DSpace at VNU: On a two-fluid model of two-phase compressible flows and its numerical approximation

Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

Contents lists available at ScienceDirect

Commun Nonlinear Sci Numer Simulat
journal homepage: www.elsevier.com/locate/cnsns

On a two-fluid model of two-phase compressible flows
and its numerical approximation
Mai Duc Thanh
Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam

a r t i c l e

i n f o

Article history:
Received 22 September 2010
Accepted 7 May 2011
Available online 13 May 2011
Keywords:
Two-fluid
Two-phase flow
Conservation law
Source term
Lax–Friedrichs
Numerical scheme

a b s t r a c t
We consider a two-fluid model of two-phase compressible flows. First, we derive several
forms of the model and of the equations of state. The governing equations in all the forms

contain source terms representing the exchanges of momentum and energy between the
two phases. These source terms cause unstability for standard numerical schemes. Using
the above forms of equations of state, we construct a stable numerical approximation for
this two-fluid model. That only the source terms cause the oscillations suggests us to minimize the effects of source terms by reducing their amount. By an algebraic operator, we
transform the system to a new one which contains only one source term. Then, we discretize the source term by making use of stationary solutions. We also present many numerical
tests to show that while standard numerical schemes give oscillations, our scheme is stable
and numerically convergent.
Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction
We consider in the present paper the stratified flow model for the two-fluid model in one-dimensional space variable. The
model with gravity consisting of 6 governing equations is given by (see Staedtke et al. [13] and García-Cascales and Paillère
[6]):

@ t ðag qg Þ þ @ x ðag qg ug Þ ¼ 0;
@ t ðag qg ug Þ þ @ x ðag ðqg u2g þ pÞÞ ¼ p@ x ag þ ag qg g;
@ t ðag qg Eg Þ þ @ x ðag ðqg Eg þ pÞug Þ ¼ Àp@ t ag þ ag qg ug g;
@ t ðal ql Þ þ @ x ðal ql ul Þ ¼ 0;

ð1:1Þ

@ t ðal ql ul Þ þ @ x ðal ðql u2l þ pÞÞ ¼ p@ x al þ al ql g;
@ t ðal ql El Þ þ @ x ðal ðql El þ pÞul Þ ¼ Àp@ t al þ al ql ul g;
where ai is the volume fraction, qi is the density, ui is the velocity, ei is the internal energy and

1
Ei ¼ ei þ u2i ;
2
is the total energy, g is the gravity constant in the model with gravity, and g = 0 in the model without gravity, and the subscript ‘‘i’’ can be ‘‘g’’ or ‘‘l’’, representing the gas or liquid phase of fluids respectively. The volume fractions of the fluid satisfy

ag þ al ¼ 1:
E-mail addresses: ,
1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.cnsns.2011.05.010


196

M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

The source terms in non-conservative form on the right-hand side (1.1) are the interphase interaction terms, indicating the
momentum and energy exchanged between phases. The system of equations is closed by supplementing the equations of state
of gas and liquid phases. In this paper, we assume that the gas phase has the equation of state of the perfect gas for the air

p ¼ ðcg À 1Þqg eg ;

ð1:2Þ

and we use the stiffened gas equation of state for the liquid phase

p ¼ clcÀ1 ql C pl T l À p1
l

el ¼

C pl

cl

T l þ pq1

ð1:3Þ

l

p ¼ ðcl À 1Þql el þ cl p1 :
For the tests, we will take (see Chang and Liou [2], for example)
kJ
cg ¼ 1:4; C gp ¼ 1:012 kg:K
;
kJ
cl ¼ 1:9276; C lp ¼ 8:07673 kg:K
; p1 ¼ 11:5968 Â 103 at:

ð1:4Þ

Compressible multi-fluid flow models such as (1.1) has been widely used to model multi-phase flows, see for example
Ishii [7], Stewart and Wendroff [14], and Chang and Liou [2]. However, there are concerns for this modeling. First, multi-fluid
models are often not to be hyperbolic, and this would lead to an ill-posed initial-valued problem. Moreover, analytical form
of the characteristic fields are not available. Hence, it is hard to use the conventional Roe or Godunov type of schemes to
calculate the numerical fluxes. Therefore, additional equations or correction terms must be included to make it well-posed
and analytical form of characteristic fields can be derived. Various types of corrected source terms of models of two-phase
flows were presented by Garcià-Cascales and Paillère [4]. Second, the system (1.1) is not in conservative form and can be
understood in the sense of nonconservative product, which was introduced by Dal Maso et al. [3]. For related works, but rather
for simpler models, were done by Marchesin and Paes-Leme [12], LeFloch and Thanh [10,11], and Thanh [16], where the
Riemann problem is solved. Third, but may not the last, source terms cause lots of inconveniences in approximating physical
solutions of the system as the errors may become larger as the meshes are refined. Therefore, constructing a stable scheme
will be important for study and applications.
Numerical treatments for systems of balance law with source terms that do not rely on the analytic forms of the characteristic fields have been attracted many authors. In the case of a single conservation law, well-balance schemes which are stable
were constructed by Greenberg and Leroux [5], and Botchorishvili et al. [1]. In the system case, a well-balance and stable

scheme for the model of fluid flows in a nozzle with variable cross-section was constructed and its properties were established
by Kröner and Thanh [9], Kröner et al. [8]. A well-balance scheme for shallow water equations was constructed by Thanh et al.
[15]. Recently, a well-balance scheme for a one-pressure model of two-phase flows, where one phase is compressible, the other
phase is incompressible, was established by Thanh and Ismail [17], where the sources are reduced to one phase.
In this paper we present several forms of the system (1.1) and the equations of state, and then using these forms of equations of state we construct a numerical scheme that is numerically stable for the system (1.1). We observe that only the nonconservative terms cause the oscillations for standard schemes. This suggests us to minimize the effects of source terms by
reducing the number of source terms involving in the system. By algebraic addition, we reduce the system to the new one
containing exactly one source term. The conservative equations can be dealt with using a convenient standard numerical
scheme. Motivated by our earlier works, we discretize the source term in the non-conservative equation using stationary
waves in the gas phase.
The paper is organized as follows. In Section 2 we provide results which describes basic properties of the two-fluid model.
We solve present basic facts that are useful for the computations of the next sections. In Section 3 we will adapt a standard
scheme and construct a stable scheme for (1.1). In Section 4 we present several tests which show that the adaptive scheme is
numerically stable. Finally, in Section 5 we provide some discussions and conclusions.
2. Several forms of the model and properties
2.1. Other forms of the two-fluid model
First, we derive several equivalent forms for the two-fluid model (1.1) for smooth solutions. Each of the equivalent form
differs from each other by the equation of balance of energy. There are three equivalent models where the equation for the
balance of energy in each phase is written in terms of the total energy as in (1.1), in terms of the internal energy, or in terms
of the specific entropy.
2.1.1. Model involving equation for internal energy
First, we check that the system (1.1) is equivalent to the following system

@ t ðai qi Þ þ @ x ðai qi ui Þ ¼ 0;
@ t ðai qi ui Þ þ @ x ðai ðqi u2i þ pÞÞ ¼ p@ x ai þ ai qi g;
@ t ðai qi ei Þ þ @ x ðai qi ei ui Þ ¼ Àpð@ t ai þ @ x ðai ui ÞÞ;

ð2:1Þ
i ¼ g; l:

197

M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

Indeed, let us rewrite the equations of balance of momentum in each phase as

ui @ t ðai qi Þ þ ai qi @ t ui þ ai qi ui @ x ui þ ui @ x ðai qi ui Þ þ ai @ x p ¼ ai qi g;
for i = g, l. Multiplying both sides of the last equation by ui, i = g, l, and then applying the chain rule, we obtain

u2i @ t ðai qi Þ þ ai qi @ t

 2
 2
ui
u
þ ai qi ui @ x i þ u2i @ x ðai qi ui Þ þ ai ui @ x p ¼ ai qi ui g;
2
2

or





u2i
u2
u2
ð@ t ðai qi Þ þ @ x ðai qi ui ÞÞ þ @ t ai qi i þ @ x ai qi ui i þ ai ui @ x p ¼ ai qi ui g;
2

2
2

i ¼ g; l:

Due to the conservation of mass, this yields





u2
u2
@ t ai qi i þ @ x ai qi ui i þ ai ui @ x p ¼ ai qi ui g;
2
2

i ¼ g; l:

ð2:2Þ

Adding Eq. (2.2) to the equation for internal energy in (2.1), we get






 
u2

u2
þ @ x ai qi ei þ
ui þ p@ t ai þ ðp@ x ðai ui Þ þ ai ui @ x pÞ ¼ ai qi ui g;
@ t ai qi ei þ
2
2

i ¼ g; l;

or






 
u2
u2
@ t ai qi ei þ
þ @ x ai qi ei þ
ui þ p@ t ai þ @ x ðai ui pÞ ¼ ai qi ui g;
2
2

i ¼ g; l:

The last equation gives

  





 
u2
u2
@ t ai qi ei þ
þ @ x ai qi ei þ
þ p ui þ p@ t ai ¼ ai qi ui g;
2
2

i ¼ g; l;

or

@ t ðai qi Ei Þ þ @ x ðai ðqi Ei þ pÞui Þ þ p@ t ai ¼ ai qi ui g;

i ¼ g; l;

ð2:3Þ

which gives the equations of balance of energy in (1.1). Thus, the system (2.1) and the system (1.1) are equivalent.
2.1.2. Model involving equation for the specific energy
Furthermore, the equation for internal energy in (2.1) can be written as

@ t ðai qi Þei þ ðai qi Þ@ t ei þ ei @ x ðai qi ui Þ þ ðai qi ui Þ@ x ei þ pð@ t ai þ @ x ðai ui ÞÞ ¼ 0;

i ¼ g; l;

or

ei ð@ t ðai qi Þ þ @ x ðai qi ui ÞÞ þ ai qi ð@ t ei þ ui @ x ei Þ þ pð@ t ai þ @ x ðai ui ÞÞ ¼ 0;

i ¼ g; l:

Therefore, equations for conservation of mass imply that



ai qi ð@ t ei þ ui @ x ei Þ þ p @ t ai þ @ x ðai ui Þ À

1

qi


ð@ t ðai qi Þ þ @ x ðai qi ui ÞÞ ¼ 0;

i ¼ g; l;

or



ai qi ð@ t ei þ ui @ x ei Þ þ p @ t ai þ @ x ðai ui Þ À

1

qi


ðai @ t qi þ qi @ t ai þ qi @ x ðai ui Þ þ ai ui @ x qi Þ ¼ 0:

Cancel the terms to get

ai qi ð@ t ei þ ui @ x ei Þ À

pai

qi

ð@ t qi þ ui @ x qi Þ ¼ 0;

or



ai qi @ t ei þ ui @ x ei À


ð@
q
þ
u
@
q
Þ
¼ 0:

t i
i x i
2

p

qi

Re-arranging terms, we get

ai qi



p
p
¼ 0;
@ t ei À 2 @ t qi Þ þ ui ð@ x ei À 2 @ x qi

qi

qi


198

M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

or

ai qi


 

 
1
1
þ ui @ x ei þ p@ x
¼ 0;
@ t ei þ p@ t

qi

qi

i ¼ g; l:

Using thermodynamical identity

T i dSi ¼ dei þ pd

1

qi

;

i ¼ g; l;

we can rewrite the above equation as

ai qi ðT i @ t Si þ ui T i @ x Si Þ ¼ 0;
or

ai qi ð@ t Si þ ui @ x Si Þ ¼ 0; i ¼ g; l:

ð2:4Þ

Adding the multiple of equations for conservation of mass to Eq. (2.4):

ai qi ð@ t Si þ ui @ x Si Þ þ Si ð@ t ðai qi Þ þ @ x ðai qi ui ÞÞ ¼ 0;
we deduce the equations for entropy:

@ t ðai qi Si Þ þ @ x ðai qi Si ui Þ ¼ 0;

i ¼ g; l:

The above calculation shows that the system (2.1) can be written as

@ t ðai qi Þ þ @ x ðai qi ui Þ ¼ 0;
@ t ðai qi ui Þ þ @ x ðai ðqi u2i þ pÞÞ ¼ p@ x ai þ ai qi g;
@ t ðai qi Si Þ þ @ x ðai qi Si ui Þ ¼ 0;

ð2:5Þ

i ¼ g; l:

2.2. Equations of states
Each fluid is characterized by its equation of state. In this section, we will establish the equations of state in terms of the

pressure and the entropy as the thermodynamical independent variables. For the gas phase the polytropic ideal gas has the
equation of state of the form

p ¼ qg Rg T g ¼ ðcg À 1Þqg eg ;
R T

eg ¼ C gv T ¼ c gÀ1g ;

ð2:6Þ

hg ¼ C pg T g

g

where Rg is the specific gas constant; Rg ¼ gg R, where g is the mole-mass fraction, and R is the universal gas constant. Let
C gv ¼ c

Rg
g À1

R c

be the specific heat at constant volume and C gp ¼ c gÀ1g be the specific heat at constant pressure, so that Cgp = cgCgv.
g

Using thermodynamical identity, we have

dqg
deg
dSg ¼ C gv

À ðcg À 1Þ
eg
qg

!

This gives

eg

Sg À S ¼ C gv log
gÃ

!

c À1

qgg

or
c À1

eg ¼ qgg

exp



Sg À Sg Ã
:

C gv

Therefore,
c

p ¼ ðcg À 1Þqgg exp



Sg À Sg Ã
:
C gv

ð2:7Þ

Thus,

qg ¼ qg ðp; Sg Þ ¼

p
cg À 1

!1=cg
exp



Sg à À Sg
:
C gp

ð2:8Þ


M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

199

Furthermore, by definition, the enthalpy is

p

hg ¼ eg þ

qg

¼ e þ ðcg À 1Þe ¼ cg e:

So that

hg ¼ cg exp



Sg À Sgà cg À1
qg ;
C gv

or, for short,
c À1

hg ¼ hg ðqg ; Sg Þ ¼ GðSÞqgg

where GðSÞ ¼ cg exp

;



Sg À SgÃ
:
C gv

ð2:9Þ

Substituting qg from (2.8) into (2.9), and re-arranging terms, we get

hg ¼ hg ðp; Sg Þ ¼

cg
ðcg À 1Þ

ðcg À1Þ=cg

pðcg À1Þ=cg exp




Sg À Sg Ã

:
C gp

ð2:10Þ

Next, we consider the liquid phase where the equations of state are given by

p ¼ ðcl À 1Þql C lv T l À p1 ;

el ¼ C lv T l þ

p1

ql

ð2:11Þ

;

cl
l
where C lv ¼ c RÀ1
be the specific heat at constant volume and C lp ¼ cRlÀ1
be the specific heat at constant pressure, so that
l
l
Clp = clClv, where Rl is the specific gas constant.
Substituting el from (2.11) into the thermodynamical identity del = TldSl À pdvl, vl = 1/ql, we obtain

dðC v l T l þ p1 v l Þ ¼ T l dSl À pdv l :

This yields

dSl ¼ C lv

dT l p þ p1
þ
dv l
Tl
Tl

or

1
dT l
dv l
dSl ¼
þ ðcl À 1Þ
:
C lv
Tl
vl
Hence,

Sl À Sl Ã
c À1
¼ logðT l v l l Þ
C lv
so that
c À1

T l ¼ ql l

exp



Sl À SlÃ
:
C lv

ð2:12Þ

Substituting Tl from (2.12) into (2.11), we obtain

p ¼ ðcl À 1ÞC lv exp



Sl À Slà cl
ql À p1 :
C lv

This implies

ql ¼ ql ðp; Sl Þ ¼


1=cl



p þ p1
S Ã À Sl
:
exp l
Rl
C lp

ð2:13Þ

Moreover, the enthalpy in the liquid phase is given by

hl ¼ el þ

p

ql

¼ C lv T l þ

p1

ql

þ ðcl À 1ÞC lv T l ¼ cl C lv T l :

Substituting Tl from (2.12) into the last equation gives
c À1

hl ¼ C lp exp ðSl À Slà C lv Þql l
or

c À1

hl ¼ hl ðql ; Sl Þ ¼ LðSÞql l ;

where LðSÞ ¼ C lp exp



Sl À Sl Ã
:
C lv

ð2:14Þ


200

M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

Substituting ql from (2.13) into (2.14), we obtain

hl ¼ hl ðp; Sl Þ ¼ C lp


ðc À1Þ=cl


p þ p1 l
Sl À Sl Ã
:

exp
Rl
C lp

ð2:15Þ

2.3. Non-hyperbolic system of balance law
Let us choose the pressure p and the entropies Sg, Sl as the thermodynamical independent variables. Then, the unknown
function, or the state variable, can be found under the form

V ¼ ðag ; p; ug ; Ss ; ul ; Sl ÞT :

ð2:16Þ

To find the Jacobian matrix of (1.1) or (2.5) for this variable V, we need to re-write (2.5) in the form

@ t V þ AðVÞ@ x V ¼ 0;

ð2:17Þ

for which A(V) is the Jacobian. First, it is derived from (2.5) that the equations for entropy can be written as

@ t Si þ ui @ x Si ¼ 0;

i ¼ g; l:

ð2:18Þ

Since qi = qi(p, Si), i = g, l, the chain rule implies that the equations for mass can be written as

ai ð@ p ðqi Þ@ t p þ @ Si ðqi Þ@ t Si Þ þ qi @ t ai þ ai ui ð@ p ðqi Þ@ x p þ @ Si ðqi Þ@ x Si Þ þ qi @ x ðai ui Þ ¼ 0; i ¼ g; l
or

ai @ p ðqi Þ@ t p þ qi @ t ai þ ai @ Si ðqi Þð@ t Si þ ui @ x Si Þ þ ai ui @ p ðqi Þ@ x p þ qi @ x ðai ui Þ ¼ 0; i ¼ g; l:
Using (2.18), we get from the last equation

ai @ p ðqi Þ@ t p þ qi @ t ai þ ai ui @ p ðqi Þ@ x p þ qi @ x ðai ui Þ ¼ 0; i ¼ g; l:
The last two equations (i = g, l) yield an equation for the volume fraction

ðal @ p ðql Þqg þ ag @ p ðqg Þql Þ@ t at þ ap @ p ðqg Þal @ p ðql Þðug À ul Þ@ x p þ qg al @ p ðql Þ@ x ðag ug Þ À ql ag @ p ðqg Þ@ x ðal ul Þ ¼ 0;

ð2:19Þ

and an equation for the pressure

ðql ag @ p qg þ qg al @ p ql Þ@ t p þ ðql ag ug @ p qg þ qg al ul @ p ql Þ@ x p þ ql qg @ x ðag ug þ al ul Þ ¼ 0:

ð2:20Þ

Next, the equations for mass can be written as

ai qi @ t ui þ ui @ t ðai qi Þ þ ui @ x ðai qi ui Þ þ ai qi ui @ x ui þ ai @ x p þ p@ x @ x ai ¼ p@ x ai ;
for i = g, l. Canceling the terms and using the equations of mass, we obtain

@ t ui þ ui @ x ui þ

1

qi

@ x p ¼ 0;

i ¼ g; l:

ð2:21Þ

Thus, we obtain a system from (2.18, 2.21):

@ t at þ a @ p ðq Þq
l

l



1
g þag @ p ðqg Þql



1

@ t p þ q ag @ p q

g þqg al @ p ql

l



ap @ p ðqg Þal @ p ðql Þðug À ul Þ@ x p þ qg al @ p ðql Þ@ x ðag ug Þ À ql ag @ p ðqg Þ@ x ðal ul Þ ¼ 0;


ðql ag ug @ p qg þ qg al ul @ p ql Þ@ x p þ ql qg @ x ðag ug þ al ul Þ ¼ 0;

@ t ug þ ug @ x ug þ q1 @ x p ¼ 0;

ð2:22Þ

g

@ t Sg þ ug @ x Sg ¼ 0;
@ t Sl þ ul @ x Sl ¼ 0;
@ t ul þ ul @ x ul þ q1 @ x p ¼ 0:
l

Setting

a1 ¼

qg ug al @ p ql þql ul ag @ p qg
al @ p ðql Þqg þag @ p ðqg Þql

;

a4 ¼ a @ p ðql Þqg

q q ðu Àu Þ

;

b2 ¼

;

b4 ¼

b1 ¼ a @ p ðq lÞqg þgag @ pl ðq
l

l

g

g Þql

qq a

b3 ¼ a @ p ðq Þq l þgagg@ p ðq
l

ag @ p ðqg Þal @ p ðql Þðug Àul Þ
al @ p ðql Þqg þag @ p ðqg Þql ;

q a @ p ðql Þag
;
l g þag @ p ðqg Þql

a3 ¼ a @ p ðqgÞql
l

a2 ¼

l

g

g Þql

l

q a @ p ðqg Þal
;
l g þag @ p ðqg Þql

ql ag ug @ p qg þqg al ul @ p ql
al @ p ðql Þqg þag @ p ðqg Þql ;
ql qg al
al @ p ðql Þqg þag @ p ðqg Þql ;


201

M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

we can write the Jacobian matrix of the system (2.22) as

0

a1

Bb
B 1
B
B0
B
AðVÞ ¼ B
B0
B
B0
@

a3

0 a4

b2

b3

0 b4

1

u

0

0

0

0

u

0

1

0

0

ul

0C
C
C
0C
C
C:
0C
C
0C
A

0

0

0

0

0

ul

qg

ql

0

1

a2

ð2:23Þ

A straightforward calculation shows that the characteristic polynomial det (A(V) À kI) for the matrix (2.23) is given by

PðkÞ ¼ ðug À kÞðul À kÞQ ðkÞ

ð2:24Þ

where

Q ðkÞ ¼ ða1 À kÞðb2 À kÞðug À kÞðul À kÞ À a2 b1 ðug À kÞðul À kÞ À

þ

a4 b1

ql

b3

qg

ðul À kÞ þ

b4

ql

!
ðug À kÞ ða1 À kÞ þ

a3 b1

qg

ðul À kÞ

ðug À kÞ:

The polynomial P(k), however, may or may not have a complete set of real zeros. This can be seen as the polynomial Q(k) may
have or may not have four real zeros, as illustrated by the Fig. 1. Consequently, the system is hyperbolic in certain regions of
the phase domain and is not hyperbolic in other regions.

3. Construction of the stable numerical scheme
Since the gravity terms agqgg, alqlg are regular terms and do not play any significant role, in the following we restrict our
consideration to the case without gravity to simplify our argument. This can be formally done by letting g = 0.
Let Dt and Dx be the given uniform time step and the spacial mesh size, respectively. Set

xj ¼ jDx;
Denote by

U nj

k¼

Dt
:
Dx

j 2 Z;

tn ¼ nDt;

n 2 N:

the approximate value of the value U(xj, tn) of the exact solution U at the point (x, t) = (xj, tn). We also set

Adding up the equations of balance law of momentum and re-arranging the equations, we obtain the following equivalent
system

@ t ðag qg Þ þ @ x ðag qg ug Þ ¼ 0;
@ t ðag qg Sg Þ þ @ x ðag qg Sg ug Þ ¼ 0;
@ t ðal ql Þ þ @ x ðal ql ul Þ ¼ 0;

@ t ðal ql Sl Þ þ @ x ðal ql Sl ul Þ ¼ 0;
@ t ðag qg ug þ al ql ul Þ þ @ x ðag ðqg u2g þ pÞ þ al ðql u2l þ pÞÞ ¼ 0;
@ t ðag qg ug Þ þ @ x ðag ðqg u2g þ pÞÞ ¼ p@ x ag :

Fig. 1. Graphs of Q (k) defined by (2.25) show two cases: (a) there are four zeros (left); (b) there are only two zeros.

ð3:1Þ


202

M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

The first five equations are conservative. Thus, we can apply an appropriate standard numerical scheme for the first five conservative equations of (3.1). The last equation of (3.1) is nonconservative. Motivated by our earlier works, [9,17], we discretize the source term of the last equation using the stationary waves in the gas phase. Precisely, we denote

1

0

1

0

ag qg ug
ag qg
C
C
B
B
ag qg Sg ug

ag qg Sg
C
C
B
B
C
C
B
B
C;
C; f ðUÞ ¼ B
a
q
u
a
q
U¼B
l
l
l
l
l
C
C
B
B
C
C
B
B

al ql Sl ul
al ql Sl
A
A
@
@
2
2
ag qg ug þ al ql ul þ p
ag qg ug þ al ql ul
V ¼ ðag qg ug Þ; gðVÞ ¼ ðag ðqg u2g þ pÞÞ:

ð3:2Þ

We now define a ‘‘composite’’ scheme



U nþ1
¼ U nj À k FðU nj ; U njþ1 Þ À FðU njÀ1 ; U nj Þ ;
j


V nþ1
¼ V nj À k GðV nj ; V njþ1;À Þ À GðV njÀ1;þ ; V nj Þ ;
j

ð3:3Þ

where F and G are convenient standard numerical fluxes. The vectors V njþ1;Æ in (3.3) will be determined as follows, relying on

the previous result that the entropy Sg is conserved across stationary waves. We thus compute qng;jþ1;À ; ung;jþ1;À from the
equations

ang;j qng;jþ1;À ung;jþ1;À ¼ ang;jþ1 qng;jþ1 ung;jþ1 ;
ðung;jþ1;À Þ2
2

þ hg ðqng;jþ1;À ; Sng;jþ1 Þ ¼

ðung;jþ1 Þ2
2

ð3:4Þ

þ hg ðqng;jþ1 ; Sng;jþ1 Þ;

and compute qng;jÀ1;þ ; ung;jÀ1;þ from the equations

ang;j qng;jÀ1;þ ung;jÀ1;þ ¼ ang;jÀ1 qng;jÀ1 ung;jÀ1 ;
ðung;jÀ1;þ Þ2
2

þ hg ðqng;jÀ1;þ ; Sng;jÀ1 Þ ¼

ðung;jÀ1 Þ2
2

ð3:5Þ

þ hg ðqng;jÀ1 ; Sng;jÀ1 Þ:

For simplicity we omit the indices for the grids (j, n), j 2 Z, n 2 N in (3.4) and (3.5). In both cases we are led to resolve for
(qg, ug) from the simultaneous nonlinear equations

ag qg ug ¼ ag;0 qg;0 ug;0 :¼ mg ;
ðug Þ2
2

þ hg ðqg ; Sg;0 Þ ¼

ðug;0 Þ2
2

þ hg ðqg;0 ; Sg;0 Þ :¼ qg ;

ð3:6Þ

i ¼ g; l;

Substituting ug = mg/agqg from the first equation of (3.6) to the second equation, after re-arranging terms, we obtain the following nonlinear equation for the density equation

F g ðqg Þ :¼ ðu2g;0 þ 2hg ðqg;0 ; Sg;0 ÞÞq2g À 2q2g hg ðqg ; Sg;0 Þ À



ag;0 ug;0 qg;0
ag

2

¼ 2q2g ðqg À hg ðqg ; Sg;0 ÞÞ À

 2
mg

ag

¼ 0:

ð3:7Þ

Let us investigate properties of the function Fg. It is derived from (3.7) that

F g ðqg Þ
¼ 2ðu2g;0 þ 2hg ðqg;0 ; Sg;0 ÞÞqg À 4qg hg ðq; Sg;0 Þ À 2q2g hg;qg ðqg ; Sg;0 Þ
dqg
¼

2ðu2g;0

þ 2hg ðqg;0 ; Sg;0 ÞÞqg À 4qg hg ðqg ; Sg;0 Þ À 2qg pq ðqg ; Sg;0 Þ ¼ 2qg

¼ 2qg u2g;0 þ 2

Z

qg;0

pqg ðs; Sg;0 Þ

s

qg

!

u2g;0

þ2

Z

qg;0

qg

!
hqg ðs; Sg;0 Þds À pqg ðqg ; Sg;0 Þ

ds À pqg ðqg ; Sg;0 Þ ;

which has the same sign as

Gg ðqg Þ :¼

U0 ðqg Þ
2qg

¼ u2g;0 þ 2

Z

qg;0

pqg ðs; Sg;0 Þ

qg

s

ds À pqg ðqg ; Sg;0 Þ:

Moreover, it is not difficult to check that

2pqg ðqg ; Sg;0 Þ þ qg pqg qg ðqg ; Sg;0 Þ > 0;

i ¼ g; l;

ð3:8Þ


M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

203

This implies that


Gg ðqg Þ
1 

:¼ Àð2hqg ðqg ; Sg;0 Þ þ pqg qg ðqg ; Sg;0 ÞÞ ¼ À
2pqg ðqg ; Sg;0 Þ þ qg pqg qg ðqg ; Sg;0 Þ < 0 8 qg :
dqg
qg

ð3:9Þ

Moreover, it is easy to check that

Gg ðqg Þ > 0 as qg ! 0þ;
Gg ðqg Þ < 0 for large qg ;

i ¼ g; l:

Hence, there exists exactly one value qg = qg,max such that Gg(qg,max) = 0, Gg(qg) > 0 iff qg < qg,max, and Gg(qg) < 0 iff qg > qg,max
for i = g, l. Moreover, a straightforward calculation shows that

qg;max ¼
ql;max ¼



2qg
cg ðcg þ1Þ



2ql
C lp

exp

exp




Sg à ÀSg;0
C gv

Slà ÀSl;0
C lv

c 1À1

c 1À1
l

g

;
ð3:10Þ

:

Consequently,
F g ðqg Þ
dqg

> 0;

qg < qg;max ;

F g ðqg Þ
dqg

< 0;

qg > qg;max ;

F g ðqg Þ
dqg

¼ 0;

qg ¼ qg;max ; i ¼ g; l:

ð3:11Þ

Observe that

F g ðqg ¼ 0Þ < 0;
F g ðqg Þ ! À1 as qg ! þ1:

ð3:12Þ

Thus, (3.7) has a solution iff

F g ðqg;max Þ P





ag;0 ug;0 qg;0 2
;
ag

ð3:13Þ

or, equivalently,

ag;0 jug;0 jqg;0
ag P amin ðU g;0 Þ :¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
F g ðqg;max Þ

ð3:14Þ

In this case, we can easily see that (3.7), and, therefore, system (3.6), has two roots, denoted by ng,1(Ug,0, ag) 6 ng,2(Ug,0, ag),
which coincide iff ag = ag,min(Ug,0). The fact that ag = ag,0 still gives us a solution of (3.6) and, therefore, of (3.7), ag,0 has to
satisfy

ag;min ðU g;0 Þ 6 ag;0 :

Fig. 2. Test 1: usual discretization of the right-hand side does not give a satisfactory result.


204

M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

To make sure that the scheme always works, we practically assign the new value for ag by

anew
¼ maxfag ; amin ðU g;0 Þg:
g

ð3:15Þ

4. Test cases
In this section, we will present several numerical tests. For simplicity, we use the standard Lax–Friedrichs scheme for the
two numerical fluxes F and G in (3.3):

U nþ1
¼ 12 ðU njþ1 þ U njÀ1 Þ À 2k ðf ðU njþ1 Þ À f ðU njÀ1 ÞÞ;
j
V nþ1
¼ 12 ðV njþ1;À þ V njÀ1;þ Þ À 2k ðgðV njþ1;À Þ À gðV njÀ1;þ ÞÞ:
j

ð4:1Þ

For comparison purposes, we take the classical Lax–Friedrichs scheme with a usual discretization of the right-hand side,
says, the central difference formula. The solution will be computed on the interval [À1, 1] of the x-space for 300 mesh points
and at the time t = 0.1. All the tests show that the classical scheme is unstable, but our adaptive scheme is stable.
In fact, let us consider the Riemann problem for the system (1.1) with the Riemann data

ðag;0 ; p0 ; ug;0 ; Sg;0 ; ul;0 ; Sl;0 ÞðxÞ ¼



if

x<0

U R ; if

x > 0;

UL ;

where UL, UR are given for each test.

Fig. 3. Test 1: adaptive scheme (3.3) can give a stable solution.

Fig. 4. Test 2: usual discretization of the right-hand side does not give a satisfactory result.


M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

205

Test 1. We can take

Sgà ¼ 0:2;

Slà ¼ 0;

and

U L ¼ ð0:7; 1; 1; 1; 1; 1Þ;
U R ¼ ð0:8; 1:1; 1:1; 1:1; 1:2; 1:1Þ:

ð4:2Þ

Fig. 2 shows that the Lax–Friedrichs scheme with the usual discretization of the right-hand side does not give a satisfactory
result.Fig. 3 indicates that our adaptive scheme (3.3) gives stable solutions.
Test 2. We may choose

Sgà ¼ 0;

Slà ¼ 0:1;

and

U L ¼ ð0:6; 1:1; 1; 0:5; 0:5; 0:5Þ;
U R ¼ ð0:5; 1; 1:2; 0:6; 0:8; 0:8Þ:

Fig. 5. Test 2: adaptive scheme (3.3) can give a stable solution.

Fig. 6. Test 3: usual discretization of the right-hand side does not give a satisfactory result.

ð4:3Þ


206

M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

Fig. 4 shows that the Lax–Friedrichs scheme with the usual discretization of the right-hand side does not give a satisfactory result.
Fig. 5 indicates that our adaptive scheme (3.3) gives stable solutions.
Test 3. We choose

Sgà ¼ 0;

Slà ¼ 0;

and

U L ¼ ð0:2; 0:9; 1; 0:3; 0:8; 0:5Þ;
U R ¼ ð0:3; 1; 0:3; 0:2; 1; 0:4Þ:

ð4:4Þ

Fig. 6 shows that the Lax–Friedrichs scheme with the usual discretization of the right-hand side does not give a satisfactory result.
Fig. 7 indicates that our adaptive scheme (3.3) gives stable solutions.
Test 4. We can take

Sgà ¼ 2;

Slà ¼ 1;

Fig. 7. Test 3: adaptive scheme (3.3) can give a stable solution.

Fig. 8. Test 4: usual discretization of the right-hand side does not give a satisfactory result.


M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

207

and

U L ¼ ð0:5; 1; 1; 0:5; 0:1; 0:5Þ;
U R ¼ ð0:4; 1; 0:9; 0:6; 0:2; 0:4Þ:

ð4:5Þ

Fig. 8 shows that the Lax–Friedrichs scheme with the usual discretization of the right-hand side does not give a satisfactory result.
Fig. 9 indicates that our adaptive scheme (3.3) gives stable solutions.
Test 5. We can take

Sgà ¼ 0:2;

Slà ¼ À0:1;

and

U L ¼ ð0:5; 2; 1; 0:5; 1; 0:5Þ;
U R ¼ ð0:6; 2:1; 1:1; 0:6; 1:2; 0:4Þ:

ð4:6Þ

Fig. 10 shows that the Lax–Friedrichs scheme with the usual discretization of the right-hand side does not give a satisfactory result.
Fig. 11 indicates that our adaptive scheme (3.3) gives stable solutions.

Fig. 9. Test 4: adaptive scheme (3.3) can give a stable solution.

Fig. 10. Test 5: usual discretization of the right-hand side does not give a satisfactory result.

208

M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

Test 6. We can take

Sgà ¼ 0:2;

Slà ¼ À0:1;

and

U L ¼ ð0:5; 2; 1; 0:5; 1; 0:5Þ;
U R ¼ ð0:4; 2:2; 0:9; 0:6; 1:1; 0:4Þ:

ð4:7Þ

Fig. 12 shows that the Lax–Friedrichs scheme with the usual discretization of the right-hand side does not give a satisfactory result.
Fig. 13 indicates that our adaptive scheme (3.3) gives stable solutions.
Test 7. In this test, let us take

Sgà ¼ À0:1;

Slà ¼ 0:1;

and

U L ¼ ð0:5; 1:2; 1; 0:5; 0:5; 0:5Þ;

U R ¼ ð0:4; 1; 1:1; 0:6; 0:8; 0:8Þ:

ð4:8Þ

We compute the solution at t = 0.1 and x 2 [À1, 1] with different meshes: 300, 1000, and 2000 mesh points. The test
shows that the approximate solutions are numerically stable, see Figs. 14–17.

Fig. 11. Test 5: adaptive scheme (3.3) can give a stable solution.

Fig. 12. Test 6: usual discretization of the right-hand side does not give a satisfactory result.


M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

Fig. 13. Test 6: adaptive scheme (3.3) can give a stable solution.

Fig. 14. Test 7: the volume fraction computed at t = 0.1 and x 2 [À1, 1] with 300, 1000, and 2000 mesh points.

Fig. 15. Test 7: the pressure computed at t = 0.1 and x 2 [À1, 1] with 300, 1000, and 2000 mesh points.

209


210

M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

Fig. 16. Test 7: the gas velocity computed at t = 0.1 and x 2 [À1, 1] with 300, 1000, and 2000 mesh points.

Fig. 17. Test 7: the liquid velocity computed at t = 0.1 and x 2 [À1, 1] with 300, 1000, and 2000 mesh points.

Thus, we have seen by the above tests that our numerical scheme is stable. It generates approximate solutions which are
numerically convergent.
5. Conclusions
Two-fluid models of two-phase flows possess complicated properties and raise challenging problems in both theoretical
study of the analytic solutions as well as in numerical treatments. This is mainly because the system is not hyperbolic and
the analytic form of characteristic fields is not available. The Riemann problem cannot be solved by the standard way and the
conventional Godunov-type schemes cannot be applied. In this paper, we derive several equivalent forms of the two-fluid
model, and we build an adaptive scheme for the two-fluid model of two-phase compressible flows. Many numerical tests
show that our scheme is stable and numerically convergent. This gain gives a convenient way to deal with numerical approximations of two-fluid model of two-phase flows, since standard numerical schemes cannot efficiently work for these models.
References
[1] Botchorishvili R, Perthame B, Vasseur A. Equilibrium schemes for scalar conservation laws with stiff sources. Math Comput 2003;72:131–57.
[2] Chang CH, Liou MS. A conservative compressible multifluid model for multiphase flow: shock-interface interaction problems. In: 17th AIAA
computational fluid dynamics conference, June 6-9, 2005, Toronto, Ontario, Canada, American Institute of Aeronautics and Astronautics Paper, 20055344, 2005.
[3] Dal Maso G, LeFloch PG, Murat F. Definition and weak stability of nonconservative products. J Math Pures Appl 1995;74:483–548.


M.D. Thanh / Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211

211

[4] Garcià-Cascales JR, Paillère H. Application of AUSM schemes to multi-dimensional compressible two-phase flow problems. Nucl Eng Design
2006;236:1225–39.
[5] Greenberg JM, Leroux AY. A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J Numer Anal
1996;33:1–16.
[6] García-Cascales JR, Paillère H. Application of ausm schemes to multi-dimensional compressible two-phase flow problems. Nucl Eng Design
2006;236(12):1225–39.
[7] Ishii M. Thermo-fluid dynamic theory of two-phase flow. Paris: Eyrolles; 1975.
[8] Kröner D, LeFloch PG, Thanh MD. The minimum entropy principle for fluid flows in a nozzle with discontinuous cross-section. ESAIM: Math Mod
Numer Anal 2008;42:425–42.

[9] Kröner D, Thanh MD. Numerical solutions to compressible flows in a nozzle with variable cross-section. SIAM J Numer Anal 2005;43(2):796–824.
[10] LeFloch PG, Thanh MD. The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Commun Math Sci 2003;1(4):763–97.
[11] LeFloch PG, Thanh MD. The Riemann problem for shallow water equations with discontinuous topography. Commun Math Sci 2007;5(4):865–85.
[12] Marchesin D, Paes-Leme PJ. A Riemann problem in gas dynamics with bifurcation. Hyperbolic partial differ equat III. Comput Math Appl (Part A)
1986;12:433–55.
[13] Staedtke H et al. Advanced three-dimensional two-phase flow simulation tools for application to reactor safety (astar). Nucl Eng Design
2005;235:379–400.
[14] Stewart HB, Wendroff B. Two-phase flow: models and methods. J Comput Phys 1984;56:363–409.
[15] Thanh MD, Fazlul KMd, Ismail AIMd. Well-balanced scheme for shallow water equations with arbitrary topography. Int J Dyn Syst Differ Equat
2008;1(3):196–204.
[16] Thanh MD. The Riemann problem for a non-isentropic fluid in a nozzle with discontinuous cross-sectional area. SIAM J Appl Math
2009;69(6):1501–19.
[17] Thanh MD, Ismail AI Md. A well-balanced scheme for a one-pressure model of two-phase flows. Phys Scr 2009:79.