures and the two me an
pressures are evaluated by searching approximations of solutions of a coupled
set of seven partial diﬀerential equations (PDE). These equations correspond to
the mass balance, the momentum balance and the energy balance within each
phase, and is supplemented by the governing equation for the statistical liquid
fraction.
This paper is devoted to the simulation of water-gas ﬂow models belonging
to the second class. Though this approach was introduced approximately thirty
years ago (see [33]), few workers have been investigating this class until the
late 90’s. Within the framework of gas-particle ﬂow s, and more precisely when
studying Deﬂagration to Detonation Transition, these models gained a consid-
erable interest within a small community. Among other studies, one should at
least point out the contributions [4, 28, 5, 27, 32] which are concerned with the
modeling aspects. Fewer papers tackle the problem of water-gas or water-vapour
ﬂows, among which we must quote [33, 34] and more recently the article [12] ,
that examines a medium of small oscillating bubbles in a liquid medium, and
also provides a general formalism in order to derive meaningful governing equa-
tions. A classiﬁcation of closure laws related to the interface pressure P
I
and
the interface velocity V
I
was proposed in [8, 11] , which provides a general
framework relying on two m ain ingredients:
(H1) the interface velocity V
I
, which gove rns the evolution of the sta-
tistical void fraction, should be such that the ﬁeld associated with the
eigenvalue λ = V
I
were linearly degenerate;
(H2) a physically relevant entropy inequality should control smooth solu-
tions.
Based on these two keystones, it has been shown in [8, 11] that one may re-
trieve the well-known Baer-Nunziato model, that c orresponds to the particular
choice P
I
= P
l
, V
I
= U
g
, among other possibilities (where P
l
and U
g
respec-
tively refer to the liquid pressure and the gas velocity). We emphasize that this
approach was recently extended to the framework of two-phase ﬂow in porous
media [19, 13] . T he same procedure provides some way to tackle the modeling
of three-phase ﬂows[18] . It also gives a relevant approach for the modeling of
dense granular ﬂows[10] . These extensions conﬁrm the relevance of the whole
2

modeling approach. However, one must be aware that at least two diﬃculties
are hidden in these sets of PDE.
First, the convective part of the system of PDE is hyperbolic with no con-
straining condition on the physical states, but it contains two (or three, de-
pending on the closure law for V
I
) linearly degenerate ﬁelds. A straightforward
consequence is that the asymptotic rate of convergence of so-called ﬁrst-order
(respectively second-order) Riemann solvers is 1/2 (respectively 2/3). This has
recently motivated great eﬀorts in order to build accurate enough Riemann
solvers for the Baer-Nunziato system (see at least [1, 2, 7, 31, 35, 36] ). Sec-
ond, the system contains stiﬀ source terms which are linked with the pressure-
velocity-temperature relaxation eﬀects. These require speciﬁc algorithms, and
the main part of the present paper is actually dedicated to this work. This is
basically motivated by the fact that few available articles discuss these tricky
problems. In the sequel, the two-ﬂuid two-pressure model accounts for velocity,
pressure and temperature relaxation, each one b eing associated with a non-zero
time scale. Actually, most of the papers in the literature focus on some speciﬁc
situations. Dealing with the velocity relaxation eﬀects is not really challenging,
and a lot of schemes have been proposed in order to cope with it. On the con-
trary, there are few articles dealing with the numerical treatment of the pressure
relaxation. In the case of instantaneous relaxation (i.e. with “zero time scales”)
one can refer to [34] (which deals with instantaneous velocity and pressure
relaxation) or [21] (which only deals with instantaneous pressurerelaxation)
among others. For non-zero pressure relaxation time scales, a scheme has been
proposed in [11], with an important drawback : the total energy of the mix-
ture is not conserved by the scheme. Moreover, most of the references in the
literature are based on models where the temperature relaxation source terms
are neglected, and consequently there are no numerical schemes accounting for
them.
Hence the paper is organized as follows. We ﬁrst recall the set of PDEs that
governs the two-ﬂuid model, and recall its main properties. Next we present
approximate Riemann solvers and algorithms used to compute approximations
of solutions of the coupled ODEs arising when taking relaxation eﬀects into
account. The most diﬃcult task dwells in the building of suitable algorithms in
the pressure relaxation step. Of course this diﬃculty vanishes when the pres-
sure relaxation time scale is set to zero, but in that case convergence diﬃculties
-with respect to the mesh size- may be expected (see [21, 17, 10]) when focusing
on unsteady computations. The ﬁrst section of numerical results focuses on the
practical estimation of the rate of convergence when looking for approximate
solutions of the convective subset. The second section gives emphasis on the
approximation of velocity-pressure-temperature relaxation eﬀects. The last sec-
tion is devoted to the two-dimensional simulation of the ﬂow close to a heate d
wall.
3

2 Governi ng equations and main properties of
the two -ﬂuid m odel
2.1 The two-ﬂuid model
Throughout the paper, indexes l, g refer to the liquid and gas phases; the sta-
tistical void fractions of gas and liquid are noted classically α
g
and α
l
, which
should agree with:
α
l
+ α
g
= 1
The mean pressures, mean velocities and mean densities of the two phases are
denoted P
φ
, U
φ
and ρ
φ
respectively, for φ = l, g. The total energy within each
phase is:
E
φ
= ρ
φ
e
φ
(P
φ
, ρ
φ
) + ρ
φ
U
2
φ
2
, φ = g, l (1)
Internal energy functions e
φ
are provided by users.
The so-called conservative variable W will be deﬁned as:
W = (α
l
, α
l
ρ
l
, α
l
ρ
l
U
l
, α
l
E
l
, α
g
ρ
g
, α
g
ρ
g
U
g
, α
g
E
g
)
Moreover, P
I
(W ) and V
I
(W ) respectively denote in this paper the interfacial
pressure and velocity, and will precised afterwards. These interface terms V
I
and P
I
will be such that:
jump conditions are well deﬁned within each isolated ﬁeld;
a physically relevant entropy inequality holds for smooth solutions of (2).
Given these notations, the governing set of equations for ﬁrst-order moments
may be written as follows in a multi-dimensional framework:
t
(α
l
) + V
I
α
l
= S
1,l
t
(α
l
ρ
l
) + .(α
l
ρ
l
U
l
) = S
2,l
t
(α
l
ρ
l
U
l
) + .(α
l
ρ
l
U
l
U
l
+ α
l
P
l
I) P
I
(α
l
) = S
3,l
t
(α
l
E
l
) + .(α
l
U
l
(E
l
+ P
l
)) + P
I
t
(α
l
) = S
4,l
t
(α
g
ρ
g
) + .(α
g
ρ
g
U
g
) = S
2,l
t
(α
g
ρ
g
U
g
) + .(α
g
ρ
g
U
g
U
g
+ α
g
P
g
I) P
I
(α
g
) = S
3,l
t
(α
g
E
g
) + .(α
g
U
g
(E
g
+ P
g
)) + P
I
t
(α
g
) = S
4,l
(2)
where right-hand side terms S
k,l
(W ) represent the source terms (for k = 2, 3, 4),
which enable to account for mass transfer, momentum and energy transfer
through the inte rface between the two phases. The term S
1,l
will also be in-
troduced later on. External sources might be included but are not considered
herein. A derivation of the ﬁrst governing equation of the statistical liquid frac-
tion α
l
can be found in[20] . Standard viscous contributions may be included,
which of course comply with the entropy inequality that will be detailed in the
4

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Abstract: Part I Fundamental of two-phase flow.- Introduction.- Local Instant Formulation.- Part II Two-phase field equations based on time average.- Basic Relations in Time Average.- Time Averaged Balance Equation.- Connection to Other Statistical Averages.- Part III. Three-dimensional model based on time average.- Kinematics of Averaged Fields.- Interfacial Transport.- Two-fluid Model.- Interfacial Area Transport.- Constitutive Modeling of Interfacial Area Transport.- Hydrodynamic Constitutive Relations for Interfacial Transfer.- Drift Flux Model.- Part IV: One-dimensional model based on time average.- One-dimensional Drift-flux Model.- One-dimensional Two-fluid Model.- Two-Fluid Model Considering Structural Materials in a Control Volume.

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• ...2 Closure laws for interfacial transfer terms The interfacial transfer contributions have been studied in [4, 5, 17, 10, 28, 29], and a summary can be found in appendix A....

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• ...with “zero time scales”) one can refer to [34] (which deals with instantaneous velocity and pressure relaxation) or [21] (which only deals with instantaneous pressurerelaxation) among others....

[...]

• ...Fewer papers tackle the problem of water-gas or water-vapour flows, among which we must quote [33, 34] and more recently the article [12] , that examines a medium of small oscillating bubbles in a liquid medium, and also provides a general formalism in order to derive meaningful governing equations....

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