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Los Alamos National Laboratoryis operated bythe Universityof Californiafor the United States Departmentof Energy undercontractW-7405-ENG.36i i1.,. i i iii i.i, i iiiii i

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TITLE: A CELL-CENTERED ICE METHOD FOR MULTIPHASEFLOW SIMULATIONS

__ _ _ __= _ _ _

mrHOR(S) B(ryan) A. Kashiwa, T-3 _ _,_>,__ _e___"=oN(ely) T. Padial, T-3 _,_=o i _ "=_.__._, __o__o_._R. M. Rauenzahn, Molten Metal Technology, Inc. >, _>"_""-_ _ _ ,W. B. V anderHeyden, Amoco Oil Company m _ ._ _ _

====_i _SUBMITTEDTO: ASME Symposium on Numerical Methods for Multiphase _ _ _ _ _ _ = _ _ _=

Flows, Lake Tahoe, Nevada, June 19-23, 1994 _ .. _ ,_"E .__ .

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The Los AMmos NationalLaboratoryrequeststhat the publisheridentifythis articleas workperformedunder the auspicesof the U.S. Departmentof Energy.

L(__ ___(__ Los Alamos National LaboratoryLos Alamos, New Mexico 87545

MASTERST. NO. 2629 5/81 CL

BI$"tfItBUlION OF THIS DOCUMENTIS IllqLI_IF_

I

A Cell-Centered ICE Method for Multiphase Flow Simulations

B. A. Kashiwa, and N. T. Padial, Los Alamos National Laboratory _ ': ,_,,j

R. M. Rauenzahn, Molten Metal Technology _::i: _ 3 _'_W. B. VanderHeyden, Amoco Oil Company .

ABSTRACT

The Implicit Continu0us:fluid Eulerian (ICE)_ethod is'a finite-volume •scheme that is stable for any value of the Courant number based on the soundspeed. In the incompressible limit, the ICE method becomes essentially identicalto the Marker and Cell (MAC) method, so the two schemes are closely related.In this article, the classical ICE method is extended to multiple interpenetratingphases, and employed with a single control volume (nonstaggered) mesh frame-work. The incompressible limit is preserved, so that problems involving equationsof state, or those exhibiting constant material densities, can be addressed withthe same computer code. The scheme reduces properly to a single--fluid method,enabling benchmarking using well-known test cases. Thus, the numerical issuesfocus only on those aspects unique to problems having multiple density, velocityand temperature fields.

The discussion begins with a derivation of the exact, ensemble-averagedequations. Examples of the most basic closures are given, and the well-posednessof the equations is demonstrated. The numerical method is described in operatornotation, and the discretization is sketched. The flow patterns in a bubble col-umn are computed as an incompressible flow example. For a compressible flowexample, the expansion and compression of a bubble formed by high-explosivegases under water is shown. In each case, comparison to experimental data ismade.

Introduction. The goal of this article is to introduce a nume_c_ method u_eful forsolving problems in multiphase flow. This method is an. ez*;ension _xld generalization ofthe so-called semi-implicit procedure known as the ICE scheme ('for).mplicit, C_ntinuous-fluid, Eulerian), for single-fluid flow. (See Harlow and Am:;2er, 1965. 1971; and alsoCasulli and Greenspan, 1984). The extension is to a nomtar;_._d rae,a_i 'amework, andthe generalization is to multiple fields of conservation laws_ _':,e _,_i_l ,:_rl:_-implicit is usedhere in the popular sense that the implicitness is limited i_<,_._ Lag_gian part of theequation system; the purpose of which is to remove any sta_:,Jity _:_,_<_!,:_io_1on the timestep,due to the propagation of pressure waves.

The present scheme belongs to the class of numerical! p_::_:_.,_du_r_esknown as Finite--Volume methods. In these, the integral form of the conse_.'_:!..',__" :'_,_is d]scretized on amesh of arbitrary 7olumes. The scheme given here is for the _._:,_ ,,:!.ependent equations, ifa steady state exists, it is found by stepping through time until the field variables cease tochange from step to step.

Multiphase flow problems span a wide area of science and engineering. Because ofthis, a great variety of equation sets may be encountered. In order to set the stage prop-erly, the discussion begins in Sec. I with a deri_ttion of the exact, ensemble-averaged

1

equations. The meaning of equilibration pressure is defined there, as is the meaning of theequilibrium and nonequilibrium model relative to that pressure. In See. II the character ofthe linearized equations is discussed, and the "physical" well-posedness of the equationsis demonstrated. Section III is a brief description of the simple closure models used inthe sample calculations. The numerical method is described in See. IV using operatornotation, and the discretization is sketched. The final section contains a set of examples

. selected to demonstrate the versatility of.the scheme. . .

I. A M ultimaterial Formalism. This section contains a systematic procedure fordeveloping averaged equations for the general multimaterial case. The formalism has thesingle--material problem as a natural subset. The reader may recognize the procedure usedhere as an analog to the statistical method for nonequilibrium flow, beginning with theBoltzmann equation. This procedure has some similarities to the process for a compositegas but exhibits important differences, which will be noted below.

The focal point for the derivation is the statistical distribution function f(t,x, r0),defined such that fdFo is the probability that the center-of-mass state at (t, x), in a volumeV, is in the interval dI'0 about the state given by the vector F0. Hence, the appropriatenormalization is, f fdI'0 = 1 , where integration is over all of the phase space of f, exceptfor time and space. This just means that the state in V falls within the set of possiblestates, given by the limits of the physical parameters making up the vector F0.

For example, let us suppose that the state of a pure material in V is completely !described by the mass m0, velocity u0, and internal energy e0. (This means that anyrelationships needed to give pressure and temperature as functions of the state, are alreadyknown.) Then F0 = (m0, u0,e0), where x = u0, and for which dI'0 = dmoduodeo. Withthis, we shall run through the averaging process for a single material; some of the resultsof which are useful for the more general multimaterial case, considered shortly.

The function f is obtained by the conceptual process of recording the system statevector F0, at time-space positions (t,x), for a large number of experiments, each havingidentical initial conditions, on average. The differences among members of the "ensemble"are in the multitude of microscopic arrangements that can produce the same average initialcondition. As the number of experiments approaches infinity, the frequency of observationsapproaches f. Here the observations are made in a collection of fixed volumes V, withposition x, and state P0, all measured in center-of-mass coordinates relative to a fixedmass m0 in V.

The observation volumes are selected so that the center-of-mass positions representonly ordinary points. By this we mean that the subdivision into observational volumes isperformed in such a way that properties within the volumes var3r continuously. To makesuch a procedure meaningful, we require that V should be specified of such a size thata large number of particles (molecules, atoms, electrons, neutrons, protons, or anythingthe user decides to call indivisible, but nevertheless identifyable as to material type). Inother words, the materials in V satisfy the usual conditions for a valid continuum. This,of course, implies that an averaging process has already occurred, and we are interested intaking a second average, by means of a statistical ensemble.

.... .... _ ....... , ............ . ............ ..... _.................. _., s ..................... ,_ ........................,.......... _ ._,,,_._,.,_._ _._._=,.=_._._ .............. _

I

The equation satisfied by f is obtained by taking the total variation of f in the phasespace (t, x, F0). That is

Of Of Df+ u0. Vf . ['0 • _ ° = D--T'

The right side of this is just the total change in f along the phase space. In the Boltzmannequation this is the collision integral. Here it is undefined, except that it must havethe same essential properties as the cMlision integral in kinetic theory, (Moredetailedmodeling of Df/Dt remains as a significant research project and represents what may bean important approach to closure modeling in turbulent single-phase and multiphase flowtheory.)

We are interested in moments of f defined by f Qofdro = (Q0) which gives theaverage of any function Q0, of F0, in V. The general result for the f equation is

O(Q0}0t ( OQo > / Qo._ff_CIFo .DfFV.(Q0u0)= I'0.-_0 + (1.1)

1 2Now let Q0 = m0, m0u0, rn0(e0 + -_Uo) = rnoEo, the mass, momentum, and total

energy of the material in V. The moment equations are, with po = tooV,

N +v. ouu+ (poUu;) = (moiJo)/vpE pEu + (poEou'o> (m0(_0 + Uo' rio))/V

where

p = (too} IV, pu = (rn0u0) IV, and pE = (moEo} IV.

Here we define the velocity fluctuation such that u0 = u + u_, which means

(m0u_) = (m0(u0 - u)) = 0, so that (m0uu_} = (m0u_/u = 0.

The terms on the left involving u_ are the well-known Reynolds stress, and fluctuationalenergy flux. They arise as a consequence of the nonlinearity of the momentum and energyadvection terms, respectively. This is exactly the same way that stress and heat flux ariseout of moments of the Boltzmann equation. Notice that terms involving DrDr do notappear because we have chosen mass, momentum and total energy, the so-called collisionalinvariants, for the averages. That is,

/mo-_tdFo=O

and so on for rnouo, and rnoEo, but not for things like e0 or for u0, or anything that isnot defined as a conserved quantity. The proof is easy. Let us use the so-called singlerelaxation time model

OfD-'-t= _'( f_ - f)

3

i

t

where w and fe are respectively a constant frequency and equilibrium distribution function.Now look at any of the invariant moments, say, the mass moment. That is

_mo(fe - f)dro = _ ((too), - (mo))

= ')), - ((pv+ 'w(((pV + m o m0)))

• =w((m'o),-(m'o))

=0

which follows because the fluctuations rrt_ average to zero, no matter what the functionis that does the averaging. Thus the reason m0, m0u0, and moEo are invariants here, isbecause they have, in effect, been defined to be invariant. The same would be true forinternal energy if we had chosen to define pe = (moeo)/V, for example. Thus the choiceof quantities to be averaged becomes just that, and the best choice may depend on theanalysis problem at hand. An important point is therefore, that once the choice is made,not all quantities will be invariants. (So the analyst had better keep it all straight, andtake care not to confuse invariant quantities with those that are not!)

This latitude in the choice of invariants in peculiar to the statistical method employedhere, and is not a latitude available when averaging over the dynamics of individual paxti-cles. In the particle case (the composite gas, say) the interactions are visualized such thattwo particles interact in a "collision" for which mass, momentum, and kinetic energy areconserved. In that case, these are the only invaxia: :s one may choose.

To complete this example, let us take the Euler equations for the variation along x.Hence

m°=0, m0u0=-VVp0, mo_o=-poVV'uo=TVi6o,

for which a perfect 7-gas is assumed. These equations are themselves the result of astatistical average over many molecules, and represent a closed set of continuum equations,valid in V. With these we may write

<mo ,o)/V= -<VVpo)/V

= - f fVpodFo

= -V _[ fpocff'o

-- -Vp

in which the second step follows because V is constant, the third step follows because f isconstant in V, where there gradient is taken, and the last step defines p = (p0). Similarlywe have, taking 7 to be constant,

(m0_0)/V = (_P0) = _P + ('y(P0 - P)')

4

and

<mouo.uo/IV= -(uo. Vpo)= -u. vp- (u;. Vpo).Now the equations are, using E = e + ½u2, gter some rearrangement,

:u][ [ o ]__o + v puu+ <po-;u_) = -vp .. ot p_ • p_u+,(p0E0u_> -_p+-(_(p0--p)')- (u_.Vp0) .....These now have a familiar form, and exhibit the basic problem of closure. Everything inthe brackets requires a model of some sort, which is the subject of ongoing research inalmost every application area of fluid dynamics today. For now we leave the closure issueunresolved, and turn to the question of what the equations look like if we are concernedabout accounting for the possibility that material of more than one type may be in V.

To accomplish this, we extend the state vector by one observation. That is the obser-vation of what material is in V when the mass, velocity, and internal energy are measured.Let ak be the volume fraction of material k observed in V. In each observation ak will

either be zero (material k is not in V) or unity (material k is in V). This is becauseobservational volumes are selected so that conditions at ordinary points can be measured;we have expressly chosen V to contain pure materials only. Material interfaces, if theyexist, coincide with the boundaries of V.

In the general multimaterial case, each observational volume may contain up to Ndifferent material types (but only one material), the definition of which is established priorto making the observations that lead to the statistical distribution function. Note that thestate may include any descriptive quantity one may wish, such as size, orientation, rate ofrotation, color, stress, etc. Using this latitude enables the definition of material "type" tobe very general. For example, one might declare the following:

A material is of type k when its phase is liquid, and the point lies in a discreteentity characterised by a size lying between 0.1 and 0.3 cm.

or

A material is of type k when the point lies in an area of pure gaseous products ofcombustion resulting from burning propellent material of type I.

So the definition of material type is limited only by the imagination of the theorist. Fornow let us examine the simplest multimaterial problem, and suppose that we have onlyobserved ak in addition to the mass, velocity, and internal energy in V. Now we have

f(t, x, m0, u0, eo, al , a2, . . . ,a N )dmoduodeodal dot2.., da N

which is the probability that the state at (t,x), is in the interval dFN centered about FN.Notice that drN = dFodalda2.., daN, which is important. It means that for material kmoments like

f al, Qofdry = (m, Qo)

the "collision integrals" in Eq. (1.1) become

/ Df / / Df da2 daNakOo--ff[drN = _ Qo-ff[drodal ... ,

i

which are zero if Q0 is a one of our invariants, just like they are in the single-material case.It is noteworthy that such is not the true for a composite gas. In the present case thereis a single distribution function f which contains all of the statistical knowledge includingthe likelihood that a given material is at a point. In a composite gas, one typically usesseparate distribution functions for each species of the gas, and the collision integrals leadto exchanges between the average species fields.

. The equations for f and.for the typical moment are still-the same as.before, but.nowwe want the moments akrno, akrn0u0, and akmoeo. (Where we choose the internal energyto illustrate the idea that the choice of invariants is open at this point, and because thisis a good choice for many problems of interest in engineering.) Now the general equationbecomes

N +V. puuk = (, kmoao)/v+(oouoa) ,

where we define p_, = (akmo>/V, pkuk = (a_mouo)/V, and pkek. = (a_moeo)/V. We alsodefine the material k velocity fluctuation u0 - utc + u_ so that

(a_m0u_) = (a_rn0(u0 - u_)/= 0, and (a_,m0uu_) = (aa,m0u_) u_ = 0,

in a fashion similar to the single-material case. Right away we identify the u_ terms asthe multiphase Reynolds stress, and fluctuational internal energy flux terms, respectively.In addition, we now have terms involving &t, which clearly represent the net rate at whichmaterial k is created. As can be expected, these are exchange terms for mass, momentumand internal energy, that result from the conversion of material k into, and from, the othermaterials. By definition, _'_ ak = 1, so the &k terms, summed over all materials, each goto zero.

Now let us take the same 7-gas as before, alid look at the/10 term. That is

<o_krnoflo>/V = --<akVp0) = --V<akp0> + <p0Vak)

= -Vpk +pVO + ((po-

= -v(p - Okp)- OkVp+ ((p0- p)Va >,

where we introduce the expected material k volume fraction (ak) = 0_, and the expectedmaterial k pressure (akp0) = pk. This is an important decomposition. The first lineseparates the pressure acceleration into a conservative part relative to field k, plus a con-servative part relative to the mizture. The second part, summed over all ma_er';als, is zero.It is the momentum exchange term. The second line decomposes the exchange term into

' mean and fluctuating parts, and the third line combines part of the mean exchange withthe conservative k momentum acc(,leration term.

This is the most popular expression for the pressure acceleration term, but the partinvolving (pk -Okp) is often forgotten, and it can be crucial. Before discussing this term

6

................................................. " ................ ....... "...... ................................... _ ........... _,_,_ *_ ....... _..... _ ...... _.................. ,_,,_ ,i _

we need to look at the energy equation, then define what we mean by 0k, and what wemean by the pressure p.

For the internal energy term we have

<_mo_o>/V= <_po>= -,_.p+ <_(po-p)'>,

- -where 7k = (c_kT>,and the right.side of the equatio.ns becomes . . ,

(po,_) ]-V(p_ - e_p)- 0kVp+ ((p0- p)Va,) + (p0u0_) .

-y_p+ <_'y(po-p)') + <poeo,_h)

In the usage above, it would appear that definition of mean pressure is arbitrary. Thatis because we are just adding and subtracting the same value. Suppose for now that we arefree to choose a definition of p, and we elect to use the so-called "equilibration" pressure;a choice which is motivated by physical reasoning alone. The equilibration pressure is justthe pressure that enables specified masses of separated materials to fill an entire volumewith no ongoing compression or expansion of the materials. The mathematical statementis

1 - _pkvk(p, Tk) = 0 (1.2a)

in which vk is the volume occupied by a unit mass of pure material k at the equilibrationpressure p, and mean material k temperature T_. This can be thought of as the asymptoticvalue of mixture pressure, as time becomes large, after packing arbitrary amounts of mul-tiple materials into a box, and allowing them to expand and compress isentropically. Forincompressible materials, which come to this equilibrium infinitely fast, it would appearthat this pressure must be very important indeed; which suggests that this may be a goodchoice for defining the mean pressure.

Now notice that the product pkvk has units of material k volume per unit total volume,which suggests the obvious definition for the equilibrium ezpec_ed volume fraction Oke. Thatis

O_" = pkvk(p, T_ ) , (1.2b)

which provides the most popular closure for this part of the equations, namely

0k = 0_e . (1.2c)

We shall call the equilibrium pressure model. In this, the term involving the pressure dif-ference (pk-Skp) vanishes for the Euler equations used here. The pressure of a componentof a perfect gas is given by

pk = p_(RT)_ = 8k(RT/v)_

in which Rk is the material k gas constant. At equilibrium, (RT/v)_ is the same for all k,so that

Ok[(RT/v)k -p] =0.

7

.°

This form suggests that we model pk = Okp_,,where p0(t,k, Tk ) is just the equation of statepressure for pure material k, the specific volume for which comes from a nonequilibriumvolume equation given shortly. In general, (pk -8kp) = 0k(p_ -p) is nonvanishing, es-pecially for flows containing solid particles that may reach a close--packed state as theirvolume fraction increases to a limiting value. In this situation p_ can exceed p by a largeamount, and is the mechanism by which the limiting volume fraction for solid fragmentsis maintained by the equation system. . .

The nonequilibrium case requires an equation for 0k whicla we can immediateiy writedown by allowing Q0 = ak in Eq. (1.1). The result is

0"7 + V. 0ku_ + V(a_u_) = (&_) (1.3a)

where it just happens to turn out that a_ is an invariant, so the integral over Df/Dtvanishes. This equation is, of course, constrained by the condition 1 - _"]_k0_ = 0, andis also ezactly the material k mass equation if v_ is a constant; that is, incompressiblematerial k. Therefore we conclude that not only is the equilibration pressure a goodchoice for the mean pressure, if all materials are incompressible, it is the only choice.

In the compressible case the solution to the Okequations (properly constrained) deter-mines the separate material pressures, on average, given a state relation for specific volumeof each material,

Ok = p_vt,(pk, Tk ) (1.3b)

which we shall call the nonequilibrium model. Notice that not all materials need to be inthe same state of equilibrium or nonequilibrium, but 1 - _k 0_ = 0 always holds, reqardlessof the respective material states.*

So what about the nonequilibrium incompressible case? Clearly if the equations formass and volume are redundant, we are lacking closure for pk. Fortunately it is not adifficult task to devise one, and the analog to a stiffened-gas equation of state is an easyone to use. For the case of solid particles approaching close--packing one can use

pT<= p + max[O, - 07<)]

which gives the increase of the material k pressure over that of the equilibration value,in a measure given by the excess of 0_ over some critical value 0_ associated with theclose-packed limit. In this the quantity ck establishes the speed at which the solid fieldtends toward 0_ when that value is exceeded. As ck ---,oo, the value 0_ becomes the largestpossible value for 0_ (and the equations get mighty stiff).

Before leaving this section, we take the Navier-Stokes equations, written for a fixedmass

m0 u0 = - VVp0 + VV. 1"0+ p0Vg

m0_o = _'0V/5o + V(ro ' e0)/2 - VV. q0

* Actually one could use the condition _Ok < 1, which means that the materials may not fill the wholespace, on average. This would allow the separate material pressures to have local nonzero values; in fact,one material may be in tension, while another may be in compression, both in the presence of a void.

in which 1"0, c0, q0, and g are respectively the deviatoric stress, rate of strain tensor,heat flux, and body force. We also define a quantity by analogy to the perfect gas "_0=po/(poC2o), where Cois the isentropic sound speed, and also the material k average is definedby (akVo/(poC2o)) = (pkok/c_).

With these, and the procedures just shown, we can summarize the equations, andannotate the meanirtg of each term. _ ..... "

Opt,--_ + V. p_uk Rate of change in expected k mass at a point;

= (po&k} net source of k mass, due to conversion from

and into other material types.

0_pt_uk + V. pkUkUk Rate of change in expected k momentum at a point;

= (p0u0&k) net source of k momentum due to k mass conversion;

-V. (akp0u_:u_} multiphase Reynolds stress;

-0_Vp acceleration by the equilibration pressure;

+Pkg acceleration by the body force;

-VO_(p°k -p) acceleration by the nonequilibrium pressure;

+V. (akv0} acceleration due to average material stress;

+ ([(P0 - p)I - r0]. Vak) momentum exchange by the action of equilibration

pressure fluctuations, and the deviatoric stress.

0O"_pkek+ V'pkekuk Rate of change in expected k energy at a point;

= (poeo&k) net source of k internal energy due to k mass conversion;

__r. (a_poeou_) multiphase fluctuational transport of internal energy;

+(pkvk/c_)p average multiphase work;

+ (akT0(p0 - p)') fluctuational work',

+ (ak'r0 ' Co)/2 average viscous dissipation;

__r. (akq0) thermal transport due to molecular conduction;

+ (q0' Va_) energy exchange due to molecular conduction.

(1.4)

9

....... , ....................................... .................... ,_, ........... _,_ ................ _.............. _ _. _.:_,_ _ . ....

These equations are exact, but unfortunately unclosed. Each term in the angle brack-ets needs a model. Simple forms for models used in our sample calculations will be givenlater. Before that, however, we must develop the equation for/5.

Recall that we define p to be the pressure satisfying 1 - _ pkvk, which we differentiate

. in time, denoted by the subscript t

Z:p,(_,), + E_,_(p,),= 0.

Now use the identities(_'k),= -(vh/c_)2pt

(p_)t = -V. p_u_ + (p0dk)

Vt'l_ = --(Vlt/Ck)2Vp ,

which allows us to eliminate gradients in v_ in favor of gradients in pressure. The result is

= p, + u_.Vp= Z:v_(poa_)- E V0_ukEC0_,_/_,) ' (1.5)

in which the tranport velocity is up = __,(ul,Okt,k/c_)/__,(O_vk/c_).This is the equation for the pressure that we need, and it has a beautiful form. It says

that as mass is converted from one type to another, in V, the equilibration pressure mustrespond if the specific volumes of the materials is different. It also says that if the volumeof one material displaces another due to flow in and out of V, the equilibration pressuremust respond, and does so in proportion to the sound speeds of the various materials.

It is interesting to note two limiting cases for Eq. (1.5), the first of which is thesingle--fluid case. Then 0 = 1 and we have

i_= (d /v)V. u,

which is an identity, and turns out to be a key expression in connection with the ICEmethod. The second interesting limit is for a single fluid having c _ e¢, corresponding tothe incompressible fluid. Then this just says

V.u=0,

which is all that it should say.

II. Character of the Equations. Equations of the form developed in Sec. I haveappeared in the literature, with minor differences [see for example, Wallis (1969), Harlowand Amsden (1975), Lahey and Drew (1984), and the references cited therein). Therehas been an ongoing debate regarding the well-posedness of these equations since it wasobserved that the equilibrium model is ill-posed (Gidaspow et al., 1973). An enormousliterature has grown from this debate; examples are Stuhmiller (1977), and Harlow andBesnard (1985). The issue is presently unresolved (Stewart and Wendroff, 1987).

10

Our goal here is to show that the equations are well-posed in the physical sense; thatis, the equations exhibit a special'character that enables solutions to be obtained thatdepend ,'ontinuously on the inital data.

_V, consider the "equilibrium model" equations, written for two isothermal compo-nents, in one space dimension,

apk + Opk c3uk

auk auk op = K(u_- ul,)/pk + F---g+ u -g2 + v

I- p,v,(v)=o,

for k = 1, 2, l = 2, 1 and where K is a momentum exchange coefficient, and F is a functioncontaining source terms like Reynolds stress, and other physical processes other than theacceleratxon due to the equilibration pressure. Now we eliminate the pressure gradient bytaking the gradient of the third equation. This gives

Op Opl Op2O"_= avl _ + av2

where the coefficient a = (01vl/c_ + 02v2/c])-1, and has dimensions of pressure. In systemform we have

0 u2 0 p2 0 p2 G,0 P2 + av_ avlv2 ul 0 -_x ulOt ul

U2 aV2Vl av 2 0 u2 u2

where now G is vector containing the K and F terms. This has the form Ut + CU,_ = G(U),and the matrix C has complex eigenvalues if ul -u2 _ 0, so the system is normally regardedas ill-posed as an initial value problem. The system is not hyperbolic, as physicists wouldlike, and is suspect of being incapable of exhibiting solutions that depend continuouslyon the initial conditions. However, notice that if K ---}oo then ul ---*u2, and these justbecome single-velocity equations, and are therefore well-posed. What is interesting is thatthe value of K has no bearing on the character ,_f the equations as being discussed here.This leads us to suspect that the true character is hidden, and we want to examine a wayto reveal that character.

Notice that the matrix C (known as the Jacobian of the flux vector, with respect tothe state vector) can be expressed as the sum of two operators, C = A + B,

0 u2 0 p2 0 u2 0 0 0 0 0 p2

av_ avl v2 u l 0 = 0 0 Ul 0 + av 2 avl vs 0 0 "

av2vx av_ 0 us 0 0 0 us avsvl av_ 0 0

11

Mi.iller and Ruggeri (1993) use this splitting to look at the linear characteristics of a single-velocity system. The extension here is obvious. The matrix B gives the change in U dueto physical processes (in this case just acceleration due to the equilibration pressure), andthe matrix A is the mapping of that change onto a point fixed in space (an Eulerian point).The eigenvalues of A are, by inspection, (ul,u2,ul,u2), which of course are real. So themapping part is clearly hyperbolic and presents no real problem to the physicist. Althoughthe repeated eigenvalues mean that this part is not pure hyperbolic, the ..ason is simple;the information is just coming along separate trajectories for each material.

The eigenvalues of B are (0, 0, :t=ccs)where

Wallis (1969, p. 143) showed that c_, the speed of a compression wave in _¢ratified flow(mulitphase flow, but without communication through friction). These are real, therfore Bis also an hyperbolic operator. So by themselves, the two parts of the Jacobian are okay,and we expect this to be an important reason solutions to the equilibrium model will bewell behaved. This is, of course, the experience of many engineers who have obtainednumerical solutions to these equations for over 20 years, and who have regarded thesesolutions as a valid representation of the differential equations.

This demonstrates the well-posedness of the equilibrium model in the "physical" sense;that is, each part of the operator C is hyperbolic. The parts correspond to the processesof Eulerian transport and Lagrangian change. What remains is to show the manner inwhich this very special property of the Jacobian may be capable of allowing solutions tobe obtained that depend continuously on the initial data. Some steps are being taken inthis direction (Burton Wendroff, private communication, 1993).

The stability of the equation system delc ads greatly on the nature of the right side(Miiller and Ruggeri, 1993, p. 143); that is, stability depends on momentum exchange, andon turbulence effects. This too is to be expected on physical grounds. It is fairly well knownthat the turbulence energy in multiphase flow expen'ences a positive growth as a result ofany nonzero relative motion (Kashiwa and Gore, 1990). Physically one would expect sucha growth to goveri_ a _elf-limiting process that will keep multiphase instabilities (such asRayleigh-Taylor, or Kelvin-Helmholtz modes) from obtaining infinite growth rates.

So the equations look fine so far, and before going into *he discretization we need toestablish the closure models, and that is the subject of the next section.

III. Sample Models. In this section we give examples of the most simple closure modelsfor momentum exchange, heat exchange, and multiphase Reynolds stress. The purpose isto close the equations in a minimal way, that is sufficient to allow performing some samplecalculations that will illustrate the general character of the solution method.

Consider the momentum exchange term first. That is, the averaged term

<[(Po - p)I- "o] " Wk>

which we identify with the process of exchange because the sum of this over all materialsvanishes. This is a force per unit of volume that is equal and opposite for the two materials,

12

in a two-fluid problem. The exact form of the force indicates that there are two maincomponents. The first is a force due to pressure fluctuations, and the second is a viscousstress. Pressure fluctuations (deviations from the mean) are created anytime there is arelative mean motion, which creates a growth in the mixture turbulence energy. Thisgrowth is accompanied by viscous dissipation of the fluctuational energy, and in steadystate the two are in balance. The fluctuation is an important part of the force that onematerial exerts on the other, and can be modeled as follows.

The force on a single sphere, with radius s, moving relative to a fluid with speed u is

lp°u2CDA

where, pO = 1Iv is the density of the pure fluid, and A is the area of the sphere. If thereare N particles of the same type, per unit of volume, then the volume fraction of particlesis

Od = (4/3)lrs3N ,

which means that the force per unit volume is, at least,

Here we use the subscripts d and c for discontinuous and continuous fields, respectively.Deciding which field is discontinuous (for a bubbly flow, say) is a modeling issue by itself,but for now we can just assume the d stands for the field having the smallest volumefraction, and the converse for c. For reasons that will become clear in the discussion of thenumerical scheme, we prefer to express this part of the force in the form

= OkOtK t(ut- ,!

where the sum is over all fluids. Notice that fkk = 0. For the case at hand, we see thatthe coefficient is

p0 3 I I

K12 = K21 = ---Co _ (3.1)0c8 s '

which is nonzero if there is any mean reltive motion at all.The heat exchange term (q0' Vak) is handled in a similar fashion. The preferred form

is

E Ok_lRkl(Tt -- Tk)!

and a typical form of the exchange coefficient isr

p0 J 'lR12 =R_I =_Hcms ' (3.2)

I

in which the parameter He is a heat transfer coefficient, per unit of area. Notice thatfor both of these forms, the condition for equilibrium is a constant mean velocity, and aconstant mean temperature, which is to be expected.

13

This is not to say that a force between materials could not exist if the mean relativemotion is zero. It just means that We have not yet considered such a force, and there aretwo (at least) that are known. One is the virtual mass force, which depends on the relativeacceleration and is nonzero as soon as any relative motion occurs, even if the mean relativemotion is zero. Another is the Basset force, which is due to the time-unsteady growth ofa viscous boundary layer. We mention these other forces not because they are central tothe discussion here, but to remind the reader that they exist, and that they are readilyincluded into the modeling process. Furthermore, these effects are easily incorporated intothe numerical solution procedure.

The last term we will model here is the multiphase Reynolds stress, and we do so byanalogy to Prandtl's mixing length model. That is

pkl lVu,lt , (3.3)

where Ik is a constant length associated with material k momentum diffusion resultingfrom material k velocity fluctuations, and where tk is the rate of strain tensor based onthe mean material k velocity. We shaU have more to say about this model when we exhibitits use in the sample problems later on.

IV. The Finite-Volume Discretization. We are finally in a position to discussthe computational scheme. The method discussed here is a so-called split Arbitrary-Lagrangian-Eulerian (ALE) scheme in the sense that the mesh is allowed to move in anarbitrary fashion (Hirt et al., 1974). This feature allows the user to use adaptive meshing;the process of moving the mesh in response to the solution, in order to improve resolutionnear rapid spatial changes in the state. The process of moving the mesh, or the advectionoperator used in the Eulerian phase of the calculation is not new, and has been described.(See for example, Addessio et al., 1992). What is new here is the particular discretizationof the Lagrangian phase.

The computational cycle is a procedure that is executed over and over as the time-dependent state is carried forward in time increments of At. Here the cycle is dividedinto three parts: (1) a Primary phase, in which auxiliary quantities are computed; (2) aLagrangian phase, in which the effects of physical processes are computed; and (3) anEulerian phase, in which the state is mapped to a common control volume. In this methoda single control volume is used for all elements of the state vector. Hence, all state quantitiesare referred to as being cell-centered.

The Primary phase is the part that is new, the Lagrangian phase is only somewhatnew, and the Eulerian phase is fairly standard. To give a clear motivation to the Primaryphase, we describe the computational cycle working backwards; that is, we look at theEulerian phase first.

"!o begin, we need to establish the frame of reference in which the computation isto take place. As is always the case, the choice of reference frame is arbitrary. Wechoose an inertial (non-accelerating) frame fixed relative to "laboratory" coordinates. Allcalculations will be performed on a finite-volume mesh of cells whose position is measuredrelative to the fixed coordinate system. In general this mesh will be allowed to move withan arbitrary velocity urn; a feature that is used for adapting the mesh to changes in thefield variables or for following the motion of a submerged body.

14i

* I, i_lll.............................................................................................................................

L, qk(x,t) be any field variable associated with the volume V(t.x) and materialspecies k. The appropriate equatioa of change is the generalized Liebnitz rule

---_dI'" + fl "qkukdS = -_ qkdV . (4.1)

In this entire article, the quantity qk is considered to be the average in V so that fv qkdV =qkV . Also, the velocity uk at the surface S(x, t) is the material k velocity. The readermay recognize this as the so-called Reynolds Transport Theorem, in which the right sideis the change of qk in V, due to physical processes. To emphasize that the right side is thechar_,ge due to physical processes we write

d Iv A(q_V)d"t qkdV = At '

which is just the fight side of our Eq. (1.4), times V.The interest here is the total change in the volume due to material transport as well

as due to physical processes. Let this change be denoted by the temporal operator dindr.

With this we may write a second form of Liebnitz rule,

-_- + fa . q_umdS = dmdt 7kdV • (4.2)

The total change on the computational volume is just the difference between Eq. (4.1) andEq. (4.2), which is

d,n A(qkV) (43)d----_qkV+ VV. qk(uk - urn) = At '

and is a rule that relates the total change in qk on the computational mesh moving atUrn, to the total change in q_ due to physical processes. We also use the shorthandnotation VV = f rids to indicate the surface integral; our Finite-Volume scheme will usea discretization of the integral form. The VV is just more compact to write down.

Let us write a discrete form of Eq. (4.3), using the superscript notation to indicatetime level, for a typical control volume V

q'_+lV"+l - q'_V" + AtA(q_) - qLvL -- q'_' V n ,

from which the time--n values of the material species state element qk, and the volume I'n,clearly drop out in favor of the "Lagrangian" values indicated by the superscript L. Herethe operator AtA is the advection operator, defined shortly.

The state vector can be any function of I." that we like, and we choose here to .,ethe species mass density, species velocity, and the species temperature. That is qk =(Pk, Uk, Tk )T, which means that we are "mapping" the total mass, along with the velocityand temperature; rather than the total mass, total momentum, and total energy. Thereason for this choice is one of taste. It happens that the algorithm is more robust thisway; one does not have to divide out the total mass to define the velocity, or convert

15

from totalenergy to get the temperature. This avoids numerical noise from creeping intothe data as the material species mass approaches zero. In multiphase problems involvingseparation of materials (simple sedimentation, for example) a vanishing species mass occursall the time. Also we axe most interested in problems with chemical reactions, or phasechange, constituting the conversion of mass from one material species to another. Thesereactions are typically temperature driven, so an accurate resolution of the temperature isessential. Therefore if makes sense to declare temperature as the independent variable.

The advection operator, acting on the state element q_ is

AtA(q_) = _],AtSi. (u_: - u,n)i((q_))_' , (4.4)

in which the sum is over all surfaces defining I,'n, determined by vertex positions on themesh, whose vertex velocity is urn, and whose surface vector of side i is 8i (area timesoutward normal). The quantity in double angle brackets is an upwind-centered value of q_,expanded in a spatial Taylor series, to a point just upstream of the face i. This Taylorseries is of second-order in space, and uses gradients that are limited in the fashion ofvan Leer (1984). The advection operator is not the focal point of this method, but it isimportant to the accuracy of the overall integration. In principal, any advection operatormay be used here. We just happen to use one that requires information from the Primaryphase, and that is the quantity (u_)i which is a face-centered velocity, and we will call itthe fluxing velocity. It is the rate at which volume of material species k passes the face i.

The fluxing velocity will be fully defined later; it is the principal product of the Primaryphase. It also gets used in computing the Lagrangian volume V L, which comes from adiscrete form of the kinematic law for volume change, as does the new mesh volume I/""+1.Hence,

"V"n+l = V n + At_"_,,(S. um)i (4.5a)

Y n = V" + At_,(S. ul)i, (4.5b)

which give us all we need to find q_+l, except for the fluxing velocity. To summarize sofar we have, using our chosen state vector

= [pv L- AtA(pk)llV

= [uV L-AtA(u)]/Y "+' (4.6)

T: +' = [TLV L - AtA(Tk )I/V n+' ,

so we need the Lagrangian values, described next. Notice that no time level has beenindicated on the state vector operated on by .A. This is because the time level can be theadvanced time, the old time, or anywhere in between including the Lagrangian time. Thedifference among the possible choices is only a term of OAt 2. Typically, we use time Lvalues, because they are readily available.

The Lagrangian values come from the right side of our conservation laws, multipliedby l/nAt. These are just &ink, Amkuk, and Arnkct,, and are in conservative form. To get

1G

the Lagrangian values of the state vector, we just expand these using Aqk = qL _ q?,. Theresult is

= [oy" + ,m llV L

u L = u_[1 -(Amk)/rn_] + (Arnkuk)/m L (4.7)

r L = T_[1 - (e_./c,,kr_)(Arnk)/m_l + (Amke_)irn kL,

in which we use the identity Aek = cvkATk, where cvk is the constant volume specific heatof material species k.

The A terms come directly from Eq. (1.4), and to write them down we need notationfor the spatial discretization, since so far we have mostly needed temporal discretization.There will be two spatial operators that we need. The first is a cell-centered integral overthe sides of the computational cell. We have been using the shorthand VV to denote suchan integration, so we will use vnv c to denote the discrete form. Notice that the operatormust act on a face-centered quantity, like it does in the advection operator defined earlier.The other spatial operator that we need will be the face-centered gradient, which willbe called V f. This is any spatially second-order finite difference expression, operatingon surrounding cell-centered data, and located at the cell face. Thus, for example, ouradvection operator can be expressed

AtA(qk) = AtVnV c" ((qk))U(U_ - urn).

Now, the A quantities in Eq. (1.4) are simply

Arnk = At(rnk&k)

(Am uk)/m = At(rn ukak)/m + gAt

-- (t'kAtVn_Cp*)/W L "4"vkOtAtI'nK(ut - uk)/V L + (AtV"V c' "r'k)/m L

(Arnkek)/rn_ = At(rntek&k)/m L + (vk/ck)2(p°kAp)/V t" + vkO AtV"R(TL - T )/V c .(4.8)

Here we ignore most of the turbulence terms, and write the two--fluid case for illustration,with k = 1,2, 1 = 2, 1, in order to point out some special features. The first thing is, weneed the face--centered pressure p* for evaluation of the pressure acceleration integral inthe momentum equation; and the second thing is, we need Ap for the energy equation.Both of these items are from the Primary phase, discussed shortly.

The other important feature to notice is the implicitness embodied in the combinationof Eq. (4.7) and Eq. (4.8), for the velocity and temperature. These form separate point-wise implicit linear systems for u L and T_ which can be inverted easily, and doing soyields a method which is free of any stability condition based on the values of momentumexchange coefficient K and heat exchange coefficient R. In addition, now we can explainour important reason for expressing the exchange functions as a product of the volumefractions times a nonzero coefficient. That is, if species k is vanishing and species l is not,

17iJ

the velocity and temperature of species l is unaffected by tile exchange terms. Howeversince K and R are nonzero, species k experiences velocity and temperature changes: thevalues tend toward the velocity and temperature of the nonvanishing species, at ratesdetermined by K and R, respectively.

Now we are all the way back to the Primary phase of the computational cycle, and itis time to compute the auxiliary quantities needed in the Lagrangian and Eulerian phases.These quantities are the face-centered fluxing velocity u_,, the face-centered equilibrationpressure p*, and the cell-centered change in the equilibration pressure/iAt = Ap. Herewe get to the heart of what is new, and that is the cell-centered implementation of theICE method, and its generalization to multiple velocity and temperature fields. To get theidea, we go through the single fluid case first, and the multifluid case second.

To begin we define the fluzing velocity to be the solution to the Lagrangian equation(mu)" = VVp +mg ezpressed at the cell-face, at time t" + At2, assuming the time-nstates as the initial conditions. So the integration involves the logical left and right cell-centered data, denoted by the subscripts I and r, with the positive direction defined aspointing from left to right. The solution is

u" = ( p'u' + p'u')p,+ Or ptAt+PrVlP+ g&t/2 '

where we assume a constant mesh spacing. The guidance for this definition comes fromlong ago when Lax and Wendroff (1960) showed that a forward-time integration schemewill be stable if the fluxes are centered in space and time. Since u" is the flux of volume, wereckon that it had better satis{y that condition. Our definition, and the resulting solutionabove gives a leading term that looks like a mass average of left and right states, plus atime-advancing term proportional to a local value of the pressure gradient; so the fluxingvelocity looks space--time centered. To make the notation more compact, we write

u* = ((u")) p - vlAtVlp+ gAt/2, (4.9)

where the double angle bracket, superscripted by p indicates the mass-weighted averageof left and fight states, and v ! = 1/(pt + pr).

In the ICE method, the fluxing velocity is used in the equation for the pressure, whichfor a single fluid is just p = -(c2/v)V.u, and the trick is to use pL,an estimate of theadvanced-time pressure, in u*. With pL = p,+ Ap, and _ = ((u"))°-vlAtVlp n +gAt/2,this is

[v"(v/c 2) - _x?V"V c . vsVl]Ap = -_tv"v c. fi, (4.10)

which is a linear system of equations for the pressure change that involves every com-putational cell of the domain, and is the key to the stability of the ICE method. Ourcolleague J. U. Brackbill (Private communication, 1988) has put it the best way: that is,

"the pressure given by pL is a good enough estimate of the ad_mced-time pressure, sothat when used to perfo: the acceleration, linear stability is guaranteed". Furthermore,that estimate becomes better and better for large sound speeds, and that is because forc2 --, _z_this becomes exactly the equation for the pressure in the incompressible case.

18

Having solved for Ap, and "back substituted" the result to get u*, we push on tofind the last auxiliary quantity p*, the face-centered pressure. For this we use a one-dimensional di,qcretization of the momentum equation, aton 9 a line normal to the cell faceof interest. We suppose that the cell-centered pressure pL is known at the "boundaries";these are the positions to the logical left and right of the face for which the face pressureis needed. The divergence of the momentum equation produces a potential equation forthe pressure, and the only unknown is the item of interest, namely p'.

We obtain the potential equation as follows. Using subscript notation for the partialderivative, we have [ut + uuz + vpz]x = 0. So far this is just

(vpz)z = --Utz --(UU_) z ,

so we need to eliminate utx, which can be done by appealing to the pressure equation,differentiated by time. That is,

[p,+up_= -(?/_)u_], ,

which may be linearized to produce uzt = -(uv/c2)([9)_. Now our potential equation is

(vp_).= (uv/c2)(b)_- (uu_)_, (4.11)

a discrete form of which is

1 1

1

a_ [u_(u_- ,,') - u_(_"- _,,)1.

The angle brackets represent an average, the form of which is suggested by the solutionitself. That is,

( ) Az/u._t,) (Ap_-Ap_) [ur(ur-u')-u,(u"-u,)]_p_ + _'_vf + -S_\ _ _ + __ _._ _

V r "4" t'! Or "[" Oi

where the denominator suggests a specific volume weighted average. Hence our face pres-sure has three parts; a volume--weighted average of pL, plus a term proportional to thegradient in Ap, plus a term that looks clot like a bulk viscosity. The second term vanishesin the incompressible limit. If c2At >> [ulAx, which occurs for low-speed, implicit calcu-lations, this term doesn't do much. However it can be important in high-speed problems.such as an infinite strength shock wave, wherein the local roach number may be changingrapidly in space. (Recall that Ap goes like c2&t, so the formulation remains consistent,even for vanishing c2 or At.)

Now we recognize that the second two terms in our equation for p" provide an impor-tant "filtering" mechanism. That is, the velocity field may exhibit a (non-physical) spatialdistribution that is the consequence of noise in the data. This noise may come from anon-monotone advection operator, for example. Since this velocity noise can be invisible

19

to the pressure equation, these terms serve as a way of removing the noise; hence they in-troduce a diffusive effect in our method, and we want to minimize that effect. To do so we

multiply the second two terms by a "limiter" ¢ that is designed such that 0 _<¢ _<1, withvalues tending toward zero if the velocity field is smooth. One such limiter is described byKashiwa and Lee (1990).

To summarize this, let the notation (< / )v represent the v-weighted average of left andright states. Thus,

_xx u_(u_- u')- u!!u"- u_)]. (4,12)"_ (ApL - ApL) - ¢[ ....... vr+vt

p* = ((pL>)v + ¢((u/c2))

This concludes the single--fluid case, Primary phase. The multiphase case followsdirectly and, of course, contains the foregoing results for u° and p° as a subset. There areonly a couple of things that stand out as particularly important, and the first is that weneed to define u_ as the time tn + At/2 solution to

(rnkuk)' = -OkV_p + O_OtVK(ut - uk) +mag,

that is, we need the exchange term to be evaluated at the cell face. This is unfortunate,because it means that the momentum exchange calculation must be done twice in thecomputational cycle; once at the cell face, and again later on at the cell center. The reasonfor the need is clear. The pressure pL has to know that momentum exchange is going on,and this seems to be the only way to let it know.

The discrete form may be written

u_,= <<u_>)p- - <<vk>>P'_tVlpL+ <<v_OiK))P'At(u7- ul) + gAt/2,

which is clearly an implicit expression involving u" for both fields, and we have to solveit. To do so, let

fi_ = ((u_>) p' -((vj+)>P'AtVlp" +gAt/2,

and let u_ = _ + Au_, and define 13_= ((v_OtK)) p*At. Now we have the system

-32 1+_2 Au_ = as+_2(_ -_52)-<(v2>>P'Z_tV!AP'

If we let A be the linear operator on the left of this, the solution can be written,

ul = ah - _ _tv Izxr,

u2 + &(fit - u2) _ (4.13)

<<v2>>°' h'This is to be placed into the discrete form of our pressure equation

v- =/,tEvm - tv"v . •

2O

I

i

The final result is another sparse system,

°, = - tv"v °. i4 4/Notice that the advectiol, operator is a used in this discretiztion. This is important beca_,_ein the incompressible limit, the sum of the mass equations, and the pressure equation, areredundant. Therefore, they had better say the same thing, which is guaranteed here.Unfortunately the system is now nonlinear, because the upwind centered volume fractiondepends on the value of Ap by way of the fluxing velocity u_. In the incompressiblecase it is crucial to recognize this nonlinearity; satisfying this equation is the only thingthat guarantees the condition 1 - _ 0k = 0. in practice the nonlinear nature of thesystem is only a minor irritation, because the process of successive substitution, with aninner implicit solution (with ({0k//_' held fixed) typically converses in at most three outeriterations, and more typically one or two.

The last thing we need to do is find the multiphase equation for the face pressure p*.For this there are a couple of tricks. The first is that we solve a potential equation forOkuk, summed for all k,

from which the effects of mass and momentum exchange have been omitted. We find thatthey are not necessary, since these effects are built into the equation for p/'. The equationfor pressure has been used, in a linear form like before in the single--fluid case. Now wediscretize once again along a line normal to the cell face, with the left and right statesknown, and with u_, being the projection of u_, on the cell face, and similarly for the leftand right velocities. The final result is, with the averaging parameter t, = __,81,vk,

p"= ((vL}}+ "-A'i +(4.16)

So far we have not mentioned much about the mass exchange terms (the ones with&k in brackets) appearing in Eq. (1.4). The general formulation for chemical reactions andphase changes, plus the numerical solution of the equation system, will require anotherfull-sized article. Therefore we must postpone the full description of those items. We willnote here, however, that the mass exchange terms by themselves form a system of ordinarydifferential equations, and can be solved separately from the rest of the system given here.The result for Amk, the change in material k mass due to conversion of type in a time At,becomes a source of volume in the pressure equation. In this way, the whole field feels theeffect of mass conversion in any particular control volume. If the specific volume of theconverted materials is greatly different, such as with a solid and a gas, this effect can beenormous, and Eq. (4.14) is designed to handle it.

At this point in the computational cycle, we suppose "hat the mass-exchange termshave been properly evaluated, and Amk is known. The x_,xt item in the Primary phase.working backward still, is evaluation of the equilibration pressure. Again, that is thevalue of p that satisfies our volume condition 1 - _ pkvl,(p, T_) = 0, and from which the

21

equilibration volume fractions O__ = pkv_ are determined. Typically we have expressionsof the form p(v, T) rather than the inverse, so the evaluation of p from the volume conditionmay not be a simple matter. Furthermore if the compressibility of the various materials iswidely different, the solution of the volume condition can be difficult to find. Fortunately aniterative procedure such as Newton-Raphson works quite well, when implemented carefully.For the case of incompressible materials the solution is trivial; the ok are all constants,and Ok = pt_/v_.

This concludes our backward presentation of the computational cycle, and we sum-marize by stepping forward through the process, beginning with the Primary phase.

The Primary Phase:

• Evaluate the equations of state. This means that for any materials that are not inpressure equilibrium, the 0k's are known from the initial conditions, and the sub-sequent solution of Eq. (1.3). It also involves finding the equilibration pressure p,all p_ measured relative to p, and the setting of 0_ = 0_e for all materials that arein pressure equilibrium.

• Integrate the system of ordinary differential equations for the mass-exchange terms.The result is Amk, plus the mass-exchange parts of Am_uh and Amkek.

• Evaluate Eq. (4.14) for the change in equilibration pressure _p, and back-substituteinto Eq. (4.13) for u_, using pt,. This is the nonlinear problem.

• Evaluate Eq. (4.16) for p*, and Eq. (4.5b) for V L.

The Lagrangian Phase:• Evaluate the Lagrangian values, using Eq. (4.7).

The Eulerian Phase:

• Determine the mesh velocity, urn and evaluate Eq. (4.5a) for V "+1.

• Finally, update the state using Eq. (4.6).

This concludes the discussion of the numerical scheme. The analysis of accuracy andstability will be reserved for a future article. For now the foregoing can be regarded as arecipe that makes some sense, and provides a way to obtain solutions to hard problems,as will be demonstrated next.

V. Sample Calculations. We illustrate the performance of the method by two illus-trative calculations, both are time-unsteady multifield cases in two dimensions. The first abubble column, with a constant specific volumes for both gas and liquid, and with bubblesrepresented as having a constant size. The second is a compressible case involving theexpansion of a single gas bubble under water, caused by the expansion and compression ofgaseous combustion products from a detonator.

A first test of the code was performed by simulating vortex streets observed exper-imentally in two-dimensional bubble columns by Chen et al. (1989), who examined theeffect of column depth on circulation patterns. Their test Column B, was 180 cm high,17.5 cm wide with a plate spacing of 1.5 cm. The air distributor was wire gauze placed atthe bottom of the column. In all of the experiments, a superficial gas velocity of 3.5 cm

22

II ....

was used. In each experiment, the column was initially filled with water to varying depths.Air flow was initiated and the flow patterns were recorded by photography.

Chen et al. observed that for shallow columns in which the depth of the gas-spargedliquid was less than or equal to the the width of the column, two symmetrical, side-by-sidevortices formed in the column from gas streaming up the center of the column. For greaterliquid depths, however, the vortices appeared in a staggered fashion like in a classical yonKhrmhn vortex street due to the channeling gas.

We simulated the cases corresponding to depth-to-width ratios of about 0.5 and 2.5for Column B test cases described by Chen et al. For this simulation, we assumed a fixedscale s = O.lcm in our momentum exchange formula, and a fixed turbulence mixing lengthfor each fluid of It = 2cm.

Free-slip boundary conditions for the side walls, uniform inlet boundary conditionsfor the air on the bottom, free slip for the water on the bottom and outflow boundaryconditions for both phases at the top of the column were used. The initial condition ofthe column simulation was pure liquid to a depth of 2.5cm and 15cm in the two cases,respectively, with pure air above. Both the water and the air had zero initial velocity, sothe startup is impulsive.

The results of the calculations are shown in Figs. 1 and 2. The velocity vectors showthe velocity of the water phase. The contour lines show the contours of constant liquidvolume fraction. As seen experimentally, the simulation predicts twin symmetric vorticiesin the case of the shallow column and a staggered vortex street in the case of the deepercolumn.

Itisinterestingtonotethebehaviorwithouttheturbulenceterm.That isthenon-

viscousproblem,forwhich we noticea highlychaoticmotioninforthedeepcolumn case,ratherthanany sortoforganizedmotionthatisshown inFig.2.Thissuggeststhattur-bulencetranportisa key partofthephysicsleadingto the motionobserved.What is

surprisingisthatsucha simplemodel can be adequatetoenablecapturingtheessenceofthemotion.

The secondcalculationisforan underwaterdetonationof a high-explosive.The

experiment was performed by Boyce (1990), in which four grams of PETN explosive wasexploded at the center of a water bath measuring 4x4x4 m. Boyce recorded (vs time) thedeflection of a drum-like structure beneath the point of explosion, and pressure at fixedpoints. The bubble was photographed with a high-speed camera.

In this experiment, the initial pressure wave emitted by the explosion reflects off thewalls of the bath, and is dissipated rapidly, leaving a more--or-less hydrostatic pressurefield, outside of a gas bubble. The high pressure combusion gases cause this gas bubble toexpand in the water. The inertia of the water moving away from the bubble cause it toover-expand, and to reach an inner pressure less than the exterior hydrostatic field. Thisreverses the expansion process and results in an implosion, followed by a second expansion,and so on.

We illustrate the utility of the ICE method with this problem by computing theexpansion and implosion through two full cycles. To do so requires use of a time step suchthat cAt/Az >> 1, in order to keep the computational time to a reasonable limit. Thisis because the typical Mach number is small, like 0.01 or less; the bubble moves with a

23

fluid speed that is much smaller than the so_md speed in the water. Despite the low Machnumber, compressibility is an important factor, since the implosion of the gas determinesthe nature of secondary pressure waves that impact surrounding structures. The impactof these waves on the structures is the item of interest.

To model this single bubble, using the two-fluid equations, we simply set K and R tovery large, fixed numbers. This causes momentum and energy exchange to occur infinitelyfast, resulting in velocity and temperature equilibrium. By initializing the problem so thatthe combustion gases and water are separated, this coupling k_ps the materials separatedfor all time. The only mixing that occurs is due to numerical errors in the advectionoperator.

For the simulation, we use axisymmetric geometry with constant radial and vericalmesh spacing of Ar = &z = lcm. A rigid obstacle on with free-slip boundaries is placedbelow the burst point, and all other sides of the problem use a zero gradient boundarycondition. The burst point is modeled as if the high-explosive has all burned, in a volumecorresponding to two mesh ceils, at time zero. This produces an initial pressure of about54Kbar in those two mesh cells, which contain material k = 1 (combusior products)whose pressure is given by the so-called BKW equation of state (Mader, 198o, p. 412).The water (material k = 2) is initially at standard conditions, and is modeled with theso-called USUP equation of state (Mader, 1988, p. 391).

Figure 3 shows the results at three successive times in the simulation. Notice that thecontours of volume fraction are very close, indicating that the "interface" between gas andliquid is kept sharp, and the definition remains good over time. Figure 4 is a time plot ofthe maximum bubble radius vs time, along with the Boyce measurements, indicating thatthe dynamics over a very long period have been captured well.

Conclusions. A multimaterial formalism is used to derive the general, exact equationsfor the conservations laws involving multiple materials. Multiphase flow is a subset of thegeneral case considered here. The character of the equations is shown to be well-posed forinitial values problems, in the "physical" sense, independent of the closure. Examples ofthe most simple closures are given, and results for two sample problems have been shown.

Closure modeling for these equations will be ongoing, but the formalism used herecan be expected to serve as a guide to modeling assumptions, as demonstrated here forthe momentum exchange term. The numerical method is found to be robust and accurate,and of sufficient generality to handle a wide spectrum of problems in engineering. Someextensions are underway, such as development of an implicit algorithm for the momentumdiffusion, and for the advection operator.

Chemical reactions and phase change have been omitted from this work, except toshow the general form the terms and where it is in the computation that these termsare evaluated. A detailed discussion of the mass-exchange physics, and aspects of thenumerical solutions, will be the subjects of future publications.

24

ACKNOWLEDGMENTS

Support for this work has come in part from a mtmber of sources, notably the U. S.Departments of Energy, and Defense, The Amoco Oil Company, and Molten Metal Tech-nology, Inc. There are numerous colleagues to thank for helpful discussions over a periodof many years. Notable among them axe (alphebetically) Jerry BrackbiU, John Dukowicz;Frank Haxlow, and Burton Wendroff. We also thank our many users of the Los AlamosHydrocode Library CFDLIB for the interesting variety of problems that have provided thetestbed for the methods shown here.

REFERENCES

Addessio, F. L., J. R. Baumgardner, J. K. Dukowicz, N. L. Johnson, B. A. Kashiwa, R. M.Rauenzahn, and C. Zemach, " CAVEAT: A Computer Code for Problems with LargeDistortion and Internal Slip", Los Alamos National Laboratory Report LA-10613-MS,Rev. 1, 1992.

Boyce, P., " Experimental Study of the Effects of an Underwater Explosion on an E1-emetaxy Structure", Direction des Constructions Navales; Toulon, France, Report no.06/90 GERPy, January 1990.

CasuUi, V., and D. Greenspan, "Pressure Method for the Numerical Solution of Transient,Compressible Fluid Flows", In_. J. Num. Meth. Fluids, 4, 1984, 1001-1012.

Chen, J.J.J., M. Jamialahmadi and S. M. Li, "Effect of Liquid Depth on Circulation inBubble Columns: A Visual Study", Chem. Eng. Res. Des., 67, 1989, pp. 203-207.

Lahey Jr., R. T., and D. A. Drew, "The Current State--of-the-Art in Modelling of Va-por/Liquid Two-Phase Flows", ASME paper 90-WA/HT-13, 1990.

Gidaspow, D., R. W. Lyczkowski, C. W. Solbrig, E. D. Hughes, and G. A. Mortnesen,"Characteristics of Unsteady One-Dimensional Two-Phase Flow", Amer. Nuc. Soci-ety Trans., 17, 1973, pp. 249-250.

Haxlow, F. H., and A. A. Amsden, "Numerical Calculation of Almost IncompressibleFlow", J. Comput. Phys., 3, 1968, pp. 80-93.

Harlow, F. H., and A. A. Amsden, " A Numerical Fluid Dynamics Calculation .Xlethodfor All Flow Speeds", J. Comput. Phys., 8, 1971, pp 197-213.

Harlow, F. H., and A. A. Amsden, "Numerical Calculation of Multiphase Fluid Flow", J.Comput. Phys., 17, 1975, pp. 19-52.

Harlow, F. H., and D. Besnaxd, "\Veil-Posed Two-Phase Flow Equations with TurbulenceTransport", Left. Math. Phy.,.. 10, 1985, pp. 161-_.66.

Hirt, C. W., A. A. Amsden, and J. L. Cook, "An Arbitrary Lagrangian-Eulerian Com-

puting Method for All Flow Speeds", J. Compu_. Phys., 14, 1974, pp. 227-253.

Kashiwa, B. A., and W. H. Lee, "Mesh Lagrangian Hydrodynamics", in H. E. Trease..kI.J. Fritz, and W. P. Crowley, eds., Advances in the Free-Lagrange Method, Springer-

Verlag, Berlin, 1991.

Kashiwa, B. A., and R. A. Gore. "A Four Equation Model for Multiphase Turbulent

Flow", First Joint ASME/JSME Fluids Engineering Conference, Portland. OR, June1991.

25

Lax, P., and B. Wendroff, "Systems of Conservation Laws", Comm. Pure Appl. Math.,13, 1960, pp. 217-237.

Mader, C. L., Numerical Modeling of Detonations, University of California Press, Berkeley,1979.

Miiller, I., and T. Ruggeri, Eztended Thermodynamics, Springer-Verlag, New York, 1993.

Stewart, H. B., and B. Wendroff, "Two--Phase Flow: Models and Methods", J. Comput.Phys., 56, 1984, pp. 363-409.

StuhmiUer, J. H., "The Influence of Interracial Pressure Forces on the Character of Two-Phase Flow Model Equations", Int. J. Multiphase Flow, 3, 1977, pp. 551-660.

WaUis, G. B., One-dimensional Two-phase Flow, McGraw-Hill, New York, 1969.

Figure Captions

Figure 1. Two--Dimensional Bubble Column, low depth case. Velocity and volumefraction are shown for the liquid and gas, on the left and fight respectively. A liquid surfaceis apparent at a height of about -15cm. The gas rises rapidly through the shallow poolof liquid, causing the liquid to for two counter-rotating vortices, in a so-called "coolingtower" pattern. The gas exits a small opening in the upper right corner of the vessel.

Figure 2. Two-Dimensional Bubble Column, high depth case. The liquid surface isat about 3.5cm. A central gas-rich plume is apparent in the center of the column, withliquid rich down flow at the sides. The flow is highly unsteady, but exhibits a regularityresembling a yon Khrmhn vortex street.

Figure 3. Axisymmetric Blast Wave Under Water. Velocity vectors for every othermesh point, and volume fraction contours are shown, for three successive times. An ob-stacle sits beneath the point of detonation. Notice that the transition width from gas toliquid stays approximately constant as the bubble expands. Dimensions are in crn.

Figure 4. Bubble Radius vs Time. Maximum bubble radius w time is shown comparedto the data of Boyce (1990). The timing and amplitude show very good agreement.

26

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