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

On a two-fluid model of two-phase compressible flowsand its numerical approximation Mai Duc Thanh Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc D

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

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 min-imize 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 discret-ize 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]):

@tðagqgÞ þ @xðagqgugÞ ¼ 0;

@tðagqgugÞ þ @xðagðqgu2

@tðagqgEgÞ þ @xðagðqgEgþ pÞugÞ ¼ p@tagþagqgugg;

@tðalqlÞ þ @xðalqlulÞ ¼ 0;

@tðalqlulÞ þ @xðalðqlu2

l þ pÞÞ ¼ p@xalþalqlg;

@tðalqlElÞ þ @xðalðqlElþ pÞulÞ ¼ p@talþalqlulg;

ð1:1Þ

whereaiis the volume fraction,qiis the density, uiis the velocity, eiis the internal energy and

Ei¼ eiþ1

2

2

i;

is the total energy, g is the gravity constant in the model with gravity, and g = 0 in the model without gravity, and the sub-script ‘‘i’’ can be ‘‘g’’ or ‘‘l’’, representing the gas or liquid phase of fluids respectively The volume fractions of the fluid satisfy

agþal¼ 1:

1007-5704/$ - see front matter Ó 2011 Elsevier B.V All rights reserved.

E-mail addresses: mdthanh@hcmiu.edu.vn , hatothanh@yahoo.com

Contents lists available atScienceDirect Commun Nonlinear Sci Numer Simulat

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c n s n s

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

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

p ¼cl 1

c l qlCplTl p1 el¼Cpl

c lTlþp1

q l

For the tests, we will take (see Chang and Liou[2], for example)

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 character-istic 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 equa-tions of state we construct a numerical scheme that is numerically stable for the system(1.1) We observe that only the non-conservative 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 Section2we 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 Section3we will adapt a standard scheme and construct a stable scheme for(1.1) In Section4we present several tests which show that the adaptive scheme is numerically stable Finally, in Section5we 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

@tðaiqiÞ þ @xðaiqiuiÞ ¼ 0;

@tðaiqiuiÞ þ @xðaiðqiu2

i þ pÞÞ ¼ p@xaiþaiqig;

ð2:1Þ

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

ui@tðaiqiÞ þaiqi@tuiþaiqiui@xuiþ ui@xðaiqiuiÞ þai@xp ¼aiqig;

for i = g, l Multiplying both sides of the last equation by ui, i = g, l, and then applying the chain rule, we obtain

u2

i@tðaiqiÞ þaiqi@t

u2 i 2

 

þaiqiui@x

u2 i 2

 

i@xðaiqiuiÞ þaiui@xp ¼aiqiuig;

or

u2

i

2ð@tðaiqiÞ þ @xðaiqiuiÞÞ þ @t aiqiu

2 i 2

þ @x aiqiui

u2 i 2

þaiui@xp ¼aiqiuig; i ¼ g; l:

Due to the conservation of mass, this yields

@t aiqiu

2

i

2

þ @x aiqiui

u2 i 2

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

@t aiqi eiþu

2 2

2 2

ui

þ p@taiþ ðp@xðaiuiÞ þaiui@xpÞ ¼aiqiuig; i ¼ g; l;

or

@t aiqi eiþu

2 2

2 2

ui

þ p@taiþ @xðaiuipÞ ¼aiqiuig; i ¼ g; l:

The last equation gives

@t aiqi eiþu

2 2

2 2

þ p

ui

þ p@tai¼aiqiuig; i ¼ g; l;

or

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

@tðaiqiÞeiþ ðaiqiÞ@teiþ ei@xðaiqiuiÞ þ ðaiqiuiÞ@xeiþ pð@taiþ @xðaiuiÞÞ ¼ 0; i ¼ g; l;

or

eið@tðaiqiÞ þ @xðaiqiuiÞÞ þaiqið@teiþ ui@xeiÞ þ pð@taiþ @xðaiuiÞÞ ¼ 0; i ¼ g; l:

Therefore, equations for conservation of mass imply that

aiqið@teiþ ui@xeiÞ þ p @taiþ @xðaiuiÞ 1

qið@tðaiqiÞ þ @xðaiqiuiÞÞ

or

aiqið@teiþ ui@xeiÞ þ p @taiþ @xðaiuiÞ 1

qiðai@tqiþqi@taiþqi@xðaiuiÞ þaiui@xqiÞ

¼ 0:

Cancel the terms to get

aiqið@teiþ ui@xeiÞ pai

qi ð@tqiþ ui@xqiÞ ¼ 0;

or

aiqi @teiþ ui@xei p

q2 i ð@tqiþ ui@xqiÞ

¼ 0:

Re-arranging terms, we get

aiqi @tei p

q2@tqiÞ þ uið@xei p

q2@xqi

¼ 0;

aiqi @teiþ p@t

1

qi

 

þ ui @xeiþ p@x

1

qi

 

Using thermodynamical identity

qi

we can rewrite the above equation as

aiqiðTi@tSiþ uiTi@xSiÞ ¼ 0;

or

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

aiqið@tSiþ ui@xSiÞ þ Sið@tðaiqiÞ þ @xðaiqiuiÞÞ ¼ 0;

we deduce the equations for entropy:

@tðaiqiSiÞ þ @xðaiqiSiuiÞ ¼ 0; i ¼ g; l:

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

@tðaiqiÞ þ @xðaiqiuiÞ ¼ 0;

@tðaiqiuiÞ þ @xðaiðqiu2

i þ pÞÞ ¼ p@xaiþaiqig;

@tðaiqiSiÞ þ @xðaiqiSiuiÞ ¼ 0; i ¼ g; l:

ð2:5Þ

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 ¼qgRgTg¼ ðcg 1Þqgeg;

eg¼ CgvT ¼Rg T g

c g 1; hg¼ CpgTg

ð2:6Þ

where Rgis the specific gas constant; Rg¼ggR, wheregis the mole-mass fraction, and R is the universal gas constant Let

Cgv¼ Rg

c g 1be the specific heat at constant volume and Cgp¼Rg c g

c g 1be the specific heat at constant pressure, so that Cgp=cgCgv Using thermodynamical identity, we have

eg

 ðcg 1Þdqg

qg

!

This gives

qcg 1 g

!

or

eg¼qcg 1

g exp Sg Sg

Cgv

:

Therefore,

p ¼ ðcg 1Þqcg

g exp Sg Sg

Cgv

Thus,

c  1

!1= cg

C

Furthermore, by definition, the enthalpy is

qg

So that

Cgv

qcg 1

or, for short,

hg¼ hgðqg;SgÞ ¼ GðSÞqcg 1

Cgv

Substitutingqgfrom(2.8)into(2.9), and re-arranging terms, we get

ðcg 1Þðcg 1Þ= cgpð cg1Þ= cgexp Sg Sg

Cgp

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

p ¼ ðcl 1ÞqlClvTl p1; el¼ ClvTlþp1

ql

where Clv¼ Rl

c l 1be the specific heat at constant volume and Clp¼Rl c l

c l 1be the specific heat at constant pressure, so that

Clp=clClv, where Rlis the specific gas constant

Substituting elfrom(2.11)into the thermodynamical identity del= TldSl pdvl,vl= 1/ql, we obtain

dðCvlTlþ p1vlÞ ¼ TldSl pdvl:

This yields

dSl¼ ClvdTl

Tl

Tl

dvl

or

1

ClvdSl¼dTl

Tl

þ ðcl 1Þdvl

vl :

Hence,

Sl Sl

Clv ¼ logðTlvcl1

so that

Tl¼qcl 1

l exp Sl Sl

Clv

Substituting Tlfrom(2.12)into(2.11), we obtain

p ¼ ðcl 1ÞClvexp Sl Sl

Clv

qcl

l  p1:

This implies

Rl

Clp

Moreover, the enthalpy in the liquid phase is given by

ql¼ ClvTlþp1

ql þ ðcl 1ÞClvTl¼clClvTl:

Substituting Tlfrom(2.12)into the last equation gives

hl¼ Clpexp Sð l Sl ClvÞqc l 1

l

or

hl¼ hlðql;SlÞ ¼ LðSÞqc l 1

Substitutingqlfrom(2.13)into(2.14), we obtain

hl¼ hlðp; SlÞ ¼ Clp

Rl

exp Sl Sl

Clp

2.3 Non-hyperbolic system of balance law

Let us choose the pressure p and the entropies Sg, Slas the thermodynamical independent variables Then, the unknown function, or the state variable, can be found under the form

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

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

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

aið@pðqiÞ@tp þ @S iðqiÞ@tSiÞ þqi@taiþaiuið@pðqiÞ@xp þ @S iðqiÞ@xSiÞ þqi@xðaiuiÞ ¼ 0; i ¼ g; l

or

ai@pðqiÞ@tp þqi@taiþai@SiðqiÞð@tSiþ ui@xSiÞ þaiui@pðqiÞ@xp þqi@xðaiuiÞ ¼ 0; i ¼ g; l:

Using(2.18), we get from the last equation

ai@pðqiÞ@tp þqi@taiþaiui@pðqiÞ@xp þqi@xðaiuiÞ ¼ 0; i ¼ g; l:

The last two equations (i = g, l) yield an equation for the volume fraction

ðal@pðqlÞqgþag@pðqgÞqlÞ@tatþap@pðqgÞal@pðqlÞðug ulÞ@xp þqgal@pðqlÞ@xðagugÞ qlag@pðqgÞ@xðalulÞ ¼ 0; ð2:19Þ

and an equation for the pressure

Next, the equations for mass can be written as

aiqi@tuiþ ui@tðaiqiÞ þ ui@xðaiqiuiÞ þaiqiui@xuiþai@xp þ p@x@xai¼ p@xai;

for i = g, l Canceling the terms and using the equations of mass, we obtain

@tuiþ ui@xuiþ1

qi

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

a l @ p ð qlÞ qgþ a g @ p ð qgÞ qlap@pðqgÞal@pðqlÞðug ulÞ@xp þqgal@pðqlÞ@xðagugÞ qlag@pðqgÞ@xðalulÞ

¼ 0;

q l a g @ p q g þ q g a l @ p q lðqlagug@pqgþqgalul@pqlÞ@xp þqlqg@xðagugþalulÞ

¼ 0;

@tugþ ug@xugþ1

q g@xp ¼ 0;

@tSgþ ug@xSg¼ 0;

@tSlþ ul@xSl¼ 0;

@tulþ ul@xulþ1

q l@xp ¼ 0:

ð2:22Þ

Setting

a1¼qg u g a l @ p q l þ q l ula g @ p q g

al@ p ð qlÞ qgþ a g @ p ð qgÞ ql; a2¼ag @ p ð q g Þ a l @ p ð q l Þðu g u l Þ

al@ p ð qlÞ qgþ a g @ p ð qgÞ ql;

a3¼ qg a l @ p ð qlÞ a g

a l @ p ð q l Þ q g þ a g @ p ð q g Þ q l; a4¼ ql a g @ p ð qgÞ a l

a l @ p ð q l Þ q g þ a g @ p ð q g Þ q l;

b1¼ ql q g ðu g u l Þ

a l @ p ð q l Þ q g þ a g @ p ð q g Þ q l; b2¼ql a g u g @ p q g þ q g a l ul@ p q l

a l @ p ð q l Þ q g þ a g @ p ð q g Þ q l;

;

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

AðVÞ ¼

q l 0 0 ul 0

0

B

B

B

B

B

1 C C C C C

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

where

qgðul kÞ þb4

qlðug kÞ

!

ða1 kÞ þa3b1

qg ðul kÞ

þa4b1

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 theFig 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 termsagqgg,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

LetDt andDx be the given uniform time step and the spacial mesh size, respectively Set

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

Dx:

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

@tðagqgÞ þ @xðagqgugÞ ¼ 0;

@tðagqgSgÞ þ @xðagqgSgugÞ ¼ 0;

@tðalqlÞ þ @xðalqlulÞ ¼ 0;

@tðalqlSlÞ þ @xðalqlSlulÞ ¼ 0;

@tðagqgugþalqlulÞ þ @xðagðqgu2

gþ pÞ þalðqlu2

@tðagqgugÞ þ @xðagðqgu2

ð3:1Þ

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

U ¼

agqg

agqgSg

alql

alqlSl

agqgugþalqlul

0

B

B

B

@

1 C C C A

agqgug

agqgSgug

alqlul

alqlSlul

agqgu2

gþalqlu2

l þ p

0 B B B

@

1 C C C A

;

V ¼ ðagqgugÞ; gðVÞ ¼ ðagðqgu2

ð3:2Þ

We now define a ‘‘composite’’ scheme

Unþ1j ¼ Unj  k FðU nj;Unjþ1Þ  FðUnj1;UnjÞ

;

jþ1;Þ  GðVnj1;þ;Vn

jÞ

;

ð3:3Þ

where F and G are convenient standard numerical fluxes The vectors Vnjþ1;in(3.3)will be determined as follows, relying on the previous result that the entropy Sgis conserved across stationary waves We thus computeqn

g;jþ1;;un g;jþ1; from the equations

an

g;jqn

g;jþ1;un

g;jþ1;¼an

g;jþ1qn g;jþ1un g;jþ1;

ðu n

g;jþ1; Þ2

2 þ hgðqn

g;jþ1;;Sng;jþ1Þ ¼ðu

n g;jþ1 Þ2

2 þ hgðqn

and computeqn

g;j1;þ;un

g;j1;þfrom the equations

an

g;jqn

g;j1;þun

g;j1;þ¼an

g;j1qn g;j1un g;j1;

ðu n

g;j1;þ Þ2

2 þ hgðqn

g;j1;þ;Sng;j1Þ ¼ðu

n g;j1 Þ2

2 þ hgðqn

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

agqgug¼ag;0qg;0ug;0:¼ mg;

ðu g Þ2

2 þ hgðqg;Sg;0Þ ¼ðug;0 Þ2

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

FgðqgÞ :¼ ðu2

g;0þ 2hgðqg;0;Sg;0ÞÞq2

g 2q2

ghgðqg;Sg;0Þ  ag;0ug;0qg;0

ag

gðqg hgðqg;Sg;0ÞÞ  mg

ag

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

FgðqgÞ

2 g;0þ 2hgðqg;0;Sg;0ÞÞqg 4qghgðq;Sg;0Þ  2q2

ghg; q gðqg;Sg;0Þ

g;0þ 2hgðqg;0;Sg;0ÞÞqg 4qghgðqg;Sg;0Þ  2qgpqðqg;Sg;0Þ ¼ 2qg u2

g;0þ 2

Z qg;0

qg

hq gðs;Sg;0Þds pqgðqg;Sg;0Þ

!

g;0þ 2

Z qg;0

q g

pqgðs;Sg;0Þ

s ds pqgðqg;Sg;0Þ

!

;

which has the same sign as

GgðqgÞ :¼U

0

ðqgÞ

2 g;0þ 2

Z qg;0

q g

pqgðs;Sg;0Þ

Moreover, it is not difficult to check that

2pqðq;Sg;0Þ þq pqqðq ;Sg;0Þ > 0; i ¼ g; l;

This implies that

GgðqgÞ

dqg

:¼ ð2hq gðqg;Sg;0Þ þ pqgqgðqg;Sg;0ÞÞ ¼ 1

qg 2pqgðqg;Sg;0Þ þqgpqgqgðqg;Sg;0Þ

Moreover, it is easy to check that

Hence, there exists exactly one valueqg=qg,maxsuch that Gg(qg,max) = 0, Gg(qg) > 0 iffqg<qg,max, and Gg(qg) < 0 iffqg>qg,max

for i = g, l Moreover, a straightforward calculation shows that

qg;max¼ 2qg

cgð cgþ1Þexp Sg S g;0

C g v

c g 1

;

ql;max¼ 2ql

C lp exp Sl S l;0

C l v

cl 1

:

ð3:10Þ

Consequently,

F g ð q g Þ

d q g >0; qg<qg;max;

F g ð qgÞ

d q g <0; qg>qg;max;

F g ð q g Þ

d q g ¼ 0; qg¼qg;max; i ¼ g; l:

ð3:11Þ

Observe that

Fgðqg¼ 0Þ < 0;

Thus,(3.7)has a solution iff

Fgðqg;maxÞ P ag;0ug;0qg;0

ag

or, equivalently,

agPaminðUg;0Þ :¼ag;0jug;0jqg;0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Fgðqg;maxÞ

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 iffag=ag,min(Ug,0) The fact thatag=ag,0still gives us a solution of(3.6)and, therefore, of(3.7),ag,0has to satisfy

ag;minðUg;0Þ 6ag;0:

To make sure that the scheme always works, we practically assign the new value foragby

anew

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):

j ¼1ðUnjþ1þ Unj1Þ k

2ðf ðUnjþ1Þ  f ðUnj1ÞÞ;

Vnþ1j ¼1ðVnjþ1;þ Vnj1;þÞ k

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 [1, 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;0ÞðxÞ ¼ UL; if x < 0



where UL, URare given for each test

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