The Riemann problem and a high-resolution Godunovmethod for a model of compressible two-phase flow

D.W. Schwendeman 1, C.W. Wahle 2, and A.K. Kapila 1

Department of Mathematical Sciences, Rensselaer Polytechnic Institute,Troy, New York, 12180

Abstract

This paper considers the Riemann problem and an associated Godunov method fora model of compressible two-phase flow. The model is a reduced form of the well-known Baer-Nunziato model that describes the behavior of granular explosives. Inthe analysis presented here, we omit source terms representing the exchange of mass,momentum and energy between the phases due to compaction, drag, heat transferand chemical reaction, but retain the non-conservative nozzling terms that appearnaturally in the model. For the Riemann problem the effect of the nozzling terms isconfined to the contact discontinuity of the solid phase. Treating the solid contactas a layer of vanishingly small thickness within which the solution is smooth yieldsjump conditions that connect the states across the contact, as well as a prescriptionthat allows the contribution of the nozzling terms to be computed unambiguously.An iterative method of solution is described for the Riemann problem, that deter-mines the wave structure and the intermediate states of the flow, for given left andright states. A Godunov method based on the solution of the Riemann problem isconstructed. It includes non-conservative flux contributions derived from an integralof the nozzling terms over a grid cell. The Godunov method is extended to second-order accuracy using a method of slope limiting, and an adaptive Riemann solveris described and used for computational efficiency. Numerical results are presented,demonstrating the accuracy of the numerical method and in particular, the accu-rate numerical description of the flow in the vicinity of a solid contact where phasescouple and nozzling terms are important. The numerical method is compared withother methods available in the literature and found to give more accurate resultsfor the problems considered.

Key words: Multiphase flow, hyperbolic PDEs, Riemann problems, Godunovmethods, shock waves.

1 Research support was given by NSF under grant DMS-0312040.2 Post-doctoral research support was given by NSF under VIGRE grant DMS-9983646.

Preprint submitted to Elsevier Science 27 June 2005

1 Introduction

This paper considers the Riemann problem and an associated high-resolution Go-dunov method for a system of nonlinear, hyperbolic partial differential equationsmodeling compressible, two-phase flow. While models of this kind arise in a numberof applications, the context of deflagration-to-detonation transition in high-energycondensed-phase explosives provides the motivation for the present effort. A two-phase continuum description of granular explosives has been provided by Baer andNunziato [1]; also see the contemporaneous study of Butler and Krier [2], the earlierwork of Gokhale and Krier [3], and the later papers of Powers, Stewart and Krier[4,5]. The model treats the explosive as a mixture of two phases, the unreactedgranular solid and the gaseous product of combustion. Each phase is assigned aset of state variables such as density, velocity, pressure, etc., which are assumed tosatisfy balance laws of mass, momentum and energy. A compaction equation and asaturation constraint for the volume fractions of the phases complete the system ofequations. The balance laws for each phase are similar to those for an isolated gas,i.e., the Euler equations, except for two important differences. First, the exchangeof mass, momentum and energy between the phases appears as source terms inthe balance equations. Second, the governing equations, although hyperbolic, areincapable of being cast in a conservative form. Non-conservative terms (also callednozzling terms by analogy with similar terms arising in equations that govern flowwithin a variable-area duct) appear in the equations, and their treatment requiresspecial consideration.

The aim of this paper is two-fold. First, we consider the Riemann problem forthe homogeneous portion of the governing equations (i.e., with the source termsomitted), and describe an iterative procedure that produces exact solutions forgeneral left and right states of the initial flow. In the Riemann problem the effect ofthe nozzling terms is confined to the contact discontinuity of the solid phase, acrosswhich the volume fraction of each phase changes discontinuously. It is assumed thatthe discontinuity can be replaced by a layer of finite but vanishingly small thicknesswithin which the volume fractions vary smoothly and the phases interact. Thisregularization was first proposed in the context of permeation through a poroussolid by Asay, Son and Bdzil [6]. An analysis of the layer yields jump conditions forthe states of the flow across the solid contact, and allows the contribution of thenozzling terms to be computed in a straightforward and unambiguous fashion. Awayfrom the solid contact the volume fractions are constant so that balance equationsfor the phases decouple and reduce to Euler equations for the individual phases.In these regions the usual jump conditions across shocks, rarefactions and the gascontact discontinuity apply, and may be used together with the conditions acrossthe solid contact where the phases are coupled to construct an exact solution of theRiemann problem for the mixture.

Next, the solution of the Riemann problem is employed in the development of a high-resolution Godunov-type method [7]. In addition to providing a means to computea numerical flux at the boundary between neighboring grid cells, the solution ofthe Riemann problem provides a natural approach to the numerical treatment of

2

the non-conservative terms. The governing equations are integrated over a grid cell.The numerical flux at the boundary emerges from this integral in the standard wayfollowing the usual Godunov flux construction. In the case of the non-conservativeterms, the integral reduces to a contribution about the solid contact in the solution ofthe Riemann problem from each cell boundary. With the thin layer structure of thecontact discontinuity at hand, this contribution can be computed unambiguously.Thus the resulting numerical method incorporates both the wave structure at cellboundaries and the appropriate behavior of the solution near the solid contact.

A high-resolution method is obtained using a second-order, slope-limited extensionof the Godunov method. The approach follows the usual description (see [8] and[9], for example) except for the treatment of the non-conservative terms which isnew. Essentially, the extension for the non-conservative terms involves two parts,one coming from a contribution to the integral of the nozzling terms about thesolid contact and the other coming from the integral away from the jump at thesolid contact, which arises from the slope correction of the left and right states ofthe Riemann problems. An improvement in efficiency in the numerical method ismade by employing various levels of approximation in the solution of the Riemannproblem. Here, we describe an adaptive Riemann solver designed to select a suitableapproximation, or perhaps the full solution, in order to achieve a sufficient accuracyat a lower computational cost.

Since its introduction, the theory of Baer and Nunziato [1] has received considerableattention in the literature. A mathematical analysis of the structure of the governingequations, including a description of the wave fields of the hyperbolic system, itsRiemann invariants and simple wave solutions, was described by Embid and Baer[10]. In this paper, a degeneracy of the hyperbolic system, which occurs when therelative flow between phases becomes sonic, is identified. The authors observe thatthis degeneracy is analogous to choked flow in a duct, and provides a constraint onthe admissible states for the Riemann problem. Modeling issues, certain physically-motivated reductions and numerical solutions were presented in a series of papersby Bdzil, Kapila, Menikoff, Son and Stewart [11–13]. These papers built upon ananalysis of a simpler approach described earlier by Bdzil and Son [14] and Asay,Son and Bdzil [6]. In a recent paper [15], Andrianov and Warnecke revisited theRiemann problem for the model. Using jump conditions across the solid contact,derived in [10], they explicitly constructed inverse exact solutions correspondingto prescribed states on either side of the solid contact. Our approach is similar totheirs but differs in one important respect: we describe an iterative procedure toobtain exact solutions of the Riemann problem directly for arbitrary left and rightstates. Such a procedure is necessary as a building block for our construction of ahigh-resolution Godunov scheme as indicated above and described in detail later.

The regularization of the solid contact into a thin layer across which the solutionis smooth no longer applies when the sonic condition is met, i.e., when the gasvelocity relative to the solid phase equals the sound speed in the gas. The governingequations exhibit a degeneracy in this case that requires a modified treatment, andthis situation is currently under study. However, such a circumstance is unlikelyto arise in the context of granular explosives, the application that motivates this

3

study, and is therefore not of concern here. In this application the drag betweenthe phases is invariably large, and serves to keep the relative velocity between thephases at a moderate level so that the system remains subsonic. Consequently, thepresent study emphasizes the subsonic case.

Finite volume methods for the model equations, or similar forms, have been consid-ered by Gonthier and Powers [16,17], Saurel and Abgrall [18], Andrianov et al [19],Saurel and Lemetayer [20], Gavrilyuk and Saurel [21] and by Abgrall and Saurel[22], among others. Gonthier and Powers, for example, develop a Godunov-typemethod for a two-phase flow model and use it to compute solutions to a number ofreactive flow problems. Their model equations, however, omit the non-conservativenozzling terms as a modeling choice. As a result, their scheme handles a conser-vative system with non-differential source terms and thus may be regarded as astraightforward extension of Godunov’s method for the Euler equations. The workby Saurel and co-workers consider two-phase models with non-conservative terms.In [18] and [19], for example, they introduce discrete approximations for the noz-zling terms according to a “free-streaming” condition. The basic idea, which is ageneralization of the condition given by Abgrall in [23], is that a numerical approx-imation of a two-phase flow with uniform velocity and pressure should maintainuniform velocity and pressure for all time. The high-resolution scheme developedhere, based either on the solution of the Riemann problem or an approximationthereof, satisfies this free-streaming condition naturally. It is also shown that thepresent method provides better agreement near the solid contact layer than methodsbased on the free-streaming conditions for several test problems.

The remaining sections of the paper are organized as follows. We introduce themodel and briefly describe its characteristic framework in §2. The Riemann problemis discussed in §3. There we describe a thin-layer analysis of the model equationswhich applies near the solid contact, and describe a two-stage iterative procedurethat may be used to obtain exact solutions of the Riemann problem. We considerproblems in which the left and right states of the flow consist of a mixture ofthe phases, as well as problems in which one of the phases vanishes in one of theinitial states. Our basic first-order Godunov method is described in §4, followedby a discussion of an adaptive Riemann solver in §5. A high-resolution, second-order extension of the basic Godunov method is presented in §6, and numericalresults for both the first-order and second-order methods are given in §7. Therewe compute numerical solutions of Riemann problems for both the mixture andvanishing phase cases, and we compare solutions given by the present Godunovmethod with numerical methods suggested by the work in [18] and [19]. Concludingremarks are made in §8.

4

2 The Governing Equations

Assuming one-dimensional flow, the governing equations of the two-phase model,without exchange terms, may be written in the form

ut + fx(u) = h(u)αx, (1)

where

u =

α

αρ

αρv

αρE

αρ

αρv

αρE

, f(u) =

0

αρv

α(ρv2 + p)

αv(ρE + p)

αρv

α(ρv2 + p)

αv(ρE + p)

, h(u) =

−v

0

+p

+pv

0

−p

−pv

.

Here, α, ρ, v and p denote the volume fraction, density, velocity and pressure ofthe gas phase, respectively, and α, ρ, v and p denote the analogous quantities ofthe solid phase. (The bar superscript will be used throughout the paper to indicatesolid phase quantities.) The total energies are given by

E = e +12v2, E = e +

12v2,

where e = e(ρ, p) and e = e(ρ, p) are internal energies of the gas and solid, respec-tively, assuming some equations of state. Finally, the volume fractions satisfy thesaturation constraint,

α + α = 1,

which closes the system of equations.

The governing equations in (1) involve both conservative and non-conservativeterms. The conservative terms are included in the flux f(u), which is similar tothe flux terms in the Euler equations for a single-phase flow. The non-conservativeterms appear on the right-hand-side of (1). These terms appeared in Baer and Nun-ziato’s original derivation and additional arguments justifying their presence weregiven in Bdzil et al [12]. They are often referred to as nozzling terms, as notedpreviously, since they are similar to the terms that appear in the one-dimensional(hydraulic) formulation of flow in a channel with variable cross-sectional area. Here,the volume fraction of the solid plays the role of the area A(x) of the channel. If

5

αx > 0, then the flow of the gas sees a converging channel, whereas if αx < 0, thenthe channel diverges.

The governing equations are hyperbolic, but not strictly hyperbolic as discussed in[10]. The seven real eigenvalues are

λ1 = v − a, λ2 = λ3 = v, λ4 = v + a,

λ5 = v − a, λ6 = v, λ7 = v + a,

where a and a are the sound speeds for the gas and solid, respectively. The C− andC+ characteristics for each phase given by (λ1, λ5) and (λ4, λ7), respectively, aregenuinely nonlinear, while the characteristics given by (λ2, λ3, λ6) which representparticle paths of the solid and gas phases are linearly degenerate. The eigenvectorsassociated with these seven eigenvalues are linearly independent provided that

α 6= 0, α 6= 0, (v − v)2 6= a2.

The first two conditions are met if the flow is a mixture of both phases, while thelast condition is met if the C± characteristics of the gas do not coincide with theparticle paths of the solid. Following [10], we shall refer to the latter constraint asthe sonic condition, as it corresponds to choked flow in a one-dimensional channel.Violating the sonic condition leads to a parabolic degeneracy as discussed in [10]and [15]. High drag between the phases in the granular-explosives context makesthe “subsonic” case,

(v − v)2 < a2, (2)

more relevant. We consider this case in detail in our analysis and numerical method,but provide a brief description of the supersonic case for completeness. We usuallyassume a mixture so that α 6= 0 and α 6= 0, but special cases in which one of thephases vanishes in the flow will also be treated.

3 The Riemann Problem

The Riemann problem for the two-phase model is

ut + fx(u) = h(u)αx, |x| < ∞, t > 0, (3)

with initial conditions

u(x, 0) =

uL if x < 0,

uR if x > 0,

6

where uL and uR are given left and right states of the flow. The general structureof the solution, as discussed in [15], consists of shocks and/or rarefactions in the C±characteristic fields, and contact discontinuities along particle paths. For example,the solution shown in Figure 1 has shocks in the C− characteristic field of the gasphase and the C+ characteristic field of solid phase, and has rarefactions in the C−characteristic field of the solid phase and the C+ characteristic field of the gas phase.The volume fractions of the solid and gas remain constant to the left and right of thecontact discontinuity in the solid phase as indicated in the figure. In these regions ofconstant α, the governing equations reduce to Euler equations for the solid and gasphase variables separately. The coupling between phases, and where the nozzlingterms play a role, is confined to a vanishing thin region about the solid contact,which we discuss in §3.1 below.

t

x

uL

uR

αL α

R

−−

R−

− S−

+C−

CS−R+

Fig. 1. A representative solution of the Riemann problem consisting of shocks, rar-efactions and contact discontinuities indicated by the symbols S, R and C, respec-tively.

The intermediate states of the solid and gas phases are indicated in Figures 2(a)and 2(b), respectively. For the solid phase, we denote the states to the left andright of the solid contact with subscripts 1 and 2, respectively. A similar notationis used for the intermediate states of the gas, but with an additional intermediatestate appearing between the solid and gas contacts. This additional state is givensubscript 0, and we note that v0 = v2 and p0 = p2 for the configuration shown sincethe velocity and pressure are continuous across the gas contact (where α and αare constant and the contact behaves as a usual contact discontinuity for the Eulerequations).

Even though the solution structure of the solid phase (as shown in Figure 2(a))is similar to that of the usual Euler equations, it is important to note that theintermediate states of the solid are coupled to those of the gas in accordance withthe solid contact jump conditions as discussed below. Thus, the solution for thesolid phase cannot be determined independently of the gas phase in general.

For the present discussion, we assume that neither phase vanishes in the flow andthat the subsonic condition in (2) is met. The latter condition implies that thesolid contact lies between the shocks/rarefactions of the gas as is the case shown inFigure 2(b). The gas contact, on the other hand, may lie to the left or to the right of

7

x

αL αR

−−

R−

− S−

+C−

pL

vL

ρL

−

−

−

p1

v1

ρ1−

−

− p2

v2

ρ2−

−

−

pR

vR

ρR

−

−

−

(a)

x

pL

αL αR

C−

CS− R+

vL

ρL

p0

v0

ρ0

p1

v1

ρ1

p2

v2

ρ2

pR

vR

ρR

(b)

Fig. 2. Intermediate states of the (a) solid and (b) gas phases for a subsonic solidcontact.

the solid contact, or it may lie on either side of the shocks/rarefactions of the solid.Figure 2(b) shows the solid contact to the left of the gas contact. If the positionsare reversed, then v0 = v1 and p0 = p1 according to the continuity of velocity andpressure across the gas contact as mentioned above.

On either side of the solid contact where α is constant, the equations reduce to apair of Euler equations for each phase separately. Accordingly, the jump conditionsacross shocks and rarefactions are the same as those for the Euler equations withthe chosen equation of state. Following the discussion in Toro [24], we choose toparameterize the states across shocks/rarefactions with pressure, and set

v1 = vL − FL(p1), ρ1 = GL(p1), v1 = vL − FL(p1), ρ1 = GL(p1), (4)

in region 1, and

v2 = vR + FR(p2), ρ2 = GR(p2), v2 = vR + FR(p2), ρ2 = GR(p2), (5)

in region 2. For an ideal gas with polytropic exponent γ, the functions Fs(p) andGs(p), s = L or R, are given by

Fs(p) =

(p− ps)

[As

p + Bs

]1/2

if p > ps (shock),

2as

(γ − 1)

[(p

ps

)(γ−1)/2γ

− 1

]if p < ps (rarefaction),

(6)

8

and

Gs(p) =

ρs

[(γ − 1)ps + (γ + 1)p(γ − 1)p + (γ + 1)ps

]if p > ps (shock),

ρs

(p

ps

)1/γ

if p < ps (rarefaction),

(7)

where

As =2

(γ + 1)ρs, Bs =

(γ − 1)(γ + 1)

ps, as =√

γps

ρs. (8)

For the solid, we assume a stiffened gas equation of state with polytropic exponentγ and constant stiffening pressure p0. For this choice, Fs(p) and Gs(p), s = L or R,become

Fs(p) =

(p− ps)

[As

p + p0 + Bs

]1/2

if p > ps (shock),

2as

(γ − 1)

[(p + p0

ps + p0

)(γ−1)/2γ

− 1

]if p < ps (rarefaction),

(9)

and

Gs(p) =

ρs

[(γ − 1)(ps + p0) + (γ + 1)(p + p0)(γ − 1)(p + p0) + (γ + 1)(ps + p0)

]if p > ps (shock),

ρs

(p + p0

ps + p0

)1/γ

if p < ps (rarefaction),

(10)

where

As =2

(γ + 1)ρs, Bs =

(γ − 1)(γ + 1)

(ps + p0), as =

√γ(ps + p0)

ρs. (11)

Similar expressions to those in (6), (7), (9) and (10) may be derived for otherequations of state, but the choice of equation of state is not significant for theanalysis of the Riemann problem or for the development of the numerical methodthat follows, and so the functions above are sufficient for the purposes of this paper.

The state of the flow in regions 1 and 2 is determined completely by the pressures(p1, p2) of the gas and (p1, p2) of the solid. These pressures also determine whetherthe transitions are shocks or rarefactions, and they specify the positions of thesewaves. The density ρ0 of the gas in region 0 is the remaining unknown. The jumpconditions across the solid contact provide the necessary equations to determinethese five quantities. An analysis of the solid contact which results in the jump

9

conditions is discussed next, and a method of iteration to solve the equations ispresented in §3.2.

3.1 Solid contact

We now consider the governing equations in the vicinity of the solid contact. It ishere that the nozzling terms play a role and the phases couple in the solution ofthe Riemann problem. Since the nozzling terms are not conservative, an analysisbased upon an integral conservation does not apply here and we turn instead toa thin-layer analysis similar to that described in [6]. For this analysis, we requiretwo rather mild assumptions, namely, that the layer is vanishingly thin, and thatthe solution through the layer is smooth. Accordingly, we consider traveling wavesolutions of the governing equations in terms of the independent variable ξ = x−Ut,which measures scaled distance across the layer traveling with a constant velocityU . In terms of the variable ξ, the model equations become

−Uuξ + fξ(u) = h(u)αξ. (12)

The first component of (12) reads

−Uαξ = −vαξ,

which implies that αξ = 0 or U = v. The former case implies α = constant acrossthe layer and leads to a decoupling of the equations for the gas and solid as notedpreviously. Here, we concentrate on the latter case which leads to jump conditionsacross the solid contact.

With U = v, the three equations in (12) for the solid become

−v(αρ)ξ + (αρv)ξ = 0,

−v(αρv)ξ + (αρv2 + αp)ξ = pαξ,

−v(αρE)ξ + (αvρE + αvp)ξ = pvαξ.

The first of these three equations is satisfied identically, while the second and thirdequations both give

(αp)ξ = pαξ. (13)

The three remaining equations in (12) for the gas are

10

−v(αρ)ξ + (αρv)ξ = 0, (14)

−v(αρv)ξ + (αρv2 + αp)ξ =−pαξ, (15)−v(αρE)ξ + (αvρE + αvp)ξ =−pvαξ. (16)

A straightforward manipulation of (13), (14), (15) and (16) leads to a set of jumpconditions involving a balance of mass, momentum, enthalpy and entropy across thelayer. An equation for the mass flux of the gas across the layer follows immediatelyfrom (14), and takes the form

αρ(v − v) = K1, (17)

where K1 is a constant of integration. An equation for momentum may be obtainedfrom (13), (14) and (15), and is given by

αρ(v − v)2 + αp + αp = K2, (18)

where K2 is another constant of integration. Next, we may use (14), (15) and (16)to obtain

αρ(v − v)[h +

12(v − v)2

]ξ

= 0, (19)

where h(ρ, p) = e + p/ρ is the enthalpy of the gas. If the mass flux across the layeris not zero, then (19) reduces to

h +12(v − v)2 = K3, (20)

where K3 is a constant. The final jump condition involves the entropy of the gas.Upon manipulation of the original four independent layer equations, we find

αρ(v − v)

[eξ + p

(1ρ

)ξ

]= 0,

or

αρ(v − v)Sξ = 0, (21)

using the thermodynamic identity de+pd(1/ρ) = TdS, where S(ρ, p) is the entropyof the gas. As before, if the mass flux across the layer is not zero, then we have S =K4 = constant which provides the last jump condition from the four independentlayer equations. These four jump conditions, along with the constraint that v =constant across the layer, are the same as those used in [15] following the work in[10], and they provide enough equations to complete the solution of the Riemann

11

problem as we discuss below. The layer equations also provide a useful guide to thenumerical treatment of the non-conservative terms in our Godunov method as wediscuss in §4.

3.2 Iterative solution

For the solution configuration of Figure 2, the jump conditions at the solid contactimply the equations

v1 = v2,

αLρ1(v1 − v1) = αRρ0(v2 − v2),

αLp1 + αLp1 + αLρ1(v1 − v1)2 = αRp2 + αRp2 + αRρ0(v2 − v2)

2,

γp1

(γ − 1)ρ1+

12(v1 − v1)

2 =γp2

(γ − 1)ρ0+

12(v2 − v2)

2,

p1

ργ1

=p2

ργ0

,

assuming an ideal gas. Recalling the relations in (4) and (5), we may regard thesejump conditions as five equations for the unknowns (ρ0, p1, p2, p1, p2), or, preferably,we may eliminate ρ0 from the equations using the entropy equation, and then viewthe remaining four equations as a system of nonlinear algebraic equations for thefour pressures. These equations have the vector form

N(p1, p2, p1, p2) = 0, (22)

where the four components of N may be taken as

N1 = v2 − v1,

N2 = αR

(p2

p1

)1/γ

(v2 − v2)− αL(v1 − v1),

N3 = αRp2 + αRp2 − αLp1 − αLp1 + αLρ1(v1 − v1)(v2 − v1),

N4 =γp2

(γ − 1)ρ1

(p1

p2

)1/γ

+12(v2 − v2)

2 − γp1

(γ − 1)ρ1− 1

2(v1 − v1)

2.

(23)

The nonlinear algebraic system (22) with components (23) may be solved iterativelyfor given left and right states using Newton’s method. The iteration requires startingvalues (p(0)

1 , p(0)2 , p

(0)1 , p

(0)2 ) for the pressures of the intermediate states. We use

p(0)1 = p

(0)2 = p∗, p

(0)1 = p

(0)2 = p∗,

12

where p∗, p∗ are defined implicitly by the equations

v∗ = vL − FL(p∗) = vR + FR(p∗), v∗ = vL − FL(p∗) = vR + FR(p∗), (24)

respectively. These “star” values correspond to the velocities and pressures for thegas and solid in the intermediate states assuming a decoupled flow in which α =constant. Thus, this choice of starting values for Newton’s method is particularlyrelevant for the case when the difference between αR and αL is small, which oftenoccurs if the left and right states are given by neighboring grid cells in a Godunov-type scheme (see §4). However, our numerical experiments have shown that thischoice is suitable even when the difference in the volume fractions is not small aswe show in a representative Riemann problem below.

Our iterative scheme proceeds in two stages. The first stage involves two independentNewton iterations to obtain p∗ and p∗ approximately. These iterations are equivalentto finding the solutions of the Riemann problem for the Euler equations for eachphase separately (see [24] for a detailed discussion of this iteration). The secondstage involves a Newton iteration to solve the solid contact jump conditions for thecoupled problem assuming αL 6= αR. If v∗ > v∗ as determined by the solution ofthe first stage, then we initially assume the configuration shown in Figure 2 for thesecond stage and solve (22) with components given by (23) for the intermediatepressures (p1, p2, p1, p2). If, on the other hand, v∗ < v∗, then we first assume thatthe gas contact lies to the left of the solid contact and consider the iterative solutionof (22) but with components

N1 = v2 − v1,

N2 = αR(v2 − v2)− αL

(p1

p2

)1/γ

(v1 − v1),

N3 = αRp2 + αRp2 − αLp1 − αLp1 + αRρ2(v2 − v2)(v2 − v1),

N4 =γp2

(γ − 1)ρ2+

12(v2 − v2)

2 − γp1

(γ − 1)ρ2

(p2

p1

)1/γ

− 12(v1 − v1)

2,

(25)

suitable for this solution configuration. During the subsequent steps of the New-ton iteration the sign of v1 − v1 is checked to determine which set of equations isappropriate.

As an example, consider the Riemann problem with left and right states given inTable 1. In this example, the initial two-phase mixture is at rest with a decreasein the volume fraction of the solid at x = 0 initially from αL = 0.8 to αR = 0.3.The solid density and pressure are equal on either side of x = 0 at t = 0. The gaspressure is high on the right which drives a shock wave in the gas towards the leftwith x/t = −1.982 and a rarefaction towards the right for 1.044 ≤ x/t ≤ 1.183. Theresponse of the solid is a shock wave moving to the right with x/t = 1.225 and ararefaction moving to left for −1.183 ≤ x/t ≤ −1.101. The intermediate states ofthe flow are computed using our iterative procedure and are included in Table 1.

13

-0.5 0 0.50

0.05

0.1

0.15

0.2

t

S−

+R−

− C−

-0.5 0 0.50

0.05

0.1

0.15

0.2

t

C−

S− C R

+

-0.5 0 0.5

0.94

0.96

0.98

1

1.02

1.04

1.06

ρ−

-0.5 0 0.5

0.2

0.4

0.6

0.8

1

ρ

-0.5 0 0.5

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

v−

-0.5 0 0.5

-0.8

-0.6

-0.4

-0.2

0

v

-0.5 0 0.50.9

0.95

1

1.05

1.1

p

x

−

-0.5 0 0.50.2

0.4

0.6

0.8

1

p

x

Fig. 3. Solution of the Riemann problem for the left and right state given in Table 1.The solid variables appear in the plots on the left and the gas variables appear onthe right.

An x-t diagram for each phase and the solution at t = 0.2 is shown in Figure 3.

The solution of the Riemann problem shown in Figure 3 involves one of manypossible combinations of shocks and/or rarefactions in the solid and gas phases. Ourtwo-stage iterative procedure makes no initial assumption of the solution structure,but rather determines it as part of the process. The only essential restriction isthat the sonic condition is met. Assuming that the sonic condition is satisfied, itcan be shown (see §5.2) that the solution of the solid contact jump conditions, andthus the solution of the Riemann problem, is unique for flows in which |αR − αL| issufficiently small. This branch of solutions may be followed for Riemann problems

14

Table 1Left and right states and the computed intermediate states for a sample Riemannproblem with γ = γ = 1.4 and p0 = 0.

Region L Region 1 Region 0 Region 2 Region R

α 0.8 0.8 0.8 0.3 0.3

ρ 1.0 0.9436 0.9436 1.0591 1.0

v 0.0 0.0684 0.0684 0.0684 0.0

p 1.0 0.9219 0.9219 1.0837 1.0

ρ 0.2 0.3266 0.6980 0.9058 1.0

v 0.0 -0.7683 -0.7683 -0.1159 0.0

p 0.3 0.6045 0.6045 0.8707 1.0

in which |αR− αL| increases (holding the other quantities in the left and right statesfixed) until |αR − αL| = 1 or until the sonic condition is violated at which pointsolutions cease to exist according to the analysis given in [15]. Thus, for a given leftand right state, we regard the unique solution of the Riemann problem to be theone that satisfies the solid contact jump conditions (along the branch of solutionsdescribed) and satisfies the usual (entropy-satisfying) jump conditions at shocks,rarefactions and the gas contact discontinuity away from the solid contact.

3.3 Vanishing phase cases

An exact solution of the Riemann problem may be constructed for the special casesin which the solid or the gas phase vanishes in one of the initial states. For thesecases, the solid contact separates the mixture from the single phase region. Thus, weneed to revisit the thin-layer equations for the solid contact for the vanishing phasecases to sort out the appropriate structure of the solution and the correspondingjump conditions required to determine it.

For example, consider the case in which 0 < αL < 1 and αR = 0 as shown in Figure 4so that the left state is a mixture while the right state consists of only the gas phase.Here, we need only specify the solid primitive variables (ρL, vL, pL) in the left state,as shown in the figure, whereas we need to supply both the left and right primitivestates for the gas. The solution for the solid phase involves a shock or rarefactionin the C− characteristic field (e.g. a rarefaction is shown) and an intermediate statewith subscript 1 to the left of the solid contact. The structure of the solution forthe gas phase remains the same as that shown in Figure 2(b), although the contactdiscontinuity of the gas may lie on either side of the solid contact (and shocks maybe replaced by rarefactions and vice versa). If the contact discontinuity of the gaslies to the right of the solid contact, then the jump conditions at the solid contact

15

become

αLρ1(v1 − v1) = ρ0(v2 − v1),

αLp1 + αLp1 + αLρ1(v1 − v1)2 = p2 + ρ0(v2 − v1)

2,

γp1

(γ − 1)ρ1+

12(v1 − v1)

2 =γp2

(γ − 1)ρ0+

12(v2 − v1)

2,

p1

ργ1

=p2

ργ0

.

These jump conditions determine the values for (ρ0, p1, p2, p1), or simply (p1, p2, p1)if the entropy equation is used to eliminate ρ0 as before.

x

αLαR

−−

R−

− C−

pL

vL

ρL

−

−

−

p1

v1

ρ1

−

−

−

=0<10<

Fig. 4. A vanishing solid case in which the left state is a mixture while the right stateconsists of only the gas phase. (The C− characteristic field may be a rarefaction asshown or a shock.)

We use a slight variation of the two-stage iterative procedure described earlier tosolve the jump conditions for (p1, p2, p1). The first stage uses Newton’s method tocompute p∗ and v∗ for the gas as before. We then set p∗ = p∗ and v∗ = vL−FL(p∗) forthe solid, and apply Newton’s method to the three equations for the jump conditionsat the solid contact. These equations have the form

N(p1, p2, p1) = 0, (26)

with components

N1 =(

p2

p1

)1/γ

(v2 − v1)− αL(v1 − v1),

N2 = p2 − αLp1 − αLp1 + αLρ1(v1 − v1)(v2 − v1),

N3 =γp2

(γ − 1)ρ1

(p1

p2

)1/γ

+12(v2 − v1)

2 − γp1

(γ − 1)ρ1− 1

2(v1 − v1)

2,

16

if v∗ > v∗, or with components

N1 = (v2 − v1)− αL

(p1

p2

)1/γ

(v1 − v1),

N2 = p2 − αLp1 − αLp1 + αRρ2(v2 − v1)(v2 − v1),

N3 =γp2

(γ − 1)ρ2+

12(v2 − v1)

2 − γp1

(γ − 1)ρ2

(p2

p1

)1/γ

− 12(v1 − v1)

2,

otherwise. Similar equations apply if the left and right states are reversed so thatthe solid phase vanishes in the left state.

Table 2Left and right states and the computed intermediate states for a sample Riemannproblem with γ = 1.4, γ = 3 and p0 = 0, and for which the right state consists ofonly the gas phase.

Region L Region 1 Region 0 Region 2 Region R

α 0.5 0.5 0.5 0.0 0.0

ρ 2.0 1.7829 1.7829 — —

v 0.0 0.2972 0.2972 — —

p 5.0 3.5422 3.5422 — —

ρ 1.0 1.3941 1.5341 1.7010 1.8

v 0.0 -0.5819 -0.5819 -0.0992 0.0

p 2.0 3.1978 3.1978 3.6956 4.0

Figure 5 shows the solution of a Riemann problem representative of the case whenthe solid phase vanishes in the right initial state. The values for the left and rightstates are given in Table 2 along with the computed values for the intermediatestates. For this example, the solution consists of a rarefaction in the C− character-istic field of the solid phase for −2.739 ≤ x/t ≤ −2.144 and a solid contact whichmoves with speed x/t = 0.297. The solution for the gas phase consists of a shockfor x/t = −2.058, a contact discontinuity for x/t = −0.5824 and a rarefaction for1.645 ≤ x/t ≤ 1.764.

The opposite situation occurs when the gas phase vanishes in the left or rightstates. For example, let us consider the case when αL = 0 and 0 < αR < 1 so thatthere is no gas phase in the left state while the right state is a mixture. Since thegas phase vanishes through the solid contact, it follows that the mass flux of gasacross the solid contact is zero. Accordingly, the contact discontinuity for the gascollapses upon that of the solid and region 0 disappears as shown in Figure 6. Thereis a shock or rarefaction in the C+ characteristic field (e.g. a rarefaction is shown)and an intermediate state with subscript 2 to the right of the solid/gas contact. Thesolution structure for the solid phase remains the same as that shown in Figure 2(a).

17

0

0.05

0.1

0.15

-0.5 0 0.5

No solid phaset

C−

R−

−

0

0.05

0.1

0.15

-0.5 0 0.5

t

C−

R+

S− C

1.75

1.8

1.85

1.9

1.95

2

-0.5 0 0.5

No solid phaseρ−

1

1.2

1.4

1.6

1.8

-0.5 0 0.5

ρ

0

0.05

0.1

0.15

0.2

0.25

0.3

-0.5 0 0.5

No solid phasev−

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

-0.5 0 0.5

v

3.5

4

4.5

5

-0.5 0 0.5

No solid phasep

x

−

2

2.5

3

3.5

4

-0.5 0 0.5

p

x

Fig. 5. Solution of the Riemann problem for the left and right state given in Table 2.The solid variables appear in the plots on the left and the gas variables appear onthe right. The solid phase vanishes to the right of the solid contact.

Thus, the jump conditions at the solid contact reduce to

N(p2, p1, p2) = 0, (27)

where

N1 = v2 − v1,

N2 = v2 − v2,

N3 = αRp2 + αRp2 − p1,

18

which determine the pressures (p2, p1, p2). The jump conditions involving the en-thalpy and entropy of the gas do not apply since the mass flux of the gas across thesolid contact is zero.

x

αL αR

C=C−

R+

p2

v2

ρ2

pR

vR

ρR

=0 0< <1

Fig. 6. A vanishing gas case in which the left state consists of only the solid phasewhile the right state is a mixture. (The C+ characteristic field may be a rarefactionas shown or a shock.)

In order to solve (27), we first determine p∗ and v∗ using Newton’s method asdescribed in §3.2, and then determine p∗ iteratively from the equation v∗ = vR +FR(p∗). From here, we use p

(0)2 = p∗ and p

(0)1 = p

(0)2 = p∗ as starting values for a

Newton iteration to solve (27). A representative solution is shown in Figure 7 forthe left and right states given in Table 3. Here, we consider a mixture at rest in theright state at t = 0 and a solid at rest in the left state. There is a large pressuredifference between the left and right states of the solid phase initially which leads toa shock traveling to the right with speed x/t = 2.336 and a rarefaction on the left for−2.739 ≤ x/t ≤ −1.816. The solid contact moves to the right with speed x/t = 0.461which creates a compression in the gas and a shock with speed x/t = 1.752.

Table 3Left and right states and the computed intermediate states for a sample Riemannproblem with γ = 1.4, γ = 3 and p0 = 100, and for which the left state consists ofonly the solid phase.

Region L Region 1 Region 2 Region R

α 1.0 1.0 0.6 0.6

ρ 120.0 99.786 124.61 100.0

v 0.0 0.4613 0.4613 0.0

p 200.0 72.496 117.75 10.0

ρ — — 2.7146 2.0

v — — 0.4613 0.0

p — — 4.6166 3.0

19

0

0.05

0.1

0.15

-0.5 0 0.5

t

C−

R−

− S−

+

0

0.05

0.1

0.15

-0.5 0 0.5

t

C−

S+

No gas phase

100

105

110

115

120

125

-0.5 0 0.5

ρ−

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

-0.5 0 0.5

No gas phaseρ

0

0.1

0.2

0.3

0.4

-0.5 0 0.5

v−

0

0.1

0.2

0.3

0.4

-0.5 0 0.5

No gas phasev

0

50

100

150

200

-0.5 0 0.5

p−

x

3

3.5

4

4.5

-0.5 0 0.5

No gas phasep

x

Fig. 7. Solution of the Riemann problem for the left and right state given in Table 3.The solid variables appear in the plots on the left and the gas variables appear onthe right. The gas phase vanishes to the left of the solid contact.

3.4 Supersonic case

We close our discussion of the exact solution of the Riemann problem by consideringbriefly the case in which the relative gas velocity is supersonic through the solidcontact. In this case, the solid contact lies to one side of all of the characteristicfields of the gas phase. For example, Figure 8 shows the intermediate states forthe solid and gas for a supersonic case in which all of the characteristic fields ofthe gas lie to the right of the solid contact. As before, the solid pressures p1 andp2 determine the density and velocity of the solid to the left and right of the solidcontact, respectively, so that in this case the jump conditions across the solid contactmay be regarded as equations for the states (p1, p2, ρ0, v0, p0). Once these equations

20

are solved (using Newton’s method for example), then the remaining intermediatestates of the gas in regions 1 and 2 may be found iteratively by considering (ρ0, v0, p0)and (ρR, vR, pR) as left and right state for a Riemann problem for the gas phasealone. A similar construction occurs when the characteristic fields of the gas lie tothe left of the solid contact.

x

αL

αR

−

−

R−

− S−

+C−

pL

vL

ρL

−

−

−

p1

v1

ρ1−

−

−p2

v2

ρ2−

−

−

pR

vR

ρR

−

−

−

(a)

x

C−

CS− R+

p0

v0

ρ0

p1

v1

ρ1

p2

v2

ρ2

pR

vR

ρR

(b)

αL

αR

pL

vL

ρL

Fig. 8. Intermediate states of the (a) solid and (b) gas phases for a supersonic solidcontact.

For the reasons mentioned earlier, we are interested primarily in the subsonic case.The supersonic case is described here for completeness, but we will not consider itfurther. However, it is worth noting that the subsonic case considered previously andthe supersonic case discussed here are the only admissible solutions of the Riemannproblem in which the solution in the solid contact layer is smooth according to theselection criterion discussed in [15].

4 A Godunov Method

We now turn our attention to a description of a Godunov method for the two-phasemodel. The method requires solutions of the Riemann problem, which we have de-scribed in the previous section and now consider to be known. The basic descriptionof our numerical method follows the usual course (see, for example, the discussionsin [25] or [24]) except for our numerical treatment of the non-conservative nozzlingterms which is new and will be the main focus of our attention. Essentially, themethod combines the standard conservative Godunov flux with a non-conservativecontribution arising from an integral of the nozzling terms over a grid cell. Both ofthese contributions are exact for a piecewise constant initial state (and for a suffi-ciently small time step so that neighboring Riemann problems do not interact). The

21

numerical approximation lies in the piecewise constant interpolation of the solutionat each time step, and as a result the numerical method is first-order accurate, butmay be extended to second-order accuracy (for smooth regions of the flow) as wedescribe later in §6.

Let us begin by assuming a uniform grid, xj = (j − 12)∆x, j = 1, 2, . . . , N , with

mesh spacing ∆x, and define the cell average

Unj =

1∆x

xj+1/2∫xj−1/2

u(x, tn) dx.

An integral of (1) over a grid cell Ωnj = [xj−1/2, xj+1/2] × [tn, tn+1] results in the

exact formula

Un+1j = Un

j −∆t

∆x

(Fn

j+1/2 − Fnj−1/2

)+

1∆x

∫∫Ωn

j

h(u)αx dxdt, (28)

where ∆t = tn+1 − tn and

Fnj±1/2 =

1∆t

tn+1∫tn

f(u(xj±1/2, t)) dt. (29)

If we assume a piecewise constant approximation for u(x, t) at t = tn, i.e.

u(x, tn) ≈

...

Unj−1 if x ∈ [xj−3/2, xj−1/2)

Unj if x ∈ [xj−1/2, xj+1/2)

Unj+1 if x ∈ [xj+1/2, xj+3/2)

...

then we may evaluate the fluxes Fnj±1/2 in (29) and the integral of the nozzling terms

in (28) using the solution of the Riemann problem with initial states given by Unj

and the corresponding values in the neighboring cells. For example, let u∗(uL,uR)be the (constant) value of the self-similar solution of the Riemann problem in (3)along the line x = 0 for t > 0, and set

Fnj−1/2 =

1∆t

tn+1∫tn

f(u∗(Unj−1,U

nj )) dt = f(u∗(Un

j−1,Unj ))

22

and

Fnj+1/2 =

1∆t

tn+1∫tn

f(u∗(Unj ,Un

j+1)) dt = f(u∗(Unj ,Un

j+1)).

This part follows the usual description of the Godunov method and the functionf(u∗(uL,uR)) may be evaluated based on the construction of the solution of theRiemann problem.

It remains to evaluate the integral of the nozzling terms in (28). In view of thecomponents of h(u), we must examine the three integrals

I1 =1

∆x

tn+1∫tn

xj+1/2∫xj−1/2

vαx dxdt, I2 =1

∆x

tn+1∫tn

xj+1/2∫xj−1/2

pαx dxdt,

and

I3 =1

∆x

tn+1∫tn

xj+1/2∫xj−1/2

pvαx dxdt.

Each integration is carried out over the grid cell Ωnj as shown in Figure 9. For

example, suppose that the solid contact from the Riemann problem centered atxj−1/2 (labeled Cj−1/2 in the figure) enters the cell from the left with constantvelocity vn

c,j−1/2 > 0. There are other waves associated with this Riemann problem(and the Riemann problem centered at xj+1/2), but they are not essential to theevaluation of the integrals since α is constant across them. We suppose furtherthat the solid contact from the Riemann problem centered at xj+1/2 has a positivevelocity so it is outside the grid cell of interest. In Ωn

j , α(x, t) equals αnj−1 to the

left of a thin layer about Cj−1/2 and it equals αnj to the right of the layer. Since

v = constant = vnc,j−1/2 across the layer, the scalar integrals reduce to

I1 =∆t

∆xvnc,j−1/2(α

nj − αn

j−1), I2 =∆t

∆x

ξ+j−1/2∫

ξ−j−1/2

pαξ dξ,

and

I3 = vnc,j−1/2I2,

where ξ = ξj−1/2 denotes the center of the thin layer about Cj−1/2. The final integral

23

may be evaluated using (13). We find

ξ+j−1/2∫

ξ−j−1/2

pαξ dξ =

ξ+j−1/2∫

ξ−j−1/2

(pα)ξ dξ = pn2,j−1/2α

nj − pn

1,j−1/2αnj−1,

where pn1,j−1/2 and pn

2,j−1/2 are the solid pressures on either side of Cj−1/2 as deter-mined by the solution of the Riemann problem centered at xj−1/2 (e.g., the pres-sures p1 and p2 in Figure 2(a)). The integral of the nozzling terms becomes pathindependent locally about each solid contact layer within the piecewise constantconstruction of the Godunov method (and our thin-layer assumption).

Uj

nUj+1

nUj−1

n

Uj

n+1

t

x

Fj−1/2

nFj+1/2

n

Cj−1/2 Cj+1/2

xj+1/2xj−1/2

nt

n+1t

− −

Ωj

n

Fig. 9. Grid cell Ωnj with solid contact Cj−1/2 from the Riemann problem at xj−1/2.

The integral of the nozzling terms in (28) with components given by I1, I2 and I3 pro-vides non-conservative contributions to the standard Godunov flux f(u∗(Un

j−1,Unj ))

depending on the sign of vnc,j−1/2. We find it convenient to define

H(Unj−1,U

nj ) =

−vnc,j−1/2(α

nj − αn

j−1)

0

+(pn2,j−1/2α

nj − pn

1,j−1/2αnj−1)

+vnc,j−1/2(p

n2,j−1/2α

nj − pn

1,j−1/2αnj−1)

0

−(pn2,j−1/2α

nj − pn

1,j−1/2αnj−1)

−vnc,j−1/2(p

n2,j−1/2α

nj − pn

1,j−1/2αnj−1)

, (30)

and then set

FL(Unj−1,U

nj ) =

f(u∗(Un

j−1,Unj ))−H(Un

j−1,Unj ) if vn

c,j−1/2 < 0,

f(u∗(Unj−1,U

nj )) if vn

c,j−1/2 > 0,(31)

24

and

FR(Unj−1,U

nj ) =

f(u∗(Un

j−1,Unj )) if vn

c,j−1/2 < 0,

f(u∗(Unj−1,U

nj ))+H(Un

j−1,Unj ) if vn

c,j−1/2 > 0.(32)

Here, FL(Unj−1,U

nj ) is the combined numerical flux applied to cell Ωn

j−1 to the left ofxj−1/2 while FR(Un

j−1,Unj ) is the combined numerical flux applied to cell Ωn

j to theright. One of these two fluxes picks up the contribution from the non-conservativeterms depending on the direction of Cj−1/2 as determined by the sign of vn

c,j−1/2.

We may now write our finite-volume Godunov scheme for the model equations. Ittakes the compact form

Un+1j = Un

j −∆t

∆x

(FL(Un

j ,Unj+1)− FR(Un

j−1,Unj ))

, (33)

where the numerical flux functions FL and FR are given by (31) and (32), respec-tively. The evaluation of the numerical flux functions relies on solutions of the Rie-mann problem. A procedure to obtain the full solution involves a two-stage iterativeprocedure as discussed previously. While this full solution is required for accuracyfor certain left and right states, an approximate solution requiring much less com-putational cost is suitable in most cases. In §5, we describe an adaptive Riemannsolver which handles both cases, and is used to evaluate FL and FR accurately andefficiently.

Before leaving this section, we note the behavior of (33) corresponding to two specialsolutions of (1). The first solution is the decoupled phase case when α = constant.Here, if αn

j is constant for all j, then the first components of both FL and FR

vanish so that αn+1j = αn

j . Further, p1 = p2 = p∗ when αL = αR (see §3.2) so thatH = 0 and the fluxes FL and FR reduce to the standard Godunov fluxes for thediscrete solid phase and gas phase variables separately. A second special solutionoccurs when p = p = constant = P and v = v = constant = V , the so-calledfree-streaming solution. In this solution, α, αρ and αρ each evolve according tolinear advection equations with characteristic velocity equal to V . For this case,if we consider pn

j = pnj = P and vn

j = vnj = V for all j (at some n), then all

Riemann problems would be solved with p = p = P and v = v = V for all states,and (α∗, ρ∗, ρ∗) equal to (αL, ρL, ρL) if V > 0 and equal to (αR, ρR, ρR) if V < 0.Assuming the case V > 0, for example, it is straightforward to show that (33)reduces to pn+1

j = pn+1j = P and vn+1

j = vn+1j = V , and

αn+1j = αn

j − σ(αn

j − αnj−1

),

αn+1j ρn+1

j = αnj ρn

j − σ(αn

j ρnj − αn

j−1ρnj−1

),

αn+1j ρn+1

j = αnj ρn

j − σ(αn

j ρnj − αn

j−1ρnj−1

),

25

where σ = V ∆t/∆x. Thus, uniform pressure and velocity is maintained exactly inthe numerical approximation of the free-streaming solution and αn

j , αnj ρn

j and αnj ρn

j

evolve according to first-order upwind methods as expected.

5 An Adaptive Riemann Solver

The solution of the Riemann problem requires an iterative procedure (in general)and we have described a two-stage process based on Newton’s method in §3.2.This solution provides the basis for our Godunov method in (33), but it is usuallydesirable to consider approximate solutions in order to reduce computational cost asmentioned previously. In this section, we discuss suitable approximations for bothstages of the iteration and a simple adaptive procedure which chooses whether theapproximate solution is sufficient or whether a full iterative solution is desirable. Webegin in §5.1 with a brief description of an approximation of the decoupled “star”states followed in §5.2 with a discussion of a linear approximation of the jumpconditions at the solid contact and a simple criterion which we use to determinewhen the approximate solution is suitable

5.1 Approximate star states

The first stage of the iterative process involves the solution of Riemann problems forthe Euler equations for each phase separately. Since the solution of these decoupledproblems is needed only as an initial guess for the Newton iteration in the secondstage, an approximate solution is sufficient. There are many approximate Riemannsolvers for the Euler equations that may be used for this task, but it is convenientto use the approximate-state Riemann solver suggested by Toro since it involvesseveral formulas and constants used in the approximation of the second stage. Weoutline briefly the relevant formulas for completeness here, a full description appearselsewhere (c.f. [24], chapter 9).

Let us consider the Riemann problem for the Euler equations with left and rightstates given by (ρL, vL, pL) and (ρR, vR, pR), respectively, either for the gas or solidvariables. If the states are close to one another, then a linearization is appropriatewhich results in the formula

p∗,lin =12

(pL + pR) +18(ρL + ρR)(aL + aR)(vL − vR). (34)

In order to determine whether or not the linearization in (34) is suitable, Torosuggests the checks

pmax

pmin< Quser and pmin < p∗,lin < pmax, (35)

26

where pmin = minpL, pR, pmax = maxpL, pR and Quser is a user-defined param-eter whose value is taken to be 2. If both conditions in (35) are satisfied, then p∗,linis considered to be an accurate approximation for p∗. If either of the conditions isviolated, then alternate approximations are used. For example, Toro suggests thevalue

p∗,tr =

[aL + aR − 1

2(γ − 1)(vR − vL)

aLp−(γ−1)/2γL + aRp

−(γ−1)/2γR

]2γ/(γ−1)

(36)

given by the two-rarefaction solution of the Riemann problem if p∗,lin < pmin, andthe value

p∗,ts =pL

(AL

p∗,lin+BL

)1/2+ pR

(AR

p∗,lin+BR

)1/2− vR + vL(

ALp∗,lin+BL

)1/2+(

ARp∗,lin+BR

)1/2(37)

given by an approximation of the two-shock solution if p∗,lin ≥ pmin. In the two-shockformula, As and Bs, s = L or R, are given in (8) and p∗,lin is given in (34).

An approximate value for p∗ is thus given by (34), (36) or (37) depending on thevarious conditions mentioned above. Using this value, we may compute the starredvelocity using

v∗ =12(vL + vR) +

12(FR(p∗)− FL(p∗)). (38)

Analogous formulas are used for the solid phase, but with p replaced by p + p0 forall pressures in (34), (35), (36) and (37) and with bar superscripts added to theremaining variables, which give approximate values for p∗ and v∗.

If |αR−αL| is sufficiently small (less than 10−3 in our calculations), then we may stophere and use the approximate values for (p∗, p∗) and (v∗, v∗) to obtain f(u∗(uL,uR))for the (decoupled) Godunov fluxes in (33). If the difference is not approximatelyzero, then we must move on and consider an approximation of the coupled problemas is discussed next.

5.2 Linearized solid contact

The coupling between phases in the Riemann problem occurs at the solid contactwhich requires the solution of a set of nonlinear algebraic equations given by (22)for the case when both the left and right states contain a mixture of the two phases.If the left and right states come from neighboring grid cells in a numerical solutionof a smooth flow, then the jump in the volume fraction between the states wouldtypically be small. Thus, it is reasonable to consider an approximation of the solid

27

contact jump conditions in the limit of small |αR − αL| and then use the result-ing approximate values for the intermediate pressures (p1, p2, p1, p2) as a means toobtain f(u∗(uL,uR)). This exercise also provides useful information regarding thesolvability of the solid contact jump conditions in general.

When αL = αR, a solution of (22) is given by

p1 = p2 = p∗, p1 = p2 = p∗,

where p∗ and p∗ are the exact star states defined by the pair of equations in (24). For|αR − αL| small, we seek a solution to the jump conditions about an intermediatevolume fraction αm, say, in the form

pk = p∗ + δk, pk = p∗ + δk, k = 1 and 2, (39)

where δk and δk are small perturbations to the starred pressures. We may pick αm tobe the left or right value of the volume fraction of the solid, or perhaps an average.For our calculations, we take αm to be αL, αR or 1

2(αL + αR) whichever value isclosest to 0.5. Substituting (39) into (22) gives

F ′L(p∗)δ1 + F ′

R(p∗)δ2 = 0, (40)

αm

[(F ′

L(p∗)δ1 + F ′R(p∗)δ2

)+

∆v∗γp∗

(δ2 − δ1)]

+ ∆v∗(αR − αL) = 0, (41)

αm(δ2 − δ1)−∆p∗(αR − αL) = 0, (42)

ρ∗∆v∗(F ′

L(p∗)δ1 + F ′R(p∗)δ2

)+ δ2 − δ1 = 0, (43)

to leading order. Here,

αm = 1− αm, ∆v∗ = v∗ − v∗, ∆p∗ = p∗ − p∗,

and

ρ∗ =

GL(p∗) if v∗ > v∗,

GR(p∗) if v∗ < v∗.(44)

Also,

F ′s(p∗) =

(1− p∗ − ps

2(Bs + p∗)

)[As

p∗ + Bs

]1/2

if p∗ > ps (shock),

1ρsas

(p∗ps

)−(γ+1)/2γ

if p∗ < ps (rarefaction),

(45)

28

for s = L or R according to (6), and

F ′s(p∗) =

(1− p∗ − ps

2(Bs + p∗ + p0)

)[As

p∗ + p0 + Bs

]1/2

if p∗ > ps (shock),

1ρsas

(p∗ + p0

ps + p0

)−(γ+1)/2γ

if p∗ < ps (rarefaction),

(46)

according to (9).

We note that the four linear equations separate into a pair of equations for thesolid pressure perturbations (δ1, δ2) and a pair of equations for the gas pressureperturbations (δ1, δ2). The pair, (40) and (42), for the solid pressure perturbationsis uniquely solvable if

αm 6= 0 and F ′L(p∗) + F ′

R(p∗) 6= 0.

The former is satisfied if the solid phase does not vanish and the latter is alwayssatisfied since F ′

s > 0, s = L or R. Assuming a non-vanishing solid phase, a uniquesolution exists and is given by

δ1 =−F ′

R(p∗)∆p∗F ′

L(p∗) + F ′R(p∗)

(αR − αL

αm

),

δ2 =F ′

L(p∗)∆p∗F ′

L(p∗) + F ′R(p∗)

(αR − αL

αm

).

(47)

Equations (41) and (43) determine the gas pressure perturbations. This pair ofequations is uniquely solvable if

αm 6= 0 and (γp∗ − ρ∗∆v2∗)(F ′

L(p∗) + F ′R(p∗)

)6= 0.

The first is satisfied if the gas phase is present in the flow, and the second is satisfiedif

M2∗ =

∆v2∗

γp∗/ρ∗6= 1,

which is true if the subsonic condition is met. Assuming these conditions are satis-fied, the unique solution is given by

δ1 =(1 + ρ∗∆v∗F

′R(p∗))∆v∗

(1−M2∗ ) (F ′

L(p∗) + F ′R(p∗))

(αL − αR

αm

),

δ2 =(1− ρ∗∆v∗F

′L(p∗))∆v∗

(1−M2∗ ) (F ′

L(p∗) + F ′R(p∗))

(αL − αR

αm

).

(48)

29

The perturbations in the pressures given by (47) and (48) reflect a linearized cou-pling between the phases and correspond to the corrections provided by the firststep in the Newton iteration of the nonlinear equations for the solid contact. Theseformulas are derived based on an exact star state, but are used in practice with theapproximate star state given by (34), (36) or (37) for the pressures and by (38) forthe velocities. We may now check the size of the residual,

ε = ‖N(p∗ + δ1, p∗ + δ2, p∗ + δ1, p∗ + δ2)‖,

to determine whether more Newton steps are required. (We assume that the prim-itive variables of the flow are dimensionless and properly scaled so that we neednot scale the components of N for this check.) If ε < 10−3, then we accept thelinearized values and use them to determine u∗(uL,uR). We also use these valuesto determine the velocity of the solid contact, vc, and the solid pressures on eitherside (p1, p2), which are needed in our numerical treatment of the nozzling terms in(30). If ε is too big, then we take more Newton steps to obtain more accurate val-ues for (p1, p2, p1, p2), which requires a higher computational cost. In our numericalexperiments, we monitor the number of Newton steps required and find that thelinear approximation is sufficient for the majority of Riemann solutions (see §7).

Finally, we note that in extreme cases, either when |αR − αL| is close to 1 or whenone of the phases nearly vanishes, the Newton iteration may not converge for theinitial guess given by the decoupled star states. For such cases, we employ a simplecontinuation procedure in either the left or right value for α. For example, supposeαL is closest to 0.5. We fix this value and adjust αR closer to αL, while holdingthe other primitive values in the left and right states fixed, until Newton’s methodconverges. Once convergence occurs, we then slowly move αR back to its originalvalue during the Newton iteration. While this situation rarely occurs (and if sorequires only a tiny fraction of the overall computational cost of the method), wefind it helpful for a robust implementation of the method.

6 A High-Resolution Method

The Godunov method described in §4 is first order accurate, and may be extendedto second order using slope-limited corrections of the left and right states in theRiemann problem. This approach to generate a high-resolution method is well es-tablished in the literature for the Euler equations, see [8] and [9] for example, andhas been used in other numerical schemes for compressible multi-phase flow, see[18] and [19] for example. The description of the second-order extension here fol-lows similar lines except for our treatment of the non-conservative terms which isnew. Essentially, the extension for the non-conservative terms involves two parts,one coming from a contribution to the integral of h(u)αx about the solid contactand the other coming from the integral away from the jump at the solid contact.The latter contribution arises due to the slope correction for α as we describe below.

30

It is convenient to obtain the corrections in terms of the primitive variables w =(α, ρ, v, p, ρ, v, p)T . This may be done using the quasi-linear form of the model whichtakes the form

wt + A(w)wx = 0,

where

A(w) =

v 0 0 0 0 0 0

0 v ρ 0 0 0 0

−∆pαρ 0 v 1

ρ 0 0 0

0 0 ρa2 v 0 0 0

−ρ∆vα 0 0 0 v ρ 0

0 0 0 0 0 v 1ρ

−ρa2∆vα 0 0 0 0 ρa2 v

,∆p = p− p,

∆v = v − v.

Let us consider the grid cell Ωnj defined earlier in §4 and let wn

j denote the prim-itive variables corresponding to the cell average Un

j . First order approximationsare needed at cell boundaries. For example, a first order approximation for w at(xj+1/2, tn+1/2) is given by

wnj,+ = wn

j +∆x

2(wx)n

j +∆t

2(wt)n

j = wnj +

12

(I − ∆t

∆xAn

j

)∆wn

j , (49)

where Anj = A(wn

j ) and ∆wnj is a discrete approximation of (wx)n

j ∆x. The slopecorrection in (49) is limited in order to suppress numerical oscillations, and this isdone in characteristic variables. Let

Λ(w) = diag(v, v − a, v, v + a, v − a, v, v + a)

and

R(w) =

1 0 0 0 0 0 0

0 ρ 1 ρ 0 0 0

0 −a 0 a 0 0 0∆pα ρa2 0 ρa2 0 0 0

− ρ∆v2

α(a2−∆v2)0 0 0 ρ 1 ρ

a2∆vα(a2−∆v2)

0 0 0 −a 0 a

− ρa2∆v2

α(a2−∆v2)0 0 0 ρa2 0 ρa2

,

31

so that Λ and R solve AR = RΛ and the characteristic variables are z = R−1w. Aslope-limited correction at (xj+1/2, tn+1/2) is

wnj,+ = wn

j +12Rn

j

(I − ∆t

∆xmaxΛn

j , 0)

∆znj , (50)

where

∆znj = minmod

(R−1(wn

j+1 −wnj ), R−1(wn

j −wnj−1)

). (51)

Here, the maximum and minimum-modulus functions are performed component-wise. Similar steps give

wnj,− = wn

j −12Rn

j

(I +

∆t

∆xminΛn

j , 0)

∆znj , (52)

which is a slope-limited correction at (xj−1/2, tn+1/2). Finally, let Unj,± denote the

conservative variables corresponding to wnj,±.

The slope-limited corrections Unj,± may now be used to extend (33) to second order.

These values provide corrections at cell boundaries which may then be used as leftand right states for the Riemann problems. They also provide a slope correctionto the integral of the non-conservative terms. For example, consider the Riemannproblem about the boundary xj−1/2 with left and right states given by Un

j−1,+ andUn

j,−, respectively. The adaptive Riemann solver provides the necessary informationto evaluate the numerical fluxes FL(Un

j−1,+,Unj,−) and FR(Un

j−1,+,Unj,−) defined by

(31) and (32), respectively. These fluxes contain the second-order Godunov flux atthe boundary of the cell, f(u∗(Un

j−1,+,Unj,−)), and the second-order contribution to

the integral of the non-conservative terms about the solid contact, H(Unj−1,+,Un

j,−).

It remains to determine the second-order contribution to the integral away fromthe solid contact. This contribution arises in the second-order extension since αinvolves a slope correction, and thus away from the solid contact αx 6= 0 in general.In order to specify this contribution, let us consider the solution of the Riemannproblem about the cell boundary xj−1/2 and define the pairs (v`,j−1/2, p`,j−1/2) and(vr,j−1/2, pr,j−1/2) as follows:

(v`,j−1/2, p`,j−1/2) = (vnc,j−1/2, p

n1,j−1/2)

(vr,j−1/2, pr,j−1/2) = (v∗j−1/2, p∗j−1/2)

if vnc,j−1/2 < 0,

and

(v`,j−1/2, p`,j−1/2) = (v∗j−1/2, p∗j−1/2)

(vr,j−1/2, pr,j−1/2) = (vnc,j−1/2, p

n2,j−1/2)

if vnc,j−1/2 > 0.

32

(See Figure 10.) These velocities and pressures may now be used to approximatethe integral of the non-conservative terms away from the jump at the solid contact.For the grid cell Ωn

j , the contribution to the integral is given by

Hnj =

−12(vr,j−1/2 + v`,j+1/2)∆αn

j

0

+12(pr,j−1/2 + p`,j+1/2)∆αn

j

+12(pr,j−1/2vr,j−1/2 + p`,j+1/2v`,j+1/2)∆αn

j

0

−12(pr,j−1/2 + p`,j+1/2)∆αn

j

−12(pr,j−1/2vr,j−1/2 + p`,j+1/2v`,j+1/2)∆αn

j

, (53)

where ∆αnj = αn

j,+ − αnj,−. This contribution applies only in smooth regions of the

flow (due to the slope limiting) so that the averages used in (53) are suitable.

uR=Uj,−uL=Uj−1,+

n

t

x

Cj−1/2

xj−1/2

−

n

pr, j−1/2

vr, j−1/2−

pl, j−1/2

vl, j−1/2−

(b)

uR=Uj,−uL=Uj−1,+

n

t

x

Cj−1/2

xj−1/2

−

n

pr, j−1/2

vr, j−1/2−

pl, j−1/2

vl, j−1/2−

(a)

Fig. 10. Values for (v`,j−1/2, p`,j−1/2) and (vr,j−1/2, pr,j−1/2) about xj−1/2 for (a)vnc,j−1/2 < 0 and (b) vn

c,j−1/2 > 0.

We now have all of the ingredients for our second-order extension of (33). It has theform

Un+1j = Un

j −∆t

∆x

(FL(Un

j,+,Unj+1,−)− FR(Un

j−1,+,Unj,−)

)+

∆t

∆xHn

j , (54)

where FL and FR are defined by (31) and (32), respectively, and Hnj is given in (53).

As with the first-order method, we note that (54) reduces to a standard slope-limitedextension of Godunov’s method for the Euler equations for each phase separatelywhen αn

j is constant. Also, it can be shown that the pressure and velocity remainconstant in the free-streaming case, and that αn

j , αnj ρn

j and αnj ρn

j evolve accordingto second-order finite difference approximations of their respective linear advectionequations (except near local extrema where the approximations become first orderdue to the slope limiter).

33

7 Numerical Results

In this section, we consider a variety of problems for the governing equations (1) inorder to illustrate the behavior and accuracy of our first order Godunov method (33)and its second-order extension (54). We begin by illustrating numerical solutionsof the Riemann problems described in §3. These problems involve the case whereboth the left and right states consist of a mixture of the phases and cases wherethe solid or gas phase vanishes in the left or right states. For the mixture case withleft and right states given in Table 1, we perform a careful check of the behaviorof the numerical solution through the thin solid contact layer. Next, we computenumerical solutions of the model equations for smooth initial conditions and thenuse these solutions to perform a grid refinement study. This is done as a means toverify the order of accuracy of the methods. Finally, we compute solutions usingtwo alternate numerical methods suggested by the work in [18] and [19]. We thencompare solutions obtained using these two alternate methods with that given by(33). It is found that the present method gives more accurate results for the testproblem used, which we attribute to a more accurate numerical treatment of thenon-conservative terms about the solid contact.

Figure 11 shows numerical solutions and the exact solution at t = 0.2 for theRiemann problem with left and right states given in Table 1. The blue curve ineach plot is obtained using the first order method while the red curve is given byits second-order extension. Both of these numerical solutions are computed usingN = 200 grid cells and a ∆t determined by

∆t = 0.8∆x

λmax, λmax = max

1≤j≤N|vn

j |+ anj , |vn

j |+ anj ,

at each time step. The black curve shows the exact solution in each plot. Bothnumerical solutions are in good agreement with the exact solution, and, as expected,the second-order solution shows less smearing at shocks and contacts, and at thecorners of the rarefactions. In particular, we note that both numerical solutions arein good agreement with the exact solution near the solid contact.

The behavior of the numerical solutions near the gas contact, shocks and rarefac-tions is typical of first-order and (slope-limited) second-order methods for the Eulerequations which applies in regions of the flow where |αn

j − αnj−1| is approximately

zero. Near the solid contact, on the other hand, where αnj varies, we may compare

the behavior of the numerical solutions with that determined by the jump con-ditions for the solid contact layer (see §3.1). Figure 12 shows the first-order andsecond-order solutions at t = 0.2 for p, ρ, v and p versus α through the layer. Theblack curve is given by layer equations

αρ(v − v) = K1, αρ(v − v)2 + αp + αp = K2,

γp

(γ − 1)ρ+

12(v − v)2 = K3,

p

ργ= K4, α + α = 1,

(55)

34

0 0.2 0.4 0.6 0.8 1

0.3

0.4

0.5

0.6

0.7

0.8

x

α−

0 0.2 0.4 0.6 0.8 1

0.94

0.96

0.98

1

1.02

1.04

1.06

ρ−

x0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

x

ρ

0 0.2 0.4 0.6 0.8 1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

v−

x0 0.2 0.4 0.6 0.8 1

- 0.8

-0.7

-0.6

-0.5

-0.4

- 0.3

- 0.2

-0.1

0

x

v

0 0.2 0.4 0.6 0.8 10.9

0.95

1

1.05

1.1

x

p−

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

x

p

Fig. 11. First-order (blue) and second-order (red) numerical solutions at t = 0.2 ofthe Riemann problem with left and right states given in Table 1. The exact solutionis shown in black.

35

0.3 0.4 0.5 0.6 0.7 0.80.9

0.95

1

1.05

1.1

p−

α−

0.3 0.4 0.5 0.6 0.7 0.8

0.7

0.75

0.8

0.85

0.9

α−

ρ

0.3 0.4 0.5 0.6 0.7 0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

v

α−

0.3 0.4 0.5 0.6 0.7 0.8

0.6

0.65

0.7

0.75

0.8

0.85

α−

p

Fig. 12. Behavior of p, ρ, v and p versus α through the solid contact layer for thesolution at t = 0.2 of the Riemann problem with left and right states given inTable 1. The blue circles are the first-order solution and the red crosses are thesecond-order solution.

for α between 0.3 and 0.8 using the exact state at α = 0.3 to determine the fourconstants, Ki, i = 1, . . . , 4. The plots show that the numerical solutions agree verywell with the solution of the thin layer equations, and this is a result of our numericaltreatment of the integral of the non-conservative terms over each the grid cell, andin particular across the solid contact layer.

The adaptive Riemann solver described in §5 is used to perform the Godunov fluxcalculations for both the first-order and second-order methods. The solver is de-signed to detect and handle both coupled (αx 6= 0) and decoupled (αx = 0) flowsand take advantage of the computational savings in the flux calculation for de-coupled (Euler) flows. In the coupled case, the adaptive solver first considers thelinearized equations for the solid contact and checks whether a solution based on thissimpler set of equations is sufficient. If the solution is not sufficiently accurate, thenadditional Newton iterations are performed at an increased computational cost. Forexample, the second-order solution shown in Figures 11 and 12 required 24120 fluxcalculations. The majority of these flux calculations, approximately 97.6%, were re-garded as decoupled. Of the remaining 2.4%, 339 flux calculations were performedbased on the linearized equations for the solid contact, while only 246 flux cal-culations required further Newton iterations (with no continuation in α). Thesepercentages are representative of numerical solutions in which the coupling betweenphases is confined to a thin region of x and the jump in α is moderate. For other

36

0 0.2 0.4 0.6 0.8 1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

α−

0.2 0.3 0.4 0.5 0.6 0.7 0.81.6

1.7

1.8

1.9

2

2.1

2.2

0.9

α−

ρ

0 0.2 0.4 0.6 0.8 11800

1850

1900

1950

2000

2050

ρ−

x

0 0.2 0.4 0.6 0.8 1

1

1.2

1.4

1.6

1.8

2

2.2

x

ρ

Fig. 13. First-order (blue) and second-order (red) numerical solutions at t = 0.15of the Riemann problem with left and right states given in (56) and with γ = 1.35,γ = 3 and p0 = 3400.

calculations, such as the smooth solution considered later, the coupled flow regionis broader, in which case the numerical fluxes may be computed based on the de-coupled equations or the linearized coupled equations alone.

As an added check of the numerical approach, we consider a case in which there is alarge variation in the values between the left and right states and between the solidand gas phases. Figure 13 shows the exact solution and numerical solutions for theRiemann problem with left and right states given by

αL = .2, ρL = 1900, vL = 0, pL = 10, ρL = 2, vL = 0, pL = 3,

αR = .9, ρR = 1950, vR = 0, pR = 1000, ρR = 1, vR = 0, pR = 1.(56)

For this case, the equation of state for the gas is specified by γ = 1.35, while γ = 3and p0 = 3400 are used in the stiffened equation of state for the solid. The valuesfor density and pressure are representative of the ones used in [11–13] and in therecent paper by Powers [26], and the parameters for the equations of state arerepresentative of ones for granular explosives. The plots of α, ρ and ρ versus x showvery good agreement between the numerical solutions and the exact solution. Theplot of the density of the gas, ρ, versus α through the solid contact layer shows verygood agreement as well. Overall, the numerical scheme performs very well for thiscase in view of the large variation in the values of the left and right states includingthe large jump in the volume fraction of the mixture.

37

A solution in which the solid or gas phase vanishes may be handled numerically bycomputing a corresponding approximate solution where the phase nearly vanishes.We may illustrate this by considering the solutions of the two Riemann problemsdescribed in §3 in which the solid phase vanishes in one of the solutions and thegas phase vanishes in the other. For example, Figure 14 shows the first-order andsecond-order numerical solutions at t = 0.15 of the Riemann problem with the leftstate given in Table 2 and a right state given by

αR = 10−6, ρR = 1.7829, vR = 0.2972, pR = 3.5422,

ρR = 1.8, vR = 0, pR = 4.(57)

A very small positive value for αR is taken in the right state to maintain a mixtureof the phases as required by the numerical method and to approximate the exactsolution where αR = 0. No values are needed for ρR, vR and pR in the exact problembut some values are needed by the numerical method, and the choice given above istaken from the exact solution for the solid phase in the constant state to the left ofthe solid contact (Region 1 in Table 2). This choice is not essential but is made herefor convenience as it (approximately) eliminates the variation in ρ, v and p throughthe solid contact in the numerical solution and any shock or rarefaction in the C+

characteristic field to the right of the solid contact. Other choices for ρR, vR andpR could be made, but this would have a negligible effect on the numerical solutionfor the solid phase variables to the left of the solid contact and on the numericalsolution for the gas phase variables on either side of the solid contact. In Figure 14,the exact position of the solid contact separating the mixture on the left from thegas on the right is marked by a dashed line. Here, we note that the numericalsolutions for the solid and gas phase variables agree well with the exact solutionon the mixture side of the solid contact, and the gas phase variables agree well tothe right of the solid contact. There are small errors in the solid phase variablesto the right of the solid contact where the volume fraction is approximately zero.These errors are generated in the first few time steps of the numerical solutions dueto the initial jump in the state, but are not considered significant since the volumefraction of the solid is approximately zero there.

An exact solution of the Riemann problem for a case in which the gas phase vanishesin left state is given in Table 3. Numerical solutions for this case may be computedusing the left state given by

αL = 1− 10−6, ρL = 120, vL = 0, pL = 200,

ρL = 2.7146, vL = 0.4613, pL = 4.6166,(58)

and the right state given in Table 3. Here, we choose the gas variables in the leftstate to be those given by the exact solution to the right of the solid contact. Asbefore, this choice is made for convenience but is not essential. Figure 15 shows thefirst-order and second-order numerical solutions at t = 0.15 for this case, and againwe note good agreement between the numerical solutions and the exact solution

38

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

α−

x

0 0.2 0.4 0.6 0.8 11.75

1.8

1.85

1.9

1.95

2

ρ−

x

0 0.2 0.4 0.6 0.8 1

1

1.2

1.4

1.6

1.8

ρ

x

0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

0.2

0.25

0.3

v−

x

0 0.2 0.4 0.6 0.8 1

- 0.6

- 0.5

- 0.4

- 0.3

- 0.2

- 0.1

0

v

x

0 0.2 0.4 0.6 0.8 1

3.5

4

4.5

5

p−

x0 0.2 0.4 0.6 0.8 1

2

2.5

3

3.5

4

p

x

Fig. 14. First-order (blue) and second-order (red) numerical solutions at t = 0.15of the Riemann problem with left state given in Table 2 and right state given in(57). The exact solution for αR = 0 is shown in black and the position of the solidcontact is indicated by the dashed line.

39

(apart from small errors in the gas phase variables where α is approximately one). Ingeneral, we find good agreement for all of our numerical experiments with solutionsof Riemann problems.

Numerical solutions with smooth initial conditions may be computed using the first-order and second-order methods and then used to verify the order of accuracy of themethods. This is done primarily as a further check of our numerical approximationof the non-conservative terms. For example, we take γ = γ = 1.4 and p0 = 0, andconsider the initial conditions

α(x, 0) =12

+25

tanh(20x− 8), v(x, 0) =12

+12

tanh(20x− 10),

and

ρ(x, 0) = ρ(x, 0) = 1, p(x, 0) = p(x, 0) = 1, v(x, 0) = 0.

For these initial conditions, α has a smooth transition from 0.1 to 0.9 initiallycentered at x = 0.4, and the solid velocity varies smoothly from 0 on the left to1 on the right which creates an expansion in the flow initially. The choice of theinitial conditions is somewhat arbitrary for the purpose of our grid refinement study,although it is desirable for α(x, 0) to vary so that the phases are coupled and forthe velocity and/or pressure to vary so that the flow is not free-streaming. Unlikethe Riemann problems, an exact solution is not available for this problem, butit may be approximated with sufficient accuracy using the second-order numericalmethod with a very large number of grid cells, e.g. 12800 grid cells on x ∈ [0, 1] withnumerical boundary conditions Un

0 = Un1 and Un

N = UnN−1. Numerical solutions

with N = 100, 200, 400 and 800 may be computed using our first-order Godunovmethod (33) and its second-order extension (54) and compared with the highly-resolved numerical solution, un

j , restricted to the coarser grids. Numerical errorsand convergence rates are computed at tn = 0.1 using

EN =N∑

j=1

‖Unj − un

j ‖∆x, rN =lnEN/2 − lnEN

ln 2,

respectively, and are presented in Table 4. The convergence rates indicate that ourGodunov method, labeled G1 in the table, is indeed first order accurate. The con-vergence rates given by (54), labeled G2, are slightly smaller than 2 due to theminimum-modulus slope limiter and the presence of local extrema in the solutionwhere the method reduces to first order. If the minmod function in (51) is replacedby an average, then second-order accuracy is achieved as indicated by the conver-gence rates in the last column of the table. For this problem no shocks or contactdiscontinuities appear, but in general the slope-limited method would be preferredin order to suppress numerical oscillations when these sharp features occur.

Finally, it is interesting to compare the present Godunov method with alternate

40

0 0.2 0.4 0.6 0.8 1

0.6

0.7

0.8

0.9

1

α−

x

0 0.2 0.4 0.6 0.8 1

100

105

110

115

120

125

ρ−

x

0 0.2 0.4 0.6 0.8 1

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

ρ

x

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

v−

x

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

v

x

0 0.2 0.4 0.6 0.8 10

50

100

150

200

p−

x0 0.2 0.4 0.6 0.8 1

3

3.5

4

4.5

p

x

Fig. 15. First-order (blue) and second-order (red) numerical solutions at t = 0.15 ofthe Riemann problem with left state given in (58) and right state given in Table 3.The exact solution for αL = 1 is shown in black and the position of the solid contactis indicated by the dashed line.

41

Table 4Numerical errors and convergence rates for the first-order, G1, and second-order,G2, methods.

G1 G2 G2 (no limiter)

N EN rN EN rN EN rN

100 1.05× 10−2 1.12× 10−4 3.56× 10−4

200 5.31× 10−3 0.98 3.14× 10−4 1.83 9.75× 10−5 1.87

400 2.75× 10−3 0.95 9.16× 10−5 1.78 1.71× 10−5 1.92

800 1.38× 10−3 1.00 2.72× 10−5 1.75 6.61× 10−6 1.97

methods available in the literature. One such method, discussed in [18], employsan HLL approximate Riemann solver to determine the conservative contributionto the numerical flux functions. The discretization of the non-conservative terms,on the other hand, is guided by the condition that constant velocity and pressureis maintained exactly in the numerical solution of free-streaming flow. Both first-order and second-order methods are described in [18], but we will focus on thesimpler first-order method for purposes of comparison. The first-order method maybe written in the form of (33), but with flux functions FL and FR given by

FL(Unj−1,U

nj ) = FHLL(Un

j−1,Unj )− h(Un

j−1)

(λ+αn

j−1 − λ−αnj

λ+ − λ−

)

and

FR(Unj−1,U

nj ) = FHLL(Un

j−1,Unj )− h(Un

j )

(λ+αn

j−1 − λ−αnj

λ+ − λ−

),

where

FHLL(Unj−1,U

nj ) =

λ+f(Unj−1)− λ−f(Un

j ) + λ+λ−(Unj −Un

j−1)λ+ − λ−

has the form of the usual HLL flux function [27] with approximate wave speedstaken to be

λ+ = maxk=j−1,j

0, vnk + an

k , vnk + an

k, λ− = mink=j−1,j

0, vnk − an

k , vnk − an

k.

We shall refer to this method as GHLL. Another method, described by Andrianov,Saurel and Warnecke in [19], determines the conservative contribution to the nu-merical fluxes based on the exact solution of a linearized Riemann problem. Thediscretization of the non-conservative terms is guided again by a condition based

42

0 0.2 0.4 0.6 0.8 1

0.3

0.4

0.5

0.6

0.7

0.8

α−

x

(a)

0.3 0.4 0.5 0.6 0.7 0.80.9

0.95

1

1.05

1.1

α−

p−

(b)

0 0.2 0.4 0.6 0.8 1

0.94

0.96

0.98

1

1.02

1.04

1.06

ρ−

x

(c)

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

x

ρ

(d)

Fig. 16. Numerical solutions at t = 0.2 of the Riemann problem with left and rightstates given in Table 1. The curves and marks are given by methods GHLL (blue),GASW (green) and G1 (red) all with 200 grid cells. The exact solution appears inblack.

on preserving free-streaming flow. A version of this method takes the form of (33),and uses flux functions

FL(Unj−1,U

nj ) =

f(u∗(Un

j−1,Unj ))−h(Un

j−1)αnj if vn

c,j−1/2 < 0,

f(u∗(Unj−1,U

nj ))−h(Un

j−1)αnj−1 if vn

c,j−1/2 > 0,

and

FR(Unj−1,U

nj ) =

f(u∗(Un

j−1,Unj ))−h(Un

j )αnj if vn

c,j−1/2 < 0,

f(u∗(Unj−1,U

nj ))−h(Un

j )αnj−1 if vn

c,j−1/2 > 0.

Here, we have replaced a linearized Godunov flux with the exact one given byf(u∗(Un

j−1,Unj )), but the numerical approximation of the non-conservative terms

follows that described in [19]. This latter method will be referred to as GASW.

Figure 16 provides a comparison of the numerical solutions given by the first-ordermethods GHLL, GASW and G1 of the Riemann problem with left and right statesgiven in Table 1. Figure 16(a) shows the behavior of α versus x at t = 0.2. Here,

43

we note that both GASW and G1 resolve the solid contact with minimal numericaldissipation while there is significantly more smearing seen in the solution given byGHLL. A closer look at the behavior in the solid contact layer is shown, for example,in the plot p versus α given in Figure 16(b). In this plot, we see that the numericalsolutions given by GHLL and GASW differ from that given by the layer equationsin (55). This is due to the numerical approximation of the non-conservative terms,which is guided by a free-streaming condition for both of these methods. While thiscondition is effective in many cases, it ignores the details of the layer and thus suffersfrom a loss of accuracy in others. The present method, G1, on the other hand, is inbetter agreement with the layer solution. The effect of the disagreement through thesolid contact layer may be seen in Figures 16(c) and 16(d) where we plot numericalsolutions for ρ and ρ, respectively, versus x. Aside from the significant smearingin the solution given by GHLL, we note an error in the states on both sides of thesolid contact as determined by GHLL and GASW. The example shown in Figure 16provides an indication of the effect of the solid contact layer on the numericalsolution, and the accuracy of the numerical approach presented in this paper. Ingeneral, we find that the present numerical approach shows similar accuracy in thelayer for any solution configuration, and for cases with a large jump in the volumefraction through the layer it gives equal or better accuracy than the solutions givenby GHLL or GASW for all test cases we have considered.

8 Conclusions

We have considered the structure of the Riemann problem for the Baer and Nun-ziato equations modeling compressible two-phase flow (without exchange sourceterms). The solution consists of shocks, rarefactions and contact discontinuities inthe various characteristic fields of the separate phases with the coupling betweenphases being confined to a (infinitesimally) thin region about the solid contact. Inthis thin region, the non-conservative terms in the governing equations contributeand are handled by assuming a quasi-steady thin layer approximation. An analysisof this thin layer results in nonlinear equations that determine the jump in the stateacross the solid contact. These jump conditions agree with those obtained by Embidand Baer [10] using generalized Riemann invariants. A two-stage iterative procedurehas been described to solve the Riemann problem for given left and right states ofthe flow. This procedure makes no initial assumption regarding the wave structureof the flow, but rather computes it as part of the iterative solution procedure. Theiterative procedure was used to compute solutions for a representative case in whichthe left and right states consist of a mixture of the phases, and for special cases inwhich one of the phases vanishes in the left or right states.

The exact solution for the Riemann problem was used to develop a Godunov methodfor the model equations. Here, we considered an integration of the governing equa-tions over a grid cell and used the exact solution of the Riemann problem to evaluateboth the integral of the conservative fluxes and the integral of the non-conservativeterms. The latter integration reduced to a contribution across the solid contact that

44

forms from each cell boundary and was evaluated using the thin layer equations. Thecontribution from the non-conservative terms was combined with the conservativepart of the fluxes to form a first-order approximation of the governing equations.The numerical method was shown to reduce to the standard Godunov method forthe Euler equations in the special case when the phases decouple and was shown tomaintain constant pressure and velocity exactly in the special case of free-streamingflow. An adaptive Riemann solver was presented and used to reduce the computa-tional cost in the calculation of the combined fluxes and a second-order extensionof the Godunov method was developed based on a slope-limiting approach.

Various numerical experiments were performed in order to study the behavior andassess the accuracy of the first-order and second-order methods. Numerical solutionsof the model equations subject to piecewise constant initial data was shown toagree well with the corresponding exact solutions of the Riemann problem. Goodagreement was obtained for cases in which the left and right states consisted of amixture of the phases, and for cases in which one of the phases vanished in theflow. Of particular interest in these flows was the behavior of the numerical solutionnear the solid contact, where the phases couple and the non-conservative terms areimportant. A close study of the numerical results near the solid contact showedthat both the first-order and second-order solutions agreed well with the behaviordetermined by the thin-layer equations. This suggests that the present numericalmethod accurately captures the assumed behavior and jump conditions across thesolid contact layer.

Smooth solutions of the governing equations were considered and solved numericallyin order to verify the order of accuracy of the first-order and second-order methods,and, finally, two other numerical methods of solution suggested by the work in [18]and [19] were compared with the present Godunov scheme. The former scheme em-ploys an HLL approximate Riemann solver to evaluate the conservative contributionof the numerical fluxes while the latter scheme uses an exact solution of a linearizedRiemann problem to determine the conservative portion of the flux, but this wasreplaced by an exact solution of the Riemann problem in our implementation of themethod. Both schemes use a condition based on preserving free-streaming flow inorder to discretize the non-conservative terms. A close study of these two methodsfor a problem involving a moderate jump in α at the solid contact showed that theGodunov method developed here performed better through the solid contact layerwhich resulted in better agreement with the states of the solution on either side ofthe layer.

The numerical method considered here provides a useful foundation for furtherstudies of the full model upon inclusion of the exchange source terms. In addition,the method may be extended to handle more general equations of state and problemsin more space dimensions. These paths are under consideration for future research.

45

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