FED-Vol. 185, Num.rlcal Methods In Multlph... Flow.ASME 1994

ON THE NUMERICAL TREATMENT OF INTERPHASEFORCES IN TWO-PHASE FLOW

Paulo J. Oliveira

Departamento de ElectromecânicaUniversidade da Beira Interior

Covilhã, Portugal.

!I.::;<i

Raad I. Issa

Department of Mechanical EngineeringImperial College of Science, Technology and Medicine

London, United Kingdom

ABSTRACTIn Eulerian computations of two-phase flow,

iterative methods are usually employed to solve thediscretised sets of equations (comprised of themomentum and continuity equations for each phase)that govern the flow. The iterative process is oftentroublesome in its convergence behaviour andsometimes fails to converge at all. A major role inaffecting the robustness and stability characteristicsof the solution algorithm is known to be played bythe magnitude of the interphase force term in themomentum equations and by how it is treatednumerically.

In this paper different schemes for treating theinterphase force term in an iterative procedure basedon the pressure-correction principIe are examined;they range from the fully explicit (where the wholeof the term is evaluated at the previous iteration) toan almost fully implicit treatment based on theelimination of phase velocities from the momentumequations of the counterpart phase.

The schemes are applied to the computation ofdispersed bubbly flow in one- and two-dimensionalconfigurations for a wide range of loading (i.e.average phase fraction) and of bubble diameter. Theperformance of the schemes is compared against eachother. It is shown that in some cases only the fullyimplicit treatment is able to produce convergedresults.

1. INTRODUCTlONOne of the two to thebasic approaches

j

mathematical modelling of two-phase flows is theEulerean two-fluid model (Ishii 1975), where bothphases are treated as inter-penetrating continuawhose motion is governed by appropriately averagedconservation equations written for each phase. Afteraveraging, interphase terms arise in the momentumequations, representing forces due to transfer ofmomentum and mass between the phases, such asdrag, virtual mass etc., and these terms must bemodelled. When the partial differential equationscomprising these forces are solved numerically, bymeans of iterative finite-difference or finite-volume

methodology, convergence problems may arise due tothe magnitude of the interphase force term (Stewart& Wendroff 1974). It is the purpose of the presentpaper to analyse different ways of handling this termnumerically.

In general the interphase force (FD) may belinearised by writing it as proportional to therelative velocity (ur); thus, for example for the liquidphase in a dispersed gas (G) - liquid (L) mixture:

FD = FD(UG - uL)'L

where the parameter FD is in general a function ofthe volume fraction (O'==O'G)' the physicalproperties of the continuous phase (the liquid) andalso, in most cases, of the relative velocity itself. Inthe case when the interphase force is due to dragonly:

131

The drag factor Cf is defined as Cf ==18JiLfd~ (Ji:viscosity; db: bubble diameter) and g is a function of

the bubble Reynolds number (Reg,=PLdbUr/ Jid,given for example by g=I+0.15Reg.6 7 as in Wallis(1969). f(O') is a function of the volume-fraction awhich can be used as a corrective factor for highconcentrations; the symmetric equation f(a )=O'(l-a)is adopted here as the base f(a )-function since ittends to the right limits whenever a tends to O or to1. For low a, f(O') becomes equal to a, as it should

for monodispersed spheres. For low Reb the functiong is approximately 1, and the drag is linearly related

to the relative velocity ur =="uG-uL 11.For high Rebthe drag becomes non-linear in Ur'

It is also useful to define a time scale for the draginteraction as:

TD =PLd~/18JiL'Lwhich is called the relaxation time (in this case forthe liquid) and is related to the drag factor via

Cf=pLfTDL' Note that the relaxation. timeassociated with the gas phase is smalIer than TDL bya factor pLf PG :::::103 for an air-water mixture.

The problems associated with the numericalimplementation of the drag force occur because therelaxation time (especialIy for the gas) is usualIysmalI compared with the time-step bt used in the

computations, and this implies that FD should betreated implicitly. As an illustration, for air-water

flow with 1 mm bubbles TDL= 1000 xx (10-3)2/18/10-3 = 0.055 s, TDG -10-5 s, andCf=1.8 104. The value of the drag is given by theproduct of a large number (Cf) by a smalI number(ur:::::0.1 m/s). As a consequence any error in ur'which is likely to occur during the iterativeprocedure, would be magnified by a factor equal to

C f' and this lies at the root of the conver,genceproblems.

There are a number of ways to treat the dragterm numerically and this is the subject. of thispaper. In section 2, the equations to be solved aregiven and the numerical method is described. This isfolIowed by a description of the different algorithmsused to deal with the drag term (section 3), and theresults of application of these algorithms is given insection 4. From a comparison of the algorithmperformance, conclusions about suitability of eachmethod can be drawn (section 5).

2. EQUATIONS

In this section the governing equations are stated

and the solution procedure is explained. This will berestricted to the necessary details, since most of it iswelI-established; the method is based on the finite-volume approach with its extension to non-staggeredmeshes (see Peric, 1985).

2.1 Di:fferential EquationsThe two-fluid model equations, restricted to

incompressible flow without phase change, can bewritten in terms of phase-averaged velocity andpressure as:

Pk(!tak + v. O'kUk) = O,

Pk(~tO'kUk+ v. O'kUkUk)= -O'k vp + v. O'kTt

(1)

+ PkO'kg + FD .k

In the equations of continuity (1) and momentum(2), the index k denotes the phases (c for thecontinuous phase and d for the dispersed one); it hasalso been assumed that the same pressure acts onboth phases and that interfacial-averaged values ofpressure and viscous stress equal the correspondingbulk phase values. The turbulent stress is given by aNewtonian-like expression:

(2)

where the eddy viscosities (Jif) are obtained from aturbulence model, similar to what is used in single-phase modelling. Since the objective of the presentpaper is the numerical formulation of the interphaseforce term, details pertaining to turbulencemodelling are excluded (this is given in Oliveira1992).

2.2 DiscretiseJ EquationsThe above differential equations are discretised by

integration over hexahedral celIs forming thecomputational mesh. A non-staggered type of meshis used, whereby alI main variables are stored at celIcentres. In what folIows the nomenclature illustratedin Fig. 1 is adopted. Locations where variables, ordifferences of variables (e.g. .6.p), are computed are.denoted with superscripts (either "P" for the celIcentre, or "f" for the face along direction 1=/). jFolIowing standard methods for non-staggeredmeshes (e.g. Peric 1985) alI the resulting algebraicequations may be cast in the generallinearised form:

6

where S is the source term and the A's arecoefficients obtained from the discretisation of theconvection and diffusion terms in the diferrentialequations. When the upwind scheme is used, theexpression for Ais:

Af = Df + Max(Ff'O)=Df - Min(F f'O)

for f=f-,for f=f+, (5)

with the diffusion flux defined by-f

D = (J:Jl )f ~Bf .Bf. = (aJl) B2f 'Y 7' fJ fJ 'Yf f'

and the convective flux defined by:

(6)

.

F -f" Bf -ff = O' P L,. f jU j .J

In these equations, the overbar denotes arithmeticaverage, Bli is the i-component of the area-vectororientated along l-direction, Bf is the face area,- 'Y isthe cell volume, 'Yf is calculated as 'Yf == E l::lXjHBf j' and (ti is an upwinded volume-fraction atpoint "/" obtained from either ap or O'F (Fig. 1)according to the sign of Fí == E jBf }l~. The cell"face-velocities" li are interpolated at cell facesusing a special interpolation practice so that pressuredecoupling on the non-staggered mesh is avoided.The actual expression used for the face fluxes issomewhat involved and can be found in Oliveira

(1992). Hereafter the contribution of surrounding

cells will be denoted as H(

accurate. Thealgorithm falls in the fully-implicitclass, with the pressure being obtained from apressure-correction equation derived from acombination of the continuity and momentumequations. The explanation of the algorithm givenbelow adopts the splitting concept and terminologyintroduced by Issa (1986). Here, for clarity, one ofthe formulations used to deal with the drag term isadopted; this will be called the base algorithm. Theother interphase force formulations, which aredescribed in the next sub-section, will not producesubstantial alterations to the solution algorithmdescribed here. The steps in the algorithm are:

1- Solve the continuous phase momentumequation (from (8) with k=c):

(Ac O:cPcf"\.", Hn(

'") P ~ Bp[~ n ]P

o I c5t ric = c uic - O:c t li P I(

n n) S

c O:cPcf" n( )+ FD ui -ui f" + u.+ ---r-"t ui' 13d c ,(I C

where ""," denotes intermediate values. Analogy with

Eq. (4) yields: A~c == A~+(O:cPcf")/6t.

2- Solve the dispersed phase momentum equation(from (8) with k=d):

(A~Io:~~f"+FDf")uid = H:I(ui) - O:P~BK[~pn]r'" d O:Pdf" n

)+ FDf"ui + Su.+ ---r-t ui' (14c , u d

with A~d ==A~+(O:Pdf")/c5t+FDf". Notice that thedrag term in (13) is treated explicitly, whereas it isimplicitly incorporated in Eq. (14). This practice isanalysed in the next sub-section.

3- Assemble the pressure correction (p') equationbased on the overall continuity (10). The pr~ssureand velocities are updated according to the equationsformulated by Issa & Oliveira (1993). These are:

(18)

The fluxes F'" are corrected in the same way as the

nodal velocities (Eqs. (16) and (17)) and thecorresponding expressions are (see Issa & Oliveira1993):

(19)

(20)

The pressure-correction coefficients appearing in Eqs.(15) - (20) are given by:

I ~~d) B} == AIJc + AIJd, (21)dAp = 2:AIJ,

6 /S~= - 2: (-1)/ (Fj / Pc+ Fj /Pd)'/ c d

In Eq. (2U the velocity coefficients Ak are defined asAk = Apk- A~, being therefore (from (13) and(14)):

(22)

(23)

(24)

4- Solve for all additional scalar equations. In thepresent case these are the turbulence quantities to besolved for, k and (. With new values of k and f theliquid and gas effective viscosities are updated.

5- The dispersed phase continuity equation (11) issolved implicitly in order to obtain an updated void-fraction (o: == O:d):

(Ag+Pt7+Max[-div(Ud)'O]) 0:'"= H~(o:"'). Pdf"

+ Max[dlv(ud)'O]o:n+ To:n. (25)

The updated void-fraction and gas flux direc.tions areused to determine upwinded cell-face void-fractions,ê;f, which are stored.

With this the algorithm for two-phase flowcomputations is complete. The solution will beadvanced in time until the normalised residuaIs ofalI the equations are smalIer than a specified value(we use 10-4). At this point the solution is said to beconverged and, since overall continuity for the sumof gas and liquid is satisfied together with that ofthe gas, continuity will also be satisfied for the liquidphase.

..'

..~;~

3.2 Algorithm VariantsTo clarify subsequent explanation, the discretised

134

equations of motion of the two phases (1. forcontinuous and 2 for dispersed) are re-written in aconcise manner as (from Eq. (8»:

AI uI =FD(u2-udY + BI(26)

where the B,s contain all terms on the rhs of Eq. (8)except interphase force. The different algorithmsdealing with this force are outlined in what follows.

1- Base algorithm. Here the force is incorporatedexplicitly in the continuous phase momentumequation and implicitly in the dispersed phaseequation, as outlined in section 3.1:

(27)

(The superscript "n" denotes previous time-leveI.)

2- Variant 2. The drag term is treated implicitlyin both momentum equations. thus:

(AI+FD'Y)UI =FD'Y u~ + BI,(A2+F D 'Y)u2 = F D 'Y u~ + B2.

(28)

3- Variant 3. Equations (26) are pre-arrangedalgebraically so that the second-phase velocity iseliminated from the first-phase momentum equation.and vice-versa. Starting from:

(AI+FD'Y)UI = FD'Yu2+ BI,

(A2+FD'Y)u2 = FD'YuI + B2'

eliminate u2 from the first equation to obtain:

(29)

where FACI and FAC2 are generally denoted byFACi which is given by:

(30)

and Ai denotes either AI or A2'

4- Variant 4. In the equation for each phase, the'li

H.

velocity of the second phase is linearised about itsprevious time-Ievel ,:,alue; an algebraic pre-elimination, as in variant 3, is then applied. Thederivation for phase-l is as folIows. The velocity u2present in the drag term is linearised as:

so that the phase-l equation becomes:

The increment bU2 is obtained from an approximatemomentum equation for phase-2, where theint~rphase force is the only retained term:

bU2 = FD'Y bUI / (A2+FD'Y)=

= FD'Y(UI-u~) / (A2+FD'Y).

The equation for phase-l becomes, after replacingbU2from expression above:

or,

(F D'Y)2(AI + FD'Y- A + F 'Y) uI =2 D F 'Y

= BI + FD'Y (u2 - A J 'Yu~).2 DThis expression can be written in a form similar tothe previous variant by making use of FAC definedabove,

and similarly for the second phase: (31)

(A2+FACI.AI) u2= B2

+ FAC1 ( Alu~ + FD'Y(u~-U2) ).

The pressure-correction equation is derived from themomentum equations and must therefore bemodified for each of the variants mentioned. Theway to do this follows closely the derivation of thepressure equation outlined in section 2.1, themodifications being self-evident except perhaps forvariant 3. That derivation is based on a splitting

135

.

procedure where the only retained terms are thepressure gradient and the inertia term (here denotedE ==o:p'f'Ibt); as a consequence, the diagonalcoefficients of the u equations in variant 3 (i.e.

Aj+FACj.Aj in Eq. (29)), will appear in thedenominator of the pressure coefficients, with the A,sreplaced by E,s (cf. Eqs. (13) and (14) against Eqs.(16) and (17)). Hence the coefficients of the pressurecorrection equation in variant 3, instead of beinggiven by Eq. (21), are given by:

p - "" j 2 ( FD'f'E2 )-1

(Ajh = (0:10:1) Bj E1+E +F 'f'2 D

p - "" j 2 ( FD'f'E1 )-1

(Ajh = (0:20:2) Bj E2+E +F 'f' .1 DIn these expressions f denotes face values, and O:i isthe volume fraction of phase i. which can be thearithmetic average value (a) or the upwinded value(ã ) at the face.

(32)

3.3 Remarks

The simplest. and perhaps mosto commonalgorithm to treat the interphase force is variant 2,where the force is implicitly incorporated in bothphase momentum equations (see Stewart. &Wendroff 1984). Looney et aI. (1985) argue that this

treatment leads to slow convergence when FD islarge and show that variant 1 when implement.ed inconjuction with the PISO method of Issa (1986)improves the convergence rate. The authors pointedout that improvement can be expected mainly if thevolume fraction of the dispersed phase remains smalIover the whole region of interest as was the case withtheir solid-liquid calculations. Otherwise thedispersed phase may become a continuous one (say if0:>0.5) and the implicit/explicit treatment acçordedto the phase equations would have to beinterchanged. For these authors, such a problem didnot arise because they were dealing with solid-liquidflow at low solid-fraction. The resulting equations invariant 4 show some similarities with those of the

full elimination (compare (29) and (31)), but somedegree of approximation is brought in. Theadvantage of this variant is that less computerstorage is required than with variant 3. As for thefulI elimination, it was first used in conjunction withthe two-fluid model by Harlow & Amsden (1975)and extended later by Spalding (1980). The formgiven above is slightly different from the one used bySpalding with the attendant differences in the

~...~1~

'1 :~~

formulation of the pressure-correction equation.1

.~....4. RESULTS

Two sets of cases were used to test the differentalgorithm variants. In the first a simple one-dimensional bubbly flow in a vertical channel wasdevised. The purpose of this case was to eliminatethe algorithms that are fundamentalIy flawed; if themethod does not work welI in a simpIe 1-D situationit would be very unlikely that it could cope withcomplex multi-dimensional flow.

The second case was a more realistic one in whicha complex flow pattern is obtained and where phasesegregation and flow recirculation occur. The casechosen was the two-dimensional bubbly flow througha T-junction.

For both of the above test cases, calculations were

performed over a wide range of realistic values of (' jthe drag factor defined in section 1. It should bestressed, however, that the results of this study arenoto restricted to the drag force as ali interphaseforces can be accorded the same treatment. It is onlybecause drag is usualIy the dominant force that it issingled out for treatment here.

.;'.

One-O;mens;onal Tests

Details of the case are:

Fluid densities: P1=1000 Kg/m3; P2=1 Kg/m3.Pipe dimensions: diameter D=50 mm, length L=500mm.

Inlet. coriditions: u1=u2= U = 1.0 ml s; 0:2=0.10 .Initial conditions: zero field for all dependentvariables (0:,u1,u2'P).Time-step bt=O.l s., corresponding to a localCourant number (ubtjbx) of 2.

The linear drag is given by FD=C jO:(1-0:), thatis f(o:)=o:(1-0:).

The results of calculations for several drag

coefficients Cj (non-dimensional Cj == Cj UIfj.pg.varying from 10-2 to 105) are shown in Fig. 2 wherethe number of iterations to convergence is plotted

against Cj' It is seen that the only method able tocope with very high Cj is variant 3, Í.e. the fullelimination. For this method the number of time-

steps to convergence (Nt) remains fixed at 43 whileCj varies from 102 to infinity. Ali the other methodsfail for large Cj' variant 2 being the more robustowith convergence for Cj=103 but requiring asmalIer time step of 0.01 seco The first variant to fail

136

CJ)300ZO

~e:::w200I-

IJ...O

ffi 100aJ~::::>z

IB&I varo

~gas i.rn~Jicit. .cit-varo - 4; '11. 1mli- varo_RI~r eJlmJ~atlo:r- varo lineanaation

o1Õ-z1Õ-t1ÕO 1Õ' 1Õz 1ÕJ 104105

DRAG PARAMETER:CF

FlG. 2 NUMBER OF ITERA TIONS TO CONVERGE

DIFFERENT DRAG FORMULA TIONS (LINEARDRAG).

10

(f) 1.-J« 1O-1:::>O 10-2(f) 1O -3,WO:::1O -4,I

:::> 1O-5,10 -I,

10-7,0.0

- variant1n_- variant2

- variant 3

0.5 1.0TIME

1.5 2.0 2.5

(sec)FlG. 4 RESIDUAL HISTORY FOR THREE DRAG

FORMULA TIONS (Cf = 102; ét = 0.01 s).

is no. 4, the linearisation procedure, and this may beexplained by the presence in Eq. (31) of termsFD(u~-u~), which adversely affects stabilitywhenever FD is large.

Further illustration of the aforementioned trends

is provided by Figs. :3and 4 which present the decayof urresiduals as a function of time (which can beseen as an iteration counter). Figs. 3 (a) and 3 (b)show the residual history for variant 2 and 3,

respectively, for Cf=I01,102,103 and 104 with bt=O.1seco The fulI elimination (variant 3) shows identical

behaviour for alI values of Cf' whereas for the

2 4TIME

6 8

(sec)10

a) Variant 2: implicit drag in both gas and liq. equations.

102

(f) 10.-J« 1:::>010 -1

(f) 1 O -2WO:::1O-3,I

:::> 1O-4,10-5.

10-1.O

- C,=10'-- 102103104

(the curves coincide)

2 4TIME

6 8

(sec)b) Variant 3: fuI! elimination

10

FlG. 3 RESIDUAL HISTORY FOR 4 DRAG

FACTORS (C! = 101,102, lrP, 104).

standard implicit treatment (variant 2) convergence

is not obtained for Cf greater than 103. Figure 4compares the residual decay for variants 1,2 and 3,

at Cf=102 and with a time step smaller than before,8t = 0.01 seco Here it can be seen that variant 1,although eventually converges, is on the verge ofinstability.

Two-Dimensional Tests

The previous results revealed the good stabilisingeffect of fulI elimination (variant 3). This is nowtested in more complex situations, whererecirc.ulation may be present, and strong phase

137

.

FIG. 5 CONTOURS OF THE DRAG PARAMETER

FD (Kgjsm3) FOR BUBBL Y FLOW IN A T-JUNCTlON.

segregation occurSj the case chosen is that of bubblyflow in a two-dimensional T-junction (Issa &Oliveira 1990). The improved linearisation algorithm(variant . 4) has also been tested but showed littledifference compared with variant 1 (base method);hence it is not discussed further. Variant 2 wasrejected because of its poor initial performance,showing little improvement over the base method,and is therefore excluded from the tests.

In this test-run the simulation is that of a low-

quality bubbly air - water mixture, flowing along astraight main-branch and, at the T-junction, part ofthe flow is diverted into a side-branch forminganangle at 90 degrees with the main-branch. The area-averaged inlet conditions were: phase velocities,ua =1.60 mfs and uL = 1.53 mfs; void-fraction,ao= 2.08 %. The bubble diameter was assumed to be

ver 1: Ges Implicitver 3: Eliminetiont( a)=a( 1-a)

..~

246 8

TIME (se c)10

(f)--.J« 10-1=>

010""(f)Wa::: 10""

--- ver 1: Ges Implicit- Ver 3: Eliminetionf(a)=a(1-a)

(f) 10~.(f)«~ 10"

10 ..Ó 2 4 6 8

TIME (sec)FIG. 6 COMPARISON OF TWO DRAGFORMULATIONS USING A TWO-PHASE FLOW INA T-JUNCTION, WITH f(a)=a(l- a).

10

1 mm. For the cases considered here the overallextraction ratio at the junction was flxed atQ3fQl = 0.38,whereQ is the volumetricflow-rate,subscript 1 is inlet arm and 3 is side-branch armo Inthe recirculation zone (located at the side-branch ofthe Tee) there are high local void-fractions. Figure 5shows contours of the drag parameter FD (kgfsm3)for this flow, with f(a) = af(l- a)2. These contoursserve to illustrate the non-uniformity of the drag-force fleld and the flow pattern. It is known that theconcentration of air bubbles follows approximatelythe same contour-shape (Issa & Oliveira, 1990), with

138

_1.0It)

It)11 0.8

"-"

.5 0.6I-Ü« 0.4D::I..L..I

00.2

0.0Ó

,,0.6I{)'r""

I{) 0.51/'-"

0.4ZOI- 0.3Ü«a:::0.2u..I

00.1O> 0.0

0.5

(f)--l« 1O -1:::>O 10-1(f)

~ 10'"(f) 1O -..(f)«~ 10-

prediction of phase separation in two-phase flowthroughT-junctions," Computers and Fluids, VoI.23, No. 2.

Looney, M.K., Issa, R.I, Gosman, A.D., andPolitis, 5., 1985, "A two-phase flow numeriealalgorithm and its applieation to solid/liquidsuspension flows," Imperial College Rep. FS/85/31.

Oliveira, P.J., 1992, "Computer modelling ofmultidimensional multiphase flow and applieation toT-junctions," Ph.D. Thesis, Imperial College, Univ.of London.

Perie, M., 1985, "A finite volume method for theprediction of three-dimensional fluid flow in eomplexgeometries," Ph.D. Thesis, Imperial College, Univ. ofLondon.

Spalding, D.B., 1980, "Numerieal eomputations ofmultiphase flow and heat transfer," Reeent Advaneesin Numerical Methods in Fluids, VoI. 1, Ed. C.Taylor and K. Morgan, Pineridge Press.

Stewart, H.B., and Wendroff, B., 1984, "Two-phase flow: models and methods," J. Comp. Phy.,V01. 56, pp. 363-409.

Wallis, G.B., 1969, One Dimensional Two-Phase.Flow, MeGraw-Hill, New York.

Zuber, N., 1964, "On the dispersed two-phase flowin the laminar flow regime," Chemical Engng. Sei,VoI. 19, pp. 897-917.

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