hybrid flux splitting schemes for numerical resolution of

Hybrid Flux Splitting Schemes for Numerical Resolution

of Two-Phase Flows

Tore Flatten

NTN U

□Thesis submitted for the Degree of

Doktor Ingeni0r

Department of Energy and Process Engineering Norwegian University of Science and Technology

September, 2003

Cover image “20th century ruin” is Copyright (c) 2000 by Johannes Ewers, reprinted with kind permission from the artist.

It was the winning entry of the January-February 2000 Internet Ray Tracing Competition

http://www.irtc.org/.

The topic of the competition was “Ruins”.

Yo soy un hombre sincero De donde crece la palma,Y antes de morirme quiero Echar mis versos del alma.

Mi verso es de un verde claroY de un carmin encendido,

Mi verso es un ciervo heridoQue busca en el monte amparo

Con los pobres de la tierra Quiero yo mi suerte echar.

El arroyo de la sierra Me complace mas que el mar.

Jo&6 Martf, 1891.

Abstract

This thesis deals with the construction of numerical schemes for approximating solutions to a hyperbolic two-phase flow model. Numerical schemes for hyperbolic models are commonly divided in two main classes: Flux Vector Splitting (FVS) schemes which are based on scalar computations and Flux Difference Splitting (FDS) schemes which are based on matrix computations. FVS schemes are more efficient than FDS schemes, but FDS schemes are more accurate. The canonical FDS schemes are the approximate Riemann solvers which are based on a local decomposition of the system into its full wave structure.

In this thesis the mathematical structure of the model is exploited to construct a class of hybrid FVS/FDS schemes, denoted as Mixture Flux (MF) schemes. This approach is based on a splitting of the system in two components associated with the pressure and volume fraction variables respectively, and builds upon hybrid FVS/FDS schemes previously developed for one-phase flow models.

Through analysis and numerical experiments it is demonstrated that the MF approach provides several desirable features, including

• Improved efficiency compared to standard approximate Riemann solvers

• Robustness under stiff conditions

• Accuracy on linear and nonlinear phenomena.

In particular it is demonstrated that the framework allows for an efficient weakly implicit implementation, focusing on an accurate resolution of slow transients relevant for the petroleum industry.

Acknowledgements

I would like to thank my supervisor, professor Ole Jprgen Nydal, for inviting me to work as his PhD student at the department of Energy and Process Engineering, the university of Trondheim, for the period October 2000 - October 2003. I am also grateful to the Norwegian Research Council for providing the funding.

Kristin Falk of ABB Corporate Research offered valuable advice and in particular words of encouragement when they were most needed. In addition, I am grateful to ABB Corporate Research for inviting me as a visiting student to their facilities in Oslo for the period June-August 2001. Those were days of inspiration during which much of the foundation for this work was made.

Kjell Rare Fjelde of RE Rogaland Research unselfishly offered encouragement, hospitality and much appreciated assistance in guiding me in the right direction. Hopefully I will have the pleasure of working more closely with you academically one day, my old friend.

I have enjoyed discussions, advice, encouragement and constructive critisism in various combinations from Jan Addicks, Vidar Alstad, Matthieu Audibert, Better Andreas Berthelsen, David Bouvier, Cato Christiansen, Gael Chupin, Gisle Otto Eikrem, Olivier Favre, Morten Hammer, Bin Hu, Pascal Klebert, Olav Kristiansen, Kjetil Lilleby, Ole Jprgen Nydal, Marie-Laure Olivier, Fa- bien Renault, Even Solbraa, Espen Storkaas, Sivert Vist, Trygve Wangensteen and Tor Ytrehus.

Steinar Evje of RE Rogaland Research played an integral part in shaping the ideas and results of this thesis. With his unique skills and dedication, Steinar has served as my mentor and inexhaustible source of inspiration.I have enjoyed our collaboration immensely.

Tore Flatten

Summary

This thesis consists of an introductory part accompanied by five papers. The introductory part motivates the papers and introduces the underlying modelling and mathematical concepts. In addition, the main results and conlusions are given.

The main part of the thesis consists of the following papers:

• Paper I: Hybrid Flux-Splitting Schemes for a Common Two-Fluid Model, Steinar Evje and Tore Flatten. Journal of Computational Physics 192(1):175-210.

• Paper II: A Mixture Flux Approach for Accurate and Robust Resolution of Two-Phase Flows, Tore Flatten and Steinar Evje. Preprint.

• Paper III: Weakly Implicit Numerical Schemes for the Two-Fluid Model, Steinar Evje and Tore Flatten. Submitted to SIAM Journal on Scientific Computing.

• Paper IV: Comparison of Various AUSM Type Schemes for the Two- Fluid Model, Tore Flatten and Steinar Evje. Preprint.

• Paper V: CFL-Free Numerical Schemes for the Two-Fluid Model, Steinar Evje and Tore Flatten. Submitted to Journal of Computational Physics.

Introductory Part

Contents

1 Introduction 11.1 Background................................................................................... 11.2 Objective of the Thesis................................................................. 2

2 Systems of Conservation Laws 42.1 Basic Definitions.......................................................................... 42.2 Variable Transformations.............................................................. 5

2.2.1 The Canonical Transformation........................................ 52.3 Classifications................................................................................ 7

2.3.1 Scalar Equations.............................................................. 82.3.2 Wave Propagation in Diagonalizable Systems............... 9

2.4 Discontinuous Solutions .............................................................. 112.4.1 The Riemann Problem.................................................... 11

3 Two-Phase Flow Models 153.1 The Euler Equations.................................................................... 15

3.1.1 Eigenstructure of the model........................................... 163.2 The Two-Fluid Model ................................................................. 16

3.2.1 Conservation of Mass....................................................... 173.2.2 Conservation of Momentum........................................... 173.2.3 Conservation of Energy.................................................... 19

3.3 Other Models................................................................................ 203.3.1 The Saurel-Abgrall Model.............................................. 203.3.2 The Drift-Flux Model .................................................... 22

4 Numerical Schemes 244.1 Conservative Schemes.................................................................... 24

4.1.1 Semi-Discrete Schemes.................................................... 244.1.2 Fully Discrete Schemes.................................................... 254.1.3 Numerical Flux Functions.............................................. 27

4.2 Approximate Riemann Solvers.................................................... 294.2.1 Godunov’s Method........................................................... 304.2.2 Rough Godunov Schemes .............................................. 314.2.3 The Roe Scheme.............................................................. 32

4.3 Hybrid FVS/FDS Schemes........................................................... 324.3.1 AUSM Schemes.................................................................. 34

i

11 CONTENTS

4.3.2 The Mixture Flux Method............................................. 374.4 Higher Order Accuracy.....................................................................444.5 Boundary Treatment........................................................................45

4.5.1 Characteristics and Compatibility Relations.....................464.5.2 The Ghost Cell Approach.................................................. 46

5 Comments on the Papers 475.1 Paper I................................................................................................. 475.2 Paper II.............................................................................................. 485.3 Paper III.............................................................................................. 495.4 Paper IV.......................................................................................... 505.5 Paper V.......................................................................................... 525.6 Conclusions and Further Work.................................................... 52

5.6.1 AUSM type schemes for the Two-Fluid Model................ 525.6.2 Mixture Flux Methods for Two-Phase Flows.....................535.6.3 Extensibility to Other Models............................................ 54

A Pressure Gradients 55A. l Hydrostatics.................................................................................. 56

A. 1.1 Liquid Phase ................................................................... 56A. 1.2 Gas Phase......................................................................... 58A. 1.3 An Equivalent Formulation .......................................... 58

B Eigenstructure Analysis 60B. l The Equal Pressure Model.......................................................... 60

B. 1.1 Quasilinear Form............................................................. 62B.l. 2 Eigenvalues...................................................................... 62B.l.3 Eigenvectors...................................................................... 64

B. 2 An Extended Model ................................................................... 65B.2.1 Eigenvalues...................................................................... 66B.2.2 Some Remarks ................................................................ 68

C Entropy and Energy Equations 70C. l The Two-Phase Energy Equations.............................................. 70C.2 Eigenstructure of the Full Model .............................................. 71

Chapter 1

Introduction

In this chapter a brief introduction to the topic of the thesis is given. We provide some background for the research project, and briefly state the objectives of the thesis. Then in Chapter 2 we define some basic mathematical concepts related to partial differential equations. In Chapter 3 different two- phase flow models are described in some detail. In particular the two-fluid model that will be the focus of this thesis is described. In Chapter 4 we describe numerical solution algorithms for hyperbolic systems in general, and the two-fluid model in particular. In Chapter 5 the papers are commented. In Section 5.6 we summarize the results of this thesis and dicuss some possible directions for further work. Finally, some mathematical properties of the two-fluid model are discussed in the appendices.

1.1 B ackgroundIn order to maintain a high level of oil production in Norway, it becomes increasingly more important to devise new solutions for optimizing the oil production of existing fields [24]. Starting in the year 2000, ABB, Norsk Hydro ASA, and 5 departments at the Norwegian University of Science and Technology (NTNU) received funding from the Norwegian Research Council (NFR) in order to jointly run a doctoral research programme (PETRONICS) with focus on optimized production and automatic control of oil wells and fields.

An important aspect of the programme is the understanding of and ability to simulate the behaviour of oil and gas flow in production pipelines. Multiphase flow is a complex phenomenon, where depending on the physical parameters of the flow, several different flow regimes [91] may occur. Flow regimes are commonly divided into separated (stratified, annular) and mixed (bubbly, dispersed) flows.

A flow regime of particular interest for pipeline operations is the slug flow regime [4, 6]. It is characterised by a sequence of liquid “slugs” (i.e. one-phase liquid regions) separated by large gas bubbles. The phenomenon of severe slugging, occuring when the gas flow rate is not sufficient to continuously lift

1

2 Chapter 1. Introduction

the liquid up the riser, is a significant cause of strain for the receiving process facilities [12, 31].

Several strategies for reducing the severe slugging problem have been tried. Examples include seabed separation [72], gas injection [29], and choking [69, 46]. One of the main purposes of the PETRONICS project is to develop methods for automatic control of transport systems and wells. So far, the project has yielded several promising results [76, 75, 16, 33].

1.2 Objective of the ThesisResearch in the field of multiphase flow may be divided in two parts:

i) Modelling. The derivation of appropriate mathematical models describing the time and space development of physical variables (pressure, volume fractions, velocities, temperatures etc.) for the physical system. These models may be mathematically derived from first principles (conservation of mass, energy etc) as well as empirically correlated. In particular, several closure relations, like friction terms and terms representing momentum exchange between the phases, must be spesified. These closure relations present the main difficulties in the model formulation.

ii) Simulation. The development and implementation of numerical solution algorithms for the models. In particular, for the application of automatic control, it is important that the solution algorithm is sufficiently efficient to provide real-time information, sufficiently accurate to correctly predict developing instabilities, and sufficiently robust to operate reliably.

The focus of this thesis is on point ii) above.A fundamental approach towards establishing a physical model for multi

phase flow would be to start with a single continuous description of each phase given by the Navier-Stokes equations. However, the presence of topologically complicated interfaces makes this approach difficult and various averaging techniques are commonly used to achieve more practical models.

For the description of two-phase flow of gas and liquid, the most common model resulting from such averaging techniques is the two-fluid model, involving conservation equations for mass, momentum and energy for each fluid. Industrial application of this model was initiated by the nuclear industry, who developed numerical codes like TRAC [84], RELAP [63] and CATHARE [3]. An overview of such early schemes is given in [85].

Later, similar computer codes like OLGA [5] and PeTra [41] were developed for the petroleum industry. Traditional usage of these computer models is for the design of pipelines and top side facilities [92].

A simpler version of the two-fluid model, termed the drift-flux model, arises from adding the two momentum conservation equations obtaining a

1.2. Objective of the Thesis 3

mixture momentum equation. In addition, an algebraic slip relation must be spesified for obtaining the gas and liquid velocities from the mixture momentum. A commercial computer simulator termed TACITE [60, 59] has been developed, based on the drift flux model. In addition, the kick simulator RF-Kick [88] of RF-Rogaland Research is based on the drift-flux model.

In this thesis, we focus on the two-fluid model. As applications turn towards automatic control, the performance of the numerical methods become of ever greater importance. The goal of this thesis is to construct accurate, efficient and robust numerical schemes that may be used for real-time simulations.

Towards this aim, ideas developed by Lion et al [49, 47, 89] for a one-phase flow model are refined and adapted to the two-fluid model. To validate the methods, numerical simulations are performed on benchmark tests known from the literature.

Chapter 2

Systems of Conservation Laws

In this chapter, a compact description of the most important concepts relevant for the understanding of partial differential equations in general, and hyperbolic systems in particular, is given. A more complete and pedagogical presentation may be found in [45, 80].

Basic concepts and properties are defined in Sections 2.1-2.3. In particular we discuss how information propagation in the system can be described in terms of characteristic, or canonical, variables.

Hyperbolic systems have the property of supporting discontinuous solutions, for which the differential formulation breaks down. In this case, care must be taken to define exactly what we mean by a “solution”. This issue is discussed in Section 2.4. The concepts introduced in this chapter are essential for our later discussion of numerical schemes (Chapter 4).

2.1 Basic DefinitionsA system of conservation laws can generally be written on the following conservative form

where U = (ui, u2,.. is the vector of conserved variables (e.g. mass,energy, momentum), F = (fi, /2,..., /at)t is the vector of fluxes and Q = (<7i, <?2,..., Qn)t is the vector of sources.

By introducing the N x N jacobi matrix of F given by9F(U)

A(U) =

the system (2.1) may be rewritten as9U

fr+a(u)^ = q(u).

(2.2)

(2.3)

The form (2.3) is termed the quasi-linear formulation of the system, because of the resemblance to a fully linear system

(2-4)

4

2.2. Variable Transformations 5

where S and A are constant matrices.

2.2 Variable TransformationsAlthough a conservation law is mathematically most naturally expressed in terms of the quantities U that are physically conserved, it may sometimes be convenient to express the system in terms of other primitive variables W that are more easily physically observed, like pressures and velocities instead of masses and momentums.

We consider a variable transformation U H- W where we define«9Uow ~ ‘ (2.5)

Applying this transformation to (2.3) we obtain

B^ + AB^ = Q.% or

(2.6)

Now multiplying on the left by B-1 we recover the quasilinear form in W

7+a(wC=qiw>’ (2.7)

whereA = B_1AB (2.8)

andQ = B_1Q. (2.9)

Remark 1. The formulations (2.3) and (2.7) are mathematically equivalent for smooth solutions.

2.2.1 The Canonical TransformationA special kind of variable transformation gives rise to the concept of characteristics, which is fundamental for the understanding of the mathematical properties of hyperbolic systems.

Jordan Canonical Form

We let R be a matrix that reduces A to its Jordan canonical form through a similarity transformation as follows

J = R_1AR. (2.10)

The Jordan matrix J may be expressed on the block form

J\ 0 • • •0 J2 • • •

J = . .

0 0 • • •

00

JK

(2.11)

6 Chapter 2. Systems of Conservation Laws

where each of the K Jordan blocks J* are given by

' Xi 1 0 ••• 0 0 "0 Ai 1 0 ••• 0

(2.12)

0 0 ............ Aj 10 0 ............ 0 Xi

An essential property of the Jordan canonical form is that it is unique, up to permutation of the blocks [50].

Canonical Form of the Equation System

Multiplying (2.3) on the left by R 1 we obtain

R_1f+ JR_1f = Tu). (2.13)

whereQ = R Q (2.14)

Introducing the canonical variables

dC = R1 dU, (2.15)

we can write (2.13) as^ = 3(0- (2.16)

C haracter ist ics

We observe that each Jordan block Ji has exactly 1 row (the last) consisting of only the diagonal entry A*. Hence there are exactly K such rows in the Jordan matrix (2.11). From (2.16), we find that these rows yield the following diagonalized componentwise equations

q- (2.17)

The variables q that satisfy (2.17) are termed characteristic variables. We may now state the following lemma

Lemma 1. Along the curves 7, in the {x,t)-plane, given by

d'T -̂ = Aj(U(iq)), (2.18)

denoted as characteristic curves, the characteristic variables satisfy the ordinary differential equation

dq (2.19)

2.3. Classifications 7

Proof.

dt dt dx dt dt 1 dx ^□

The importance of Lemma 1 lies in the fact that only the source terms % affect the characteristic variable c* along the path xt. Hence xt is the path of propagation of q, and the velocity of propagation is A,.

By (2.15), we can express the characteristic variable q in terms of conservative variables as follows

dci = R”1 dU, (2.21)

which means that (2.19) can be written as a compatibility relation in terms of the conservative variables

_xdXJ_ 'l dt «.(U) (2.22)

along the characteristic

Remark 2. Compatibility relations are often used in conjunction with the numerical treatment of boundary conditions, see also Section f.5.

Following Toro [80], we now make the following definitions:

Definition 1. The variables carried along the characteristic corresponding to Ai constitute a characteristic field, the Xi-field.

Definition 2. A A*-characteristic field is said to be linearly degenerate if

VuVR*(U) = 0 V U e Rw, (2.23)

where R* is the right eigenvector corresponding to A*, and Vu is the operator

Vu 4> = (2.24)

Definition 3. A A*-characteristic field is said to be genuinely non-linearif

VuAi-Ri(U) #0 VUe Rn. (2.25)

2.3 ClassificationsThe uniqueness of the Jordan canonical form (2.16) allows us to classify first- order systems of partial differential equations by the structure of the Jordan matrix J. More precisely, we have


Definition 4. A system (2.3) is said to be hyperbolic for the state U if the matrix A(U) (2.2) is diagonalizable in the real numbers. A system (2.3) is said to be strictly hyperbolic for the state U if in addition, all the eigenvalues o/A(U) are distinct.

Definition 5. A system (2.3) is said to be elliptic for the state U if the matrix A(U) (2.2) is diagonalizable, and none of the eigenvalues of A(U) are real In general, if A(U) has both real and complex eigenvalues, the system is said to be of mixed elliptic-hyperbolic type for the state U.

Definition 6. A system (2.3) is said to be parabolic if the Jordan form J of the matrix A(U) does not have any non-degenerate lxl Jordan blocks J% = [A*]. In general, if A(U) has both non-degenerate and degenerate real eigenvalues, the system is said to be of mixed parabolic-hyperbolic type. If A(U) also has complex eigenvalues, the system may even be of mixed elliptic-parabolic-hyperbolic type.

Remark 3. For a diagonalizable system, all the Jordan blocks are simply J± = [Aj] and K = N. For a hyperbolic system, all effects modelled are necessarily local — disturbances flow along characteristic curves with velocities A*.

2.3.1 Scalar EquationsIt is instructive to discuss how the above classifications are related to the common classifications of scalar equations. We consider the second-order equation

allAA + W/(!'y) + + ddffv) + edffy) = F(X, y).dx2 dxdy dy2

Now introducing the variables

and

=

if =

df{x,y)

df(x,y)

(2.26)

(2.27)

(2.28)

the equation (2.26) can be written as a system of first order equations as follows

' 0 -1 ' 9U ' 1 0 ' 9U _ 0 "00"a 0 iW _ b c #3/ . -F(%, 2/) . d e U,

whereU = if

(2.29)

(2.30)

The system (2.29) can be written in more compact quasilinear form as follows

2.3. Classifications 9

where' 0 -1 ' -1 "10" 5/a c/a

p o b c 1 O

The eigenvalues of A are

b ± y/b2 — 4ac 2a

with corresponding eigenvectors

to± 1—

(2.32)

(2.33)

(2.34)

Hence we may classify the equation (2.26) in terms of its discriminant b2—4ac as follows

• b2 — 4ac >0: A is diagonalizable in the real numbers and the system is hyperbolic by Definition 4.

• b2 — 4ac <0: A is diagonalizable in the complex numbers and the system is elliptic by Definition 5.

• b2 — 4ac = 0: A is not diagonalizable as it does not have a base of linearly independent eigenvectors. The system is parabolic by Definition 6.

The analogy with the conic sections of the same names is now apparent.

2.3.2 Wave Propagation in Diagonalizable SystemsParabolic equations typically represent a smooth spreading of an initial disturbance, caused by diffusion or viscosity. As we have seen in the previous section, hyperbolic equations represent a propagating disturbance moving along characteristic curves in the (x, t)-plane. What happens in the elliptic case, when the characteristics are complex, is not so intuitive.

In fact, for a system with elliptic components, the temporal evolution U(or, t) from initial data U(x, 0) may not be calculated as a unique, well- behaved mathematical solution.

A linear analysis may shed some light on this issue and provide some intuition into the concept of complex characteristics. Loss of hyperbolicity, by the occurence of complex eigenvalues, has been discussed more generally by Lax [43, 44].

Lemma 2. We consider the linear equation

f + A?H = SU.ot ox (2.35)

Let initial data U (x, 0) for x € M be given as a sum of Fourier components

U(z,0) = ^LW=*. (2.36)k


ThenU(z, t) = 5] gWtA+sXy^i^ (2.37)

kprovided the sum exists, is a solution of (2.35) for the given initial data.

Proof

dU d\J "aT + A9? (2.38)

53 (~ikA + S) e(-ikA+s)t^jkeikx + A^2ike(-ikA+s)ttikeikxk k

^2Se(-ikA+S)t-(jkeikx = gyk

□

We now writeu (x,t) = 53u*om)

k(2.39)

whereU*(z,t)=e^LW*= (2.40)

and= —ikA T R. (2.41)

We assume that can be diagonalized by

H k = P-1AfeP. (2.42)

ThenXJk(x,t)=J>-1eAktI>XJk(x,0). (2.43)

The system is stable if all the real parts of the eigenvalues of H& are negative, corresponding to exponentially decaying modes.

Long Wavelength Limit

In the limit k —»■ 0 we find = R. Hence in the long wavelength limit onlysource terms affect the dynamics.

Short Wavelength Limit

As k —> oo, Hfe —y —ikA. Now we separate between two cases:

1. A has only real eigenvalues (the hyperbolic case). Then all eigenvalues of Hfc will be purely imginary, and the waves will propagate without change in magnitude.

2.4. Discontinuous Solutions 11

2. A has some complex eigenvalues (the elliptic case). By A £ M.NxN, this implies that A has at least one pair of complex conjugate eigenvalues. Then Hfc will have at least one eigenvalue with a positive real part, and the system will be unstable. What is worse, as k —> oo at least one of the eigenvalues of will have a real part that tends to positive infinity. Hence the system will be infinitely unstable for infinitely small wavelengths, independent of the value of the source terms.

2.4 Discontinuous SolutionsA feature particular to the hyperbolic class of systems is that they allow fordiscontinuous solutions. As follows from the analysis in Section 2.2.1, an initial discontinuity may simply propagate along characteristic curves without smearing. For nonlinear systems, discontinuous solutions (shock waves) may arise even if the initial data are smooth.

The derivative does not exist across a discontinuity, hence the differential formulation (2.1) locally breaks down and we must find a way to generalize our concept of a solution. For this purpose we introduce an integral formulation. Choosing a region V = [x,, x?] the integral form of (2.1) reads (assuming Q = 0)

U(x,t) dx = F(U(^i,f)) - F(U(x2,t)). (2.44)

The differential and integral formulations are equivalent if the data U(x) are smooth. If not, the integral formulation must be applied to obtain a solution in the classical sense. Also, such weak solutions may not be unique, and we have to impose additional conditions to pick out the “correct” physical solution.

Often, conservation laws represent simplified physical systems where diffusive effects like viscosity are neglected. The natural way to define the correct weak solution would then be to consider a more fundamental model with viscous forces included, and consider the limit where the viscosity tends to zero.

A more practical and common approach is to use a selection principle based on entropy considerations. We will briefly discuss these issues below.

2.4.1 The Riemann Problem

A conservation law with initial constant data separated by only a single discontinuity is known as the Riemann problem. That is, we consider

for x < x0,for x > xq. (2.45)


The Scalar Case

In the event that U is a scalar variable, the solution to the Riemann problem has a particularly simple structure provided that F(U) is strictly convex (or equivalently, strictly concave). That is, we assume that for X(U) given by

A([/)

the derivative X'(U) does not change sign.

(2.46)

Remark 4. Note that the assumption of convexity is the scalar equivalent of genuine nonlinearity (Definition 3).

According to the Lax entropy principle [45], the two possible solutions to the scalar strictly convex Riemann problem are

• Shock wave. If A(%) > X(U-r). the entropy satisfying weak solution is a shock wave given by

[7l for x < st,[7-r for x > st (2.47)

where s is the shock speed given by the Rankine-Hugoniot condition

F(%) - F(%) = s(% - %). (2.48)

• Rarefaction wave. If A(Z7l) < A([/r). the entropy satisfying weak solution is a rarefaction wave given by

( UL for x < X(UL)t,U(x,t) = < H(x/t) for A(UL)t < x < X(t/R)t, (2.49)

{ Un for x > X(Un)t

where H{x/t) solvesF'(#(z/t)) = z/L (2.50)

Remark 5. Note that the structure of the rarefaction wave depends only on the parameter x/t, and not upon x and t independently.

For a flux function that is neither convex nor concave, the solution will generally be a combination of a shock and a rarefaction. Shocks and rarefactions are elementary building blocks also for solutions to Riemann problems for systems.

Remark 6. The Rankine-Hugoniot condition (2.48) is not invariant under the variable transformations described in Section 2.2. Although the mathematical structure of the differential equations is invariant under such transformations, this is not the case for the weak solutions to the integral equations. Different choices of variables lead to different shock speeds s from the Rankine-Hugoniot conditions.

It is therefore clear that a solution scheme based on variables other than the conserved variables U will not converge to the correct weak solutionswAen ore present /#<!/.

2.4. Discontinuous Solutions 13

t

Figure 2.1: Schematic representation of the wave structure associated with the general Riemann problem. Here A% and Xp represent shocks or contact discontinuities, whereas A2 and Aat are associated with rarefaction fans.

Nonlinear Systems

We now consider a strictly hyperbolic system of conservation laws where U and F are iV-vectors. The solution to the Riemann problem (2.45) consists of N + 1 constant states separated by N elementary waves corresponding to the characteristic fields. If the Ay-field is genuinely non-linear, the wave connecting the states Up_i and Up is either a shock or a rarefaction, where a shock is a valid physical solution if the generalized Lax entropy principle

Ay(U„_i) > Sp > A„(U„) (2.51)

is satisfied.

Definition 7. If the Xp-field is linearly degenerate by Definition 2, the resulting weak solution is termed a contact discontinuity. Then A& = Aji, and characteristic curves neither enter nor leave the discontinuity (across which Xp is constant).

The general solution to the Riemann problem may now be obtained by the appropriate combination of shocks, contacts and rarefactions. A schematic illustration of the solution is given in Figure 2.1.

The following theorem, the proof of which may be found in [70], states that under certain conditions (in particular that ||UR — Ul|| is sufficiently small) a solution to the Riemann problem is guaranteed to exist.

Theorem 1. Let Ul E Af, where Af C RA'. and suppose that the system (2.45) is hyperbolic and that each characteristic field is either genuinely nonlinear or linearly degenerate in Af. Then there is a neighbourhood Af C Af of UL such that if UR E Af, the Riemann problem (2.45) has a solution. This solution consists of at most (N +1) constant states separated by shocks, rarefaction waves or contact discontinuities. There is precisely one solution of this kind in Af.


Linear Systems

A special case of interest is the linear Riemann problem

UL for x < xq, UR for x > xQ (2.52)

where A is a constant matrix. The solution to this problem consists only of contact discontinuities and can be easily computed. Furthermore, practical methods to compute solutions to non-linear Riemann problems are commonly based on some linearization procedure. We will return to this in Chapter 4.

To solve (2.52), we first transform to characteristic variables (see Section 2.2.1), obtaining the composition and velocities of the elementary waves directly. Then we transform the solution back to conservative variables. The jump across the discontinuity may be decomposed as

N

(2.53)p= i

where Vp is component p of the vector

V = R"1(Ul-Ue) (2.54)

and Rp is the pth eigenvector of A (i.e. the pth column of R). The solution may now be expressed as

u (x,t) = uL + 53P | (x—Xpt)>0

= Ur - 53P | (x-Xpt)<0

(2.55)

Chapter 3

Two-Phase Flow Models

For deriving mathematical models for multiphase flow phenomena, a natural approach would be to start with the Navier-Stokes equations for each phase, supplied with jump conditions across the interfaces. Ideas along this line have been developed by Ishii [34] and others.

However, such local instant approaches easily lead to complex and intractable models and few applications currently exist. Instead, the common approach is to apply different averaging techniques to obtain simpler and more practical models.

In particular, we are often interested in modelling large-scale features of two-phase flow phenomena in pipelines. Here the dynamics of interest are commonly associated with the direction of the flow, and it is desirable to obtain one-dimensional averaged equations.

In this chapter, we describe some common such one-dimensional models. The averaging procedures leading to these models are described in more detail by Ishii [34].

3.1 The Euler EquationsThe two-phase flow models we are considering can be viewed as natural extensions of the Euler model for gas dynamics, which has been extensively studied in the literature (see [45, 80] and the references therein).

The Euler equations will be briefly described in this section. This will provide the basis for introducing the different two-phase flow models in the following sections.

The model consists of conservation of mass, energy and momentum for one-dimensional flow of a single fluid, described by an equation of state

P = p(p,T). (3.1)

More precisely, we have

• Conservation of mass:

(3.2)

15

16 Chapter 3. Two-Phase Flow Models

• Conservation of momentum:

(pv) + (pv2 +p) = Qm (3.3)

• Conservation of energy:

TF + iL{viE+p)) = QE- (3-4)

Here p is the density of the fluid, p is the pressure, v is the velocity and E is the total energy

E = ^pv2 + pe, (3.5)

where e = e(p, p) is a thermodynamic variable, the spesific internal energy. The variables Qm. and Qe represent the non-differential momentum and energy source terms.

3.1.1 Eigenstructare of the modelWriting the model on quasilinear form and performing an eigenvalue analysis, we find the characteristic wave velocities to be [45]

Ap = v ± c, (3.6)

andA* = %. (3.7)

Here c is the sound velocity

where the compressibility is evaluated at constant entropy.The eigenvalues Ap represent sonic waves carrying rapid pressure vari

ations. The associated characteristic fields are genuinely non-linear in the sense of Definition 3.

The eigenvalue Aa represents density and temperature variations that are carried along with the flow under constant pressure. The associated characteristic field is linearly degenerate in the sense of Definition 2.

3.2 The Two-Fluid ModelWe now proceed to describe a rather standard extension of the Euler model to two separate fluids, yielding the model commonly denoted as the two-fluid model [74].

Here each phase is treated separately in terms of three sets of conservation equations; conservation of mass, energy and momentum for each phase. More precisely, for a gas (g) and a liquid (1) phase, we have the following

3.2. The Two-Fluid Model 17

Conservation of Mass0 d^ (/>g#g) + ^ (pg#g%) = r, (3.9)

^ (a«0 + ^ {pmvi) = -r. (3.10)

Here pk and % are respectively the density and velocity of phase k. The parameters are the volumetric phase fractions. They are defnined at each point as the fraction of the total cross sectional area of the pipe that is occupied by the phase k. The phase fractions satisfy

ttg + «i = 1. (3.11)

Moreover, F is a (often negligible) source term representing interphase mass transfer.

3.2.2 Conservation of Momentum

(AWg) + ^ (#Wg + <%) " + ^g = A + (3-12)

— {picxiVi) + — {p\a\v( + QiiPi) — pi-7^- + = f\ + Mf*. (3.13)

Here p* is the pressure of phase k, Pg is the pressure at the gas-liquid interface, and p\ is the pressure at the liquid-gas interphase. Furthermore, represents sources (frictional or gravitational) acting on phase &, represents interfacial momentum exchange terms satisfying M® + Mf* = 0, and 4 represent interfacial forces that contain differential terms.

Here some additional issues are introduced compared to the Euler equations, we will address them in turn.

Pressure Terms

As for the Euler equations, we assume that the pressure can be found by an equation of state

Pk = PkXpki'Tk)- (3.14)

An effect not present in the Euler equations is the non-conservative product p^dxttk. This term represents momentum transfer due to pressure forces on an inclined interface, as in general a* is not constant. This effect is described in more detail in Appendix A.

The pressure p& represents the average pressure of phase k and may differ from the pressure jfk at the interface, due to hydrostatics, surface tension, or related effects. Some approaches for modelling the interface pressure terms are described by Cortes et al [11] and the references therein.

For some applications, hydrostatic pressure gradients in the transversal direction (y-axis) is an important effect, and it deserves special mention. Due to instabilities as the relative velocity between the phases becomes large, a


sudden mixing of the phases (flow regime change) may occur. This phenomenon is often referred to as hydrodynamic slugging and may cause operational problems for two-phase transport pipelines.

Such instabilities are predicted by the two-fluid model when hydrostatic pressure gradients are included. A theoretical analysis has been performed by Barnea and Taitel [2], and numerical investigations have peen performed by Issa et al [35, 90, 36].

Interface pressure correction terms due to hydrostatic gradients are described in more detail in Appendix A.l.

Additional Differential Terms

In the context of (3.12) and (3.13) above, the simplest choices of submodels would be

Pg = P\ = Pg = Pi = P (3.15)as well as

= 0. (3.16)

The biggest problem with such a simplification lies not so much in the neglect of important physical effects, but rather in the fact that this “equal- pressure” model has complex eigenvalues, and therefore elliptic components as described in Appendix B. Hence it makes little or no sense to talk about a mathematical solution to such a model.

However, a numerical solution may still be relatively well-behaved, provided there is a sufficient amount of numerical dissipation. The interpretation of such numerical solutions is questionable [73, 74]. It is well documented in the literature (see for instance [57], as well as paper II of this thesis) that numerical solution schemes will fail to converge to a stable solution for such a non-hyperbolic model.

For this reason, it is common to modify the equal pressure model in some fashion. We have already mentioned hydrostatic pressure gradients. A different approach is to make the simplification (3.15), and instead introduce differential terms 5^ not associated with the pressure. We will briefly describe some such approaches.

• Virtual mass force. Hydrostatic pressure gradients as discussed in Appendix A.l are valid for stratified flow regimes, but are not applicable to mixed flows. On the other hand, for bubbly flows accelerating gas bubbles will induce a momentum transfer between the phases. This effect, termed the virtual mass force, has been analysed by Drew and Lahey [14, 39], who suggested the following expression

dg — — c^gpiCymi^dtiyg Ui) T Vgdx(vg ui) (3.17)+(vg — ui)((A — 2)dxvg + (1 — A)<9xui)),

where A(ag) is a volume-fraction dependent parameter and Cvm is the coefficient of virtual mass. The value of Cvm is 1/2 for non-interacting

3.2. The Two-Fluid Model 19

spheres, whereas for other shapes Cvm <1/2. A discussion of the virtual mass force can also be found in [82].

• Momentum flux parameters. The velocities % that appear in the mass and momentum equations (3.9)-(3.13) are averaged quantities, where the average is taken over the cross-sectional area perpendicular to the flow. Hence for general velocity distributions

^ Vk A 7^ A Vk > ■> (3.18)

but ratherCm < Vk > =< Vfc > (3.19)

for some momentum flux parameter Cki■ In this context, the assumption = 0 amounts to Cj,i = 1. In general, we have

8k — dx ((C&f T)phOikVk). (3.20)

The two-fluid model with momentum flux parameters has been analysed by Song and Ishii [71].

Remark 7. A rather common simplification is to assume 8k equality of the phasic pressures, that is

= 0 as well as

Pg=P\= P1 (3.21)

andPg=Pi= P,

but maintain a pressure jump across the interface

(3.22)

Ap = p — p1 7^ 0. (3.23)

Depending on the size of Ap, the resulting model will be conditionally hyperbolic, as is demonstrated in Appendix B.

This approach is used by [7, 11, 10, 57], among others, as well as the papers included in this thesis. It should be noted that the physical interpretation of such a correction term is debatable [61].

3.2.3 Conservation of EnergyIn their simplest form [9], the energy conservation equations may be expressed as:

ddt

PgttgUg + PgOlgCg +A

ddt

d2 P\a\v\ + Aaiei I + Vi

^PgOigVg + pgOLgCg + (XgPg

~Picx\vf + PiOi\e\ + QiiA J

+Pt _ c%

(3.24), _ c+p>~m ~ Sv

(3.25)


Here e*,{pk,Pk) is the spesific internal energy of phase k, and 5* represents source terms such as wall heat fluxes, interface heat exchange and gravitational energy.

For a full description of the source terms, a large number of submodels (closure models) are needed, to calculate friction factors, interface momentum exchange terms, heat fluxes etc. Such modelling is of highest importantance from a physical point of view. However, from a numerical point of view the exact functional form of the source terms do not normally affect the choice of solution method.

Remark 8. We will not focus on the modelling of non-differential source terms in this thesis, but refer instead to [5] and the references therein.

The Isentropic Model

A common simplification is neglecting the energy conservation equations, reducing the model to a set of four partial differential equations for mass and momentum conservation. As is demonstrated in Appendix C.l, the energy conservation equations are trivially satisfied for flows of uniform entropy. Hence the 4-equation model is often referred to as the isentropic model.

As can be seen from the analysis of Appendix C.2, the energy equations do not add significant complexity to the mathematical structure of the model. This has the following consequence:

Remark 9. Numerical schemes for the isentropic model may often be rather directly extended to the full 6-equation model involving energy equations [83, 57]. For reasons of simplicity and clarity, the studies in this thesis have been made exclusively on the isentropic model.

3.3 Other ModelsAlthough the two-fluid model seems the most natural extension of the Euler equations to a two-phase flow system, it does not correctly represent physical reality for all situations. Furthermore, the need to introduce regularization parameters to obtain a hyperbolic model is unsatisfactory. Several ways of extending or simplifying the two-fluid model have been investigated [53, 15, 58, 8], here we will briefly describe two such approaches.

3.3.1 The Saurel-Abgrall ModelAs is demonstrated in Appendix B, the two-fluid model predicts two pressure waves moving with the velocities

up ± c, (3.26)

where up is a mixture fluid velocity and c is a mixture sound velocity.

3.3. Other Models 21

However, for stratified flows we expect sound waves to exist in each fluid separately, moving with the velocities

vit ± c&, (3.27)

where is the one-phase sonic velocity

Ck (3.28)

Recently, Saurel and Abgrall [67, 68, 40] have proposed a two-phase flow model possessing the following features:

• unconditional hyperbolicity for the entire range of physical parameters,

• eigenvalues of the form (3.27).

For isentropic flow, their two-fluid model can be written as

9 9^ (/>g<%g) + ^ (/%%) = 0 (3.29)

A +A ° (3.30)

(3.31)

^ (Pgag%) + ^ (Pg^^ + agPg) p ^ - /g + %) (3.32)

& (pmvi) + (piaivf + aipi) -p1^ = A - A(m - vg). (3.33)

Here the parameters A and p, determine the relaxation rates of the phasic pressures and velocities. The following definitions for the interface pressure

and velocity v-ml are used

P1 = agPg + «iPi, (3.34)

PgCXgVg + p\CX\V\ yint — :

Pgttg + P\(X\

Compared to the standard two-fluid model, there is an additional degree of freedom in that we allow the phasic pressures to evolve independently. Closure is achieved by the introduction of an advection equation for the volume fraction (3.31).

A straightforward computation yields the following five eigenvalues:

Ag = vg ± cg (3.36)

A^ = v\ ± ci (3.37)

Aa = Vint- (3.38)A further discussion on the properties of the Saurel-Abgrall model may

be found in [40] and the references therein.


3.3.2 The Drift-Flux Model

As we have seen, the Saurel-Abgrall model gives more physically realistic sonic velocities for stratified flows than the two-fluid model. For mixed flows the two-fluid model predicts a mixture sound velocity (see appendix B)

c %Plttg + pgOi\

A<%g +(3.39)

The extremal values of c(ag) are found to be the one-phase limits ag —> 0,1

lim cctg —^0dp#A

(3.40)

and

lim cag-sUdp9/>g

(3-41)

with a monotone increase/decrease in between.This is not in accordance with physical experiments, where the mixture

sound velocity typically satisfies [20]

min(c) « mindp / dp#Pg' y api

(3.42)

This discrepancy is commonly explained by incorrect modelling of the differential terms representing interphasic momentum exchange (see Section 3.2.2). These terms are notoriously difficult to model, and a common approach is to simplify the model so that they cancel out.

We assume equal phasic pressures as stated in Remark 7, and add the gas and liquid momentum conservation equations. We obtain the drift-flux model

^ (/%) + ^ (/%<%) = 0

Qj. (a<t) + fa (AQWi) = o (3.44)

d d— (PgttgUg + f>\a\V\) + — (pgQZgUg + A°W +p) = Qm, (3.45)

which must be supplemented with a non-differential relation (state relation)

F(p, ah vg, ui) = 0. (3.46)

The closure relation (3.46) is commonly denoted as the slip relation.

3.3. Other Models 23

Eigenstructure of the Model

The slip relation (3.46) is commonly expressed as

vg = K (a\V\ + ttg-Ug) + S, (3.47)

where K and S are flow-dependent parameters. Assuming that pgag « p] o, the following approximate eigenvalues may be obtained [21]

Ai — V\ u), Ag — Ug, A3 — T uj (3.48)

wherew = ] (1 - *r«g) (3‘49)

is an approximate sound velocity. It should be noted [20] that for many cases of interest, (3.49) may be a better approximation to the observed sonic velocity than (3.39).

There are other advantages to the drift-flux model besides better modelling of pressure propagation in mixed flows. In particular we note that the model (3.43)-(3.45), as opposed to the two-fluid or Saurel-Abgrall model, can be written on conservative form

dU <9F(U)dt dx

Q(U). (3.50)

Unlike the two-fluid model, the drift-flux model is hyperbolic for most physically relevant flow conditions without additional regularizing terms. Also the slip relation (3.46) may be more directly determined from experiments than the interphasic momentum exchange terms that appear in the two-fluid model.

Remark 10. Both the Saurel-Abgrall model and the drift flux model yield only one wave associated with volume fraction propagation, hence they are expected to be unable to properly model surface waves in stratified flows. When such waves are of importance, e.g. for modelling of hydrodynamic slugging, the two-fluid model is expected to be superior (see [2, 94] and Section 3.2.2).

Regarding more large-scale (long wavelength) phenomena related to mass transport, there is yet no final assessment regarding the applicability of the different models. A comparison between the drift-flux and two-fluid model on transient flow phenomena was performed by Masella et al [53]. It should be noted that for industrial applications, the two-fluid model [5, fl] is currently

Chapter 4

Numerical Schemes

In this chapter, standard numerical schemes for hyperbolic systems of conservation laws are described (see also [45, 80]). In particular the emphasis is on first-order schemes of the upwind class, the properties of which motivate the mixture flux method that is the focus of papers II-V.

In Section 4.1 basic concepts related to stability and consistency of numerical schemes are introduced. In Section 4.2 the generalisation of upwind schemes to systems of conservation laws is discussed. In Section 4.3 we discuss some recent attempts at achieving the benefits of upwind schemes, without having to resort to a full wave structure decomposition of the system. In particular the mixture flux method is motivated and described.

Finally, issues related to extensions to higher order accuracy and implementation of numerical boundary conditions are briefly discussed.

4.1 Conservative SchemesOur starting point is the system of N conservation laws on the form

9U <9F(U) dt dx Q(U). (4.1)

We consider a spatial mesh of size Ax, where each grid cell is indexed by the letter j. By convention, we place the mesh so that the center of grid cell j is located at the position

Xj = jAx (4.2)

and the interface between cell j and j + 1 is located at the position

ab+1/2 = U + l/2)Az. (4.3)

4.1.1 Semi-Discrete SchemesStarting with an integral form of the equations, we first derive an approximation to (4.1) that is discrete in space only. We write (4.1) on the integral

24

4.1. Conservative Schemes 25

formQ rxi+1/2— / U(x, t) cZa: + F(U(^j+i/2, f)) — F(U(xj_i/2,t)) (4.4)OT J*3-1/2rx/+1/2

= / Q(U(a;,Z)) dx.xj-1/2

We define the index j on U to mean the ceZZ average of U

1 rxi+1/2U j (f) = ~r— / U(ic, t)dx. (4.5)

We further define the approximations

1 rxi+1/2Qj(Z) = Q(Uy(t)) «/ Q(U(x,t)) cZa; (4.6)

and the discrete numerical flux function

Fj+1/2(U?-p(t), Uj_p+i(Z),..., Uj-|-9(f)) (4-7)

which approximates

Fj+i/2(f) ~ F(U(®j+i/2,t)). (4.8)

Inserting into (4.4) we obtain a semi-discrete scheme for approximating Uj

~df + ~Ax (FZ+1/2(*) “ Fj-i/2(Z)) = QjOO- (4.9)

Definition 8. A scheme that can be written on the form (4.9) with a numerical flux function Fj+1/2 is termed a conservative scheme. Here the numerical flux of U from cell j into cell j + 1 balances the flux from j + 1 into j, and the variable U is numerically conserved.

4.1.2 Fully Discrete SchemesHaving discretized in space, we proceed to create a mesh in time

t = nAt. (4.10)

For the purposes of this thesis, we will limit ourselves to a first-order temporal discretization of (4.9) as follows

U"+1 — U” 1J At—1 + ^ (Fi+i/2 - Fj-i/2) = Qj- (4.11)

Remark 11. Higher order accuracy may be achieved by using Runge-Kutta integration and Strang splitting of the source terms. Such techniques have been applied to two-phase flow models in [79, 17].

26 Chapter 4. Numerical Schemes

We may classify schemes on the form (4.11) according to the time level discretization of the fluxes and sources, as given below.

Definition 9. If all the flux and source terms are discretized at time level tn

jn+l _^ 4-

Ar

■jjn+l _ Tjn -I' ’ + XT (F"+i/2 " F"-i/2) = Q*.

A t

the scheme is termed an explicit scheme.

(4.12)

Definition 10. If all the flux and source terms are discretized at time level tn+1

Ui+1 - UiAt

+ (% - F/-b) = Q?+1, (4.13)

the scheme is termed a fully implicit scheme.

Definition 11. We use the notation n +1/2 to indicate that some terms are discretized at time level tn, others at tn+1. A scheme on the form

Tjn+l _ U« X

(FK+1/2 _ pH+1/2^ _ j-^n+1/2At Ax x ■?'+1/2 i-1/2/

is termed a semi-implicit scheme.

= Q“- (4.14)

Implicit schemes involve couplings across the computational domain, and the resulting difference equations must be solved simultaneously. Fully implicit schemes generally yield non-linear equations that must be solved by iterative techniques. However, it is common to perform some linearization procedures, obtaining a semi-implicit scheme where the resulting difference equations become a sparse linear system.

Explicit schemes are far more easy to implement, but suffer from the well-known CFL restriction (see for instance [45])

f\7*

> max |A|. (4.15)

That is, the timestep is limited by the largest (in terms of absolute value) wave velocity existing in the computational domain.

Remark 12. For the two-fluid model, the wave structure consists of rapid sonic waves and slow volume fraction waves (See Appendix B). The sonic waves may be 10-1000 times larger in magnitude than the volume fraction waves. Hence the CFL criterion (4.15) may be considered severe if we are primarily concerned with accurate modelling of the volume fraction waves. Implicit schemes for the two-fluid model is the topic of papers III and V of this thesis, where efficiency and accuracy considerations are discussed.

4.1. Conservative Schemes 27

4.1.3 Numerical Flux FunctionsBy far the most important aspect of a numerical scheme is the construction of the numerical flux Fj+i/2. The complex interaction between the space and time components of the system (i.e. the “partial” nature of the equations) is fully contained in the flux function F, as described in Chapter 2.

Definition 12. The numerical flux Fj+i/2 is said to be consistent if it

1. is Lipschitz continuous,

2. satisfies Fj+i/2(U, U,..., U) = F(U) for all U.

Here F is the physical flux function.

Consistency is an obvious minimal requirement for the scheme (4.11) to make sense.

In the simplest case, Fj+i/2 depends only on the neighbouring values

Fj+i/2 = Fj+i/^LL,-, Uj+i). (4.16)

In this case, the scheme (4.12) is termed a three-point scheme as the value of Uj+1 depends upon values from the three grid cells (j — 1 ,j,j + 1) at time level tn. In general we have the following definition

Definition 13. When the numerical flux function F j+1/2 takes M arguments, the numerical scheme (4.12) is said to be a (M + 1 )-point scheme.

Convergence and Stability

0. (4.17)

In the following we will simply neglect the source term and consider the homogeneous equation

9U <9F(U)&t dx

For linear systems the Lax Equivalence Theorem applies (see [45]), which basically states that if a numerical method is stable (in a sense that can be made precise), it converges if and only if it is consistent.

For non-linear systems convergence can be proved only for some special cases. In general, our confidence in numerical methods is to a large degree based on practical experience. However, in the event that a conservative and consistent numerical method does converge to some function U(ar, f), the Lax-Wendroff theorem [42] guarantees that U(:r, t) will be some weak solution (although not necessarily one that satisifies the entropy condition) of the conservation law.

To ensure convergence, at the least we need some form of stability. An important class of numerical methods are the TVD (Total Variation Diminishing) methods, where the total variation of the numerical solution Un is defined as

TV (CP) = |U-■nj+1 U?l (4.18)


Definition 14. A numerical method is said to be Total Variation Diminishing (TVD) if

TV(Un+1) < TV(Un). (4.19)

The TVD condition represents a particularly strong form of stability. Not only does it provide a bound on the amplitude of numerical oscillations, it also severely limits the possibility of numerical oscillations even occuring (as such oscillations tend to increase the total variation of the data).

The TVD theory for scalar equations has been well developed. An important theorem is stated by Tadmor [77]

Theorem 2. Consider the scalar equation

&u df(u) dt dx (4.20)

solved by the numerical scheme

-l/W+l _ n,n -i+ AS 7K, <+i) - *K-i. «“)) = 0 (4.21)

where the numerical flux F(u”,u"+1) is written on viscous form

1 1 AtFj+1/2 = F(uj,Uj+1) = - (f(Uj) + f (u^+1)) --—Uj+y2(u]+1-u^). (4.22)

The scheme (4-21) is total variation nonincreasing provided its numerical viscosity coefficient z/^1/2 = u(Uj,Uj+1) satisfies

At f{uf+1) - f(v%) Ax Uj+l - Uj — ^i+1/2 A 1. (4.23)

This theorem has several consequences. In particular we note that choosing a naive central differencing of the flux term

*j+i/2(%, «i+i) = \ (/(«”) + /(«?+,)) (4.24)

will simply not work, and some numerical dissipation (also termed numerical viscosity) must be added to stabilize the scheme.

This numerical viscosity must be chosen between the two following extremes:

1. The upwind scheme. Choosing the minimal required viscosity as statedby (4.23)

n _ At |

^'+1/2 - |V+V2| , (4.25)

4.2. Approximate Riemann Solvers 29

with

Aj+1/2

we recover the upwind scheme

1

f(u"+ i) ~ f(u")(4.26)

At + (/(Mj) - /(Mi-i)) - 0 f°r Ai+1/2 > 0, (4.27)

un+l _ y tl -I

j j + (/(Mj+i) - /(«”)) = 0 for Aj+i/2 < 0. (4.28)At

2. The Lax-Friedrichs scheme. Choosing the maximal allowed viscosity as stated by (4.23)

= 1, (4.29)^'+1/2

we recover the Lax-Friedrichs scheme

,n+lU- 1/2 «-i + “"«) + J_ (/(u»+i) _ /(^J) = o. (4.30)At 2Aa;

Theorem 2 basically states that of the stable, first-order three-point schemes, the upwind scheme based on one-sided differences is the least dissipative (most accurate). From a physical point of view, this makes sense. We have seen in Section 2.2.1 that information propagates with the characteristic velocity A, so it is natural to calculate the flux derivative in the “upwind” direction with respect to information flow. The “downwind” direction has no influence on the variable Uj whatsoever, and would only pollute the calculation.

4.2 Approximate Riemann Solvers

In the event that all eigenvalues of the system are of the same sign, the upwind method may be trivially generalised to systems as follows

Un+1 - U7? 1’ ' + 5 (f(u?) - Ffuy,)) = q

Atfor positive eigenvalues,

(4.31)U«+! _ u» j—I-------- L j-------

At Ax(4.32)

However, the eigenvalues are generally of mixed signs, and each characteristic field should be examined separately to determine the upwind direction. This is precisely the idea behind the Godunov scheme [25, 28].

(F(LT+1) — F(LF)) = Q” for negative eigenvalues.


4.2.1 Godunov’s MethodThe method may be described by the following steps:

1. From the numerical data Un, construct a piecewise constant approximation to U(x,t) as follows

U(x,tn) = Uj for x £ (^_i/2,^j+i/2]- (4.33)

Note that J U dx = J U dx (4.34)

because Uj is defined as the cell average of U.

2. Observe that the approximation U(x,tn) defines a series of Riemann problems at the cell interfaces Xj+i/2, as described in Section 2.4.1.

3. Calculate the solution to these Riemann problems to evolve the function U(%,t), t£

4. Calculate new cell averages U”+1 as

1 rxj+1/2 „^ = — / U(%,f+i)dz. (4.35)

*xj-1/2

Repeat from step 1.

The difficulty, of course, lies in step 3. However, this step may be significantly simplified by the following proposition:

Proposition 1. Provided that waves from neighbouring Riemann problems do not interact, the cell interface value \J(xj+i/2,t) is constant for t £ (tn,tn+1 ]. By using the integral form of the conservation law, Godunov’s method can be written on conservation form

TJn+1 _ TJn 1~J~At—L + (F"+i/2 - F"-i/2) = Q? 0.36)

with numerical fluxes

F”+1/2 = F(U(i,-+1/2,t”)). (4.37)

The proof may be found in [80].

Remark 13. R could be argued [78] that Godunov’s essential insight was the interpretation ofXJj as cell averages instead of point values, allowing for the construction of a conservative scheme. For this reason, some authors (see for instance [22]) like to refer to the original Godunov scheme as an upwind Godunov-type scheme, whereas a conservative scheme based on central differencing (like the Lax-Friedrichs scheme) is termed a central Godunov- type scheme. In this thesis, when we refer to Godunov schemes, we will mean the scheme of Proposition 1.

4.2. Approximate Riemann Solvers 31

4.2.2 Rough Godunov SchemesAlthough Proposition 1 significantly simplifies the implementation of the Godunov scheme, we are still left with the task of computing the intermediate state U”+1/2 = \J(xj+i/2,tn). For nonlinear systems, this may require some iteration. On the other hand, we note that by making some approximationtO U;+l/2

Uj+i/2(Uj, Uj+i) ~ Uj+i/2) (4.38)subject to the consistency criterion

U;+1/2(U,U) = U, (4.39)

the numerical fluxF"+i/2 = F(U*+1/2) (4.40)

still yields a conservative and consistent scheme. Of course the quality of such a scheme depends on the accuracy of the approximation U*+1/,2.

A natural way to obtain U*+1/2 is t° replace the original conservation law with a local linear approximation as follows

where Aj+i/2(Uj, U/+i) is subject to the consistency criterion

9Udt + Aj+i/2^r-

9Udx

A,-+1/2(U,U) = (4.42)

This may be achieved by associating Aj+i/2 with the exact jacobi matrix A(U) evaluated at some average state U, e.g. the arithmetic average

Aj+i/2 = A 2 + Uj+i) (4.43)

The linear Riemann problem on (4.41) directly yields (see Section 2.4.1)

u-+1/2 = U" + J2 rPR, = u?+1- 0-44)p | Ap<0 p | Ap>0

Remark 14. Such Rough Godunov Schemes as described in this section may suffer from some drawbacks: (i) They may calculate entropy violating numerical solutions, (ii) The numerical flux may be discontinuous with respect to its arguments when some eigenvalue changes sign across the cell interface j + 1/2, possibly leading to numerical instabilities.

These issues are discussed in [52].

A rough Godunov scheme has been implemented for the drift-flux model by Faille and Heintze [19], and for the two-fluid model by Tiselj and Petelin[79]. The Riemann problem for the two-fluid model has been investigated by Zeidan et al [93].


4.2.3 The Roe SchemeA widely used linearized approximate Riemann solver was proposed by Roe [64, 65]. The Roe scheme has the desirable property that if Ul and Ur are connected by a single shock wave or contact discontinuity, the solution to the linearized problem agrees with the solution to the original exact problem.

We locally replace the original system with a linearized conservation law

OTT OTT— + A(U_f, Uj+l) (4-45)

where the following conditions are imposed on A:

Rl: A(Uf, U/+i)(Uf+i - Uf) = F(Uj+i) - F(Uy)

R2: A(Uj, Uj+i) is diagonalizable with real eigenvalues

R3: A(Uj, Uj+i) —>• A(U) smoothly as U,, Uy+i —>• U.

Condition Rl ensures conservation across discontinuities. Condition R2 ensures that the linearized system is hyperbolic, and Condition R3 is the consistency requirement.

The numerical flux for the Roe scheme may now be written as

F,+1/2(U”,UJ+1) = i(F(U") + F(UJ+1))-i A(UJ,U“+1) (UJnj+1

where given A = RAR , the “absolute value” of A is defined as

•up,(4.46)

A = R R (4.4T)

withA diag(|Ai|, |A2|, j |Av|), (4.48)

where A* are the eigenvalues of A.The Roe scheme only allows discontinuities as elementary wave solutions

to the Riemann problem, possibly leading to entropy violating numerical solutions when there is a change of sign in an eigenvalue. An entropy fix may be applied to cure this problem, see [45, 80] for details.

A Roe scheme has been implemented for the drift-flux model by Romate[66] and Fjelde and Karlsen [23]. A Roe scheme for the two-fluid model has been developed by Toumi et al [81, 82, 11].

4.3 Hybrid FVS/FDS SchemesHaving discussed generalisations of the upwind scheme to nonlinear systems, it will be instructive to reexamine the classical Lax-Friedrichs scheme discussed in Section 4.1.3.

4.3. Hybrid FVS/FDS Schemes 33

For systems, the Lax-Friedrichs scheme can be written as

Ui■n+l - 1/2 (U?_t + uy+1) F(U,+1) - F(Uj_t)At ^ 2Aa; " (4.49)

This may be written on the flux-conservative form (4.12) with the numerical fluxes

1 1 At

F^(U?, U^) = - (F(U,) + F(IW) + (U,- - U,-+i). (4.50)

It is now worth noting that this flux can be written as a direct sum of left and right contributions as follows

F,+1/2(U”, U»+1) = F+(U”) + F-(U"+i), (4.51)

withF±(U) = 5 (f(U) ± ^u) . (4.52)

Definition 15. A conservative scheme where the numerical flux Fj+i/2 can be splitted as

Fj+i/2(Uj, Ui+i) = F+(Uj) + F (Uj+i) (4.53)

is termed a Flux Vector Splitting (FVS) scheme.

Remark 15. In the literature, one usually also insists that an FVS scheme should reduce to the upwind scheme in the case where all eigenvalues are of the same sign [80]. Under this requirement a central scheme like the Lax- Friedrichs scheme is not an FVS scheme. For the purposes of this thesis, this may be considered a technicality.

Generally, the Roe flux (4.46) can not be expressed on FVS form. However, by using the Roe condition Rl, the Roe scheme can be rewritten as

U?+1 = U? - ^ (V-i/M" - U”_.) + A-+1/2(U»+1 - u-

whereA± = RA±R_1, A± = diag(A±, Af,..., A^),

withA+= max(0, Aj), X[ = min(0, Xf).

+ Q-At,(4.54)

(4.55)

(4.56)

Hence the Flux Difference (Fj+1/2 — Fj_i/2) is splitted in left and right components as

Fj+1/2 - F,-1/2 = At_1/2(U” - U”_,) + A-+1/2(U”+1 - UJ). (4.57)3+1/2''■'2+1

Definition 16. A conservative scheme tha,t can be written on the form (4.54) is termed a Flux Difference Splitting (FDS) scheme.


Note that the class of FDS schemes is not very exclusive. In fact, bychoosing A+ and A such that

Fj+i/2(Uj, Uf+i) — F(Uj) + A (Uj+i - Uj) — F(UJ+i) - A+(Ui+i - Uj),(4.58)

any scheme reduces to the form (4.54).It is nevertheless a useful classification to consider

• FVS schemes, where the flux is obtained from scalar computations.

• FDS schemes, where the flux is obtained from matrix computations.

Hence FVS schemes are easier to implement and more efficient than FDS schemes. On the other hand, FVS limits our ability to apply upwind techniques on each characteristic field separately. In practice, the numerical dissipation for FVS schemes must be adapted to the fastest moving waves. A consequence of this is that FVS schemes are generally more diffusive than FDS schemes.

In the following, we will describe some strategies for combining the simplicity of FVS with the accuracy of FDS. In particular, we will describe the mixture flux method that is the focus of this thesis.

4.3.1 AUSM SchemesFor the Euler equations (Section 3.1), van Leer [87] suggested an FVS scheme where the convective <3>-fluxes (mass, energy and momentum) are given as follows:

($v)j+l/2 ~ + Vj+1$j+1. (4.59)

Here the velocity is splitted according to the van Leer splitting formulas

(4.60)

This splitting has the desirable property of being differentiable at the sonic points. However, it yields a poor resolution of contact discontinuities, a behaviour typical of FVS schemes.

The Liou-Steffen Scheme

To improve the approximation properties of the van Leer FVS scheme, Liou and Steffen [49] suggested retaining the van Leer splitting formulas, but applying an uwpind principle to the splitting of the convective fluxes. For this reason, Liou and Steffen denoted their scheme as AUSM (Advection Upstream Splitting Method).

We first split the Euler system in convective and pressure components asfollows

dV , 8FC , dFp—W---1-------- 1------dt dx dx

Q(U), (4.61)


wherepv ' 0 '

Fc =_ v(E +p) _

5 Fp — P_ 0 _

We calculate the cell interface velocity

^+1/2 = c) + %(u,-+i, c)

(4.62)

(4.63)

which determines the upwinding of the convective fluxes as follows

FcJ+1/2 - vj+1/2P

pu(F7+p)

for vj+1/2 > 0, (4.64)

FcJ+1/2 — yj+l/2P

pu(^ + P)

for vj+1/2 < 0.

-I j+iFor the pressure flux an FVS-type splitting is applied

Fpj+1/2 - ‘Pj'Fpj + Vj+1Fpj+i

where

/P±{v,c) = V±(v,c) ■ { f±i (±2 — if M < c otherwise.

(4.65)

(4.66)

(4.67)

Remark 16. Lion [f.7] demonstrated that the stability properties of the A USM scheme could be improved by using higher order polynomials for the pressure and velocity splitting formulas, denoting the resulting scheme as AUSM+. Furthermore, the AUSM schemes were demonstrated to possess desirable properties such as (i) Exact resolution of stationary shocks or contacts, (ii) Conservation of positivity for physical variables like pressure and density. (Hi) Improved stability compared to approximate Riemann solvers on various difficult test cases.

AUSM schemes are currently popular for industrial applications, for instance in aeronautical engineering [48, 62].

Two-phase flow generalisations of AUSM schemes include a homogeneous equilibrium model with phase transitions [15], as well as the drift-flux [17, 18] and two-fluid model [57].

Remark 17. Note that the AUSM schemes are not FVS anymore as the numerical fluxes (4.64) and (4.65) can not be written on the form (4.53). Nor are they typical FDS schemes as the flux-splitting is performed on each conservation equation separately. Hence it makes sense to classify the A USM scheme as a hybrid FVS/FDS scheme.


AUSMD and AUSMV schemes

A related approach was suggested by Wada and Lion [89], who went back to the original FVS scheme

P PFcj+1/2 = V+ pu + V3 +1 /W

. (2+p). 3 . (2+P) .(4.68)

i+i

FpP+i/2 - 'Pj'Fp,j + ^i+iFp,i+i- (4-69)

Now, instead of applying upwinding directly to the numerical mass fluxes, Wada and Lion suggested modifying the velocity splitting formulas of the FVS scheme in such a way that

1. The FVS splitting formulas are recovered in the limit (Uj — U?+i) —> 0.

2. The upwind flux is recovered for a contact discontinuity moving with uniform pressure and velocity.

They achieved these goals by introducing the following modification of the van Leer splitting formulas:

■r±f., „ _ J xV±{v,c) + (1 - x)^ \v\<c

whereXL

Uv± M)

2(p/p)iXn

otherwise,

2(p//))n

(4.70)

(4.71)(p//))n + (p//?)n' ^ (p//))n + (p/p)n

They further derived schemes denoted as AUSMV and AUSMD, given by:

• Mass flux:

(pv)j+1/2 = PjV+(vj, cj+1/2, Xl) + pj+iV (uj+i, Cj+1/2, Xr)- (4.72)

• Momentum flux:AUSMD (FDS-like flux splitting):

= I kr Wj+i/2 > 0,^ J+ ! \ (pv)j+i/2Vj+i otherwise.

AUSMV (FVS-like flux splitting):

(4.73)

(P^)j+i/2 = (/w)^-V+(ry, c^+1/2, Xn) + (^+i, 9+1/2, Xa)-(4.74)

• Energy flux:

= i Mj+i/2#j for W;+i/2>o,^ 0+ / \ (pv)j+1j2Hj+i otherwise (4.75)


where H = E + p/p is the entalpy, and Cj+1/2 = max(cr Cj_i) is a common velocity of sound at the cell interface.

Niu has investigated the AUSMD/V approach for a multicomponent flow model [55] and the Saurel-Abgrall model [56]. Evje and Fjelde [17] proposed an AUSMV scheme for the drift-flux model. The adaption of the AUSMD/V approach to the two-fluid model is the topic of paper I of this thesis. Properties of the AUSMD/V and AUSM+ splittings are discussed in paper IV.

Remark 18. The results of papers I and IV as well as [17, 57] indicate the following: For the two-fluid model, the AUSMD/V approach is more robust than the AUSM(-h) approach in the resolution of sonic waves. This may be a consequence of AUSMD/V representing a greater step in the FVS direction

It should be noted that even the AUSMD/V mass flux is not fully FVS, as the splitting formulas are functions of both Uj and U)+:L.

4.3.2 The Mixture Flux MethodThe motivation behind the AUSMD/V approach of Wada and Lion [89] may be summarized as follows:

1. For numerical schemes, linear contact discontinuities are harder to resolve accurately than non-linear shocks, where the nonlinearities tend to carry the inaccuracies “into the shock” (see for instance [45, 78]).

2. On the other hand, it is much easier to calculate the exact solution to the Riemann problem for linear contacts.

Issue (1) is clearly demonstrated in Figure 4.1, taken from paper I. This figure depicts discontinuities in the volume fraction variable for the separation problem (see paper I for details). Here both discontinuities move with the same absolute velocity, but oppositely directed. The right discontinuity is a shock whereas the left discontinuity behaves as a contact discontinuity. The difference in the numerical resolution of these two phenomena is striking.

Such results suggest that the complexity of an exact Riemann solver may not be generally required. What we may aim for is a scheme that is robust on non-linear shocks, and accurate on linear contacts. Then robustness on contacts and accuracy on shocks should more or less follow.

So a strategy (and one could argue, the one followed by Wada and Lion [89]) would be

1. Devise a numerical scheme that is robust on all waves (e.g. the FVS scheme).

2. Devise a numerical scheme that is accurate on contact discontinuities (e.g. the upwind scheme).


analytical solution 1000 cells 100 cells 25 cells

Distance (m)

Figure 4.1: Numerical resolution of a shock and contact discontinuity. Contact to the left, shock to the right. Taken from Figure 11 of paper I.

3. Hybridize (1) and (2) such that we recover (2) on contact discontinuities while generally retaining as much of (1) as possible. Wada and Lion [89] accomplished such a hybridization of the FVS and upwind scheme through their splitting formulas (4.70).

The mixture flux method is now motivated by the following observation:

Remark 19. The results of paper I and [57] illustrate that the AUSM type schemes for one-phase flow suffer from spurious oscillations when generalised to the two-fluid model. A more refined dissipative mechanism is needed to achieve a fully non-os dilatory approximation of discontinuous waves. The motion of the phases are strongly coupled, an effect that is not taken into consideration by the basic A USM type schemes. It is therefore of interest to investigate further the mathematical properties of the relevant contact discontinuities in the two-fluid model, to see how phasic couplings can be naturally integrated in the dissipation mechanism.

The Two-Fluid Model

Our starting point is the two-fluid model given by (see Section 3.2)

ddt (pga g) +

d_dx

(pgagvg) — 0, (4.76)

ddt (A« 0 +

d_etc (a«Ti) = 0, (4.77)

8

dt(pgCXgVg) +

d_&r

/(pgOgfg) + Oig— + Ap'etc &r — Q& (4.78)


5(Aoiti)+A +a,%.+-Qi’ <4-79)where we assume that Ap —>■ 0 as vg —> v\.

For convenience in notation we express the model on the following form:

dtmk + dxfk = 0,+ (Ap)a^at = Q&,

where k =g,l and

fk — Pk^k^k and mk — Pk^k 9k — Pk^-k^k and Ik — pk^k^k-

(4.80)

(4.81)(4.82)

A General Formulation

We write the numerical approximation to (4.80) as

mn+1k,j At—— + ~Kx (Ffcd+i/2 - Fk,j-1/2) - 0, (4.83)

Contact Discontinuities

We consider a contact discontinuity given by

Pl = Pk = P (4.85)al ^ «R

(yg)n = (wi)l = {vs)r = (vi)r = v-

All pressure terms vanish from the model (in particular Ap = 0 because vg = v{), and it is seen that the exact solution to this initial value problem is that the discontinuity will propagate with the velocity v. This exact solution yields the numerical mass flux

(pav)j+1/2 = ^p(aL + aR)u - ^/>(aR - aL)\v\ (4.86)

for both phases.

Definition 17. A numerical flux Fk that satisfies (4-86) for the contact discontinuity (4-85) will in the following be termed a “mass coherent” flux.

In accordance with Remark 19, we should now aim to obtain a scheme that is robust on pressure (which is constant across the contact) and accurate on volume fraction (which is discontinuous across the contact). In particular, for the contact discontinuity (4.85), we want our scheme to be mass coherent.


A Robust Pressure Splitting

For the pressure splitting Pj+i/2(Vj, U:/ ,) we wish to take advantage of the stability properties of the Lax-Friedrichs scheme (see Section 4.1.3).

Performing a variable transformation as described in Section 2.2, we obtain the differential

dp = K,(pidmg + pgdm\) (4.87)

where

k = (4.88)

By applying (4.87) to the mass conservation equations we obtain the pressure evolution equation

dp ( d , . d , A~dt+K (pgCXgVg)+ pgdx (pl0llV^y °-

Discretizing (4.87) at the cell interface j + 1/2 we obtain

p"tlv-Wj+pU) = _ (KPg)?+i/2

(4.89)

7"n _ TnAj+l

Ax(4.90)

where, by convention, we obtain cell interface values k, p from the definitions

Pk,j+1/2 — PkiPj+1/2), Otj+1/2 — - (oq + «i+l) • (4.91)

The pressure splitting is naturally obtained from (4.90) as

f>j+1/2(U”,U”+1)=p”«/2. (4.92)

Remark 20. The cell interface pressure Pj+1/2 can be written on consistent, viscous form

o+i/2(u",u"+1) =Py;/2

n (f? +P”+.) - %+./2%+. - %) + »U+1/2(^+1 - %)(4.93)

w/iere the numerical viscosity coefficients D^+1j2 are given by

^LKn nn/\^^+l/2Aj+l/2-Dn^gd+1/2 (4.94)

and#v+i/2 = (4-95)

Hence, although basing the splitting on a non-conservative evolution equation, there is no associated loss of consistency or global conservation. It is only the numerical viscosity, not the cell average, that is computed by the non-conservative equation.


Robust Mass Fluxes

Our hypothesis, which seems confirmed by the numerical experiments performed in the accompanying papers, is that the problems observed for the basic AUSM type schemes are mainly related to the dicretization of the mass equations (4.83).

The “mass coherency” property (Definition 17) ensures an accurate resolution of a contact discontinuity. A weaker criterion for robustness can be derived by imposing the condition that the pressure should remain constant during the temporal evolution of (4.85).

We write (4.87) asdp = ndp (4.96)

wheredp = pgdnii + pidmg. (4.97)

To maintain a constant pressure we must have dp = 0. Assuming constant pressure, (4.97) can be integrated to yield

p — PgWii T pprig — pgf}\{oi\ Ofg) — pgp\.

To maintain constancy of p and hence p we now insist that the flux F is a consistent numerical flux when applied to the mix mass p. That is, we impose

PgFi + piFg = pgpiv. (4.98)

for the contact discontinuity (4.85).

Definition 18. A pair of numerical fluxes F\ and Fg that satisfy U-98) for the contact discontinuity (4-85) will in the following be termed “pressure coherent” fluxes.

One can easily show that the class of mass coherent fluxes is a proper subset of the class of pressure coherent fluxes. Consequently, we have greater freedom to introduce numerical dissipation (i.e. stability) within a pressure coherent scheme than a mass coherent scheme. Generally, a pressure coherent scheme is less accurate than a mass coherent scheme.

Spesification of Pressure Coherent Fluxes

Going back to the pressure evolution equation (4.90), we see that it naturally defines a conservative scheme for calculating masses at j + 1/2 as

<+1/2 - I (< + <+i) %+i - %At Ax

0. (4.99)

If we now compute the simple average

1mk,j — 2 1/2 + mk,j+i/2) (4.100)


and substitute (4.99), we obtain the following difference equation for

- i (2™^ + + ™L+1) + ^ (Jj _ = o, (4.101)At 2A%

which can be written on flux-conservative form (4.12) with the numerical fluxes

F?s 1 Axk,j+1/2 - Fk (Uj, Uj+i) - -{h,j + h,j+i) + 4^(Wfcb - mfc,j+i)- (4.102)

Here the fluxes (4.102) ensure an additional numerical coupling between the cell center pressure pj and the cell interface pressure Pj+1/2, allowing them to develop in a concurrent manner. The fluxes (4.102) can easily be shown to be pressure coherent in the sense of Definition 18. However, they are quite diffusive (a fact that is demonstrated in paper II of this thesis), hence the superscript “D”.

Remark 21. The fluxes (4.102) yield a strict CFL-like criterion on the timestep Ax/At if the TVD condition (Theorem 2) is to be satisfied. This issue is discussed in paper IV, where a fix is proposed.

The Mixture Fluxes

Our goal is to hybridize the pressure coherent flux FD with a more accurate (i.e. mass coherent) flux FA such that the hybrid flux retains the mass coherency property of FA while incorporating robustness properties from FD.

We aim for a linear combination of FT) and FA such that FT) affects the component of the masses that is constant across the contact discontiuity (pressure), and the FA component similarly affects the component that is discontinuous across the contact (volume fraction).

The splitting of the mass variables into pressure and volume fraction components is expressed through the differentials

dp = K(pidmg + pgdm\), da 1 = K{-â\dmg + ^âgdm\) (4.103)

and

dmg = aj^^dp — pgda\, dm\ = aS^dp + p\da\. (4.104)

Hence a “pressure” and “volume fraction” component of the fluxes can be obtained from (4.103) as

Fp = KpgF[D + KpiF^ (4.105)

Fr, -- K - Kâ,Fhand

(4.106)


By transforming back to conservative variables through (4.104) we obtain the final hybrid mass fluxes

Fg — ag Qp Fp pgFa, Fi — a\-^Fp + p\Fa (4.107)

which can be written out as

fi - % ^ FI + #(%g ^ Fi + ^ (F^ F^r )j| (4.108)

and

fg - K ^ F^ 4- PgOigp fg+ Psas dp (fi )) • (4.109)

These are cell interface values, so a subscript j + 1/2 is implicitly assumed on both the fluxes and the coefficients. Consistent with the treatment of the pressure evolution equation (4.90) we suggest obtaining the cell interface coefficients from the cell interface pressure Pj+1/2, as well as the relation

aO+1/2 = g (aj + aj+1)- (4.110)

Remark 22. The consistency criterion

Fk(U, U) = fk(U) = pkOikVk-, (4.111)

relating the numerical flux Fk to the physical flux fk, is satisfied for the hybrid fluxes (4.108) and (4-109) provided the fluxes Fjf and Ffl are consistent. In particular if Fk = Fjf the expressions (4-108) and (4-109) reduce to the trivial identity

Fk = FtA = FtD. (4.112)

The following proposition is proved in paper II:

Proposition 2. Let the mixture fluxes (4-108) and (4-109) be constructed from pressure coherent fluxes F®, and fluxes Ff) that reduce to the upwind fluxes (4-86) on a contact discontinuity of the form (4-85). Then the hybrid fluxes (4-108) and (4-109) also reduce to the upwind fluxes (4-86) on the

The Gk,j+i/2 Term

In the accompanying papers, the discretization of the Gk,j+1/2 term has been assumed to be consistent with the mass flux component FA in the following sense:

For a flow with velocities which are constant in space for the time interval [tn,tn+1], that is,

vk,j{t) — vk,j+i{t) — vk{t)i t<=[tn,tn+1l (4.113)


we assume that Gk,j+1/2(^) takes the form

Gk,j+i/2{t) = (4.114)

where F^._hl/2(t) is the numerical flux component introduced above and assumed to be consistent with the physical flux fk = pkCXkVk-

Generally, we could consider applying a mixture flux strategy also to the Gk,j+1/2 terms. However, the numerical examples of the accompanying papers illustrate that a straightforward discretization satisfying (4.114) generally gives satisfactory results.

The Afcj+i/2 Term

The A&J+1/2 term is associated with the interfacial pressure correction, and is expected to be significant in the short wavelength domain. For many applications of interest, such terms are of little physical relevance and may be considered as correction terms. For the purposes of this thesis, we have followed [9, 57] and chosen a simple central discretization:

Afej+i/2 = ^ + Afej+i) • (4.115)

For cases where low wavelengths are of importance (e.g. studies of hydrodynamic instabilities, see Section 3.2.2), more sophisticated ways of discretizing this term could be considered.

Remark 23. A standard stability criterion for numerical methods for two- phase flows is due to Abgrall [If:

A flow, uniform in pressure and velocity, must remain uniform in the same variables during its time evolution.

A straightforward proof that the mixture flux methods satisfy the Abgrall principle is given in paper II.

Remark 24. The mixture flux method may be viewed as a splitting of the system into two components, associated with pressure and volume fraction respectively. This is a simpler splitting than the one employed by the approximate Riemann solvers, who would split the two-fluid model into its full eigenstructure of four waves.

On the other hand, the mixture flux method is less simple than the two- fluid A USM type schemes of paper I and [57]. Hence, it makes sense to classify the mixture flux method as a hybrid FVS/FDS scheme where there is more emphasis on FDS than for the basic A USM type schemes.

4.4 Higher Order AccuracyAll the schemes we have discussed so far have been of first order in space. The construction of TVD schemes (see Section 4.1.3) of higher order spatial

4.5. Boundary Treatment 45

accuracy has been a highly active research area for many years, see [45, 78] and the references therein.

With the Godunov method (and other first-order conservative methods) the Riemann problem is solved (approximately or exactly) with the assumption that the data are piecewise constant, equal to Uj across the cell j. The main idea is to replace these piecewise constant data with some interpolation (piecewise linear or higher order polynomials) such that the total variation of the interpolant does not exceed the total variation of the original data. Notable examples include the MUSCL strategy of van Leer [86] and the higher order central schemes of Tadmor et al (see for instance [54, 37]).

Investigation of higher order numerical schemes for the drift-flux model was the main topic of a thesis by Fjelde [22]. Furthermore, the MUSCL strategy has been applied to numerical schemes for the drift-flux model by Evje and Fjelde [17,18] and Faille and Heintze [19]. For the two-fluid model, a related second order strategy was adopted by Tiselj and Petelin [79]. Another work of interest is the thesis of Lorentzen [51].Remark 25. To construct higher order TVD schemes, it is a minimal requirement that the first order versions of the schemes possess the TVD property. In this thesis, the emphasis is on the construction of efficient such first order schemes — with the intent that they allow for second order extensions, following the approach of for instance [47, 17].

The extension of the mixture flux scheme to second order spatial accuracy is briefly touched upon in paper IV.

4.5 Boundary TreatmentCharacteristic analysis (see Section 2.2.1) is an essential tool in determining the boundary conditions. For a strictly hyperbolic system of N equations, we will have N independent waves moving with separate characteristic velocities. Hence, to achieve well-posed boundary conditions the following must apply:

• Inlet: If p of the characteristic wave velocities are positive, exactly p boundary conditions must be imposed (physical boundary conditions) and (N—p) boundary conditions must be calculated from the available data in the computational domain (numerical boundary conditions).

• Outlet: Similarly, if q of the characteristic wave velocities are negative, exactly q boundary conditions must be imposed (physical boundary conditions) and (N — q) boundary conditions must be calculated from the available data in the computational domain (numerical boundary conditions).

Several different methods may be applied to determine the numerical boundary conditions. Hirsch [30] discusses various treatments of boundary conditions for both one-dimensional flow (with particular reference to the Euler equations) and multidimensional flow. Stability and accuracy of different boundary treatments is discussed by Gottlieb and Turkel [26].


4.5.1 Characteristics and Compatibility RelationsAs we have seen in Section 2.2.1, the following compatibility relations apply

a;1^=9p(u) (4.116)

along the p-characteristic. The numerical boundary conditions may now be found by a finite difference upwind discretization of the compatibility relations

=»(u) (4.ii7)for negative eigenvalues at the inlet and positive eigenvalues at the outlet. Further details may be found in [5, 23]. Such a characteristic boundary scheme was also used by Lage et al [38] for the drift-flux model.

4.5.2 The Ghost Cell ApproachAlthough characteristic boundary methods are most rigorous from a mathematical point of view, simpler approaches may often yield acceptable results. A popular strategy is the ghost cell approach, where we imagine cells existing outside the computational domain (ghost cells) and simply extrapolate the numerical boundary conditions from the computational domain, see for instance [51].

Remark 26. The ghost cell approach with simple zeroth order extrapolation has been used for all simulations performed in the accompanying papers. No related stability problems were observed.

Chapter 5

Comments on the Papers

5.1 Paper I

Hybrid Flux-Splitting Schemes for a Common Two-Fluid Model

Coauthor: Steinar Evje

In Paper I the AUSMD and AUSMV strategy of Wada and Lion [89] is adapted to the two-fluid model. The paper also builds rather directly upon an earlier paper by Evje and Fjelde [17], who considered the drift-flux model.

Related to [17], some additional difficulties are faced in paper I:

1. The two-fluid model involves non-conservative pressure terms which must be handled with care in the numerical discretization.

2. The motion of the phases is weakly coupled compared to the drift-flux model, and instabilities are more easily induced. For this reason, a more refined mechanism for removing numerical dissipation from the mass equations than the one used in [17] had to be developed (See Section 4.4 of paper II).

Non-conservative products have not been given much focus in this thesis, where most of our discussions are based on fully conservative models. It is nevertheless a well-known fact that the two-fluid model can not be written on such conservative form. As a consequence of this, the integral formulation of the system is not well-defined, as the resulting integrals are path-dependent in variable space. Attempts to derive ways to define weak solutions for nonconservative systems are described in [82] and the references therein. In paper I, it is the approach of [82] that is followed; here we define the solution to the weak integral over a stationary discontinuity, and impose the condition that the numerical scheme must agree with the weak integral formulation under stationary conditions.

47

48 Chapter 5. Comments on the Papers

Remark 27. From an engineering perspective, the discussion on consistent treatment of the non-conservative term may be argued to be academic. In particular, for smooth solutions we expect no inconsistencies. Generally, only when we have a significant jump in both volume fraction and pressure may we expect different treatments of the non-conservative term to cause convergence to different numerical solutions. Such a situation is unlikely to occur for practical cases.

Remark 28. In this paper, we follow the approach of [17] and use a velocity of sound common to both phases (mixture sound velocity) to define the numerical sound velocity used in the van Leer and FVS fluxes. Paillere et al considered using the one-phase sonic velocities (dpp)1^2 for each phase respectively. The apparent freedom in choosing the numerical sound velocity may indicate that for two-phase flows, the exact value used for the parameter c is not critical for the performance of the AUSM schemes (see also [18]). However, a straightforward calculation shows that the two-phase FVS (van Leer) mass fluxes are pressure coherent in the sense of Definition 18 if and only if the same value Cj+1/2 is used for both phases. Consequently these schemes will be unstable unless a common velocity of sound is used.

The A USM scheme is mass coherent independent of the choice of numer- ffioZ aotmd Woc%, mW Aemce oko preaawe coAeremt (aec

Remark 29. To achieve a stable transition to-one phase flow, it was found necessary (see Section 5.3 of paper I) to abandon the ‘A USM” property (exact resolution of contact discontinuities) in favor of increased numerical dissipation near one-phase regions.

For the Euler equations, it has been demonstrated [27] that there is an intrinsic inconsistency between conservation of positivity and accurate resolution of contact discontinuities for FVS-type schemes.

5.2 Paper II

A Mixture Flux Approach for Accurate and Robust Resolution of Two-Phase FlowsCoauthor: Steinar Evje

In paper II the mixture flux (MF) strategy (see Section 4.3.2) is introduced. The motivation for the paper is the following observation made in paper I and [57]:

Although the AUSM type schemes (AUSM+, AUSMD, AUSMV) are able to provide accurate and efficient solutions to several benchmark two-phase flow cases, some problems are observed. In particular the schemes have a tendency to produce spurious oscillations around discontinuities in the volume fraction waves.

5.3. Paper III 49

In Section 4.3.2 the mixture flux strategy is presented in a general setting, allowing for a freedom of choice of convective fluxes. In this paper the AUSMD fluxes are used, yielding the spesific MF scheme termed MF- AUSMD. The basic conclusion is that MF-AUSMD solves the problems observed for AUSMD in paper I.

Remark 30. In this paper, a rather straightforward discretization of the nonconservative pressure terms is performed. In particular we do not discuss how to define weak integrals and make our numerical scheme consistent with such a definition, like we did in paper I. Nevertheless, as the numerical examples in the paper show, the schemes of paper I generally seem to converge to similar solutions as the mixture flux scheme.

It would be of interest to perform further work, both theoretical and numerical experiments, to shed more light on the effect of the discretization of the non-conservative term in the presence of discontinuities in both pressure and volume fraction. Ideally, we would like to impose a principle that guarantees the existence of a unique solution, and a principle that guarantees that o%r mmrenc&Z .scheme com;en?&9 Zo ZMs wwgwe

5.3 Paper III

Weakly Implicit Numerical Schemes for the Two-Fluid ModelCoauthor: Steinar Evje

When in paper II we dropped the “upwind” splitting based on the polynomials T± [49, 47] for the benefit of the Lax-Friedrichs splitting, we also introduced a temporal coupling between the pressure and velocity fields. That is, as the cell interface pressure Pj+1/2 is calculated from an evolution equation, we can consider the momentum equations as applying a backwards time integration for the calculation of the pressure terms.

This situation allows for impliciting the scheme in a very natural manner. By changing the time index n to (n + 1) for the fluxes in the pressure evolution equation, the pressure-momentum fields become coupled across the computational domain.

As is demonstrated in the paper, this simple change, denoted as weakly implicit MF-AUSMD (WIMF-AUSMD) results in

• Freedom from the CFL restriction for sonic waves, allowing for significantly increased computational efficiency.

• Increased accuracy on moving contact discontinuities (in fact, as is proved in paper V, the scheme allows for exact capture of such discontinuities).

Conservation and consistency are retained in the scheme.


Remark 31. As is demonstrated in the Appendix of paper III, a rescaling of the numerical sound velocity is required to apply a “transition fix” similar to the one used in paper I. This confirms the belief stated also in Remark 28; it is not essential that the numerical parameter c is associated with a physical sound velocity.

The basic conclusion of paper III is that WIMF-AUSMD is superior to MF-AUSMD in accuracy and efficiency on slow transients responible for mass transport. However, the WIMF scheme is highly diffusive on the rapid pressure waves.

5.4 Paper IV

Comparison of Various AUSM Type Schemes for the Two-Fluid ModelCoauthor: Steinar Evje

When the mixture flux method has been discussed in papers II and III, it has been exclusively in the context of AUSMD convective fluxes. In paper IV, the mixture flux method is investigated in a more general setting, and several aspects of the method are illuminated. However, in this paper only explicit schemes are considered.

In particular the following issues are studied:

1. In papers II and III linear relations were used for the thermodynamic density models. In paper IV we demonstrate that the MF method is extensible to non-linear state relations.

2. We shed some light on what kind of convective fluxes may work well within the MF context. In particular, it seems that the convective fluxes have little effect on the sonic waves, which are handled by the basic MF framework. Rather it is important to choose convective fluxes that work well on volume fraction waves.

3. We demonstrate that the explicit versions of the MF schemes place a rather strict stability criterion on the timestep. A fix is suggested that places more of the responsibility for sonic waves on the convective fluxes. Within this framework, denoted as RMF (Rescaled Mixture Flux), the RMF-AUSMD seems most promising because of the good stability properties of the basic AUSMD scheme on sonic waves (as demonstrated in paper I).

4. We demonstrate that a simple application of the MUSCL strategy of van Leer [86] allows for increased accuracy in the resolution of volume fraction waves.

5.4. Paper IV 51

Issue (4) deserves special mention here. By applying the minmod limiter to the primitive variables (p, a\,vg,v\) we achieve higher accuracy at the price of introducing slight numerical oscillations. Even though the numerical simulations indicate that these oscillations decay with grid refinement, they are undesirable.

As the first order version of the scheme is demonstrated to be non- oscillatory, it would seem that the observed TVD violation could be associated with a poor choice of limiting variables (i.e. the primitive variables).

The first order simulations of papers II, III and IV indicate that a simple convective momentum splitting (e.g. the AUSMD convective splitting of paper I) is sufficient to achieve stability of the scheme. However, in general one could consider mixture momentum variables associated with pressure waves and volume fraction waves respectively. A natural choice for such momentum variables would be to start with the differentials

dp = n(pidmg + pgdrni), da i — k{———a\drfiv dp + Vm0 (5-1)

and replace the mass differentials with corresponding mass flux differentials as follows:

dip — /v (pi dig ~\~ pgd/i), dJ* = + ^agd/i). (5.2)op op

In general, one could explore different choices for Ip and Ia and then proceed to

1. Consider the possibility of constructing mixture momentum fluxes Gk,j+1/2 from a diffusive component G^-+1^2 associated with pressure waves and an accurate component Gf^J+1//2 associated with volume fraction waves.

2. Apply slope-limiters to the variables (p, op Ip, Ia) with the aim of obtaining a fully non-oscillatory scheme of higher order spatial accuracy.

This may be an interesting direction for further work.

To summarize, the main conclusions of this paper are:

• The mixture flux method may be used in conjunction with the AUSM+ scheme suggested by Paillere et al [57], and the resulting MF-AUSM+ is an improvement of the basic AUSM+ scheme.

• A rescaling technique denoted as RMF may be applied, making the explicit ME framework stable for larger timesteps.


5.5 Paper V

CFL-Free Numerical Schemes for the Two-Fluid ModelCoauthor: Steinar Evje

The contributions of paper V may be summarized as follows:

1. By applying linear convective fluxes for the FA-component of the MF- method, a strongly implicit (i.e. free from any CFL criterion) scheme may be constructed. Such a scheme, denoted as SIMF-AUSM, is proposed.

2. By a natural modification of the WIMF-AUSMD scheme considered in paper III, a WIMF-AUSM scheme is obtained. A comparative study of WIMF-AUSM and SIMF-AUSM is made. In particular, it is observed that WIMF-AUSM is more accurate on slow transients than SIMF- AUSM.

3. In particular, we prove that by choosing the timestep optimally, schemes of the WIMF class are able to capture a moving contact discontinuity exactly. Neither explicit schemes or schemes of the SIMF class can possess this property.

SIMF schemes however, are demonstrated to be highly efficient in calculating solutions when accurate resolution of volume fraction fronts is not essential, for instance for steady state calculations.

5.6 Conclusions and Further WorkThe most important conclusions of the thesis may be summarized by the following points:

5.6.1 AUSM type schemes for the Two-Fluid ModelPaillere et al [57] investigated an extension of the AUSM 1 scheme of Lion[47] to the two-fluid model. In paper I, we investigated an extension of the AUSMD scheme of Wada and Lion [89] to the same two-fluid model.

These investigations were made independently, and both illustrate some stability problems in the resolution of volume fraction waves. This suggests that such problems are likely to be associated with the basic AUSM strategy itself, rather than the spesihc implementation of AUSM.

Regarding the differences between AUSM(+) and AUSMD/V, it would seem (in particular by the results of paper IV) that the latter approach has an advantage over AUSM(+) in robustness on the sonic waves. Such an observation was also made by Evje and Fjelde for the drift-flux model [17] as well as Wada and Lion [89] for the Euler Model.

5.6. Conclusions and Further Work 53

Conclusions

The work of this thesis seems to demonstrate that the naive extension of AUSM type schemes to two-phase flow models (i.e. treating the two-phase flow model basically as two separate single-phase models) does not yield fully robust schemes, and phasic couplings should be taken into account to obtain a numerical scheme that is both accurate and free of oscillations.

Hence we may make the preliminary conclusion that standalone AUSM type schemes may not be candidates of choice for use in practical applications of the two-fluid model. Modifications along the lines of the MF schemes investigated in this thesis seem more promising.

5.6.2 Mixture Flux Methods for Two-Phase FlowsThe Mixture Flux methods introduced in this thesis have been demonstrated to possess desirable efficiency, accuracy and robustness properties. In particular we obtain a level of robustness and accuracy comparable to the Roe scheme with a significant gain in efficiency.

The Weakly Implicit Mixture Flux (WIMF) scheme looks very promising when it comes to efficient and accurate resolution of slow transients, which is the problem of relevance to the oil industry. This is an interesting observation in view of the fact that the most commonly used industrial simulators [5, 41] should be classified as Strongly Implicit schemes, and are consequently not as accurate.

Further Work

• The possibility of taking the mixture flux concept even further, by applying it also to the convective momentum flux Gj+i/2, should be explored (see Section 5.4).

• Extensions to higher order spatial accuracy should be systematically explored, both for the pressure waves (FD-component) and volume fraction waves (FA-component).

• Extensions to a two-fluid model involving energy equations should be considered.

• The MF strategy, in particular the WIMF strategy, should be evaluated for industrial applicability by testing on large-scale field cases. For this purpose, more physically realistic friction models should be incorporated.

• Extensions to 2D and 3D-models should be considered.

• Extensions to related two-phase flow models (in particular, the drift- flux model) should be explored.


5.6.3 Extensibility to Other ModelsThe main steps in the MF philosophy can be described as follows:

1. Identify a class of contact discontinuities for the system that we wish to resolve accurately.

2. Identify a variable that is constant across the contact.

3. Construct a flux FD that is stable in the variable that is constant across the contact.

4. Construct a flux FA that is accurate across the contact.

5. Hybridize FA and FD such that we recover an accurate resolution of the contact, while incorporating additional stability properties from FD.

Further Work

It would seem that several physical models have a mathematical representation in terms of conservation laws that allow for an analysis as given above. It would be of interest to investigate whether the ideas presented in this thesis are applicable to other systems of conservation laws.

Appendix A

Pressure Gradients

%(%)

P\{x)---------

dx'

pg(x + dx)

Pi (re + dx)

Figure A.l: The pressure forces on a pipe volume element.

In the following, we let pg be the pressure of the gas phase, pi the of the liquid phase, and p1 the pressure on the gas-liquid interface.

As can be seen from Figure A.l, the differential terms involving forces are given as

_ d(aipi) i&xi

pressure

pressure

(A.l)

where /pi is taken to be negative along x (i.e. /pi is a negative term if placed on the right hand side of the liquid momentum equation). The first term represents the rate of change of the net force on vertical segments of the element, whereas the second term is the component in the ^-direction of the interface pressure force on the inclined surface.

Symmetry yields the equivalent expression for the gas phase

, _ a(agPg)PS" (A-2)

IntroducingApt = Pt - (A.3)

55

56 Appendix A. Pressure Gradients

the result can be written for phase k as

^ = dx + °'kW (A-4)

A.l HydrostaticsIn this section, expressions for the hydrostatic pressure terms are derived, assuming a circular pipe. We recover expressions commonly used in the literature [13, 53].

Figure A.2: Pipe cross section.

A.1.1 Liquid PhaseAs can be seen from Fig. A.2, the wetted area A\ is given by

R2A\ = -^-(cv — sin tv), (A.5)

where tv is the wetted angle and R is the radius of the pipe.The liquid level h is given by

h = R(l — cos0 . (A.6)

The hydrostatic contribution phi to the pressure at any point in the liquid section is

Phi = M{h - h!) cos 9, (A.7)

taking the liquid level as the zero point, h! is the height of the point in question and 9 is the inclination of the pipe.

A.l. Hydrostatics 57

The area averaged height in the liquid is

f h'dAi f h'dAih

JdA, A,From Eq. (A.5) we find

R2dA\ — ^ (1 — cos uj^duj.

Thus

h'dA\r ^ ,I II — cos I (1 — COS UJ jdco

R? ( . . w 1 . 3w— I u - sinw — sm - + - sm —

which yields for h

h = R ( 1 + R?_2Ai

R?_ 2Ai

1 . 3a; . uj- sm — - sm -

uj 9 uj 1 , q a; . uj^2^" 2-3^ 2-^2

= R 1 +a;/ g a; \ 1 . 3

sin — cos — — 1 — - sin2 \ 2 / 3

UJ

Introducing the liquid area fraction cq given byA\ _ _Ai_ _ 4Ai

ttR2 7tD2 ’cq A ■^■pipewe find

h = R ( 1 -"(j A- sm ! -37rcq 2

and for the average hydrostatic pressure in the liquid phase

Phi — pig cos 9{h — h) — pi ( —- cos — +UJ

(A.8)

(A.9)

(AAO)

(ATI)

sin3 ^ ) gDcos 9. (AA2)

With p\ as the interface pressure, we find for the total average pressure pi in the liquid

Pi = Pi + Phi- (AA3)Defining Api as

Api = Pi - Pi, (A.14)We find

Ap1 = -phl = -p1^-5cos- + —

This is the result as presented by [13].

UJ sm” % 1 gD cos 9. (AA5)

58 Appendix A. Pressure Gradients

A.1.2 Gas PhaseThe derivation for the gas phase is similar. Exchanging the phases and turning the pipe upside down (i.e. reversing the direction of g) we find by- symmetry

cv„ft*--M-5=oSy + — sin' y ) gD cos 9,

where tvg is the dry angletVg — 2ir tv .

Which means that Eq. (A.16) can be written as

/ 1 tv 1 . 3 tv .Phs = -Pg -5cos--—sm ¥]SDcos9,

(A.16)

(A.17)

(A.18)

and we find

Apg = —Phg = —Pg cos — + sin3 — ^ gD cos 9. (A.19)

We have made the simplifying assumption that the gas density pg is constant across the pipe cross section.

A. 1.3 An Equivalent FormulationUsing Eq. (A.5) and Eq. (A.10) we find

cq tv — sm tv 2tt

Now a\Api can be written as

cq tv : 1 . 3 tvcqApi — — pi ( —— cos — + — sin — ) gD cos 9

~P isin tv — tv tv 1 . q tv . ^

cos — + — sm — gD cos 94tt 3tt

(A.20)

/ 1 .tV 2 tV 1 tV 1 . 3 tvX_ -p^—sm-cos ___wcos_ + _sm -jjDcosO

/ 1 . 3tv 1 tv 1 . tv\- -A Wsm 2 "SWC0S2+2^Sm2)9'DcOS'>-

We find

^(cqApi) / 1 . 2W tV 1 .tV.■_P,WSm Is0008"' (A.21)

From Eq. (A.6) we finddu) _ 4dfi Dsin(tv/2)

(A.22)

A.l. Hydrostatics 59

We have

d(aiApi) d(aiApi) duodh duj dh

fl.u u 1 . „= —pi — sin — cos — — —u ) g cos 9

2tt= pictig cos 9.

For the gas phase we similarly find

d(agApg) = dh

2 2 2%-

127TC

7r

— —Pi I %— sin u) — ——uj 1 g cos 9

PgOg^COS^,

as the reversals of direction of h and g cancel out.These results can be written as

<9((%kApk) #((%kApk) gA—§r~ = si^di =

Substituting in (A.4), we get

,'dpl dh+"cosS&

(A.23)

(A.24)

(A.25)

which is a mathematically equivalent result as given by [2], valid for more general pipe geometries.

A drawback of the simplification (A.25) is that it involves the transformation of a conservative product into a non-conservative one through (A.24), which may be an inconvenience when devising numerical schemes.

Remark 32. It should be kept in mind that these expressions rely heavily on the assumption that the phases flow in separate layers (stratified flow), as indicated in Figure A.2. For mixed flows (bubbly flows, annular flows) the expressions (A.25) have no physical relevance whatsoever.

Appendix B

Eigenstructure Analysis

In this appendix, an eigenvalue analysis is performed on the isentropic two- fluid model. Although the model formally allows for exact results to be derived in algebraic form, the resulting expressions are of enormous complexity and this is not a practical approach. Instead we follow a suggestion of Toumi [81, 82], and use a perturbation technique to derive a power series approximation.

First we consider the basic equal-pressure model, and recover the well- known fact that this model yields complex wave velocities for vg ^ v\. We then proceed to demonstrate that, by the introduction of an interface pressure correction term, a modified model is hyperbolic for a wider range of physical parameters.

The extensibility of these isentropic results to the full 6-equation two-fluid model is discussed in Appendix C.

B.l The Equal Pressure ModelWe consider first the basic 1-dimensional isentropic two-fluid model, givenby

• Mass conservation

(B.l)

(B.2)

• Momentum balance

60

B.l. The Equal Pressure Model 61

Where the nomenclature is as follows

pg - gas density pi - liquid density Vg - gas velocity V\ - liquid velocity ag - gas fraction a\ - liquid fractionp - pressureQg - non-differential gas momentum sources Qi - non-differential liquid momentum sources

Closure of the system is achieved by the relations

ag + a i = l (B.5)

and thermodynamic relations for the pressure

P = PW = P(Pg)

with inverses

andPg — Pg(p)

a = &&)-

(B.6)

(B.7)

(B.8)We are now left with a set of 4 independent primitive variables given by the vector

" p " Wi«1 W2Vg Wg

. yl . to4

W =

with the corresponding vector of conservative variables

U

(B.9)

pgOLg Uip\a\ u2

Pg%%plOi\V\

(B.10)

Remark 33. In transforming between the variables W and U, we will find the transformation matrix

dVB =<9W (B.ll)

useful.We find

B(W)

-pg 0 0

pi 0 0

—pgvg pgGg 0

PlV\ 0 pia\ _

(B.12)

62 Appendix B. Eigenstructure Analysis

which has the inverse

where

B1

Kpg 0 0

%ng 0 0

0 l 0Psas0

Pgag0 1

pm

1

pm

dp ( dui^ " r + pg

(B.13)

(B.14)

(B.15)

B.1.1 Quasilinear FormThe quasilinear form in conservative variables is given by

w + A(u)^ = q(u)- (ai6)

The Jacobi matrix for the conservative variables, but written in terms of primitive variables, is

A(U)

0 0 100 0 0 1

Kp\ag — v2 KyOgttg 2ug 0Kp\a\ npga\ — v2 0 2ui

(B.1T)

B.1.2 EigenvaluesWriting z = Kpiag, the eigenvalues of the matrix A (B.17) are the roots of the polynomial equation

(z — (A — ug)2) {zk - (A — ui)2) - z2k = 0 (B.18)

where& = (B.19)

We now observe that writing the eigenvalue as

A = v\ + ac, (B.20)

and substituting in (B.18), we obtain the following equation for a

(l — (1 + k)(a — e(l + k))2) (k — (1 + k)a2) = k (B.21)

c(l + A;)

where(B.22)

B.l. The Equal Pressure Model 63

and c is a sound velocity given by

c = y1 z( 1 + k). (B.23)

Hence a depends only on e and k. A power series expansion yields

00

a(e, k) = ^2 Pi(k)el. (B.24)i-o

The coefficients can now be found by repeatedly solving equation (B.21) to the corresponding order in e. We will here present the third order accurate results.

Downstream Pressure Wave

From (B.21) we get

• A) = 1

• A = 1

• A = #

• /% = 2k{k — 1).

This yields the eigenvalue

W+ = + C ^ (B.25)

Upstream Pressure Wave

From (B.21) we get

• A) = — i

• A = i

' A = -#

• /?3 = 2k (k — 1).

This yields the eigenvalue

AP- = hiii -c( 1 + ye2 - 2 k(k - l)c3 + 0(e4)) . (B.26)


Downstream Volume Fraction Wave

From (B.21) we get

• /?o = 0

• pi = k + iVk

• /32 = 0

• /% = §(! — i'/k)i\/k.This yields the eigenvalue

A"+ = - «,) +a/i(1_ + 0(£4)) . (B.27)1 T k \ 2 J

Upstream Volume Fraction Wave

From (B.21) we get

• A) = o

• A = k — i'/k

• A> = 0

• /?3 = t(1 + iVk)4Vk-This yields the eigenvalue

Av— = + kv‘ ~ ~ -a(hl+ iVib)4Vfc3 + o(e4)). (B.28)IT k \ 2 y

The eigenvalues corresponding to volume fraction waves are complex, having the physical interpretation that the equal pressure model can not sustain stable travelling volume fraction waves (see Section 2.3.2).

We note that compressibility has only a third order effect on the wave velocity.

B.1.3 EigenvectorsLeft Eigenvectors

Using equation (B.20) and solving for w in

uA = Xu

we obtain

where

u

\ — 2vg(A-2%)f

15

£ — (l — (1 + k)(a — (IT k)e)2).

(B.29)

(B.30)

(B.31)

B.2. An Extended Model 65

Right Eigenvectors

Using equation (B.20) and solving for u in

Ao; = Xu

we obtain

where

1

.AC.

C = —— (l — (1 + k)(a — (1 + A:)e)2) .Pg

(B.32)

(B.33)

(B.34)

B.2 An Extended ModelOften the basic equal pressure model is modified in order to make it hyperbolic, i.e. having real eigenvalues, as described in Section 3.2.2. Generalizations of the method applied here may then be used. In particular, we will now analyse the model being studied in the accompanying papers.

We use the equal pressure model of the previous chapter as a starting point and assume that the pressure at the interface is modified with a correction term Ap, yielding the set of equations

• Mass conservation9 9^ (pg%) + (/%%%) = o,

<9 9— (a«0 + (pmvi) = o.

• Momentum balance9 , ,— (PgWgUg) +

^ (A«m) +

d_

d_9a;

/ 2\ 9p / jv 0oi& „(pgCKgUg) + (%g— + (p - P )-^ = Qg,

(piaivf) + ai-^ + (p-p1)-^ = Qi-

The interface pressure p1 is given by

(B.35)

(B.36)

(B.37)

(B.38)

p — p1 = Ap(W).

The modified jacobi matrix is

(B.39)

A(U)

00

K(piag + Apozi|^) - u2 n(piai — Apai^)

0 1 00 0 1

^{pg^g ~ Apttg-^) 2vg 0/((pgcri + Ap«g^) - 0 2n%

(B.40)


B.2.1 EigenvaluesThe eigenvalues of the matrix A (B.40) are the roots of the polynomial equation

k (pittg + Apai^j - (A - vg)z

-k2 (^pgCtg - Apa.

(B.41)

dpsK gp

k ^pgai + Apag^^j - (A - v\Y

Aai — = 0-

Cortes et al [11] note that a natural requirement for Ap is that

Ap —v 0 as (vg — ui) —y 0,

so we assumeApdpgA

TV

for some V(W). We also introduce the new variables

aia

The eigenvalue equation can now be written as

[z (1 + paPs2) - (A - ug)2] [z (k + Vs2) - (A - i>i)2] —z2 (k — aVs2) (1 — pVs2) = 0.

(B.42)

(6.43)

(B.44)

(6.45)

(6.46)

Again writing the eigenvalue as

A = v\ + ac (6.47)

yields the following equation for a

[l + paVs2 — (1 + k) (a — s{ 1 + k))2] [k + Vs2 - (1 + k)a2] (B.48)= (k — aVe2) (1 - pVe2).

A power series expansion yields

a(s, V, k, p, a) = ^ j5i{k, p, d, V)s\ (B.49)i

Solving for the coefficients yields


Downstream Pressure Wave

From (6.48) we get• A) = i

• A = i

e A = # + (k—a)(l—fi) sp2(l+fc)2 '

• A = - 1) + (^+M)(l-*)+2*(A««-l)p

Upstream Pressure Wave

From (6.48) we get• A) = — i

• At = i

• *=-t -

• A = 2&(& - 1) + («+Mfe)(i-fc)+2fc(^~i):pj

Now writing the pressure eigenvalues as

Ap = up ± c (6.50)

we findkv i + Vo•”p — ——-+c

1 + k2k{k — 1) +

{a + /ik)(l — k) + 2k(jia — 1) \ 3 4

(1 +

andc = c 3k {k — Q')(l — /x).

2 + 2(1 +A;)2a+(w+

(6.51) "

(6.52)

Volume Fraction Waves

We will find it convenient to introduce the shorthand

9 =(1 + (J<k) (1 + dz)

(TTW2P-A;. (6.53)

Downstream Volume Fraction Wave

From (6.48) we get

» A) = o• Pi = k + g

• A = 0a _ {l+k){g-l)2{k+g)2 -({g-l)2 +ag,{k+g)2)v

• ^ - 2(l+t)g


Upstream Volume Fraction Wave

From (B.48) we get

• A) = o

• Pi=k-g

• /32 = 0Q _ (l+k)(g+l)2 (k-g)2 -((g+l)2 +&ii(k-g)2)v

• ~ 2(1+k)g

Writing the volume fraction eigenvalues as

A* = ± 7, (B.54)

we find

v\ + kv gva = fic1 + k

Here 7 is found to be

2 k(k — 1)(d + pk)( 1 — k) + 2k(pa — 1)^ ^ 3

(1 + A;)2) d* + O(^)

(B.55)

-y/Ap(i0ga1+/?1ag)-(t)g-^)2/>1pga1ag I ~ (PgOl+pittg)2 (B.56)

+c (s3 + (V - 4t + 1)5 - p + j) + C(^)

B.2.2 Some RemarksRemark 34. In the limit V —> 0 the coefficients reduce to the previously calculated coefficients for the equal pressure model.

Remark 35. All higher order corrections for va must exactly cancel the corresponding corrections for up due to the following Lemma

Lemma 3.va + VP = V\ + Vg. (B.57)

Proof. We note thattrace (A) = 2(ug + ui). (B.58)

From fundamental algebra we have

A j = trace (A). (B.59)

By definition of va and up

y^A i = 2(va + vp) (B.60)

and the result follows. □


Remark 36. The third order coefficient for 7 diverges as g 0, casting doubts on the general convergence properties of the method.

Remark 37. We see that the eigenvalues are read only if g is real, or equivalently

Ap > lhPia'°g («„ - t',)2.Pg0t\ +

(B.61)

T/ie pressure correction g = 0 is used for some physical parameters by the CATHARE code [7], presumably because it in some sense is the minimal correction that will make the equal pressure model hyperbolic.

Remark 38. For the volume fraction waves, we note that to zeroth order ine we /m%e

Av- = Av+ = vh (B.62)

The eigenvector is

We obtain

Hence

Pg-PiPg^i

~P\V\

d\ _ 1 <9U piai

-Vi01

0

(B.63)

(B.64)

(B.65)

and, by Definition 2, the associated characteristic field is linearly degenerate to zeroth order in epsilon.

Appendix C

Entropy and Energy Equations

We consider for simplicity the homogeneous equal pressure model, written as

• mass:ddt (pa) +

d_ckc

(pav) = 0,

• momentum:ddt

(pav) +d_dx

/ 9 . do(pav + ap) = p—,

(Cl)

(C.2)

• energy:

a /i 2 \ a^t-pav +pae 1+^ v ( -pav +pae + ap +p— — 0. (C.3)

Here we assume a phase index (g or 1) on each variable.

C.l The Two-Phase Energy EquationsWe now focus on the energy equation. Isolating the terms involving the internal energy we obtain

IG'””2)+Mv (w"2+ap+ p^-RHS, (C.4)

where RHS isRHS = ——(pae) — (paev). (C.5)

Expanding derivatives and cancelling some terms by use of the mass equation, (C.4) can be written out as

dv 2dv d , . da T.TTr,paV- + pay—+ — (yap) +p—- RHS, (C.6)

or

vaa~di + pavdi + di(ap\ a# crt (C.7)

70

C.2. Eigenstructure of the Full Model 71

Using the mass and momentum equations this can be simplified to

da dv da r,TTn vpte+apai+p-M~ms-

We now write this as

f d , . da\ d , x d , , np (a£(m,) +m) + + Tx(paev) = °-

Using the mass equation we obtain

( d , . da\ f de de\p{diiav)+m)+pa U + Wj=0'

We now take advantage of the thermodynamic potential

Pdc = Tds H—-dp,P2

(C.8)

(C.9)

(C.10)

(C.ll)

where s is the spesific entropy. Eq. (C.10) can now be written as

-(£'*•'*6) *"($*•£) *7@ *•$)-■ ™

which can be written more simply as

l ijt(pa) + + paT (fm+w)_ °’ (c-13)

or by the mass equationds dsm+vdi 0. (C.14)

Remark 39. We observe that (C.14); and hence the energy equation, is trivially satisfied for ds = 0. This justifies terming the model resulting from removing the energy equations the isentropic model.

C.2 Eigenstructure of the Full ModelWe observe that s becomes a characteristic variable in (C.14), advected by the fluid velocity v. We now introduce the vector of variables

U =

Pgttg

PgQJgUgP\<X\Vi

%Si

(C.15)

72 Appendix C. Entropy and Energy Equations

and write the full system on quasilinear form

dUdt

+ A(U)f = 0.

Here the Jacobi matrix A is

(C.16)

A(U)

0 0 1 0 0 0

0 0 0 1 0 0

KpiCXg — Vg KPgCXg 2vg 0 % w, “s W,

Kp\a\ KpgOi\ - if 0 2v\ ^ w, (*),0 0 0 0 % 0

0 0 0 0 0 V\(C.17)

whereK = --- r----------- t-—r--------. (C.18)

(%) ^ % + ("^) ,

The eigenvalues of this matrix are now seen to be vg and v\ as well as the eigenvalues of the submatrix

A'(U)

0 0 10 0 0 0 1

KpiOig — Ug KPgOg 2ug 0Kp\a\ Kpga\ — v\ 0 2v\

(C.19)

which is precisely the Jacobi matrix (B.17) for the standard isentropic system. We can therefore conclude that

Remark 40. For the standard nonhyperbolic equal pressure model, the addition of energy equations does not change the basic eigenstructure of the system. However, entropy waves are added moving with the fluid velocities Vk- We note that as opposed to the isentropic case, the compressibilities no longer depend only on p and should be evaluated at constant entropies to yield values appropriate for calculating the mixture sound velocity. In this respect we obtain the same result as for the Euler equations [80].

We note that including an interface pressure correction Ap complicates the picture somewhat. However, this situation has been analysed by Toumi [81], and the results of this appendix hold to first order in s as given by(B.22).

Bibliography

[1] R. Abgrall. How to prevent pressure oscillations in multicomponent flow calculations. J. Comput. Phys, 125:150-160, 1996.

[2] D. Barnea and Y. Taitel. Kelvin-Helmholtz stability criteria for stratified flow: Viscous versus non-viscous (inviscid) approaches. Int. J. Multiphase Flow, 19(4):639-649, 1993.

[3] F. Barre et al. The CATHARE code strategy and assessment. Nucl.124(3)=257-284, 1990.

[4] K. Bendiksen and M. Espedal. Onset of slugging in horizontal gas-liquid pipe flow. Int. J. Multiphase Flow, 18:237-247, 1992.

[5] K. H. Bendiksen, D. Malnes, R. Moe, and S. Nuland. The dynamic two-fluid model OLGA: Theory and application. In SPE Production Engineering, volume 6, pages 171-180, 1991.

[6] K. H. Bendiksen, D. Malnes, and 0. J. Nydal. On the modelling of slug flow. Chem. Eng. Comm, 141-142:71-102, 1996.

[7] D. Bestion. The physical closure laws in the CATHARE code. Nucl. Eng. Design, 124:229-245, 1990.

[8] F. Bouchut, Y. Brenier, J. Cortes, and J.-F. Ripoll. A hierarchy of models for two-phase flows. J. Nonlinear Sci, 10:639-660, 2000.

[9] F. Coquel, K. El Amine, E. Godlewski, B. Perthame, and P. Rascle. A numerical method using upwind schemes for the resolution of two-phase flows. J. Comput. Phys, 136:272-288, 1997.

[10] J. Cortes. On the construction of upwind schemes for non-equilibrium transient two-phase flows. Comput. Fluids, 31:159-182, 2002.

[11] J. Cortes, A. Debussche, and I. Toumi. A density perturbation method to study the eigenstructure of two-phase flow equation systems. J. Comput. Phys, 147:463-484, 1998.

[12] A. Courbot. Prevention of severe slugging in the dunbar 16” multiphase pipeline. In Offshore Technology Conference, Houston, Texas, May 1996.OTC 8196.

73

74 BIBLIOGRAPHY

[13] V. De Henau and G. D. Raithby. A transient two-fluid model for the simulation of slug flow in pipelines-I. Theory. Int. J. Multiphase Flow,21(3):335-349, 1995.

[14] D. Drew, L. Cheng, and R. T. Lahey Jr. The analysis of virtual mass effects in two-phase flow. Int. J. Multiphase Flow, 5:233-242, 1979.

[15] J. R. Edwards, R. K. Franklin, and M.-S. Lion. Low-diffusion fluxsplitting methods for real fluid flows with phase transition. AIAA Jour-W, 38:1624-1633, 2000.

[16] G. 0. Eikrem, L. Imsland, and B. Foss. Stabilization of gas lifted wells based on state estimation. In IFAC-symposium Adchem ’2003, Hong Kong, January 2004.

[17] S. Evje and K. K. Fjelde. Hybrid flux-splitting schemes for a two-phase flow model. J. Comput. Phys, 175:674-701, 2002.

[18] S. Evje and K. K. Fjelde. On a rough AUSM scheme for a onedimensional two-phase flow model. Comput. Fluids, 32:1497-1530, 2003.

[19] I. Faille and E. Heintze. A rough finite volume scheme for modeling two-phase flow in a pipeline. Comput. Fluids, 28:213-241, 1999.

[20] K. Falk. Pressure Pulse Propagation in Gas-Liquid Pipe Flow. Dr. ing. thesis, Department of Petroleum Engineering, NTNU, Norway, 1999.

[21] K. K. Fjelde. Numerical Methods for Simulating a Kick. Gaud, sclent. thesis, Department of Mathematics, University of Bergen, Norway, December 1995.

[22] K. K. Fjelde. Numerical Schemes for Complex Nonlinear Hyperbolic Systems of Equations. Dr. sclent, thesis, Department of Mathematics, University of Bergen, Norway, April 2000.

[23] K. K. Fjelde and K. H. Karlsen. High-resolution hybrid primitive- conservative upwind schemes for the drift flux model. Comput. Fluids,31:335-367, 2002.

[24] Institute for Energy Technology. Annual report, http://www.ife.no/,2002.

[25] S. K. Godunov. A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb, 47:357-393, 1959.

[26] D. Gottlieb and E. Turkel. Boundary conditions for multistep finite- difference methods for time-dependent equations. J. Comput. Phys,26:181-196, 1978.

BIBLIOGRAPHY 75

[27] J. Gressier, P. Villedieu, and J.-M. Moschetta. Positivity of flux vector splitting schemes. J. Comput. Phys, 155:199-220, 1999.

[28] A. Harten, P.D. Lax, and B. Van Leer. On upstream differencing and Godunov schemes for hyperbolic conservation laws. SIAM Review, 25:35- 61, 1981.

[29] T. J. Hill. Gas injection at riser base solves slugging, flow problems. Oil k, Gas Journal, Feb. 26:88-92, 1990.

[30] C. Hirsch. Numerical Computation of Internal and External Flows, volume 2 of Computational Methods for Inviscid and Viscous Flows. John Wiley & Sons, England, 1990.

[31] J. F. Hollenberg, S. de Wolf, and W. J. Meiring. A method to suppress severe slugging in flow line riser systems. In 7th BHR Group Ltd et al Multiphase Prod. Int. Conference, pages 89-103, Cannes, France, 1995.

[32] T. Y. Hon and P. G. LeFloch. Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comp, 62(206):497-530, 1994.

[33] L. Imsland, B. Foss, and G. O. Eikrem. State feedback control of a class of positive systems: Application to gas lift stabilization. In ECC’03, Cambridge (UK), September 2003.

[34] M. Ishii. Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eymiles, France, 1975.

[35] R. I. Issa and P. J. Woodburn. Numerical prediction of instabilities and slug formation in horizontal two-phase flows. In Third international conference on multiphase flow. ICMF, June 1998.

[36] R.I. Issa and M.H.W. Kempf. Simulation of slug flow in horizontal and nearly horizontal pipes with the two-fluid model. Int. J. MultiphaseMow, 29:69-95, 2003.

[37] G. S. Jiang, D Levy, C. T. Lin, S. Osher, and E. Tadmor. High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws. SIAM J. Numer. Anal, 35:2147-2168, 1998.

[38] A. C. V. M. Lage, K. K. Fjelde, and R. W. Time. Underbalanced drilling dynamics: Two-phase flow modeling and experiments. In 2000 IADC/SPE Asia Pacific Drilling Technology, Malaysia, September 2000. IADC/Society of Petroleum Engineers. SPE 62743.

[39] R. T. Lahey Jr, L. Y. Cheng, D. A. Drew, and J. E. Flaherty. The effect of virtual mass on the numerical stability of accelerating two-phase flows. Int. J. Multiphase Flow, 6:281-294, 1980.

76 BIBLIOGRAPHY

[40] M.-H. Lallemand and R. Saurel. Pressure relaxation procedures for multiphase compressible flows. Rapport de Recherche n* 4038, INRIA Rocquencourt, France, October 2000.

[41] M. Larsen, E. Hustvedt, P. Hedne, and T. Straume. Petra: A novel computer code for simulation of slug flow. In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, October1997. SPE 38841.

[42] P. Lax and B. Wendroff. Systems of conservation laws. Communication on Pure and Applied Mathematics, 13:217-237, 1960.

[43] P. D. Lax. Asymptotic solutions of oscillatory initial value problems.m&e Moth. J, 24:627-648, 1957.

[44] P. D. Lax. On the notion of hyperbolicity. Comm. Pure Appl. Math,23:395-397, 1980.

[45] R. J. LeVeque. Numerical Methods for Conservation Laws. Lectures in Mathematics. Birkhauser, ETH Zurich, 1992.

[46] H. Linga and P. Hedne. Suppression of terrain slugging with automatic and manual riser choking. SINTEF Report no. STF 11 A91006, January 1991.

[47] M.-S. Liou. A sequel to AUSM: AUSM(+). J. Comput. Phys, 129:364-382, 1996.

[48] M.-S. Liou. Recent progress and applications of AUSM+. In Sixteenth International Conference on Numerical Methods in Fluid Dynamics, pages 302-307, Berlin, 1998.

[49] M.-S. Liou and C. J. Steffen. A new flux splitting scheme. J. Comput.107:23-39, 1993.

[50] L. Livshits, G.W. MacDonald, B. Mathes, and H. Radjavi. The Jordan-form proof made easy. Preprint, available online at http://www.colby.edu/personal/l/llivshi/*Jordan .pdf.

[51] R. J. Lorentzen. Higher Order Numerical Methods and Use of Estimation Techniques to Improve Modelling of Two-Phase flow in Pipelines and Wells. Dr. scient. thesis, Department of Mathematics, University of Bergen, Norway, June 2002.

[52] J. M. Masella, I. Faille, and T. Gallouet. On an approximate Godunov scheme. Int. J. Comput. Fluid. Dyn, 12:133-149, 1999.

[53] J. M. Masella, Q. H. Tran, D. Ferre, and C. Pauchon. Transient simulation of two-phase flows in pipes. Int. J. Multiphase Flow, 24:739-755,1998.

BIBLIOGRAPHY 77

[54] H. Nessyahu and E. Tadmor. Non-oscillatory central differencing for hyperbolic conservation-laws. J. Comput. Phys., 87(2):408-463, 1990.

[55] Y-Y. Niu. Simple conservative flux splitting for multi-component flow calculations. Numer. Heat Trans, 38:203-222, 2000.

[56] Y-Y. Niu. Advection upwinding splitting method to solve a compressible two-fluid model. Int. J. Numer. Meth. Fluids, 36:351-371, 2001.

[57] H. Paillere, C. Corre, and J.R.G Cascales. On the extension of the AUSM+ scheme to compressible two-fluid models. Comput. Fluids,32:891-916, 2003.

[58] H. Paillere, A. Kumbaro, C. Viozat, and S. Clerc. Development of an AUSM+ type scheme for the homogeneous equilibrium two-phase flow model. Seminaire Schemas de Flux pour la Simulation Numerique des Ecoulements Diphasiques — Reunion de Groupe de Travail CEA-EDF- CMLA, Cargese, September 1999.

[59] C. Pauchon, H. Dhulesia, G. B. Cirlot, and J. Fabre. TACITE: A transient tool for multiphase pipeline and well simulation. In SPE 69th Annual Technical Conference and Exhibition, New Orleans, USA, September 1994. Society of Petroleum Engineers. SPE 28545.

[60] C. Pauchon, H. Dhulesia, D. Lopez, and J. Fabre. TACITE: A comprehensive mechanistic model for two-phase flow. In BHRG Conference on Multiphase Production, Cannes, June 1993.

[61] H. Pokharna, M. Mori, and V. H. Ransom. Regularization of two-phase flow models: A comparison of numerical and differential approaches. J. Comput. Phys, 134:282-295, 1997.

[62] R. Radespiel. Efficient computation of compressible flows. In Sixteenth International Conference on Numerical Methods in Fluid Dynamics, pages 237-253, Berlin, 1998.

[63] V. H. Ransom et al. RELAP5/MOD1 Code Manual Volume 1: Code Structure, System Models, and Numerical Method. NUREG/CR-1826, U.S. Nuclear Regulatory Commision, 1982.

[64] P. L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys, 43:357-372, 1981.

[65] P.L. Roe. The use of Riemann problem in finite difference schemes. Lectures Notes of Physics, 141:354-359, 1980.

[66] J. E. Romate. An approximate Riemann solver for a two-phase flow model with numerically given slip relation. Comput. Fluids, 27(4) :455-477, 1998.

78 BIBLIOGRAPHY

[67] R. Saurel and R. Abgrall. A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys, 150:425-467,1999.

[68] R. Saurel and R. Abgrall. A simple method for compressible multifluid flows. SIAM J. Sci. Comput, 21:1115-1145, 1999.

[69] Z. Schmidt, J. P. Brill, and H. D. Beggs. Choking can eliminate severe pipeline slugging. Oil & Gas Journal, Nov. 12:230-238, 1979.

[70] J. Smoller. Shock Waves and Reaction-Diffusion Equations. Springer Verlag, 1994.

[71] J. Song and M. Ishii. The one-dimensional two-fluid model with momentum flux parameters. Nucl. Eng. Design, 205:145-158, 2001.

[72] S. Song and G. Kouba. Fluids transport optimization using seabed separation. In Proceedings of ETCE/OMAE2000 Joint Conference, New Orleans, February 2000. ETCE2000/PROD-10051.

[73] H. B. Stewart. Stability of two-phase flow calculation using two-fluid models. J. Comput. Phys, 33:259-270, 1979.

[74] H. B. Stewart and B. Wendroff. Review article; two-phase flow : Models and methods. J. Comput. Phys, 56:363-409, 1984.

[75] E. Storkaas and S. Skogestad. Cascade control of unstable systems with application to stabilization of slug flow. In IFAC-symposium Adchem ’2003, Hong Kong, January 2004.

[76] E. Storkaas, S. Skogestad, and J.-M. Godhavn. A low-dimensional dynamic model of severe slugging for control design and analysis. In Multiphase ’03, San Remo, Italy, June 2003.

[77] E. Tadmor. Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comp, 43:369-381, 1984.

[78] E. Tadmor. Approximate solutions of nonlinear conservation laws. In Lecture Notes in Mathematics 1697, 1997 C.I.M.E. course in Cetraro, Italy, June 1997, pages 1-149, Berlin, 1998. Springer Verlag.

[79] I. Tiselj and S. Petelin. Modelling of two-phase flow with second-order accurate scheme. J. Comput. Phys, 136:503-521, 1997.

[80] E.F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer Verlag, second edition, 1999.

[81] I. Toumi. An upwind numerical method for two-fluid two-phase flow models. Nucl. Sci. Eng, 123:147-168, 1996.

BIBLIOGRAPHY 79

[82] I. Toumi and A. Kumbaro. An approximate linearized riemann solver for a two-fluid model. J. Comput. Phys, 124:286-300, 1996.

[83] I. Toumi, A. Kumbaro, and H. Paillere. Approximate Riemann Solvers and Flux Vector Splitting Schemes for Two-Phase Flow. Commisariat a l’Energie Atomique, France, 1999. Notes presented at the 30th VKI Computational Fluid Dynamics Lecture Series, 8-12 March 1999.

[84] TRAC-PF1 /MODI. An advanced best-estimate computer program for pressurized water reactor thermal-hydraulic analysis. NUREG/CR- 3858, LA-10157-MS, Los Alamos National Laboratory, 1986.

[85] J. A. Trapp and R. A. Riemke. A nearly-implicit hydrodynamic numerical scheme for two-phase flows. J. Comput. Phys, 66:62-82, 1986.

[86] B. van Leer. Towards the ultimate conservative difference scheme. V. a second-order sequel to Godunov’s method. J. Comput. Phys, 32:101- 136, 1979.

[87] B. van Leer. Flux vector splitting for the Euler equations. In Proceedings of the 8th Internationa,l Conference on Numerical Methods in Fluid Dynamics, pages 507-512, Springer-Verlag, 1982.

[88] E. H. Vefring, Z. Wang, S. Gaard, and G. F. Bach. An advanced kick simulator for high angle and horizontal wells - part 1. In 1995 IADC/SPE Drilling Technology, The Netherlands, February 1995. LADC/Society of Petroleum Engineers. SPE 29345.

[89] Y. Wada and M.-S. Lion. An accurate and robust flux splitting scheme for shock and contact discontinuities. SIAM J. Sci. Comput, 18:633-657, 1997.

[90] P.J. Woodburn and R. I. Issa. Well-posedness of one-dimensional transient, two-fluid models of two-phase flows. In Proceedings of the ASME Fluids Engineering Division Summer Meeting, Washington, DC, USA, June 1998. American Society of Mechanical Engineers. ASME Paper FEDSM98-5023.

[91] H. Wu, F. Zhou, and Y. Wu. Intelligent identification system of flow regime of oil-gas-water multiphase flow. Int. J. Multiphase Flow, 27:459- 475, 2001.

[92] Z. G. Xu. Solutions to slugging problems using multiphase simulations. In 3rd IBC UK Multiphase Meeting Int. Conference, Aberdeen, Scotland, March 1997.

[93] D. Zeidan, E. F. Toro, and A. Slaouti. On the Riemann problem for a hyperbolic two-phase flow model. In Proceedings of the ACOMEN Conference, Belgium, May 2002.

80 BIBLIOGRAPHY

[94] J. Zhou and M.Z. Podowski. Modelling and analysis of hydrodynamic instabilities in two-phase flow using two-fluid model. In Ninth International Topical Meeting On Nuclear Reactor Thermal Hydraulics (NURETH-9), San Francisco, California, October 1999.

Paper I

Hybrid Flux-Splitting Schemes for a CommonTwo-Fluid Model

Steinar Evje and Tore Flatten

Journal of Computational Physics

ELSEVIER

Available online at www.sciencedirect.comSCIENCE DIRECTS

Journal of Computational Physics 192 (2003) 175-210

JOURNAL OF COMPUTATIONAL

PHYSICS

www.elsevier.com/locate/jcp

Hybrid flux-splitting schemes for a common two-fluid modelSteinar Evje a’* *, Tore Flatten b

a RF-Rogaland Research, Thormohlensgt. 55, Bergen N-5008, Norway b Department of Energy and Process Engineering, Norwegian University of Science and Technology,

Kolbjorn Hejes vei IB, Trondheim N-7491, Norway

Received 2 December 2002; received in revised form 10 July 2003; accepted 10 July 2003

Abstract

The aim of this paper is to construct hybrid flux vector splitting (FVS) and flux difference splitting (FDS) schemes for a commonly used two-fluid model consisting of two separate momentum equations. This is done by refining ideas previously applied to develop hybrid FVS/FDS schemes for a simpler two-phase model consisting of a mixture momentum equation [J. Comput. Phys. 175 (2002) 674]. More specifically, we seek to construct upwind type of schemes which are not based on calculations of the full eigenstructure of Jacobi matrices as needed by approximate Riemann solvers like the Roe scheme. Based on a crude approximation of the eigenstructure of the model, we derive schemes of the van Leer and FVS type. We demonstrate that these schemes possess desirable stability properties, but are excessively diffusive. By adapting ideas originally suggested by Wada and Liou [SIAM J. Sci. Comput. 18 (1997) 633] for the Euler equations, we suggest a mechanism for removing numerical dissipation. We present numerical simulations where we compare the performance of the resulting schemes with that of the Roe scheme, and by that shed light on the issues of accuracy, efficiency, and robustness of the proposed schemes. Particularly, we consider the classical water faucet problem as well as a stiff separation problem which locally involves transition from two-phase to single-phase flow. Results from these test cases show that we are able to construct hybrid FVS/FDS schemes which properly combine the accuracy of FDS in the resolution of sharp mass fronts and the robustness of FVS which ensures stability under stiff conditions.© 2003 Elsevier B.V. All rights reserved.

AMS: 76T10; 76N10; 65M12; 35L65

Keywords: Two-phase flow; Two-fluid model; Hyperbolic system of conservation laws; Flux vector splitting; Flux difference splitting; Hybrid scheme; Numerical dissipation

1. Introduction

Accurate resolution of the dynamics related to two-phase flow phenomena is of high importance, for instance, to the oil industry. Among several two-phase flow models there are two fundamentally different

* Corresponding author. Tel.: +47-55-54-38-50; fax: +47-55-54-38-60.E-mail addresses: [email protected] (S. Evje), [email protected] (T. Flatten).

0021-9991/$ - see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jcp.2003.07.001

176 5. Evje, T. Flatten / Journal of Computational Physics 192 (2003) 175-210

formulations of the macroscopic field equations for the two-phase flow system; namely the two-fluid model and the mixture model [30]. Here we focus on the two-fluid model. This is considered to give the most general and detailed description of transient two-phase flows. In the two-fluid model each phase is treated separately in terms of two sets of conservation equations; one for each phase. The interaction terms between the two phases appear in the basic equations as transfer terms across the interfaces (source terms).

It seems that many authors agree on the basic form of the two-fluid model, see for instance [5], However, due to the lack of some physical properties as well as the appearance of complex eigenvalues (loss of hyperbolicity), the derivation and inclusion of additional terms in the two-fluid model has been widely studied. In this work, we consider a common variant of the two-fluid model. Our main concern is to develop simple schemes whose numerical dissipation mechanism allows for producing stable and accurate predictions of two-phase flow phenomena relevant for the oil industry. A main issue is then accurate resolution of sharp mass fronts as well as stable and accurate calculations of flows where one of the two phases may disappear locally. More precisely, following [6], we have: If TJ denotes the vector of unknowns, the equations for the averaged two-fluid flow model are given by the system

8,H + &J(H) + G(H)8,H + 8,D(H, 8,(7) = 3(H).

Hence, the evolution of U is governed by convection, diffusion, and source terms. To a large extent the issue of accurate and stable approximation of typical mass transport problems is tightly connected to the convective part of the model, consequently, we focus on the first order variant of the above system given by

8,H + + G(H)8,H = 0. (1)

In particular, we do not discuss how to incorporate source terms and diffusion terms in the numerical schemes. However, an essential ingredient in the construction of numerical schemes for the two-fluid model, from our point of view, is that they should naturally allow for incorporation of more terms without changing the basic solution method. In this respect, we follow along the same line as Coquel et al. [5]. Another important aspect is that the numerical algorithms we study naturally can be used together with more complex equation of states (EOS).

Due to the wide range of fundamental and industrial applications of the two-fluid models, there has been a long-time interest in the development of efficient numerical algorithms for solving these models. The first computer models like CATHARE [2] were originally used to describe steam and water flow in nuclear reactors. It was based on pure advective upwinding, using a staggered grid together with implicit time integration to achieve stability. This approach was later adopted by the oil industry, resulting in computer codes like OLGA [3] and the more recently developed Petra [14]. These schemes are known to be robust, but diffusive and front-tracking methods have been incorporated to accurately resolve liquid slugs. In addition, they are not considered as being well suited for complex geometries.

During the last decade, various upwind type of schemes have been proposed for solving two-phase flow models, mixture models as well as two-fluid models. Many of these schemes, often categorized as flux difference splitting (EDS) schemes, are based on suitable modifications of classical upwind schemes like Godunov-type schemes [12,22] and Roe-type schemes [25,26], Such schemes are accurate and robust, however they tend to be time consuming due to the need for repeated calculation of the Jacobian of the system with respect to the conservative variables. Examples of such upwind schemes for two- phase flow models include implementations of the Roe scheme by Toumi et al. [6,32,33], Romate [27], Tiselj and Petelin [31], Fjelde and Karlsen [11]. A rough Godunov scheme was implemented by Masella et al. [19]. Coquel et al. [5] studied kinetic upwind schemes, which do not make use of the eigenstructure, for the approximation of a general two-fluid model. They also demonstrated that these schemes could handle phase separation where fronts propagate and one of the two phases disappears locally.

S. Evje, T. Flatten / Journal of Computational Physics 192 (2003) 175-210 111

Simpler schemes of the FVS type are based on dividing the numerical flux function F into positive and negative parts

F(U) =F"(U) + F+(U).

The numerical flux at the cell interface j + 1/2 is now given as

F/+1/2(U^, Uy+i) = F^(Uy) + F-(U,+i). (2)

FDS is based on matrix calculations, while FVS is based on scalar calculations. Consequently, FVS is more efficient than FDS, however at the price of introducing excessive numerical dissipation. During the last years, a lot of research has been done for the Euler equations motivated by the desire to combine the efficiency of FVS and the accuracy of FDS. These schemes are not FVS anymore, since their numerical flux typically no longer can be expressed in the splitting form (2). They are a hybrid of FVS and FDS. We refer to the works of Liou et al. [7,17,18,35] for more background on these methods, commonly denoted as Advection Upstream Splitting Methods (AUSM).

Recently, some of these ideas have been adapted to two-phase flow models. Here, we mention the works of Niu [20] and Edwards et al. [8]. Niu explored hybrid flux-type flux splitting schemes for a multicomponent flow model, whereas Edwards et al. studied a homogeneous equilibrium two-phase model with phase transitions. Characteristic for these models is that they are very similar to the Euler systems in structure and mathematical character. Evje and Fjelde [9,10] considered a simplified isothermal two-phase model consisting of separate mass conservation equations and a mixture momentum equation. This model is more difficult to solve, since no analytical expression for the Jacobian is available. This is due to the fact that the model has to be supplemented with a more or less complicated slip relation leading to unequal fluid velocities. In [9,10] a rough estimate of the sound velocity of the two-phase model was employed in the construction of hybrid FVS/FDS schemes. Basically, it was found that the corresponding rough AUSM scheme was suitable for simulating typical mass transport flow cases relevant for the oil industry. The scheme gave accurate and non-oscillatory resolution of mass fronts (comparable with Roe scheme), also for flows where more general slip relations were used. In particular, it was demonstrated that the AUSM scheme possesses a positivity-preserving property which ensures that it is well suited for handling the case where one of the two phases may disappear locally. It was also observed that by introducing a hybrid FVS/FDS scheme denoted as AUSMV, which combines AUSM and FVS in an appropriate way, we obtained results comparable with the Roe scheme in the resolution of rapid pressure waves generated by this two-phase model.

In this paper, we consider a two-fluid model which possesses one momentum conservation equation for each phase. The model represents added complications in several ways.• Motions of the two phases are no longer coupled in the same way as for the drift-flux model considered

in [9,10] leading to flows with highly unequal phasic velocities.• The two-fluid model involves non-conservative terms (the term G(U)dxlI in (1)) which must be handled

in a consistent manner by the numerical discretization.• The two slowest eigenvalues are approximately equal and the model is very close to being parabolic. In

deed, additional terms must be added to the model to maintain hyperbolicity, i.e., avoid complex eigenvalues.An extension of the AUSM+ scheme for the Euler equations [17] was investigated by Paillere et al. [23]

for this model. They based their approach on treating the model basically as two separate Euler models coupled through the pressure. They demonstrated that by including a pressure diffusion term, they were able to obtain stable and accurate solutions to several mass transport problems involving a local transition from two-phase to single-phase flow.

In the present work, we follow the approach of [9,10] and base the hybrid FVS/FDS schemes on a crude approximation of the eigenstructure of the model. First, we derive schemes of the van Leer and FVS type.


We demonstrate that these schemes possess desirable stability properties, but are excessively diffusive. By adapting ideas originally suggested by Wada and Liou [35] for the Euler equations, we suggest a mechanism for removing numerical dissipation. This leads to hybrid FVS/FDS schemes which will be denoted as AUSMV and AUSMD, which is in accordance with the notation used in [9,35]. In particular, we demonstrate:• The AUSMD scheme is comparable with the Roe scheme for the classical water faucet problem [24] with

respect to accuracy in the resolution of the discontinuity of the gas volume fraction. The AUSMV scheme is more diffusive than AUSMD.

• By introducing a slight modification of AUSMV, which ensures that the corresponding numerical flux coincides with that of the FVS scheme locally in the transition zone where the flow changes from two- phase to single-phase, this scheme produces stable and non-oscillatory solutions for the stiff separation problem considered in [5], The AUSMD scheme is much less diffusive than AUSMV in the approximation of sharp mass fronts. This motivates us to construct a hybrid of AUSMV and AUSMD, denoted as AUSMDV, which produces excellent results for this stiff test case, when it is compared with an approximate analytical solution. The Roe scheme is not able to produce stable solutions for this problem due to the change from two-phase to single-phase flow.

• The hybrid FVS/FDS schemes presented here do not give non-oscillatory approximations of all waves for typical shock tube problems. However, it is observed that AUSMV converges to the same solution as the Roe scheme as the discretization parameters are taken to zero.

In view of the applications we have in mind, the important observation made in this work is that AUSMV/ D give non-oscillatory, stable, and accurate approximations for typical mass transport problems, even when transition from two-phase to single-phase flow appears locally.

Finally, we would like to mention that Saurel and Abgrall [28] have suggested a non-conservative, unconditionally hyperbolic two-fluid model. Their model involves a separate pressure for each phase and an additional differential equation for the evolution of the volume fraction. A hybrid FVS/FDS scheme for this model has also been proposed by Niu [21].

Our paper is organized as follows: In Section 2, we present the two-fluid model that forms the basis for this work. In Section 3, we discuss some mathematical properties of this model and describe an implementation of a Roe scheme. The various flux-splitting schemes are introduced in Section 4. First, we briefly describe a van Leer scheme and an FVS scheme for the current two-fluid model. In particular, we describe a discretization of the non-conservative term that is consistent with how our Roe scheme treats this term. Then, we describe how to remove numerical dissipation from the van Leer and the FVS scheme, giving rise to two hybrid FVS/FDS type of schemes denoted as AUSMD and AUSMV, respectively. Section 5 is devoted to numerical experiments whose purpose is to highlight the stability and accuracy properties of the various schemes as observed when they are tested on several well-known flow cases.

2. The two-fluid model

The model we will be concerned with is formulated by stating separate conservation equations for mass and momentum for the two fluids, which we will denote as a gas (g) and a liquid (1) phase. For simplicity, we will assume an isentropic model and no energy equation will be taken into account.

We let U be the vector of conserved variables

---1

■»ru2

PgOCgfg %. Pl<X\V\ _ %4

(3)


The system of equations is given by • Conservation of mass

dt

d_8%

(PgtXgVg) — 0, (4)

— (pfx i) + — (PiCavi) — 0.

• Conservation of momentum

(Pg%^g) + ^ + ^g " %) + ^g

(5)

(6)

0 0 dp1(plalL’l) + fa (Pla‘Vl + (pi - Pi) a0 + — Qh8t 8% (7)

where for phase k the nomenclature is as follows: pk is the density; pk is the pressure; vk is the velocity; ak is the volume fraction; is the pressure at the gas-liquid interface; and Qk is the momentum sources (due to gravity, friction, etc.).We here treat Qk as a pure source term, assuming that it does not contain any differential operators. To

close the system we use the basic relation

oci + ag = 1. (8)

In addition, appropriate thermodynamical submodels must be specified.

2.1. Thermodynamic submodels

For phase k, we assume the simplified linear thermodynamic relations

The compressibilities are constant, given by

8/%8Pr

— 6Tk'

Throughout this work, for the liquid phase we use the parameters

pip = 1 bar = lO3 Pa, Pro = 1000 kg/m3,

(9)

(10)

and

ai = 103 m/s.

For the gas phase we set

Pg.o — 0, Pg.o — 0,

180 S. Evje, T. Flatten / Journal of Computational Physics 192 (2003) 175-210

and

flg = 105 (m/s)2.

This constitutes a rough model of the behavior of air and water.

2.2. Interface pressure modelling

The pressure corrections pk — fk represent effects such as hydrostatics or surface tension and can effect the low wavelength dynamics of the system. We recall the well-known fact that the simplest assumption of equal pressure, pt=p\ andpk = jik, will lead to a non-hyperbolic model whose applicability is questionable [29,30],

For a description of some common pressure correction models, we refer to the work of Cortes et al. [6] and the references therein. For the purposes of this paper, we will assume the equality of the phasic pressures, p = ps = pi, as well as p, = /)' = p\. In this respect we follow the footsteps of [6,23].

For the interface pressure pi, we choose the simple model

Ap = p-Pl3 QigOCiPgA

PgOCi + PjOCg(Ug - %i)2, (11)

using <5 = 1.2. This choice ensures that the system is hyperbolic if the relative velocity vr = vs — V\ does not approach the sound velocity of the mixture. This assertion will be justified in Section 3.1 to follow.

We remark that the expression (11) is based on mathematical considerations and has little physical justification. However, a similar approach for achieving hyperbolicity was used for the CATHARE code [4], see also [6], Paillere et al. [23] also based their investigations on this approach. Consistency with these previous works is our main motivation for choosing the model (11). We emphasize that we believe that the numerical techniques outlined in this paper are extensible to handle more general pressure correction models.

The assumption of equal pressure p= pg= p\ allows us to write the volume fraction Eq. (8) in terms of the conserved variables as

«1 »2 _ ,

yielding the relation p = p(u\,u2).

(12)

3. Eigenstructure and an approximate Riemann solver

3.1. Eigenstructure of the model

Writing the system in quasilinear formA+a(u)A = q(u),

the Jacobi matrix A can be found to be

A(U)

00

0 1 0 "0 0 1

K(pg% + Ap%g^)-:f 0 2%_

(13)

(14)


where1

K = ■(8pi/8^)«i/)g + (8pg/8f)%gPi'

The eigenvalues of the matrix A are the roots of the polynomial equation

k( pya.g + Apcty-^ ) - (a - vg)2 K( Pgai + Apag-^ ) — (a — l>l)

— K2 ( PgCtg ~ Apxg ^ f PyOCi — Apcty ) — 0.8;, y ,

These eigenvalues correspond to the wave velocities of the four eigenmodes of the model.

(15)

(16)

3.1.1. A perturbation methodSolving Eq. (16) exactly leads to highly complicated expressions and is not a practical approach. Instead,

we adopt a technique suggested by Toumi and Kumbaro [33], who obtained approximate eigenvalues for a two-fluid model with a virtual mass force term. In the following, we use this technique to derive approximate eigenvalues for the current model involving the interface pressure correction term (11).

We introduce the perturbation parameter s given by

■ ve ~ **c(i+ty (17)

where k is defined as

_Pg%k =A«g'

and c is an approximate mixture sound velocity given by

We further write Ap as

AWe also introduce the new variables

%i 0A / 8 A0C = —, P= — / Z = KPytXg.

«g 8p / 8pIn particular, by using (18) we obtain the relation

1 + k = —. z

The eigenvalue equation (16) can now be written as

(18)

(19)

z( 1 + pal^e2) - (a z(t + ^e^) - (A - %)2 z2(k - a.5Pt?){l - pSPr?) = 0. (20)


Writing the eigenvalue as

X — V[ + ac

and invoking (19) and (17) yields the two relations

(A - %)2 _ afc2—-----— ci (1 -\-k),z Z

(/. — fg)2 (X — V\ — cc)2 c2 A — V\ e ) = (1 +k)(a - e(l +k))2.z z z \ c

Combining these with (20) yields the following equation for a:

1 + fictsPe2 — (1 + k)(a — e(l + k))2 [k + #e2 — (1 + k)a2] = (k — o.^e2)(l — /u^e2).

A power series expansion yields

Inserting (22) into (21) and solving for the coefficients A we obtain:El: Downstream pressure wave

• A) = i>• A = A• A = (3k/2) + ((k — oc)(l — p)/2(l + k) .

E2: Upstream pressure wave• A = -i>• A = A• A = — {'ik/l) — {{k — o)(l — A/2(l +k)2)fP.

E3: Downstream void wave• A = A• Pi=k + g,• A = 0-

E4: Upstream void wave• A =• Pi=k-g,• A = 0;

where we have used the shorthand

g =/(I + pk)(l + 5)

(i+^r

(21)

(22)

(23)

3.7.2. Approximate eigenra/wefWe write the eigenvalues corresponding to pressure waves as

^ ^ ± c, (24)

whereas the eigenvalues corresponding to volume fraction waves are written as

r = tT ± y. (25)


By the results in Section 3.1.1, we obtain the approximations

(26)PgV-l + p l«g

(27)Pg%l -I-

c = c(l + (P(e^)), (28)

y = \Ap(Pg«i + #2g) - PiPg2i%g(„g - %)2 |

(29)\ (Pg%i +

where c is given by (18).From (28) we see that the use of c as an approximate sound velocity is justified, while (29) clearly

demonstrates that a zero pressure correction (Ap = 0) will render the model non-hyperbolic with complex eigenvalues. Note that the expression (11) with 5 = 1 corresponds exactly to y = 0. For 5 > 1 we obtain y > 0, whereas for <5 < 1 the parameter y becomes imaginary. We remark that the analysis does not guarantee hyperbolicity if the higher order terms in s become significant, i.e., if e « 1.

Remark 1. To zeroth order in e, i.e., the limit vs = n = v, we have that

(30)

and

zv = v, (31)

where 2V now becomes a degenerate eigenvalue. It can be shown that the characteristic field corresponding to /T has the properties of a linearly degenerate field as long as this limit holds. Consequently, the eigenstructure of the equations, in this limit, becomes similar to the structure of the Euler equations [16]. This approximation will form the basis for our extension of numerical schemes for the Euler equations to the current two-fluid model, as will be described in Section 4.

3.2. Treatment of non-conservative integrals

In this section, we deal with the mathematical difficulties associated with the non-conservative terms in the momentum equations in the form

(32)

Eqs. (6) and (7) are perfectly valid for smooth flows where the derivatives exist. However, in the presence of discontinuities the differential formulation breaks down and the equations must be replaced with corresponding integral equations. For conservative systems, the corresponding integrals are well defined, but this is unfortunately not the case for non-conservative systems.

In this paper, we treat this issue largely following the approach of Toumi and Kumbaro [33]. We consider a discontinuity separating two states (UL,UR), where ct\ f aR and f The integral of (32) across this discontinuity is


oq — as,/uL 8^ (33)

where s(U) is a path Unking the states UL and UR. This integral is path-dependent and additional physical assumptions must be made to single out a unique path that will define the “correct” mathematical solution.

Toumi and Kumbaro [33] suggested writing (33) as

defining the path s indirectly through the choice of an averaging function a(of, aR). In the case of incompressible liquid phase, they derived the harmonic average

2«1LaR aj- + af- (35)

by showing that the resulting non-conservative system has an equivalent conservative formulation.Unfortunately this result relies heavily on the non-compressibility of the liquid phase and is not valid for the

more general case where both phases are compressible. With no a priori difference between the gas and liquid phase, the basic equations (4)-(7) are symmetric under the interchange of phase labels. Hence, we insist that the averaging function x(of, aR) must possess the same kind of phasic symmetry, which we express as

= 1-cf)- (36)Note that the harmonic average (35) does not satisfy the requirement (36). The arithmetic average, however, does satisfy (36). Consequently, for the purposes of this paper, we propose to use the averaging function

+ (37)to define the non-conservative integrals of the form (34).

We emphasize that this choice is only one of many that satisfy the symmetry requirement (36). Here, we do not wish to advocate a particular strategy for dealing with the non-conservative term. Our concern is to ensure that the numerical schemes we investigate are mutually consistent in their treatment of the nonconservative integrals, making sure that the same momentum change is induced by a discontinuity in pressure and volume fraction. We stress that the hybrid flux-splitting schemes we developed in Section 4 are derived without making any assumptions of the particular functional form of a(aR, aR) and are trivially extensible to other choices of averaging functions.

3.3. Derivation of an approximate linearized Riemann solver

We are now in a position to derive a Roe scheme in the weak sense of Toumi and Kumbaro [33], where the Roe matrix A satisfies the following conditions:Rl: A(U1,U2)(U2-U1)=AF(U1,U2),R2: A(Ui,U2) is diagonalizable with real eigenvalues,R3: A(Ui,U2) —>• A(U) smoothly as UbU2 —>• U.Here

{/Wg}(AW

{Pg*s^} + {VW + - Af}{/wf} + + Si {p - Ap}

AF(U1;U2) (38)


and {•} denotes the operation

{%} — x2— Xi.

Moreover, a is given by (37) and A is the Jacobi matrix (14). ^We now wish to obtain an average state U(U1t U2) having the property that A(Ui, U2) = A(U) satisfies

the conditions R1-R3. The condition R1 gives rise to a set of coupled algebraic equations for U which may be solved to yield the result

v\\/{p<x)i +^2v/(p«)2

V(Mi + VW2

a = - (ai + a2),

and

P = 2(^1 + Pi)

for each phase. We treat the pressure correction as an independent variable that is averaged as

We assume constant compressibilities as described in Section 2.1. We may now easily check that the matrix

A(Ui,U2) = A(U)satisfies the weak Roe conditions R1-R3 when U is in the hyperbolic region.

3.3.7. AwfMun'ca/ n/gon'fAmLetting A be diagonalized as

A = RAR

we write

A£ = RA±R"1

whereA^ = diag(;.^,^,A^,A4),

with

At =max(0,A!-), Xt = min(0, A,).

We now can write the scheme in the non-conservative form as

(39)

where

F±(U/)Uy+l) — Ay+1/2(U/+1 - Uy). (40)


The numerical results produced by the Roe scheme, presented in Section 5, were obtained using a numerical algorithm to compute the eigenstructure of the Roe matrix.

4. Hybrid flux-splitting schemes

We will now describe an FVS and a van Leer scheme for the above two-fluid model, by adapting the schemes considered by Wada and Liou [35] for single-phase flow. In doing so, we will closely follow the approach that was previously used in [9] for a certain two-phase mixture model. Only the discretization of the non-conservative pressure term requires some special treatment, which will be discussed in more detail in Section 4.3.

We recall that for FVS the flux is split into upstream and downstream components as

F(U) =F+(U) + F"(U).

The numerical flux at the interface j + 1/2 is given as

F/+i/2(Ul,Ur) = F+(Ul)+F"(Ur). (41)

The van Leer scheme is slightly different from the FVS scheme, since it introduces an upwind principle in the discretization of the momentum convective flux terms.

In the following, we will find it convenient to split the fluxes into convective and pressure parts and deal with each term separately. We write the system (4)-(7) as follows:

9U 8FC 6FP 8ft8f 8% 8% 8%

where

Q,

' 0 ' 'O' 00 , H = 0 , Fp = 0 and Fc = fWi

Pg«g^%LaJ _«i. <%|Ap

We now consider discrete schemes in the form

U'77 + 1 ■u; , [fc]"+1/2 — [f. V-l/2 v+i/2 - [Fp]”-py-i/2At Ax Ax

H %8%

(42)

(43)

4.1. Definition of numerical convective flux [Fc]J+I/2

A main feature of the splitting of the convective fluxes is the introduction of a local “convective” speed which will be defined such that the effects of sonic waves are included. That is, we define a splitting of the velocity v as

v — V+ (v, c) + V~ (v, c). (44)

This splitting should satisfy a set of natural requirements as given by Liou [17]. We restate theserequirements as


Assumption 1. Let the split velocity functions V± be chosen such that they satisfy the following requirements:VI: Consistency. V+(v,c) + V~(v,c) = v.V2: Symmetry. V+(v, c) = — V~(v, c).V3: Left upwinding. V+(v, c) — v for v ^ c.

Right upwinding. V~(v,c) = v for vfl — c.V4: Differentiability. V~ are continuously differentiable.V5: Positivity. V+(v,c) > 0 and V~(v, c) < 0.V6: Monotonicity. V± are monotone increasing functions of v.

We note that for our model, the limits vk = ±c do not generally correspond to sonic points where all eigenvalues become of the same sign. However, this correspondence is achieved in the limit 8 = 0 where the eigenvalues (30) become valid approximations. We therefore propose to introduce the approximate eigenvalues (30) as basis polynomials for the splitting formulas, as stated in Remark 1. We remark that practical applications of the two-fluid model deal mainly with the low Mach number domain and this approximation seems to work well in practice.

We then arrive at a direct generalization of the splitting formulas for the Euler equations

±^(u±c)2 if|v|<c,j(r±|r|) otherwise. (45)

Following the standard set by earlier works [9,35], we chose a common sound velocity

A/+1/2 = max(c,, cy+i) (46)

at the cell interface. The concept of a common velocity of sound will later allow us to modify the schemes to remove numerical dissipation at moving discontinuities associated with the volume fraction waves. This will be described in Section 4.4.

We are now in a position to define the numerical convective fluxes for our model.(1) Mass flux. We let the numerical mass flux (par)/+1/2 be given as

W)/+l/2 = Wyk+(^,C;+i/2) + (%,'+!, C/+I/2) (47)

for each phase.(2) Momentum flux. We let the numerical convective momentum flux (pav2)j+^2 be given as

• TVS:

(pow%i/2 = E+(8y,C;+i/2)(p%r). + F (%+i,C/+i/2)(p%i;)^i,

• van Leer:

(pm/)/+i/2 = (P%")/+i/2%./ ^ (/My+1/2 ^ 0,(pxv)j+l/2Vj+l otherwise

or equivalently

(48)

(49)

(50)

We remark that the momentum flux constitutes the only difference between the TVS and van Leer schemes.


4.2. Definition of numerical pressure flux [FP]j+i/2

Similar to the previous splitting of velocities, we introduce a weighting factor P±(v,c) designed to distribute the pressure waves into upstream and downstream travelling components. The weighting factor is normalized by

P+(v,c) +P~(v,c) = 1.

Following Liou [17], we restate a set of natural requirements on such a weighting factor as follows.

Assumption 2. Let the split pressure functions P± be chosen such that they satisfy the following requirements:PI: Consistency. P+(v,c) +P~(v,c) = 1.P2: Symmetry. P+(v,c') = P~(—v,c).P3: Left upwinding. P+(v,c) = 1 for v > c.

Right upwinding. P~ (v, c) = 1 for v < —c.P4: Differentiability. P± are continuously differentiable.P5: Positivity. P±(v,c) > 0.P6: Monotonicity. P± are, respectively, monotone increasing and decreasing functions of v.

Again, using the approximate eigenvalue expression (30), we obtain the direct generalization of the splitting for the Euler equations

[ 4 otherwise........ , .otherwise.

Using this weighting factor, we split the conservative pressure flux (ot,Ap)J+lf2 as follows:(%Ap)/+l/2 = f + ay+|/2)(«Ap)y + f-(%y+,, Cy+pz)(%Ap)^.

(51)

(52)

4.3. The non-conservative term

Now we focus on the non-conservative pressure term of (6) and (7) given by

dp' _ 8 , . s (53)

4.3.1. Consistency with non-conservative integrals We propose discretizing the term (53) as

dfi - fflx (ft(U„ Uy+i) - FL(Uy_i,Uj)),

where we, as opposed to the conservative case, allow

^L(U/, Uy+l) f PR(Uy, Uy+l),

(54)

subject to the condition

FL(U,U)=FR(U,U)=0. (55)


For a moment, let us assume that we have a stationary discontinuity in the pressure and volume fraction variable. We now aim to obtain expressions for Fy and If that will induce the “correct” momentum change over the assumed discontinuity in pressure and volume fraction. Correct in the sense that it is consistent with the description in Section 3.2 as expressed through the relations (34) and (37). For simplicity in notation we drop the indices k and / in the following.

Integrating (54) over a box containing only a discontinuity (Ui,U2) we obtainfu2 QD

Fr(Ui,U2) — FL(Ui,U2) = J c/.— ds = a(«i, a2)(/>2 — Pi), (56)

relating the integral and discrete formulation of the model.

4.3.2. An FVS-like splittingWe propose to use an FVS-type splitting of the fluxes of the form

fk(u,,U2)=^(u,,a)+^(U2,a) (57)

and

FL(Ui,U2)=^(Ui,«)+FT(U2,«). (58)Inserting (57) and (58) into (56) we obtain the relation

/£(Ui,a) +Fr (U2,a) — F]^(Ui,a) — FL (U2,a) — a(«i, a2)(p2 ~Pi),

which suggests that FR and FR should satisfy the following relations:

F+ (U, a) - F+ (U, a) = ap,

fl (u, «) - fr (U, a) = -ap.

In addition, in view of (55), FR and Fp should also satisfy

^r"(U, a) = ~Fr (U, a),^(U,a) = -F[(U,«).

The following assumption summarizes these concerns and guarantees a treatment of a discontinuity in pressure and volume fraction consistent with the description in Section 3.2.

Assumption 3. Let the split non-conservative pressure fluxes F^ and FR be chosen such that they satisfy the following requirements:Cl: F+(U,a)+FL-(U,a) = 0.C2: Fr (U, a) + FR (U, a) = 0.C3: F^(U,a) +F£(U,a) = up.

Working within the framework of the pressure splitting functions which were applied in the discretization of the conservative pressure term as described in Section 4.2, some natural candidates for FR and FR are given by

7^(U, a) = c)o^ (59)

and

7^(U,«) = Tf-(z),c)^, (60)


where p1 =p — Ap and where we use the same splitting formulas P± as given by (51). Note that the requirements Cl and C2 are trivially satisfied, whereas C3 is a consequence of the property PI of Assumption 2 possessed by P±. To sum up, we split the non-conservative pressure flux (<xdxjf)j as follows:

— ^ (^r(U/, Uy+l, a/+l/2) - CL(Uy-l, Uy, Xj-l/l))

= J-fAyU

(C, Cy-l/z)^ + «y+l/2f (fy+1, C/+I/2M+I

%y_,/2f+(Uy-l,Cy-|/2)^._, +ay+l/2f"(fy,Cy+|/2)^ (61)

where we have used that CR and FL are given by (57)—(60), whereas a/+i/2 is the average (37) discussed in Section 3.2

°9+i/2 — 2 (%/ + a/+0- (62)

Note that the last equality of (61) expresses that the discretization of y.dxf on cell j is formed by using the weights Xy_i/2 and ay+i/2 to defining appropriate averages of p at cell interface j - 1/2 as well as j + 1/2, and by that reflects the fact that a stands left of the differentiation operator. We also observe that the weighting of the discrete pressure values by means of the pressure splitting functions P± is similar to what we find in the discretization of the conservative pressure term as described by (52).

Remark 2. This straightforward analysis based on consistency with the definition of non-conservative integrals does not take velocities into account. The discontinuity is basically treated as stationary. However, an interesting property of the proposed splitting (61) is that it correctly yields a vanishing contribution for a case of uniform pressure (pj = pj+\) and velocity (v,- = 1,7+1). As remarked in the end of the next section, this property ensures that the resulting schemes obey Abgrall’s principle [1,28].

4.4. Removal of numerical dissipation

It is a well-known fact that the discretization of the FVS and van Leer scheme is excessively diffusive on the slow waves mainly responsible for mass transport, as too much emphasis is put on the sonic waves in the splitting formulas as given by (45). We refer to the previous analysis of the mixture model [9], where the same mass conservation equations are considered whereas a mixture momentum equation is used instead of two separate momentum equations. Consequently, we propose to use a similar mechanism for removing the excessive numerical dissipation as the one employed in [9],

In order to depict the main idea, we consider a contact discontinuity given by

Pl =Pk =P,

*L f %R, (63)(vb)l = (v0l = (^s)r = (^i)r = v-

Now e = 0 as defined by (17) and the approximate eigenvalues (30) and (31) become exact. All pressure terms vanish from the model (4)-(7) and it is seen that the solution to this initial value problem is simply that the discontinuity will propagate with a velocity corresponding to the eigenvalue v. The exact solution of the Riemann problem will then give the numerical mass flux


Following Wada and Lion [35], we now let the splitting formulas V± be replaced by a more general pair V which also involve the eigenvalue v in the polynomial expansion. That is, we write

(65)

We now restate the following lemma, which is proved in [9].

Lemma 1. Let the velocity splittings (65) be used for the numerical mass fluxes (47) where a common velocityof sound C/+1/2 is assumed. Then, the exact Riemann solver flux (64) is recovered for the contact discontinuity given by (63) provided that the parameter % of (65) satisfies

(66)Xr*R - %I%L = 0.

In our previous work [9] on the mixture model, the simple choice

(67)Xl = %R, Xr = «L

was made. However, we observe that (66) allows for a degree of freedom in the choice of % and the choice (67) may not be optimal. In particular, numerical investigations show that (67) does not work as well for the two-fluid model as it did for the more strongly coupled mixture model. A refinement of (67) will be presented in the following:

Scaling. First we wish to recover the FVS flux for the case UL = Ur, to achieve maximum stability for continuous flow. That is, we want Xl = Xr = 1 for UL = UR. We may achieve this by the following rescaling

where and are functions satisfying (66).Pressure-dependent term. The analysis leading to Lemma 1 assumes uniform pressure. This means that

we are free to introduce a pressure-dependent weighting factor into the expressions and yR, writing them in the following form:

The purpose of the weighting factor w{p) is to stabilize the scheme in the presence of pressure oscillations.Finding a theoretical basis for the derivation of w(p) is difficult. Wada and Liou [35] suggested the

straightforward w(p) = p for the Euler equations. This choice ensures a relative increase in the splitting velocity, and hence in the mass flux, from the cell containing the larger pressure. Other choices that could be considered are w(p) = 1 and w(p) = pjp. However, for the two-fluid model we consider here, we observed that better results are achieved by the weighting

w(p) =Xf). (70)

This weighting factor ensures that only compressibility effects, as expressed by density differences across a pressure jump, are taken into account. It is justified by its performance in numerical experiments.

To summarize, we choose the following expressions for yL and /R:

2(p/«) l 2(p/q) R

(p/a)L + (p/%)%' (p/%)L + W^R 'Xl = (71)


Definition 1. Using the terminology of Wada and Lion [35], we will henceforth refer to the FVS scheme modified with the splittings (65) and the choice of % described by (71) as the AUSMV scheme. That is, the AUSMV scheme is described by• Mass flux.

(m^+i/2 = 9+1/2, 9+1/2, %&). (72)

• Momentum flux.

(pw/%+1/2 = 9+1/2,%L)(m^L + 9+1/2,%&)(/)%%)&. (73)

The pressure terms are discretized as described in Sections 4.2 and 4.3.

Definition 2. Similarly, we will henceforth refer to the van Leer scheme modified with the splittings (65) and the choice of % described by (71) as the AUSMD scheme. That is, the AUSMD scheme is described by• Mass flux.

(m%)/+i/2 = 9+1/2, Xu) + Wn 9+1/2, Xu). (74)

• Momentum flux.

1 1(/)%% )y+l/2 = 2 ^^/+V2 + ^) - 2 I W)/+l/2l(% - t).

The pressure terms are discretized as described in Sections 4.2 and 4.3.

(75)

As for the FVS and van Leer schemes, the only difference between AUSMV and AUSMD is their treatment of the convective momentum flux term. We note that neither the AUSMV nor the AUSMD scheme is a flux vector splitting (FVS) scheme as the numerical mass flux cannot be written in the form of (41).

Remark 3. According to the principle due to Abgrall [1,28], we want numerical schemes to obey the following physical principle: A flow, uniform in pressure and velocity, must remain uniform in the same variables during its time evolution.

In other words, if we had constant pressure and velocity everywhere in a flow at the time level f, then we will get the same pressure and velocity at the time t"+l.

We now check if the AUSMV and AUSMD schemes obey Abgrall’s priciple. Consequently, we assume that we have the contact discontinuity given by (63) and that it remains unchanged during the time interval [tn,tn+l]. In view of Definitions 1 and 2, we immediately can conclude that the mass equations and the momentum equations take the form

(Wy^ - (Wy - ^ ((PCw)y+i/2 " (PWO/-1/2),

^ ((m%)J+i/2 - (m^_,/2) AtAx (KAf)J+l/2 (%Ap)J_i/2) - Af a dp'

8%

where (par)"+1/,2 is in the form (64). From (52) and (11) we see that (aAp)nj+i/2 = 0, whereas it follows from(61) that [a|£] =0. Consequently, the pressure terms vanish and we conclude that the AUSMV andAUSMD schemes satisfy Abgralfs principle.


5. Numerical simulations

In the following, some selected numerical examples will be presented. As our main concern will be to demonstrate the inherent accuracy and stability properties of the different schemes, we limit ourselves to first order accuracy in space and time. Explicit time integration is used.

5.1. Shock tube problems

Shock tube problems are interesting for the following reasons:• They test the ability of numerical schemes to handle initial data that are far removed from an equilib

rium state.• The existence of discontinuities in both volume fraction and pressure provides a test that numerical

schemes converge to the same weak solutions in the presence of the non-conservative terms.In this section, we will investigate a couple of shock tube problems where the relative velocity between the phases is rather large. This provides a test of the validity of using approximate eigenvalues as basis polynomials for the splitting formulas, as described in Remark 1.

5.1.1. Shock tube problem 1We consider an initial Riemann problem also investigated by Cortes et al. [6] for a similar two-fluid

model. The initial states are given by

~ p~ "265,000 Pa"«1 0.71h 65 m/s

_Vi_ 1 m/s

and' P' "265,000 Pa"

«i 0.7ve 50 m/s

_Vi_ 1 m/s

(76)

(77)

We used the timestep Ax/At — 103 m/s and a computational grid of 100 cells. The results, plotted at the time T — 0.1 s, are given in Figs. 1 and 2. The reference solution was computed using the Roe scheme on a fine grid of 10,000 cells. The existence of two separate volume fraction waves can be seen from the small wedge in liquid fraction at x = 50 m, which appears clearly only in the reference solution.

We make the following observations:• The FVS and van Leer schemes are able to produce stable and non-oscillatory approximations. As ex

pected, they are excessively diffusive on the slow volume fraction waves. The van Leer scheme is more accurate than FVS on liquid velocity.

• The AUSMV and AUSMD produce a resolution of sonic waves which is comparable to that of FVS and van Leer. However, the slow volume fraction waves are reproduced with less numerical diffusion. The price to pay is that some oscillations around the volume fraction discontinuities are introduced by AUSMV. More severe oscillations, which are particularly visible for the liquid velocity, occur for AUSMD.

For this problem, it was noted that the oscillations produced by AUSMD developed into instabilities as the grid was refined. However, the oscillations observed for AUSMV would decay with grid refinement. This is demonstrated in Fig. 3, where AUSMV on a grid of 50,000 cells is compared to the Roe reference solution. The two solutions are virtually identical.


Fig. 1. Shock tube problem 1. FVS and AUSMV scheme on a grid of 100 cells. Top left: liquid fraction. Top right: pressure. Bottom left: liquid velocity. Bottom right: gas velocity.

5.1.2. Shock tube problem 2We now consider a problem similar to shock tube problem 1, but with a bigger volume fraction jump

and a jump also in the liquid velocity. The initial states are given by' p' "265,000 Pa"a i 0.7

65 m/s_V\_ 10 m/s

and

"265,000 Pa"«i 0.1VB 50 m/s

_Vl_ 15 m/s

(78)

(79)

S. Evje, T. Flatten I Journal of Computational Physics 192 (2003) 175-210 195

Distance (m)Distance (m)


Fig. 2. Shock tube problem 1. Comparison between van Leer and AUSMD scheme on a grid of 100 cells. Top left: liquid fraction. Top right: pressure. Bottom left: liquid velocity. Bottom right: gas velocity.

Using a timestep of Ax/At = 750 m/s, results for AUSMV are plotted in Fig. 4 for gradually finer grids. Similar results for AUSMD are plotted in Fig. 5. As for shock tube problem 1, the reference solution was computed by the Roe scheme using a grid of 10,000 cells.

The volume fraction variable is largely unaffected by the pressure waves. We have therefore magnified the volume fraction plots, focusing on the slow-moving volume fraction waves instead. The number of grid cells in the legend refers to the number of cells visible in the plots.

This problem does not display an essential difference in the stability properties of AUSMV and AUSMD. However, we note that the AUSMD scheme produces more accurate solutions for liquid velocity than AUSMV.

It is interesting that both AUSMV and AUSMD seem to produce the same wave structure in the volume fraction variable as the Roe scheme. This structure arises due to the existence of two separate volume fraction waves, as described in Section 3.1. This is a nonlinear effect that is not taken into consideration by the splitting formulas, which are based on the approximate eigenvalues stated in Remark 1.

196 S. Evje, T. Flatten I Journal of Computational Physics 192 (2003) 175-210

Distance (m) Distance (m)


Fig. 3. Shock tube problem 1. Convergence of Roe and AUSMV schemes. Top left: liquid fraction. Top right: pressure. Bottom left: liquid velocity. Bottom right: gas velocity.

5.1.3. Preliminary conclusionsTo summarize the results of this section, we have observed:

• The FVS and van Leer schemes provide non-oscillatory numerical solutions around discontinuities.• AUSMV and AUSMD are less diffusive on volume fraction waves than the FVS and van Leer schemes.

Stability problems may occur for AUSMD. AUSMV is stable.• The AUSMV and Roe scheme seem to converge to the same solutions. This is also in accordance with

observations made in [9] for a two-phase mixture model. Particularly, these numerical tests provide a justification of the discretization of the non-conservative pressure term as described in Section 4.3.

Remark 4. The oscillations observed for AUSMV and AUSMD indicate that these schemes do not have the “Total Variation Diminishing" property. As described in Section 3.1, the wave structure of the model involves strong couplings between the phasic variables. Such couplings are naturally incorporated in approximate Riemann solvers like the Roe scheme, which take the full eigenstructure into account to


258000 -


Distance (m)

I 13 -

Distance (m)

Fig. 4. Shock tube problem 2. Grid refinement for the AUSMV scheme. Top left: liquid fraction. Top right: pressure. Bottom left: liquid velocity. Bottom right: gas velocity.

determine the flux splittings. However, the splitting formulas given by (65) and (71) involve phasic couplings only in the common sound velocity. This simplification may partly explain the loss of monotonicity observed for AUSMV and AUSMD.

5.2. Water faucet

We now consider a simplified faucet flow problem proposed by Ransom [24]. This problem has previously been used by several authors for testing the ability of numerical schemes to accurately resolve volume fraction fronts [5,21,23,33,34] and has become a standard benchmark. We consider a vertical pipe of length 12 m with the initial uniform state

>" " 105 Pa'ai 0.8

0_V\_ 10 m/s_

W = (80)


Distance (m)

<8 256000 -

Distance (m)

reference - 20 000 cells -

800 cells - 200 cells 100 cells -

0 10 20 30 40 50 60 70 80 90 100Distance (m) Distance (m)

Fig. 5. Shock tube problem 2. Grid refinement for the AUSMD scheme. Top left: liquid fraction. Top right: pressure. Bottom left: liquid velocity. Bottom right: gas velocity.

Gravity is the only source term taken into account, i.e., in the framework of (6) and (7) we have

Qk = SPk^k") (81)

with g being the acceleration of gravity. At the inlet, we have the constant conditions oq = 0.8, v\ = 10 m/s and pg = 0. At the outlet, the pipe is open to the ambient pressure p = 105 Pa. We determined the remaining variables at the boundaries by simple extrapolation.

Ransom noted that an analytical solution for volume fraction and liquid velocity can be found assuming that the pressure variation in the vapor phase can be ignored. The procedure is described by Trapp and Riemke [34], here we provide only the result

vl(x,t) = { ^ + 2gx for x<vot + i2St2, (82)( v0 + gt otherwise.

ai(x,t) «o(l + 2gxv02) 1/2 for x < vQt + \gt2, a0 otherwise.

(83)

The parameters a0 = 0.8 and v(l = 10 m/s are the initial states.


Remark 5. This solution is not "analytical" in the sense that the model (4)-(6) is solved exactly by analysis. Rather, it is "analytical" in the sense that the approximate solutions (82) and (83) are given in terms of analytical expressions. However, the simplifying assumption leading to these analytical expressions is valid to a high degree of accuracy and we expect these approximate solutions to be virtually inseparable from the real solutions to the full model.

5.2.1. Test of accuracy of the different schemes on volume fractionIn Fig. 6, the gas volume fraction is plotted for T — 0.6 s for a grid of 120 computational cells. The

different schemes are plotted together for the sake of comparison. We observe that• The van Leer scheme is more accurate than TVS. This is consistent with our findings for shock tube

problem 1.• AUSMV and AUSMD are both more accurate than the van Leer scheme. AUSMD is more accurate

than AUSMV. The accuracy of AUSMD is comparable to that of the Roe scheme.

5.2.2. Results for hybrid flux-splitting schemesFor the AUSMD/V and Roe schemes, the full set of variables are plotted in Fig. 7. For the reference

solution the approximate analytical solution was used to calculate volume fraction and liquid velocity, while the Roe scheme on a finer grid (1200 cells) was used to calculate the pressure and gas velocity. A notable fact is that severe oscillations, as were observed for AUSMD on shock tube problem 1, do not occur. In fact, AUSMD gives a resolution comparable to the Roe scheme also on the velocities.

5.2.3. Convergence comparisonA more detailed look at the accuracy of the schemes regarding the volume fraction is given in Fig. 8. This

figure demonstrates the fundamental difference in the dissipative mechanism of AUSMD and AUSMV. In particular, we note that AUSMD is approximately as accurate as AUSMV calculated on a grid size whose

----- analytical solution— - Roe

□ AUSMD- - AUSMV

van Leer - FVS

0.35 -

0.25 -

Distance (m)

Fig. 6. Water faucet. T = 0.6 s. Resolution of volume fraction for Roe, AUSMD, AUSMV, van Leer, and FVS schemes on a grid of 120 cells.

200 S'. Evje, T. Flatten I Journal of Computational Physics 192 (2003) 175-210


— reference

Distance (m)

Fig. 7. Water faucet. T = 0.6 s. Roe, AUSMD and AUSMV schemes on a grid of 120 cells. Top left: liquid fraction. Top right: pressure. Bottom left: liquid velocity. Bottom right: gas velocity.

magnitude differs by one order. We also note that for the case with 1200 cells, AUSMD has introduced a slight overshoot in the approximation of the volume fraction. This is a manifestation of the weaker dissipative mechanism of AUSMD.

Remark 6. One may wonder whether simpler choices for the splitting formulas would work, for instance, for the water faucet problem. In [10], it was observed that simpler choices produced good results for typical mass transport problems described by a two-phase model consisting of two mass conservation equations and a mixture momentum equation. However, for the current two-fluid model where the coupling between the phasic velocities is much looser, it seems that the introduction of the sound velocity in the splitting formulas (45) and (51) is very essential. Neglecting the sound velocity and using pure advective up winding

^(y,c) = \(v± MX


------ analytical solution

Distance (m)


- AUSMD 24 cells

Distance (m)


— AUSMV 24 cells

Distance (m)

Fig. 8. Water faucet. T = 0.6 s. Accuracy on volume fraction. Top: grid refinement for Roe scheme. Center: grid refinement for AUSMD scheme. Bottom: grid refinement for AUSMV scheme.


as well as central pressure splitting

P=(v,c)

were found to cause severe instabilities.

5.3. Transition to one-phase flow

It is seen that the model (4)—(7) becomes singular in the limits ak —> 0 (k = g, 1), corresponding to the transition to one-phase flow. Stability problems are commonly encountered as this limit is approached. We observed that our implementation of the Roe scheme could not handle this in a satisfactory manner, and Coquel et al. [5] report instabilities for their Riemann-free upwind scheme based on kinetic considerations, occurring for values of oq near 10-6. They suggested a modification of the discretization of the nonconservative term to solve this problem.

Indeed, a similar problem was also observed for the AUSMD and AUSMV schemes and a fix is required. We suggest a method consistent with the framework we are working within, leaving the nonconservative term unaffected. Our suggestion is based on the following observations:• The resolution of sonic waves is very similar for the FVS/van Leer and the AUSMV/D schemes. The van

Leer and TVS schemes seem to be able to deal with the transition to one-phase flow in a stable manner, whereas AUSMV/D typically become unstable.

• The volume fraction waves disappear in the one-phase limit of the system. Hence, the effect of volume fraction waves is expected to disappear as a phase fraction tends to zero and the dynamics will be dominated by pressure waves.

This naturally suggests removing the advective velocity contribution to (65), falling back to the splitting (45) in near one-phase regions. We achieve this by replacing (71) by the following expressions:

Xl = (i — 2(/V«) L

W%)L+ (/>/*)% + 4>l (84)

and

Xk = (1 - <K) 2{p/%)n (85)

for both phases. Here, 0 is a smooth symmetric function 0(a) = 0(1 — a) designed to be 1 near one-phase regions and 0 otherwise. A simple expression having this property is

^ = <K%g) =1 1

gir(l-ag) ’ (86)

where the parameter k determines the degree of smoothness of 0. We found the value

A: = 200 (87)

to be a good compromise, providing both a smooth and accurate transition mechanism.

Definition 3. The modification of the AUSMV scheme obtained by replacing (71) in Definition 1 by (84) and (85) will be denoted as the AUSMV* scheme. Similarly, the modification of the AUSMD scheme obtained by replacing (71)in Definition 2 by (84) and (85) will be denoted as the AUSMD* scheme.


Remark 7. The property of AUSMD and AUSMV of reproducing the exact Riemann solver mass flux is formally lost by this modification. However, the flux modification is significant only for near one-phase regions where we expect little loss of accuracy. In particular, for the shock tube and water faucet examples considered so far, cf> < 10-17 and the results are unchanged to plotting accuracy.

JJ.L zHAS'MDUWe observed that the van Leer and AUSMD* schemes could handle a smooth transition to one-phase

flow. We also observed that if the transition is very abrupt, the van Leer and AUSMD* schemes could fail. However, the FVS and AUSMV* schemes seem stable also for such situations.

This demonstrates the need for a more sophisticated hybridization where we want to combine the accuracy of AUSMD* with the stability of AUSMV*. Based on our observations so far, we note the following:• The FVS scheme possesses outstanding stability properties. The AUSMV scheme largely keeps these

properties, and with the transition fix (84) and (85), AUSMV* seems to be able to handle very general flow conditions without introducing instabilities.

• The AUSMD has a weak dissipation mechanism allowing it to resolve discontinuities with an accuracy comparable to the Roe scheme. However, it is more prone to produce instabilities and overshoots.

This naturally suggests combining the AUSMD and AUSMV fluxes as follows:ÂUSMDV ^7AUSMV _|_ __ ^^7AUSMD (88)

where s is some parameter. We remark that only the convective flux in the momentum equations will be affected by this modification.

For the parameter 5 many choices are possible. As the previous examples show, the optimal choice might be problem-dependent. For one-phase flow, Wada and Liou [35] suggested letting s depend on the local pressure gradient, and similar ideas may be fruitful here.

We will not discuss this issue in full depth, but proceed to demonstrate that a simple choice for s will make the AUSMDV able to handle a stiff transition to one-phase flow in a stable and accurate manner. We observe that for a typical interface problem the strong gradients are associated only with the transition points between one-phase and two-phase flow. We consequently propose to use

a = max((A,<W, (89)where </> is given by (86). In particular, we note that this choice of s will make AUSMDV reduce to AUSMD for the water faucet and shock tube problems.

Definition 4. The scheme obtained by combining the AUSMV* and AUSMD* convective momentum fluxes is as follows:

ÂUSMDV* _ ÂUSMV* + (1 _ ÂUSMD*

where s is given by (89), will be denoted as the AUSMDV* scheme.

(90)

We note that AUSMDV* basically reduces to the stable FVS scheme near one-phase regions and the accurate AUSMD scheme elsewhere.

It seems that this approach can provide a good basis for methods aiming to resolve practical problems related to mass transport of oil and gas in pipelines. For such problems, the main dynamics are associated with slow transients and strong discontinuities are expected to occur only at the transition points to one-phase flow. Such discontinuities will commonly be induced by the buildup of liquid slugs due to gravity.


5.4. Separation problem

To illustrate the effect of the transition fix described above, we consider a simplified gravity-induced phase separation problem proposed by Coquel et ah [5], For this problem the transition from two-phase to one-phase flow occurs under stiff conditions, providing a good test for the stability of the schemes.

We consider a vertical pipe of length 7.5 m, where, as for the water faucet problem, gravity is the only source term taken into account. Initially, the pipe is filled with stagnant liquid and gas with a uniform pressure of p= 105 Pa and a uniform liquid fraction of cti = 0.5. The pipe is considered to be closed at both ends, he., both phasic velocities are forced to be zero at the end points. Assuming that the pressure variation can be neglected, an analytical solution can be derived in a similar manner as for the water faucet. We assume that the liquid is accelerated by gravity only until it is abruptly brought into stagnant conditions at the lower part of the tube. This yields the following approximate analytical solution for liquid velocity and volume fraction:

for x <

0 for L - \gt1 < x,(91)

f 0 for x < jgt2,«i(x, t) = < 0.5 for jgt2 <L- jgt2,

( 1 for L - f gt2 < x,

where L = 7.5 m is the length of the tube. After the time

(92)

(93)

we expect the phases to be fully separated and the liquid fraction will reach a stationary state. The other variables will slowly converge towards a stationary solution.

5.4.1. Results for the separation problemFor this problem, we used a constant timestep of Ax/At = 2 x 103 m/s. The pressure and liquid volume

fraction at the boundaries were determined by simple extrapolation.In the following, the AUSMDV* and AUSMV' schemes are compared. Snapshots of the simulations at

T = 0.6 s and T = 1.0 s are shown in Figs. 9 and 10. A grid of 100 cells was used. The reference solution was calculated using AUSMDV* on a grid of 1000 cells, except for Fig. 9 and volume fraction in Fig. 10, where the approximate analytical solutions given by (91) and (92) were used.

Fig. 9 shows the solution in the transient period where two volume fraction fronts, one upward and another downward directed, have been formed. In particular, we observe that both AUSMV* and AUSMDV* are able to handle the transition from two-phase to single-phase flow without loss of positivity. Comparison with the approximate analytical solution both in the liquid volume fraction and the liquid velocity variable clearly reveals that AUSMDV* is strongly superior to AUSMV* in the resolution of the discontinuous waves.

Fig. 10 shows the solution when the steady-state conditions have been reached, where the two-phase mixture is separated into a liquid part located at the bottom and a gas part located at the top. The plots clearly show the importance of the weak dissipative mechanism possessed by AUSMDV*: AUSMDV* is to a large extent able to reproduce the exact steady-state solution in the volume fraction variable, whereas AUSMV* performs much poorer on this fairly coarse grid.


----- reference- - AUSMDV*- - AUSMV*

0.8

c 06

0.2 - x / / /

01 -

0 1 2 3 4 5 6 7Distance (m)

AUSMDV*

Distance (m)

Fig. 9. Separation problem, snapshot at T = 0.6 s. Left: liquid fraction. Right: liquid velocity. The plots show the approximate solutions in the transient period on a grid of 100 cells.

For completeness, we have included the plots of the pressure and gas velocity as well. From the plot of the gas velocity, we observe that it becomes very large as the gas phase is disappearing. This is a result of our unphysical neglection of friction terms, which implies that no forces will balance the relatively strong hydrostatic pressure gradients induced by the heavy liquid phase.

5.4.2. ConvergenceA further investigation of the accuracy of AUSMDV* on the volume fraction is made in Fig. 11, where

the effect of grid refinement is illustrated. Although we have no stable Roe scheme to compare with, it seems that the effect of increased diffusion due to the transition fix is minimal. We also note that the AUSMDV* scheme gives good results compared to the upwind scheme of Cortes et al. [5] for this flow case.

5.5. Oscillating manometer problem

For our last numerical test, we consider the oscillating manometer problem introduced by Ransom [24]. This problem involves a moving liquid plug where the flow direction is time dependent. We hence believe that the numerical challenges presented by this problem are representative for typical transport pipeline simulations.

We consider a U-shaped tube of total length 20 m. The geometry of the tube is reflected in the x-component of the gravity field

( g for 0<x< 5 m,gx(x) - < geos ( n) for 5 m < 15 m, (94)

l — g for 15 m < x<20 m.

Initially, we assume that the liquid fraction is given by

( 10-6 for 0<x< 5 m,(x) = < 0.999 for 5 m < x< 15 m,

I 10-6 for 15 m < x< 20 m.(95)


;°.S-

J 0.4 -

0.3 -

0.2 -

0.1 -

0

----- reference- - AUSMDV*- - AUSMV*

3 4Distance (m) Distance (m)

- - AUSMV*

1.05-

Distance (m)

— reference

- - AUSMV*

Distance (m)

Fig. 10. Separation problem, snapshot at T = 1.0 s on a grid of 100 cells. Top left: liquid fraction. Top right: liquid velocity. Bottom left: pressure. Bottom right: gas velocity. At this time the phases are fully separated.

The initial pressure is assumed to be equal to the hydrostatic pressure distribution. The initial velocities of both phases are uniformly vk = to, where V0 = 2.1 m/s.

We treat the manometer as a closed loop, so that the left and right edges are connected to each other. Hence, there are no boundary conditions for this problem. We assume that the liquid column will move with uniform velocity under the influence of gravity, giving the following approximate analytical solution for the liquid velocity [23]

V\(t) = to cos(W), (96)where

and L = 10 m is the length of the liquid column.

(97)


analytical solution ' 1000 cells

100 cells 25 cells

0.7

0.3

✓/

' // / *//

3 4Distance (m)

Fig. 11. Separation problem, T = 0.6 s. Liquid volume fraction. Grid refinement for the AUSMDV* scheme.

5.5.1. Numerical resultsWe used the AUSMDV* scheme on a grid of 100 cells with a timestep Ax/At = 3000 m/s. The time

development of the liquid velocity is given in Fig. 12. The velocity is sampled at the lowest point (middle grid cell) of the manometer. We note that good accordance with the approximate analytical solution is obtained, although a small phase difference seems to develop. We also observe some numerical damping.

referenceAUSMDV*

1 - \

Time (s)

Fig. 12. Oscillating manometer, 100 cells. AUSMDV* scheme, time development of the mid-cell liquid velocity.

208 S'. Evje, T. Flatten I Journal of Computational Physics 192 (2003) 175-210

We remark that even though the scheme is flux-conservative, damping is expected as the gravity field is discretized using a simple Euler integration.

The distribution of the physical variables at the time T — 20.0 s is plotted in Fig. 13. There are no signs of any numerical oscillations. As expected, a hydrostatic pressure distribution is reproduced. There is very little dissipation in the volume fraction variable.

Remark 8. All the simulations we have considered have been first order accurate in space and time. In principle, second order accuracy may be achieved using Runge-Kutta time integration and MUSCL interpolation [15] of the primitive variables. This approach was successfully applied to similar flux-splitting schemes for the mixture model in [9]. The simulations in Sections 5.2-5.5 essentially demonstrate monotonicity of the numerical solutions, suggesting that for such cases similar strategies may be successful also for the current model.

However, the shock tube simulations in Section 5.1 show that more refined dissipation mechanisms must be developed before higher order techniques can be applied. In this respect, the good stability properties of the simpler TVS and van Leer schemes become of interest. As is demonstrated in for example [13], highly accurate solutions may be obtained by higher order techniques even if the basis schemes are diffusive. Of

ussoooooooqpjocooocoqpcaxaxxxpooooooocqpoex -̂----r

Distance (m)

135000

130000

125000

120000

115000

110000

105000

100000

Distance (m)


Fig. 13. Oscillating manometer, T = 20.0 s, 100 cells. AUSMDV* scheme. Top left: liquid fraction. Top right: pressure. Bottom left: liquid velocity. Bottom right: gas velocity.


particular interest here is the ability of FVS to handle the transition to one-phase flow without any kind of modification.

6. Summary

Schemes of FVS and van Leer type have been proposed for a two-phase flow model. Methods for removing numerical dissipation from these schemes have been explored. A mechanism for handling the difficult transition from two-phase to single-phase flow within this context has also been proposed. The resulting schemes, denoted as AUSMV* and AUSMD , are demonstrated to have desirable properties. In particular, the AUSMV* is stable and the AUSMD* possesses an inherent accuracy comparable to an approximate Riemann solver, with a highly reduced computational cost. A hybrid AUSMDV* scheme, taking advantage of both these properties, has been proposed with particular focus on the kind of discontinuities expected to appear for slow transients associated with mass transport in pipelines. The proposed scheme does not provide the same level of robustness as an approximate Riemann solver for strong shocks. However, the framework has been demonstrated to contain the mechanism for providing accurate and efficient solutions to several benchmark two-phase flow problems.

Acknowledgements

The second author thanks the Norwegian Research Council for financial support through the “Petronics” programme. The authors also thank the referees for their valuable remarks which led to a substantial improvement of the first version of this paper.

References

[1] R. Abgrall, How to prevent pressure oscillations in multicomponent flow calculations, J. Comput. Phys. 125 (1996) 150-160.[2] F. Barre et al, The cathare code strategy and assessment, Nucl. Eng. Des. 124 (1990) 257-284.[3] K.H. Bendiksen, D. Malnes, R. Moe, S. Nuland, The dynamic two-fluid model OLGA: theory and application, SPE Prod. Eng. 6

(1991) 171-180.[4] D. Bestion, The physical closure laws in the cathare code, Nucl. Eng. Des. 124 (1990) 229-245.[5] F. Coquel, K. El Amine, E. Godlcwski, B. Pcrthamc, P. Rasclc, A numerical method using upwind schemes for the resolution of

two-phase flows, J. Comput. Phys. 136 (1997) 272-288.[6] J. Cortes, A. Debussche, I. Fourth, A density perturbation method to study the eigenstructure of two-phase flow equation systems,

J. Comput. Phys. 147 (1998) 463^-84.[7] J.R. Edwards, M.-S. Liou, Low-diffusion flux-splitting methods for flows at all speeds, AIAA J. 36 (1998) 1610-1617.[8] J.R. Edwards, R.K. Franklin, M.-S. Liou, Low-diffusion flux-splitting methods for real fluid flows with phase transition, AIAA J.

38 (2000) 1624-1633.[9] S. Evje, K.K. Fjelde, Hybrid flux-splitting schemes for a two-phase flow model, J. Comput. Phys. 175 (2002) 674-701.

[10] S. Evje, K.K. Fjelde, On a rough ausm scheme for a one-dimensional two-phase flow model, Comput. Fluids 32 (2003) 1497-1530.[11] K.-K. Fjelde, K.H. ICarlsen, High-resolution hybrid primitive-conservative upwind schemes for the drift flux model, Comput.

Fluids 31 (2002) 335-367.[12] A. Harten, P.D. Lax, B. Van Leer, On upstream differencing and Godunov schemes for hyperbolic conservation laws, SIAM Rev.

25 (1981) 35-61.[13] G.-S. Jiang, D. Levy, C.-T. Lin, S. Osher, E. Tadmor, High-resolution nonoscillatory central schemes with nonstaggered grids for

hyperbolic conservation laws, SIAM J. Numer. Anal. 25 (1998) 2147-2168.[14] M. Larsen, E. Hustvedt, P. Hedne, T. Straume, Petra: A novel computer code for simulation of slug flow, in: SPE Annual

Technical Conference and Exhibition, SPE 38841, October 1997, pp. 1-12.[15] B.V. Leer, Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov’s method, J. Comput. Phys.

32 (1979) 101-136.


[16] R.J. LeVeque, Numerical Methods for Conservation Laws, second ed., Birkhauser Verlag, 1992, pp. 89-93.[17] M.-S. Lion, A sequel to AUSM: AUSM(+), J. Comput. Phys. 129 (1996) 364-382.[18] M.-S. Lion, C.J. Steffen, A new flux splitting scheme, J. Comput. Phys. 107 (1993) 23-39.[19] J.M. Masella, I. Faille, T. Gallouet, On an approximate Godunov scheme, Int. J. Comput. Fluid. Dyn. 12 (1999) 133-149.[20] Y.Y. Niu, Simple conservative flux splitting for multi-component flow calculations, Numer. Heat Trans. 38 (2000) 203-222.[21] Y.-Y. Niu, Advection upwinding splitting method to solve a compressible two-fluid model, Int. J. Numer. Meth. Fluids 36 (2001)

351-371.[22] S. Oshcr, Ricmann solvers, the entropy conditions, and difference approximations, SIAM J. Numer. Anal. 21 (1984) 217-235.[23] H. Paillere, C. Corre, J.R.G Cascales, On the extension of the AUSM+ scheme to compressible two-fluid models, Comput. Fluids

32 (2003) 891-916.[24] V.H. Ransom, Numerical bencmark tests, Multiphase Sci. Tech. 3 (1987) 465^-73.[25] P.L. Roe, The use of Riemann problem in finite difference schemes, Lect. Notes Phys. 141 (1980) 354-359.[26] P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43 (1981) 357-372.[27] J.E. Romate, An approximate Riemann solver for a two-phase flow model with numerically given slip relation, Comput. Fluids 27

(1998) 455^477.[28] R. Saurel, R. Abgrall, A multiphase godunov method for compressible multifluid and multiphase flows, J. Comput. Phys. 150

(1999) 425^467.[29] H.B. Stewart, Stability of two-phase flow calculation using two-fluid models, J. Comput. Phys. 33 (1979) 259-270.[30] H.B. Stewart, B. Wendroff, Review article; two-phase flow: models and methods, J. Comput. Phys. 56 (1984) 363-409.[31] I. Tiselj, S. Petelin, Modelling of two-phase flow with second-order accurate scheme, .1. Comput. Phys. 136 (1997) 503-521.[32] I. Toumi, An upwind numerical method for two-fluid two-phase flow models, Nucl. Sci. Eng. 123 (1996) 147-168.[33] I. Toumi, A. Kumbaro, An approximate linearized Riemann solver for a two-fluid model, J. Comput. Phys. 124 (1996) 286-300.[34] J.A. Trapp, R.A. Ricmkc, A ncarly-implicit hydrodynamic numerical scheme for two-phase flows, J. Comput. Phys. 66 (1986) 62-

82.[35] Y. Wada, M.-S. Liou, An accurate and robust flux splitting scheme for shock and contact discontinuities, SIAM J. Sci. Comput.

18 (1997) 633-657.

Paper II

A Mixture Flux Approach for Accurate and Robust Resolution of Two-Phase Flows

Tore Flatten and Steinar Evje

Preprint

A MIXTURE FLUX APPROACH FOR ACCURATE AND ROBUST RESOLUTION OF TWO-PHASE FLOWS

TORE FLATTEN* AND STEINAR EVJEB’°

Abstract. The aim of this paper is to construct robust and accurate hybrid FVS/FDS type of schemes for a standard four-equation isentropic compressible two-fluid model governing 1- dimensional flow of a gas (g) and liquid (1) mixture. The starting point for our investigations is a Roe scheme and a hybrid FVS/FDS scheme. The latter is an AUSMD type of scheme obtained through a natural and rather straightforward extension of the corresponding scheme for the Euler equations (single-phase model) as described by Wada and Liou (1997, SIAM J. Sci. Comput. 18, 633-657). The main advantage of such hybrid FVS/FDS schemes is that they neither require the use of Riemann solvers nor the computation of nonlinear flux Jacobians. However, we observe that the two-phase AUSMD scheme is prone to introducing oscillations and overshoots around discontinuities. Based on the belief that this deficiency is due to the loose coupling between mass and momentum equations in the discretization of the two-phase model, we propose a method for improving the approximation properties of hybrid FVS/FDS schemes by enforcing a tighter coupling between the various equations.

The method, which is denoted as a ” Mixture Flux” (MF) method, is composed of two main ingredients. First, we make use of an additional pressure evolution equation which is derived from the equation describing the conservation of the total mass. This provides us with information how to construct an appropriate numerical flux for the discretization of the pressure term of the momentum equations. Second, we introduce a consistent decomposition of the numerical mass fluxes into two components; one flux component Fv associated with the fast- moving pressure waves and another component FA associated with the slow-moving volume fraction waves. The F’D-component is designed by using information from the momentum equations and is crucial for ensuring non-oscillatory behavior around the slow-moving volume fraction waves, whereas the FA-component is responsible for the accuracy of these waves.

Particularly, by associating the flux FA with the AUSMD mass flux we demonstrate through numerical experiments that the resulting MF-AUSMD scheme possesses accuracy and stability properties on the same level as the Roe scheme while allowing for highly improved computational efficiency. In addition, by using a slight modification of MF-AUSMD we can also simulate flow cases which locally involve transition from two-phase to single-phase.

The MF-method represents a general strategy for refining hybrid FVS/FDS schemes for two-phase flow models.

subject classification. 76T10, 76N10, 65M12, 35L65

key words, two-phase flow, two-fluid model, hyperbolic system of conservation laws, flux splitting, explicit scheme

1. Introduction

Among several two-phase flow models there are two fundamentally different formulations of the macroscopic field equations for the two-phase flow system; namely the two-fluid model and the mixture model [24], which is a simplified isothermal two-phase model consisting of separate mass conservation equations and a mixture momentum equation. Here we focus on the two-fluid model, which is considered to give a more general and detailed description of transient two-phase flows.

Date: July 21, 2003.^Department of Energy and Process Engineering, Norwegian University of Science and Technology,

Kolbjpm Hejes vei IB, N-7491, Trondheim, Norway.BRF-Rogaland Research, Thormphlensgt. 55, N-5008 Bergen, Norway.Email: [email protected], [email protected]. c Corresponding author.

1

2 FLATTEN AND EVJE

This model is expressed as a set of 4 partial differential equations, one mass and one momentum conservation equation for each phase. The interaction terms between the two phases appear in the basic equations as transfer terms across the interfaces (source terms). More precisely, the basic form of the model can be written on the following vector form:

( pgag ^ ^ Pgagvg ^

dtPlOt\ + dx

p\a\v 1pgOlgVg + <*gP

\ mw\ ) X piaivf + aip J

f 00

P&xa g + Tg\ pdxa 1 + n

\ ( °0

Qg +V Qi + M?

\

)(1)

Here is the volume fraction of phase k with a\ + ag = 1, pk and n* denote the density and fluid velocities of phase k, and p is the pressure common to both phases. Moreover, t/,, represents the interfacial forces which contain differential terms (hence, is relevant for the hyperbolicity of the model) and satisfy rg + t\ = 0. represents interfacial drag force with M® + Mj° = 0 whereas Qk represent source terms due to gravity, friction, etc.

During the last decade, flux-splitting techniques denoted as “Advection Upstream Splitting Methods” (AUSM) have become popular for solving the equations of gas dynamics [13, 12, 30, 5]. An advantage of such methods is that they do not require a knowledge of the full eigenstructure of the system, and are consequently more efficient than classical approximate Riemann solvers like the Godunov [16, 11] and Roe [19, 20] schemes.

A recent trend has been to adapt such ideas to two-phase flow models [6,14,15]. Evje and Fjelde [7, 8] considered the mixture two-phase model. Basically, it was found that an AUSM scheme based on a rough estimate of the sound velocity gave accurate and non-oscillatory resolution of mass fronts comparable with the more computationally demanding Roe scheme [21, 10].

Regarding the two-fluid model, Paillere et al [17] investigated an extension of the AUSM"1" scheme of Liou [12] on a model including an energy conservation equation for each phase. Evje and Flatten [9] investigated a related approach, using an extension of the AUSMD/V scheme of Wada and Liou [30] on the two-fluid model. Results similar to the work of Paillere et al were obtained.

A feature common to both these approaches is a tendency towards introducing spurious oscillations and overshoots around discontinuities. Based on the belief that this deficiency is due to the loose coupling between mass and momentum equations in the discretization procedure, we here propose a general method for improving the approximation properties of such hybrid FVS/FDS schemes for the two-phase model. The main idea behind this novel construction can be described as follows: Assuming that the phases have equal pressure, the mass coupling can be expressed as (writing mk = pkOtk)

mg | mi _1PgW #(P) '

which is a rewritten form of the basic volume fraction equation

(2)

ag + a\ = 1.The relation (2) contains essential information about the interrelation between the masses and the pressure. The idea of this work is to derive hybrid FVS/FDS schemes which explicitly make use of this relation in the construction of appropriate numerical mass fluxes. Basically, the implementation of this idea is carried out in two steps.

• First, we couple the pressure calculation more directly to the momentum equations. For this purpose, we derive a pressure evolution equation by combining the mass conservation equations (two first equations of (1)) and the relation (2). This pressure equation is discretized at the cell interface whereas cell-centered pressure values are obtained directly from (2). We apply a discretization of the pressure evolution equation which enforces a coupling between the cell interface pressure Pj+1/2 and cell-centered pressure pj, ensuring consistency of the pressure splitting.

• Second, we couple the calculation of masses closer to the momentum equations. To achieve this, we employ (2) and derive consistent numerical mass fluxes associated with the mass conservation equations which are composed of two components; one diffusive part FD

A MIXTURE FLUX APPROACH 3

for stable (non-oscillatory) resolution of volume fraction waves and another nondissipative part Fa for accurate resolution of these waves.

The above two steps bring forth numerical fluxes for the various equations of (1) which consist of a mixture of terms from the other equations. This motivates us to denote the general algorithm as a Mixture Flux (MF) method. Consequently, we lose some of the simplicity of the original AUSM concept which basically treats the system as a set of scalar equations without accounting for the interrelation between the various equations. However, the efficiency properties of the original AUSM type schemes largely remain for the proposed MF-type schemes as we still avoid use of Riemann solvers and computation of nonlinear flux Jacobians.

Particularly, by associating the flux component FA with the AUSMD flux used in [9] we obtain a MF-AUSMD scheme. We formally demonstrate that under natural assumptions on the T13 flux component, the resulting MF-AUSMD scheme recovers the numerical flux of an exact Riemann solver for a moving or stationary contact discontinuity. This ensures that mass fronts are properly resolved. We also verify that Abgrall’s principle [1] is satisfied; that a flow, uniform in velocity and pressure, must remain uniform during its temporal evolution.

We demonstrate through numerical experiments that the proposed MF-AUSMD scheme matches the good accuracy and stability properties of the Roe scheme. More precisely, the MF-AUSMD is slightly more diffusive on the fast moving sonic waves. For the approximation of the slow volume fraction waves we see that the MF-AUSMD scheme and the Roe scheme behave very similar. In particular, the deficiencies of the AUSMD scheme studied in [9] have been removed. In addition, we may easily modify the MF-AUSMD scheme so that it can handle flow cases which locally involve transition from two-phase to single-phase flow.

Hence, the MF-AUSMD scheme, which is totally free from Riemann solvers and computation of nonlinear flux Jacobians, allows for highly improved computational efficiency compared to the Roe scheme.

Our paper is organized as follows: In Section 2 we present the two-fluid model we will be working with. In section 3 we briefly restate the flux-splitting schemes that were investigated in [9] and which will be used as a basis for the methods we develop in this paper. The purpose of Section 4 is to motivate for the Mixture Flux method by observing some ’’weak points” of the flux-splitting schemes considered in [9]. In Section 5 we introduce the Mixture Flux method, and in Section 6 we verify that the method satisfies certain ’’good” properties. In Section 7 we apply it to a set of test cases found in the literature. Comparisons are made with the Roe scheme as well as the AUSMD scheme considered in [9] and an appropriate modification is applied making the scheme able to handle the transition to one-phase flow.

2. The Two-Fluid Model

Throughout this paper we will be concerned with the common two-fluid model formulated by stating separate conservation equations for mass and momentum for the two fluids, which we will denote as a gas (g) and a liquid (1) phase. The model has been studied by several authors [28, 3, 4, 17, 9] and will be briefly stated here. We let U be the vector of conserved variables

pgag " "p\Oi\ mi

PgOigVg hp\a\v\ _ h

By using the notation Ap = p — p\ where p1 is the interfacial pressure, and Tfc = (p* — p)dxau = —Apdxa.k, we see that the model (1) can be written on the form

• Conservation of massd_dtru (Psas) + qx (Pgagvg) ~ 0? (4)

9 #Qj. W%) + (pi®ivi) = 0, (5)

4 FLATTEN AND EVJE

• Conservation of momentum

— (peagvg) + ^ (igCtgVg + agAp) + ag —(p - Ap) = Qg + Afj?, (6)

^ (piaiui) + ^ (pio-mi2 + «iAp) + ai^(p - Ap) = Q\ + MD. (7)

Since we focus on the development of numerical schemes which can handle the basic two-fluid model, we have set the interfacial drag force terms to zero, i.e. M-P = M® = 0.

In addition, other constitutive laws are needed to close the system. In particular, the volume fractions must satisfy

ag + «i = 1. (8)The system is closed by some equation of states (EOS) for the liquid and gas phase. The numerical methods we study in this work allow general expressions for the EOS. However, for the numerical simulations presented in this work we assume the simplified thermodynamic relations

. P~PoPi = Pl,0 + 2

°1(9)

and NT11 (10)

wherePo = 1 bar = 105 Pa

pi,o = 1000 kg/m3, og = 105(m/s)2

01 = 103 m/s.

The models (9) and (10) correspond to a general stiffened gas EOS of the form

P = (lk~ l)a2kpk -7fc7Tfc,where ir* = {a\pk,o —po)/2 where pk,o represents the material density and po the ambient pressure. 7fc and 7r& are constants specific for each phase. Particularly, by choosing 71 = 2 we recover (9) while (10) is obtained by choosing 7g = 2 and 7rg = 0.

Moreover, we will treat Qk as a pure source term, assuming that it does not contain any differential operators. We use the interface pressure correction

Ap = a asa^sPipgo;i + pi«g (11)

where we set a = 1.2. This choice ensures that the model is a hyperbolic system of conservation laws, see for instance [4, 9]. Another feature of this model is that it possesses an approximate mixture sound velocity c given by

c =p\OLg + PgCKl

(12)

We refer to [28, 9] for more details.Having solved for the conservative variable U, we need to obtain the primitive variables

(ag,p, vg, Vi). For the pressure variable we see that by writing the volume fraction equation (8) in terms of the conserved variables as

mg , JKL =/%(P) #(P) '

(13)

we obtain a relation yielding the pressure p(mg,mi). Using the relatively simple form of EOS given by (9) and (10) we see that the pressure p is found as a positive root of a second order polynomial. For more general EOS we must solve a non-linear system of equations, for instance by using a


Newton-Rapson algorithm. Moreover, the fluid velocities vg and v\ are obtained directly from the relations

ve = UsUx'

-S-

Remark 1. Throughout this work we will study only the isentropic 4-equation model given above. The inclusion of energy equations does not significantly alter the existing eigenstructure of the isentropic model, but adds entropy waves moving with the fluid velocities. It is our belief that the main difficulties related to the strong phasic couplings in the pressure and volume fraction waves are fully present in the isentropic model. Formally, the method investigated in this paper should be naturally extensible to the full model. This will be explored elsewhere.

Remark 2. Concerning the EOS for the liquid and gas phase, we emphasize that the methods we develop do not require simple linear relations as given by (9) and (10). In principle, the only point which is affected by using more complicated EOS is the resolution algorithm which determines the pressure from the general relation (13).

Remark 3. The Mixture Flux approach we propose in this paper is to a large extent independent of the hyperbolicity assumption since it does not rely on any Riemann solver nor calculation of flux jacobians. Thus, there are good reasons to believe that MF-schemes can be used to explore problems where the model becomes non-hyperbolic. More generally, since the dependence on the special properties of the underlying model is weak, the MF-methods should have a potential for becoming a useful tool when studying what happens when perturbation parameters reach critical values such that the nature of the model changes.

3. Two Hybrid FVS/FDS Numerical Schemes

We here briefly restate two of the flux splitting schemes we investigated in [9], the van Leer scheme and the modified version denoted as AUSMD. Both are discrete schemes of the general form

_ At Ax AtAx

(Fe(u7,u7+1)-Fc(u7_1}

(Fr(U?,U?+i)-FP(U?_i,

0, + AfT-

u?»

u?))

Here Fc and Fp are numerical fluxes assumed to be consistent with the corresponding physical fluxes fc and fp,

I pgCXgVg ^ p\tt\V\

Pg«g^ 'V fwff

( ° \0

agAp ’ \ at Ap J

and H is given by( 0 \0

OCcr

V «i /We see that the fluxes of the the model (4)-(7) consist of three different sort of terms; convective flux terms dx(pav) and dx(pav2), conservative pressure terms dx(aAp) and non-conservative pressure terms adx(p — Ap). The discretization of these terms follows closely the work of Wada and Liou [30] for Euler equations (except from the non-conservative pressure term which does not appear in their model).

3.1. Convective Flux Splitting for the Fc-component.

6 FLATTEN AND EVJE

3.1.1. Van Leer. We consider the velocity splitting formulas used in previous works [12, 30, 7, 8, 9].

if |u| < c otherwise. (14)

where the parameter c represents the physical sound velocity for the system. For the two-fluid model, we assume that it is given by the approximate expression (12). We now let the numerical fluxes be given as follows:

(1) Mass Flux. We let the numerical mass flux (pav)j+1/2 be given as

(mu);+l/2 = (m)j (%j, Cj+l/2 ) + (m) j+1 f (%j+l, <h+l/2 ) (15)for each phase.

(2) Momentum Flux. We let the numerical convective momentum flux (pav2)j+i/2 be given as

(pav2)j+1/2 = -(pav)j+1/2(vj + Uj+i) - -j\(Pav)i+1!2\(vi+1 ~ yi)- (16)

Here and in the following cj+1/2 = max(cj,Cj+i) in accordance with the practice of other works [30, 7, 9]. The van Leer scheme possesses good stability properties but is excessively diffusive, especially on the volume fraction waves. This motivates for proposing a mechanism for eliminating such numerical dissipation. This leads to the AUSMD scheme which we define next.

3.1.2. AUSMD. We consider the AUSMD scheme obtained by replacing the splitting formulas V± given by (14) and used in (15) and (16) with the less diffusive pair

i^(%,c,x) %yi(u,c) + (l-x)Z±W |u|<c^(v ± |u|) otherwise (17)

where= 2(p/a)L

(p/a)L + (/>/a)nand

XR =___2(p/a)*___(/V«)l + (pA*)r

(18)

(19)

for each phase. In order to depict the main idea of the modification leading to AUSMD we consider a contact discontinuity given by

Pl=Pr=Poil f o-r (20)(ug)L = (vi)l = (vs)k = (d)r = v.

All pressure terms vanish from the model (4)-(7), and it is seen that the solution to this initial value problem is simply that the discontinuity will propagate with a velocity corresponding to the velocity v. The exact solution of the Riemann problem will then give the numerical mass flux

(pav)j+1/2 = ^p(aL + aR)v - ^p{aR - «L)M- (21)

It is easy to check that the use of the modified splitting functions (17)-(19) ensure that AUSMD mass fluxes satisfy (21) for the contact discontinuity (20). This is not true for the van Leer scheme.

3.2. Pressure Splitting for the Fp and F“-components. The discretization of the pressure terms is the same for both schemes.


3.2.1. Conservative pressure term. We follow the approach used in [9] which is based on [30], where an upwind type of discretization was used. The conservative pressure term given by aAp is discretized as follows

(aAp)j+l/2 = P+(vL,cj+i/2)(aAp)L +P (vR,cj+1/2)(aAp)R.

where the pressure splitting formulas P±(u, c) are given by

P±(v,c) = V± J(±2 if |u| < c otherwise.

(22)

(23)

3.2.2. Non-conservative pressure term. The non-conservative term is discretized as follows

— f[aj_i/2P+(vj,Cj_1/2)pj + aj+i/2P (vj+i,cj+1/2)p)+1]ziz \ (24)

“ [&j-l/2P+(Vj-l,Cj-l/2)pj-l +d'j+1/2P-(Vj,Cj+1/2)Pj]'j,

where

aj+1/2 - 2 (ai +ai+i)-

We refer to [9] for a description of the motivation behind this particular discretization.

4. Some Observations

In the following two selected numerical examples taken from [9] will be presented. We want to compare the performance of the van Leer, AUSMD, and a Roe scheme and thereby reveal characteristic behavior. The implementation of the Roe scheme is described in [9].

As will be the case for all numerical simulations presented in this paper, our main concern will be to demonstrate the inherent accuracy and stability properties of the schemes. Consequently we limit ourselves to first order accuracy in space and time together with an explicit time integration.

4.1. A Large Relative Velocity Shock. We consider an initial Riemann problem also investigated by Cortes et al [4] for a similar two-fluid model. The initial states are given by

and

p ' 265000 '0.71

vs 65Vi 1

WR" P ' 265000 '

<*i 0.7vg 50

. yi 1

(25)

(26)

No source terms are taken into account. We used the timestep Ax/At = 103 m/s and a computational grid of 100 cells. The results, plotted at the time T = 0.1 s, are given in Figure 1. The reference solution was computed using the Roe scheme on a fine grid of 10 000 cells. The existence of two separate volume fraction waves can be seen from the small wedge in liquid fraction at x = 50m. We make the following observations:

• The van Leer scheme is able to produce stable and nonoscillatory approximations, however, it is excessively diffusive on the slow volume fraction waves.

• The AUSMD produces a resolution of sonic waves which is comparable to that of the van Leer and Roe scheme. However, the slow volume fraction waves located around x = 50 m are reproduced with less numerical diffusion. Unfortunately numerical oscillations, which are especially severe for the liquid velocity, occur for AUSMD.

FLATTEN AND EVJE

reference--------

. .V- ......................................................

0 10 20 30 40 50 60 70 SO 90 100Distance Cm)

Distance (m)

Distance (m)

Distance (m)

Figure 1. LRV shock tube problem. Comparison between van Leer, AUSMD, and Roe scheme on a grid of 100 cells. Top left: liquid fraction. Top right: pressure. Bottom left: liquid velocity. Bottom right: gas velocity.

4.2. Water Faucet. We now consider a benchmark faucet flow problem proposed by Ransom [18], which has been extensively studied [3, 9, 28, 15, 29, 17].

We consider a vertical pipe of length 12 m with the initial uniform state

" P ' 105 Pa 'a\ 0.8yg 0

. u 10 m/s(27)

Gravity is the only source term taken into account, i.e. in the framework of (6) and (7) we have

Qk — QPkOlkt (28)

with g being the acceleration of gravity. At the inlet we have the constant conditions a\ — 0.8, v\ — 10 m/s and vg = 0. At the outlet the pipe is open to the ambient pressure p = 105 Pa. The remaining variables at the boundaries are determined by simple extrapolation.

A contact discontinuity in the volume fraction will arise as the liquid falls under the acceleration of gravity. It is possible to express an approximate solution in analytical form: [9, 29, 17]

hi(#,t) + 2.W for x <v0t+ \gt2 Vo + gt otherwise. (29)

oi(z,t) ao(l + 2gxv0 2) 1/2 for x < vot + ^gt oto otherwise. (30)

The parameters ao = 0.8 and vq — 10 m/s are the initial states.A comparison between the AUSMD and the Roe scheme regarding the accuracy on volume

fraction is given in Figure 2. For coarse grids, the AUSMD and Roe scheme produce similar solutions. However, AUSMD introduces a slight overshoot for 1200 cells, which increases in amplitude


ROE

reference 12 000 cells

1 200 cells 120 cells 24 cells

Distance (m)

AUSMD

reference12 000 cells

1 200 cells120 cells24 cells

Distance (m)

Figure 2. Water Faucet. t=0.6 s. Accuracy on volume fraction. Top: Grid refinement for Roe scheme. Bottom: Grid refinement for AUSMD scheme.

with further grid refinement. This contrasts the Roe scheme where the reference solution seems to be approached in a fully monotone way.

This problem indicates the same as we observed for the previous example: A USMD has some problems with resolving the slow-moving volume fraction waves without introducing oscillations.

Remark 4. Despite the above observed deficiency of the AUSMD scheme, the scheme still seems to be a good candidate for solving two-phase problems relevant e.g. for the petroleum industry. In [9] we demonstrated that by introducing a slight modification of the basic A USMD scheme, mainly consisting of a switch to a more diffusive scheme like the van Leer scheme in the transition to single-phase flow, we could accurately solve difficult problems like separation of two phases and the oscillating manometer problem.

4.3. Conjecture. The above numerical experiments indicate that the numerical fluxes of the Roe and the AUSMD scheme possess numerical dissipation of very similar strength. However, while

10 FLATTEN AND EVJE

the Roe scheme gives stable and non-oscillatory approximations we observe that the AUSMD scheme tends to introduce oscillations and/or overshoot around volume fraction waves. We believe that this deficiency is due to the weak coupling between mass and momentum equations in the construction of numerical fluxes for AUSMD. Basically, the equations are discretized as a set of scalar equations. This strongly contrasts the Roe scheme which treats the model in a strongly coupled manner through the Jacobian matrix, hi the next section, motivated by the success of the Roe scheme, we propose a general method for modifying hybrid FVS/FDS schemes such that the numerical flux associated with the pressure term as well as the numerical mass fluxes are composed of a mixture of components from the momentum equations and by that enforce a stronger coupling between the mass and momentum equations.

5. The Mixture Flux (MF) Method

The aim of this section is to develop a modification of the AUSMD scheme presented in Section 3 which possesses the following properties:

• Riemann-free solver;• non-oscillatory approximations;• accuracy comparable to the Roe scheme on all waves.

The starting point is the model (4)-(7) written on the form

( Pgag \ ( Pgagvg \ f ® \ {+dt

p\OL\ + dxV Pia\v\ )

p\a\V\

V J

o(%gdxp

V aAp J

+

o X ( 0 \0 0(Ap)<%,ag Qg (31)

As a motivation, before introducing the modified solution method we now focus on the following two observations:

5.1. Two Observations relevant for the MF method.

Observation 1. We have already observed that the pressure is related to the masses mg and mi through the static relation (13). In the following we want to demonstrate how the pressure is related to the momentums through a certain dynamical relation.

To see this, we consider the total mass conservation equation obtained by adding the two separate mass conservation equations.

d d— (mg + m0 + (Pgagvg + Pi®-\v\) — 0. (32)

From (13) we have that

Consequently, we see that

mg = nig{m\,p) = (1 — ^ Pg-

dmg Pg dmg 1 ni{ % , "mipg"dm\ Pi ’ dp Pi.

dp +. Pi .

dfh dp" (33)

Differentiating out the first temporal term of (32) and using (33) we get the following nonconservative differential equation for the pressure

~it+K ifldx (pe0lgV^ + pgdx - 0

whereK = dpg '

!tyagPi

(34)

(35)

Having seen that the pressure is directly related to the momentums we also recall that thepressure p can be obtained from the masses m* = pkOik through the relation (13). In the following we want to include both these aspects in the calculation of the pressure. First, as before we obtain the pressure pj at the cell center from (13). In addition, we will introduce the pressure Pj+i/2 at


the cell interface, obtained through an appropriate discretization of the pressure evolution equation(34).

Observation 2. Again, noting that the relation (13) can be written on the form

m. = mg(mhp) = ( 1 - mi#(p) Pg(p),

we see that

= ——dm i [1~7]z$Pb)p mpsPt

dmg = {mg)midm\ + (mg)pdp

mPi V Pi

In other words, we have the relationdp = n(pidmg + pgdm$,

where1

^<%Pg + %^#g#Moreover, noting that the relation (13) can be written on the form

mg = mg(ai,p) = (1 - ai)pg(p),we see that

dmg = (mg)aida i + (mg)pdp

= -pgda\ + (mg)pdp.Using (36), this relation can be rewritten as

(pi)P dp.

(36)

doci — KQtg (pg)pdmi —h f oig {pg)p/i | dmgV Pg PgJ

= K^ag(pg)pdmi — a\(pi)pdm^j.

In other words, we havedot.\ — k(—-^-ayd/m.r H—J^O'^dnii).

op B dp B(37)

By combining (36) and (37) we can write the masses in terms of a “pressure” and a “volume fraction” component as follows:

and

dmg = ag^^dp — pgda\

dm\ = a\^p-dp + p\da\. op

(38)

(39)

The relations (36) and (37) reflect that differentials of the primitive variables a\ and p generally depend strongly on properties of the mixture of both masses through the differentials dmg and dm\. We recall from Section 3.1 that the AUSMD scheme is derived with the motivation of obtaining an accurate resolution of a discontinuity in the volume fraction variable, with the assumption of equal pressure. The derived mass fluxes do not take into consideration the effect of a varying pressure, and for the numerical experiments in Section 4 spurious oscillations were observed near discontinuities in the volume fraction variable. We want to take this aspect into account in the proposed modification of AUSMD. More specifically, we shall derive numerical mass fluxes which are consistent with the differential relations (36)-(39).

Related to this, we may also recall that the eigenstructure of the system is such that the pressure is commonly associated with fast-moving waves and the volume fraction is associated with slow- moving waves. This suggests that we should solve the pressure with a more robust scheme, where the numerical dissipation is increased to be in accordance with the faster velocity of the sonic

12 FLATTEN AND EVJE

waves whereas for the slow volume fraction wave we should use a numerical flux whose numerical dissipation is low. In the next section we try to implement this understanding is a specific sense.

5.2. Outline of the MF method. With the above two observations in hand we will now describe an approach consisting of basically two main steps; the first step deals with the calculation of the cell interface pressure Pj+1/2 (Observation 1) whereas the second deals with the calculation of the cell center pressure pj from the masses via (13). The essential part of the second step is to develop numerical mass fluxes which are consistent with Observation 2.

Note that step (I) deals with the pressure splittings whereas step (II) deals with the convective splittings.

(I) Derivation of an evolution equation for the pressure Pj+1/2 o,t the cell interface. The purpose of this pressure equation is to allow the pressure p and the momentum pkOtkVk to develop in a coherent manner. Particularly, we obtain a stronger coupling between the pressure and velocity fields than was the case for the pressure splittings used in the previous AUSM-type schemes [17, 9]. Increased robustness is the motivation behind our present approach.

By splitting the pressure based on an evolution equation, we honor time-tested traditions (see for instance [2]). However, by means of the mixture fluxes defined below, several new aspects are introduced.

(Ha) Derivation of mixture mass fluxes appropriate for solving for m&j from the mass equations. This step involves the construction of mixture mass fluxes which are motivated by the relations (36) and (37). In particular, we introduce flux components Fjf associated with sonic waves and Fff associated with volume fraction waves.

(lib) Specification of appropriate choices for F® and Fff• Fj?: The calculation of this mass flux component should be tightly coupled with the

calculation of the cell interface pressure Pj+1/2 obtained from the pressure equation. This step ensures that Pj+1/2 and pj will follow a concurrent time development, and this is important in order to avoid oscillations around the slow-moving volume fraction waves.

• Fjf: The construction of this mass flux component should be chosen such that an accurate resolution of volume fraction waves is ensured.

We now describe a fully discrete implementation of the above algorithm. Given a uniform grid with time step At and spatial mesh grid size Ax, we can define an approximation U”+1 of U(xj,tn+1) by the following three step algorithm:

Step I: The Pressure Evolution Equation. Discretizing the equation (34) by a staggered Lax-Friedrichs type of scheme we obtain

pRb-M+fW g.H-1 - % %+. - %AxAt

where we use the shorthand

= -(w)"+1/2- - (Kpg)j+l/2" Ax(40)

Ik — rrikVk.This cell interface pressure Pffxpi ^ then used in the momentum equations of (31) as follows

j-n+l _ rra n”-*-1 — r/1"*"1^^ = -4 - (Ap);4W - W

At

Ift1 - /,"•id At ------ - ’3

Here 5X represents the operator

Axnn+1 _ nn+1 Pj+1/2 Pj-1/2

Ax

Ax

" + Wg )j (41)

+ (Qi)?. (42)

(43)


For the numerical fluxes (pkCtkvl)j+i/2 we employ the AUSMD fluxes as described by (15)-(19). For the numerical flux (%)j+i/2 we use a central discretization as follows

(®k)j+1/2 = 2 (afcJ + ak,j+1) - (44)

In this respect we follow in the footsteps of Coquel et al [3] and Paillere et al [17]. In particular, this simple treatment is independent of the splitting formulas P± given by (23).

Note that the cell interface pressure pJ+i/2 can be written on the viscous form

P^11/2 = -P(uy,u»+1)1 r 1 (45)= 5(P? + !>”+.) - [%+./:%+. - %) + %+,„(%+. - %)]

where the numerical viscosity coefficients Dff j+1/2 are given by

Dti+l/2 = ~^K7+l/2P\,j+l/2‘ (46)

andDnuhi+i/2

^-Kn On^xKj+l/2Pg,j+l/2‘ (47)

Interface values (•)”+1/2 needed for the coefficients (46) and (47) are obtained by using P™+1/2 (which has been calculated from (40) at the previous timestep) and ct”+1y2 defined as the arithmetic average

a■7+1/2 - x(ai +al+i)-

Remark 5. We note that the role of the non-conservative pressure evolution equation (40) is simply to define an appropriate numerical fluxPj+1/2 = P(Uj,Uj+i) for the discretization of the pressure term in (41) and (42). From (45) we easily see that this numerical flux is consistent with the physical flux, i.e. P(U, U) = p for all U.

Remark 6. Other choices for the discretization of the pressure evolution equation (34) than the one given by (40) would of course be possible. One natural choice could be to consider

~(m)'3+1/2 C+i-%Ax - (Hpg)7+1/2

rn1l,j+1 %

Ax (48)

We will refer to this as the FTCS (forward time, centered space) scheme. Now the consistency relation (45) between the interface pressure Pj+1 /2 and cell center pressure pj no longer holds, and there is no obvious mechanism that drives the difference between these two pressures to zero. Note also that there is no numerical dissipation terms associated with the discretization of dxIk in (48), whereas such dissipation terms may be introduced in (45) through the term (pf +p”+1)/2. In the numerical section we explore these two different discretizations of the pressure evolution equation in order to shed light on the importance of a consistent treatment of pj and Pj+1/2.

Step Ila: Construction of Mixture Mass Fluxes. What remains to be solved for now is the masses through a proper discretization of the mass conservation equations of (31). We consider a general discretization

,n+lAt

-^(%+l/2 ■Ki- ■1/2 )■ (49)

where F&d+i/s = Fk{Uj,Uj+1) is the numerical mass flux at cell interface j + 1/2 corresponding to the physical flux fk(U) = pkCtk^k- From (38) and (39) we see that the mass differentials dm* can be splitted in a pressure component dp and a volume fraction component da. We now want to design a numerical flux which is consistent with this splitting, i.e. we introduce a flux component Fp and Fa such that the mass fluxes F\ and Fg are given by

Pi = Pi Fa (50)

14 FLATTEN AND EVJE

and^ = (51)

The flux component Fp is associated with the pressure, hence we want to assign a diffusive mass flux FD for stable approximation of pressure for all waves. Inspired by the differential relation (36) we propose to give Fp the following form

Fp = KpgFP + KpiF^ (52)Similarly, the flux component Fa is associated with the volume fraction, hence we want to assign an accurate mass flux FA. Inspired by the differential relation (37), we propose to give Fa the following form

Fa = K"5p _ K"^ai"FgA'

Here we note that a subscript j + 1/2 is assumed on the fluxes and coefficients. Substituting (52) and (53) into (51) and (50) we obtain the final hybrid mass fluxes

Fx = k (pgax-^F? + pi«g(Fg ~ Fg )j (54)

andfs = k (^as^FgD + pza^Fs + ~ (55)

The coefficient variables at j + 1/2 remain to be determined. Consistent with the treatment of the coefficients of the pressure evolution equation (45) we suggest finding these from the cell interface pressure pj+1/2 as well as the relation

aj+1/2 = g Or,- + aj+1)-Remark 7. We remark that the consistency criterion

Ffc((7, U) = fk(U) = pkakVk, (56)relating the numerical flux Fk to the physical flux fk, is satisfied for the hybrid fluxes (54) and (55) provided the fluxes F^ and Fj? are consistent. In particular if F^ = Fjf the expressions (54) and (55) reduce to the trivial identity

= (57)Step lib: Specification of Mass Flux Components Fj)) and Fj).

Fj?-component. The purpose of this flux component is to ensure consistency between calculation of masses and the pressure calculation and by that ensure stable (non-oscillatory) approximations of slow moving volume fraction waves. Going back to the pressure equation (40), we see that it naturally defines a conservative scheme for calculating masses at j + 1/2 as

mn+1W+l/2

Atlk,j+1 tM. -

Ax0. (58)

If we now compute the simple average

mk,j = 2 (mk,j-1/2 + mk,j+1/2)

and substitute (58), we obtain the following difference equation for

mn+1k,j - i (2">L - mfcd-i + rn 1=d+i)

+ ■ (4VH -4V1) =«.A t 2Azwhich can be written on flux-conservative form (49) with the numerical fluxes

1 Ax

(59)

(60)


Now we may solve for the masses mk,j using the fluxes (61), taking advantage of the stabilizing effect given by their interdependence of the cell interface pressure Pj+1/2 through and Ik,j+1-

FA-component. The purpose of this flux component is to ensure accurate resolution of slow-moving volume fraction waves (mass fronts). As demonstrated in Section 4 the AUSMD mass fluxes give a resolution of such waves comparable to that of the Roe scheme. Hence, we propose to identify FA with the AUSMD mass fluxes given in Section 3.1 and defined by (15) and (17)-(19). We expect that other choices also would be possible, e.g. the AUSM+ flux of Paillere et al [17]. This will be explored elsewhere.

Remark 8. The fluxes (61) are central and hence highly diffusive. Consequently the fluxes (61) will produce highly inaccurate solutions to slow volume fraction fronts. Therefore we wish to hybridize the flux (61) with a more accurate flux FA via the mixture fluxes (54) and (55) such that we maintain the stability of (61) for the pressure variable while falling back to the accuracy of Fa on the volume fraction waves.

5.3. MF-AUSMD. We now summarize the numerical scheme just derived, referred to as the MF-AUSMD (Mixture Fluxes based on AUSMD) scheme. Let 5X be defined as in (43).

Mass Equations. We discretize the mass equations as follows

At “ ~dxFk’F

Fi-k[ pgai-^F? + piaig-^Ff + piai-^(F® - FA)

where the mass fluxes Fkj+1/2 are given by

andfs = k - FiA)^

as described in Step Ha. Here1 Ax

Fk,j+1/2 - ^(4j + h,j+1) + 4 a# (mfcd mfc-i+i)> - Pkak,

andAUSMD

%+l/2 =as described in Section 3.1.

Ik — mkvk

Momentum Equations. We discretize the momentum equations as follows ■1 _

~Atjn+l _ jn

^ = -a, - (at)?a„ (p); + (Qg)?

Here

and(ak)j-1-1/2 — X (ak,j + Otk,j+1)

1' ■ ' ‘ ■' - h,tri~h,3 a -it~ ^ Jid+1 ~ KiPj+l/2 ~ g (PJ +Pj+0 “ At(Kpl)i+i/2 Ax--- - “ At(Kpg)j+1/2

as described in Step I. Finally

>j+1/2as described in Section 3.1.

6. Basic properties possessed by the Mixture Flux AUSMD scheme

In this section we want to verify that the proposed MF-AUSMD scheme possesses certain basic accuracy and stability properties known from the literature.

16 FLATTEN AND EVJE

6.1. Accurate approximation of steady and moving contact discontinuities. We will now investigate how this Mixture Flux method is related to the AUSM (Advection Upstream Splitting Method) framework of Liou et al [12, 13, 30] regarding the performance on a moving or stationary contact discontinuity. For this purpose we consider the contact discontinuity given by

Pl=Pr=P (62)«l # OR

(vs)l = («i)l = (vs)k = (f)r = v.

All pressure terms vanish from the model (4)-(7), and it is seen that the solution to this initial value problem is simply that the discontinuity will propagate with the velocity v. The exact solution of the Riemann problem will then give the numerical mass flux

(pav)j+1/2 = ^p(aL + aR)v - ^/>(aR - aL)M- (63)

As remarked in Section 3.1 AUSMD mass fluxes take this form for the contact discontinuity (62). Now we want to find a criterion for the Fj^ flux components that ensures that the mixture mass fluxes (54) and (55) also will possess this good feature.

In particular, we note that the pressure will remain constant and uniform as the discontinuity is propagating. Consequently a natural requirement on a “good” flux Ff for stable pressure resolution is that it preserves the constancy of pressure for the moving or stationary contact discontinuity given by (62).

We write (36) asdp = ndfi (64)

wheredp = pgdmi + pidmg. (65)

To maintain a constant pressure we must have dp = 0. Assuming constant pressure, (65) can be integrated to yield

p = pgm\ + pimg = pgpi(ai + as) = pspy.

To maintain constancy of p and hence p we now insist that the flux FD is a consistent numerical flux when applied to the mix mass p. That is, we impose

PgpF + PiFg = PgPiV- (66)

for the contact discontinuity (62).

Definition 1. A pair of numerical fluxes F\ and Fg that satisfy (66) for the contact discontinuity (62) will in the following be termed “pressure coherent” fluxes.

We note that the van Leer mass fluxes given by (15) and (14) as well as the upwind fluxes (63) are pressure coherent. However, we can easily construct a pair of perfectly valid mass fluxes that are not pressure coherent. Consider for example the stationary contact discontinuity (62) with v = 0. Let Fg be given by the upwind flux (63) and F\ be given by the van Leer flux (15). Then

pgFi + piFg = pgpx^ ((ai)L - (ai)R) 0,

not satisfying the requirement (66).We now state the following lemma relevant for schemes obtained by using the Mixture Flux

method.

Lemma 1. Let the mixture fluxes (54) and (55) be constructed from pressure coherent fluxes F(?, and fluxes F^ that reduce to the upwind fluxes (63) on a contact discontinuity of the form (62). Then the hybrid fluxes (54) and (55) also reduce to the upwind fluxes (63) on the contact discontinuity (62).


Proof. We consider the hybrid liquid mass flux (54) and assume that v > 0. Remembering that a subscript j + 1/2 is assumed on the variables, we write the flux as

Fl = K + AigD) + Pia?>~dpF^ ~ milfyFz ) ' (67)

Using the required properties of Ff and Fjf we obtainF\ = k + PiOig-^(ai)iJv — pgpiai~^(l — (q:i)l)v^ = Pi(a\)-LV, (68)

where we have used thatPj+1/2 = Pj = Pj+i (69)

which follows from the assumption of constant, uniform pressure. Spatial and phasic symmetry directly give the corresponding results for Fg and v < 0, completing the proof. □

In view of Lemma 1 we obtain the following result for the MF-AUSMD scheme.Proposition 1. The mass fluxes of the MF-AUSMD scheme described in Section 5.3 reduce to the upwind fluxes (63) on a contact discontinuity of the form (62).

Proof. In view of Section 3.1.2 we know that the Fj^ components in the MF-AUSMD scheme reduce to the upwind fluxes (63) on a contact discontinuity of the form (62). Thus, we only need to check that the F]? components given by (61) are pressure coherent in the sense of Definition 1 and then appeal to Lemma 1. Substituting constant pressure and velocities in (61) we get

PsfiD + PiF? = PgP

= PgPi

2 (“1.1 + “1.1+1) + 4^ (“i.l - “i,l+i)

+ PgPi

fa + a + ^a -1)

v. . Ax . 'gWg.l + “g,l+U + 4^(“g,l - “g,l+i)

— PgPlV-□

This result illustrates that the Mixture Flux method presented here is a close relative to the AUSMV/D philosophy of Wada and Liou [30], as it achieves the same goal of accurately resolving moving or stationary contact discontinuities. A notable difference is that the AUSMV/D framework combines velocity splitting formulas, whereas the MF method presented here combines numerical fluxes directly and thereby enforces a much stronger coupling between the various equations.

6.2. Abgrall’s principle. According to the principle due to Abgrall [1, 22, 23] it is desirable that the numerical scheme respects the following physical principle:A flow, uniform in pressure and velocity must remain uniform in the same variables during its time evolution.In other words, if we had constant pressure and velocity everywhere in a flow at the time level tn, then we will get the same pressure and velocity at the time tn+1.

We now check if the MF-AUSMD scheme obeys Abgr all’s priciple. Consequently, we assume that we have the contact discontinuity given by (62) and that it remains unchanged during the time interval [tn,tn+1]. In view of Proposition 1 and the fact that the convective fluxes of the momentum equations are based on (16), we immediately conclude that the mass equations (49) and the momentum equations (41) and (42) take the form

(pa)J+1 - (pa)™ - {(pav)™+1/2 - (/xw)j_1/2)

(Ap) At2Ax'(“j+i “i-i) • an^L(„n+l

A% '̂j+1/2 Pi-1/2).

18 FLATTEN AND EVJE

■5

reference-------- reference--------

MF-AUSMD + /f"................................

/: \270000

269000

1| 268000

f

, X,0 10 20 30 40 50 60 70 SO 90 100 0 10 20 30 40 50 60 70 SO 90 100

Distance Cm) Distance (m)

Distance Cm) Distance (m)

Figure 3. LRV shock tube problem, T = 0.1 s, 100 grid cells. MF-AUSMD vs Roe scheme. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

where (pav)™+1j2 is on the form (63). From (11) we conclude that (Ap)J = 0. Referring to (45) we also see that P^Xl/2 reduces to

Pj+1/2 = P~ Ki+1/2Xx ^>1Psv(agtj+1 ~ ag,i) + PlPgv(al,j+l ~ ai,j)] = P-

Consequently, the pressure terms vanish and we can conclude that the discretization of the MF- AUSMD satisfies Abgrall’s principle.

7. Numerical Simulations

In the first example we seek to obtain some more detailed insight into the mechanisms of the MF method. Particularly, we focus on the two following points which constitute the heart of the MF approach: (i) The effect of using the mixture mass fluxes (54) and (55) obtained by mixing the F® and the Fjf components; (ii) The discretization of the pressure evolution equation (40).

The purpose of the rest of the section is to demonstrate the overall good approximation properties of the MF-AUSMD scheme by considering the performance for a series of various flow cases. In particular, we compare with the AUSMD and the Roe scheme used in Section 4. The purpose is to demonstrate that the MF-AUSMD scheme possesses stability properties similar to the Roe scheme and at the same time keeps the accuracy of the AUSMD scheme.

7.1. Large Relative Velocity Shock. We now revisit the LRV shock studied in Section 4.1.

7.1.1. Test of stability and accuracy for MF-AUSMD. We now aim to compare the MF-AUSMD scheme with the Roe scheme under equal conditions. As in Section 4.1, we assume a grid of 100 cells and a timestep of

^ = 1Q3 m/s. (70)

The results, plotted at the time T = 0.1 s, are given in Figure 3. As the figure indicates, the schemes


Distance Cm)0 10 20 30 40 50 60 70 80 90 100

Distance (m)

Figure 4. LRV shock tube problem, T = 0.1 s, 100 grid cells. MF-AUSMD vs the purely diffusive (Ff) and accurate (F^) components. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

are virtually inseparable on the resolution of the right-going sonic wave. On the left-going sonic wave, the MF-AUSMD scheme is slightly more diffusive than the Roe scheme. A comparison with the plot in Section 4.1 reveals that the MF-AUSMD scheme introduces somewhat more diffusion on this wave also compared to the AUSMD scheme. On the other hand, we note that the oscillatory behaviour of the original AUSMD scheme is completely absent, demonstrating the fruitfulness of the MF approach.

7.1.2. Mixture mass fluxes versus single mass fluxes. We now wish to illustrate more precisely the effect of introducing the mixture mass flux obtained by combining the two different flux components F]? and F^ as described by (54) and (55). For that purpose, we consider the following 3 schemes:

(1) The MF-AUSMD scheme.(2) The scheme obtained by replacing the MF-AUSMD mass fluxes with the pure AUSMD

fluxes F^. This is denoted as the F(A)-scheme in Figure 4. Note that this is not identical to the AUSMD scheme of Section 3.1.2, as the discretization of the pressure term is different.

(3) The scheme obtained by replacing the MF-AUSMD mass fluxes with the pure diffusive mass fluxes F® given by (61). This is denoted as the F(D)-scheme in Figure 4.

Results for 100 cells are given in Figure 4. We note the following:• The F(D)-scheme is very stable but highly diffusive for the volume fraction wave, more

diffusive than the van Leer scheme (compare with Figure 1 of Section 4.1).• The F(A)-scheme is accurate on all waves, similar to the AUSMD scheme. However, the

heavy oscillations observed for the liquid velocity for the AUSMD scheme (Figure 1 of Section 4.1), have been eliminated. This clearly is an effect of the use of the pressure

20 FLATTEN AND EVJE

reference--------MF-AUSMD--------

uncoupled +

0 10 20 30 40 50 60 70 SO 90 100Distance Cm)

Distance Cm)

uncoupled

264000

Distance (m)

Distance (m)

Figure 5. LRV shock tube problem, T = 0.1 s, 100 grid cells. Lax-Friedrichs based MF-AUSMD vs FTCS based MF-AUSMD. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

evolution equation (40) in the discretization of the pressure terms of the momentum equations. However, the F(A)-scheme is not perfect as the oscillation for the pressure variable remains.

• The use of the mixture mass fluxes (54) and (55) in the MF-AUSMD scheme efficiently removes the oscillation observed for the pressure variable of the F(A)-scheme. The MF- AUSMD scheme really seems to combine the fluxes Fjf and FtP in the desired way, giving results similar to F]? for fast waves and Fjf for slow waves.

• The fact that MF-AUSMD is slightly more diffusive than AUSMD on the sonic waves seems to follow directly from the fact that Ff is slightly more diffusive than Fjf on these waves.

7.1.3. Comparison of two different discretizations of the pressure evolution equation (34). We now compare the Lax-Friedrichs discretization of the cell interface pressure used in MF-AUSMD, with a modified variation where we use the FTCS scheme of Remark 6. This latter scheme implies that the cell interface pressure pj+i/2 is uncoupled from the cell center pressure pj. We keep the mass fluxes unchanged, using the mixture mass fluxes (54) and (55). Results are given in Figure 5. We note the following:

• The lack of a consistency mechanism between the cell interface pressure Pj+1/2 and the cell center pressure pj produces a large undershoot in the pressure near x = 50 m where the volume fraction wave is located.

• The uncoupled scheme based on a FTCS pressure discretization produces a sharp resolution of the pressure waves. On the other hand, numerical oscillations are produced near the pressure discontinuities. As noted in Remark 6 the FTCS scheme has zero numerical viscosity, and will be unstable on a scalar equation (see for instance [25]). The result above indicates that a certain amount of numerical dissipation in the pressure equation is needed to stabilize the solution for sonic waves.



Figure 6. Modified LRV shock tube problem, T = 0.1 s, 100 grid cells. MF- AUSMD, AUSMD and Roe scheme. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

Remark 9. Clearly, the discretization of the pressure evolution equation at the cell interface has a strong effect on the cell center pressures produced by the MF scheme. For the purposes of this paper, we prefer to stick to the Lax-Friedrichs discretization (40), because of the simplicity and robustness of this scheme. In particular it allows for writing the cell interface pressure on a consistent, viscous form (45) in a straightforward manner. The numerical results indicate that other ways of discretizing the pressure equation could be explored with the possibility of improving the accuracy of the fast (sonic) waves.

7.2. Modified Large Relative Velocity Shock. We consider a modified version of the LRV shock, where we introduce a jump in the liquid velocity as well as a larger jump in volume fraction. This problem was studied as shock tube problem 2 in [9].

The initial states are given by

and

' p ' 265000 Pa 'ttl 0.7vs 65 m/s

. D 10 m/s

WR" P ' 265000 Pa '

a\ 0.1vs 50 m/s

. D 15 m/s

(71)

(72)

7.2.1. Comparison between AUSMD, MF-AUSMD and Roe scheme. Results after T = 0.1 s are given in Figure 6, using a grid of 100 cells and a timestep Ax/At = 750 m/s. The reference solution was calculated by the Roe scheme on a grid of 10 000 cells.

22 FLATTEN AND EVJE

Distance Cm)

Figure 7. Toumi’s shock tube problem, T = 0.08 s, 100 grid cells. MF-AUSMD vs basic AUSMD scheme. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

We note that the AUSMD scheme produces a large overshoot in the pressure variable, and is also inaccurate in the resolution of the left-moving sonic wave. MF-AUSMD is non-oscillatory, and produces here a solution that is intermediate between the Roe and AUSMD solution.

7.3. Toumi’s Water-Air Shock. We now consider an initial value problem of a kind introduced by Toumi [27] and investigated by Tiselj and Petelin [26] and Paillere et al [17]. The initial states are given by

p " 2 • 107 Pa "ai 0.75vs 0Vl 0

(73)

and

WR" p " ' 1 -107 Pa "

ai 0.9yg 0D 0

(74)

Again no source terms are taken into account. Following Paillere et al [17], we modify the interfacial pressure correction (11) for this problem, setting <7 = 2.

7.3.1. Comparison between MF-AUSMD and basic AUSMD scheme. Results after T = 0.08 s are given in Figure 7, using a grid of 100 cells and a timestep Ax/At = 1000 m/s. The reference solution was calculated by the MF-AUSMD scheme on a grid of 10 000 cells. We note that we achieve a wave structure that is highly similar to the one reported by Paillere et al [17], although slightly different submodels are used. This observation supports our belief that the wave structure of the model is largely unaffected by the inclusion of energy equations, as stated in Remark 1.

This example demonstrates overshoots for the basic AUSMD scheme whereas the MF-AUSMD scheme is fully nonoscillatory.


Distance (m) Distance (rn>

Figure 8. Toumi’s shock tube problem, T = 0.08, 200 grid cells. MF-AUSMD vs Roe scheme. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

7.3.2. Comparison between MF-AUSMD and Roe scheme. Results after T = 0.08 s are given in Figure 8, using a grid of 200 cells and a timestep Ax/At = 1000 m/s.

We observe that the Roe and MF-AUSMD schemes give a similar resolution of the sonic waves for this problem. However, tendencies for overshoots on the volume fraction waves are observed for the Roe scheme whereas the the MF-AUSMD scheme is nonoscillatory. On the other hand, MF-AUSMD seems more diffusive on the near-stationary discontinuity at x = 50 m.

7.3.3. Convergence properties of the MF-AUSMD scheme. In Figure 9 we study the convergence of the MF-AUSMD scheme as the grid is refined. The result demonstrates that the MF-AUSMD approaches the reference solution without introducing any spurious oscillations.

7.4. Water Faucet Problem. We now wish to focus more on the resolution of volume fraction waves. For this purpose we revisit the water faucet problem studied in Section 4.2.

7.4.1. Comparison between MF-AUSMD and Roe scheme. In Figure 10 the MF-AUSMD is compared to the Roe scheme for T = 0.6 s on a grid of 120 computational cells. The timestep Ax/At = 103 m/s is used. The pressure reference was calculated using MF-AUSMD on a grid of 12 000 cells, for gas fraction and liquid velocity the approximate analytical expressions were used. Only for the pressure does the plot demonstrate any notable difference between the schemes - the MF-AUSMD is somewhat more diffusive than the Roe scheme.

7.4.2. Convergence properties of the MF-AUSMD scheme. In Figure 11 we investigate how the scheme converges to the expected analytical solution as the grid is refined. In Section 4.2 we found that AUSMD produces small overshoots in the volume fraction for very fine grids. A similar behaviour was reported by Paillere et al [17] for their AUSM+ scheme. As we can see, no overshoots are produced by the MF-AUSMD scheme, and by that the improved approximation properties of the MF-AUSMD scheme over the AUSMD scheme are verified.

24 FLATTEN AND EVJE

Figure 9. Toumi’s shock tube problem. Grid refinement for the MF-AUSMD scheme. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.


Figure 10. Water faucet problem, T = 0.6 s, 120 grid cells. MF-AUSMD vs Roe scheme. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.


reference0.5 - 12 000 cells

1 200 cells120 cells24 cells

0.45 -

0.4 -

0.35 -

0.3 -

0.25 -

Distance (m)

Figure 11. Water faucet problem, T=0.6 s. Convergence properties of the MF- AUSMD scheme.

reference 600 cells 120 cells 24 cells0.45 -

0.4 -

0.35 -

0.3 -

0.25 -

0.15 -

Distance (m)

Figure 12. Water faucet problem, T=0.6s. Non-hyperbolic model. Convergence properties of the MF-AUSMD scheme.

7.4.3. Non-hyperbolic model. We now consider a non-hyperbolic model, that is in the framework of (11) we consider the choice

<7 = 0. (75)Paillere et al [17] found that for this non-hyperbolic model, oscillations were produced near the discontinuity as the grid was refined. We now want to investigate to which degree this effect is independent of the numerical scheme, and to which degree the dissipative mechanism of different schemes act differently in magnifying the expected oscillatory behaviour of a non-hyperbolic model. The effect of grid refinement for the MF-AUSMD scheme is demonstrated in Figure 12.

26 FLATTEN AND EVJE

We observe that an oscillation is produced to the right of the discontinuity for the grid of 600 cells. In fact it was observed that for 1200 cells, this oscillation would grow to the point were the scheme exploded. This confirms the assertion of Paillere et al [17] that we should expect oscillations of a mathematical nature for this test case, demonstrating the importance of using a hyperbolic model.

7.5. Separation Problem. We now consider the separation problem introduced by Coquel et al [3], previously investigated by Evje and Flatten [9] and Paillere et al [17]. We consider a vertical pipe of length 7.5 m, where gravitational acceleration is the only source term taken into account. Initially the pipe is filled with stagnant liquid and gas with a uniform pressure of po = 105 Pa and a uniform liquid fraction of a\ = 0.5. The pipe is considered to be closed at both ends, i.e. both phasic velocities are forced to be zero at the end points.

The following approximate analytical solution was presented in [9]

f ^/2gx for x < \gt2Vl(x,t) = < gt for ^gt2 < x < L — \gt2 (76)

l 0 for L — \gt2 < x

f ° for x < \gt2ai (x,t) = < 0.5 for \gt2 < x < L — \gt2 (77)

l 1 for L - \gt2 < x

where L = 7.5 m is the length of the tube. This approximate solution consists of a contact discontinuity at the top of the tube and a shock-like discontinuity at the lower part of the tube. After the time

T = — — 0.87 s (78)

these discontinuities will merge and the phases become fully separated. The volume fraction reaches a stationary state, whereas the other variables slowly converge towards a stationary solution. In particular we expect the stationary pressure to be fully hydrostatic, approximately given by

PoPo + pi9 (x ~ L/2)

for x < L/2 for x > L[2. (79)

7.5.1. Transition to One-Phase Flow. We observed that the basic MF-AUSMD scheme would produce instabilities in the transition to one-phase flow. Indeed this is a common problem for two- phase flow models, observed among others by Coquel et al [3] for their kinetic scheme, Paillere et al [17] for their AUSM+ scheme and Romate [21] for his Roe scheme. Romate suggested a scheme switching strategy for solving this problem, where the original scheme is replaced with a stable, diffusive scheme near one-phase regions.

In [9] we introduced a modification of the basic AUSMD scheme, denoted as AUSMDV*, where we took advantage of a highly robust flux vector splitting (FVS) scheme to achieve a stable transition to one-phase flow. Using a frictionless model, we observed strong velocity gradients for the disappearing phase. Such large velocities are unphysical and may also cause the pressure variable to fail to converge to a hydrostatic distribution [9].

In this paper we follow the approach of Paillere et al [17], and include an interface momentum exchange term on the form

= C<%gai/)g(ng-ri), (80)where C > 0 and Mj° = — M^, conserving total momentum.

In addition to allowing for more physically valid velocity calculations, this term also prevents numerical instabilities related to large relative velocities in the one-phase regions. This allows for a stable numerical transition to one-phase flow using only a slight modification of the MF-AUSMD scheme, as described in the following.


MF-AUSMD' o

Distance (m)6 7

Distance (m)

Figure 13. Separation problem, T=0.6 s. MF-AUSMD* scheme, 100 cells. Left: Liquid fraction. Right: Liquid velocity.

Definition 2. We consider a hybrid of the MF-AUSMD and the van Leer scheme, denoted as MF-AUSMD*, where the numerical mass fluxes are given by the following expression

pMF—AUSMD* _ ^pvaii Leer ^__^jpMF-AUSMD (81)

Here s is chosen ass = max(0L,</>R), (82)

where is an indicator function designed to be 1 near one-phase regions, 0 otherwise.

For the purposes of this paper we choose^ (83)

where we use the parameter k = 50.We note that the MF-AUSMD* scheme differs from the MF-AUSMD scheme only near one-

phase liquid regions. For the coefficient C of (80), we choose the expressionC - Co4>j, (84)

ensuring that the interface friction acts more strongly near one-phase liquid regions where we expect the gas to dissolve in the liquid. For the value Co we follow Paillere et al [17] and choose

Co = 50000 sA (85)

7.5.2. Numerical results. Results after T = 0.6 s are plotted in Figure 13, using a grid of 100 cells and a timestep Ax/At = 2000 m/s. We note that good accordance with the expected analytical solution is achieved.

Results after T = 1.5 s are plotted in Figure 14, using the same grid. Now fully stationary conditions are reached. Again we observe good agreement with the approximate analytical solutions.

7.5.3. Convergence properties of the MF-AUSMD* scheme. In Figure 15 the effect of grid refinement on resolution of volume fraction is illustrated for the MF-AUSMD* scheme at the time of T—0.6 s. The figure indicates that the expected analytical solution is approached in a monotone way.

8. Summary

We have presented a framework, the Mixture Flux (MF) method, for constructing accurate and robust numerical schemes for the two-fluid model. The framework may be viewed as a refinement of previously studied flux-splitting schemes, involving a stronger coupling between the phasic variables in accordance with the mixture nature of the model. Particularly, we have constructed a numerical scheme on the basis of the AUSMD flux used in [9], demonstrating that we keep the accuracy properties of the basic AUSMD while significantly improving its stability properties. In particular we have demonstrated that the resulting MF-AUSMD scheme compares very well

28 FLATTEN AND EVJE

cpL0XCCOXOXO3OXU.

0 1 2 3 4 5 6 7Distance (m> Distance (m)

Figure 14. Separation problem, T=1.5 s. MF-AUSMD* scheme, 100 cells. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity.

1

0.8

0.6

0.4

0.2

0

reference--------1 000 cells--------

100 cells -------25 cells ..........

0 1 2 3 4 5 6 7Distance (m)

Figure 15. Separation problem, T=0.6s. Convergence properties of the MF- AUSMD* scheme.

with the Roe scheme in terms of accuracy and robustness for several different test cases. Most importantly the MF-AUSMD does not involve solving a local Riemann problem by eigenstructure decomposition and is therefore superior to the Roe scheme when it comes to computational efficiency.

Acknowledgements. The first author thanks the Norwegian Research Council for financial support through the “Petronics” programme.

References

[1] R. Abgrall, How to prevent pressure oscillations in multicomponent flow calculations, J. Comput. Phys. 125, 150-160, 1996.

[2] K. H. Bendiksen, D. Malnes, R. Moe, and S. Nuland, The dynamic two-fluid model OLGA: Theory and application, in SPE Prod. Eng. 6, 171-180, 1991.

[3] F.Coquel, K. El Amine, E. Godlewski, B. Perthame, and P. Rascle, A numerical method using upwind schemes for the resolution of two-phase flows, J. Comput. Phys. 136, 272-288, 1997.


[4] J. Cortes, A. Debussche, and I. Toumi, A density perturbation method to study the eigenstructure of two-phase flow equation systems, J. Comput. Phys. 147, 463-484, 1998.

[5] J. R. Edwards and M.-S. Liou, Low-diffusion flux-splitting methods for flows at all speeds, AIAA Journal 36, 1610-1617, 1998.

[6] J. R. Edwards, R. K. Franklin, and M.-S. Liou, Low-diffusion flux-splitting methods for real fluid flows with phase transition, AIAA Journal 38, 1624-1633, 2000.

[7] S. Evje and K. K. Fjelde, Hybrid flux-splitting schemes for a two-phase flow model, J. Comput. Phys. 175, 674-701, 2002.

[8] S. Evje and K. K. Fjelde, On a rough ausm scheme for a one-dimensional two-phase flow model, Comput. & Fluids 32, 1497-1530, 2003.

[9] S. Evje and T. Flatten. Hybrid flux-splitting schemes for a common two-fluid model. J. Comput. Phys, to appear.

[10] K.-K. Fjelde and K.H. Karlsen, High-resolution hybrid primitive-conservative upwind schemes for the drift flux model, Comput. & Fluids 31, 335-367, 2002.

[11] A. Harten, P.D. Lax and B. Van Leer, On upstream differencing and Godunov schemes for hyperbolic conservation laws, SIAM Review. 25, 35-61, 1981.

[12] M.-S. Liou, A sequel to AUSM: AUSM(+), J. Comput. Phys. 129, 364-382, 1996.[13] M.-S. Liou and C. J. Steffen, A new flux splitting scheme, J. Comput. Phys. 107, 23-39, 1993.[14] Y. Y. Niu, Simple conservative flux splitting for multi-component flow calculations, Num. Heat Trans. 38,

203-222, 2000.[15] Y.-Y. Niu, Advection upwinding splitting method to solve a compressible two-fluid model, Int. J. Numer.

Meth. Fluids 36, 351-371, 2001.[16] S. Osher, Riemann solvers, the entropy conditions, and difference approximations, SIAM J. Num. Anal. 21,

217-235, 1984.[17] H. Paillere, C. Corre and J.R.G Cascales, On the extension of the AUSM+ scheme to compressible two-fluid

models, Comput. & Fluids 32, 891-916, 2003.[18] V. H. Ransom, Numerical bencmark tests, Multiphase Sci. Tech. 3, 465-473, 1987.[19] P.L. Roe, The use of Riemann problem in finite difference schemes, Lectures Notes of Physics 141, 354-359,

1980.[20] P.L. Roe. Approximate Riemann solvers, parameters vectors and difference schemes, J. Comput. Phys. 43,

357-372, 1981.[21] J. E. Romate, An approximate riemann solver for a two-phase flow model with numerically given slip relation,

Comput. & Fluids 27, 455—477, 1998.[22] R. Saurel and R. Abgrall, A multiphase godunov method for compressible multifluid and multiphase flows, J.

Comput. Phys. 150, 425-467, 1999.[23] R. Saurel and R. Abgrall. A simple method for compressible multifluid flows. SIAM J. Sci. Comput. 21,

1115-1145, 1999.[24] H. B. Stewart and B. Wendroff, Review article; two-phase flow : Models and methods, J. Comput. Phys. 56,

363-409, 1984.[25] E. Tadmor. Numerical viscosity and the entropy condition for conservative difference schemes. Mathematics of

Computation. 43, 369-381, 1984.[26] I. Tiselj and S. Petelin, Modelling of two-phase flow with second-order accurate scheme, J. Comput. Phys.

136, 503-521, 1997.[27] I. Toumi, An upwind numerical method for two-fluid two-phase flow models, Nuc. Sci. Eng. 123, 147-168,

1996.[28] I. Toumi and A. Kumbaro, An approximate linearized riemann solver for a two-fluid model, J. Comput. Phys.

124, 286-300, 1996.[29] J. A. Trapp and R. A. Riemke, A nearly-implicit hydrodynamic numerical scheme for two-phase flows, J.

Comput. Phys. 66, 62-82, 1986.[30] Y. Wada and M.-S. Liou, An accurate and robust flux splitting scheme for shock and contact discontinuities,

SIAM J. Sci. Comput. 18, 633-657, 1997.

Paper III

Weakly Implicit Numerical Schemes for the Two-Fluid Model


Submitted toSIAM Journal on Scientific Computing

WEAKLY IMPLICIT NUMERICAL SCHEMES FOR THE TWO-FLUID MODEL

STEINAR EVJEA’c AND TORE FLATTEN3

Abstract. The aim of this paper is to construct semi-implicit numerical schemes for a two- phase (two-fluid) flow model, allowing for violation of the CFL-criterion for sonic waves while maintaining a high level of accuracy and stability on volume fraction waves. Based on the results of a previous work [12], we here present a general framework for constructing such weakly implicit schemes without making use of any Riemann solver nor referring to any calculation of flux jacobians.

One important step of the proposed methods is the introduction of a pressure evolution equation. This equation, which is discretized at cell-interfaces, naturally defines a consistent numerical flux for the discretization of the pressure term of the two momentum equations. This step is crucial for the stability of the solutions when the CFL-criterion for sonic waves is violated. Another major step is the decomposition of the numerical mass fluxes , corresponding to the physical mass flux fk = into two components Ff’ and F^ respectively. The purpose ofthe Ff-component is to ensure stability (non-oscillatory behavior) of solutions when the time step is dictated by the fluid velocity and not the sonic velocity, whereas the F^-component is designed such that accurate resolution of volume fraction waves is ensured. Our techniques, which we refer to as ’’Mixture Flux” (MF) methods, are based on the above two steps, but give room for different choices in the discretization of the pressure evolution equation as well as the construction of the Fjf and Ff flux components. Particularly, by using an AUSMD type of discretization for the FA-component (originally proposed for the Euler equtions in [32]) we obtain a Weakly Implicit Mixture Flux AUSMD scheme.

We present several numerical simulations, all of them indicating that the CFL-stability of the resulting WIMF-AUSMD scheme is largely governed by the velocity of the volume fraction waves and not the rapid sonic waves. Comparisons with an explicit Roe scheme indicate that the scheme presented in this paper is highly efficient, robust and accurate on slow transients. In fact, by exploiting the possibility to take much larger time steps it outperforms the Roe scheme in the resolution of the volume fraction wave for the classical water faucet problem. On the other hand it is more diffusive on pressure waves. Although conservation of positivity for the masses is not proven, we demonstrate that a transition fix may be applied making the scheme able to handle transition to one-phase flow while maintaining a high level of accuracy on volume fraction fronts.

subject classification. 76T10, 76N10, 65M12, 35L65key words, two-phase flow, two-fluid model, hyperbolic system of conservation laws, flux splitting, implicit scheme

1. Introduction

Accurate resolution of the dynamics related to two-phase flow phenomena is of high importance for a number of engineering applications, including nuclear reactor safety analysis and petroleum engineering. Among several two-phase flow models there are two fundamentally different formulations of the macroscopic field equations; namely the two-fluid model and the mixture model [26]. Here we focus on the two-fluid model. This is considered to give the most general and detailed description of transient two-phase flows. In the two-fluid model each phase is treated separately

Date: August 7, 2003.ARF-Rogaland Research, Prof. Olav Hanssensvei 15, Stavanger, Norway.BDepartment of Energy and Process Engineering, Norwegian University of Science and Technology,

Kolbjom Hejes vei IB, N-7491, Trondheim, Norway.Email: [email protected], [email protected] Corresponding author.

1

2 EVJE AND FLATTEN

in terms of two sets of conservation equations; one for each phase. The interaction terms between the two phases appear in the basic equations as transfer terms across the interfaces (source terms). More precisely, the basic form of the model can be written on the following vector form:

( pgag ^ ( PgOigVg Xpiai + dx

piam2 iPgOtgVg Pg«g< +

\ piam } \ p\ol\v] + mp J

( 0 \0

pdxag + Tg

V pdxm + n

( 0 X0

(1)

Here a* is the volume fraction of phase k with cti+ag = 1, and denote the density and fluid velocities of phase k, and p is the pressure common to both phases. Moreover, T& represents the interfacial forces which contain differential terms (hence, is relevant for the hyperbolicity of the model) and satisfy Tg + r; = 0. Mf? represents interfacial drag force with + Mf = 0 whereas Qk represent source terms due to gravity, friction, etc.

The model we will be concerned with is classified as a hyperbolic set of differential equations, with the implication that information flows in the system along characteristic curves with a certain velocity. For such models explicit numerical schemes are commonly used, advantage being taken of the fact that the time development of the state at some point depends only on points within the span of the characteristic curves in time and space. Explicit schemes are simple to implement and may give more flexibility in the treatment of complex pipe networks. However, they are subject to the CFL constraint

~fa > | Xmax | (2)where Amax is the largest eigenvalue for the system. For the two-fluid model we are concerned with, the four eigenvalues are pairwise associated with sonic and volume fraction waves [9]. The sonic waves may be several orders of magnitude faster than the volume fraction waves, although the latter may often be of greater interest to the researcher. For this reason the CFL criterion (2) may severely limit the computational efficiency of explicit schemes.

To remedy the situation, a step in a more implicit direction, i.e. coupling one or more variables throughout the computational domain, may be made. Such approaches may be classified as follows:

• Weakly implicit. The original CFL criterion (2) may be broken for sonic waves, but a weaker CFL criterion for volume fraction waves still applies

~fa > I Xmax I, (3)where A^v is the largest of the two eigenvalues corresponding to volume fraction waves.

• Strongly implicit. No CFL-like stability criterion applies and the equations may be integrated with arbitrary timestep. However, stability could still be affected by other issues such as inherent stiffness of the equations.

The majority of engineering computer software for two-fluid simulations seems to be based on some implicit approach. Examples include the CATHARE code [2] developed for the nuclear industry, and OLGA [3] aimed towards the petroleum industry. The recently developed PeTra [13] is largely based on the OLGA approach, being strongly implicit in the sense of the classification above. On the other hand, the TACITE code [21] was originally based on an explicit approach.

Trapp and Riemke [31] describe some of the earlier computer simulation tools for two-phase flow. It seems to be long known that a weak pressure-momentum coupling may be enough to break the sonic CFL-criterion, making the scheme weakly implicit in the sense of the classification above. However, due to the lack of computer power even weakly implicit schemes were often too inefficient and strongly implicit schemes were favored in the early days. In addition, most early schemes where based on upwinding based on the advective velocities, and the effect of sonic waves was not naturally integrated in the discretization. Such advective splitting schemes are intrinsically unstable and a staggered donor-cell approach was typically needed to stabilize the numerical solution. This lack of stability seems also to have been a motivation in moving from weakly to the more stable strongly implicit schemes.

In recent years there has been several new applications of different upwind techniques for the equations of two-phase flow. Examples include implementations of the Roe scheme by Toumi et

WEAKLY IMPLICIT NUMERICAL SCHEMES 3

al [30, 29, 5], Romate [23], Tiselj and Petelin [27], Fjelde and Karlsen [11]. A rough Godunov scheme was implemented by Masella et al [17]. Coquel et al [4] studied kinetic upwind schemes, which do not make use of the eigenstructure, for the approximation of a general two-fluid model. Saurel and Abgrall have studied a general compressible unconditionally hyperbolic two-phase model with a wide range of applications, see [24, 25].

For one-phase flow, Wada and Lion [32] suggested a hybrid flux difference splitting (FDS) and flux vector splitting (FVS) scheme with good accuracy and stability properties. Their idea has been extended to two-phase flow models by Edwards et al [6], Niu [18, 19], and Evje et al [7, 8, 9].

Based on these recent results and the rapid development of computer speed we believe that weakly implicit schemes may provide several advantages compared to strongly implicit or explicit schemes. This is based on the following considerations:

• Efficiency. Many industrial applications require that simulations must be performed in real time, for instance if the numerical model is used to produce input to an automatic choke controller. This limits the applicability of explicit schemes due to the strict CEL criterion (2). However, computers are now so fast that timesteps below the volume fraction CEL criterion (3) may give real time performance.

• Robustness. Modern discretization techniques are more robust, implicating that the difference in stability between weakly and strongly implicit schemes may not be significant.

• Accuracy. As opposed to strongly implicit schemes, weakly implicit schemes more easily allow for high resolution methods like the MUSCL approach of van Leer [14] to achieve improved accuracy on the volume fraction waves.

A weakly implicit numerical scheme for the mixture model has been presented by Faille and Heinze[10]. They used a rough finite volume method based on an eigenvalue decomposition of the jacobi matrix for the system. A weakly implicit scheme for the two-fluid model was studied by Masella et al [16].

The aim of this work is to develop a general methodology for constructing numerical schemes for the two-fluid model which possesses the following important properties:

• No use of riemann solver or computation of nonlinear flux jacobians;• Accurate and non-oscillatory resolution of mass fronts, i.e. slow-moving volume fraction

waves, comparable with the resolution given by upwind type of schemes like the Roe scheme;

• Stability under the weak CEL condition (3).We first describe the discretization procedure in a general semi-discrete setting where a system of DDEs replaces the continuous model (1). A special feature of the proposed method is that we systematically make use of the following pressure evolution equation (see Section 2 for details)

^ + K (Psasvs) + Psq^ (piam)j = 0 (4)

wherek = -r------------------- . (5)

!§aiPg + ~d^agPiThe main mechanism of the proposed semi-discrete scheme can be described by the following two steps:

(I) The pressure calculation is coupled to the momentum equations. This is achieved by discretizing (4) at the cell interface j + 1/2. We then directly obtain a numerical flux for the pressure term of the momentum equations which is consistent with the model under considerations. This turns out to be an essential step in order to redeem the scheme from the strong CFL condition (2).

(II) As regards the mass conservation equations, we introduce a decomposition of the numerical mass fluxes Fk, which is an approximation to the physical mass flux into twocomponents F® and F^ respectively. The purpose of the F)?-coniponent is to couple the mass and momentum equations in a consistent manner and thereby ensure that the scheme yields stable mass calculations also for timesteps dictated by the weak CFL condition (3).

4 EVJE AND FLATTEN

The purpose of the FA component is to ensure that accurate resolution of volume fraction waves is obtained.

Our techniques, represented by the above two steps, will be referred to as ’’Mixture Flux” (MF) methods. A delicate issue is to derive an appropriate balancing of the two components F® and FA in the formulation of the numerical mass fluxes Fk. This balancing is achieved by systematically using the important relation

+ = 1, rrik = pkUk, (6)

obtained from the volume fraction equation ag + a; — 1. More precisely, we obtain mixture mass fluxes Ffc of the form

Fi = k (pgCn-^F? + piag-^-F^ + pm-^(F^ - FA

Fg = k (piag-^F£ +pgai-^F^ + pgag-^(F^ - F,A)j ,

where k is given by (5). We also verify that under natural assumptions on the FA flux component and the Fj? flux component, the resulting MF schemes possess certain ’’good” properties relevant for the approximation properties of numerical schemes in general for the two-fluid model. These results are stated in Lemma 1 and Lemma 2.

More specifically, Lemma 1 states that the MF mass fluxes recover the numerical fluxes of an exact riemann solver for a moving or stationary contact discontinuity. Lemma 2 ensures that Abrall’s principle [1] is satisfied; a flow, uniform in velocity and pressure must remain uniform during its temporal evolution. The fact that this principle is obeyed, ensures that the use of the pressure evolution equation (4) in the discretization of the non-conservative pressure term is consistent with basic physcial understanding of two-phase flow phenomena.

Based on the semi-discrete MF scheme, we then proceed to the construction of fully discrete schemes. Motivated by previous investigations of the current two-fluid model (1) for certain hybrid FVS/FDS type of schemes, see [9, 12], we propose here to use an AUSMD/V-type discretization (originally proposed for the Euler equations in [32]) for the numerical mass flux component FA as well as for the discretization of the convective terms gk = of the momentum equations. Inthis sense the present work can be considered as an extension to a weakly implicit version of the MF-AUSMD scheme we developed in [12]. The Mixture Flux scheme studied in [12] was developed in the same framework as presented in this work, however, only pure explicit time discretizations was considered there. We emphasize that the approach presented in this paper should be general enough to apply for other flux-splitting schemes as well.

hr particular, we perform numerical experiments for this weakly implicit MF-AUSMD scheme, denoted as WIMF-AUSMD, which indicate that the scheme in fact is subject to the weak CFL condition (3). More precisely, we observe:

• For a typical shock tube problem the WIMF-AUSMD scheme give non-oscillatory approximations of all waves, as opposed to the explicit AUSMD scheme investigated in [9] but similar to the explicit MF-AUSMD scheme in [12]. Comparison with a Roe scheme shows that the resolution of the volume fraction waves is very similar for both schemes whereas the resolution of sonic waves is more diffusive for the WIMF-AUSMD scheme.

• For a typical mass transport problem, like the classical water faucet problem, the strong features of the proposed WIMF-AUSMD scheme is clearly observed. Exploiting the possibility to take timesteps determined by (3), which in this case implies that the timestep is chosen to be more than 50 times larger than the timestep we use for the explicit Roe scheme, the WIMF-AUSMD scheme outperforms the Roe scheme in the resolution of the volume fraction wave.

• Numerical tests demonstrate that by employing a minor modification, similar to the one used in [9, 12], the good features of the WIMF-AUSMD scheme carries over to more difficult flow cases which locally involve transition from two-phase to single-phase flow.


Most importantly, the MF approach allows us to unify two different aspects of two-phase flow calculation; namely producing a high level of accuracy on volume fraction waves while allowing for violation of the sonic CFL criterion.

Our paper is organized as follows: In Section 2 we present the two-fluid model we will be working with. In Section 3 the MF approach is presented in a semi-discrete setting where the pressure evolution equation is introduced as well as the construction of mixture mass fluxes. These two steps constitute the main components of the Mixture Flux (MF) methods. In Section 4 we present a straightforward analysis, similar to the one presented in [12], demonstrating that the MF schemes possess some desirable properties relevant for their approximation properties.

Based on the semidiscrete scheme of Section 3, we then in Section 5, 6, and 7 proceed to construct fully discrete schemes which possess the properties identified in Section 4. In Section 8 we present numerical simulations where we attempt to shed light on the issues of stability, robustness and accuracy for the scheme. Particularly, we investigate how the scheme can handle a transition to one-phase flow using a transition fix similar to the one introduced in [9].


Throughout this paper we will be concerned with the common two-fluid model formulated by stating separate conservation equations for mass and momentum for the two fluids, which we will denote as a gas (g) and a liquid (1) phase. The model is identical to the model previously considered by Evje and Flatten [9] and will be only briefly restated here. For a closer description of the terms and their significance, we refer to the previous work and the references therein.

2.1. Generally. We let U be the vector of conserved variables

Pgag " ^ "pm rri\

pgagvg hpmv\ . h

(7)

By using the notation Ap = p — p% where pl is the interfacial pressure, and 7% = (p1 —p)dxau, we see that the model (1) can be written on the form

• Conservation of mass8dt (Pgag) + Qx (Pg°-gVg) ~ 0?

ol (/4«i) + {p\0t\vi) = 0,

(8)

(9)• Conservation of momentum

-Qt (pgasvg) + (Psasvl + <*gAp) +ag]fa(P- Ap) = Qg + > (10)

(p\ot\'V\) + (piaivf + aiAp) + ai^(p - Ap) = Qi + Mf, (11)

where for phase k the nomenclature is as follows

pk - densityp - pressureVk - velocityotk - volume fractionAp - pressure correction at the gas-liquid interface Qk - momentum sources (due to gravity, friction, etc)

- interfacial drag force.

The volume fractions satisfyag + a\ = 1. (12)

6 EVJE AND FLATTEN

For the numerical simulations presented in this work we assume the simplified thermodynamic relations

pi = pi,o + P 2° (13)°i

andPg = ^ (14)

wherepo = 1 bar = 105 Pa

pi,o = 1000 kg/m3,

and= 105(m/s)2

a\ = 103 m/s,The models (13) and (14) correspond to a general stiffened gas EOS of the form

P — ilk 1 )®fcPfc 'TfcTTfc;

where = {a\pk,o-po)/2 where pk,o represents the material density and po the ambient pressure. 7fc and are constants specific for each phase. Particularly, by choosing 71 = 2 we recover (13) while (14) is obtained by choosing 7g = 2 and 7Tg = 0.

Moreover, we will treat Qu as a pure source term, assuming that it does not contain any differential operators. We use the interface pressure correction

Ap = Ap ([/, a) = a (ug - %)2, (15)PgOii + piag

where we set S = 1.2. This choice ensures that the model is a hyperbolic system of conservation laws, see for instance [30, 5]. Another feature of this model is that it possesses an approximate mixture sound velocity c given by

c =piag + pgai (16)


(a9,p,vg,vi). For the pressure variable we see that by writing the volume fraction equation(12) in terms of the conserved variables as

™g , = 1Pg(p) #(P) '

(17)

we obtain a relation yielding the pressure p(ms,mi). Using the relatively simple form of EOS given by (13) and (14) we see that the pressure p is found as a positive root of a second order polynomial. For more general EOS we must solve a non-linear system of equations, for instance by using a Newton-Rapson algorithm. Moreover, the fluid velocities vg and vy are obtained directly from the relations

'On =UstV

Remark 1. Concerning the EOS for the liquid and gas phase, we would like to emphasize that the methods we develop do not require simple linear relations as given by (13) and (14). Formally, the only point of the algorithm which is affected by using more complicated EOS is the resolution algorithm which determines the pressure from the general relation (17).


2.2. Some useful differential relations. Noting that the relation (17) can be written on the form

mg = mg(mi,p) = pg(p),

we see thatdmg = (mg)TOlefmi + (mg)pdp

= ——dm\ Pi

In other words, we have the relation

where

mpg L pf

(p\)p dp.

dp = n(pidmg + pgdmi),

1K =

(18)

(19)

Similarly, noting that

we see thatmg = mg(a\,p) = (1 - cq)pg(p),

dmg = (mg)aida\ + (mg)pdp

— pgda\ (mg)pdp.

Using (18), this relation can be rewritten as

doc\ — (pg) j) din-i "f- | Og (pg) p/v ) dfitn

V Pg Pg,— Kôtg(ypg^pd'iYi\ O] (yi]) rdmg ^.

(20)

In other words, we haved/t <9/%-â.drng + -^-

By combining (18) and (20) we can write the masses m& in terms of a “pressure” and a “volume fraction” component as follows:

ddi\ — k{———Oi\dfng H—-^Lo:gdiV[).

and

drng = ag ~^dp — pgda\

dm\ = a\^p-dp + A dor op

(21)

(22)

The relations (18) and (20) reflect that differentials of the primitive variables a\ and p generally depend strongly on properties of the mixture of both masses through the differentials dmg and dm\. Later we will derive numerical mass fluxes which are consistent with the differential relations(18)-(22).

2.3. A pressure evolution equation. The relation (17) gives the pressure p = p(mg, riq) through a state relation. Now we describe another procedure for determining the pressure through a dynamic relation.

Multiplying the gas mass conservation equation with Kpi and the liquid mass conservation equation with Kpg and adding the two resulting equations, we get

d d d ( . d . .K/3ldim9 + Kf>9~dimi + Kpl~dx ^^g%) + KP-?v^; WîU) — 0.

In view of (18) we get the following non-conservative pressure evolution equation

~it+K ifldx ^gagVg) + Psdx = 0 (23)

EVJE AND FLATTEN

where k is given by (19). Coupling this pressure evolution equation to the momentum equation will be an important ingredient in allowing us to break the CFL-criterion (2).

3. A SEMI-DISCRETE SCHEME

In this section we construct semidiscrete approximations of solutions to (8)-(ll). In the Sections 5, 6, and 7 we describe fully discrete approximations, and finally in Section 8 we explore properties of these fully discrete schemes for several well known two-phase flow problems.

3.1. General form. It will be convenient to express the model (8)-(ll) on the following form:

diWik + dxfk = 0,"h 0x9k "h akdxp ~\~ (Ap)^u^ — Q&?

where k = g,l and

fk = PkakVk and mk = pkak

9k — Pk^k^k and Ik — pk^k^k^

We assume that we have given approximations % (mfcj(tn),/fcj(f1)). Approximations mk,j(t) and Ik,j(t) for t € (tn,tn+1] are now constructed by solving the following ODE problem:

W'kJ -\-$xFk,j — 0,Ik,j -k$xGkj + akjSxPj + (Ap)jSxAkj = Qk,j,

subject to the initial conditions

rnk,j(tn) = mt,j > Ik,j(tn) = IjHere Sx is the operator defined by

(25)

nk,j‘

Wj+1OxWj-1-1/2 —

WjAx

and (Ap)j(t) = (Ap) (Uj(t),5) is obtained from (15). Moreover, Fktj+1/2(t) = Fk(Uj(t), Uj+i(t)), Gk,j+i/2(t) = Gk(Uj(t), Uj+1(t)), Pj+i/2(t) = P(Uj(t), Uj+1(t)), and Ak>j+1/2(t) = Ak(Uj(t), Uj+1(t)) are assumed to be numerical fluxes consistent with the corresponding physical fluxes, i.e.

Pk(Ui U) — fk — Pk^k^k

Gk(Uj U) — 9k — Pk^k'^k

Ak(U,U) = ak.

The purpose now is to derive these numerical fluxes.

3.2. The numerical flux A*j+1/2(f). We first start with the numerical flux A,,j+1/2(/). Following the approach taken by others, see for example Faille [20] and Coquel et al [4], we discretize this term centrally. Thus we use the numerical flux

A„„+1/2(f) = + (26)

hr the following we seek to discretize the remaining fluxes so that they are consistent with the underlying dynamics of the model. Essential information about the interplay between masses mk and pressure p is given by the relation (17). We shall exploit this systematically when we devise numerical fluxes Fkj+i/2(t) and P,+i/2(i).


3.3. The numerical flux P_,+i/2(t). We suggest to associate the numerical flux Pj+1/2{t) with the solution of the pressure evolution equation (23) and (19) discretized at the cell interface j + 1/2. More precisely, given the cell centered pressure pj % p(xj,tn) we determine P/+1/2(*) for t G (tn,tn+1] by solving the ODE

Pj+l/2 +[Kj+l/2Pl,j+l/2]<>xIg,j+l/2 + [^j+l/2Pg,j+l/2]^xIl,j+l/2 ~ 0(27)

where the interface values Kj+1/2 and Pk,j+1/2 are computed from Pj+1/2 (t) together with the arithmetic average (26) which defines akj+1/2(t).

Remark 2. The numerical flux Pj+1/2 (f) = P(Uj(t), Uj+i(t)) is consistent with the physical flux. This follows easily since assuming that Uj(t) = Uj+i(t) = U(t) for t G [tn,tn+1], implies that we shall solve the ODE

0, p,-+i/2(4) = = p(r),

i.e. Pj+i/2(t) =p(tn) =p(t) for t G [tn,tn+1].

3.4. The numerical flux Fkj+1/2(t). We first recall that from the masses mkj(t), which in turn depend on the numerical mass fluxes Fkj+1/2(t) via the mass conservation equations of (25), we obtain the pressure pj(t) as well as the volume fraction akj(t) by using the relation (17). In order to give more room for incorporating several properties which are relevant for accurate and non-oscillatory approximations of the pressure pjit) and the volume fraction akj(t), we suggest to describe the numerical massfluxes Fk(t) as a combination of two different flux components F®(t) and Fjf(t) respectively.

More precisely, we associate the mass flux component Fk with the pressure calculation p = p(mg,mi) via the relation (17) while the Fk component is associated with the volume fraction calculation ak — mk/pk(p('tng,'mi))- An important point here is to give an appropriate description of the balance between the two components F® and Fff as well as to develop the F®- and Fff- components themselves. The first point is discussed in the following while the latter is postponed until Section 6 and 7 respectively.

From (21) and (22) we see that the mass differentials dmk can be split in a pressure component dp and a volume fraction component da. We now want to design numerical fluxes which are consistent with this splitting, i.e. we introduce a flux component Fp and Fa such that the mass fluxes E] and Fg are given by

(28)and

(29)The flux component Fp is associated with the pressure, hence it is natural to assign a diffusive mass flux FD for stable approximation of pressure for the various waves. Inspired by the differential relation (18) we propose to give Fp the following form

Fp = npgF^ + KpiFV (30)Similarly, the flux component Fa is associated with the volume fraction. Hence we seek to assignan mass flux FA such that an accurate resolution of the volume fraction variable can be obtained. Inspired by the differential relation (20), we propose to give Fa the following form

(31)

Here we note that a subscript j + 1/2 is assumed on the fluxes and coefficients. Substituting (30) and (31) into (29) and (28) we obtain the final hybrid mass fluxes

(32)

10 EVJE AND FLATTEN

andFs = K {^az~ltyF& + pzai-^Fz + psas-^(Ff - F1A)) ■ (33)

The coefficient variables at j + 1/2 remain to be determined. We suggest finding these from the cell interface pressure Pj+i/2(t) as well as the relation

aj+i/2(t) = 7}(uj(t) + aj+i (t))

which is consistent with the treatment of the coefficients of the pressure evolution equation (27).

Remark 3. We remark that the consistency criterion

Fk(U,U) = fk(U) = pkO-kVk,

relating the physical flux fk to the numerical flux Fu, is satisfied for the hybrid fluxes (32) and (33) provided the fluxes Fjf and Fk are consistent. In particular if Fk = Fjf the expressions (32) and (33) reduce to the trivial identity

Fk = f£ = FkD.

3.5. The numerical flux Gk,j+i/2 (*)• Based on the belief that the difficult and critical part is to obtain a numerical mass flux Pfcj+i/2 (t) which is consistent with the discretization of the pressure term of the momentum equations, we seek a more straightforward construction of Gk,j+1/2 (f)- In particular, we want to couple this convective flux to the mass flux Fk. In order to emphasize this we use the superscript ”A”, i.e.

Gk,j+i/2(t) = Gkj+l/2(t). (34)

More precisely, we choose Gk-+l^2 (t) to be consistent with the flux component Pj(j+1/2(*) in the following sense: For a flow with velocities which are constant in space for the time interval [tn,tn+1], that is,

Vk,j(t) = vk,j+i(t) = vk{t), t G [f\tn+1], (35)we assume that G^j+1/2(f) takes the form

^k,j+1/2W =vkif)Fk,j+i/2W> C3^)

where Fk-+1j2(t) is the numerical flux component introduced above and assumed to be consistent with the physical flux fk = pk&kVk-

Remark 4. We remark that the consistency criterion

Gk(U, U) = pk(U) = pkCKkVfo,

relating the numerical flux Gk to the physical flux gk, is satisfied for Gk as given by (36) provided the numerical flux Fk is consistent with the physical flux fk ■

4. Further Development of the Mass Flux Fk>j+i/2(t)

A main issue in the resolution of two-phase flow as described by the current model is to obtain an accurate resolution of mass fronts, i.e. slow-moving volume fraction waves. Hence, in the following we want to ensure that the mass fluxes Fk{t) and Fk(t) are constructed so that certain ’’good” properties in this respect are ensured for the resulting mass flux Fk(t). Particularly, we shall identify a simple characterization of some properties which Fk and Fk should possess.

In order to identify this characterization, we consider the contact discontinuity given by

Pl=Pr=P

«l OR(us)l = (vi)l = K)r = (vi)k = v,

(37)


for the time period [tn,tn+1]. All pressure terms vanish from the model (8)-(ll), and it is seen that the solution to this initial value problem is simply that the discontinuity will propagate with the velocity v. The exact solution of the Riemann problem will then give the numerical mass flux

(pav)j+1/2 = ^p(aL + aR)u - ^p{ar - «L)M- (38)

Definition 1. A numerical flux F that satisfy (38) for the contact discontinuity (37) will in the following be termed a “mass coherent” flux.

4.1. A "mass coherent” flux Ffl. The purpose of the flux component Ffl is to ensure accuracy at volume fraction waves. A natural requirement for F() is then that it should be mass coherent in the sense of Definition 1. We shall return to a more detailed specification in Section 7 but at this stage it might be instructive to briefly mention two examples of numerical mass fluxes studied before for the two fluid model [9], one which is mass coherent and one which is not mass coherent.

Two examples. In [9] we studied a FVS type of scheme for the current two-phase model whose mass fluxes are given by

(Pav)j+1/2 = {pa)LV+(vL,Cj+1/2) + {pa)rV (vr,cj+i/2) for each phase where Cj+i/2 = max(cl,cr) and V± are given by

V± i(u±M)if |u| < c otherwise.

(39)

Here the parameter c controls the amount of numerical diffusion, and is normally associated with the physical sound velocity for the system. This flux is not mass coherent according to Definition 1 and leads to poor resolution of mass fronts, as was clearly observed in [9].

In [9] we also studied a modification of the mass fluxes (39) obtained by replacing by

t^(%,c,%) %yi(„,c) + (l-x)^ |„|<c^(v ± |u|) otherwise

where Xl and xr satisfy the relation%R(%R - %L(%L = 0. (40)

It is easy to verify that the resulting mass flux is mass coherent in the sense of Definition 1, and we observed in [9] that the level of accuracy was similar to that of a Roe scheme in the resolution of mass fronts.

Knowing that the total flux component F& given by (32) and (33) also should be accurate at volume fraction waves, i.e. mass coherent, we way ask: What is a minimal condition satisfied by the T)?-component which ensures that F/. still becomes mass coherent?

4.2. A "pressure coherent” flux Fjfl. We note that the pressure will remain constant and uniform as the discontinuity (37) is propagating. Consequently, a natural requirement on a “good” flux Ffl for stable pressure resolution is that it preserves the constancy of pressure for the moving or stationary contact discontinuity given by (37).

We write (18) asdp = ndp

wheredp = pgdmi + pidrrig. (41)

To maintain a constant pressure we must have dp = 0. Assuming constant pressure, (41) can be integrated to yield

p = pgin\ + pimg = Pgpflai + ag) = pgp\.

To maintain constancy of p, and hence p, we now insist that the flux F^ is a consistent numerical flux when applied to the mix mass p. That is, we impose

Ps^j+l/2 + P^^s,j+1/2 = Pgplv-

for the contact discontinuity (37).

(42)

12 EVJE AND FLATTEN

Definition 2. A pair of numerical fluxes (Ft, Fg) that satisfy (42) for the contact discontinuity (37) will in the following be termed “pressure coherent” fluxes.

In particular, we note that the FVS mass fluxes (39) as well as the upwind fluxes (38) are pressure coherent. Thus, the class of mass coherent fluxes is contained in the class of pressure coherent fluxes. However, it should be noted that we can easily construct a pair of perfectly valid mass fluxes, in the sense that they are consistent with the physical flux, that are not pressure coherent. Consider for example the stationary contact discontinuity (37) with v = 0. Let Fs be given by the upwind flux (38) and F\ be given by the FVS flux (39). Then

PgFi,j+i/2 + /h-Fgj+i/2 = ((<*i)l - («i)r) # 0,

defying the requirement (42). Thus, this mass flux is neither pressure nor mass coherent in the sense of Definition 1 and 2.4.3. Construction of mass coherent fluxes Fk{t). We now state the following important lemma:Lemma 1. Let the mixture fluxes (32) and (33) be constructed from pressure coherent fluxes Ffl in the sense of Definition 2, and mass coherent fluxes F^ in the sense of Definition 1. Then the hybrid fluxes (32) and (33) reduce to the upwind fluxes (38) on the contact discontinuity (37), i.e. they are mass coherent.

Proof. We consider the hybrid liquid mass flux (32) and assume that v > 0. Remembering that a subscript j + 1/2 is assumed on the variables, we write the flux as

Fl = K (ai 5^+ 2iFg )+mg~fyfFiA ~ Piai ffyFt) (43)

Using the required properties of F^ and Ffl given by Definition 1 and Definition 2 respectively, we obtain

F = k ^aii-^pgpiv + p\Oig-!^{ai)lu - pgp\a\-j^(l — («i)l)^ = Pi(cu)lv, (44)

where we have used thatPj+1/2 = PL= PR (45)

which follows from the assumption of constant, uniform pressure. Spatial and phasic symmetry directly give the corresponding results for Fg and v < 0, completing the proof. □Remark 5. The importance of Lemma 1 lies in the fact that it allows us to search for an appropriate fl,ux component Fj? outside the class of mass coherent fluxes, and still, as long as Fj? is pressure coherent and F^ is mass coherent, we obtain mass coherent fluxes F^. This is the crucial mechanism of the decomposition (32) and (33).4.4. The class of Mixture Flux (MF) methods. Motivated by the mixture mass fluxes (32) and (33) as well as the use of the pressure evolution equation (27), we propose the following definition:Definition 3. We will use the term Mixture Flux (MF) methods to denote numerical algorithms which are constructed within the above semi-discrete framework, that is: (i) the numerical mass flux Fktj+i/2(t) is given by the mixture fluxes (32) and (33) where F^ is pressure coherent in the sense of Definition 2 and Fjf is mass coherent in the sense of Definition 1: (ii) the numerical pressure flux Pj+i/2(t) is obtained as the solution of (27); (in) the convective flux Gf^+1 /2(t) satisfies (36) for flow with uniform velocity (35).

Next, we apply Lemma 1 to verify that the MF methods satisfy the following principle due to Abgrall [1, 24, 25]:A flow, uniform in pressure and velocity must remain uniform in the same variables during its time evolution.

Lemma 2. The MF methods given by Definition 3, obey AbgralT-s principle.


Proof. We assume that we have the contact discontinuity given by (37) and that it remains unchanged during the time interval [tn,tn+1]. In view of Lemma 1 and the fact that the convective fluxes j+1j2(t) of the momentum equations of the MF methods satisfy (36), we immediately conclude that the semidiscrete model (25) takes the form

iTikj Jr&x{pkakvk)j = 0, ^40 _)f ih'Cj Fv6x(f>i,-o:kVk) j T o/i: j6XPj T (ApjjdxAj,j — 0,

where (pk®kVk)j+i/2 is on the form (38). In view of (15) we conclude that (Ap)j = 0. Moreover, we see that (27) reduces to

Pj+1/2 - ~[Kj+l/2Pl,j+l/2]dxIg,j+l/2 + [Hj+l/2Pg,j+l/2]^xIl,j+l/2

= —Kj+if2{piPgvSxag:j+if2 + pgpiv8xaij+i/2\ — 0,

since ag+ai = 1. In other words,

Pj+l/2(t) = Pj+l/2(t+) =pj+pj+l

= p,

for all j. Consequently, SxPj — 0, and we can conclude that Abgrall’s principle holds for the MF methods. □

Remark 6. We may consider the class of schemes introduced in this paper, which all employ mass fluxes of the form (32) and (33), as genuine two-phase flux splitting schemes. This flux splitting is based on a decomposition of the mass fluxes into several phasic components, i.e. one specific mass flux involves components from both the liquid and gas phase. In this sense the class of schemes we study is fundamentally different from the solution method used in e.g. [4, 19, 20, 9] where the underlying philosophy is to solve the two-phase model basically as two single-phase problems.

In the next sections (Section 5,6 and 7) we shall specify fully discrete schemes based on the semi-discrete scheme presented in Section 3 and 4. In particular, we will develop a flux component F}? which is pressure coherent, but not mass coherent. This flux component is constructed so that it allows us to obtain a stable pressure p = p(mg,mi) via (17), even for time-steps which obey only the weak CFL condition (2). The fact that it is pressure coherent, i.e. satisfies (42) for a contact discontinuity (37), ensures that it does not introduce undesirable numerical dissipation at volume fraction waves. For the construction of appropriate flux components Fjf and G) we are going to use the AUSMV/D framework developed by Wada and Lion [32] for Euler equations and adapted to the two-phase flow model in [9], see also [20] for similar type of schemes for the two-fluid model.

5. Fully Discrete Numerical Schemes

We now consider a fully discrete scheme corresponding to the semi-discrete scheme given by(25), (26), (27), (32), (33), and (34).

General form.• Gas Mass

= -<5,CW2 (47)At

• Liquid Mass

= (48)At

• Pressure at cell interfaceppi/2-l(pi+p"+i)

At

14 EVJE AND FLATTEN

• Gas Momentum

Atpn+1 _ pn+1

= -4(CX " * '"'a, " (AP)?»-A1, + (M-

• Liquid Momentum

Atj>n+l _ p>n+l

= -4to% " <+ w/a. - (Ap)y{»A,”j + to,)?.

Here we have introduced the shorthands

mk = Pkakt Ik = mfcUfe.

(50)

(51)

In accordance with (26) we use

A'k,j+1/2 (52)

and where (Ap)J = (Ap) (11^,6) is evaluated from (15). For the discretization of the pressure evolution equation (27) as given by (49), we keep the coefficients npk fixed at timelevel tn whereas the massfluxes Ik are given an implicit treatment as they are discretized at timelevel tn+1. Particularly, this enforces a coupling between the equations (49), (50), and (51). We end up with solving a linear system Ax = b where A is a sparse banded matrix with 2 superdiagonals and 2 subdiagonals.

For the numerical mass fluxes Fk++12/2 the purpose of the ”n +1/2” notation is to indicate that we shall discretize some terms at time level tn, others at time tn+1. More precisely, we propose to use the following time discretization for the mass fluxes (32) and (33) (for simplicity we have again dropped the subscript j + 1/2):

and

F^+1/2 = +[K/,g%w,r - (^r).(54)

In other words, the flux component is kept at the timelevel tn whereas the flux component F® involves terms at timelevel tn+1. Particularly, we want to make use of the updated momentums Ik+1 obtained from solving (49)-(51) in the expressions for Fj?. We describe the details in the next section.

It turns out that this implicit treatment is crucial in order to maintain the stability of the scheme for large time steps. This aspect is explored in more detail in Section 8.1. Note that we shall not need to solve any linear system here as will become clear from Section 6. In view of (53) and (54), we see that what remains, is to specify the numerical flux components (F’/i)”+1^2 and(G'k),j+i/2i 38 well as We start with the latter.

Remark 7. The discretization of the pressure equation at the cell interface can be viewed as a staggered Lax-Friedrichs scheme. We assume that the pressure pj is found from the masses mj by (17). The interdependence between Pj+1/2 and the couple (pj,pj+1) through the proposed discretization (49) ensures that the numerical flux Pj+1/2 is consistent with the physical flux, as pointed out in Remark 2.


6. Specification of the pressure coherent convective flux {Ff?)n+1/2

Due to the fact that the mass flux component Fj? is associated with the pressure calculation as described in Section 3.4, it is natural to choose a discretization of this flux which is consistent with the discretization of the pressure evolution equation. On the semi-discrete level, in view of (27), we therefore propose to consider the following discretization of the mass conservation equations

™'k,j+1/2 +5xIk,i+1/2 — 0, t € (tn ,tn+1](55)mk,j+lTTlVl

mkj+i/2(t+) = —— 2 We now suggest to average as follows:

^ +mtj+i/2(t)),

which implies that"lk,j (t) = 2 (™k,j-l/2 (i)+ ™k,j+l/2 (*)) •

By substituting (55) into (56) we obtain the following ODE equation for m,h,j(t):

™k,j +2Ax^fc,i+1 ~~ = * e (tn,tn+1]

(56)

(57)

To achieve conservative mass treatment while maintaining CFL-stability, it is clear that we somehow should take advantage of the already implicitly calculated mass fluxes /^t1 obtained from solving (49)-(51). A fully discrete version of (57) which employs this updated mass fluxes 1 is then given by

ton+1 k,j

~ i (2mk,j mk,j-1 - mfcj+i)

At(58)

which can be written on flux-conservative form (47) and (48) with the numerical fluxes

(Ffc))"+i//2 = 2^^ + Jfc,l+i) + 4 At (mfcd " mh+1)- (59)

Now we may solve for the masses mj’j1 using the fluxes (59), taking advantage of the fact that they emerge through an implicit coupling to the pressure. We found that by doing this we were able to violate the CFL-criterion for sonic waves. This is explored in more detail in the numerical Section 8.1.

Next, we check that the proposed flux Fj? possesses the ’’pressure coherent” property of Definition 2.

Proposition 1. The flux component Fjfl given by (59) is pressure coherent in the sense of Definition 2.

Proof. We just need to check that Ff? satisfies the relation (42). Using the constants of (37), a direct calculation gives

,T\n+1/2 _ n+1V .2^ + Oil

A 7*U+i) + 4At%' “ alj+1)n+l

+ Pg.Pl

= Pgpi Id + D-b^d-D= PgPlV-

□Note however, by direct calculation, that this Ff mass flux component is not mass coherent in

the sense of Definition 1.

16 EVJE AND FLATTEN

Remark 8. Our experience is that it is essential to use a discretization of the mass equations, represented by the F/f flux component (59), which is consistent with the one used for the pressure evolution equation in order to obtain non-os dilatory (stable) approximations for the pressure when large time steps governed by (2) are employed. However, this leads to mass fluxes Fj? which are not mass coherent according to Definition 1.

Consequently, by using Fj? only as mass fluxes, i.e. F^ = Ff?, we must expect that a strong smearing of volume fraction waves is introduced. However, Lemma 1 states that by the introduction of the mixture mass fluxes (32) and (33) we only needF® to satisfy the weaker ’’pressure coherent”- condition given by Definition 2, and still we retain mass fluxes Fk which are mass coherent as long as we use a ’’mass coherent” F^ component.

7. Specification of the mass coherent convective fluxes (Fjf)n and corresponding

CONVECTIVE MOMENTUM FLUXES. k

In this section we look for appropriate choices for the numerical flux components Fff and by considering so-called hybrid FDS/FVS type of schemes. Such schemes have been explored for the present two-fluid model more recently [9, 12]. We here briefly restate the numerical convective fluxes (pav)j+i/2 and (pav2)j.|_i/2 corresponding to the flux splitting schemes we investigated in [9].

7.1. FVS/van Leer. We consider the velocity splitting formulas used in previous works [15, 32,

y>.=) = {||±Hf “1141 («°)

Here the parameter c controls the amount of numerical diffusion, and is normally associated with the physical sound velocity for the system. Following [9] we here assume that the sound velocity is given by (16). Following the standard set by earlier works [32, 7, 9] we choose a common sound velocity

cj+1/2 = max(cL,cR)at the cell interface.

(1) Mass Flux. We let the numerical mass flux (pav)j+i/2 for FVS and van Leer be given as{pa v)j+1/2 = (pa)-LV+(vL,cj+1/2) + {po)rV~ {vR,cj+1/2) (61)

for each phase.(2) Momentum Flux. We let the numerical convective momentum flux (pav2)j+1/2 be given

as• FVS:

(/**^)j+l/2 = V+(%,Cj+i/2)(/X*v)L + y-(FR,Cj-t_i/2)(/X%v)R (62)

• van Leer:(pav2)j+1/2 = ^{pav)j+1/2(ul + vr) - ^\{pav)j+1/2\{vR - vl) (63)

7.2. AUSMV/AUSMD. Following [9], we consider the convective fluxes associated with the AUSMV and AUSMD scheme obtained by replacing the splitting formulas V± used in (61)-(63) with the less diffusive pair

xV±{v,c) + (1 -x}2^ M < c ^{v ± |v|) otherwise (64)

where

and

for each phase.

2(p/o) l

{p/a) l + {p/a) R

2(p/q)r(/V«)L + (/>/<%)R

(65)

(66)


Definition 4. Using the terminology of Wada and Lion [32], we will henceforth refer to the FVS scheme modified with the splittings (64) and the choice of % described by (65) and (66) as the AUSMV scheme. That is, the convective fluxes of A USMV are described by

• Mass Flux:(67)

• Momentum Flux:(/*%^)^^ = F+(%,Cj+i/2,xi;)(/)m;)L + ^"(%a,9+1/2,Xa)(pmOa. (68)

Definition 5. Similarly, we will henceforth refer to the van Leer scheme modified with the splittings (64) and the choice of x described by (65) and (66) as the AUSMD scheme. That is, the convective fluxes of A USMD are described by

• Mass Flux:(m^)^T = (m)if+(%L, 9+1/2, X^) + (Maf" (%a, 9+1/2, Xa) (69)

• Momentum Flux:(/>av2)f+iS/fD = \(pav)j+1/2(vL + vR) - ^\(pav)j+lf2\(vR - vL). (70)

We note that xl and xr given by (65) and (66) satisfy the relation (40). Consequently, as remarked in Section 4.1, it is easy to check by direct calculation that the AUSMV and AUSMD convective fluxes hold the following property, see also [9, 12].Proposition 2. The convective fluxes (pav)A^J^v and (pau)^™13 are mass coherent in the sense of Definition 1.

7.3. WIMF-AUSMD and WIMF-AUSMV. We are now in a position where we can give a precise definition of fully discrete MF schemes. We shall consider the following two different choices for (Fjf)n and (G^)n leading to two different MF schemes:Definition 6. We will use the term WIMF-AUSMV to denote the numerical scheme given by (47)-(54) where (FjP)j^j2 given by the pressure coherent component (59) whereas (Fjf)f+1/2

and (Gf)J+1/,2 are given by

(%i/2 =Definition 7. We will use the term WIMF-AUSMD to denote the numerical scheme given by (47)-(54) where (FjP)™^j2 is given by the pressure coherent component (59) whereas (F^)"+1^2 and (Gf)f+1/2 are given by

%i/2 = (*<%%'", «%i/2 = W)^'"The following result holds for WIMF-AUSMV and WIMF-AUSMD:

Proposition 3. WIMF-AUSMV and WIMF-AUSMD satisfy the following properties:(i) The mass fluxes of WIMF-AUSMV and WIMF-AUSMD are mass coherent in the sense of Definition 1. (ii) Both schemes obey Abgrall’s principle.

Proof. In view of Lemma 1, result (i) follows directly from Proposition 1 and Proposition 2.Result (ii) follows by observing that the flux component Gf of both schemes (see Definition

6 and 7) satisfy the relation (36) for flow with uniform velocity (35), and then applying Lemma 2. □

Remark 9. We observed in [9] that the convective fluxes of AUSMV were considerably more diffusive on volume fraction waves than those of AUSMD. Thus, for numerical simulations we prefer to use the WIMF-A USMD scheme which applies A USMD mass and momentum fluxes for F£ and G^ respectively. However, we will take advantage of the robustness of the convective fluxes of A USMV and apply these in combination with the convective fluxes of A USMD in an appropriate manner when we consider flows which locally involve transition to single-phase flow. We refer to Section 8.3 for details.

18 EVJE AND FLATTEN


In the following some selected numerical examples will be presented. We will consider the performance of the WIMF-AUSMD scheme given by Definition 7. In order to ensure that this scheme can handle flow cases which involves transition to single-phase flow, we introduce a slight modification whose purpose basically is to introduce more numerical dissipation near the singlephase zone. This is explained in detail in Section 8.3.

As our main concern will be to demonstrate the inherent accuracy and stability properties of the WIMF-AUSMD scheme, we limit ourselves to first order accuracy in space and time. The boundary conditions are implemented using a simple “ghost cell” approach, where the variables are either imposed or determined by simple (zeroth order) extrapolation from the computational domain.

In the first example we explore more carefully central mechanisms of the WIMF-AUSMD scheme.

8.1. A Large Relative Velocity Shock. We consider a Riemann initial value problem investigated by Cortes et al [5] for a similar two-fluid model. Our primary motivation for studying this problem is to investigate the performance of WIMF-AUSMD on sonic waves. The initial states are given by

and

WL

Wr

" p ' 265000 Pa 'ai 0.71vs 65 m/sv\ 1 m/s

" P " ' 265000 Pa 'ai 0.7% 50 m/s

_ Vi 1 m/s

(71)

(72)

8.1.1. Comparison with explicit scheme. We here aim to compare the WIMF-AUSMD with an explicit Roe scheme at the same spatial and temporal grid. We refer to [9] for a description of the implementation of the Roe scheme. We assume a grid of 100 cells and use the timestep

A^=400 m/s. (73)

The results, plotted at the time t = 0.1 s, are given in Figure 1. The reference solution was computed using the Roe scheme on a grid of 10 000 cells.

We note that the implicit pressure-momentum coupling used in WIMF-AUMSD causes a stronger numerical dissipation associated with the sonic waves as compared to the explicit Roe scheme whereas the approximation of the volume fraction waves located at about 50 m seem to be very similar. The approximation properties regarding the slow volume fraction waves for WIMF-AUSMD is explored in more detail in the next example (water faucet).

8.1.2. Test of timestep sensitivity for calculation of pressure using the WIMF-AUSMD scheme. We now investigate what happens when the timestep is increased beyond the sonic CFL criterion. The two-fluid model possesses an approximate mixture velocity of sound given by (see [30, 9] for details)

c =m g + pga i (74)

Hence the mixture sound velocity is approximately given by the sound velocity of the gas phase, giving

c to 317 m/s. (75)Hence for timesteps satisfying

Ax


reference -

MMF-AUMSD

Figure 1. LRV shock tube problem. WIMF-AUSMD vs Roe scheme for a grid of 100 cells. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

the sonic CFL criterion is broken. For a grid of 1000 cells, the results of the pressure calculation for several different values of Ax/At is given in Figure 2. We observe that increasing the timestep beyond the sonic CFL criterion (2) does not induce instabilities. However, a significant increase of the numerical dissipation of the sonic waves follows the increased timestep.

8.1.3. Test of stability and convergence for the WIMF-AUSMD scheme under violation of sonic CFL condition. Using the timestep Ax/At = 100 m/s, the effect of grid refinement for the WIMF- AUSMD scheme is demonstrated in Figure 3. We observe that the Roe reference solution is approached in a monotone way and by that verifies that the stability of the WIMF-AUSMD scheme is not governed by the maximal speed of the sonic waves.

8.1.4. Test of using purely explicit mass fluxes Fk ■ We now wish to illustrate the need for using the implicitly calculated mass fluxes IJ/+1 as given by (59) when we approximate the mass equations. We consider a slight modification of the flux component Fj? given by (59), where we instead use the momentum from the previous timestep as follows

(-Ff>)"+i/2 = 2% +Ik,i+1) + 4

Results are given in Figure 4 for the timesteps Ax/At = 1000 m/s and Ax/At = 100 m/s using a grid of 1000 cells. We observe that this works well for Ax/At = 1000 m/s when the sonic CFL condition is satisfied. However, increasing the timestep by an order of magnitude leads to CFL-like instabilities, despite the fact that the pressure-momentum coupling still is implicit. It seems to be a crucial step to use information from time level tn+l to achieve stable mass calculations.

Remark 10. In particular, these results illustrate that the combination of using the pressure evolution equation (49) and the mixture mass fluxes (53) and (54), where {F/fl)n+1/2 is given by (59), makes the pressure calculation independent of any sonic CFL condition.

20 EVJE AND FLATTEN

£

0-

271000

270000

269000

268000

267000

266000

265000

referencei i i

1000 m/s --------100 m/s -------50 m/s ..........25 m/s -------15 m/s -------

<;i

i 1

0 20 40 60 80 100Distance (m)

Figure 2. LRV shock tube problem. Pressure is shown for a grid of 1000 cells. Different timesteps are considered by considering different values for Ax/At for the WIMF-AUSMD scheme. * 2

reference--------10 000 cells--------2 500 cells ..........

800 cells ..........200 cells -------

269000

e11

smx»

reference - 10 000cells -2 500 cells -

800 cells 200 cells -

reference - 10 000 cells -2 500 cells -

800 cells 200 cells -

Distance (m)

Figure 3. LRV shock tube problem. Grid refinement for the WIMF-AUSMD scheme. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity.


274000reference 1000 m/s

100 m/s273000 -

272000 -

271000 -

270000 -

269000 -

268000 -

267000 -

266000 -

265000

Distance (m)

Figure 4. LRV shock tube problem, 1000 cells. Modfied WIMF-AUSMD. Purely explicit mass fluxes are used.

The strength of the mixture fluxes (53) and (54) lies in their ability to properly combine the stability of an implicit scheme with the accuracy of an explicit scheme, at least for the resolution of volume fraction waves. This is the central issue in the next example.

8.2. Water Faucet Problem. We now wish to focus more on the resolution of volume fraction waves. For this purpose, we study the faucet flow problem of Ransom [22], which has become a standard benchmark [31, 30, 4, 19, 20].

We consider a vertical pipe of length 12 m with the initial uniform state" p " ' 105 Pa '

a\ 0.8v& 0

. D 10 m/s(77)


Qk — 9PkQki (78)

with g being the acceleration of gravity. At the inlet we have the constant conditions «i = 0.8, v\ = 10 m/s and vg = 0. At the outlet the pipe is open to the ambient pressure p = 105 Pa.

We restate the approximate analytical solution presented in the references [20, 31]+ 2.W for x <v0t+ igt2

Vo + gt otherwise. (79)

ao(l + 2gxv02) for x <vot + ^gt2cto otherwise

where the parameters = 0.8 and Vq = 10 m/s are the initial states.

(80)

8.2.1. Comparison with explicit Roe scheme. We now compare the WIMF-AUSMD scheme with the explicit Roe scheme under equal conditions. That is, we assume a grid of 120 cells and use the timestep

AxAt

= 103 m/s. (81)

22 EVJE AND FLATTEN

Figure 5. Water faucet problem, 120 cells, T=0.6 s. Roe vs WIMF-AUSMD, Ax/At = 103 m/s. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity.

Results are given in Figure 5 after t = 0.6 s. We note that there is hardly any visible difference between WIMF-AUSMD and the Roe scheme on the volume fraction wave. However, the WIMF- AUSMD is somewhat more diffusive on pressure. This is consistent with our observations in Section 8.1.1.

8.2.2. Effect of increasing the timestep for WIMF-AUSMD. An eigenvalue analysis (see [30, 9]) reveals that the velocities of the volume fraction waves are approximately given by

A± = PgQgUg + piagui ± I Ap(pgai + piag) - pipgaiag{vg - uQ2 pgOti + piag V (pg®i T /h^g)2

For a weakly implicit scheme as defined by (3) we must then haveAr— > max(Av ). (83)Lit J,n

Having p\» pg we obtain from (82)At D, (84)

hence we expect a weakly implicit scheme to encounter CFL related stability problems near timesteps corresponding to the liquid velocity.

We now study the effect of increasing the timestep for the WIMF-AUSMD scheme. We consider the following timesteps:

• Ax/At = 1000 m/s.• Ax/At = 25 m/s.• Ax/At = 17 m/s.• Ax/At = 14 m/s.

Results for these timesteps are given in Figure 6. We observe that increasing the timestep towards



Figure 6. Water faucet problem, 120 cells, T=0.6 s. Different timesteps for the WIMF-AUSMD scheme. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity.

the timestep corresponding to the liquid velocity significantly improves the accuracy of WIMF- AUSMD on the volume fraction wave, as seen on the plots of velocities and volume fraction. The rate of improvement in accuracy is largest near the optimal timestep Ax/At = v\. Increasing the timestep further violates the weak CFL criterion (83) and instabilities occur. The increased accuracy in volume fraction is accompanied by increased numerical dissipation in the pressure variable, consistent with our observations in Section 8.1.2.

8.2.3. Optimal WIMF-AUSMD vs Roe scheme. To emphasize the increased accuracy in volume fraction that is allowed by increasing the timestep beyond the sonic CFL criterion, the explicit Roe scheme at Ax/At = 1000 m/s is plotted together with the optimal WIMF-AUSMD scheme (Ax/At = 17 m/s) in Figure 7. The improvement of the WIMF-AUSMD scheme is rather striking and is equivalent to an increase in the number of grid cells by an order of magnitude for the Roe scheme.

8.2.4. Test of convergence for WIMF-AUSMD. In Figure 8 we investigate how the scheme converges to the expected analytical solution as the grid is refined. The optimal timestep Ax/At = 17 m/s is used. As we can see, the expected analytical solution is approached in a nonoscillatory way.

8.3. Separation Problem. We consider a gravity-induced phase separation problem introduced by Coquel et al [4], also investigated by Paillere et al [20]. This problem tests the ability of numerical schemes to handle the transition to one-phase flow under stiff conditions.

We assume a vertical pipe of length 7.5 m, where gravitational acceleration and possibly interfacial friction are the source terms taken into account. Initially the pipe is filled with stagnant liquid and gas with a uniform pressure po = 105 Pa and a uniform liquid fraction «i = 0.5.

24 EVJE AND FLATTEN

Figure 7. Water faucet problem, T=0.6s, 120 grid cells. WIMF-AUSMD with optimal timestep vs explicit Roe scheme.

reference 1200 cells

120 cells 60 cells 24 cells 12 cells 6 cells

0.45 -

0.4 -

0.35 -

0.3 -

0.25 -

Distance (m)

Figure 8. Water faucet problem, T=0.6s. Grid refinement for the WIMF- AUSMD scheme.

Assuming that the liquid column falls freely under the influence of gravity, the following approximate analytical solution can be derived for the transient period

>/2~gx for x < \gt2gt for ^gt2 < x < L —0 for L — |gt2 < x

Ui(x,t) — (85)


<%l(z,t)0 for x < \gt2

0.5 for \gt2 <x<L— \gt21 for L - \gt2 < x

(86)


T — ^ = 0.87 s (87)

these discontinuities will merge and the phases become fully separated. The volume fraction reach a stationary state, whereas the other variables slowly converge towards a stationary solution. Assuming hydrostatic conditions the pressure will approximately be given by

p(z,t) = Popo + p\g (% - i/2)

for x < i/2 for x > i/2. (88)

8.3.1. Transition to one-phase flow. We observed that the basic WIMF-AUSMD scheme would produce instabilities in the transition to one-phase flow. Indeed this is a common problem for two- phase flow models, observed among others by Coquel et al [4] for their kinetic scheme, Paillere et al [20] for their AUSM+ scheme and Romate [23] for his Roe scheme. Romate suggested a scheme switching strategy for solving this problem, where the original scheme is replaced with a stable, diffusive scheme near one-phase regions. Here we will follow a similar approach, using a strategy that has been previously applied with success [9, 12]. We proceed as follows:

8.3.2. Modification of basic AUSMV and AUSMD splitting formulas. We modify the parameters x used in the splitting formulas (64) corresponding to the AUSMV and AUSMD schemes as follows

and

xl = (l - <k)

Xr = (1 - <h.)

2W<*)iW«)l + (p/a) R

2(/)/<i)r(p/a) l + (p/a) r.

for each phase. Here (j> is the transition fix function

+ <k (89)

+ 4> R (90)

ag) (91)where F& is a parameter controlling the diffusive effect of the transition fix. This fix ensures that we recover the more stable FVS/van Leer fluxes, as given by (60)-(63), in one-phase regions.

We observe that the transition to one-phase liquid flow (the denser phase) more easily induces instabilities than the transition to one-phase gas flow (the less dense phase). For the purposes of this paper, we choose the parameters

Fg = 50 (92)

andFi = 500. (93)

Definition 8. The modified A USMD scheme as described by (89) and (90) will be denoted as the AUSMD* scheme. Similarly, the modified AUSMV scheme as described by (89) and (90) will be denoted as the AUSMV* scheme.

8.3.3. WIMF-AUSMDV*. We consider convective fluxes which are a hybrid of those employed by AUSMD* and AUSMV*, and denoted as AUSMDV*. More precisely, the numerical convective fluxes (apv)j+i/2 and (apv2)j+1/2 are given by the following expression:

+ (1 -

(W)jTiT^"=*(W)^r+(i - s)(W)^,rHere s is chosen as

s = max(</>L, </>r), (95)

26 EVJE AND FLATTEN

where (f> is the transition fix function given by (91). Note that this hybridization only affects the momentum convective fluxes since — (apv)^™0*. The construction (94) ensuresthat AUSMDV* uses the accurate AUSMD* fluxes in two-phase regions and switches to the more stable AUSMV* fluxes in one-phase regions.

The WIMF-AUSMDV* scheme is now constructed straightforwardly by associating the fluxes Fjf and with the corresponding AUSMDV* fluxes as follows.

Definition 9. We will use the term WIMF - AU SMD V* to denote the numerical scheme given by (47)-(54) where ^ given by the pressure coherent component (59) whereas (Fjf)h+l/2and (G£)]+1/2 are given by

= (pav)AUSMDVfcJ+1/2 i+i/2 =(pav2)AUSMDV*,n

fcJ+1/2

Remark 11. The idea of increasing the numerical dissipation near one-phase regions may be explored more systematically with the aim of obtaining more general relations that do not involve free parameters. Paillere et al [20] used a related approach, introducing a diffusion term depending on the pressure gradient to improve the performance of their AUSM+ scheme near one-phase liquid regions. We stress that the above modified schemes are still fully conservative and consistent with the original basic two-fluid model. In particular the WIMF-AUSMDV* scheme differs from the WIMF-A USMD scheme only for one-phase regions.

8.3.4. Numerical Results. We now consider two different formulations of the two-fluid model:• Frictionless flow. We assume that gravity is the only source term taken into account. In

this case, the lack of friction terms causes the gas velocity to become large as the gas phase is disappearing. We note that for one-phase liquid flow we have a\ » as and the volume fraction velocities (82) are dominated by this large gas velocity. Hence the weak CFL criterion (83) becomes very restrictive here. With this model we use the relatively low timestep

A T-^-=500 m/s. (96)

For stability of the FVS scheme, which AUSMDV* employs in the transition to single phase flow, we rescale the sound velocity c as described in the Appendix, using

c = 750 m/s (97)instead of the sound velocity determined from (16). This choice was based on the fact that we observed that the gas velocity could become as high as approximately 400 m/s. According to (124) in the Appendix, we should then choose c such that 200 < c < 800. We consistently have chosen c in the upper region.

• Interfacial momentum exchange. The low timestep needed for the frictionless model is undesirable. In addition the assumption of frictionless low is unphysical. In reality we expect the last remnants of the disappearing phase to be completely dissolved, and we expect vg ps v\ near one-phase regions. To more realistically model this situation, we consider an interfacial momentum transfer model also used by Paillere et al [20]. For the gas momentum equation, we introduce the source term

MjP = Caga\pg(vg - v\), (98)where C is a positive constant. Likewise the liquid momentum source term is given as

MP = -MtP = -Cagaipg(vg - vi), (99)conserving total momentum. We write

C = (W, (100)making the exchange term kick in more strongly near one-phase regions. Following Paillere et al [20], we set

Co = 50000 s™1. (101)


Distance (m>


Figure 9. Separation problem, T=0.6 s, 100 grid cells. WIMF-AUSMDV* scheme with and without interfacial momentum exchange terms. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity

To avoid stability problems related to stiffness in this term, we use a semi-implicit implementation as follows

C?(«g<*Pg)" wr1 (102)

We found that we could now increase the timestep toAt*

= 75 m/s, (103)

consistent with the largest gas (volume fraction) velocities during the transient period. The sound velocity is rescaled as

c = 150 m/s. (104)Again, this choice is based on the criterion (124) where we now can assume that the fluid velocity becomes zero in the transition to single-phase flow (due to the inclusion of the interfacial momentum transfer model). This gives us that c should be chosen in the interval 0 < c < 2A — 2 Ax/At.

Results after t = 0.6 s are plotted in Figure 9, using a grid of 100 cells. The approximate analytical solutions (85) and (86) are used for reference. We note that good accordance with the expected analytical solutions is achieved. The most notable effect of the interfacial momentum exchange term is the reduction of the gas velocity in the one-phase liquid region.

Although the phases will be separated for t < 1.0 s, it takes some seconds before the excess momentum has been dissipated at the endpoints. Results for fully stationary conditions (t = 5.0 s) are plotted in Figure 10. We note that the frictionless model does not exactly yield the expected hydrostatic pressure distribution. This seems to be due to the strong velocity gradients at the

28 EVJE AND FLATTEN

Figure 10. Separation problem, t = 5.0 s, 100 grid cells. WIMF-AUSMDV* scheme with and without interfacial momentum exchange terms. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity.

separation point, and hydrostatic conditions are never fully reached. The inclusion of the interfacial friction term removes these gradients.

In Figure 11 the effect of grid refinement on the resolution of volume fraction is illustrated for the WIMF-AUSMDV* scheme with momentum exchange terms. The timestep Ax/At = 75 m/s is used. The expected analytical solution is approached in a monotone way.

8.4. Oscillating Manometer Problem. Finally, we consider a problem introduced by Ransom [22] and investigated by Paillere et al [20] and Evje et al [9]. This problem tests the ability of numerical schemes to handle a change in the flow direction.

We consider a U-shaped tube of total length 20 m. The geometry of the tube is reflected in the ^-component of the gravity field

{g for 0 < x < 5 mycos (^T5T7^7r) for 5 m < x < 15 m (105)

—g for 15 m < x < 20 m.Initally we assume that the liquid fraction is given by

( 10-6 for 0 < x < 5 ma\(x) = < 0.999 for 5 m < x < 15 m (106)

10-6 for 15 m < x < 20 m.The initial pressure is assumed to be equal to the hydrostatic pressure distribution. We assume

that the gas velocity is uniformly vg = 0, and the liquid velocity distribution is given by0 for 0 < x < 5 m

v\(x) = ^ Vq for 5 m < x < 15 m 0 for 15 m < x < 20 m,

(107)


reference --------1000 cells --------

100 cells25 cells

- -

- -

-

0.8 -

0.6 -

0.4 -

0.2 -

3 4Distance (m)

Figure 11. Separation problem, T=0.6s. Convergence properties of the WIMF- AUSMDV* scheme with interfacial momentum exchange terms.

where Vo = 2.1 m/s.Ransom [22] suggested treating the manometer as a closed loop. We will follow the approach

of Paillere et al [20], assuming that both ends of the manometer are open to the atmosphere. We assume that the liquid column will move with uniform velocity under the influence of gravity, giving the following approximate analytical solution for the liquid velocity [20]

v\ (t) = Vo cos (cot), (108)where

(109)

where L = 10 m is the length of the liquid column.In order to exploit the possibility of taking large timesteps, we include the interfacial momentum

exchange term as described in Section 8.3.4. The sound velocity is rescaled to c = 30 m/s which is consistent with (124) where we use that the fluid velocity becomes zero in the transition to single-phase flow.

8.4.1. Numerical results. We consider the following grids• 100 cells. We use a timestep corresponding to Ax/At = 50 m/s.• 500 cells. We use a timestep corresponding to Ax/At = 15 m/s.

For the fine grid with 500 cells, the critical timestep was found to be consistent with the weak CFL criterion (83). For the coarse grid consisting of 100 cells, a lower CFL number was needed to ensure stability. The evolution of the center cell liquid velocity is given in Figure 12. We note that the results for 100 and 500 cells are virtually identical, indicating that the resolution of the liquid velocity is not grid sensitive. We observe a slight phase difference from the approximate analytical solution as was also observed in [20, 9].

The distribution of all variables after t = 20 s is given in Figure 13 for the grid of 500 cells. We observe that the variables are approximated without any numerical oscillations. In particular there is little numerical diffusion for the volume fraction variable. The strong gradients in the velocities are a consequence of the sudden volume change at the transition points between the

30 EVJE AND FLATTEN

reference 500 cells 100 cells

Time (s)

Figure 12. Oscillating manometer, WIMF-AUSMDV* scheme. Time development of the liquid velocity.

wimf-aJsmdv* -

I*= 0.5 -

Figure 13. Oscillating manometer, t=20.0 s, 500 grid cells. WIMF-AUSMDV* scheme. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity


phases. We remark that the gas velocity was extrapolated at the boundaries, whereas the liquid velocity was forced to zero at the boundaries to avoid liquid mass leakage.

9. Summary

We have proposed a general framework for constructing weakly implicit methods for the two- fluid model. Particularly, we have constructed a weakly implicit numerical scheme, denoted as WIMF-AUSMD, that allows the CFL criterion for sonic waves to be violated. All the numerical experiments indicate that a weaker CFL criterion applies with relation to the slow-moving volume fraction waves.

The scheme is based on a previously developed “Mixture Flux” approach [12] which properly combines diffusive and nondissipative fluxes to yield an accurate and robust resolution of sonic and volume fraction waves on nonstaggered grids. The sonic CFL criterion is violated by enforcing a coupling between the “pressure wave” component of the mixture flux, the cell center momenta and the cell interface pressure. In particular all convective (mass and momentum) fluxes are treated in an explicit manner. Hence we believe that higher order versions of the scheme may be implemented, for instance by using the MUSCL technique of van Leer [7, 14].

The numerical evidence indicates that the WIMF-AUSMD is highly robust and efficient, and gives an accuracy potentially superior to the explicit Roe scheme on volume fraction waves. An added advantage of the WIMF-AUSMD scheme is that it does not require a full eigenstructure decomposition of the jacobi matrix for the system. However, the scheme is diffusive on pressure waves, especially for large timesteps.

By increasing the numerical dissipation near one-phase regions, we have demonstrated that the framework allows for accurate, efficient and robust solutions also for flow cases which locally involve the transition from one-phase to two-phase flow.

Acknowledgements. The second author thanks the Norwegian Research Council for financial support through the Petronics program.

References


[2] F. Barre et al. The cathare code strategy and assessment. Nucl. Eng. Des. 124, 257-284, 1990.[3] K. H. Bendiksen, D. Malnes, R. Moe, and S. Nuland, The dynamic two-fluid model OLGA: Theory and

application, in SPE Prod. Eng. 6, 171-180, 1991.[4] F. Coquel, K. El Amine, E. Godlewski, B. Perthame, and P. Rascle, A numerical method using upwind schemes

for the resolution of two-phase flows, J. Comput. Phys. 136, 272-288, 1997.[5] J. Cortes, A. Debussche, and I. Toumi, A density perturbation method to study the eigenstructure of two-phase

flow equation systems, J. Comput. Phys. 147, 463-484, 1998.[6] J. R. Edwards, R. K. Franklin, and M.-S. Lion, Low-diffusion flux-splitting methods for real fluid flows with

phase transition, AIAA Journal 38, 1624-1633, 2000.[7] S. Evje and K. K. Fjelde, Hybrid flux-splitting schemes for a two-phase flow model, J. Comput. Phys. 175,

674-701, 2002.[8] S. Evje and K. K. Fjelde, On a rough ausm scheme for a one-dimensional two-phase flow model, Comput. &

Fluids 32, 1497-1530, 2003.[9] S. Evje and T. Flatten, Hybrid flux-splitting schemes for a common two-fluid model, J. Comput. Phys, to

appear.[10] I. Faille and E. Heintze, A Rough Finite Volume Scheme for Modeling Two-Phase Flow in a Pipeline, Computers

& Fluids 28, 213-241, 1999.[11] K.-K. Fjelde and K.H. Karlsen, High-resolution hybrid primitive-conservative upwind schemes for the drift flux

model, Comput. & Fluids 31, 335-367, 2002.[12] T. Flatten and S. Evje, A Mixture Flux Approach for Accurate and Robust Reolution of Two-Phase Flows,

Submitted, July 2003.[13] M. Larsen, E. Hustvedt, P. Hedne, and T. Straume. Petra: A novel computer code for simulation of slug flow.

In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, October 1997. SPE38841.

[14] B. V. Leer, Towards the ultimate conservative difference scheme. V. a second-order sequel to godunov’s method, J. Comput. Phys. 32, 101-136, 1979.

[15] M.-S. Liou, A sequel to AUSM: AUSM(+), J. Comput. Phys. 129, 364-382, 1996.

32 EVJE AND FLATTEN

[16] J. M. Masella, Q. H. Tran, D. Ferre and C. Panchon, Transient simulation of two-phase flow in pipes, Int. J. Multiphase Flow 24,739-755, 1998.

[17] J. M. Masella, I. Faille, and T. Gallouet, On an approximate godunov scheme, Int. J. Comput. Fluid. Dyn., 12:133-149, 1999.

[18] Y. Y. Niu, Simple conservative flux splitting for multi-component flow calculations, Num. Heat Trans. 38, 203-222, 2000.

[19] Y.-Y. Niu, Advection upwinding splitting method to solve a compressible two-fluid model, Int. J. Numer. Meth. Fluids 36, 351-371, 2001.

[20] H. Paillere, C. Corre and J.R.G Cascales, On the extension of the AUSM+ scheme to compressible two-fluid models, Comput. & Fluids 32, 891-916, 2003.

[21] C. Pauchon, H. Dhulesia, D. Lopez and J. Fabre, TACITE: A Comprehensive Mechanistic Model for Two-Phase Flow, BHRG Conference on Multiphase Production, Cannes, June 1993.

[22] V. H. Ransom, Numerical bencmark tests, Multiphase Sci. Tech. 3, 465-473, 1987.[23] J. E. Romate, An approximate riemann solver for a two-phase flow model with numerically given slip relation,

Comput. <k Fluids 27, 455—477, 1998.[24] R. Saurel and R. Abgrall, A multiphase godunov method for compressible multifluid and multiphase flows, J.

Comput. Phys. 150, 425-467, 1999.[25] R. Saurel and R. Abgrall, A simple method for compressible multifluid flows, SIAM J. Sci. Comp. 21, 1115-

1145, 1999.[26] H. B. Stewart and B. Wendroff, Review article; two-phase flow: Models and methods, J. Comput. Phys. 56,

363-409, 1984.[27] I. Tiselj and S. Petelin, Modelling of two-phase flow with second-order accurate scheme, J. Comput. Phys.

136, 503-521, 1997.[28] E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp.

43,369-381, 1984.[29] I. Toumi, An upwind numerical method for two-fluid two-phase flow models, Nuc. Sci. Eng. 123, 147-168,




SIAM J. Sci. Comput. 18, 633-657, 1997.

Appendix

Rescaling the Sound Velocity. A problem with the original FVS scheme is that it can produce instabilities for large timesteps if the discretization parameter A = Ax/At is chosen much smaller than the sound velocity. For an explicit scheme this will never be a problem as the CFL criterion limits the timesteps we can take. For a semi-implicit method however, we wish to use a value for A that may be several orders of magnitude smaller than the physical sound velocity and the issue becomes of relevance. To describe the problem, we consider the mass conservation equation

du d(uv) dt dx

where u = We now consider the FVS scheme

= 0 (110)

(uv)j+i/2 = V+ (Vj, c)Uj + V (vj+1, c)uj+1.

where we use the splitting formulas (60), assuming v < cV±(v,c) = ±^-Jv±cf.

(111)

(112)

Total Variation Stability. We now take advantage of the following theorem due to Harten, as stated by Tadmor [28]

Theorem 1. Consider the scalar equation

du df(u)

solved by the numerical scheme

,»+l — Uj

Ax(F(u?,w?+J-F(u?_i,w?))=0

(113)

At(114)


where the numerical flux F(uf ,uf+l) is written on viscous form

1 1 AtPj+1/2 = F(uj,Uj+1) = 2 (/(uj) + /(ui+1)) _ 2 At^i+V2^^-1 ~uj^- (115)

The scheme (Ilf) is total variation nonincreasing provided its numerical viscosity coefficient

Qj+1/2 = Q(ujiuj+i) satisfies

At

Ax - ^i+1/2 — (116)

For the scheme (111) using the splitting formulas (112) we obtain the numerical viscosity coefficient

«?»/* = (U7) Using this and assuming uniform velocity we can write the requirement (116) as

At At v2 + c2Ax'° ~ Ax 2c ’ (118)

which yields the following lemmaLemma 3. Let the mass equation (110) be solved using the numerical fluxes given by (111) and (112). Then the resulting scheme is total variation nonincreasing if

Ax > (m,At 2c

andc > 0.

The criterion (119) attains its minimum value for v = c, for which we obtain

Ax ^

At

(120)

(121)

which is the standard CFL criterion.

To further investigate how c should be chosen, we now assume that. Ax X= At (122)

is known and investigate which criteria govern the possible choices for c. From (119) we obtainc2 — 2cA + v2 < 0.

Solving this equation we obtain the following corollary(123)

Corollary 1. Let the linear advection equation (110) be solved using the numerical fluxes given by (111) and (112). Assume the time-step A = Ax/At is known. Then the resulting scheme is total variation nonincreasing if the “sound velocity” c satisfies

A - \/A2 - v2 < c < A + \/A2 — v2. (124)This result is confirmed by numerical experiments and illustrates that if c >> A the FVS

scheme is unstable. We hence propose to rescale the sound velocity used in the flux-splitting schemes such that the requirement (124) is satisfied also for large timesteps. We stress that this step is necessary to achieve stability on the advective effects for the FVS scheme. Stability of the sonic waves is an independent problem that we wish to achieve through taking advantage of the implicit pressure-momentum coupling together with the decomposition of Fk into F/f and Ff).

Paper IV

Comparison of Various AUSM Type Schemes for the Two-Fluid Model

Tore Flatten and Steinar Evje

Preprint

COMPARISON OF VARIOUS AUSM TYPE SCHEMES FOR THETWO-FLUID MODEL

TORE FLATTEN* AND STEINAR EVJE8

Abstract. In this paper we make further investigations of the Mixture Flux (ME) method for two-phase flows originally developed in the framework of the AUSMD scheme [12]. Here we use the method in conjunction with nonlinear state relations. We address stability and accuracy issues related to the discretization of the pressure terms, and suggest a modification that allows for a less strict stability criterion on the timestep. We further apply the framework to the two-phase AUSM+ scheme of Paillere et al (2003, Comput. Fluids 32, 891-916), denoting the resulting scheme as MF-AUSM+. Comparisons between the previously developed MF-AUSMD [12] as well as the original AUSM+ are made through numerical experiments. In particular, we observe that the MF-AUSM+ offers significant improvements in robustness over the original AUSM+.

subject classification. 76T10, 76N10, 65M12, 35L65

key words, two-phase flow, two-fluid model, hyperbolic system of conservation laws, flux splitting, explicit scheme

1. Introduction

During the last years a class of upwind schemes for the Euler equations have emerged, not being based on the characteristic field decomposition typical of approximate riemann solver schemes like the methods of Godunov and Roe. This new class of schemes, denoted as Advection Upstream Splitting Methods (AUSM) is instead based on simplified velocity and pressure splittings where the sonic waves are taken into account in the upwinding. We refer to the works of Lion et al [16, 15, 28, 7] where these methods are elaborated.

The adoptation of such schemes has been a recent and successful trend among multi-phase flow researchers. Examples include the works of Edwards et al [6] and Niu [17]. Niu explored hybrid flux-type flux splitting schemes for a multicomponent flow model, whereas Edwards et al studied a homogeneous equilibrium two-phase model with phase transitions. Characteristic for these models is that they are very similar to the Euler systems in structure and mathematical character.

Evje and Fjelde [8, 9] considered the mixture two-phase model (drift-flux model), which is a simplified isothermal two-phase model consisting of separate mass conservation equations and a mixture momentum equation. Accurate and non-oscillatory resolution of mass fronts was achieved, comparable with the Roe scheme.

In this paper we will consider a more general two-fluid model where each phase is treated separately in terms of two sets of conservation equations; one for each phase. The interaction terms between the two phases appear in the basic equations as transfer terms across the interfaces (source terms). More precisely, the basic form of the model can be written on the following vector

Date: September 25, 2003.^Department of Energy and Process Engineering, Norwegian University of Science and Technology,

Kolbjom Hejes vei IB, N-7491, Trondheim, Norway.BRF-Rogaland Research, Thormphlensgt. 55, N-5008 Bergen, Norway.Email: [email protected], [email protected] Corresponding author.

1

2 FLATTEN and evje

form:( pgplg ^ ( pgCtgVg ^

dtP\Oi\ + dx

p\OL\V\

Pg#g% PgOgfg + OgPX pmv\ / x piaivf + aip /

( o \ 0

pdxQ!g + Tg\ pdxai + ti /

/

V

00

% + MD Qi + Mp

\

)(1)

Here a* is the volume fraction of phase k with a\ + ag = 1, pk and Vk denote the density and fluid velocities of phase k, and p is the pressure common to both phases. Moreover, T& represents the interfacial forces which contain differential terms (hence, are relevant for the hyperbolicity of the model) and satisfy rg + n = 0. represents interfacial drag force with M® + Mj° = 0 whereas Qk represent source terms due to gravity, friction, etc.

Paillere et al [19] investigated an extension of the AUSM+ scheme of Liou [15] on the full two-fluid model, including an energy conservation equation for each phase. They found that the AUSM+ scheme was able to handle a wide range of two-phase flow problems in a stable and nondissipative manner. However, the AUSM+ scheme displayed a tendency towards introducing spurious oscillations and overshoots around discontinuities.

Evje and Flatten [10] investigated a related hybrid flux-splitting approach denoted as AUSMD/V on the isothermal two-fluid model. The advantage of this approach is that robust resolution of sonic waves may be achieved without the inclusion of additional low Mach number pressure diffusion terms [7, 8, 19].

However, the AUSMD/V approach suffers from the same problems as AUSM+ regarding accurate and robust resolution of discontinuities associated with the volume fraction waves. As far as the current two-fluid model is concerned, the wave phenomena depend strongly on properties of the mixture and involve expressions where the phasic variables are tightly coupled [5, 10]. The AUSM class of schemes solves each phasic set of equations independently and these couplings are not fully taken into account.

The AUSMD scheme was later refined [12] by enforcing a stronger coupling between the phasic variables in the numerical resolution algorithm, leading to the concept of Mixture Flux (ME) methods. Using this approach numerical oscillations were removed, and the resulting MF-AUSMD scheme was demonstrated to possess accuracy and robustness properties on level with the Roe scheme.

The starting point of the present work is two basic AUSM+ type schemes similar to those studied by Paillere et al [19]. A main objective of this work is to understand more precisely where the MF class of schemes stand when they are compared to these two basic AUSM+ schemes.

The MF methods are first presented in a semidiscrete setting, similar to the one introduced in [11]. Particularly, the MF methods are constructed so that they satisfy the following ’’good” properties: (i) The numerical mass fluxes reduce to upwind type of fluxes for a linear contact discontinuity similar to those produced by an exact Riemann solver; (ii) Abgrall’s principle is satisfied; that is, a flow uniform in velocity and pressure, must remain uniform during its temporal evolution.

A special feature of the MF approach is that one systematically makes use of the following pressure evolution equation

!&+K\pidx (piaivi)j - 0, (2)

where

for the construction of a suitable numerical flux associated with the pressure.The original MF approach, as described in [12], employed a straightforward Lax-Friedrichs-like

discretization of the pressure evolution equation (2). In [12] the resulting MF-AUSMD scheme was compared with a Roe scheme and we observed that the resolution of the sonic waves was slightly more diffusive for the MF-AUSMD scheme. One of the purposes of this work is to eliminate this drawback. More precisely, the main contributions of this work can be summarized as follows:

AUSM SCHEMES FOR TWO-PHASE FLOWS 3

(1) We provide more insight into mechanisms which are important for accurate and robust resolution of the various waves by comparing two basic AUSM+ schemes, similar to those studied by Paillere et al [19], to corresponding Mixture Flux (MF) type AUSM schemes derived within the framework of [12].

(2) We demonstrate that the MF approach does not depend strongly on the particular form of the basis flux used for the discretization of the convective fluxes. Particularly, we construct an AUSM+ based mixture flux scheme, denoted as MF-AUSM+. We perform numerical experiments indicating that the mixture flux method acts upon AUSM+ by reducing numerical oscillations while maintaining the desirable nondissipative properties around discontinuities. Hence the picture observed for the MF-AUSMD [12] is maintained, demonstrating the general applicability of the MF class of schemes.

(3) We show that the MF approach presented in [12] leads to a stability criterion for the timestep which is more strict than the standard CFL criterion

^x •> \At ~ Amax’ (4)

where Amax is the fastest characteristic velocity for the system.By way of an argument based on simplified assumptions, a modification of the MF

approach is suggested where we rescale the numerical diffusion coefficients to act more as an “upwind” type of numerical viscosity. We demonstrate that this “upwind” rescaling fixes the above problem, i.e. the resulting MF-schemes are stable under the CFL condition(4). Within this framework, denoted as Rescaled Mixture Flux (RMF), the poorer stability properties of AUSM+ on sonic waves resurface. We observe that the accuracy in the resolution of sonic waves for RMF-AUSMD is very similar to a Roe scheme whereas RMF- AUSM+ tends to produce small overshoots, demonstrating that the AUSMD seems to be the most promising candidate for the convective flux splitting.

The paper is organized as follows: In Section 2 we state the two-fluid model we will be working with. In Section 3 we restate two basic AUSM+ schemes for the isothermal two-fluid model similar to those presented in [19] for the full non-isothermal two-fluid model. In Section 4 we give a general presentation of the class of Mixture Flux (MF) schemes in a semi-discrete setting. In Section 5 we construct three fully discrete MF schemes denoted as MF-AUSMD, MF-AUSM+, and MF-CVS. The only difference between these schemes lies in the choice of the numerical convective fluxes associated with the apv and apv2 terms. In Section 6 we introduce a viscosity rescaling refinement to the MF framework, improving the accuracy and efficiency on sonic waves. In particular, this brings forth rescaled versions of the MF schemes, denoted as RMF type schemes. In Section 7 we perform numerical simulations comparing the performance of the various schemes. Finally the basic results and conclusions of the paper are summarized.


Throughout this paper we will be concerned with the common two-fluid model formulated by stating separate conservation equations for mass and momentum for the two fluids, which we will denote as a gas (g) and a liquid (1) phase. The model is identical to the model previously considered by Paillere et al [19] and will be briefly stated here. We let U be the vector of conserved variables

U =Pgag "mg"piai mi

pgCXgVg h

P\Ot\Vl _ h

(5)

By using the notation Ap = p — pz, where pl is the interfacial pressure, and T& = (p* —p)dxotk, we see that the model (1) can be written on the form

• Conservation of massd d

4 FLATTEN and evje

# 0ol (piai) + (piaiVi) = 0,

Conservation of momentum

— (pgagvg) + — (pgagv2g + agp) + (Ap - p)-^ = <?g + M°,

(7)

(8)

^ (pmvi) + (pmvf + aip) + (Ap-p)^ = Qi + Mi0, (9)

Alternatively, the momentum conservation equations may be written on the equivalent form

— OgCtgUg) + ^ (^gtig-Vg) + ag-^ + (AP)~of - Qg + M°, (10)

^ ^ = Qi + (11)

2.1. Submodels. For the numerical simulations presented in this work we follow Paillere et al [19] and use thermodynamic relations representative of water and air, derived under the assumption of constant entropies.

For the gas phase we havePg(p) = Pg (^) ^, (12)

where C = 105 Pa, pg = 1 kg/m3 and 7 = 1.4. The sound velocity is given by„2 _dp__lP

« %>g pg'For the liquid phase we have

where pi = 103 kg/m3, B = 3.3 • 105 Pa and n = 7.15. The sound velocity is given by

(13)

(14)

(15)

Moreover, we will treat Qk as a pure source term, assuming that it does not contain any differential operators. We use the interface pressure correction

/Ip =PgUl + pie. (%g-%)2, (16)

where throughout this paper we use a = 1.2, ensuring a hyperbolic model.Having solved for the conservative variables U, we need to obtain the primitive variables

(ag,p,vg,v1). For the pressure variable we see that by writing the volume fraction equation ag + «i = 1 in terms of the conserved variables as

mg ml

Pg(p) pi(p)(17)

we obtain a relation yielding the pressure p(rog,mi). This is a nonlinear equation which does not easily allow for an algebraic solution. Instead we use an iterative numerical algorithm to obtain the pressure from (17).

Moreover, the fluid velocities vg and v\ are obtained directly from the relationsU3Z7i’7T> Vl = 7T

Uiu2‘

Paillere et al [19] considered a more general model where conservation of total energy for each phase was included. Throughout this work we will study only the isentropic 4-equation model given above. The inclusion of energy equations does not significantly alter the existing eigenstructure of the isentropic model, but adds entropy waves moving with the fluid velocities. It is therefore our belief that the main difficulties related to the strong phasic couplings in the pressure and volume fraction waves are fully present in the isentropic model.


2.2. Wave Phenomena. The phasic sonic velocities are given by og (13) and a\ (15), satisfying

(18)

However, we note that the model possesses two characteristic sonic wave velocities approximately given by

Ap = vp ± c, (19)where

p PgOZlVl + Pl<XgVgPg«l + PKKg

and c is a mathematical mixture sound velocity approximately given by

(20)

c = pl<* g + Pg<Xl

lyfpla g + 1(21)

In addition, the model possesses two characteristic volume fraction wave velocities approximately given by

A* =tf =L-y, (22)

where

and

,a = pgai'Qg + piagVi pgai + piag

(23)

7 =Ap(pgai + piag) - pipgaiag(vg - vx)2

(/)gOi + pt«g)2 (24)

These approximations are derived under the assumption that |ug — ui| << c. We refer to [26, 10] for more details.Remark 1. Using an interface pressure correction term of the form (16), we see that the approximate volume fraction velocity (24) becomes imaginary if a < 1. Choosing a > 1 we obtain real-valued wave velocities and a hyperbolic model.

3. Two AUSM+ Schemes

We now consider the basic system (6)-(9), and assume it is discretized on the following form:

u i+1 = U? - ^ MW'W+i) - mu?-i,u?))

- ^ MU?,U?+i) - F"(U?_1,U?))

-At([Ap-p]dxlI)] + AtQ].

(25)

Here Fc and Fp are numerical fluxes assumed to be consistent with the corresponding physical fluxes fc and fp,

( pgagvg ^ ( 0 \fc =

p\tt\V\2 , fp =

0PgOgUj «gP

V j V «ip /and H and Q is given by

f ° \H = 0

Oi&

V «1 /

( o \0

Qg\ <Pi /

We see that the fluxes of the the model (6)—(9) consist of three different sort of terms; convective flux terms dx(pav) and dx(pav2), conservative pressure terms dx(ap) and non-conservative pressure terms [Ap — p]dxa. The discretization of these terms as described below closely follows the work of Paillere et al [19].

6 FLATTEN and evje

3.1. Convective fluxes. For phase k we assume a numerical velocity of sound = \ck]j+1/2

at the cell interface, to be defined below. Following Lion [15] and in turn Paillere et al [19], we further consider the velocity splitting formulas

i ’ ' \ f (u ± |u|) otherwise.For each phase we now define the cell interface velocity Vj+i/2 as

Vj+1/2 = V+(vj,cj+1/2) + y-(uj+i,cj+i/2), obtaining the convective mass flux

(pav) / Wl^+1/2 ^ ^'+1/2 >0>3+1/2 | (pa)_,+iUj+i/2 otherwise.

From this we construct the convective momentum flux_ J (m^)j+i/2^ if (paw);+i/2 > 0(pav ‘)j+1/2 = {

(pav)j+1/2Vj+i otherwise.

(26)

(27)

(28)

(29)

3.2. Pressure fluxes. We discretize the conservative pressure term as(ap)j+1/2 = P+(vj,cj+i/2)(ap)j + p-(vj+i,Cj+i/2)(ap)j+i

where the pressure splitting formulas are given aspi = T i c)^(2 f ^) ± - cT if 1^1 < c

\ §(1 ± sgn(u)) otherwise.

(30)

(31)

3.3. Definition of cell interface sound velocity. To define the sound velocities at the cell interface we use the expression

[cfe]j+i/2 = \J \ck\j\ck\j+1 (32)for both phases. Here cg = ag and c\ = a\ given by (13) and (15), in accordance with Paillere et al [19].Remark 2. Another natural choice, more in accordance with the mixture nature of the model, is to use the mixture sound velocity cmix given by (21). That is, for the splitting formulas (26) we could use

Cg — Cl — Cniix* (33)Such a choice was considered in [8] for the mixture model and in [10] for the two-fluid model. For two-phase flows, the numerical performance of the A USM scheme may not be strongly dependent on the choice of expression for the numerical sound velocity [9].

3.4. Low Mach number flows. Paillere et al [19] noted the need to couple the pressure and velocity fields for low speeds, as the AUSM+ scheme acts much like a central difference scheme in the low Mach number limit. This may lead to oscillations due to odd-even decoupling. The problem is particularly relevant for the near incompressible (i.e. liquid) phase, as the sound velocity may here become very large. To remedy the situation, Paillere et al suggested following the approach of Edwards [7], modifying the liquid mass flux as described by the steps below.

(1) Calculate the basic AUSM+ mass flux (pia\vf)j+i/2 as given by (28).(2) Define a cell interface Mach number Mj+1/2 as follows:

M,Vi\

‘i-M/2 = j

(3) Define a scaling factor f(M) as

^ y(l-M^M2+4M2^ ^ 1 + M2

where the parameter M0 is a “cut-off” Mach number.

(34)

(35)


(4) Rescale the cell interface sound velocity Cj+1/2 as followsD = /(^fi+i/2)[ci]i+i/2- (36)

(5) Compute a rescaled interface liquid velocity correction as

Mj+1/2 = V+([v\]j,ci) - + |Mil) _ ^_(Mi+i,ci) - ^(Ni+i + IMi+i|)> (37)

where the splitting formulas are the AUSM+ splitting formulas (26).(6) Evaluate the pressure diffusion term as

c’+i/2=Hi ~ 0 (38)(7) Modify the liquid mass flux

(piam)j+1/2 = (pmvi)j+1/2 + Dj+i/2- (39)The diffusion term .Dj+1/2 has a stabilizing effect on oscillations in the pressure variable, and vanishes for smooth flows. The disadvantage of this approach is that the parameter M0 may require some tuning.

3.5. Non-Conservative Differential Terms. For the spatial term on the form pdxa we follow the approach of Coquel et al [4] and Paillere et al [19] who suggested a central differencing. That is, we write

(40)

3.6. Two AUSM type schemes. Based on the above specifications we introduce the two following definitions:Definition 1. We will use the term AUSM+ to denote the numerical algorithm obtained from (25) by using the convective fluxes (28) and (29), the conservative pressure flux (30), and the non-conservative pressure flux (40).Definition 2. We will use the term PD-AUSM+ to denote the numerical algorithm obtained as for the A USM+ scheme but where pressure diffusion has been introduced for the liquid mass flux as described by (34)-(39).

4. The MF (Mixture Flux) Class of schemes

4.1. General form. In this section we first present the MF class of schemes in a semi-discrete setting. Fully discrete approximations of the model (6)-(9) are then obtained in Section 5. The starting point is the model (6)-(9) on the following form:

dtmk + dxfk = 0,(41)

dflk T &x9k T ^k^xP T (^p)dx&k — Qfc)where fc = g, 1 and

fk - PkO-kVk and mk = pkOtk

9k = PkOikv\ and Ik = pkO-k^k-

We assume that we have given approximations « (rrik,j(tn),Ik,j(tn)^. Approximations mk,j(t) and Ik,j{t) for t E (tn,tn+1] are now constructed by solving the following ODE problem:

rnk,j =0,(42)

Ik,j ASxGk,j T OLk,jdxPj T {^p)j^xA-k,j = Qk,ji


FLATTEN and evje

Here 8X is the operator defined by

SxWi =Wj+l/'2 — Wj -1/2

Ax 8xWj+1/2 = w3+1 Wi

Axand (Ap)j(t) — (Ap) (Uj(t),5) is obtained from (16). Moreover, Fk^+1/2(t) — Fk(Uj(t),Uj+i(t)), Cjfcj+i/2(t) = Gk(Uj(t), Uj+i(t)), Pj+1/2{t) = P(Uj(t), Uj+i(t)), and Ak,j+i/2(t) = Ak(Uj(t), Uj+i(t))are assumed to be numerical fluxes consistent with the corresponding physical fluxes, i.e.

Fk(U, U) = fk= pkakvk

Gk(U, U) = Qk = pk&kVju

Ak(U,U) = ak.

4.2. The class of Mixture Flux (MF) methods. Before we describe the MF approach it will be useful to introduce some basic concepts consistent with those used in [11, 12]. Assume that we consider a contact discontinuity given by

Pl=Pr=P (43)«l # or

(vs)l = M)n = (vs)r = (ui)r = v,

for the time period [tn, tn+1]. All pressure terms vanish from the model (6)-(9), and it is seen that the solution to this initial value problem is simply that the discontinuity will propagate with the velocity v. The exact solution of the Riemann problem will then give the numerical mass flux

{pav)j+1/2 = ^p(aL + aR)v - ^p(aR - «l)M- (44)

DeHnition 3. A numerical flux F that satisfies (44) for the contact discontinuity (43) will in the following be termed a “mass coherent” flux.

Definition 4. A pair of numerical fluxes (Fi,Fg) that satisfy the relation

PgF\,j+i/2 + PiFg,j+i/2 — PgPiv- (45)for the contact discontinuity (43) will in the following be termed “pressure coherent” fluxes.

For a more detailed presentation of the motivation behind these definitions we refer to [12, 11]. With these definitions in hand we can proceed to a more precise definition of the Mixture Flux methods.Definition 5. We will use the term Mixture Flux (MF) methods to denote numerical algorithms which are constructed within the semidiscrete frame of (42) where fluxes are given as follows:

(1) The numerical flux Akjj+1/2{t) is obtained as

A„,+1/2(i) = atj(l|+;w». (46)

(2) We determine Pj+i/2(t) for t G (tn,tn+1] by solving the ODE

Pj+1/2 +[Kj+l/2pl,j+l/2]8xIg,j+l/2 + [Kj+l/2pg,j+l/2]8xh,j+l/2 ~ 0

Pj+l/2 (*+) —(47)

where the interface values Kj+1/2 and pk,j+i/2 are computed from Pj+i/2{t) together with the arithmetic average (46) which defines Ofcj+1/2 (t). Here n is given by

k = (48)


(3) We consider hybrid mass fluxes Fk^+i/2(t) of the form

F\,j+i/2(t) = 1/2(*) (*) + Piag~^FiA(t) + _ ^(49)

and

Fgj+i/2(t) = Kj+1/2(t) (piatg-j^Fg (t) + pga\-^Fjf (t) + pgag-^(Fy> - F]A^ • (50)

Tfte coefficient variables at j + 1/2 ore determined from the cell interface pressure Pj+i/2(t) as well as the relation

Oij+i/2{t) = -(aj(t) + aj+i(t))

which is consistent with the treatment of the coefficients of the pressure evolution equation (47).

(a) The flux component Fk(t) is assumed to be consistent with its physical flux (pav)k(t) as well as ’’mass coherent” in the sense of Definition 3.

(b) The flux component Ff(t) is assumed to be consistent with its physical flux (pav)k(t) as well as ’’pressure coherent” in the sense of Definition 4-

(4) We choose Gkj+i/2(t) to be consistent with the flux component F^.+1j2{t) in the following sense: For a flow with velocities which are constant in space for the time interval [tn,tn+1], that is,

Vkj(t) = vk,j+i(t) = vk(t), t G [tn,tn+1], (51)we assume that Gk,j+1/2 (i) takes the form

Gk,j+i/2(t) = £7fcj+i/2 W = vk (t)Fkj+i/2 (t), (52)

where F^j+1^2(t) is the numerical flux component introduced above.

It is easy to check that the above numerical fluxes Afcj+]y2, Pj+1/2, -F&J+1/2, and Gk,j+1/2 are consistent with the corresponding physical fluxes. We refer to [12] for more details. We now state the following important lemma whose proof can also be found in [12]:Lemma 1. Let the mixture fluxes (49) and (50) be constructed from pressure coherent fluxes F() in the sense of Definition 4, and mass coherent fluxes F^ in the sense of Definition 3. Then the hybrid fluxes (49) and (50) reduce to the upwind fluxes (44) on the contact discontinuity (43), i.e. they are mass coherent.

It follows directly from Definition 5 and Lemma 1 thatCorollary 1. The mass fluxes of the MF methods given by Definition 5, o,re mass coherent in the sense of Definition 3.

Moreover, by application of Lemma 1 and Definition 5, we can verify that the MF methods satisfy the following principle due to Abgrall [1, 21, 22]:A flow, uniform in pressure and velocity must remain uniform in the same variables during its time evolution. We refer to [12] for its straightforward proof.Corollary 2. The MF methods given by Definition 5, obey Abgrall’s principle. More precisely, for the contact discontinuity (43) the semidiscrete approximation (42) takes the following form

Fik,j F3x(pkakvk)j = 0, (53)

v rnkj +v8x(pkoikvk)j — 0,

where (pkOtkVk)j+i/2 is on the form (44). Consequently, no momentum change is introduced and the contact discontinuity remains unchanged except from experiencing a convective transport.

10 FLATTEN and evje

In conclusion, Corollary 1 states that the MF mass fluxes recover the numerical fluxes of an exact Riemann solver for a moving or stationary contact discontinuity. Corollary 2 ensures that Abgrall’s principle [1] is satisfied. The fact that this principle is obeyed, ensures that the use of the pressure evolution equation (47) in the discretization of the non-conservative pressure term is consistent with basic physical understanding of two-phase flow phenomena.Remark 3 (Mixture mass fluxes). The following differential relations are obtained from the basic relation (17) (see [12, 11] for more details):

dp = K,(pidmg + pgdmi)

dcti — k(—-—a\dmg -|—dp op

(54)

where k is given by (48) and

dmg = ag^^dp — pgda\

dm\ = a\^p-dp + p\da\. op

(55)

The mixture mass fluxes (49) and (50) are obtained by first introducing a flux component Fp (associated with the pressure) and Fa (associated with the volume fraction) such that the mass fluxes F\ and Fs, inspired by (55), are given by

F\ — + p\Fa

Fg = ag-Q^Fp ~ PgFa-

Inspired by the differential relations (54) we propose to give Fp and Fa the following form

(56)

Fp = npgF[D + npiFg

Fa = K^dpazF^ ~ K^ttlFgA ’ (57)

where should possess the ’’pressure coherency” property whereas F^ should possess the ’’mass coherency” property. Combining (56) and (57) yields the mixture mass fluxes (49) and (50). The purpose of the F® component is to ensure that stable (non-oscillatory) pressure calculations based on (17) is obtained whereas the purpose of the F£- component is to ensure accurate resolution of volume fraction contact discontinuities.

Remark 4 (Pressure evolution equation). Since the pressure calculation is based on the masses mu through the relation (17), we want the pressure p to be consistent with the mass equations

dtmk T Hxfk — 0. (58)

This equation can be recast in terms of the pressure variable as follows: Multiplying the gas mass conservation equation by npi and the liquid mass conservation equation by rtpg and then adding the two resulting equations, yields the equation

(pg«g%) + «/)g^ = 0.

In view of the first relation of (54), the following pressure evolution equation is obtained

dtp T vpiOxF T KpgdxI\ — 0.

We want to consider a discretization of this equation at the cell interface in order to obtain an appropriate numerical pressure flux Pj+i/2(t) for t € (tn,tn+1]. This is the motivation leading to (47).


Remark 5 (The mass flux Fjf). The mass flux component Fj? is associated with the pressure calculation as described in Remark 3. Therfore it is natural to choose a discretization of this flux which is consistent with the discretization of the pressure evolution equation. On the semi-discrete level, in view of (47), we therefore propose to consider the following discretization of the mass conservation equation (58)

WfcJ+l/2(i+) :mh+mh+i

2(59)

We now suggest to average as follows:

mk,j(f) = 7} (™kj-i/2 (t) +mkJ+1/2(t)) ,

which implies that

™k,j (t) = - (m*j_i/2 (t)+ mkj+1/2 (t)). (60)

By substituting (59) into (60) we obtain the following ODE equation for mkj(t):

"*~2Ax _ ^4-1) = 0) t € (tn, tn+ ]

This equation is the basis for designing the flux component F^j+1^2

5. Three fully discrete MF schemes

The purpose of this section is to construct fully discrete schemes based on the general class of MF schemes given by Definition 5. We first describe how to construct appropriate candidates for the mass flux components Fj^ and F® which were introduced in Definition 5. Then we apply these components to propose fully discrete schemes, denoted respectively as MF-AUSM+, MF-CVS, and MF-A USMD. The difference between them lies in the choice of the convective fluxes only, that is, the Ffc and Gf components.

5.1. A pressure coherent convective mass flux F^. As explained in Remark 5 we shall define the numerical flux Fj? from the following ODE equation for mk,j(t):

rnk,j + 2Ar ^fcx,+1 Ik,j-i) — 0, t G (tn,tn+ ]

mk,j{fl\-) = ^ (mfcJ-l + ^mk,j + mfc,j+l) •(61)

A fully discrete version of (61) is given by

m"+1 - 3k,j + ™W+l'•+0 +. 1

At 2 AxThis equation can be written on the flux-conservative form

- AfcLJ?'”

(4”,1+1 1) - 0- (62)

whereIT’D ,n _ t(Tn 1 jn \*k,j+1/2 - 2 0 W + Jfc,j+lJ 4 At

(63)

We can easily check that the proposed flux F® possesses the ” pressure coherent” property of Definition 4, see [12, 11].Proposition 1. The flux component F]? given by (63) is pressure coherent in the sense of Definition 4-

5.2. Convective fluxes F^ and Gjf.

12 FLATTEN and evje

5.2.1. Van Leer. Our starting point is a simpler (lower order) version of the splitting formulas (26) used in the AUSM+ scheme

y^(%,c) =l 2(V±M)

if |u| < c otherwise.

We now let the numerical fluxes be given as follows:(1) Mass Flux. We let the numerical mass flux (pav)j+1/2 be given as

(Pav)j+1/2 = (p®)-LV+(vL,cj+i/2) + (pa)-R.V~(vR,cj+1/2)

(64)

(65)

for each phase.(2) Momentum Flux. We let the numerical convective momentum flux (pav2)j.|_i/2 be defined

by employing upwinding based on the cell interface momentum (pav)j.|_i/2

(pav2 (/X^)j+l/2% if (pav)j+i/a > 0 otherwise. (66)

The van Leer fluxes possess good stability properties but are excessively diffusive on the volume fraction waves. This motivates for proposing a mechanism for eliminating numerical dissipation, along the lines of Wada and Lion [28].

5.2.2. AUSMD. We consider the AUSMD scheme [10] obtained by replacing the splitting formulas V± in the van Leer scheme with the less diffusive pair

i^(%,c,%) xV±(v,c) + (1 -x)^l \v\<c](®1 I'D |) otherwise (67)

where= 2{p/a)L

(p/a)L + (p/a) rand

XR =__ 2^/a___(p/a) l + (p/a) R

for each phase. That is,(1) Mass Flux.

(pav)j+1/2 = (pa)j/V+ (vl , cj+1/2, Xl ) + (m)n^"(%, 9+1/2, An)

for each phase.(2) Momentum Flux.

(68)

(69)

(70)

We remark that for the van Leer and AUSMD schemes, it is essential to use a common velocity of sound to both phases to avoid oscillations in the pressure variable. We refer to [10, 12] for details. Here we use the mixture sound velocity given by (21), and define

Cj+i/2 — max(cj, cy+i). (72)

5.2.3. AUSM+. We define a cell interface velocity Vj+i/2 as

vj+l/2 = V+(vLiCj+i/2) + V (vr ,Cj+1/2), (73)where the splitting functions V± now are given by (26) and obtain the convective fluxes as follows:

(1) Mass Flux.(pav)H1/2 = { 0 <74)

(2) Momentum Flux.

(T5>


5.2.4. CVS. We define a cell interface velocity Vj+1/2 as

vj+l/2 = g(fL + |%|) + - 1-%!)- (76)

This corresponds to (73) with chosen asV±{v7c) = ^(u ± M).

Then we obtain convective fluxes as for the AUSM+ scheme described by (74) and (75). Hence, the upwinding in CVS is based on pure advection. In particular, CVS does not make use of a numerical sound velocity to determine the upwind direction.

In the following we use AUSMD, AUSM+, and CVS convective fluxes as bases to define MF type of schemes. In this connection it is relevant to note that AUSMD, AUSM+, and CVS mass fluxes possess the ’’mass coherent” property.Proposition 2. The convective fluxes (pav)^^0, (pav)j^^+, and (pav)^^2 are mass coherent in the sense of Definition 3.

5.3. Three Mixture Flux (MF) Schemes. We are now ready to describe fully discrete MF schemes.

5.3.1. General form. We use the shorthands = pkOik and J/; = mk'Vk and consider a fully discrete scheme based on (42) given as follows.

• Gas Mass

Liquid Mass

• Gas Momentum

At

miT - mti At = -<v%

At

• Liquid Momentum

At

= - (Ap)?a=A%, + (Qi)?.

(77)

(78)

(79)

(80)

5.3.2. MF-AUSMD.

Definition 6. We will use the term MF-AUSMD to denote the numerical algorithm which is constructed within the discrete frame of (77)-(80) where fluxes are given as follows:

(1) The numerical flux A£ j+1/2 w obtained as

A'k,j+l/2 (81)

(2) We determine by considering the following discretization of the pressure evolutionequation (47)

pn+lW+l/2 &(p?+p?+l)

At

— (K/,l)j+l/2gd+i -I.g

Ax(K/)g)y+i/2

TilWj+i

Ax

(82)

where the interface values K"+]y2 and p'f j+1/2 are computed from PJY1/2 together with the arithmetic average (81) which defines af. J.+1^2.

14 FLATTEN and evje

(3) We consider hybrid mass fluxes F£j+1/2 of the form

and

The coefficient variables at j + 1/2 are determined from the cell interface pressure P^+1/2 as well as the relation

al+1/2 = g (<%? + aj+i)which is consistent with the treatment of the coefficients of the pressure evolution equation(82).(a) For the flux component jy2 we refer to Section 5.2 and use

(b) For the flux component i',^)j”1y2 we refer to Section 5.1 and use

(85)

t,j+1/2 - n(% + %+i) + 2Wl(mk,j ~mk,j+i)-Fff:1 Ax.4 At

(86)

(4) The flux component G% j+i/2 chosen to be consistent with the flux component ly/2 by using

<%,+!/, = %Hl/2 = m

DeHnition 7. We will use the term MF-AUSM+ to denote the numerical algorithm which is identical to MF-AUSMD except from the convective flux terms ^2 and which aredefined as follows:

(a) For the flux component F^F+1j2 we use

Af+i/, = (K»C'

(b) For the flux component G^F+1j2 we use

Gk!j+1/2 = (PaV‘2)kVj+1/2

(88)

(89)

DeHnition 8. We will use the term MF-CVS to denote the numerical algorithm which is identical to MF-AUSM except from the convective flux terms F^F+1j2 and G^+l^2 which are defined as follows:

(a) For the flux component F^F+ lj2 we use

(b) For the flux component GA ,n we usek,j+1/2

(90)

(91)

In view of Definitions 6, 7, and 8 and Proposition 1 and Proposition 2 it follows that MF- AUSMD, MF-AUSM+, and MF-CVS are MF schemes in the sense of Definition 5. Consequently, Corollary 1 and 2 are applicable, and we immediately conclude thatProposition 3. MF-AUSMD, MF-AUSM+, and MF-CVS satisfy the following properties:(i) The mass fluxes are mass coherent in the sense of Definition 3. (ii) All schemes obey Abgrall’s principle.


6. Viscosity Rescaling

6.1. Motivation. Although the MF class of schemes allows for excellent accuracy and robustness, it is observed that a timestep stability criterion applies that is more strict than the standard CFL criterion

At> I'Mmax- (92)

To shed some light on the mechanism behind this feature, we write (63) as

nDJw2 = CM) = + <rn) + (93)

where the numerical viscosity coefficient is simply Rh+1y2 =1/2. Assuming that the pressure waves travel with the velocity of sound c (i.e. assuming << c), the following stability criterion applies [24]

^ (94)

where we consider the mass equation as an independent, scalar equation. Hence, using the value A+i/2 = 1/2 we obtain the CFL-like criterion

which is twice as strict as the expectedAx ^A?-C-

(95)

(96)

Here the mass equations are viewed as scalar equations and stabilizing effects related to couplings between the other equations are not taken into account. However, this simple analysis may to a large extent explain why a restrictive stability criterion applies for the explicit (in time) MF class of schemes.

6.2. Rescaled MF schemes. To improve matters, we suggest scaling the numerical viscosity of the flux (63) as follows

1 At+ Tk,j+1) + 4^n+i/2^«,,- - m;k,j+1 ), (97)

whereVj+l/2 = 2c/+ 1/2 7^' (98)

Here we use the mathematical mixture sound velocity (21) to define ch+1^2.The choice (98) ensures that the numerical viscosity behaves as an upwind viscosity for a wave

which travels with the velocity c. That is, assuming that |n£j| = |v£j+11 = \v^j+1^21 = c"+i/2 we

see that lj2 takes the form

pT),n _^k,j+l/2 ~

mk,jVk,j+1/2 lf Vk,j+1/2 - 0^L+i^W+i/2 otherwise.

(99)

hi view of (93) and (94) we see that the stability condition now reads

^j+l/22 <1,

which is in accordance with the standard CFL criterion (96).The interplay between the pressure evolution equation and the FD flux suggests that we should

also rescale the viscosity for the cell interface pressure (82). We modify as follows:

Pj+1/2 “ 2 (P? +^i+i) + Ax [£>sd+i/2(/gd ~ Jg,i+i) + A”+1/2(70 “ i0+1)_ > (10°)

where FW+]y2 — (Kpi)h+1y2 and Aj+i/2 — (kPe)^+i/2-

16 FLATTEN and evje

To determine £”+i/2 we consider the model equation

du du(101)

which describes advection of u with the velocity c > 0. Discretizing (101) at the cell interfaces with a staggered Lax-Friedrichs scheme, similar to (100), we obtain

^+1/2 - 2 (Ui + Ul+l) + 5j+l/2C"+l/2^: (wi - Uj+l)

Treating as a flux, we wish to recover the upwind formn+l

-*1+1/2 1 ’which suggests that £"+i/2 should be chosen as

1 Ax 1c1+1/2 - 2 Ate

1+1/2

-11+1/2;

(102)

(103)

(104)

Remark 6. An essential property of the kind of flow model we are studying is that the pressure and momentum are inversely related to each other. More precisely, the pressure appears as a flux term in the momentum equations and the momentum appears as a flux term in the pressure equation. This relationship is numerically expressed by the fluxes (63) and (82). The inverse relationship between the rescaled viscosities (104) is natural in view of this inverse relationship between the pressure and momentum.

Definition 9. We will use the term RMF-AUSMD (Rescaled Mixture Flux AUSMD) to denote the numerical algorithm which is constructed within the discrete frame of (77)-(80) where fluxes are given as follows:

(1) The numerical flux j+1 ^2 is obtained as in (81).(2) We determine Fjî/2 by considering the following discretization of the pressure evolution

equation (47)

Pj+~l/2 +Pj+1)A t

— (KP\fYj+l/2rn-*g,l’+l ~ 7g,l

Axr

(K/>g^)j+l/2 —1+1 A,iAx

(105)

where the interface values «?+]y2 and />£j+1/2 are computed from Pjî/2 together with the arithmetic average (81) which defines «)T,+1/2, and $”+i/2 9iven by (104).

(3) We consider hybrid mass fluxes Fffj+1/2 °f the same form as given by (83) and (84).(a) For the flux component we refer to Section 5.2 and use

riA ,nrfc,j+l/2 = (pav)AUSMD,n

fcJ+1/2 (106)

(b) For the flux component F^j” we use

pD,nrk, 1+1/2 = 2% lk,j+1) + 4î+l/2^(mfc,i - mk,j+1)’

Ax, (107)

(4)

where V’"+i/2 given by (98).The flux component G'/T,+i/2 is chosen to be consistent with the flux component F^F+1j2 by using

rmUk,j+1/2 n A,n

Ufc,j+l/2 = (pav2)AUSMD, n kJ+1/2 (108)

Definition 10. We will use the term RMF-AUSM+ to denote the numerical algorithm which is identical to RMF-AUSMD except from the convective flux terms Fkj+1/2 an<^ ^k’j+1/2 which are defined as follows:


(a)

(b)

For the flux component F^F+1j2

pA,nfc,j+l/2

For the flux component

-xA,n Jk,j+l/2

we «se

(109)

(110)

To illustrate the basic effect of the RMF strategy, we will also consider a more elementary scheme where the mass fluxes consist of only the FD component.Definition 11. We will use the term MF-F(D) to denote the numerical algorithm which is identical to MF-AUSMD except from the mass flux terms F(\" which are defined as follows:

TpA-,n _ r»D,nfc,j+l/2 ~~ rk9j+1/2- (ill)

Similarly, we will use the term RMF-F(D) to denote the numerical algorithm which is identical to RMF-AUSMD except from the mass flux terms F^t” which are defined as follows:

TpA-,n _ rpD,n*kj+1/2 - rk>j+1/2' (112)


7.1. AUSM+ vs PD-AUSM+. The purpose of this section is to investigate the effect of the pressure diffusion term as described in Section 3.4. We explore the performance for two shock tube problems as well as for the classical water faucet flow problem.

7.1.1. Toumi’s Water-Air Shock. We consider an initial value problem of a kind introduced by Toumi [25] and investigated by several authors [23, 19, 12]. The initial states are given by

and

WL =p ' 2 -107 Pa "

0.75vs 0

. F 0

WR =" P ' 1•107 Pa "

a\ 0.9W 0Vl 0

(113)

(114)

No source terms are taken into account.Using the timestep Ax/At = 1200 m/s, results for AUSM+ and PD-AUSM+ are given in Figure

1. The computation was performed on a grid of 100 cells for a time of t = 0.04 s. For the pressure diffusion term, the value

Mq = 0.2 (115)was used. The MF-AUSMD scheme was used to compute the reference solution, using a fine grid of 10 000 cells.

We observe that severe oscillations are produced using the basic AUSM+. For PD-AUSM+, these oscillations are significantly reduced. However, a slight overshooting behaviour still persists.

The results for PD-AUSM+ are qualitatively consistent with the results reported by Paillere et al [19] for this problem, who considered a full 6-equation model assuming an initial temperature corresponding to athmospheric conditions. The isentropic model with the state equations (12) and (14) we use here assumes an entropy corresponding to atmospheric conditions. These approaches give somewhat different results as the conditions of the problem are not atmospheric. In particular we observe a much less dense gas phase as compared to Paillere et al [19].

18 FLATTEN and evje

reference -

PD-AUSM+ -

Distance (m)

Figure 1. Toumi’s shock tube problem, 100 cells. AUSM+ vs PD-AUSM+. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

7.1.2. A Large Relative Velocity Shock. We now consider the Riemann problem given by the initial states

p " 265,000 Pa "«1 0.71vs 65 m/s

1 m/s(116)

andP " 265,000 Pa "Oil 0.7vs 50 m/s

. W 1 m/s(117)

Again, no source terms are taken into account. This initial value problem has previously been studied by Cortes et al [5] and Evje and Flatten [10, 11, 12]. In particular, this problem tests the ability of numerical schemes to handle a large velocity difference between the phases.

Using the timestep Ax/At = 1000 m/s, results for AUSM+ and PD-AUSM+ are given in Figure 2. The computation was performed on a grid of 100 cells for a time of t = 0.08 s. The RMF-AUSMD scheme was used to compute the reference solution, using a fine grid of 10 000 cells. For the pressure diffusion term, the value

Mo = 0.005 (118)

was used.We observe that the basic AUSM+ scheme is stable on the sonic waves. However, severe

oscillations are produced around the volume fraction waves. These oscillations are removed for PD-AUSM+, at the price of slightly smearing out the solution.


reference--------AUSM+ +

PD-AUSM+ ..........

+

-

A

J\X

reference -

PD-AUSM+ -

Figure 2. LRV shock tube problem, 100 cells. AUSM+ vs PD-AUSM+. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

7.1.3. Water faucet. We now study the classical faucet flow problem of Ransom [20], which has become a standard benchmark [4, 10, 26, 18, 27, 19, 12].

We consider a vertical pipe of length 12 m with the initial uniform state" p ' 105 Pa '

a\ 0.8vs 0

. 10 m/s(119)


Qk — SPk^-ki (120)

with g being the acceleration of gravity. At the inlet we have the constant conditions cq = 0.8, v\ = 10 m/s and vs = 0. At the outlet the pipe is open to the ambient pressure p — 105 Pa.

We restate the approximate analytical solution presented in the references [10, 27, 19], derived from a simplified model where the pressure variation is neglected.

i%(%, f) \/vq + 2gx for x <vot+ \gt2Vo + gt otherwise. (121)

<*i(%,t) a0(l + 2gxv0 2) G2 for x <v0t+ \gt2 a0 otherwise. (122)

Here the parameters = 0.8 and vq = 10 m/s are the initial states.In Figure 3 we compare the basic AUSM+ scheme (that is, in the framework of Section 3.4 we

consider M0 = 1) and the PD-AUSM+ scheme with M0 = 10-5. A grid of 1200 cells and the timestep Ax/At = 550 m/s was used.

We observe that the schemes are inseparable to plotting accuracy, and both schemes produce a slight overshoot in the gas fraction. A similar behaviour was reported by Paillere et al [19], and the

20 FLATTEN and evje

referenceM=1M=0.0001

Distance (m)

Figure 3. Water faucet problem, 1200 cells. AUSM+ (M0 = 1) vs PD-AUSM+ (M0 = 10-5).

basic AUSMD scheme also suffers from this [10]. For this problem the pressure is approximately uniform, hence the pressure diffusion term has little effect.

7.1.4. Conclusions. These flow cases illustrate that the basic AUSM+ scheme suffers from a tendency to produce numerical oscillations near discontinuities. The modified PD-AUSM+ is better, altough some oscillatory behaviour still persists. A weakness of the PD-AUSM+ approach is that the diffusion parameter M0 requires some tuning and may be problem-dependent.

7.2. Comparison of the MF schemes, hi this section, we compare the performance of MF- AUSMD, MF-AUSM+ and MF-CVS. In particular, it is demonstrated that MF-AUSM+ is a better alternative than PD-AUSM+ for removing numerical oscillations from the basic AUSM+ scheme.

7.2.1. Toumi’s Water-Air Shock. In Figure 4 the MF-AUSM+, the MF-AUSMD and the MF-CVS schemes are compared. A grid of 100 cells and a timestep Ax/At = 2000 m/s was used.

We see that the results represent a significant improvement as compared to the basic AUSM+ scheme of Figure 1. Notably the MF-CVS produces results comparable to the PD-AUSM+ scheme without incorporating free parameters nor a numerical sound velocity.

We observe no oscillatory behaviour around sonic waves. However, a slight difference between the different schemes in the resolution of volume fraction waves is visible. The MF-CVS scheme produces a significant overshoot in the liquid velocity, and we also observe some spurious oscillations for MF-AUSM+. The performance of MF-AUSMD is best.

7.2.2. Large Relative Velocity Shock. As can be seen from Figure 2, the volume fraction waves appear as a small wedge near x = 50 m. As opposed to the water faucet problem, the volume fraction waves here split into two genuinely non-linear waves moving with different velocities. We now wish to focus more strongly on this phenomenon, and thereby illustrate a basic difference in the dissipative mechanism of MF-CVS and the MF-AUSM schemes.

Focusing the plot around the voulme fraction waves, the MF-AUSM+ and MF-AUSMD schemes are compared in Figure 5, using a grid of size Ax = 0.01 m. We observe little difference between the schemes.

The MF-CVS scheme is compared with the MF-AUSM+ scheme in Figure 6, using the same computational grid Ax = 0.01 m. Strong oscillations occur for the MF-CVS scheme.

It is worth noting that the pure advective upwinding (76) used by CVS is the correct upwind form for a contact discontinuity of uniform pressure and velocity. Hence we expect MF-CVS to


0,08 0 10 20 30 40 50 60 70 80 90 100

Distance (m)

Distance (m)

Distance (m)

Distance (rn>

Figure 4. Toumi’s shock tube problem, 100 cells. MF-AUSM+, MF-AUSMD, MF-CVS. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

II

I CM,

49.9 49.95 50 50.05 50.1 50.15 502 50.25 502Distance (m)

49.9 49.95 50 50.05 50.1 50.15 50.2 5025 502Distance (m)

Figure 5. LRV shock tube problem, volume fraction waves. 50 cells. MF- AUSM+ vs MF-AUSMD scheme. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

22 FLATTEN and evje

49.85 49.9 9.95 50 50.05 50.1 50.15 502 50.25 502Distance (m)

50.05 50.1 50.15Distance (m)

502 50.25 502

49.85 49.9 49.95 50 50.05 50.1 50.15 50.2 50.25 502Distance (m)

49.85 49.9 50 50.05 50.1 50.15 50.2 502 5 502Distance (m)

Figure 6. LRV shock tube problem, volume fraction waves. 50 cells. MF- AUSM4" vs MF-CVS scheme. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

work well for such contact discontinuities (or weakly non-linear waves). However, as this last example shows, the more refined dissipative mechanisms of the MF-AUSM schemes are needed to properly resolve strongly non-linear phenomena in the volume fraction waves.

7.2.3. Water Faucet. In Figure 7 we investigate how the different schemes converge to the expected analytical solution for the gas fraction as the grid is refined. There is no significant difference between any of the MF schemes, which all produce non-oscillatory behaviour for this problem. In particular we observe that the overshoot of Figure 3 is removed.

7.2.4. Conclusions. MF-AUSM4" represents a better way of introducing the right amount of dissipation for a robust resolution of pressure than PD-AUSM+. No significant differences between MF-AUSM"1" and the related MF-AUSMD scheme have been demonstrated. The simpler MF-CVS scheme performs well when the volume fraction waves are essentially linear, as is the case for the water faucet problem, but is inferior to the MF-AUSM schemes in resolving non-linear volume fraction waves.

7.3. The RMF Strategy. In this section we apply the rescaling technique descried in Section 6 to the MF-AUSM4" and MF-AUSMD schemes, enabling us to use larger timesteps and achieve a more accurate resolution of sonic waves. In order to demonstrate clearly the impact of the rescaling on the resolution of the sonic waves, we first compare the performance of the MF-F(D) and the RMF- F(D) schemes given by Definition 11. For these schemes the flux component F)'1 = F®, hence, we cannot expect an accurate resolution of the volume fraction contact discontinuity. Comparison with a Roe scheme is made. We refer to [10, 26] for a description of the implementation of the Roe scheme.

Next, we compare the RMF-AUSMD and RMF-AUSM"1". We shall observe that the rescaling technique reveals a difference in the dissipative mechanisms of AUSM+ and AUSMD regarding their behaviour on sonic waves.


MF-OVS


120 cells 24 cells

Distance (m)

MF-AUSM+


120 cells 24 cells

Distance (m)

MF-AUSMD


120 cells 24 cells

Distance (m)

Figure 7. Water faucet problem, T=0.6s. Grid refinement for different schemes. Top: MF-CVS scheme. Middle: MF-AUSM+ scheme. Bottom: MF-AUSMD scheme.

24 FLATTEN AND EVJE

Distance (m)

265000 0 10 20 30 40 50 SO 70 80 90 100Distance (m)

48 0 10 20 30 40 50 SO 70 80 90 100Distance (m)

Figure 8. LRV shock tube problem, 100 cells. Roe, MF-F(D) and RMF-F(D) scheme. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

7.3.1. Large Relative Velocity Shock. For the LRV shock, the maximum wave velocity is approximately

|A|max ~ 500m/s, (123)representing the right-moving sonic wave.

Assuming a grid of 100 cells, numerical investigations reveal the critical timestep to be

Ax/At = 525 m/s (124)

for the RMF schemes, and more strictly

Ax/At — 660 m/s (125)

for the basic MF schemes. Hence the RMF schemes allow for larger timesteps and more efficient integration.

In Figure 8 we compare the Roe scheme, the MF-F(D) scheme and the RMF-F(D) scheme using a grid of 100 cells. Here the timestep (124) is used for the Roe and RMF-F(D) scheme, whereas (125) is used for the MF-F(D) scheme. We observe that RMF-F(D) produces a non-oscillatory approximation of the sonic waves, while noticably improving the approximation properties of the basic F(D) flux. We obtain an accuracy on sonic waves comparable with the Roe scheme, and the RMF strategy works as intended.

In Figure 9, RMF-AUSMD and RMF-AUSM+ are compared to the RMF-F(D) scheme using a grid of 100 cells and the timestep (124). The RMF-AUSMD is able to preserve the basic stability properties of RMF-F(D), while being significantly more accurate on the volume fraction waves. However, RMF-AUSM+ introduces small overshoots as compared to RMF-F(D). This comparison reveals that the AUSMD fluxes seems to be more consistent with the MF approach than the AUSM+ fluxes.


Distance (m)


Figure 9. LRV shock tube problem, 100 cells. RMF-F(D), RMF-AUSM+ and RMF-AUSMD scheme. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

7.3.2. Toumi’s Water-Air Shock. Results are given in Figure 10 for the RMF-AUSM+ scheme and RMF-AUSMD scheme, using a grid of 100 cells and a timestep

At— = 1140 m/s. (126)

We observe that the sonic waves are reproduced with less numerical diffusion as compared to Figure 4. There is an overshoot in the right-going (fastest) sonic wave for RMF-AUSM+, whereas RMF-AUSMD has no such problems.

7.3.3. Water Faucet. As observed for the shock tube problems, we found that the viscosity rescaling allows us to use a bigger timestep for the numerical integration. In particular, we found the critical timestep to be Ax/At = 540 m/s for the basic MF schemes and

At= 390 m/s (127)

for the RMF schemes.In Figure 11 the RMF-AUSM+ scheme is compared to the RMF-AUSMD scheme for T = 0.6

s on a grid of 120 computational cells, using the timestep Ax/At = 390 m/s. The schemes are inseparable to plotting accuracy.

7.3.4. Conclusions. The viscosity rescaled RMF-AUSM+ and RMF-AUSMD allow for higher integration timesteps than the basic MF-AUSM+ and MF-AUSMD schemes. A difference between the inherent robustness properties of AUSM+ and AUSMD manifests itself on the resolution of sonic waves. More precisely, we observe that the RMF-AUSMD scheme is robust whereas the RMF-AUSM+ scheme may induce oscillations around sonic shocks.

Liqu

id v

eloc

ity (m

/s)

26 FLATTEN and evje

10 20 30 40 50 60 70 10 20 30 40 50

Figure 10. Toumi’s shock tube problem, 100 cells. RMF-AUSM+ vs RMF- AUSMD. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

Figure 11. Water faucet problem, 120 cells. RMF-AUSM+ vs RMF-AUSMD scheme. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.


RMF-AUSMD (first order) RMF-AUSMD (minmod)

Distance (m)

1 (first older) ' (minmod)

Distance (m)

RMF-AUSMD (minmod)' (first order) 1 (minmod)

Figure 12. Water faucet problem, 120 cells. First order vs minmod RMF- AUSMD. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

7.4. Extensions to Higher Order Spatial Accuracy. In this section, we illustrate the possibility of achieving higher order spatial accuracy by adapting the MUSCL strategy of van Leer [14]. Following [15, 8], we apply a slope-limiting procedure to the primitive variables

W =POilVg

V\

(128)

to calculate new interface values tff (Wj_i, W,, W,+i) and Wj, W,+i). We nowmodify the convective fluxes Fj^ and G£ of Definition 5 as follows:

(129)and

%i/2 = G^,%i). (130)where we use the “minmod” slope-limiter.Remark 7. In general, one could explore applying a slope-limiting procedure also on the pressure evolution equation (47) and the flux component F]f, aiming for higher order accuracy on the pressure waves. In this paper, we restrict ourselves to demonstrating that the above simple procedure allows for improved accuracy on the volume fraction waves without sacrificing stability on the pressure waves.

7.4.1. Water Faucet. In Figure 12 the first order and minmod RMF-AUSMD schemes are compared, using a grid of 120 computational cells.

We observe that the minmod strategy allows for significantly improved accuracy, at the price of introducing slight overshoots. These overshoots seem to decay with grid refinement, as is illustrated in Figure 13.

28 FLATTEN and evje

reference 1 200 cells

120 cells 24 cells

Distance (m)

Figure 13. Water faucet problem. Grid convergence for the minmod RMF- AUSMD scheme.

7.4.2. Separation Problem. We now consider the separation problem introduced by Coquel et al [4], previously investigated by Paillere et al [19] and Evje and Flatten [10, 11, 12]. The problem consists of a vertical pipe of length 7.5 m, initially filled with stagnant liquid and gas with a uniform pressure of po = 105 Pa, and a uniform liquid fraction of a\ = 0.5. The pipe is considered to be closed at both ends, i.e. both phasic velocities are forced to be zero at the end points.

Assuming that the phases are accelerated by gravity only, the following approximate analytical solution was presented in [10]

( 0 for x < \gt2a\ (x,t) = < 0.5 for ^gt2 <x<L— |gt2 (131)

( 1 for L — |gt2 < x

where L = 7.5 m is the length of the tube. This approximate solution consists of a “contact” wave at the top of the tube and a shock wave forming at the bottom. After the time

T =L 0.87 s, (132)

these discontinuities will merge and the phases become fully separated. The volume fraction reaches a stationary state, whereas the other variables slowly converge towards a stationary solution. In particular we expect the stationary pressure to be fully hydrostatic, approximately given by

PoPo + Pi3 (x ~ L/2)

for x < L/2 for x > L/2.

(133)

7.4.3. Transition to One-Phase Flow. It has previously been observed that the basic MF-AUSMD scheme may produce instabilities as the limit aSti = 0 is approached [12]. The minmod RMF- AUSMD scheme has a similar behaviour, and a modification is required for a stable numerical transition to one-phase flow.

We will here follow an approach similar to the one used in [12]. The approach may be described by two steps:

(1) Removal of numerical stiffness. The idea is to replace the RMF-AUSMD scheme with a more dissipative scheme near one-phase regions, as described by the following definition



Figure 14. Separation problem, T=1.5 s. RMF-AUSMD* with minmod limiter, 100 cells. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity.

Definition 12. We consider a hybrid of the RMF-AUSMD and the van Leer scheme, denoted as RMF-AUSMD*, where the numerical mass fluxes are given by the following expression

pRMF-AUSMD* _ gpvan Leer ^ _ g^pRMF—AUSMD (134)

Here $ is chosen as

s = max(</>L,</>R), (135)

where <j> is an indicator function designed to be 1 near one-phase regions, 0 otherwise.The momentum fluxes are unchanged.

For the purposes of this paper we choose

<j>j = e~k[a^ + e-fe[“l]" (136)

where we use the parameter k = 50. In addition, the minmod limiter is used both on the RMF-AUSMD component and the van Leer component.

(2) Removal of unphysical velocity gradients. With no friction forces acting upon the phases, the hydrostatic pressure gradients will induce an acceleration of the light gas phase as «g -l 0. The resulting large velocity gradients are unphysical, as in reality we expect the last remnants of gas to be dissolved in the liquid, yielding vg to v\.

To remedy this, we follow the approach of Paillere et al [19], and include an interface momentum exchange term on the form

MjP = Cagaipg(vg - m), (137)

where C > 0 and Mfl = —M^, conserving total momentum. For the coefficient C we choose

C = Coe-*M, (138)

where Co = 3000 s-1.Results after T = 1.5 s are plotted in Figure 14, using a grid of 100 cells and a timestep

Ax/At = 650 m/s. At this point stationary conditions are reached. The phases are separated, and the expected hydrostatic pressure gradient is recovered.

A comparison between the minmod and first order RMF-AUSMD* schemes during the transient period is given in Figure 15, where we consider the effect of grid refinement. As expected, we observe a significant improvement in accuracy for the minmod scheme.

30 FLATTEN and evje


Figure 15. Separation problem, T=0.6 s. Grid refinement for the RMF- AUSMD* scheme. Left: First order. Right: Minmod.

8. Summary

In this paper, we have investigated several aspects of AUSM type schemes for a two-phase flow model. Our conclusions may be summarized as follows:

• The basic AUSM+ scheme is highly oscillatory around discontinuities. In particular, AUSM+ lacks robustness around sonic waves. Additional dissipative terms are required to stabilize AUSM+.

• Paillere et al [19] investigated the effect of a pressure diffusion term, resulting in the scheme denoted as PD-AUSM"1" in this paper. We have demonstrated that the mixture flux (MF) strategy of Flatten and Evje [12] may be applied to AUSM+, and the resulting MF-AUSM+ is more robust than PD-AUSM+.

• There is little difference between the MF-AUSMD and the MF-AUSM+ schemes. However, they are both superior to a simple one-sided advective splitting, the MF-CVS, in robustness on volume fraction waves.

• We have introduced a viscosity rescaling technique allowing us to relax the timestep restriction for the MF schemes. Within this framework, denoted as Rescaled Mixture Flux (RMF), the poorer stability properties of AUSM+ on sonic waves resurface. We observe that the RMF-AUSMD is superior to RMF-AUSM+ in robustness on sonic waves.

• The MUSCL strategy has been successfully applied to the RMF-AUSMD scheme, allowing for higher order accuracy on the volume fraction waves.

In particular, we have generalised the mixture flux strategy introduced in [12]. Within the MF framework, the AUSMD seems the most promising candidate for the convective flux splittings.

Acknowledgements. The first author thanks the Norwegian Research Council for financial support through the “Petronics” programme.

References


[2] F. Barre et al. The cathare code strategy and assessment, Nucl. Eng. Des. 124, 257-284, 1990.[3] K. H. Bendiksen, D. Malnes, R. Moe, and S. Nuland, The dynamic two-fluid model OLGA: Theory and

application, in SPE Prod. Eng. 6, 171-180, 1991.[4] F.Coquel, K. El Amine, E. Godlewski, B. Perthame, and P. Rascle, A numerical method using upwind schemes


flow equation systems, J. Comput. Phys. 147, 463-484, 1998.[6] J. R. Edwards, R. K. Franklin, and M.-S. Liou, Low-diffusion flux-splitting methods for real fluid flows with

phase transition, AIAA Journal 38, 1624-1633, 2000.[7] J. R. Edwards and M.-S. Liou, Low-diffusion flux-splitting methods for flows at all speeds, AIAA Journal 36,

1610-1617, 1998.


[8] S. Evje and K. K. Fjelde, Hybrid flux-splitting schemes for a two-phase flow model, J. Comput. Phys. 175, 674-701, 2002.

[9] S. Evje and K. K. Fjelde, On a rough ausm scheme for a one-dimensional two-phase flow model, Comput, fc Fluids 32, 1497-1530, 2003.

[10] S. Evje and T. Flatten. Hybrid flux-splitting schemes for a common two-fluid model. J. Comput. Phys, in press.

[11] S. Evje and T. Flatten. Weakly implicit numerical schemes for the two-fluid model. Submitted for publication, August 2003.

[12] T. Flatten and S. Evje. A mixture flux approach for accurate and robust resolution of two-phase flows. Submitted for publication, July 2003.

[13] M. Larsen, E. Hustvedt, P. Hedne, and T. Straume, Petra: A novel computer code for simulation of slug flow, in SPE Annual Technical Conference and Exhibition, SPE 38841, p. 1-12, October 1997.

[14] B. V. Leer, Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov’s method, J. Comput. Phys. 32, 101-136, 1979.

[15] M.-S. Liou, A sequel to AUSM: AUSM(+), J. Comput. Phys. 129, 364-382, 1996.[16] M.-S. Liou and C. J. Steffen, A new flux splitting scheme, J. Comput. Phys. 107, 23-39, 1993.[17] Y. Y. Niu, Simple conservative flux splitting for multi-component flow calculations, Num. Heat Trans. 38,


Meth. Fluids 36, 351-371, 2001.[19] H. Paillere, C. Corre and J.R.G Cascales, On the extension of the AUSM+ scheme to compressible two-fluid

models, Comput. & Fluids 32, 891-916, 2003.[20] V. H. Ransom, Numerical bencmark tests, Multiphase Sci. Tech. 3, 465-473, 1987.[21] R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows, J.


1145, 1999.[23] I. Tiselj and S. Petelin, Modelling of two-phase flow with second-order accurate scheme, J. Comput. Phys.

136, 503-521, 1997.[24] E. Tadmor. Numerical viscosity and the entropy condition for conservative difference schemes. Mathematics of

Computation. 43, 369-381, 1984.[25] I. Toumi, An upwind numerical method for two-fluid two-phase flow models, Nuc. Sci. Eng. 123, 147-168,




SIAM J. Sci. Comput. 18, 633-657, 1997.

Paper V

CFL-Free Numerical Schemes for the Two-Fluid Model


Submitted toJournal of Computational Physics

CFL-FREE NUMERICAL SCHEMES FOR THE TWO-FLUID MODEL

STEINAR EVJEA’c AND TORE FLATTEN* 3

Abstract. The main purpose of this paper is to construct an implicit numerical scheme for a two-phase flow model, allowing for violation of the CFL-criterion for all waves. Based on the Mixture Flux (ME) approach developed in [12] we propose both a Weakly Implicit ME (WIMF) scheme, similar to the one studied in [11], and a Strongly Implicit ME (SIMF) scheme. The WIMF scheme is stable for a weak CEL condition which relates time steps to the fluid velocity whereas the SIMF scheme is unconditionally stable, at least for a moving contact discontinuity. Both schemes apply AUSM (advection upstream splitting methods) type of convective fluxes.

The SIMF scheme is obtained by enforcing a stronger implicit coupling between the mass equations than the one used for the WIMF scheme. The resulting scheme allows for sequential updating of the momentum and mass variables on a nonstaggered grid by solving two sparse linear systems. The scheme is conservative in all convective fluxes and consistency between the mass variables and pressure is formally maintained. We present numerical simulations indicating that the CFL-free scheme maintains the good accuracy and stability properties of the WIMF scheme as well as an explicit Roe scheme for small time steps.

Moreover, we demonstrate that the WIMF scheme is able to give an exact resolution of a moving contact discontinuity. Explicit schemes cannot possess this property since it closely hang on the fact that the time step can be related to the fluid velocity. This feature of the WIMF scheme explains why it is very accurate for calculation of unsteady two-phase flow phenomena, as was also observed in [11]. The SIMF scheme does not possess the ’’exact resolution” property of WIMF, however, the ability to take larger time steps can be exploited so that more efficient calculations can be made when accurate resolution of sharp fronts is not essential, e.g. to calculate steady state solutions.

subject classification. 76T10, 76N10, 65M12, 35L65key words, two-phase flow, two-fluid model, hyperbolic system of conservation laws, flux splitting, implicit scheme

1. Introduction

We consider in this paper the two-fluid model governing two-phase flow of gas and liquid in a pipeline. Here each phase is treated separately in terms of two sets of conservation equations, averaged in space to yield a one-dimensional model. The interaction terms between the two phases appear in the basic equations as transfer terms across the interfaces (source terms).

More precisely, the basic form of the model can be written on the following vector form:

dt

( Pgag ^p\0i\

pgttgVg

\ pmvi

+ dx)

^ Pgagvg ^ piarn

Pg«g%g + UgP\ p\Oi\v( + a\p )

( ° \0P%(%g +V pdxCti + 71 /

( 0 0

Og + MD\ Q\ + Mx

\

)(1)

Here a* is the volume fraction of phase k with cq +ag = 1, pk and denote the density and fluid velocities of phase k, and p is the pressure common to both phases. Moreover, 7% represents the interfacial forces which contain differential terms (hence, is relevant for the hyperbolicity of the

Date: August 18, 2003.ARF-Rogaland Research, P. O. Box 8046, N-4068 Stavanger, Norway.3Department of Energy and Process Engineering, Norwegian University of Science and Technology,

Kolbjpm Hejes vei IB, N-7491, Trondheim, Norway.Email: [email protected], [email protected]. c Corresponding author.

1

2 EVJE AND FLATTEN

model) and satisfy rg + t\ = 0. Mj? represents interfacial drag force with + M,D = 0 whereas Qfc represent source terms due to gravity, friction, etc.

The majority of computer software for such two-fluid simulations are based on implicit time integration, allowing for violation of the CFL criterion

> |Vax| (2)

where Amax is the largest eigenvalue for the system. Examples include the CATHARE code [2] developed for the nuclear industry, as well as OLGA [3] and PeTra [13] aimed towards the petroleum industry.

Following [11], we classify implicit schemes as follows:• Weakly implicit. The original CFL criterion (2) may be broken for sonic waves, but a

weaker CFL criterion for volume fraction waves still applies

> l^maxl? (3)

where is the largest of the two eigenvalues corresponding to volume fraction waves.• Strongly implicit No CFL-like stability criterion applies and the equations may in principle

be integrated with arbitrary timestep. In practice, a stability criterion applies related to the inherent stiffness of the equation system. However, by freeing themselves from CFL considerations, strongly implicit schemes may allow for larger timesteps (and hence potentially more efficient computation) than weakly implicit schemes.

To build a fully discrete numerical scheme we need a basis splitting technique for the discretization of the pressure and convective fluxes at the cell interfaces. For one-phase flow, the AUSM (Advection Upstream Splitting Method) and its derivatives [16, 15, 27, 7] have proved highly successful!. These ideas have been extended to two-phase flow models by Niu [17, 18], Edwards et al [6] and Evje and Fjelde [8, 9].

For the two-fluid model we will be concerned with here, Paillere et al [19] studied an extension of the AUSM+ scheme of Liou [15]. Evje and Flatten [10] investigated an extension of the AUSMD scheme of Wada and Liou [27]. Common to both these approaches is an inherent accuracy comparable to approximate Riemann solvers, achieved by a computationally cheap algorithm. However, spurious oscillations and overshoots are observed near discontinuities.

This problem was to a large extent solved by taking the coupling between the mass equations into account [12]. This approach, denoted as the Mixture Flux (MF) methods, involves a rough splitting of the mass fluxes into a fast-moving and a slow-moving component dependent of properties of the mixture. In a previous work [11], we developed a weakly implicit scheme, termed WIMF-A USMD, based on the MF approach combined with the use of AUSMD convective fluxes similar to those applied in [10].

The purpose of the present work is to elaborate further on the class of MF schemes for the two-fluid model. The MF methods are first presented in a semidiscrete setting, similar to the one introduced in [11]. Particularly, the MF methods are constructed so that they satisfy the following ’’good” properties: (i) The numerical mass fluxes reduce to upwind type of fluxes for a linear contact discontinuity similar to those produced by an exact riemann solver; (ii) Abgrall’s principle is satisfied; that is, a flow uniform in velocity and pressure, must remain uniform during its temporal evolution. Fully discrete MF schemes are then designed as follows:

• First, we construct a fully discrete Weakly Implicit MF scheme, denoted as WIMF-AUSM, which employs AUSM type of convective fluxes similar to those used e.g. by Paillere et al[19];

• second, we construct a fully discrete Strongly Implicit MF scheme, denoted as SIMF- AUSM, which also apply convective fluxes of the AUSM type.

In previous works dealing with construction of schemes within the MF framework [12, 11], we have applied AUSMD type of convective schemes similar to those used in [10]. Hence, this work serves to demonstrate some of the flexibility of the MF approach by replacing the AUSMD/V convective fluxes applied in [12, 11] with AUSM type. Roughly speaking, it seems that as long as

CFL-FREE NUMERICAL SCHEMES 3

we work with convective fluxes which satisfy the requirements of the MF framework, the resulting MF schemes are not very sensitive for the specific choice. The motivation for using AUSM type of fluxes in the present work is that we then naturally can enforce an implicit time discretization which allows us to produce strongly implicit schemes as described above.

Many numerical simulations are made to highlight the differences and similarities between WIMF-AUSM and SIMF-AUSM. In particular, we observe the following:

• For timesteps dictated by the sonic CFL condition (2) both schemes give a performance which is similar to an explicit Roe scheme.

• The WIMF-AUSM scheme allows exact resolution of a moving contact discontinuity. This property closely hang on the fact that WIMF-type of schemes are stable for timesteps dictated by the weak CFL condition (3). Explicit schemes, like the Roe scheme used in this work for comparison purposes, are excluded from possessing this property since the timestep must obey the strong CFL condition (2).

• The SIMF-AUSM scheme gives numerical mass fluxes similar to the WIMF-AUSM scheme, but does not possess the ’’exact resolution property” for a linear contact discontinuity due to the implicit discretization of its numerical mass fluxes. On the other hand, this scheme is unconditionally stable for a moving linear contact discontinuity, however at a price of introducing a strong smearing of the contact discontinuity.

More generally, the results when WIMF-AUSM and SIMF-AUSM are explored for many different flow cases, indicate that for several cases, the SIMF scheme allows for an increased timestep and improved computational efficiency on a given grid. In particular the SIMF scheme allows for efficient steady state calculations. However, the SIMF is inherently more diffusive than the WIMF on volume fraction waves. This limits the applicability of the SIMF scheme for accurate calculation of slow transients (mass fronts), where a weakly implicit scheme may generally be preferable.

Our paper is organized as follows: In Section 2, the particular two-fluid model we study is presented. In Section 3 we describe the MF approach as developed in [12, 11]. Then, in Section 4 we detail fully discrete numerical schemes by working within the MF frame of Section 3. In particular, we develop a WIMF-AUSM scheme and a SIMF-AUSM scheme. In Section 5 we state some important properties of the SIMF-AUSM and WIMF-AUSM schemes. Finally, in Section 6 we present numerical simulations. Particularly, we demonstrate that the SIMF scheme introduced in this paper is able to violate the CFL criterion for all waves for a wide range of problems, justifying its description as a strongly implicit scheme. In Section 6.5 we suggest a slight modification of the SIMF-AUSM scheme consistent with our framework, enabling it to handle the transition to one-phase flow in a stable and accurate manner.


Throughout this paper we will be concerned with the common two-fluid model formulated by stating separate conservation equations for mass and momentum for the two fluids, which we will denote as a gas (g) and a liquid (1) phase. The model has been studied by several authors [25, 4, 5, 19, 10] and will be briefly stated here. We let U be the vector of conserved variables

Pgttg■jj _ pioc\

pgCXgVgp\a\v\

mgmihh

(4)

By using the notation Ap = p — p1, where p% is the interfacial pressure, and T& = (p* —p)dxak, we see that the model (1) can be written on the form

• Conservation of massd dqI (Psas) + ^ (Psasvs) =

Qj. (/4«i) + (piam) = 0,

(5)

(6)

4 EVJE AND FLATTEN

• Conservation of momentum^ (/>g<*g%g) + ^ (/>g<*g%g) + ^g^ + = 0g + (7)

+ + + (8)The system is closed by some equation of states (EOS) for the liquid and gas phase. The numerical methods we study in this work allow general expressions for the EOS. However, for the numerical simulations presented in this work we assume the simplified thermodynamic relations

and

Pi = O,o + P~Poa-

(9)

(10)

where

and

Po = 1 bar = 105 Pa O,o = 1000 kg/m3,

Og = 105(m/s)2

a\ = 103 m/s.Moreover, we will treat Qk as a pure source term, assuming that it does not contain any differential operators. We use the interface pressure correction

O'gO'lpgplAp = apgai + piag (% - O) (11)

where unless otherwise stated we set a = 1.2. This choice ensures that the model is a hyperbolic system of conservation laws, see for instance [25, 5]. Another feature of this model is that it possesses an approximate mixture sound velocity c given by

c =p\ag + pgai

%^A<2g + ^pg<%(12)


(ag,p,vg,v\). For the pressure variable we see that by writing the volume fraction equation ag + «i = 1 in terms of the conserved variables as

m+W)=l- (13)

we obtain a relation yielding the pressure p(mg,mi). Using the relatively simple form of EOS given by (9) and (10) the pressure p is found as a positive root of a second order polynomial. For more general EOS we must solve a non-linear system of equations, for instance by using a Newton-Rapson algorithm.

Moreover, the fluid velocities vg and v\ are obtained directly from the relationsUz

vi =Ui

EVThroughout this work we will study only the isentropic 4-equation model given above, whereas

in general energy conservation equations for each phase could also be included. In this respect we are consistent with our previous works [10, 11].

3. A SEMI-DISCRETE SCHEME

In this section we construct semi-discrete approximations of the model (5)-(8). In Section 4 we describe fully discrete schemes obtained from the semi-discrete scheme, and in Section 5 we state basic properties possessed by these schemes. Finally, in Section 6 we explore the performance of these fully discrete schemes by studying several well known two-phase flow problems.


3.1. General form. It will be convenient to express the model (5)-(8) on the following form:dtrrik + 9xfk — 0,

dt-ffc "h dxQk "t" oifcdxP + (Ap)dx(Xk = Qk->

where k = g, 1 and

(14)

fk - PkOikVk and mk = Pk^k

9k = PkOikvl and Ik = pkOLkvk.

We assume that we have given approximations Approximations nik,j(t) and h,jfl) for t € (tn,tn+1] are now constructed by solving the following ODE problem:

^k.j P^xPk.j = 0,Ik,j ~\~hxGk,j "b ctk,j$xPj "b {Ap)jSxAktj = Qk,j,


mk,j(tn) = m'k,j) Ik,j(tn) = Ik,j-

Here 5X is the operator defined by

(15)

i(j; W j — Wj+1/2 ~ Wj-1/2^aWj+1/2 — %+l ~ %

Az ' " ,and (Ap)flt) - (Ap) (Uj(t),S) is obtained from (11). Moreover, Fktj+1/2(t) - Fk(Uj(t), Uj+i(t)), Gk,j+i/2(t) = Gk(Uj(t), Uj+i(t)), Pj+1/2(t) = P(Uj(t), UJ+1(t)), and Afcj+1/2(t) — Ak(Uj(t), Uj+i{t)) are assumed to be numerical fluxes consistent with the corresponding physical fluxes, i.e.

Fk(U, U) = fk= PkOtkVk

Gk(U,U) =gk = pkakv%

P([/,[/)=pAk(U,U) = ak.

3.2. The class of Mixture Flux (MF) methods. Before we describe the MF approach it will be useful to introduce some basic concepts consistent with those used in [11, 12]. Assume that we consider a contact discontinuity given by

Pl=Pr=P (16)«l # OR

(vs)l = (fi)L = (vs)r = (v\)r = v,

for the time period [tn, tn+1]. All pressure terms vanish from the model (5)-(8), and it is seen that the solution to this initial value problem is simply that the discontinuity will propagate with the velocity v. The exact solution of the Riemann problem will then give the numerical mass flux

(j>av)j+1/2 = %p(ai + aR)u - ^/>(aR - «l)M- (17)

DeHnition 1. A numerical flux F that satisfy (17) for the contact discontinuity (16) will in the following be termed a “mass coherent” flux.

Definition 2. A pair of numerical fluxes (F\,FS) that satisfy the relation

PgP\,j+l/2 + PlPg,j+l/2 = PgPlv- (18)for the contact discontinuity (16) will in the following be termed “pressure coherent” fluxes.

DeHnition 3. We will use the term Mixture Flux (MF) methods to denote numerical algorithms which are constructed within the semidiscrete frame of (15) where fluxes are given as follows:

6 EVJE AND FLATTEN

(1) The numerical flux Akj+1/2(t) is obtained as

A AO _ +«W+lM^■k,j+l/2\P) — 2

(2) We determine Pj+i/2(t) for t £ (tn,tn+1] by solving the ODE

Pj+1/2 +[Kj+l/2pl,j+l/2]8xIg,j+l/2 + [Kj+1/2pg,j+l/2]8xIl,j+l/2 = 0

(19)

(20)

where the interface values Kj+i/2 and pk,j+1/2 <we computed from P,+i/2(t) together with the arithmetic average (19) which defines akj+i/2(t). Here k is given by

1 (21)K —

(3) We consider hybrid mass fluxes Pfcj+i/2(l) of the form

Fi,j+i/2(t) = Kj+i/2(#) ^psai-^Ff (t) + piag-^F^it) + pia\-^(F^ - F^)(t)^ ^ (22)

and

Fs,j+i/2(t) =Kj+1/2(t) (pias-j^Fg (t) + Pg®i (*) + Psas(-^1° "" FiA)(*)) 2- (23)

The coefficient variables at j+1/2 ore determined from the cell interface pressure Pj+1 /2 (#) os we// os t/ie relation

«j+l/2 W = ^(«j(*) + «j+i(*))

which is consistent with the treatment of the coefficients of the pressure evolution equation(20).

(a) The flux component Fk(t) is assumed to be consistent with its physical flux (pav)k(t) as well as ’’mass coherent” in the sense of Definition 1.

(b) The flux component Ffffl) is assumed to be consistent with its physical flux (pav)k(t) as well as ’’pressure coherent” in the sense of Definition 2.

(4) We choose Gk^+i/2(t) to be consistent with the flux component P^+1//2(t) in the following sense: For a flow with velocities which are constant in space for the time interval \tn,tn+1], that is,

Vk,j(t) = Vk,j+i(t) = vk(t), t e [tn,tn+1], (24)we assume that Gkj+i/2(f) takes the form

Gk,j+i/2(t) = Gkj+1 j2(t) = Vk(t)Fkj+1/2(t), (25)where Pkj+i/2(t) is the numerical flux component introduced above.

It is easy to check that the above numerical fluxes A&j+i/2, Pj+1/2, Fkj+1/2, and Gk,j+i/2 are consistent with the corresponding physical fluxes. We refer to [12] for more details. We now state the following important lemma whose proof can also be found in [12]:Lemma 1. Let the mixture fluxes (22) and (23) be constructed from pressure coherent fluxes FtP in the sense of Definition 2, and mass coherent fluxes Fk in the sense of Definition 1. Then the hybrid fluxes (22) and (23) reduce to the upwind fluxes (17) on the contact discontinuity (16), i.e. they are mass coherent.

It follows directly from Definition 3 and Lemma 1 thatCorollary 1. The mass fluxes of the MF methods given by Definition 3, ore mass coherent in the sense of Definition 1.


Moreover, by application of Lemma 1 and Definition 3, we can verify that the MF methods satisfy the following principle due to Abgrall [1, 21, 22]:A flow, uniform in pressure and velocity must remain uniform in the same variables during its time evolution. We refer to [12] for its straightforward proof.

Corollary 2. The MF methods given by Definition 3, obey Abgrall’s principle. More precisely, for the contact discontinuity (16) the semidiscrete approximation (15) takes the following form

d~bx(pkakvk)j — 0, (26)V TH-Avdxi.PkQk'Vk)j — 0,

where {pkOtkVk)j+i/2 is on the form (17). Consequently, no momentum change is introduced and the contact discontinuity remains unchanged except from experiencing a convective transport.

In conclusion, Corollary 1 states that the MF mass fluxes recover the numerical fluxes of an exact riemann solver for a moving or stationary contact discontinuity. Corollary 2 ensures that Abgrall’s principle [1] is satisfied. The fact that this principle is obeyed, ensures that the use of the pressure evolution equation (20) in the discretization of the non-conservative pressure term is consistent with basic physical understanding of two-phase flow phenomena.

Remark 1. The following differential relations are obtained from the basic relation (13) (see [12, 11] for more details):

dp = K(pidmg + pgdmi)

dct\ — k(—o:[dm„ -\—J^-<Xgdm\), op op

(27)

where k is given by (21) and

dmg = a a ^7^ dp — pgda\

dm i = ai^-dp + p\da\. op

(28)

Multiplying the gas mass conservation equation by npi and the liquid mass conservation equation by Ky9g and then adding the two resulting equations, yields the equation

(/%%) + «/)g^ (#<W = 0.

In view of the first equation of (27), the pressure evolution equation (20) follows.The mixture mass fluxes (22) and (23) are obtained by first introducing a flux component Fp

(associated with the pressure) and Fa (associated with the volume fraction) such that the mass fluxes F] and Fg, inspired by (28), are given by

F ~ ai~dpFp +plFa

Fs = as~jtyFP ~ psFa'

Inspired by the differential relations (27) we propose to give Fp and Fa the following form

Fp = KpgFP + npiF®

Fa = K^dpa&F^ ~ ’

(29)

(30)

where should possess the ’’pressure coherent” property whereas F^ should possess the ’’mass coherent” property. Combining (29) and (30) yields the mixture mass fluxes (22) and (23).

It will be useful to introduce the following two definitions of the terms weakly implicit and strongly implicit

EVJE AND FLATTEN

Definition 4. Assume that we initially are given a contact discontinuity (16). A numerical scheme is said to be weakly implicit if it allows stable calculation of solutions under the CFL condition

AxAi~ \v (31)

Definition 5. Assume that we initially are given a contact discontinuity (16). A numerical scheme is said to be strongly implicit if it allows stable calculation of solutions under no restriction on the time step.

Remark 2. Note that the time step restriction (31) is consistent with the CFL condition (3). This follows from the fact that for the contact discontinuity (16) the two eigenvalues corresponding to volume fraction waves degenerate and coincide with fluid velocity v. We refer to [10] for more details concerning the eigenvalue structure of the two-fluid model under considerations.

4. Fully Discrete Numerical Schemes

The purpose of this section is to construct fully discrete schemes based on the general class of MF schemes given by Definition 3. We first describe how to construct appropriate candidates for the mass flux components Fjf and Fjf which were introduced in Definition 3. Then we apply these components to propose fully discrete schemes, one type which is denoted as Weakly Implicit Mixture Flux (WIMF) and another denoted as Strongly Implicit Mixture Flux (SIMF). Both schemes contain the mechanism which allows us to obtain stable pressure calculation for large time steps. The difference between them lies in the temporal discretization of the mass fluxes, more precisely, the F^ mass flux component.

4.1. A pressure coherent convective mass flux Fjf. Due to the fact that the mass flux component Fj? is associated with the pressure calculation as described in Remark 1, it is natural to choose a discretization of this flux which is consistent with the discretization of the pressure evolution equation. On the semi-discrete level, in view of (20), we therefore propose to consider the following discretization of the mass conservation equations

"R,j+1/2 +SxIktj+1/2 — 0, t € (tn,tn+1]

We now suggest to average as follows:

mk,j(t) ~ 2 (mk,j-l/2(t) +mkj+i/2(t)) ,

which implies that

(*) = 2 1/2 (*)+ ^fcj+1/2 (*)) •By substituting (32) into (33) we obtain the following ODE equation for mk,j(t):

+ 2Ax^k’^+1 ~ = ^ (tn,tn+ ]

^ + ^mk,j ~^mk,j+1)‘

(32)

(33)

(34)

This equation is the basis for designing the flux component F^j+1^2. A fully discrete version of (34) which employs updated mass fluxes IffA1 is given by

i + ™L+i) 1At + 2Ax

This equation can be written on the flux-conservative form

= 0.

At

(35)


where(F?)"+i/2 = 2) + 4 At(m*9 mL'+i)' (36)

We can easily check that the proposed flux FJ? possesses the ’’pressure coherent” property of Definition 2, see [12, 11].

Proposition 1. The flux component Ff given by (36) is pressure coherent in the sense of Definition 2.

Remark 3. The motivation for enforcing an implicit treatment of the terms Ik in the mass fluxes (36) is to free the resulting schemes from the strong CFL condition (2). In [12] we studied explicit MF schemes where the mass flux FtP was given the pure explicit form

(Fk )"+i/2 = 2 (% + Tkd+i ^ + 4 At - TO^i+i)• (37)The main difference between using (36) and (37) is that that (36) introduces a stronger smearing of the sonic waves, see [11]. On the other hand, (37) does not allow the resulting MF schemes to break the strong CFL condition (2).4.2. Convective fluxes F^ and Gft.

4.2.1. FVS. For an FVS type of scheme, the convective flux terms are split into upstream and downstream travelling components as

F(U) =F+(U)+F"(U),where F = (pav,pav2)T so that the numerical flux at the interface j + 1/2 is given as

9+1/2 = F(Ul,Ur) = F+(UL) + F-(UR).We consider the velocity splitting formulas used in previous works [15, 27, 8, 9, 10]

v±(v,c)={ ,

(38)

(39)

(40)

where the parameter c represents the physical sound velocity for the system. For the two-fluid model, we assume that it is given by the approximate expression (12). We define a cell interface sound velocity 9+1/2 as follows

cy+i/2 = max(cj,Cj+i). (41)We now let the numerical fluxes be given as follows

(1) Mass Flux. We let the numerical mass flux (pav)j+i/2 be given as

(/**%)j+l/2 = (/**); F+ (9, 9+1/2 ) + (m)j+l y (9+1, 9+1/2 ) (42)for each phase.

(2) Momentum Flux. We let the numerical convective momentum flux (pctv2)j+1/2 be given as

(/*w%+i/2 = 9+1/2) + (mv);+iy"(9+i,9+1/2)- (43)4.2.2. AUSM. We define a cell interface velocity 9+1/2 as

9+1/2 - V+ (9,9+1/2) + V (9+1,9+1/2), and obtain the convective fluxes as

and

('“Aw*={ K;x1/2In the following we use AUSM convective fluxes as bases to define MF type of schemes, straightforward to check that AUSM possesses the ’’mass coherent” property.

if 9+1/2 > 0otherwise

if 9+1/2 > 0 otherwise.

(44)

(45)

(46)

It is

Proposition 2. The convective flux {pav)^fj^ is mass coherent in the sense of Definition 1.

10 EVJE AND FLATTEN

The FVS convective mass fluxes are pressure coherent but not mass coherent. Consequently, they are not accurate for a moving or steady contact discontinuity [10]. However, the FVS convective fluxes are very stable and will be introduced in an appropriate manner when we study flows which involve transition to single-phase flow. We refer to Section 6.5 for details.

4.3. A Weakly Implicit Mixture Flux (WIMF) Scheme.

4.3.1. General form. We use the shorthands = pk&k and Ik = nikVk and consider a fully discrete scheme based on (15) given as follows.

• Gas Mass

• Liquid Mass

= r p»+l/2At

m^t1 - mfc■ _ n+i/2- <hTij

• Gas Momentum■l _

"At

- -4(<?A)g,j - + ,

• Liquid Momentum

At

pn+1 _ pn+11 4_L 1 /‘2 1 ■i — t /‘>

Ax- (Ap)"JzA2j + (Qg)j.

Atpn+1 _ pn+1

= _ (Ar%M,A% + (Qi)?.

(47)

(48)

(49)

(50)

4.3.2. WIMF-AUSM.

Definition 6. We will use the term WIMF-AUSM to denote the numerical algorithm which is constructed within the discrete frame of (47)-(50) where fluxes are given as follows:

(1) The numerical flux A£j+1/2 w obtained as

\n _ ah+ah+ij+1/2 - o (51)

(2) We determine Pp~i/2 by considering the following discretization of the pressure evolution equation (20)

jn+l _ jn+1 jn+1 _ jn+1 (^^)

where the interface values an<^ Pk j+1/2 are computed from Pjfx/2 together with thearithmetic average (51) which defines Ok,j+i/2-

(3) We consider hybrid mass fluxes of the form

j+1/2

(53)and

j+1/2

(54)


The coefficient variables at j + 1/2 are determined from the cell interface pressure Pffi^ as well as the relation

a"+i/2 = 2 +a]+i)

which is consistent with the treatment of the coefficients of the pressure evolution equation(52).(a) For the flux component jy2 we refer to Section f.2 and use

AUSM,n_ f Wk,Xj+l/2

(/%%) k,j+1 uk,j+1/2 otherwise.

(b) For the flux component we refer to Section 4-1 and use

FZ?+i/2 = 2^kd'1 + -mfcj+i)-

(55)

(56)

(4) The flux component G% j+1/2 is chosen to be consistent with the flux component 1//2 by using

rm _ z^fA,raUk,j+1/2 - ufcj+l/2

= i (^)L'^W+l/2 (f^W+1/2 >0’k,j+l/2

4.4. Two Strongly Implicit Mixture Flux (SIMF) Schemes.

(mv)tj+i%t,,+i/2 oAerwwe. (57)

4.4.1. General form. We use the shorthands mu = puoik and Ik = m+% and consider a fully discrete scheme based on (15) given as follows.

• Gas Mass

Liquid Mass

• Gas MomentumC1

mS‘ - mh = ™+iAt

%At

= -^(^)2T-C• Liquid Momentum

At

At

pn+l _ pn+lj+1/2 1-1/2

Ax

(58)

(59)

(60)-(Ap)^A2, + (Q«)?.

pn+l _ pn+l= -4(GA)1nti - «r -y±1/2—

(61)

fi' Aar(Ap)?4A^ + (Qi)?

4.4.2. SIMF-AUSM.

Definition 7. We will use the term SIMF-AUSM to denote the numerical algorithm which is constructed within the discrete frame of (58)-(61) where fluxes are given as follows:

(1) The numerical flux A'%j+1/2 is defined as in (51).(2) We determine P™+i/2 by considering the discretization (52) of the pressure evolution equa

tion (20)(3) We consider hybrid mass fluxes FJfof the form

1+1/2(62)

12 EVJE AND FLATTEN

and

((%)

The coefficient variables at j + 1/2 are determined from the cell interface pressure Ptf1/2 as well as the relation

aj+1/2 = + a?+i)which is consistent with the treatment of the coefficients of the pressure evolution equation(52).(a) For the flux component F^’Pf^2 we use

AUSM,n+lfcj+1/2

(b) For the flux component we use

pD,n+l/2<=,1+1/2

(asi + ce,) +k,j+

1 Ax 4 At

n • mkj+i)

(64)

(65)

(4) The flux component G%~j+1/2 is chosen to be consistent with the flux component f£~T^2

by using

nn+1 Uk,j+1/2 UW+1/2 = (pav2)AUSM,n+l

k, 2+1/2(mK)k,y«t,2+1/2 */<2+1/2 > 0

otAenawe. (66)

4.4.3. SIMF-FVS.

Definition 8. We will use the term SIMF-FVS to denote the numerical algorithm which is identical to SIMF-AUSM except from the convective flux terms F^’"^2 and G^F^2 which are defined as follows:

(a) For the flux component F^Tff^2 we use

pA,n+lrfc,i+1/2 = (pav)FVS,n+l

k, 2+1/2 (67)

(b) For the flux component G^'J^2 we use

Some comments are in order.

Remark 4. The MF approach, as reflected by the above WIMF and SIMF schemes, allows for sequential updating of the conservative variables in the following manner:

(1) For both schemes the momentum equations (49) and (50) are solved coupled with the pressure equation (52) to yield P™Xi/2 an<^ ^j1 •

(2) For WIMF the mass equations (47) and (48) with the mixture fluxes (53) and (54) are solved separately and in an explicit manner whereas for SIMF the mass equations (58) and (59) with the mixture fluxes (62) and (63) are solved coupled with each other to yield

In this respect our strongly implicit schemes (SIMF) resemble the schemes used in common industrial codes [3, 13]. Advantages of the current schemes include:

• The use of the hybrid FVS/FDS convective fluxes allows for solving the conservative variables on a nonstaggered grid.

• The central pressure flux Fj? and the stronger coupling between the mass equations allow for nonosdilatory resolution of the pressure for large time-steps.


• The conservative momentum variables h are solved for directly, and there is automatic consistency between the pressure and mass variables.

An advantage of using the AUSM and FVS fluxes described above is that they are linear with respect to their arguments (pa)/Al and (pav)1/^1. Hence only one matrix inversion (per set of equations) is required to solve the resulting system exactly.

5. Properties of the fully discrete ME schemes

In view of Definitions 6 and 7 and Proposition 1 and Proposition 2 it follows that both the WIMF-AUSM and SIMF-AUSM scheme are MF schemes in the sense of Definition 3. Consequently, Corollary 1 and 2 are applicable, and we immediately conclude thatProposition 3. WIMF-AUSM and SIMF-AUSM satisfy the following properties:(i) The mass fluxes of WIMF-AUSM and SIMF-AUSM are mass coherent in the sense of Definition 1. (ii) Both schemes obey Abgrall’s principle.

More precisely, for the contact discontinuity (16) the mass fluxes of WIMF-AUSM take the form

= (69)whereas SIMF-AUSM gives mass fluxes on the form

SIMF-AUSM,n+l _ n+1 (70)where we have assumed (without loss of generality) that v > 0. The term mass coherent of Definition 1 does not take into account the temporal aspect of the discretization and both schemes are classified as ’’mass coherent” since they produce the correct upwind form.

The purpose of the next paragraph is to focus on this temporal aspect and provide some insight into a special feature possessed by WIMF type of schemes concerning the ability to resolve a linear contact discontinuity (16) accurately.

5.1. Resolution of moving or stationary contact discontinuity. We now take a closer look at the contact discontinuity given by (16). We consider a WIMF scheme where the flux component Fj) is mass coherent in the sense of Definition 1. Then, as noted above, we obtain the mass fluxes

where we have assumed that v > 0.The discrete evolution equation for the mass at cell j is given by

(71)

(/%(%t)r^ - o%a&)2 (/%<*&)?_i - (/%(*&)A, ' A, ' (?2)

Using that pk is constant, this may be simplified to yield the discrete evolution equation for the volume fractions. For simplicity in notation we drop the phase index k and obtain

n - a'-A, -• Ax • <73>

If the contact discontinuity is exactly reproduced within the grid at time tn = nAt, the discrete representation may be expressed as

for j < i, (74), - «r for j > i

for some i. We remember that here v&j = v and pj = p. From (73) we see that for such an exactly reproduced discontinuity, only the value oq will change by stepping forward in time from nton + 1. We then obtain

XJ —I— 1

OiL — C*R

aj = aL

o7

a"+1 - aR = v-A t AxIn particular, if Ax/At = v we obtain an interesting result. Then

(75)

qT+1 — 0:|j, = 0:l — OR (76)

14 EVJE AND FLATTEN

or simplya.,n+1

whereas= OR.

(77)

(78)So we conclude that integrating the contact discontinuity (74) using the timestep Ax/At = v will simply shift the location of the discontinuity exactly one grid cell to the right. This is exactlythe distance the contact discontinuity will move in one fraction distribution is now given as

timestep, Ax = vAt. The discrete volume

a"+1 = aL for j <i + 1, (79)«"+1 = «R for j > i + 1,

and by induction

for j < i + m, (80)for j >i + m

for all m (within the boundaries of the grid). We may now state the following lemma

Lemma 2. Consider a WIMF type of scheme as described in Section f.3 which is mass coherent in the sense of Definition 1. Apply the WIMF scheme to a contact discontinuity moving with the velocity v, as described by (16). If the optimal timestep Ax/At = |u| is used, the WIMF scheme will exactly capture the contact discontinuity for all tn >t°.

Proof The above discussion proves the Lemma for v > 0. Repeating the steps for v < 0 completes the proof. □

Some remarks are now in order.

Remark 5. Notably the proof of Lemma 2 does not rely directly upon the scheme being of the WIMF class. An explicit scheme (like the basic AUSMD scheme studied in [10], or the MF- AUSMD scheme considered in [12],) which correctly reduces to the upwind scheme for the contact discontinuity (16) will also formally satisfy Lemma 2. However, in practice such schemes will not work as they are unstable under the violation of the sonic CFL criterion implied by the timestep Ax/At = v. This means that slight numerical oscillations will grow exponentially into instabilities. Even for the above case with a linear contact discontinuity, such numerical errors are expected due to the limited floating point precision of computers.

For the WIMF class of schemes however, the presence of the implicit flux component F/? as given by (56) will prevent the development of such instabilities.

The ability to exactly capture a contact discontinuity in a stable manner is a very desirable feature unique to the class of WIMF schemes. Numerical evidence of this fact will also be provided in the next section.

Remark 6. Lemma 2 does not apply to the SIMF class of mass coherent schemes as described in Section f.f. In this case, the numerical mass flux becomes

(81)

for a contact discontinuity of the form (16), and (73) must be replaced by

<l a

At= v-

,n+1i-i — a,n+1

Ax(82)

Hence SIMF operates on a contact discontinuity much the same way as an implicit upwind scheme operates on a scalar advection equation. That is, we expect the SIMF class of schemes to be stable, yet diffusive. This issue is explored in the numerical section.


2 oaI

0 20 40 60 80 100Distance (m)

Figure 1. Linear contact discontinuity, 100 cells, T=5.0 s. Roe, SIMF-AUSM, and WIMF-AUSM scheme for Ax/At = 1000 m/s.

A


In the first example we study the performance of WIMF-AUSM and SIMF-AUSM for a linear contact discontinuity. In particular,

• we want to demonstrate that WIMF-AUSM possesses the ” exact resolution property” of Lemma 2 and is ” weakly implicit” in the sense of Definition 4;

• we want to demonstrate that SIMF-AUSM is ’’strongly implicit” in the sense of Definition5.

The purpose of the rest of the examples is to demonstrate that these ’’good” properties observed for WIMF-AUSM and SIMF-AUSM for a linear contact discontinuity to a large extent carry over to more difficult flow cases. For many flow cases we also include results produced by the explicit Roe scheme considered in [10].

6.1. Linear Contact Dicontinuity. We now wish to illustrate the properties of WIMF-AUSM and SIMF-AUSM as stated in Section 5. We consider a simple linear contact discontinuity in the volume fraction, where the initial states are given by

and

WL

Wr

p ' 105 Pa '«1 0.75vs 10 m/sVi 10 m/s _

P ' 105 Pa 'm 0.25W 10 m/s•vi 10 m/s

(83)

(84)

We consider a 100 m long pipe and assume that the discontinuity is initially located at x = 0. We use a computational grid of 100 cells and simulate a time of t = 5.0 s. The discontinuity will then have moved to the center of the pipe, being located at x = 50 m.

First, in Figure 1 we have plotted the solutions produced by the Roe, WIMF-AUSM, and SIMF-AUSM scheme when the timestep corresponding to Ax/At = 1000 m/s is applied. All three schemes are mass coherent, i.e. they produce the same upwind type of mass fluxes, and for this time step the solutions are the same, practically speaking.

Results for different lower values of Ax/At are given in Figure 2 for the WIMF and SIMF scheme. For these larger values of At the Roe scheme becomes unstable since it must obey the sonic CFL condition (2). SIMF-AUSM and WIMF-AUSM behave very similarly for a low timestep (Ax/At = 1000 m/s). However, increasing the timestep increases the accuracy for WIMF-AUSM but decreases it for SIMF-AUSM.

16 EVJE AND FLATTEN

WIMFAUSM

0 20 40 60 80 100 0 20 40 60 80 100Distance (m> Distance (m>

Figure 2. Linear contact discontinuity, 100 cells. SIMF-AUSM vs WIMF-AUSM scheme for different values of Ax/At. Left: WIMF-AUSM. Right: SIMF-AUSM.

SIMF-AUSM

We observe that for the critical timestep Ax/At = vg = v\ = 10 m/s, WIMF-AUSM captures the discontinuity exactly, as stated by Lemma 2. Increasing the timestep beyond this value will make the WIMF-AUSM scheme unstable. On the other hand, we may increase the timestep beyond Ax/At = 10 m/s for SIMF-AUSM without inducing instabilities. Once we exceed this critical timestep, there is a significant increase in numerical diffusion.

Thus, we may conclude that the WIMF-AUSM scheme is weakly implicit in the sense of Definition 4 whereas the SIMF-AUSM scheme is strongly implicit in the sense of Definition 5. In addition, we have demonstrated that the WIMF-AUSM scheme possesses the ’’exact resolution property” of Lemma 2.

6.2. Water Faucet Problem. We now choose another problem which focuses on volume fraction waves. We consider the classical faucet flow problem of Ransom [20], which has become a standard benchmark [4, 10, 25, 18, 26].

We consider a vertical pipe of length 12 m with the initial uniform state" p ' 105 Pa "

ai 0.8vs 0Vl 10 m/s

(85)

Gravity is the only source term taken into account, i.e. in the framework of (7) and (8) we haveQk=gPka-k, (86)

with g being the acceleration of gravity. At the inlet we have the constant conditions — 0.8, v\ = 10 m/s and vs = 0. At the outlet the pipe is open to the ambient pressure p — 105 Pa.

We restate the approximate analytical solution presented in the references [19, 26]

m(a,t) \Jvq + 2gx for x <vot+ \gt Vo + gt otherwise. (87)

<%i(z,t)ao(l + 2gxv02) 1/2 for x < vot + \gt oto otherwise (88)

where the parameters «o = 0.8 and i>o = 10 m/s are the initial states.In Figure 3 we compare the SIMF-AUSM and the Roe scheme for T = 0.6 s on a grid of 120

computational cells. In addition, the effect of reducing the timestep to A = 17 m/s is investigated for the SIMF-AUSM and the WIMF-AUSM scheme.

We note that for the small timestep A = 1000 m/s the SIMF-AUSM scheme is virtually indistinguishable from the Roe scheme. Only for the pressure is any difference visible, here the SIMF-AUSM scheme is slightly more diffusive.

However, increasing the timestep to A = 17 m/s (approximately the liquid velocity) causes a significant increase in numerical diffusion for the SIMF-AUSM scheme, both in pressure and




Figure 3. Water faucet problem, 120 cells, T=0.6 s. SIMF-AUSM, WIMF- AUSM and Roe scheme. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity.

Figure 4. Water faucet problem, 120 cells. SIMF-AUSM vs WIMF-AUSM scheme for different values of Ax/At. Left: WIMF-AUSM. Right: SIMF-AUSM.

volume fraction. This sharply contrasts the results of the WIMF-AUSM scheme, where the lower integration velocity significantly improves the performance of the scheme on the slow waves.

6.2.1. Effect of increasing the timestep. We now investigate further how the SIMF and WIMF schemes behave under different timesteps. Results after t = 0.6 s are given in Figure 4.

We observe the same picture as for the linear contact discontinuity studied in the previous section. For low timesteps, the SIMF-AUSM and WIMF-AUSM have a similar behaviour. Increasing the timestep improves the accuracy of WIMF-AUSM but has the opposite effect on the SIMF scheme. Upon breaking the strong (volume fraction) CFL criterion, WIMF-AUSM becomes unstable whereas SIMF-AUSM merely becomes more diffusive.

18 EVJE AND FLATTEN


Figure 5. Water faucet problem, 60 cells, stationary conditions at T=5.0 s. SIMF-AUSM vs analytical solution. Left: Gas fraction. Right: Liquid velocity.

Remark 7. These results confirm the picture observed in Section 6.1 and highlight an important difference between the SIMF and WIMF class of schemes. In effect, WIMF-A USM reduces to the upwind explicit flux (69) for a contact discontinuity, whereas SIMF-AUSM reduces to the upwind implicit flux (70).

6.2.2. Stationary solution. We now investigate the performance of the SIMF-AUSM scheme for very large timesteps, where the volume fraction CFL criterion is strongly violated.

Using the timestep At = 5 s, results after 2, 4 and 7 iterations are given in Figure 5, where the results are compared to the analytical stationary solutions. We observe that the SIMF-AUSM scheme produces qualitatively correct solutions already after 2 iterations. After 7 iterations, the numerical solutions coincide with the analytical reference solutions for liquid velocity and volume fraction.

6.3. Toumi’s Water-Air Shock. We consider an initial value problem introduced by Toumi [24] and investigated by Tiselj and Petelin [23] and Paillere et al [19]. The initial states are given by

p ' 2 -107 Pa 'ctl 0.75% 0Vi 0

(89)

and" P ' 1•107 Pa "

a\ 0.9% 0

. Vl 0

(90)

No source terms are taken into account. For consistency with the work of Paillere et al [19], we modify the interfacial pressure correction (11) for this problem, setting a = 2. Results after T = 0.08 s are given in figure 6, using a grid of 100 cells and a timestep Ax/At = 1000 m/s. Here we compare an explicit Roe scheme, the WIMF-AUSM scheme and the SIMF-AUSM scheme. The reference solution was calculated by the explicit MF-AUSMD scheme described in [12], using a grid of 10 000 cells.

We observe that the implicit schemes seem slightly more diffusive than the explicit Roe scheme. On the other hand, the Roe scheme seems to overshoot on the volume fraction waves compared to the reference solution.

We also note that for this low timestep, the SIMF-AUSM scheme and the WIMF-AUSM scheme produce virtually identical solutions.

6.3.1. Effect of increasing the timestep. We now consider the SIMF-AUSM scheme on a grid of 2000 cells for varying values of the integration parameter A = Ax/At. Results for A = 1000 m/s, A = 100 m/s and A = 10 m/s are given in Figure 7.


Figure 6. Toumi’s shock tube problem, 100 cells. SIMF-AUSM, WIMF-AUSM and Roe scheme. Top left: Gas fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

Figure 7. Toumi’s shock tube problem, 2000 cells. Different timesteps for the SIMF-AUSM scheme. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

20 EVJE AND FLATTEN

Roe 10000 cells SIMF-AUSM 2000 cells SIMF-AUSM 400 cells SIMF-AUSM 100 cells



fl


Figure 8. LRV shock tube problem. Grid refinement for the SIMF-AUSM scheme. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

We observe that increasing the timestep to A = 100 m/s has the effect of increasing the numerical diffusion on pressure waves. However, CFL instabilities are not produced even if the sonic CFL criterion is violated.

Increasing the timestep even further to A = 10 m/s violates also the volume fraction CFL criterion. We note that the diffusion on the pressure waves is increased even further. CFL instabilities do not occur in the volume fraction waves, although for this high timestep spurious oscillations and overshoots are observed.

6.4. A Large Relative Velocity Shock. We consider the Riemann problem given by the initial states

p ' 265000 Pa 'Ot\ 0.71vz 65 m/s

. 'A 1 m/s(91)

and

WR' P ' 265000 Pa '

Ot\ 0.7vs 50 m/s

. yl 1 m/s(92)

Again no source terms are taken into account. This initial value problem was proposed by Cortes et al [5] and is of interest in that the initial discontinuity contains a large difference in the relative velocity between the phases.

6.4.1. Convergence of SIMF-AUSM scheme. In Figure 8, grid refinement for the SIMF-AUSM scheme is studied using a timestep of A = 1000 m/s. For reference, the Roe scheme on a grid of 10 000 cells is included. The simulation is carried out until the time T = 0.1 s is reached.


reference -

am":

0 10 20 30 40 50 70 SO 90 100

reference -

am":

V10 20 30 40 50 60

Distance (m)70 SO 90 100

Distance (m)0 10 20 30

Distance (m)

Figure 9. LRV shock tube problem, 2000 cells. Effect of large timestep for SIMF-AUSM and SIMF-FVS. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity: Bottom right: Gas velocity.

We see that the SIMF-AUSM scheme seems to converge to the reference Roe solution in a monotone way. Note that the wedge in the liquid fraction at x = 50 m is a slow moving wave structure, not a numerical oscillation.

6.4.2. Comparison with SIMF-FVS scheme. We now compare the SIMF-AUSM scheme with the SIMF-FVS scheme using a large timestep. We use a grid of 2000 cells and a timestep A = 10 m/s, results are given in Figure 9.

We oberve that the schemes are indistuingishable on sonic waves. On volume fraction waves, the SIMF-AUSM scheme produces oscillations whereas the SIMF-FVS scheme is stable, yet somewhat diffusive.

Remark 8. This is an interesting property of the SIMF-FVS scheme. Whereas the SIMF-AUSM is preferable for small timesteps due to its accuracy properties, the more diffusive SIMF-FVS has the ability of removing some stiffness from the system (58)-(61). Hence there may be cases where the SIMF-FVS is the preferable approach if large timesteps are used.

6.5. Separation Problem. We follow Coquel et al [4] and consider a vertical pipe of length 7.5 m, where gravitational acceleration is the only source term taken into account. Initially the pipe is filled with stagnant liquid and gas with a uniform pressure of po = 105 Pa and a uniform liquid fraction of ai = 0.5. The pipe is considered to be closed at both ends, i.e. both phasic velocities are forced to be zero at the end points.

Assuming that the liquid column to be incompressible and freely falling under the influence of gravity, the following approximate analytical solution was derived [10] for the transient period

>/2gx for x < \gt2 gt for Igt2 < x < L — Igt2 0 for L — Igt2 < x

v\{x,t) — (93)

22 EVJE AND FLATTEN

0 for x < \gt20.5 for \gt2 <x<L— \gt2

1 for L - \gt2 < x(94)


T = ^ = 0.87 s (95)

these discontinuities will merge and the phases become fully separated. The volume fraction reach a stationary state, whereas the other variables slowly converge towards a stationary solution. In particular we expect the stationary pressure to be fully hydrostatic, approximately given by

PoPo + Pi9 (x ~ L/2)

for x < Lj 2 for x > L/2. (96)

6.5.1. Transition to one-phase flow. As for the WIMF-AUSMD scheme [11], we observed that the basic SIMF-AUSM scheme could produce instabilities in the transition to one-phase flow.

Here we will follow a strategy successfully applied in earlier works [10,11]. We consider a hybrid of the SIMF-AUSM and the SIMF-FVS scheme, denoted as SIMF-AUSM*, where the numerical convective fluxes F = (pav,pav2) are given by the following expression

pSIMF-AUSM* _ gpSIMF-FVS _|_ (1 _ g)pSIMF-AUSM

Otherwise the SIMF-AUSM* scheme is identical to the SIMF-AUSM scheme. Here s is chosen ass = max(0L,</R), (98)

where <f> is an indicator function designed to be 1 near one-phase regions, 0 otherwise. For the purposes of this paper we follow [11] and choose

<j>j = e-tsW? + e-fel[ai1" (99)where we use the parameters kg = 50 and h = 500.

We note that the SIMF-AUSM* scheme differs from the SIMF-AUSM scheme only near one- phase regions.

6.5.2. Numerical results for the transient period. In Figure 10 the results of the SIMF-AUSM* scheme are plotted for a grid of 100 cells and a timestep A = 100 m/s. The simulation was carried out until the time T = 0.6 s was reached.

We observe good accordance with the expected analytical solutions.

6.5.3. Numerical results for the stationary state. Using the same grid of 100 cells and the timestep A = 100 m/s, results for the SIMF-AUSM* scheme are plotted in Figure 11 at the time T = 1.0 s. Now quasi-stationary conditions are reached.

We see that the phases are well separated at this point. The lack of friction terms causes the gas velocity to be large as the gas phase is disappearing, which causes the pressure distribution to deviate slightly from the expected hydrostatic distribution.

6.5.4. Convergence properties of SIMF-AUSM*. In Figure 12 we investigate the convergence of the SIMF-AUSM* scheme as the grid is refined. We use the timestep A = 100 m/s and the plot is made at the time T = 0.6 s. As we can see, the SIMF-AUSM* approximates the expected solution in a monotone way.Remark 9. An eigenvalue analysis [10] demonstrates that the volume fraction wave velocities are roughly given by

y = PgOiVg+_^Og^ ± Ap(pgai + plag) - pipgaiag(vg - vx)2Pgtti + A<*g V (pgQa + piag)2

(100)

For this particular problem we see that these are approximately given by the gas velocity as the gas phase is disappearing. As the maximum gas velocity here becomes higher than the integration


^^jj-xrxcrrCTJJxerrrjjjxcrrrjjxcrrrjjjxoa^

Distance (m)


Figure 10. Separation problem, 100 cells. SIMF-AUSM* scheme at T=0.6 s. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity.

Distance (rn> Distance (m)

Figure 11. Separation problem, 100 cells. SIMF-AUSM* scheme at T=1.0 s. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity.

24 EVJE AND FLATTEN

reference1 000 cells

100 cells25 cells

0.8 -

0.6 -

0.4 -

0.2 -

Distance (m)

Figure 12. Separation problem, T=0.6s. Convergence properties of the SIMF-AUSM* scheme.

parameter X, we conclude that the the SIMF-AUSM* scheme is able to violate the CFL criterion for both sonic and volume fraction waves for this problem.

6.6. Oscillating Manometer Problem. Finally, we consider a problem introduced by Ransom [20] and investigated in [19, 10, 11]. This problem tests the ability of numerical schemes to handle a change in the flow direction of a moving liquid plug.

We consider a U-shaped tube of total length 20 m. The geometry of the tube is reflected in the x-component of the gravity field

{g for 0 < x < 5 mgeos (f^WFT17) for 5 m < x < 15 m (101)

—g for 15 m < x < 20 m.Initally we assume that the liquid fraction is given by

{10—6 for 0 < x < 5 m0.999 for 5 m < x < 15 m (102)

10-6 for 15 m < x < 20 m.The initial pressure is assumed to be equal to the hydrostatic pressure distribution. We assume that the gas velocity is uniformly ug = 0, and the liquid velocity distribution is given by

{0 for 0 < x < 5 mVo for 5 m < x < 15 m (103)

0 for 15 m < x < 20 m,where Vo = 2.1 m/s.

Ransom [20] suggested treating the manometer as a closed loop. We will follow the approach of [19, 11], assuming that both ends of the manometer are open to the atmosphere. We assume that the liquid column will move with uniform velocity under the influence of gravity, giving the following approximate analytical solution for the liquid velocity [19]

v\ (t) = U0 cos (cot), (104)where


E

reference NF 50 m/s

WF 50 m/s WF 15 m/s

Time (s)

Figure 13. Oscillating manometer, 100 cells. SIMF-AUSM* scheme. Time development of the liquid velocity.

where L = 10 m is the length of the liquid column.

6.6.1. Interfacial momentum exchange terms. For this problem we will investigate the effect of including a source term modelling momentum exchange between the phases. For the gas momentum equation, we introduce the source term [19, 11]

MjP = Caga\pg(vg - ui), (106)where C is a positive constant. Likewise the liquid momentum source term is given as

Mj0 = = — Caganpg(vg — ui), (107)conserving total momentum. We write

C = CW, (108)

making the exchange term kick in more strongly near one-phase regions. For the purposes of this paper we choose

C0 = 1000 s-1, (109)and use a semi-implicit discretization as follows

=%<%%)?

We now consider the following models:WF (With friction). We use the momentum exchange terms MjD and M/f as described above. NF (No friction). We set Mj0 = — M® = 0.

6.6.2. Temporal evolution of the liquid velocity. Using a grid of 100 cells, the evolution of the center cell liquid velocity is given in Figure 13. For the frictionless model the timestep Ax/At = 50 m/s was used. By including the friction terms, we found we could increase the timestep to Ax/At = 15 m/s without losing stability. However, for this timestep a non-physical increase in momentum is observed due to the coarse Euler discretization of the gravity field.

(mg)? (mi)(110)

26 EVJE AND FLATTEN

Distance (m> Distance (m)


Figure 14. Oscillating manometer, t=20.0 s, 100 grid cells. SIMF-AUSM* scheme. Top left: Liquid fraction. Top right: Pressure. Bottom left: Liquid velocity. Bottom right: Gas velocity

We observe little difference between WF and NF for Ax/At = 50 m/s, where we achieve good accordance with the expected analytical solution. A slight phase difference seems to develop, which is in accordance with previous results [19, 10, 11].

6.6.3. Frictionless versus frictional flow. The distribution of all variables after t = 20.0 s is given in Figure 14 for the grid of 100 cells and the timestep Ax/At = 50 m/s.

We see that the inclusion of interfacial friction terms has the effect of reducing the gas velocity in the near one-phase liquid regions. As an effect of this, the pressure distribution approximates more accurately the expected hydrostatic distribution.

By the comments of Remark 9, we see that we are able to violate the CFL criterion for all waves also for this problem.

7. Summary

We have constructed a framework termed Strongly Implicit Mixture Flux (SIMF) which allows us to construct fully CFL-free numerical schemes for a standard two-fluid model. This class of schemes keeps the accuracy and stability properties of its explicit predecessors for small timesteps.

Within this framework we have constructed natural extensions of the schemes investigated by Evje et al [8, 10, 12, 11], resulting in the WIMF-AUSM, SIMF-AUSM, and SIMF-FVS schemes.

We have demonstrated that the SIMF-AUSM scheme possesses accuracy and stability properties comparable to the Roe scheme for small timesteps. On breaking the sonic CFL criterion, the SIMF-AUSM scheme becomes less accurate than its weakly implicit variant WIMF-AUSM in the resolution of volume fraction waves.

Based on observations in this paper as well as previous works [12, 11], we may classify schemes in terms of their applicability as follows:

• Explicit schemes. Due to their easy and efficient implementation, explicit schemes are suitable for applications where fast pressure transients are of interest. This may more


often be the case for the nuclear industry than for the petroleum industry, where slow transients related to mass transport are generally more interesting.

• Weakly implicit schemes. These schemes are superior to explicit schemes in stability for large timesteps and accuracy on the slower waves. In particular, by choosing the timestep optimally, weakly implicit schemes may capture a moving contact discontinuity exactly. Consequently these schemes may be suitable for cases where slow transients are the main focus.

• Strongly implicit schemes. These schemes are superior to weakly implicit schemes in stability for very large timesteps. However, they are more diffusive and do not easily allow for high-resolution extensions like the MUSCL strategy of van Leer [14]. For this reason they are not well suited for cases where accurate tracking of the volume fraction waves is of interest. On the other hand, strongly implicit schemes may be used as steady state solvers or for cases where a computationally cheap qualitative description of the transient is desired.

Acknowledgements. The second author thanks the Norwegian Research Council for financialsupport through the “Petronics” programme.

References


[2] F. Barre et al. The cathare code strategy and assessment, Nucl. Eng. Des. 124, 257-284, 1990.[3] K. H. Bendiksen, D. Malnes, R. Moe, and S. Nuland, The dynamic two-fluid model OLGA: Theory and

application, in SPE Prod. Eng. 6, 171-180, 1991.[4] F.Coquel, K. El Amine, E. Godlewski, B. Perthame, and P. Rascle, A numerical method using upwind schemes


flow equation systems, J. Comput. Phys. 147, 463-484, 1998.[6] J. R. Edwards, R. K. Franklin, and M.-S. Lion, Low-diffusion flux-splitting methods for real fluid flows with

phase transition, AIAA Journal 38, 1624-1633, 2000.[7] J. R. Edwards and M.-S. Liou, Low-diffusion flux-splitting methods for flows at all speeds, AIAA Journal 36,

1610-1617, 1998.[8] S. Evje and K. K. Fjelde, Hybrid flux-splitting schemes for a two-phase flow model, J. Comput. Phys. 175,

674-701, 2002.[9] S. Evje and K. K. Fjelde, On a rough ausm scheme for a one-dimensional two-phase flow model, Comput. &

Fluids 32, 1497-1530, 2003.[10] S. Evje and T. Flatten. Hybrid flux-splitting schemes for a common two-fluid model. J. Comput. Phys, to

appear.[11] S. Evje and T. Flatten. Weakly implicit numerical schemes for the two-fluid model. Submitted for publication,

August 2003.[12] T. Flatten and S. Evje. A mixture flux approach for accurate resolution of two-phase flows. Submitted for

publication, July 2003.[13] M. Larsen, E. Hustvedt, P. Hedne, and T. Straume, Petra: A novel computer code for simulation of slug flow,

in SPE Annual Technical Conference and Exhibition, SPE 38841, p. 1-12, October 1997.[14] B. V. Leer, Towards the ultimate conservative difference scheme V. A second-order sequel to godunov’s method,

J. Comput. Phys. 32, 101-136, 1979.[15] M.-S. Liou, A sequel to AUSM: AUSM(+), J. Comput. Phys. 129, 364-382, 1996.[16] M.-S. Liou and C. J. Steffen, A new flux splitting scheme, J. Comput. Phys. 107, 23-39, 1993.[17] Y. Y. Niu, Simple conservative flux splitting for multi-component flow calculations, Num. Heat Trans. 38,


Meth. Fluids 36, 351-371, 2001.[19] H. Paillere, C. Corre and J.R.G Cascales, On the extension of the AUSM+ scheme to compressible two-fluid

models, Comput. & Fluids 32, 891-916, 2003.[20] V. H. Ransom, Numerical bencmark tests, Multiphase Sci. Tech. 3, 465-473, 1987.[21] R. Saurel and R. Abgrall, A multiphase godunov method for compressible multifluid and multiphase flows, J.


1145, 1999.[23] I. Tiselj and S. Petelin, Modelling of two-phase flow with second-order accurate scheme, J. Comput. Phys.

136, 503-521, 1997.

28 EVJE AND FLATTEN

[24] I. Toumi, An upwind numerical method for two-fluid two-phase flow models, Nuc. Sci. Eng. 123, 147-168, 1996.

[25] I. Toumi and A. Kumbaro, An approximate linearized riemann solver for a two-fluid model, J. Corn,put. Phys. 124, 286-300, 1996.

[26] J. A. Trapp and R. A. Riemke, A nearly-implicit hydrodynamic numerical scheme for two-phase flows, J. Comput. Phys. 66, 62-82, 1986.

[27] Y. Wada and M.-S. Lion, An accurate and robust flux splitting scheme for shock and contact discontinuities, SIAM J. Sci. Comput. 18, 633-657, 1997.

hybrid flux splitting schemes for numerical resolution of

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