Application of Osher and PRICE-C schemes to solve compressible isothermal two-fluid models of two-phase flow

Computers & Fluids 86 (2013) 363–379

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Application of Osher and PRICE-C schemes to solve compressibleisothermal two-fluid models of two-phase flow

0045-7930/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compfluid.2013.07.018

⇑ Corresponding author. Tel.: +98 611373 8532; fax: +98 611336 9684.E-mail addresses: [email protected] (Y. Shekari), [email protected]

(E. Hajidavalloo).

Younes Shekari, Ebrahim Hajidavalloo ⇑Department of Mechanical Engineering, Shahid Chamran University, Ahvaz 61355, Iran

a r t i c l e i n f o

Article history:Received 16 May 2013Received in revised form 17 July 2013Accepted 19 July 2013Available online 29 July 2013

Keywords:Two-fluid modelFour-equation modelPath-conservativeOsher schemePRICE-C schemeTVD-MUSCL–Hancock

a b s t r a c t

In this paper, two path-conservative schemes, namely Osher and PRICE-C schemes have been used tosolve isothermal compressible two-phase flows using four-equation model. Path-Conservative Osher(PC-Osher) scheme is an upwind method using the full eigenstructure of the system but PRICE-C schemeis a central method, in which using the full eigenstructure of the system is not necessary. Different two-phase flow problems are solved using these schemes and their results are compared. The numerical effi-ciency of two schemes and their abilities in the simulation of near single phase flows are also examined.The extension of these schemes to the second order of accuracy is performed using the well-known TVD-MUSCL–Hancock (TMH) method. The results show that for the same level of accuracy, the PC-Osher ismore efficient than the PRICE-C scheme in view of computational time. However, the PC-Osher schemefails to predict near single phase flows compared to the PRICE-C scheme. The results also show that thesecond order extension of both schemes is less diffusive on the sonic waves while they show small ampli-tude oscillations on the volume fraction waves.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Mathematical modeling and numerical simulation of two-phaseflow phenomena is currently a very active research field due to therapid application of two-phase flow in the different engineeringfields. The underlying physics of the problems is very complex;therefore, the aim of mathematical models is to accurately capturethe different behavior of the phases and their interactions causedby the exchange of mass, momentum and energy.

There are several two-phase flow models in the literature topredict flow behavior including homogenous equilibrium model(HEM), Drift Flux Model (DFM) and Two-Fluid Model (TFM) [1].From a mathematical point of view, the HEM and DFM modelshave some advantages over the TFM model, since they are conser-vative and also they include fewer equations to be solved. How-ever, they also have some disadvantages as they fail to predictflow behavior accurately when the phases are not strongly cou-pled. Two-fluid models, however, are in a non-conservative form,but they give more details of the flow field and they are also appli-cable to every two-phase flow with totally different velocities.

A common type of two-fluid model is the four-equation modelin which a same pressure is considered for two phases. The four-equation model, also known as Single Pressure Model (SPM), is in

a non-conservative form and its hyperbolicity depends on the pres-sure correction term. However, there is no simple analyticalexpression for its eigenvalues. Evje and Flatten [2] computed theeigenvalues of this model using a perturbation method, but theydid not give an analytical expression for the eigenvectors of thesystem.

For the solution of compressible two-fluid models, one can useexact Riemann solvers or approximate Riemann solvers. Andrianovand Warnecke [3] were the first to propose an inverse solution tothe Riemann problem for BN model of two-phase flow. Schwend-eman et al. [4] and Deledicque and Papalexandris [5] proposed ex-act forward Riemann solvers and high resolution Godunov-typefinite volume methods for the BN model. Exact Riemann solversare accurate, but they are very time-consuming and cumbersomeas they need iterative procedure for non-conservative systems.Furthermore, there has not been an exact Riemann solver in thecase of isothermal two-fluid models yet. So, one has to look foran approximate Riemann solver for the numerical solution of thesetypes of two-phase models.

In the literature, there are several numerical methods for solv-ing two-fluid models. Basically, the numerical methods for solvingsuch systems can be divided in two general groups, i.e. centralmethods and upwind methods. The central methods are easy toimplement and they do not need the full eigenstructure of the sys-tem. However, they do not take into account the upwind structureof the flow. This class of methods has been employed by severalresearchers [6,7].

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364 Y. Shekari, E. Hajidavalloo / Computers & Fluids 86 (2013) 363–379

On the other hand, upwind methods use full eigenstructure ofthe system to predict the dynamics of the flow. For the numericalsolution of the hyperbolic equations, the approximate Riemannsolvers of Roe [8] and Osher and Solomon [9] are good attractivecandidates, since they provide an upwind resolution of all wavephenomena inherent in the models. An approximate linearizedRiemann solver was used for the simulation of two-phase flowsin Ref. [10]. This solver was based on a linearization of non-conser-vative products and used an extension of the Roe’s approximateRiemann solver. Paillere et al. [11] used an AUSM+ for the numer-ical solution four and six-equation models of the one dimensionaltwo-phase flows. A hybrid flux splitting method for numericalsolution of four-equation two-fluid model were used in Ref. [2].The authors have presented AUSMDV� scheme which has a goodability in capturing near single phase flows with sharp interfaces.Karni et al. [12] used central and upwind Godunov type methodsto solve the seven-equation model of the Saurel and Abgrall [13],by considering the pressure and velocity relaxation. Their upwindmethod was a Roe type method. In spite of its capability in the sim-ulation of two-phase flows, the Roe scheme needs an entropy fix toprevent violating the second law of thermodynamics, while Osher’sscheme is an entropy satisfying scheme. Thus, it does not producerarefaction shocks.

Dealing with a non-conservative form of the equations remainsa challenge from both mathematical and numerical point of views.Mathematically, shock waves and associated Rankine–Hugoniotconditions can be defined once a conservative form of the equa-tions exists. For non-conservative systems, according to the DalM-aso–LeFloch–Murat (DLM) theory [14] one can establishgeneralized Rankine–Hugoniot conditions involving a family ofpaths in the phase space. Based on this theory, it has been possibleto propose new numerical methodologies to solve non-conserva-tive systems [15,16]. Munkejord et al. [17] used a path-conserva-tive MUSTA scheme for isothermal and non-isothermal two-fluidmodels. Though this scheme is accurate, it is computationally inef-ficient for moving discontinuity problems [17]. The five-equationmodel of two-phase flow was studied by Castro and Toro [18]. Theyproposed a four-rarefaction approximate Riemann solver (4R) forthis two-phase flow model which was an extension of the two-rar-efaction approximation for single-phase gas dynamics. The disad-vantage of the 4R solver of Ref. [18] is its iterative characterwhich is quite time-consuming [19]. Tian et al. [19] presented anHLLC-type path-conservative Riemann solver for the numericalsolution of the five-equation system and Tokareva and Toro [20]proposed the first HLLC-type solver for the full seven equationBaer–Nunziato model. This scheme is simpler and cheaper than4R scheme used in Ref. [18].

Dumbser and Toro [21,22] presented the first path-conservativeOsher-type scheme for the numerical solution of general non-con-servative hyperbolic systems, with applications to the several non-conservative systems including the shallow water equations withvariable bottom topography and the two-fluid debris flow modelof Pitman and Le [23] and non-isothermal compressible two-fluidmodel of Baer and Nunziato [24]. In their method, the underlyingpath integral which appears already in the conservative versionof Osher’s scheme, has been evaluated numerically along astraight-line segment path. The Osher scheme is a complete Rie-mann solver that accounts for all intermediate characteristic fieldspresent in the Riemann problem. In contrast to Roe’s method, theOsher flux is entropy satisfying and therefore, it does not needany additional entropy fix [21]. The need of an entropy fix for Roe’sscheme for isothermal two-fluid models can be seen in [25]. So far,PC-Osher schemes have not yet been applied to the isothermalcompressible two-fluid models.

However, the Osher’s method needs information about fulleigenstructure of the system. This is usually a time consuming

procedure especially when there is not an analytical expressionfor eigenvectors of the system. Therefore, one may look for centralschemes in which using the full eigenstructure of the system is notnecessary. This may lead to speed up solution procedure. Recently,Canestrelli et al. [26] introduced a path conservative primitivelycentered (PRICE) PRICE-C scheme for shallow water equation withfixed and mobile bed. This scheme reduces to the FORCE schemefor conservative systems which is the optimally centered schemein the sense that it is the least dissipative of all three-point-cen-tered methods which are monotone [26].

To the best of the authors’ knowledge, the solution of isother-mal two-fluid model equations by path-conservative Osher’s andPRICE-C schemes were not reported in the open scientific resourcesyet. So the aims of the current research are twofold. The first aim isnumerical solution of the isothermal two-fluid model using path-conservative Osher and PRICE-C schemes and examining their abil-ity in the simulation of two-phase flow problems. The second oneis comparison between their results in view of the accuracy, com-putational time and costs and their ability in capturing near singlephase flows.

The rest of the paper is organized as follows; in Section 2, thegoverning equations of SPM are introduced and a brief hyperbolic-ity analysis of the systems is carried out. In Section 3, numericalmethods of solution are presented. In Section 4, numerical resultsare presented for some test cases. Conclusions are drawn inSection 5.

2. Governing equations

The four-equation or Single Pressure Model (SPM) is used to de-scribe the dynamics of two-phase flow. To derive the governingequations, it is assumed that the flow is one dimensional and iso-thermal. Besides, the interface and wall friction are neglected andit was assumed that there is no mass transfer between the phases,and the gravity force is the only source term which is considered insome of the test cases.

In this model, the dynamics of two-phase flow is described byfour PDEs including two equations for the conservation of massand two equations for linear momentum.

@akqk

@tþ @akqkuk

@x¼ 0 ð1Þ

@

@tðakqkukÞ þ

@

@xðakqku2

kÞ þ ak@pk

@xþ ðpk � pikÞ

@ak

@x¼ akqkg ð2Þ

where ak is the volume fraction of phase k (note that, k = g is consid-ered for the gas phase and k = l for liquid phase), and qk is the den-sity, uk is the velocity, pk is the pressure of phase k, and pik is theinterfacial pressure. In the right hand side, akqkg is due to gravity.Due to the term pik

@ak@x , the above equations cannot be written in

the conservative form in terms of akqk and akqkuk. Therefore, spe-cial care is needed for the spatial discretization of the system. In thismodel, an expression is needed for the relation between the pres-sures in the phases as:

pk ¼ pf þ rkf ð3Þ

where rkf is a constant pertaining to the relation between thephases k and f, which is assumed zero in this work. Furthermore,k and f are general indices which can be used for either the liquidor gas phase.

2.1. Equation of state

To close the system of equations, a relation between the pres-sure and the density of the phases is required, which can beachieved through an equation of state. For each phase of k, a simple

Y. Shekari, E. Hajidavalloo / Computers & Fluids 86 (2013) 363–379 365

linearized thermodynamic relation is assumed as the equation ofstate,

qk ¼ qk;0 þpk � pk;0

c2k

ð4Þ

where qk,0 is the reference density of the phase, pk,0 is the referencepressure of the phase k and ck is the speed of sound in the phase. Forthe gas and liquid phases, the constant parameters are shown inTable 1.

2.1.1. Primitive variablesNumerical solution of governing equations yields four conser-

vative variables. However, usually the primitive variables are themain concern. To obtain the primitive variables one additionalequation is necessary, because there are four conservative vari-ables and five primitive variables namely, ag, qg, ql, ug and ul. How-ever, in SPM two phases share the same pressure. Based on thisassumption one can obtain an equation for this share pressureusing the following geometrical constraint:

al þ ag ¼ 1 ð5Þ

By rewriting ag ¼agqg

qg ðpÞand al ¼ alql

qlðpÞand using the equation of

state for qk(p) one can obtain the following second order polyno-mial for the pressure:

p2 þ apþ b ¼ 0 ð6Þ

where

a ¼ c2l ðql;0 �mlÞ þ c2

gðqg;0 �mgÞ � ðpl;0 þ pg;0Þ ð7Þ

�c2g c2

l ðmgql;0 þmlqg;0 � qg;0ql;0Þ � c2g pl;0ðqg;0 �mgÞ

� c2l p0;gðq0;l �mlÞ þ pg;0pl;0 ð8Þ

with mk = akqk.By solving the above mentioned equation and choosing the po-

sitive root, one can obtain a common pressure and then using theequation of state, the densities of phases can be obtained. Now, thevolume fraction can be obtained using akqk, which is already avail-able from the solution of the hyperbolic part of the problem.

2.1.2. Modeling the interfacial pressureA relation for the interfacial pressure pik must be employed.

There are several models for the interfacial pressure in Refs.[2,7,13,27,28]. The following relation is mostly used, which hasbeen introduced in Ref. [29].

pk � pik ¼ Dpik ¼ cagalqgql

agql þ alqgðug � ulÞ2 ð9Þ

This formula was also used in the CATHARE code for non-strat-ified flows [30]. In this formula c is assumed 1.2, following Evje andFlatten’s work [2].

If the initial velocities of phases are equal, according to Eq. (9),the pressure correction term is zero which means that pk is equal topik. In this case, the system of governing equations has complexeigenvalues, resulting an ill-posed initial value problem, wherethere is an unphysical and unbounded growth of small wavelengthof disturbances [31]. The ill-posedness will produce instabilities ifnot balanced by the exchange terms (which damp low frequencies)and by the numerical diffusion (at high frequencies) [31–33].

Table 1Constants in the equation of state [2].

Phase qk,0 (kg/m3) pk,0 (Pa) c2k (m2/s2)

Air 0 0 105

Water 1000 100,000 106

Existence of complex eigenvalues leads to a non-diagonalizablesystem of equations which prevents employing the present numer-ical methods for this model. To avoid this, a combination of Eq. (9)and the Soo [34] model is used as follows:

pk � pik ¼ Dpik ¼ cagalqgql

agql þ alqgðug � ulÞ2 þ ð1� BkÞpk ð10Þ

where the displacement factor is set to a high value e.g.Bk = 0.999999 which results in some amount of negligible addi-tional diffusion, however, making the coefficient matrix diagonaliz-able. Furthermore, in such situations c = 2 is employed followingRefs. [11,25,35].

2.2. Quasi-linear form of the governing equations

In order to employ a path-conservative scheme and also for thehyperbolicity analysis of the model equations, the system of equa-tions should be written in quasi-linear form as follows:

@q@tþ AðqÞ @q

@x¼ SðqÞ ð11Þ

where q is the vector of conservative variables, A(q) is the systemmatrix, which is equal to the Jacobian matrix for conservative sys-tems and S(q) is the vector of source terms.

For the four-equation system the conservative variables of Eq.(11) are:

q ¼ ðagqg alql agqgug alqlul ÞT ð12Þ

and the system matrix reads as follows:

AðqÞ ¼

0 0 1 00 0 0 1

agqlþDpigal=c2l

v � u2g

agqg�Dpigag=c2g

v 2ug 0alql�Dpilal=c2

lv

alqgþDpilag=c2g

v � u2l 0 2ul

0BBBBB@

1CCCCCA ð13Þ

where v ¼ alqg

c2lþ agql

c2g

and Dpik = pk � pik. The source term is

S ¼ ð 0 0 agqgg alqlg ÞT ð14Þ

It is unfeasible to derive an exact closed-form expression for theeigenvectors and eigenvalues of A. Evje and Flatten [2] developedan approximate expression for the eigenvalues of this system byusing a perturbation method. They presented the eigenvalues asfollows:

k1;2 ¼ up � cm

k3;4 ¼ uu � vð15Þ

where

up ¼agqlug þ alqgul

agql þ alqgþ c

_

mOðe3Þ uu

¼agqlul þ alqgug

agql þ alqgþ c

_

mOðe3Þ ð16Þ

and

v ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDpiðagql þ alqgÞ � agalqgqlðug � ulÞ2

ðagql þ alqgÞ2

vuut þ c_

mOðe3Þ ð17Þ

cm ¼ c_

mð1þ Oðe2ÞÞ ð18Þ

c_

m ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

agql þ alqg@qg

@p

� �agql þ

@ql@p

� �alqg

vuut e ¼ uG � uL

c_

m 1þ aLqGaGqL

� �


If Dpik in Eq. (17) is zero, this leads to complex eigenvalues, sothe SPM will be ill-posed.

It should be mentioned that there is not an analytical expres-sion for the eigenvectors of A. Therefore, in the present calcula-tions, they have been found numerically using the RG subroutineof the linear algebra software package EISPACK.

3. Numerical methods

Two path-conservative schemes are used for numerical simula-tion of present two-fluid model, namely PC-Osher and PRICE-Cschemes. The PC-Osher, which is using an upwind approach, needsthe full eigenstructure of systems including eigenvalues and corre-sponding eigenvectors. This scheme is one of the earliest approxi-mate Riemann solvers [36]. Its bases was introduced in the papersby Engquist and Osher in 1981 [37] and Osher and Solomon [9].Although the original Osher scheme is quite difficult to implementfor general complicated nonlinear hyperbolic systems, Dumbserand Toro [21,22] presented a new simple but very general reformu-lation of the Osher scheme for general conservative and non-con-servative hyperbolic systems. The Osher flux has severalremarkable properties. This method is an entropy satisfyingscheme, therefore, it does not need any additional entropy fix incontrast to Roe’s method. Moreover, the Osher scheme is non-lin-ear and performs well near the low-density flows, in contrast tolinearized Riemann solvers [21].

The second employed numerical method is the path-conserva-tive PRICE-C scheme which is a central method and it does notneed the full eigenstructure of the systems. There are differentpath-conservative PRICE schemes, among them; PRICE-C is moreefficient than the others [26]. This scheme reduces to FORCEscheme for conservative systems which is the optimally centeredscheme in the sense that it is the least dissipative of all three-point-centered methods that are monotone [26]. In this sectionthese schemes are introduced.

3.1. Numerical details of employed schemes

Any path-conservative scheme for the non-conservative system(11) with equidistant mesh spacing Dx ¼ xiþ1

2� xi�1

2and time step

Dt = tn+1 � tn reads as:

qnþ1i ¼ qn

i �DtDx

D�iþ12þ Dþi�1

2

� �þ DtSi ð19Þ

and with the compatibility condition given in Eq. (20).

Dþiþ12þ D�iþ1

2¼Z 1

0A w q�iþ1

2;qþ

iþ12; s

� �� @w@s

ds ð20Þ

where the jump terms D�iþ12

for the Osher scheme is defined as fol-lows [21]:

D�iþ12¼ 1

2

Z 1

0A w q�iþ1

2;qþ

iþ12; s

� �� A wðq�iþ1

2;qþ

iþ12; sÞ

� �� @w@s

ds

ð21Þ

and these jump terms for the PRICE-C scheme are as follows [26]:

D�iþ12¼1

4

Z 1

02A w q�iþ1

2;qþ

iþ12;s

� �� Dx

DtI�A w q�iþ1

2;qþ

iþ12;s

� �� 2� �

@w@s

ds� �

ð22Þ

where I is the identity matrix. Here, the path w q�iþ1

2;qþ

iþ12; s

� �with

0 < s < 1 that connects the left state q�iþ1

2with the right state qþ

iþ12

in

phase-space is a Lipschitz continuous function with

w q�iþ12;qþ

iþ12;0

� �¼ q�iþ1

2w q�iþ1

2;qþ

iþ12;1

� �¼ qþ

iþ12

ð23Þ

For a first order scheme, the following piecewise-constantreconstruction is assumed q�

iþ12¼ qn

i and qþiþ1

2¼ qn

iþ1. Furthermore,for the absolute value operator of a matrix the usual conventionapplies i.e.

jAj ¼ RjKjR�1 jKj ¼ diagðjk1j; jk2j; . . . ; jkN jÞ ð24Þ

where N is the number of equations of the desired model.The available schemes presented up to now are still a function

of the path. Throughout this entire paper, the following straight-line segment path is used for the reason of simplicity as

wðsÞ ¼ wðq�;qþ; sÞ ¼ q� þ sðqþ � q�Þ ð25Þ

With this particular choice of the path, Eq. (21) yields

D�iþ12¼ 1

2

Z 1

0ðAðwðsÞÞ � jAðwðsÞÞjÞds

� �qþ

iþ12� q�iþ1

2

� �ð26Þ

and Eq. (22) reads:

D�iþ12¼ 1

4

Z 1

02A w q�iþ1

2;qþ

iþ12; s

� �� Dx

DtI� Dt

DxA w q�iþ1

2;qþ

iþ12; s

� �� 2�

ds�

qþiþ1

2� q�iþ1

2

� �ð27Þ

In order to make the integration process for the present meth-ods feasible with no analytical effort, the numerical computationof the jump term in the schemes via a high order accurateGauss–Legendre quadrature rule is suggested instead of trying acumbersome analytical integration [21]. The integral in Eq. (26)can be approximated using an M-point Gaussian quadrature rulein the [0;1] unit interval with positions sj and correspondingweights xj as follows:

D�iþ12¼ 1

2

XM

j¼1

xj AðwðsjÞÞ � jAðwðsjÞÞj� !

qþiþ1

2� q�iþ1

2

� �ð28Þ

Similarly the integral in Eq. (27) can be approximated asfollows:

D�iþ12¼ 1

4

XM

j¼1

xj 2AðwðsjÞÞ �DxDt

I� DtDx

AðwðsjÞÞ2� � !

qþiþ1

2� q�iþ1

2

� �ð29Þ

For example, one can use a three-point Gaussian quadraturerule with the following positions and corresponding weights:

s1;3 ¼12�

ffiffiffi5p

10; s2 ¼

12

w1;3 ¼5

18; w2 ¼

818

ð30Þ

3.2. Extension to the second order of accuracy

In this section the extension of path-conservative schemes tothe second order of accuracy is presented using TVD version ofthe MUSCL–Hancock method. This method includes three stepsto achieve the second order of accuracy in the space and time. Inthe first step spatial reconstruction is performed using piece-wiselinear functions. Using these functions the left and right extrapo-lated boundary values are obtained which are evolved in time bya half time step in the second step. In the last step the evolved dataare used as initial data for piece-wise constant Riemann problem.These steps are described in more details in the followingparagraphs.

For the first step, the cell average values are used to obtain apiece-wise linear function in each computational cell. The bound-ary extrapolated values have the following form [18,36]:

qLi ¼ qn

i �Di

2; qR

i ¼ qni þ

Di

2ð31Þ

x (m)

α g

u g(m

/s)

0 2 4 6 8 10 120.2

0.3

0.4

0.5

0.6

0.7

0.8

9.9999

9.99995

10

10.0001

10.0001

αg Osherαg PRICE-Cαg AUSMDV*ug (m/s) Osherug (m/s) PRICE-Cug (m/s) AUSMDV*

Fig. 1. Discontinuity moving in uniform flow: gas volume fraction and velocityvariations on a gird of 100 cells and using CFL = 0.9 for path-conservative methodsat t = 0.2 s.

u l(m

/s)

p(M

Pa)

9.99995

10

10.0001

10.0001

0.99995

1

1.00005

1.0001

p (Pa) Osherp (Pa) PRICE-Cp (Pa) AUSMDV*ul (m/s) Osherul (m/s) PRICE-Cul (m/s) AUSMDV*


where

Di ¼12ð1þxÞDi�1=2 þ

12ð1�xÞDiþ1=2; Di�1=2 ¼ qi � qi�1;

Diþ1=2 ¼ qiþ1 � qi ð32Þ

with x e [�1, 1]. These values are evolved in time using the follow-ing formula [18,36]:

�qLi ¼ qn

i �12

Iþ DtDxeAi

� �Di þ

Dt2

Si

�qRi ¼ qn

i þ12

I� DtDxeAi

� �Di þ

Dt2

Si

ð33Þ

However, this method is linear and second order accurate, soaccording to the Godunov theorem [38] one expects spurious oscil-lations near flow discontinuities. To avoid this, one can use a TVDslope limiter in the reconstruction step. In the present paper, theMonotonized central difference (MCD) [39] limiter was used forthe both schemes.

To achieve the second order of accuracy, Eq. (19) should bemodified to consider the effect of linear distribution of data as fol-lows [26]:

qnþ1i ¼ qn

i �DtDx

D�iþ12þ Dþi�1

2

� �� 1

DxAqx þ DtSi ð34Þ

where

Aqx ¼Z tnþ1

tn

Z x�iþ1=2

xþi�1=2

Aðqiðx; tÞÞ@qiðx; tÞ@x

dxdt

¼ A�qL

i þ �qRi

2

� ��qR

i � �qLi

Dx

� �DtDx ð35Þ

Note that the term Aqx vanishes for a first order accuratescheme in which oqi/ox is zero.

x (m)0 2 4 6 8 10 12

9.9999 0.9999

Fig. 2. Discontinuity moving in uniform flow: liquid velocity and pressurevariations on a gird of 100 cells and using CFL = 0.9 for path-conservative methodsat t = 0.2 s.

4. Numerical simulations

In this section, the ability of two path-conservative schemes isassessed using several two-phase flow test cases. The results foreach test case are compared against the reference solution fromother researchers. In the present research, AUSM type family espe-cially the AUSMDV� method of Evje and Flatten [2] is used to val-idate the results of PC-Osher and PRICE-C schemes.

x (m)

α g

u g(m

/s)

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

9.99999

10

10

10

10

αg ε = 0.1αg ε = 0.01ug ε = 0.1ug ε = 0.01

x (m)

α g

u g(m

/s)

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

9.99999

10

10

10

10

αg ε=10-10

ug (m/s) ε=10-10

(a) Osher using 100 cells (b) PRICE-C using 4000 cells

Fig. 3. Ability of path-conservative schemes in the simulation of near single phase flow.

Fig. 4. Schematic sketch of the water faucet test case.


4.1. Discontinuity moving in uniform flow

A very important condition that has to be satisfied by a numer-ical method for system (11) in two-phase flows is the pressure andvelocity preservation condition or the so-called Abgrall condition,which states that ‘‘A two phase flow, uniform in pressure andvelocity must remain uniform on the same variables during itstemporal evolution’’ [13,40]. Munkejord [25] used a test case tostudy the Abgrall condition for the Roe-type scheme. This test caseis used to study the Abgrall condition for the present numericalschemes. The system consists of a 12 m long horizontal tube,where the velocities are initially equal, i.e. ug = ul = 10 m/s andthe pressure is p = 106 Pa. At the middle of the tube, the gas volumefraction jumps from 0.2 to 0.8.

As already noted, the Eq. (9) for the pressure correction termleads to a non-diagonalizable system matrix, because in this testcase, the velocities of the two phases are equal, thus correspondingpressure correction term is zero. Therefore, the Osher scheme can-not be used for numerical solution of the model equations. To over-come this difficulty, Eq. (10) with 1 � Bk = 10�10 is used.

Fig. 1 shows the velocity of the gas phase and the volume frac-tion profile along the pipe. The results displayed in this figure havebeen calculated with three numerical methods including PC-Osher,PRICE-C and AUSMDV� method of Ref. [2]. The AUSMDV� is used

x (m)

α g(-

)

2 4 6 8 10 120.2

0.25

0.3

0.35

0.4

0.45

0.5

ReferenceAnalyticalOsher-100 CellsOsher-200 CellsOsher-400 CellsOsher-800 CellsOsher-1600 Cells

x (m)

p(P

a)

2 4 6 8 10 1299500

99600

99700

99800

99900

100000

ReferenceOsher-100 CellsOsher-200 CellsOsher-400 CellsOsher-800 CellsOsher-1600 Cells

(a) Gas Volume Fraction (b) Pressure

x (m)

u g(m

/s)

0 2 4 6 8 10 12-25

-20

-15

-10

-5

0

ReferenceOsher-100 CellsOsher-200 CellsOsher-400 CellsOsher-800 CellsOsher-1600 Cells

x (m)

u l(m

/s)

2 4 6 8 10 1210

11

12

13

14

15

16

ReferenceAnalyticalOsher-100 CellsOsher-200 CellsOsher-400 CellsOsher-800 CellsOsher-1600 Cells

(c) Gas Velocity (d) Liquid Velocity

Fig. 5. Water faucet test case: grid refinement study for PC-Osher scheme at t = 0.6 s using CFL = 0.9.


just to validate the results of the present path-conservative meth-ods. The computations were performed on a grid of 100 computa-tional cells and using CFL = 0.9 for the PC-Osher and PRICE-Cschemes. Data are plotted at t = 0.2 s.

As shown in Fig. 1, the gas velocity profile at t = 0.2 s. shows nodisturbance along the pipe and remains constant. This figure alsoshows that the volume fraction discontinuity advected to the rightwhile it is smeared due to numerical diffusion. As this figureshows, the accuracy of PC-Osher is close to AUSMDV�. The maxi-mum amplitude of pressure oscillation of the PC-Osher was about0.00012 Pa. However, the PRICE-C method produces more numer-ical diffusion when it compared with the AUSMDV� method. Themaximum amplitude of pressure oscillation for the PRICE-Cscheme was about 5 � 10�8 Pa. The liquid velocity and pressureprofiles at t = 0.2 s. computed from the employed numericalschemes are shown in Fig. 2. As it can be observed, the liquid veloc-ity and pressure are constant along the pipe and are equal to theircorresponding initial values at t = 0.

An important point about a numerical scheme for two-phaseflows is its ability to capture near single phase flows. To examinethe ability of the present methods in dealing with such flows, thediscontinuity moving in uniform flow test case was run using

x (m)

αg

(-)

0 2 4 6 8 10 120.2

0.25

0.3

0.35

0.4

0.45

0.5

ReferenceAnalyticalPRICE-C-1600 CellsPRICE-C-3200 CellsPRICE-C-6400 CellsPRICE-C-12800 CellsPRICE-C-25600 Cells

(a) Gas Volume Fraction

x (m)

u g(m

/s)

0 2 4 6 8 10 12-25

-20

-15

-10

-5

0

ReferencePRICE-C-1600 CellsPRICE-C-3200 CellsPRICE-C-6400 CellsPRICE-C-12800 CellsPRICE-C-25600 Cells

(c) Gas Velocity

Fig. 6. Water faucet test case: grid refinement study o

ag = e, and ag = 1 � e in the left and right part of the tube respec-tively, where e approaches a small value. In this study different val-ues of e are considered. Part (a) of Fig. 3 shows the results obtainedusing the PC-Osher scheme. As seen, this scheme can only capturenear single phase flows when e is greater or equal to 0.1. This figurealso shows the results for e = 0.01. As one can see, there is a distur-bance in the velocity profile of the gas phase. So it can be concludedthat PC-Osher scheme along with SPM is not a good candidate forsimulation of near single phase flows. As shown in Fig. 3 part (b),the PRICE-C scheme can capture near single phase flows for verysmall values of e (e.g. 10�10) without producing any disturbancein the pressure and phases velocities. The computation of PRICE-Cmethod was performed on a finer mesh because the numerical dif-fusivity of this scheme on the coarse meshes is too high and so onecannot obtain volume fractions close to zero or one.

4.2. Water faucet

This test case was suggested by Ransom [41] and has previouslybeen used by other researchers to validate their numerical results[2,7,13,27,42]. Therefore, it has become a standard benchmark forone-dimensional compressible two-fluid models.

x (m)

p(P

a)

0 2 4 6 8 10 12

99600

99700

99800

99900

100000

ReferencePRICE-C-1600 CellsPRICE-C-3200 CellsPRICE-C-6400 CellsPRICE-C-12800 CellsPRICE-C-25600 Cells

(b) Pressure

x (m)

u l(m

/s)

0 2 4 6 8 10 1210

11

12

13

14

15

16

ReferenceAnalyticalPRICE-C-1600 CellsPRICE-C-3200 CellsPRICE-C-6400 CellsPRICE-C-12800 CellsPRICE-C-25600 Cells

(d) Liquid Velocity

f the PRICE-C scheme at t = 0.6 s using CFL = 0.9.

x (m)

αg

(-)

0 2 4 6 8 10 120.2

0.25

0.3

0.35

0.4

0.45

0.5

ReferenceAnalyticalOsher-800 CellsOsher-1600 CellsPRICE-C-800 CellsPRICE-C-1600 Cells

Fig. 8. Faucet test case: comparison of the PRICE-C and PC-Osher schemes’ resultson the same grids at t = 0.6 s using CFL = 0.9.


4.2.1. Problem descriptionThis problem consists of a vertical pipe of 12 m length. A sche-

matic sketch of the problem is shown in Fig. 4. Gravity is the onlysource term taken into account. The top of the tube has a fixed vol-umetric inflow rate of water with a velocity of 10 m/s, a liquid vol-ume fraction of 0.8, and a temperature of 50 �C. The bottom of thetube is open to the ambient pressure at 105 Pa, the top of the tubeis closed to vapor flow. Initially, the flow is uniform throughout thecomputational domain and the inlet conditions are equal to the ini-tial conditions. A thinning of the liquid jet takes place due to theeffect of gravity.

When the pressure variations in the vapor phase are ignored,Ransom [41] stated that one can find an approximate analyticalexpression for volume fraction and liquid velocity. The procedureto obtain these analytical expressions was described by Trappand Riemke [43]. Here only the final expressions are presented as:

ul ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðu0

l Þ2 þ 2gx

qfor x < u0

l t þ 12 gt2

u0l þ gt otherwise

8<: ð36Þ

ag ¼1� a0

lu0

lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðu0

lÞ2þ2gx

p for x < u0l t þ 1

2 gt2

1� a0l otherwise

8<: ð37Þ

where u0l and a0

l are velocity and volume fraction of liquid phase atinitial state, respectively.

x (m)

α g(-

)

0 2 4 6 8 10 120.2

0.25

0.3

0.35

0.4

0.45

0.5

ReferenceAnalyticalPRICE-C-25600 CellsPC-Osher-1600 Cells


x (m)

u g(m

/s)

0 2 4 6 8 10 12-25

-20

-15

-10

-5

0

ReferencePRICE-C-25600 CellsPC-Osher-1600 Cells

(c) Gas Velocity

Fig. 7. Water faucet test case: comparison of th

Fig. 5 shows the results of PC-Osher scheme for water faucettest case using five computational grids including 100, 200, 400,800 and 1600 cells. As can be seen, by refining the grid, the solutionapproaches to the reference solution which shows that the PC-Osher scheme can successfully predict the solution. The reference

x (m)

p(P

a)

0 2 4 6 8 10 1299500

99600

99700

99800

99900

100000

ReferencePRICE-C-25600 CellsPC-Osher-1600 Cells

(b) Pressure

x (m)

u l(m

/s)

0 2 4 6 8 10 1210

11

12

13

14

15

16

ReferenceAnalyticalPRICE-C-25600 CellsPC-Osher-1600 Cells

(d) Liquid Velocity

e schemes at t = 0.6 s and using CFL = 0.9.

Table 2Initial states in Toumi’s shock tube problem.

Quantity Symbol (unit) Left RIGHT


solution was obtained using AUSMDV� on a grid of 3200 computa-tional cells. No significant differences were observed between theresults of 800 and 1600 cells. There are some differences in the

CPU time (s)

Nor

mof

erro

r

101 102 103

0.05

0.1

0.15

0.2

PRICE-COsherAUSMDV*

Fig. 9. Water faucet: comparison of the CPU time and norm of error of the PRICE-Cand PC-Osher schemes at t = 0.6 s using CFL = 0.9.

Gas volume fraction ag (–) 0.25 0.1Pressure p (MPa) 20 10Gas velocity ug

ms

� 0 0

Liquid velocity ulms

� 0 0

x (m)

α g(-

)

0 20 40 60 80 100

0.1

0.15

0.2

0.25

0.3

0.35

ReferenceOsher 100 CellsOsher 1000 CellsOsher 5000 CellsOsher 10000 Cells


x (m)

u g(m

/s)

0 20 40 60 80 1000

20

40

60

80

100

120


(c) Gas Velocity

Fig. 10. Toumi’s shock tube solution: grid refinement study

analytical solution of volume fraction and liquid velocity withnumerical results, which may be attributed to the assumption usedin the derivation of the analytical solution.

Fig. 6 shows the results of PRICE-C scheme for water faucet testcase using five computational grids including 1600, 3200, 6400,12,800 and 25,600 cells. As shown in this figure, the results ap-proach to the reference solution as the grids are refined enoughwhich is an indication of PRICE-C validity in the prediction of thesolution. The grid study shows that there is no significant differ-ence between the results of two finest grids.

Having shown the ability of two schemes in capturing the phys-ics of the flow, they can be compared with each other from differ-ent directions. Fig. 7 shows, the results of the PC-Osher scheme ona grid of 1600 cells and the PRICE-C scheme on a grid of 25,600cells. The results were also compared with the reference and

x (m)

p(M

Pa)

0 20 40 60 80 10010

12

14

16

18

20


(b) Pressure

x (m)

ul(

m/s

)

0 20 40 60 80 1000

2

4

6

8

10

12

14


(d) Liquid Velocity

for PC-Osher scheme at t = 0.08 s and using CFL = 0.9.

x (m)

α g(-

)

0 20 40 60 80 100

0.1

0.15

0.2

0.25

ReferencePRICE-C 100 CellsPRICE-C 1000 CellsPRICE-C 10000 CellsPRICE-C 20000 Cells

x (m)

p(M

Pa)

0 20 40 60 80 10010

12

14

16

18

20



x (m)

u g(m

/s)

0 20 40 60 80 1000

20

40

60

80

100

120


x (m)

ul(

m/s

)

0 20 40 60 80 1000

2

4

6

8

10

12

14



Fig. 11. Toumi’s shock tube: grid refinement study for PRICE-C scheme at t = 0.08 s and using CFL = 0.9.

CPU time (sec)

Nor

mof

Err

or

10-1 100 101 102 1030

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

PRICE-COsherAUSMDV*

Fig. 12. Toumi’s shock tube: comparison of the CPU-time and norm of error of thePRICE-C and PC-Osher schemes at t = 0.08 s using CFL = 0.9.

Table 3Initial states in the LRV shock tube problem.

Quantity Symbol (unit) Left Right

Gas volume fraction ag (–) 0.29 0.30Pressure p (Pa) 265,000 265,000Gas velocity ug

ms

� 65 50

Liquid velocity ulms

� 1 1


approximate analytical solutions. The reference solution was ob-tained using AUSMDV� on a grid of 3200 computational cells. Asthis figure shows, the results of the present path-conservativemethods are in good agreement with those of the reference and

analytical solution. However, while the accuracy of both schemesis the same the grid sizes used for both schemes are different. Inthis calculations, CFL = 0.9 for path-conservative methods andCFL = 0.1 for ASUMDV� method.

Numerical diffusivity and accuracy of the methods on the samegrids were examined and the results are shown in Fig. 8. Two com-putational grids including 800 and 1600 cells were used for bothmethods and the results were compared against the referencesolution. The volume fraction profile of Fig. 8 shows that on a samecomputational grid, the PRICE-C scheme is more diffusive than thePC-Osher scheme. This is due to the fact that the PC-Osher schemeuses the full eigenstructure of the system in the solution procedurewhile the PRICE-C does not employ the full eigenstructure of thesystems. However, on the same grid, PRICE-C is computationallyfaster than the PC-Osher scheme; therefore, one should determinethe CPU time usage of the methods to reach a specified accuracy toassess their ability in the simulation of two-phase flows.


To quantify the error in the computational domain X the 1-norm is used, which is calculated using the following formula:

kEk1 ¼ DxX8i2XjEij ð38Þ

The 1-norm was calculated for the water faucet case with theerror in the gas volume fraction as error measure which is:

Ei ¼ ag � ag;ref ð39Þ

The reference volume fraction ag,ref was calculated using thePRICE-C scheme on the finest grid of 25,600 computational cells.

Fig. 9 shows the norm of error versus CPU-time for both path-conservative methods and AUSMDV� method. As seen, to reach aspecified accuracy, the PC-Osher scheme consumes less CPU-timethan the PRICE-C scheme. Therefore, for this test case this schemeis more efficient than the PRICE-C scheme. However, for this prob-lem the AUSMDV� is more efficient than PC-Osher scheme.

4.3. Toumi’s shock tube

Toumi’s two-phase shock tube is another Riemann problem fortwo-fluid models. This test case was introduced by Toumi [44]and has also been studied in [11,25,35,45]. The left and right initialstates are given in Table 2. Because the initial velocities of thephases are equal, Eq. (10) should be used to model the pressure

x (m)

α g(-

)

0 20 40 60 80 100

0.29

0.295

0.3

0.305 ReferenceOsher-100 CellsOsher-1000 CellsOsher-5000 CellsOsher-10000 Cells


x (m)

u g(m

/s)

0 20 40 60 80 10050

55

60

65

ReferenceOsher-100 CellsOsher-1000 CellsOsher-5000 CellsOsher-10000 Cells

(c) Gas Velocity

Fig. 13. LRV shock tube: grid refinement study of

correction term for PC-Osher scheme. This is not necessary for thePRICE-C scheme. However, the same pressure correction term isconsidered for both schemes. In this case, the value of Bk is 0.999999.

The simulation results obtained by PC-Osher scheme are shownin Fig. 10. As this figure show, the results are in good agreementswith the reference solution which is obtained using AUSMDV� ona grid of 40,000 cells. Also, no significant difference can be seen be-tween two finer meshes. It should be noticed that Roe4 scheme ofRef. [25] produced six plateaus without using the entropy fix and itneeds an entropy fix to produce physical results. In contrast, thepresent PC-Osher scheme does not need any type of entropy fixfor the four-equation model.

The same problem was solved using the PRICE-C scheme andthe simulation results are shown in Fig. 11. As shown, by increas-ing the number of computational cells, the results approach to thereference solution. The reference solution was obtained using theAUSMDV� method of Evje and Flatten [2] using 40,000 computa-tional cells. Also, no substantial difference can be observed be-tween the results of 10,000 and 20,000 cells.

To compare the accuracy and efficiency of the PC-Osher andPRICE-C scheme for this test case, the CPU time needed to reacha specified accuracy is compared for both schemes which is shownin Fig. 12. As this figure show, in this case both schemes approxi-mately have the same efficiency and their efficiency is better thanthe AUSMDV� method. The reference volume fraction ag,ref in Eq.

x (m)

p(k

Pa)

0 20 40 60 80 100

265.5

267

268.5

270

271.5


(b) Pressure

x (m)

ul(

m/s

)

0 20 40 60 80 100

0.98

0.99

1

1.01


(d) Liquid Velocity

PC-Osher scheme at t = 0.1 s using CFL = 0.9.

x (m)

α g (-

)

49.5 50 50.5 510.29

0.292

0.294

0.296

0.298

0.3

0.302

0.304

ReferenceOsher 10000 CellsPRICE-C-40000 Cells

Fig. 15. LRV shock tube: comparison of the PRICE-C and PC-Osher schemes on thevolume fraction waves prediction. Data are plotted at t = 0.1 s using CFL = 0.9.


(39) was computed using the AUSMDV� scheme on a grid of 40,000cells (i.e. the reference solution).

4.4. Large relative velocity shock tube

The large relative velocity (LRV) shock tube is another impor-tant Riemann problem used as benchmark for compressible mul-ti-phase flows. The system consists of a 100 m long tube with adiaphragm in the middle. The initial left and right states of theproblem are given in Table 3. This test case was investigated inRef. [46] and later it has been also studied by other authors [2,25].

The results of the PC-Osher scheme are shown in Fig. 13 on thedifferent grids and they are also compared with the reference solu-tion. As the figure shows, by refining the mesh the results approachto the reference solution. The reference solution was obtained byusing the AUSMV method of Evje and Flatten [2] on a grid of50,000 computational cells. As shown in this figure, the volumefraction waves can be seen in the reference solution as well as inthe results of PC-Osher scheme on a grid of 10,000 cells. However,the reference method produces a sharp wedge in the volume frac-tion at x = 50 m which is due the fact that the reference solution isnot Total Variation Diminishing (TVD) [2].

Fig. 14 shows the results of PRICE-C scheme using different gridsizes. As shown, the results are converged to the reference solutionas the grid is refined. No substantial difference was observed be-tween two finer meshes. As the figure shows, the PRICE-C method

x (m)

αg

(-)

0 20 40 60 80 100

0.29

0.295

0.3

0.305ReferencePRICE-C 100 CellsPRICE-C 10000 CellsPRICE-C 20000 CellsPRICE-C 40000 Cells


x (m)

u g(m

/s)

0 20 40 60 80 10050

55

60

65


(c) Gas Velocity

Fig. 14. LRV shock tube: grid refinement study o

can capture the wedge in the gas velocity and volume fraction atabout x = 50 m. However, the results are more diffuse when theyare compared with the reference solution.

The close up view of the volume fraction wedge produced bydifferent methods is shown in Fig. 15. As stated before, the

x (m)

p(k

Pa)

0 20 40 60 80 100

265.5

267

268.5

270

271.5


(b) Pressure

x (m)

ul(

m/s

)

0 20 40 60 80 100

0.98

0.99

1

1.01


(d) Liquid Velocity

f PRICE-C scheme at t = 0.1 s using CFL = 0.9.

CPU time (s)

Nor

mof

erro

r

100 101 102 103

0.001

0.002

0.003

0.004

0.005

0.006

PC-OsherPRICE-CAUSMDV*

Fig. 16. LRV shock tube: comparison of the CPU-time and norm of error of thePRICE-C and PC-Osher schemes at t = 0.1 s using CFL = 0.9.

Fig. 17. Schematic sketch of the water–air separation test case.


PRICE-C scheme is more diffusive than the others while the AUSMVmethod produces a sharp wedge. However, the PC-Osher scheme

x

α g(-

)

0 1 2 30

0.2

0.4

0.6

0.8

1

(a) Gas Volum

x (m)

u l(m

/s)

0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

AnalyticalReference2500 Cells5000 Cells10000 Cells

(b) Liquid Velocity

Fig. 18. Water–air separation–validation of the results of

produces results which are very accurate when they are comparedwith those of the Roe4 method of [25].

The CPU-time usage of both methods is shown against norm of er-ror in Fig. 16. As this figure shows, to obtain a specified accuracy, thePC-Osher scheme is more efficient than the PRICE-C and AUSMDV�

schemes. In this case, the reference volume fraction in Eq. (39) wasobtained using the PC-Osher scheme on a grid of 40,000 cells.

(m)

4 5 6 7

AnalyticalReference2500 Cells5000 Cells10000 Cells

e Fraction

x (m)

p(k

Pa)

0 1 2 3 4 5 6 7100

110

120

130

140

150

Reference2500 Cells5000 Cells10000 Cells

(c) Pressure

the PRICE-C scheme at t = 0.6 s and using CFL = 0.9.


It should be noted that the eigenstructure evaluation of theproblem is the most expensive part of the computations for PC-Osher scheme because there is not an analytical expression for sys-tem eigenvectors and they should be computed numerically. A de-tailed study of the eigenstructure computations shows that theCPU-time usage of this part is about 10% of the total CPU-timeusage of the problem.

4.5. Near single phase flows treatment

In this section the ability of the employed numerical methods indealing with near single phase flows is examined. This is per-formed using the standard water–air separation test case whichis one of the most challenging two-phase flow problems. This prob-lem was proposed by [42] and later was studied by severalresearchers [2,11,31].

As shown in Fig. 17, the problem consists of a vertical pipe of7.5 m length of which the top and bottom are closed. Initially,the pipe is filled with a stagnant homogenous mixture of air andwater where the volume fraction is 0.5. The pressure is uniformalong the pipe and is 105 Pa. Water and air are separated underthe action of gravity. If the pressure variation along the pipe is ne-glected, there is an analytical solution for the gas volume fractionand liquid velocity which are as follow:

x (m)

αg

(-)

0 20 40 60 80 100

0.29

0.295

0.3

0.305 First Order-PC-OsherTMH-PC-Osher


x (m)

u g(m

/s)

20 40 60 8050

55

60

65

First Order-PC-OsherTMH-PC-Osher

(c) Gas Velocity

Fig. 19. LRV shock tube: comparison of first and second order PC-O

ag ¼1 for x < 1

2 gt2

0:5 for 12 gt2

6 x 6 L� 12 gt2

0 for L� 12 gt2 < x

8><>: ð40Þ

ulðx; tÞ ¼

ffiffiffiffiffiffiffiffi2gx

pfor x < 1

2 gt2

gt for 12 gt2

6 x 6 L� 12 gt2

0 for L� 12 gt2 < x

8><>: ð41Þ

where L is the pipe length. After the time

t ¼

ffiffiffiLg

s� 0:87

the phases are fully separated and the liquid volume fractionreaches to steady state condition. In this test case the gas velocitybecomes too high if the interfacial drag force between two phasesis neglected. To avoid numerical difficulties related to this problem,the interfacial drag force suggested by Evje and Flatten [35] wasused. It was tried to solve this test case using the PC-Osher schemebut the scheme fail to predict this near single phase flow. Thisweakness of the PC-Osher scheme was already studied for the Abgr-all principle. So it can be concluded that this scheme is not a goodcandidate for near single phase flows.

x (m)

p(k

Pa)

0 20 40 60 80 100

265

266

267

268

269

270

271


(b) Pressure

x (m)

u l(m

/s)

0 20 40 60 80 100

0.98

0.99

1

1.01

1.02

1.03


(d) Liquid Velocity

sher schemes on a gird of 1000 cells using CFL = 0.5 at t = 0.1 s.

x (m)

αg

(-)

0 20 40 60 80 1000.29

0.292

0.294

0.296

0.298

0.3

0.302TMH-PRICE-CFirst Order PRICE-C

x (m)

p(k

Pa)

20 40 60 80

265

266

267

268

269

270

271

TMH-PRICE-CFirst Order PRICE-C


x (m)

u g(m

/s)

0 20 40 60 80 10050

55

60

65


x (m)

u l(m

/s)

0 20 40 60 80 100

0.98

0.99

1

1.01



Fig. 20. The LRV shock tube: comparison of first and second order PRICE-C schemes on a gird of 1000 cells using CFL = 0.5 at t = 0.1 s.

CPU time (s)

Nor

mof

erro

r

101 102

0.001

0.002

0.003

0.004

TMH-PC-OsherTMH-PRICE-C

Fig. 21. LRV shock tube: comparison of the CPU-time and norm of error of thesecond order TMH–PRICE-C and TMH–PC-Osher schemes at t = 0.1 s using CFL = 0.5.


In contrast to the PC-Osher scheme, the PRICE-C scheme couldpredict this challenging problem. The results of this scheme areshown in Fig. 18 along with the reference and approximate analyt-ical solutions at t = 0.6 s. The reference solution was obtained usingthe AUSMDV� method of Evje and Flatten [2]. As this figure shows,by increasing the number of computational cells the results ofpresent PRICE-C scheme approach to those of the reference solu-tion. As seen, the AUSMDV� show some oscillations in the predic-tion of pressure profile while the PRICE-C method givesoscillation free results. This is due the fact that the AUSMDV�

method is not TVD. On the other hand, the PRICE-C method pro-duces more numerical diffusion than the AUSMDV�. It should benoted that the gas velocity (not shown) oscillates at the transitionboundary from pure liquid zone to the two-phase mixture zone.

4.6. Results of TVD-MUSCL–Hancock method

In this section the results of second order extension of PC-Osherand PRICE-C schemes using the well-known TVD-MUSCL–Hancock(TMH) method are presented. The large relative velocity shock tubeis solved using the first and second order methods on a grid of 1000computational cells. For both schemes the Monotonized centraldifference (MCD) limiter of van Leer [39] was used.

The simulation results of LRV shock tube problem are shown inFig. 19. As this figure shows, the TMH version of PC-Osher scheme

is less diffusive on the sonic waves however; this scheme showssome small amplitude oscillations on the volume fraction waves.The results of both methods were obtained on a grid of 1000 cellsand using CFL = 0.5.


The LRV shock tube was also solved using the first and secondorder PRICE-C schemes. The results were obtained on a grid of1000 cells using CFL = 0.5, are shown in Fig. 20 at t = 0.1 s. theMCD limiter was used in the TMH method. As this figure shows,in this case the results of the second order method are less diffu-sive on the sonic waves while there are some small oscillationson the volume fraction waves. However, using the MCD limiter,the amplitude of the oscillation in the PC-Osher is larger than thatof the PRICE-C scheme. Fig. 21 shows, the norm of error versusCPU-time for the second order extension of the PRICE-C and PC-Osher schemes. As this figure shows, here again the PC-Osher ismore efficient than the PRICE-C scheme.

5. Conclusion

One of the main problems of present two-fluid model in two-phase flow is existence of non-conservative terms in the momen-tum equations. In the present paper, the concept of path-conserva-tive schemes was used to solve the model equations numerically.The upwind Path-Conservative Osher (PC-Osher) and centralPRICE-C schemes were used for numerical solution of two-phaseflows and their ability and numerical efficiency were examined.Several two-phase flow test cases were solved using theseschemes. First of all, the pressure and velocity preservation condi-tion or the so called Abgrall condition was examined for thesemethods and it was concluded that both employed path-conserva-tive methods satisfy Abgrall conditions. The water faucet, large rel-ative velocity shock tube and Toumi’s shock tube were solvedusing both methods and the results compared against the resultsof the AUSMDV�. Comparison of the results shows that thePRICE-C is more diffusive than the PC-Osher scheme but numericalefficiency of the PC-Osher is higher. However, the PC-Osherscheme failed to predict near single phase flow zones such aswater–air separation problem where the PRICE-C scheme couldpredict this problem with sufficient degree of accuracy.

The extension of the employed path-conservative schemes tothe second order of accuracy was performed using TVD-MUSCL–Hancock scheme. The results show that the second order versionsof both schemes are less diffusive on the sonic waves while theyproduced small amplitude oscillations on the volume fractionwaves.

Acknowledgments

The authors like to acknowledge Prof. Michael Dumbser forgreat helps during this research. The authors also like to thank Sha-hid Chamran University of Ahvaz for providing finical support ofthis research. The first author likes to thank Dr. Sevend TolakMunkejord and Dr. Tore Flatten for answering scientific questionsabout two-fluid model and the AUSM type methods respectively.

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