Applied Mathematics and Computation 186 (2007) 980–991

www.elsevier.com/locate/amc

Numerical simulation for two-phase flows using hybrid scheme

Jun Zhou a, Li Cai a,*, Jian-Hu Feng b, Wen-Xian Xie c

a College of Astronautics, Northwestern Polytechnical University, You Yi Xi Lu Road 127, P.O. Box 250, Xi’an, Shaanxi 710072, PR Chinab College of Science, Chang’an University, Xi’an, Shaanxi 710064, PR China

c School of Science, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China

Abstract

The present paper is devoted to the computation of two-phase flows using two-fluid approach. Two difficulties arise forthe model: one of the equations is written in non-conservative form; non-conservative terms exist in the momentum andenergy equations. To deal with the aforementioned difficulties, the non-conservative volume fraction equation is discretizedby WENO scheme, and the CWENO-type central-upwind scheme is applied to solve the mass, momentum and energyequations. Computational results are eventually provided and discussed.� 2006 Elsevier Inc. All rights reserved.

Keywords: Two-phase flows; Weighted essentially non-oscillatory (WENO) scheme; Central weighted essentially non-oscillatory (CW-ENO) scheme; Central-upwind scheme

1. Introduction

It is widely believed that an accurate prediction of two-phase flow phenomena by means of computationalmethod is highly required in the analysis of fluids consisting several fluid components with different physical orthermodynamic properties. In general, the ability to predict these phenomena depends on the availability ofmathematical models and numerical methodology. Among several two-phase flow models, there are two fun-damentally different cases, namely the mixture model in [13,1,4,19] and two-fluid models in [17,15,20,5,16,14].

In the mixture model, the averaging procedure in the conservation formulation causing numerical mixing,often fails to maintain pressure equilibrium among different fluid components, and the pressure field oftengenerate erroneous fluctuations at the material interfaces. Of cause, great efforts have been developed to over-come numerical difficulties (see [13,1,4,19], and references therein).

Another way to solve two-phase flow problems is based on the two-fluid model. This compressible model istime-hyperbolic at all flow regimes, but allows for the phases to be out of physical or thermodynamic equilib-rium. Then equilibrium between the phases can be restored by relaxation processes. Thus, the exchange termsof momentum and energy have to take place across the interfaces. Now two avoidless difficulties arise for the

0096-3003/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.08.043

* Corresponding author.E-mail addresses: licai@mail.nwpu.edu.cn, eign@eyou.com (L. Cai).

J. Zhou et al. / Applied Mathematics and Computation 186 (2007) 980–991 981

model: one of the equations is written in non-conservative form; non-conservative terms exist in the momen-tum and energy equations.

In this paper, we have implemented WENO scheme in [6] to approximate the non-conservative volumefraction equation, and the CWENO-type central-upwind scheme in [2,7,3,8] is applied to solve the mass,momentum and energy equations.

2. Two-phase flows model: two-fluid model

The two-fluid model considers each phase separately in terms of independent sets of mass, momentum andenergy equations. The interactions between two phases are described by the coupling of volume fraction andtransfer terms of mass, momentum and energy of each phase. It involves six equations obtained from conser-vation laws for carrier phase and disperse phase, and completed by a seventh equation for the evolution of thevolume fraction and a eighth equation for compatibility. It takes the form as in [16,14]

Mass conservation

oðagqgÞot

þoðagqgugÞ

ox¼ _m; ð2:1Þ

oðalqlÞot

þ oðalqlulÞox

¼ � _m: ð2:2Þ

Momentum conservation

oðagqgugÞot

þoðagqgu2

g þ agpgÞox

¼ P i

oag

oxþ _mV i þ F d; ð2:3Þ

oðalqlulÞot

þ oðalqlu2l þ alplÞox

¼ �P i

oag

ox� _mV i � F d: ð2:4Þ

Energy conservation

oðagEgÞot

þoðugðagEg þ agpgÞÞ

ox¼ P iV i

oag

oxþ _mEi þ F dV i þ Qi; ð2:5Þ

oðalElÞot

þ oðulðalEl þ alplÞÞox

¼ �P iV ioag

ox� _mEi � F dV i � Qi: ð2:6Þ

Volume fraction equation

oag

otþ V i

oag

ox¼ 0; ð2:7Þ

and

Compatibility

ag þ al ¼ 1: ð2:8Þ

We assume that the two-phase flow be composed of gas (subscript g) and liquid (subscript l). The interfacialvariables have the subscript i. In the above equations, ak stands for the phase volume fraction and qk, uk, Ek

are density, velocity and total energy for kth (k 2 {g, l}) phase, respectively. The left-hand sides of Eqs. (2.1)–(2.6) are classical. On the right-hand sides of the same equations are terms related to mass transfer _m, dragforce Fd, convective heat exchange Qi, and the non-conservative terms P i

oag

ox and P iV ioag

ox . Each fluid is governedby the Stiffened gas equation of state

Ek ¼ qkek þ1

2qku2

k ; qkek ¼pk þ ckpk

ck � 1; ð2:9Þ

where ek is specific internal energy of kth phase, and pk is a stiffness parameter. To close the system, we follow[16,14] and choose the interface pressure and velocity as

982 J. Zhou et al. / Applied Mathematics and Computation 186 (2007) 980–991

P i ¼ agpg þ alpl; V i ¼agqgug þ alqlul

agqg þ alql

: ð2:10Þ

Enforcing a common pressure on the phases is done by imposing a pressure relaxation source terml(pl � pg). This leaves the hydrodynamic system hyperbolic and circumvents the ill-posedness associated withthe single pressure assumption. The velocity relaxation process has already been expressed in Eqs. (2.1)–(2.6).Terms related to the drag force Fd are responsible for it and Fd can be written as

F d ¼ kðul � ugÞ; ð2:11Þ

where k is a positive relaxation coefficient or a more complicated positive function. To summarize, the oversystem that omits mass and energy transfer terms can be rewritten as

oag

otþ V i

oag

ox¼ �lðpl � pgÞ; ð2:12Þ

oðagqgÞot

þoðagqgugÞ

ox¼ 0; ð2:13Þ

oðagqgugÞot

þoðagqgu2

g þ agpgÞox

¼ P i

oag

oxþ kðul � ugÞ; ð2:14Þ

oðagEgÞot

þoðugðagEg þ agpgÞÞ

ox¼ P iV i

oag

oxþ lP iðpl � pgÞ þ kV iðul � ugÞ; ð2:15Þ

oðalqlÞot

þ oðalqlulÞox

¼ 0; ð2:16Þ

oðalqlulÞot

þ oðalqlu2l þ alplÞox

¼ �P ioag

ox� kðul � ugÞ; ð2:17Þ

oðalElÞot

þ oðulðalEl þ alplÞÞox

¼ �P iV ioag

ox� lP iðpl � pgÞ � kV iðul � ugÞ; ð2:18Þ

where the dynamics compaction viscosity l is a positive coefficient or function.Our numerical method is performed by a split-step algorithm, first accounting for the hydrodynamic step

(k = 0, l = 0), then accounting for instantaneous relaxation effects (k!1, l!1).We describe our algorithm in the following sections.

3. Non-conservative flux based WENO discretization

We first consider the volume fraction equation (2.12) omitting the pressure relaxation effect, i.e. l = 0. Forsimplicity, we discretize both space and time with uniform mesh spacings of Dx and Dt respectively. Following[6], we can construct a fifth non-conservative flux based WENO discretization for

oag

ox , and a third order TVDRunge–Kutta method in [18] is applied for time integration.

Consider a specific grid point j0. If ðV iÞj0¼ 0, then setting

oag

ox

� �j0

¼ 0 will be algorithmically correct since

V ioag

ox

� �j0

¼ 0 is desired, otherwise ðV iÞj0determines the upwind direction.

The associated numerical flux function oag

ox

� �j0þ1=2

is defined as follows: If ðV iÞj0> 0, then q1 ¼

oag

ox

� �j0�2

; q2 ¼

oag

ox

� �j0�1

; q3 ¼oag

ox

� �j0

; q4 ¼oag

ox

� �j0þ1

and q5 ¼oag

ox

� �j0þ2

. If ðV iÞj0< 0, then q1 ¼

oag

ox

� �j0þ3

; q2 ¼oag

ox

� �j0þ2

; q3 ¼oag

ox

� �j0þ1

; q4 ¼oag

ox

� �j0

and q5 ¼oag

ox

� �j0�1

. Define the smoothness

IS1 ¼13

12ðq1 � 2q2 þ q3Þ

2 þ 1

4ðq1 � 4q2 þ 3q3Þ

2; ð3:19Þ

IS2 ¼13

12ðq2 � 2q3 þ q4Þ

2 þ 1

4ðq2 � q4Þ

2; ð3:20Þ

IS3 ¼13

12ðq3 � 2q4 þ q5Þ

2 þ 1

4ð3q3 � 4q4 þ q5Þ

2; ð3:21Þ

J. Zhou et al. / Applied Mathematics and Computation 186 (2007) 980–991 983

and the weights

x1 ¼a1

a1 þ a2 þ a3

; x2 ¼a2

a1 þ a2 þ a3

; x3 ¼a3

a1 þ a2 þ a3

; ð3:22Þ

with

a1 ¼0:1

ðeþ IS1Þ2; a2 ¼

0:6

ðeþ IS2Þ2; a3 ¼

0:3

ðeþ IS3Þ2; e < 10�14;

to get the flux

oag

ox

� �j0þ1=2

¼ x1

q1

3� 7q2

6þ 11q3

6

� �þ x2 �

q2

6þ 5q3

6þ q4

3

� �þ x3

q3

3þ 5q4

6� q5

6

� �: ð3:23Þ

Likewise, the associated numerical flux functionoag

ox

� �j0�1=2

is defined as follows: If ðV iÞj0> 0, then

q1 ¼oag

ox

� �j0�3

; q2 ¼oag

ox

� �j0�2

; q3 ¼oag

ox

� �j0�1

; q4 ¼oag

ox

� �j0

and q5 ¼oag

ox

� �j0þ1

. If ðV iÞj0< 0, then

q1 ¼oag

ox

� �j0þ2

; q2 ¼oag

ox

� �j0þ1

; q3 ¼oag

ox

� �j0

; q4 ¼oag

ox

� �j0�1

and q5 ¼oag

ox

� �j0�2

. Then the smoothness, weights

and flux are defined exactly as above yielding

oag

ox

� �j0�1=2

¼ x1

q1

3� 7q2

6þ 11q3

6

� �þ x2 �

q2

6þ 5q3

6þ q4

3

� �þ x3

q3

3þ 5q4

6� q5

6

� �: ð3:24Þ

Finally,oag

ox

� �j0

¼oagox

� �j0þ1=2

� oagox

� �j0�1=2

Dx .

Remark. As we know, gradients of ak become difficult to compute accurately if ak is close to 1 (or 0), anderrors that are introduced may destabilize the computation. In order to stabilize the computation, Karni et al.define a new variable in [14]

bk ¼ logak

1� ak

� �:

Then oag = ag(1 � ag)obg is used in the computations of the derivative of ag in Eq. (2.12). This way, ag is guar-anteed to be in the range (0, 1). In this paper, we implement above upwind WENO discretization, as we havefound that this gives better control of ak.

4. CWENO-type central-upwind scheme for two-fluid model

4.1. CWENO Reconstruction

Let us consider the 1-D system of hyperbolic conservation laws

ou

otþ ofðuÞ

ox¼ 0; u 2 Rd ; d P 1: ð4:25Þ

The numerical approximation of the scalar cell-average in the cell Ij = [xj�1/2,xj+1/2] :¼ [xj � Dx/2,xj + Dx/2]at time tn = nDt, is denoted by �un

j .From the cell-averages f�un

jg, we reconstruct three polynomials of degree two, Pj�1(x), Pj(x), Pj+1(x).According to [12], each of these polynomials satisfies the following interpolation requirements

1Dx

RIk�1

P kðxÞdx ¼ �unk�1;

1Dx

RIk

P kðxÞdx ¼ �unk ðk ¼ j� 1; j; jþ 1Þ;

1Dx

RIkþ1

P kðxÞdx ¼ �unkþ1:

8>><>>: ð4:26Þ

984 J. Zhou et al. / Applied Mathematics and Computation 186 (2007) 980–991

and the second degree polynomial Rj(x) of reconstruction is a convex combination of the above polynomialsPk(x),

RjðxÞ ¼ xjj�1P j�1ðxÞ þ xj

jP jðxÞ þ xjjþ1P jþ1ðxÞ; ð4:27Þ

with

P kðxÞ ¼ ~uk þ ~u0kðx� xkÞ þ1

2~u00kðx� xkÞ2; ð4:28Þ

~uk ¼ �unk �

�unkþ1 � 2�un

k þ �unk�1

24; ð4:29Þ

~u0k ¼�un

kþ1 � �unk�1

2Dx; ð4:30Þ

~u00k ¼�un

kþ1 � 2�unk þ �un

k�1

Dx2; k ¼ j� 1; j; jþ 1: ð4:31Þ

The reconstruction (4.27) is just the fourth-order CWENO-type interpolant polynomial, which can be rewrit-ten as

RjðxÞ ¼ uj þ u0jðx� xjÞ þ1

2u00j ðx� xjÞ2; ð4:32Þ

where

uj ¼ xjj�1 ~uj�1 þ Dx~u0j�1 þ

1

2Dx2~u00j�1

� �þ xj

j~uj þ xjjþ1 ~ujþ1 � Dx~u0jþ1 þ

1

2Dx2~u00jþ1

� �;

u0j ¼ xjj�1ð~u0j�1 þ Dx~u00j�1Þ þ xj

j~u0j þ xj

jþ1ð~u0jþ1 � Dx~u00jþ1Þ; ð4:33Þu00j ¼ xj

j�1~u00j�1 þ xj

j~u00j þ xj

jþ1~u00jþ1;

xjk ¼

ajk

ajj�1 þ aj

j þ ajjþ1

ðk ¼ j� 1; j; jþ 1Þ; ð4:34Þ

ajk ¼

Ck

ðeþ ISjkÞ

2ðk ¼ j� 1; j; jþ 1Þ; e < 10�6; ð4:35Þ

Cj�1 ¼ 3=16; Cj ¼ 5=8; Cjþ1 ¼ 3=16; ð4:36Þ

ISjj�1 ¼

13

12ð�un

j�2 � 2�unj�1 þ �un

j Þ2 þ 1

4ð�un

j�2 � 4�unj�1 þ 3�un

j Þ2;

ISjj ¼

13

12ð�un

j�1 � 2�unj þ �un

jþ1Þ2 þ 1

4ð�un

j�1 � �unjþ1Þ

2; ð4:37Þ

ISjjþ1 ¼

13

12ð�un

j � 2�unjþ1 þ �un

jþ2Þ2 þ 1

4ð3�un

j � 4�unjþ1 þ �un

jþ2Þ2:

It is easy to see that CWENO reconstruction, (4.32), follows the WENO methodology through making useof the existing information to select the stencil automatically. The overall order of accuracy of the resultingreconstruction and its non-oscillatory properties are retained.

4.2. CWENO-type central-upwind scheme

Recently, a new class of Godunov-type central schemes, referred to as central-upwind schemes, has beendiscussed in [11,9,10]. Those central-upwind schemes are derived in a general form which are independentof the reconstruction step, as long as the reconstructed interpolants are sufficiently accurate and non-oscilla-tory. Hence, we combine the fourth-order central weighted essentially non-oscillatory (CWENO) reconstruc-tions with the central-upwind schemes. So-called CWENO-type central-upwind schemes have non-oscillatorybehavior which have been shown in [2,7,3,8]. In this paper, we have extended our CWENO-type central-upwind schemes to simulate the two-phase flow subsystem (2.13)–(2.18) without relaxation processes

J. Zhou et al. / Applied Mathematics and Computation 186 (2007) 980–991 985

o

ot

agqg

agqgug

agEg

alql

alqlul

alEl

266666664

377777775þ o

ox

agqgug

agqgu2g þ agpg

ugðagEg þ agpgÞalqlul

alqlu2l þ alpl

ulðalEl þ alplÞ

266666664

377777775¼

0

P ioag

ox

P iV ioag

ox

0

�P ioag

ox

�P iV ioag

ox

2666666664

3777777775: ð4:38Þ

For simplicity, we denote the above subsystem as

ou

otþ ofðuÞ

ox¼ SðuÞ: ð4:39Þ

The semi-discrete central-upwind scheme for (4.39) can be written in the following form:

d

dt�ujðtÞ ¼ �

Hjþ1=2ðtÞ �Hj�1=2ðtÞDx

þ SjðtÞ: ð4:40Þ

That is,

d

dt�uðkÞj ðtÞ ¼ �

H ðkÞjþ1=2ðtÞ � H ðkÞj�1=2ðtÞDx

þ SðkÞj ðtÞ ðk ¼ 1; 2; . . . ; 6Þ; ð4:41Þ

where the fourth-order numerical fluxes H ðkÞjþ1=2ðtÞ (k = 1,2, . . . , 6) are given by

H ðkÞjþ1=2ðtÞ ¼aþjþ1=2f ðkÞðu�jþ1=2Þ � a�jþ1=2f ðkÞðuþjþ1=2Þ

aþjþ1=2 � a�jþ1=2

þaþjþ1=2a�jþ1=2

aþjþ1=2 � a�jþ1=2

ðuþjþ1=2ÞðkÞ � ðu�jþ1=2Þ

ðkÞh i

; ð4:42Þ

and integration of source terms are

Sð1Þj ¼ 0;

Sð2Þj ¼ ðP iÞjððagÞjþ1=2 � ðagÞj�1=2Þ;

Sð3Þj ¼ ðP iÞjðV iÞjððagÞjþ1=2 � ðagÞj�1=2Þ;

Sð4Þj ¼ 0;

Sð5Þj ¼ �ðP iÞjððagÞjþ1=2 � ðagÞj�1=2Þ;

Sð6Þj ¼ �ðP iÞjðV iÞjððagÞjþ1=2 � ðagÞj�1=2Þ:

ð4:43Þ

The one-sided local speeds of propagation can be estimated by

aþjþ1=2 ¼ maxnðug þ cgÞju�

jþ1=2; ðul þ clÞju�

jþ1=2; 0o; ð4:44Þ

a�jþ1=2 ¼ minnðug � cgÞju�

jþ1=2; ðul � clÞju�

jþ1=2; 0o; ð4:45Þ

where the speeds of sounds are ðckÞ�jþ1=2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffickððpkÞ�jþ1=2þpkÞðqÞ�jþ1=2

r, and u�jþ1=2 are calculated by the fourth-order CWE-

NO-type piecewise polynomial reconstruction (4.32),

u�jþ1=2 ¼ Rjðxjþ1=2Þ; uþjþ1=2 ¼ Rjþ1ðxjþ1=2Þ: ð4:46Þ

5. Relaxation processes

5.1. Instantaneous pressure relaxation

In many physical situations, it is reasonable to assume that pressure tends to equilibrium instantaneously. Itis also necessary when the pressure relaxation parameter l has not been determined physically or experimen-tally. This corresponds to an infinite value of l. The corresponding system of ODEs can be written in full as

986 J. Zhou et al. / Applied Mathematics and Computation 186 (2007) 980–991

oag

ot¼ �lðpl � pgÞ; ð5:47Þ

oðagqgÞot

¼ 0; ð5:48Þ

oðagqgugÞot

¼ 0; ð5:49Þ

oðagEgÞot

¼ lP iðpl � pgÞ; ð5:50Þ

oðalqlÞot

¼ 0; ð5:51Þ

oðalqlulÞot

¼ 0; ð5:52Þ

oðalElÞot

¼ �lP iðpl � pgÞ: ð5:53Þ

By combining Eqs. (5.51) and (5.52) we get ul = Const, which together with (5.52) imply

oðalqlu2l Þ

ot¼ 0: ð5:54Þ

Eqs. (5.47) and (5.53) show

oðalElÞot

¼ P i

oag

ot: ð5:55Þ

From Eqs. (5.54) and (2.9), Eq. (5.55) reduces to

oðalqlelÞot

¼ P i

oag

ot; ð5:56Þ

which can be integrated in [t0, t] to give

alqlel ¼ a0l q

0l e0

l � P iðal � a0l Þ; ð5:57Þ

where (*)0 corresponds to the conditions at t0, and P i ¼ P iðsÞ is an intermediate value of Pi at s 2 (t0, t1).Substituting Eq. (2.9) into (5.57) obtains

al

pl þ clpl

cl � 1¼ a0

l q0l e0

l � P iðal � a0l Þ; ð5:58Þ

and a similar results is obtained for the gas phase

ag

pg þ cgpg

cg � 1¼ a0

gq0ge0

g � P iðag � a0gÞ: ð5:59Þ

Since the final pressure is common to both phases, we denote the equilibrium pressure by p :¼ pl = pg, andapproximate P i by the midpoint rule P i � 1

2ðp þ P 0

i Þ. Hence, Eqs. (5.58) and (5.59) can be rewritten as

p1� ag

cl � 1�

ag � a0g

2

!¼ð1� a0

gÞp0l

cl � 1þ clpl

cl � 1ðag � a0

gÞ þP 0

i

2ðag � a0

gÞ ð5:60Þ

and

pag

cg � 1þ

ag � a0g

2

!¼

a0gp0

g

cg � 1�

cgpg

cg � 1ðag � a0

gÞ �P 0

i

2ðag � a0

gÞ: ð5:61Þ

Eliminating the common pressure p from (5.60) and (5.61) yields a quadratic equation for ag. One root is re-jected on physical grounds, the other may be shown to satisfy ag 2 [0,1]. Substituting ag into (5.60) or (5.61)gives p. Then qk = (akqk)0/ak and internal energy can be updated by the EOS (2.9).

J. Zhou et al. / Applied Mathematics and Computation 186 (2007) 980–991 987

5.2. Instantaneous velocity relaxation

When k is infinite (instantaneous velocity relaxation), we look at the following ODE system:

oag

ot¼ 0; ð5:62Þ

oðagqgÞot

¼ 0; ð5:63Þ

oðagqgugÞot

¼ kðul � ugÞ; ð5:64Þ

oðagEgÞot

¼ kV iðul � ugÞ; ð5:65Þ

oðalqlÞot

¼ 0; ð5:66Þ

oðalqlulÞot

¼ �kðul � ugÞ; ð5:67Þ

oðalElÞot

¼ �kV iðul � ugÞ: ð5:68Þ

We can easily obtain that

ag ¼ Const:; agqg ¼ Const:; alql ¼ Const: ð5:69Þ

Then, combining Eqs. (5.64) and (5.67) achieves

agqg

oug

otþ alql

oul

ot¼ 0; ð5:70Þ

which can be integrated to give

agqgðug � u0gÞ þ alqlðul � u0

l Þ ¼ 0: ð5:71Þ

From the above equation, u, the common velocity after instantaneous relaxation, can be expressed as

u :¼ ug ¼ ul ¼agqgu0

g þ alqlu0l

agqg þ alql

: ð5:72Þ

It remains to account for changes of the internal energies of each phase. Combination of the mass, momentumand energy equations yields

oek

ot¼ V i

ouk

ot� 1

2

ou2k

ot; k 2 fg; lg: ð5:73Þ

We make an approximate integration of this equation

ek ¼ e0k þ V iðuk � u0

kÞ �1

2ððukÞ2 � ðu0

kÞ2Þ; k 2 fg; lg: ð5:74Þ

By estimating V i as V i ¼V 0

iþu

2we get

ek ¼ e0k þ

1

2ðu� u0

kÞðV 0i � u0

kÞ; k 2 fg; lg: ð5:75Þ

After correction of the velocities and internal energies, the conservative vector can be rebuilt.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8

1

1.2

1.4

1.6

1.8

2

x

g

Density

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8

1

1.2

1.4

1.6

1.8

2

2.2

x

l

Density

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

ug / u

l

Velocity ratio

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

pg / p

l

Pressure ratio

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

αg

Volume fraction

ρ ρ

(5)

(3) (4)

(1) (2)

Fig. 1. Propagation of a void wave. (1) Density of the gas phase; (2) density of the liquid phase; (3) velocity ratio ag/al; (4) pressure ratiopg/pl; (5) volume fraction.

988 J. Zhou et al. / Applied Mathematics and Computation 186 (2007) 980–991

J. Zhou et al. / Applied Mathematics and Computation 186 (2007) 980–991 989

6. Numerical results

6.1. Propagation of a void wave

Fig. 1 describes the propagation of a void wave by our CWENO-type central-upwind scheme. The compu-tational domain is [0,1] and the initial conditions are given by

0.

0.

0.

0.

0.

0.

0.

0.

0.

αg

0.

0.

0.

0.

1.

u

Fig. 2.(3) mix

cg ¼ 1:4; ðagÞL ¼ 0:1; ðqgÞL ¼ 2; ðugÞL ¼ 1; ðpgÞL ¼ 1; ðqlÞL ¼ 1; ðulÞL ¼ 1; ðplÞL ¼ 1;

cl ¼ 1:2; ðagÞR ¼ 0:9; ðqgÞR ¼ 1; ðugÞR ¼ 1; ðpgÞR ¼ 1; ðqlÞR ¼ 2; ðulÞR ¼ 1; ðplÞR ¼ 1:

We use a mesh with 100 cells. The corresponding results are shown in Fig. 1 at time 0.06 s. A void wave inthis example is also a contact surface in each of the phases (see Fig. 1(1) and (2)). From Fig. 1(3) and (4), wesee that mechanical equilibriums between two phases are preserved. The evolution of the volume fraction ag

can be observed in Fig. 1(5).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

1

x

Volume fraction

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

x

ρ

Density

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

1

2

x

Velocity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

x

p

Pressure(3) (4)

(1) (2)

Comparison between exact and numerical solutions in air–air shock tube problem. (1) Volume fraction; (2) mixture density;ture velocity; (4) mixture pressure.

990 J. Zhou et al. / Applied Mathematics and Computation 186 (2007) 980–991

6.2. Air-air shock tube problem

In this example, we test our method at the condition of two pure fluids separated by an interface. That is tosay a region occupied by only one phase (a � 1) and a region occupied only by the other phase. Here we select(ag)L = 1 � 10�6 and (ag)R = 10�6. The initial states are

0

0

0

0

0

0

0

0

0

αg

0

0

0

0

1

u

Fig. 3.(3) mix

cg ¼ 1:4; ðqgÞL ¼ 1; ðugÞL ¼ 10�6; ðpgÞL ¼ 1; ðqlÞL ¼ 10�6; ðulÞL ¼ 10�6; ðplÞL ¼ 10�6;

cl ¼ 1:4; ðqgÞR ¼ 10�6; ðugÞR ¼ 10�6; ðpgÞR ¼ 10�6; ðqlÞR ¼ 0:125; ðulÞR ¼ 10�6; ðplÞR ¼ 0:1:

This example consists of a classical shock tube and admits an exact solution. The exact solution is repre-sented by solid lines, while the numerical solution is represented by symbols. The computation at time 0.05 s ismade on a mesh involving 100 cells. Fig. 2(1) shows the current volume fraction of the gas phase. FromFig. 2(2)–(4), we note that the numerical solutions match the exact solutions.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

x

Volume fraction

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

x

ρ

Density

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

.2

.4

.6

.8

1

.2

x

Velocity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

x

p

Pressure(3) (4)

(1) (2)

Comparison between exact and numerical solutions in air–helium shock tube problem. (1) Volume fraction; (2) mixture density;ture velocity; (4) mixture pressure.

J. Zhou et al. / Applied Mathematics and Computation 186 (2007) 980–991 991

6.3. Air–helium shock tube problem

This test problem is solved with the same initial data as the above example, except cl = 5/3. Computedresults on a 100 cells mesh are in agreement with the exact solution, which are shown in Fig. 3(2)–(4). Themultifluid limit of two pure fluids can be observed successfully (see Fig. 3(1)).

7. Conclusions

We have described an efficient numerical method, so-call hybrid scheme, for the simulation of compressibletwo-fluid model. The hybrid scheme consists of the WENO scheme, which discretize the non-conservative vol-ume fraction equation, and the CWENO-type central-upwind scheme, which solve the mass, momentum andenergy equations. As a result of our numerical experiments we conclude that the algorithm can be applied toapproximate the states of pure fluids and mixtures.

Acknowledgements

The author gratefully acknowledges the support of Youth for NPU Teachers Scientific and TechnologicalInnovation Foundation and the supports by the Center for High Performance Computing of NorthwesternPolytechnical University.

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