All Speed Multi-Phase Flow Solvers

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Te Florida State University DigiNole Commons

Electronic eses, Treatises and Dissertations e Graduate School

6-13-2005

All Speed Multi-Phase Flow SolversSamet Yucel KadiogluFlorida State University

Follow this and additional works at:h p://diginole.lib.fsu.edu/etd

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Recommended CitationKadioglu, Samet Yucel, "All Speed Multi-Phase Flow Solvers" (2005). Electronic eses, Treatises and Dissertations.Paper 3391.
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THE FLORIDA STATE UNIVERSITY

COLLEGE OF ARTS AND SCIENCES

ALL SPEED MULTI-PHASE FLOW SOLVERS

By

SAMET Y. KADIOGLU

A Dissertation submitted to theDepartment of Mathematicsin partial fulllment of the

requirements for the degree of Doctor of Philosophy

Degree Awarded:Summer Semester, 2005


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The members of the Committee approve the Dissertation of Samet Y. Kadioglu defended

on June 13, 2005.

Mark SussmanProfessor Directing Dissertation

John TelotteOutside Committee Member

Yousuff HussainiCommittee Member

Qi WangCommittee Member

Gordon ErlebacherCommittee Member

The Office of Graduate Studies has veried and approved the above named committee members.

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To my family

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ACKNOWLEDGEMENTS

I would like to acknowledge my deep gratitude to my committee members, Yousuff

Hussaini, Qi Wang, Gordon Erlebacher, and John Telotte.

I would like to thank my thesis advisor Mark Sussman. I am indebted to him for inspiring

my interest in the eld of computational uid dynamics. His readiness and willingness to

answer my questions made this work possible. His great sense of humor and nice personality

has lead us to many fruitful and benecial discussions from which I have gained tremendously.

I also would like to thank Mimi Burbank for helping me in typesetting this thesis.

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TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. FLUID EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Momentum equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Thermodynamics Relations . . . . . . . . . . . . . . . . . . . . . . . . . 15

3. ONE DIMENSIONAL ALL SPEED METHOD . . . . . . . . . . . . . . . . . 183.1 One dimensional conservation laws . . . . . . . . . . . . . . . . . . . . . 183.2 Algorithm (The Second Order Primitive Preconditioner) . . . . . . . . . 183.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4. MULTI-DIMENSIONAL ALL-SPEED AMR METHOD . . . . . . . . . . . . 494.1 A multi-dimensional all speed multi-phase AMR (Adaptive Mesh Rene-

ment) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3 Components of the new all speed AMR technique . . . . . . . . . . . . . 544.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

A. THE CLSVOF ADVECTION ALGORITHM . . . . . . . . . . . . . . . . . . 89A.1 Computation of the uxes when s represents the level set function . . . . 91A.2 Computation of the uxes when s represents the volume fraction function 92

B. DIMENSIONAL ANALYSIS OF THE GOVERNING EQUATIONS . . . . . 94

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C. EQUIVALENCY OF THE FIRST ORDER PRIMITIVE PRECONDITIONERTO THE LOW ORDER SEMI-IMPLICIT METHOD . . . . . . . . . . . . . 96

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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LIST OF TABLES

3.1 Error analysis for Sods shock tube problem based on three levels of gridrenement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Error analysis for the zero Mach test problem based on three levels of gridrenement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1 Comparison of the number of time steps with no sub-cycling versus sub-cycling

to achieve a given nal time computation. Final run time is 15 msecs . . . . 734.2 Comparison of the run time with no sub-cycling versus sub-cycling to achieve

a given nal time computation. Final run time is 15 msecs . . . . . . . . . . 73

4.3 Comparison of the number of cells advanced per second with no sub-cyclingversus sub-cycling to achieve a given nal time computation. Final run timeis 15msecs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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3.10 Numerical results of a smooth ow test with CFL=3 at zero Mach limit. . . 37

3.11 Bubble interface (the material surface) represented by the level set function. 40

3.12 A moving boundary condition representation for pressure. . . . . . . . . . . 41

3.13 Demonstration of the convergence of the all speed primitive precondioner fordifferent grid resolutions at t = 0 .5msecs . . . . . . . . . . . . . . . . . . . . . 43

3.14 Demonstration of the convergence of the all speed primitive precondioner fordifferent grid resolutions (enlarged part of the shock front) at t = 0.5msecs . . 43

3.15 Comparison of numerical computations with benchmark results, using gridsize r = 1cm. The solid line represents the benchmark bubble radius andthe dashed line represents the solution obtained by our all speed primitivepreconditioner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.16 A closed tube in which a column of water oscillates back and forth, watercolumn initially moves from left to right with speed U . . . . . . . . . . . . . 45

3.17 Time evolution of pressure coefficients at left and right boundaries . . . . . . 47

3.18 Time evolution of relative error in total mass of air. Left gure is generatedby using the conservative explicit building block, and right gure is generatedby using the primitive explicit building block. CFL=0 .8 on left and CFL=3 .0on right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.19 Water velocity for oscillating water column problem at t = 10 with x = 180 . 48

4.1 Level set and volume-of-uid representation of the bubble interface. . . . . . 53

4.2 A properly nested grid around a front. . . . . . . . . . . . . . . . . . . . . . 55

4.3 Tagging cells around the bubble interface using the level set function. . . . . 56

4.4 Rectangular grid patch with 65% efficiency. . . . . . . . . . . . . . . . . . . . 57

4.5 Illustration of the time sub-cycling AMR algorithm with space and timerenement ratio r = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.6 Illustration of the extrapolation procedure. Figure on the left represents theone dimensional direction-wise extrapolation along the x-direction. Figure on

the right represents a multi-dimensional extrapolation which is based on theclosest cell to the interface along the normal probe direction. . . . . . . . . . 66

4.7 The synchronization step between level l + 1 and level l. The synchronizationequations are solved in the underlying level l region with the appropriateboundary data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.8 Sketch of the underwater explosion test problem . . . . . . . . . . . . . . . . 72

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4.9 Evolution of the pressure eld from the bubble jetting problem with the sub-cycling procedure off. The physical times for these gures are 0.14msecs ,0.31msecs , 0.78msecs , and 0.90msecs from the upper left to the lower right. 75

4.10 Evolution of the pressure eld from the bubble jetting problem with the sub-cycling procedure on. The physical times for these gures are 0 .14msecs ,0.31msecs , 0.78msecs , and 0.90msecs from the upper left to the lower right. 76

4.11 Evolution of the material surface from the bubble jetting problem with thesub-cycling procedure off. The physical times for these gures are 3.14msecs ,7.41msecs , 9.40msecs , and 14.98msecs from the upper left to the lower right. 77

4.12 Evolution of the material surface from the bubble jetting problem with thesub-cycling procedure on. The physical times for these gures are 3 .14msecs ,7.41msecs , 9.40msecs , and 14.98msecs from the upper left to the lower right. 78

4.13 Comparison of the bubble radius: no sub-cycling (solid line) versus sub-cycling

(dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.14 Sketch of the bubble collapse problem. . . . . . . . . . . . . . . . . . . . . . 80

4.15 Comparison of the sliced pressure proles for the bubble expansion problem.Final run time is 0 .5msecs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82



4.18 Pressure prole from the bubble expansion problem generated by an explicitmethod. Final run time is 2 .9msecs . . . . . . . . . . . . . . . . . . . . . . . 85

4.19 Pressure prole from the bubble expansion problem generated by the semi-implicit method. Final run time is 2 .9msecs . . . . . . . . . . . . . . . . . . . 86

4.20 Pressure prole from the bubble expansion problem generated by the primitivepreconditioning method. Final run time is 2 .9msecs . . . . . . . . . . . . . . 87

A.1 Representation of the grid location of the interface motion variables. . . . . . 90

A.2 Representation of the volume fraction uxes. . . . . . . . . . . . . . . . . . . 92

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ABSTRACT

A new second order primitive preconditioner technique (an all speed method) for solving

all speed single/multi-phase ow is presented. With this technique, one can compute

both compressible and incompressible ows with Mach-uniform accuracy and efficiency (i.e.,

accuracy and efficiency of the method are independent of Mach numbers). The new primitive

preconditioner (all speed/Mach uniform) technique can handle both strong and weak shocks,

providing highly resolved shock solutions together with correct shock speeds. In addition,

the new technique performs very well at the zero Mach limit. In the case of multi-phase ow,

the new primitive preconditioner technique enables one to accurately treat phase boundaries

in which there is a large impedance mismatch.

When solving multi-dimensional all speed multi-phase ows, we introduce adaptive

solution techniques which exploit the advantages of Mach-uniform methods. We compute a

variety of problems from low (low speed) to high Mach number (high speed) ows including

multi-phase ow tests, i.e, computing the growth and collapse of adiabatic bubbles for study

of underwater explosions.

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CHAPTER 1

INTRODUCTION

We are interested in solving all speed ow phenomena including multi-material ow

systems. We will say that a ow is weakly compressible if the Mach number satises

0 < M 0.2, and compressible if M > 0.2. There are many applications in uid

dynamics where compressible and weakly compressible ow occur simultaneously (all speed

ow phenomena)[1]. A possible short list of such applications could be given as (1)solving

ow problems in the inlet of internal combustion engines; the ow is weakly compressible

in the bulk of the inlet channel and compressible near the valve, (2)predicting ow elds

around aircrafts during take-offs and landings; the ow shows compressible regimes around

the wings and weakly compressible regimes around the rest of the body, and (3)(the topic of

our current research) solving adiabatic bubble growth and collapse to simulate underwater

explosions; the numerical treatment in water requires a compressible treatment around theshock front, and water behaves as a weakly compressible uid elsewhere. There exist a

variety of good numerical techniques designed specically for compressible or specically for

incompressible ows. The difficulty arises when we use such numerical techniques in the

case when both compressible and weakly compressible ows exist at the same time. For

instance, if we apply a standard density-based compressible formulation to solve weakly

compressible ows, we must expect loss of accuracy and efficiency due to weak coupling

between pressure and density [ 2, 3, 4]. Such compressible methods are generally based

on explicit time integration methods that impose severe CFL (Courant Friedreichs Lewy)restrictions for stability. Applying these kinds of schemes to low Mach number ows without

special treatment will lead to impractical computations (due to the large sound speed),

especially for three-dimensional problems. The situation is even worse when we tackle multi-

phase ow problems. Unphysical pressure oscillations and other computational inaccuracies

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are often reported at or near material interfaces [ 5, 6]. This typically occurs with large density

ratios across a material interface (e.g., 1:1000 for air-water), or with a stiff equation of

state on one side of the interface. One way to overcome these difficulties is to introduce a

unied numerical procedure for computing both compressible and incompressible ows with

Mach-uniform accuracy and efficiency (i.e., the accuracy and efficiency of the scheme are

independent of the Mach numbers from subsonic to supersonic values [ 7]). The procedure

must include an effective way of eliminating unphysical oscillations associated with multi-

material discontinuities.

There are two approaches for developing a unied numerical technique for all speed ows.

The rst approach extends the functionality of explicit methods in order to effectively handle

low Mach number ows. The second approach extends the functionality of semi-implicit (e.g.

projection methods or all-speed methods) methods in order to effectively handle ows withstrong shocks.

Of the unied methods that fall under the rst approach, the most popular methods are

the so-called explicit preconditioning techniques [ 8, 9, 10, 11]. These techniques involve

multiplying the time derivative part of the system of the compressible equations by a

suitable preconditioning matrix, and then solving the resulting equations. However there

are drawbacks. First of all, since only the time derivatives are modied, temporal accuracy

is lost. And due to their explicit nature, they are still subject to severe CFL restrictions.

Furthermore, the preconditioning matrix may become singular in the limit M 0, and thisimpairs the robustness of the computation [ 12]. An alternative unied method that attempts

to extend the functionality of an explicit approach to low-Mach number ows is based on

asymptotic expansions of the governing equations in terms of the Mach number [ 13, 14, 15].

Unfortunately, this methodology is applicable only for small Mach numbers (e.g., M 0.2).

As an alternative to unied methods which extend functionality of high Mach number

schemes to low-Mach number regimes, one can extend the functionality of a low-Mach

number, semi-implicit, scheme to be able to handle a high Mach number ow. One of

the earliest examples in this class is Harlow and Amsden [ 16]. In their original work known

as ICE (Implicit Continuous Eulerian), they pointed out that one has to separate out the

incompressible part of the ow and treat it implicitly. Implicit treatment can relax the severe

CFL restriction which is based on the uid sound speed. Detailed information about ICE

methods can be found in [16, 17, 2] and the references therein. ICE methods perform well for

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material interfaces (contact discontinuities). Alternatively, she introduced non-conservative

(primitive) models to capture oscillation-free contact discontinuities [ 5]. Later Abgrall et al

[27, 28] improved Karnis work by introducing a quasi-conservative approach. Their work is

based on the GFM (Ghost Fluid Method) of Fedkiw et al [29, 30], i.e, they solve a single-uid

problem for each uid by dening ghost points on each side, and then they reconstruct the

material interface by solving the level set equation. There are several other multi-material

formulations [31, 32, 33], but none of them including Karnis and Abgralls methods are

all speed. Now we focus our attention on all speed multi-phase ow solvers with good

shock capturing ability and good multi-material discontinuity representations. Yabe et al

[6, 34, 35, 36, 4, 37] have put substantial effort in this direction. They utilize a CIP (Cubic

Interpolated Polynomial [ 38, 35]) based time splitting predictor-corrector technique. In other

words, they separate the governing equations into advection and non-advection parts by atime splitting method [ 39]. They solve the advection terms including the interface advection

by the CIP method, then they employ a pressure-based predictor-corrector method to update

the ow variables. Their early works are based on a primitive formulation of both the uid

and interface equations [ 34, 37]. These early methods [34, 37] worked well for solving low

speed ows and gave good multi-material solutions, but they failed to calculate correct

shock speeds due to their purely primitive formulations. More recently, they added articial

viscosity [6, 4] and also adopted partially or fully conservative procedures [ 36, 40] to calculate

correct shock solutions, but they still have over and undershoots (Gibbs phenomenon) atshocks and contact discontinuities. In addition, the fully conservative approach [ 40] is

untested for low speed or multi-phase ows, and more importantly these methods are only

rst order accurate in time.

A review of the literature cited above clearly indicates that an efficient and accurate

method including good shock and multi-material discontinuity representations for all speed

multi-phase ows has not been developed yet. We know that fully conservative methods

create problems at contact discontinuities [ 5], and primitive (non-conservative) methods

fail to calculate correct shock speeds [34, 37]. On the other hand, conservative techniques

have the advantage of capturing correct shock speeds and primitive techniques are good at

representing accurate multi-material discontinuities. Thus we want to use the advantages

of both representations and introduce a new unied numerical procedure that will handle

ows at any speed and provide accurate interface representations in the case of multi-phase

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CHAPTER 2

FLUID EQUATIONS

Figure 2.1: Left gure represents a nite control volume xed in space with the uid movingthrough it. Right gure represents a nite control volume moving with the uid such thatthe same uid particles are always in the same control volume.

Consider a ow eld, represented by the streamlines in Figure 2.1. Let us imagine aclosed volume drawn within a nite region of the ow. This is dened as a control volume

with volume V and surface area S . The control volume may be either xed in space with

the uid moving through it, or moving with the uid such that the same uid particles are

always inside it[45].

Consider a ow eld represented by the streamlines in Figure 2.2. Let us imagine

an innitesimally small uid element in the ow, with volume dV . The uid element is

innitesimal in the same sense as differential calculus; however, it is large enough to contain

a huge number of molecules so that it can be viewed as a continuous medium[ 45]. The uid

element may be xed in space with the uid moving through it, or it may be moving along

a streamline with velocity V equal to the ow velocity at each point.Below we shall derive integral and differential forms of the conservation equations for

inviscid ows.

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Volume, dV Volume, dV

A

B

V,M

Figure 2.2: Left gure represents an innitesimal uid element xed in space with the uidmoving through it. Right gure represents an innitesimal uid element moving along astreamline with the velocity V equal to the ow velocity at each point.

2.1 Continuity equation

The continuity equation will be derived under the physical principle stating that mass can

be neither created nor destroyed. Let us apply this principle to the model of a xed control

volume in a ow, as illustrated in Figure 2.3. The volume is V , and the area of the closed

surface is S . First, consider point B on the control surface and an elemental area dS around

B. Let n be a unit vector normal to the surface at B. Dene dS = ndS . Also, let

V = ( u,v,w ) and be the local velocity and density at B. The mass ow through any

elemental surface arbitrarily oriented in a owing uid is equal to the product of density,

the component of velocity normal to the surface, and the area. Letting m denote the mass

y

z

x

S

dS

B

n

V

Figure 2.3: A xed control volume for derivation of the governing equations.

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ow through dS , and referring to Figure 2.3,

m = (V cos)dS = V n dS = V dS, (2.1)

where the product V n is called the mass ux

, i.e, the ow of mass per unit area per unittime. The net mass ow into the control volume through the entire control surface S is the

sum of the elemental mass ows from equation ( 2.1), namely,

S V dS, (2.2)where the minus sign denotes inow (in the opposite direction of V and dS in Figure 2.3.

Consider now an innitesimal volume dV inside the control volume. The mass of this

innitesimal volume is dV . Hence, the total mass inside the control volume is the sum

of these elemental masses, namely,

V dV . (2.3)The time rate of change of this mass inside the control volume is therefore

t V dV . (2.4)

Finally, the physical principle that mass is conserved states that the net mass ow into

the control volume must equal the rate of increase of mass inside the control volume. In

terms of the integrals given above, a mathematical representation of this statement is simply

S V dS = t V dV . (2.5)This equation is called the continuity equation ; it is the integral formulation of the

conservation of mass principle as applied to a uid ow. Equation ( 2.5) is quite general;

it applies to all ows, compressible or incompressible, viscous or inviscid [45, 46, 47].

To obtain a differential equation for the conservation of mass principle, we will make useof the following vector identities:

S A dS = V ( A )dV , (2.6)8
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S dS = V ( )dV , (2.7)where A and are vector and scalar functions, respectively, of time and space.

Using equation ( 2.6) in the form

S (V ) dS = V (V )dV , (2.8)we combine equation ( 2.5) and( 2.8) to obtain

V [t + (V )]dV = 0 . (2.9)Since, the control volume is an arbitrary shape and size, and in general the only way

equation ( 2.9) can be satised is for the integrand to be zero at each point within the

volume. Hence,

t

+ (V ) = 0 . (2.10)

Equation ( 2.10) is the differential form of the continuity equation .

2.2 Momentum equation

The momentum equation will be derived under the physical principle stating that the time

rate of change of momentum of a body equals the net force exerted on it. Written in vector

form, the above statement becomes

ddt

(mV ) = F , (2.11)

where m, V , and F represent mass, velocity, and force respectively.

For constant mass, equation ( 2.11) yields

F = m dVdt

= ma , (2.12)

which is the more familiar form of Newtons second law, namely, that force = mass

acceleration . However, the above physical principle with equation ( 2.11) is a more general

statement of Newtons second law than equation ( 2.12). In here, we will put Newtons second

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law equation ( 2.11) in uid mechanic terms by employing the same control volume utilized

in the previous section and sketched in Figure 2.3.

First, we consider two types of forces acting on the control volume:

1. Body forces acting on the uid inside V . These forces stem from action at a

distance, such as gravitational and electromagnetic forces that may be exerted on the uid

inside V due to force elds acting through space. Let f represent the body force per unit

mass of uid. Considering an elemental volume, dV , inside V , the elemental body force on

dV is equal to the product of its mass and the force per unit mass, namely, ( dV )f [45, 47].

Hence, summing over the complete control volume,

Total body force = V f dV . (2.13)2. Surface forces acting on the boundary of the control volume. Surface forces in a

uid stem from two sources: pressure and shear stress distributions over the surface. Since

we are dealing with inviscid ows here (i.e, we solve problems that require inviscid model

(Appendix B)), the only surface force is therefore due to pressure. Consider the elemental

area dS sketched in Figure 2.3. The elemental surface force acting on this area is pdS,

where the minus sign signies that pressure acts inward, opposite to the outward direction

of the vector dS. Hence, summing over the complete control surface,

Total surface force due to pressure = S pdS. (2.14)We note that the sum of equation ( 2.13) and (2.14) gives F in equation (2.11). That is,at any given instant time, the total force F acting on the control volume is

F = V f dV S pdS. (2.15)Now, consider the left-hand side of equation ( 2.11). Since the control volume (Figure

2.3) is xed in space, mass ows into the control volume from the left at the same time

that other mass is streaming out toward the right. The mass owing in brings with it a

certain momentum. At the same time, the mass owing out also has momentum. With

this picture in mind, let A1 represent the net rate of ow of momentum across the surface

S . The elemental mass ow across dS is V dS. Associated with this elemental mass

ow is a momentum ow (or ux) (V dS)V . Note from Figure 2.3 that the direction

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of V is away from the control volume physically represents an outow of momentum and

mathematically represents a positive value of V dS. Conversely, the direction of V is

toward the control volume physically represents an inow of momentum and mathematically

represents a negative value of V dS. The net rate of ow of momentum, summed over the

complete surface S , is

A 1 = S (V dS)V . (2.16)In an unsteady ow, A 1 does not represent the whole contribution to the left-hand side of

equation ( 2.11). There is, in addition, a time rate of change of momentum due to unsteady,

transient effects in the ow eld inside V . Let A2 represent this uctuation in momentum.

Also consider an elemental mass of uid, dV . This mass has momentum ( dV )V . Summing

over the complete control volume V , we have

Total momentum inside V = V VdV . (2.17)Hence, the change in momentum in V due to unsteady uctuations in the local ow

properties is

A 2 = t V VdV = V (V )t dV . (2.18)

Finally, the sum A1 + A2 represents the total instantaneous time rate of change of momentum of the uid as it ows through the control volume. This is the uid mechanical

counterpart of the left-hand side of equation ( 2.11), i.e,

ddt

(mV ) = A 1 + A 2 = S (V dS)V + V (V )t dV . (2.19)Therefore, the physical principal, the time rate of change of momentum of the uid that

is owing through the control volume at any instant is equal to the net force exerted on the

uid inside the volume, gives:

S (V dS)V + V (V )t dV = V f dV S pdS. (2.20)Equation ( 2.20) is called the momentum equation . It is the integral formulation of

Newtons second law applied to inviscid uid ows [45, 46, 47].

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We obtain the differential form of the momentum equation by the following steps: rst

we write equation ( 2.7) in the form

S

pdS =

V

( p)dV , (2.21)

then we combine equations ( 2.20) and (2.21) to obtain

V f dV V ( p)dV = V (V )t dV + S (V dS)V . (2.22)Equation ( 2.22) is a vector equation; for convenience, let us consider cartesian scalar

components in the x, y, and z directions, respectively. The x component of equation ( 2.22)

is

V f x dV V px dV = V (u)t dV + S (V dS)u. (2.23)However from equation (2.6), S (V dS)u = S (uV ) dS = V (uV )dV . (2.24)

Substituting equation ( 2.24) into (2.23),

V

[f x p

x

(u)

t (uV )]dV = 0. (2.25)

By the same reasoning used to obtain the continuity equation, equation ( 2.25) yields

(u)t

+ (uV ) = px

+ f x . (2.26)

Equation ( 2.26) is the differential form of the x component of the momentum equation [45,

46, 47]. The analogous y and z components are

(v)

t

+ (vV ) = p

y

+ f y , (2.27)

(w)t

+ (wV ) = pz

+ f z . (2.28)

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2.3 Energy equation

We derive the energy equation by following the physical principle that energy can be neither

created nor destroyed; it can only change in form. Let

B1 = rate of heat added to the uid inside the control volume from the surroundingsB2 = rate of work done on the uid inside the control volumeB3 = rate of change of the energy of the uid as it ows through the control

volume.By applying the rst law of the thermodynamics, we have

B1 + B2 = B3. (2.29)

The rate of heat added to the control volume can be handled by rst dening q to be the

rate of heat added per unit mass, and then writing the rate of heat added to an elementalvolume as q (dV ). Summing over the complete control volume,

B1 = V qdV . (2.30)Now consider the elemental area dS of the control surface in Figure 2.3. The pressure

force on this elemental area is pdS. The rate of work done on the uid passing through dS

with the velocity V is ( pdS) V . Hence, summing over the complete control surface,

Rate of work done onthe fluid inside V due = S ( pdS) Vto pressure forces on S (2.31)

In addition, consider an elemental volume inside the control volume. Recalling that f is

the body force per unit mass, the rate of work done on the elemental volume due to body

force is (f dV ) V . Summing over the complete control volume,

Rate of work done onthe fluid inside V due =

V (f dV ) V

to body forces(2.32)

Thus, the total work done on the uid inside the control volume is the sum of equations

(2.31) and (2.32),

B2 = S pV dS + V (f V )dV . (2.33)13
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The energy per unit mass of the moving uid is the sum of both internal and kinetic

energies, e + V 2/ 2. The mass owing in brings with it a certain energy, while at the same

time the mass owing out also has energy. The elemental mass ow across dS is given by

V dS and therefore the elemental ux of energy across dS is (V dS)(e+ V 2/ 2). Summing

over the complete control surface,

Net rate of flowof energy across = S (V dS)(e +

V 22 )

the control surface(2.34)

However, this is not necessarily the total energy change inside the control volume. If the

ow is unsteady there is also a rate of change of energy due to local transient uctuations

of the ow eld variables inside the control volume. The energy of an elemental volume is

(e + V 2/ 2)dV , and hence the energy inside the complete control volume at any instant in

time is

V (e + V 2

2 )dV . (2.35)

Therefore,

Time rate of changeof energy inside V due = t V (e +

V 22 )dV .

to transient variationsof the flowfield variables

(2.36)

In turn, B3 is the sum of equations (2.34) and (2.36):

B3 = t V (e + V

2

2 )dV + S (V dS)(e + V

2

2 ) (2.37)

Recalling the physical principle stating that the rate of heat added to the uid plus the

rate of work done on the uid is equal to the rate of change of energy of the uid as it ows

through the control volume ( B1 + B2 = B3), i.e, energy is conserved , we have

V qdV S pV dS + V (f V )dV = V t [(e+ V 2

2 )]dV + S (e+ V

2

2 )V dS.

(2.38)

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Equation ( 2.38) is called the energy equation . It is the integral formulation of the rst

law of thermodynamics applied to an inviscid uid ow [45, 46, 47].

To obtain a differential equation for the conservation of energy, we rst write equation

(2.6) in the forms

S (e + V 2

2 )V dS = V [(e + V

2

2 )V ]dV , (2.39)

and

S pV dS = V ( pV )dV . (2.40)We then combine equations ( 2.38),(2.39), and ( 2.40) to obtain

V {q ( pV ) + (f V ) t [(e + V 2

2 )] [(e +

V 2

2 )V ]}dV = 0 (2.41)

Setting the integrand equal to zero, we obtain

t

[(e + V 2

2 )] + [(e +

V 2

2 )V ] = ( pV ) + q + (f V ) (2.42)

Equation ( 2.42) is the differential form of the energy equation .

2.4 Thermodynamics Relations

Thermodynamics is an essential ingredient in the study of compressible ow, since energy

concepts play a major role in the understanding of compressible ow phenomena. This

section gives a brief outline of thermodynamics concepts and relations.

A gas is a collection of particles (molecules, atoms, ions, electrons, etc,) which are in

random motion. Due to the electronic structure of these particles, a force eld pervades

the space around them. The force eld due to one particle reaches out and interacts with

neighboring particles is called intermolecular force . By denition, a perfect gas is one in

which intermolecular forces are neglected[ 45](A real gas is one in which intermolecular forces

are taken into account. In this section we will not consider real gases). For a large number of

engineering applications, the effect of intermolecular forces on the gas properties is negligible.

By ignoring intermolecular forces, the equation of state for a perfect gas can be derived

from the theoretical concepts of modern statistical mechanics of kinetic theory. However,

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historically it was rst synthesized from laboratory measurements. The empirical result from

the laboratory observation was

p = RT, (2.43)

where p is pressure, is the specic volume, R is the universal gas constant, and T is thetemperature. Since the density = 1/ , equation ( 2.43) becomes

p = RT. (2.44)

In the view of a gas as a collection of particles in random motion, the individual kinetic

energy of each particle contributes to the overall energy of the gas. These energies, summed

over all the particles of the gas, constitute the internal energy of the gas [45, 47]. Let e

denote the specic internal energy (internal energy per unit mass), then the enthalpy (per

unit mass) is dened as

h = e + p. (2.45)

and we have

e = e(T, )

h = h(T, p). (2.46)

If the gas is not chemically reacting, and if we ignore intermolecular forces, the resulting

system is a thermally perfect gas , where internal energy and enthalpy are functions of

temperature only, and where the specic heats at constant volume and pressure, cv and

c p, are also functions of temperature only:

e = e(T )

h = h(T )

de = cvdT

dh = c pdT. (2.47)

If the specic heats are constant, the system is a calorically perfect gas , where

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e = cvT

h = c pT. (2.48)

In many compressible ow applications, the pressures and temperatures are moderateenough that the gas can be considered to be calorically perfect. Using equations ( 2.43),

(2.45), and ( 2.48) we write,

c p cv = R. (2.49)

Equation ( 2.49) holds for a calorically perfect or a thermally perfect gas. It is not valid

for either a chemically reacting or a real gas. Some useful forms of equation ( 2.49);

c p = R 1, (2.50)

and

cv = R 1

, (2.51)

where = c p/c v is the specic heat ratio. For a calorically perfect or a thermally perfect

gas, by equation ( 2.51) and (2.48), we have

e = p

( 1), (2.52)

and using (2.52) in the denition of the total energy, we write

E = e + 12

u2. (2.53)

Yet another fundamental thermodynamics quantity is the entropy. Rougly speaking,

entropy measures the disorder in the system, and indicates the degree to which the internal

energy is available for doing useful work. The greater the entropy, the less available the

energy [48]. The specic entropy s (entropy per unit mass) is given by

s = cv log( p/ ) + constant. (2.54)

Euler equations can be manipulated to derive the relation,

s t + u s = 0, (2.55)

which says that entropy is constant along particle paths in regions of smooth ow.

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CHAPTER 3

ONE DIMENSIONAL ALL SPEED METHOD

3.1 One dimensional conservation laws

Our numerical computations will be based on inviscid (see an example that justies an

inviscid model assumption (Appendix B)) compressible Euler equations with the assumption

that the ow system is adiabatic and body forces are neglected. Then the one dimensional

conservation equations derived in Chapter 2 can be written as:

U t

+ F (U )

x = 0 , (3.1)

where

U =

u

E

,

and

F (U ) =u

u2 + p(E + p)u

,

where ,u,p, and E denote density, velocity, pressure, and the total energy per unit volume

respectively, E satises E = p 1 + 12 u

2 with denoting the specic heat ratio.

3.2 Algorithm (The Second Order PrimitivePreconditioner)

We give the general structure of our second order primitive preconditioner algorithm which

is based on an implicit correction of an explicit building block of the Euler equations.

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2 t2 [ pn +1 ,k

n +1 ,k 1]

1(cn +1 ,k 1)2n +1 ,k 1

pn +1 ,k = t[ u exp,k u n +1 ,k 1]

1

(cn +1 ,k

1)2n +1 ,k

1 pexp,k

+ 2 t2 [ pn +1 ,k 1

n +1 ,k 1 ]. (3.5)

After solving the elliptic system 3.5, we update the velocity eld. Using the newly

computed pressure and velocity elds, we then compute the other primitive variables using

the following approximations,

n +1 ,k exp,k

t = n +1 ,k 1[ u n +1 ,k u n +1 ,k 1], (3.6)

en +1 ,k eexp,k

t =

pn +1 ,k 1

n +1 ,k 1[ u n +1 ,k u n +1 ,k 1], (3.7)

and nally we update the momentum and total energy by using

mn +1 ,k = n +1 ,k u n +1 ,k , (3.8)

E n +1 ,k = n +1 ,k en +1 ,k + 1

2

n +1 ,k |u n +1 ,k |2. (3.9)

3.2.1 Example of a conservative explicit building block

In this section, we describe how to obtain the explicit terms (the ( exp,k)th values)

using the conservative explicit building block in preparation for the second order primitive

preconditioner algorithm. The conservative explicit building block is ideal for all speed ows

with shock capturing. The conservative explicit solver uses local Lax Friedrich numerical

uxes together with second order ENO in order to obtain shock solutions with high resolution.

Our method is in the same spirit as the second order ENO central schemes developed by [ 49]in that we do not use eld-by-eld decomposition in order to extrapolate the characteristic

variables. We remark that our proposed preconditioner is exible in that we can use as an

alternative explicit building block, the less diffusive ENO/WENO schemes [ 50, 51, 52, 53, 54]

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or third order central schemes [ 55]. We consider the one dimensional Euler equations ( 3.1)

and use the following conservative discretization,

U (exp,k )i U (n +1 ,0)i

t =

F LLF,( n +1 ,k 1)+( n +1 , 0)

2i+ 12

F LLF,( n +1 ,k 1)+( n +1 , 0)

2i 12

x , (3.10)where superscript (n +1 ,k 1)+( n +1 ,0)2 represents the simple average of the ( k 1)

th iterate and

the initial values of related ow variables, and F LLF i+ 12 denotes the local Lax Friedrich numerical

ux function, i.e,

F LLF,( n +1 ,k 1)+( n +1 , 0)

2i+ 12

= F LLF (U R,( n +1 ,k 1)+( n +1 , 0)

2i+ 12

, U L,( n +1 ,k 1)+( n +1 , 0)

2i+ 12

) (3.11)

= 1

2[F (U R,

( n +1 ,k 1)+( n +1 , 0)2

i+

12

) + F (U L,( n +1 ,k 1)+( n +1 , 0)

2i+

12

)]

12

i+ 12 [U R, ( n +1 ,k

1)+( n +1 , 0)2

i+ 12 U L,

( n +1 ,k 1)+( n +1 , 0)2

i+ 12].

Here

i+ 12 = max p | p(L,R )i+ 12

|, p = 1 , 2, 3 (3.12)

with p(L,R )i+ 12 s denoting the left or right values of the eigenvalues of the system, and U Li+ 12

,

U Ri+ 12 represent the left and right values of the state variables respectively. U Li+ 12

and U Ri+ 12are dened as

U Li+ 12 = U i + x

2 U x,i , (3.13)

U Ri+ 12 = U i+1 x

2 U x,i +1 , (3.14)

where

U x,i = U i +1 U i

x if |U i+1 U i | < |U i U i 1 |U i U i 1

x otherwise. (3.15)

3.2.2 Example of a non-conservative explicit building block

In this section, we describe how to obtain the explicit terms using the non-conservative

explicit building block in preparation for the second order primitive preconditioner. The

non-conservative explicit building block is ideal when we solve zero Mach limit ows which

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are dominated by acoustic waves or ows that contains no shock discontinuity. The main

idea is to give up the conservative form of the Euler equations in the explicit step whenever

there is no shock discontinuity in the ow. Instead, we solve the primitive Euler equations in

the explicit step, and then we apply the implicit correction algorithm. The primitive Euler

equations are

t + ux + ux = 0 (3.16)

u t + uu x + 1 px = 0 (3.17)

pt + upx + c2ux = 0. (3.18)

We discretize the primitive Euler equations as

exp,ki n +1 ,0i

t = u

( n +1 ,k 1)+( n +1 , 0)2

i

< >( n +1 ,k 1)+( n +1 , 0)

2i+ 12

< >( n +1 ,k 1)+( n +1 , 0)

2i 12

x (3.19)

( n +1 ,k 1)+( n +1 , 0)

2i

u( n +1 ,k 1)+( n +1 , 0)

2i+ 12

u( n +1 ,k 1)+( n +1 , 0)

2i 12

x ,

uexp,ki un +1 ,0i t

= u( n +1 ,k 1)+( n +1 , 0)

2i

< u >

( n +1 ,k 1)+( n +1 , 0)2

i+ 12 < u >

( n +1 ,k 1)+( n +1 , 0)2

i 12

x (3.20)

1

( n +1 ,k 1)+( n +1 , 0)

2i

p( n +1 ,k 1)+( n +1 , 0)

2i+1 p

( n +1 ,k 1)+( n +1 , 0)2

i 1

2 x ,

and

pexp,ki pn +1 ,0i

t = u

( n +1 ,k 1)+( n +1 , 0)2

i

< p >( n +1 ,k 1)+( n +1 , 0)

2i+ 12

< p >( n +1 ,k 1)+( n +1 , 0)

2i 12

x (3.21)

c2i ( n +1 ,k 1)+( n +1 , 0)

2i

u( n +1 ,k 1)+( n +1 , 0)

2i+ 12

u( n +1 ,k 1)+( n +1 , 0)

2i 12

x .

Here

u( n +1 ,k 1)+( n +1 , 0)

2i+ 12

= u

( n +1 ,k 1)+( n +1 , 0)2

i+1 + u( n +1 ,k 1)+( n +1 , 0)

2i

2 , (3.22)

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Let L p be the numerical operator for the pressure correction equation, then we have

L p( p, u, ) = pn +1 pexp t(cn )2n [ u n +1 u n ] 0. (3.28)

Let L p be an analytical operator corresponding to ( 3.28), and P,U, represent the

analytical solutions, then the local truncation error is dened as

= L p(P,U, ) L p( p, u, ). (3.29)

Using (3.26) and the following Taylor series in ( 3.29)

U n +1 = U n + tU nt + t2

2 U ntt + . . . , (3.30)

we obtain,

n

= O( t2

) (3.31)This shows the scheme that computes the new pressure is rst order accurate in time.Similarly, letting Lu and Lu be the numerical and analytical operators for the velocity

correction equation (equation 3.25), and using the equation ( 3.27) together with the following

series expansion,

P n +1 = P n + tP nt + t2

2 P ntt + . . . , (3.32)

we obtain,

= Lu (P,U, ) L u ( p, u, ) = O( t2). (3.33)

In similar way, it can be easily seen that the density correction scheme ( 3.6) is also rst

order in time for k = 1.Now we shall show that the primitive preconditioner is second order in time for k = 2

and = 1. Representing ( n + 1, 2) values with (n + 1) and ( n + 1 , 1) values with ( ), we

have the following equations

pn +1 pexp

t = (c )2 [ u n +1 u ], (3.34)

and

u n +1 u exp

t =

1

[ pn +1 p ]. (3.35)

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We assume that the explicit building block provides second order accurate approximations

to the analytical solutions at this stage, in other words we have

pexp = P n +1 + O( t3), (3.36)

uexp = U n +1 + O( t3). (3.37)

We know that at the rst sweep, ( ) values are rst order approximations to the analytical

solutions, i.e,

p = P n +1 + O( t2), (3.38)

u = U n +1 + O( t2). (3.39)

Using equations (3.36), (3.37), (3.38), and ( 3.39) and following the similar analysis given

above, we obtain

= L p(P,U, ) L p( p, u, ) = O( t3), (3.40)

= Lu (P,U, ) L u ( p, u, ) = O( t3). (3.41)

This shows that the schemes for calculating the new pressure and velocity are second

order accurate in time. it can be easily seen that the density correction scheme ( 3.6) is also

second order in time for k = 2.

Remark : Only few stability analyses exist for Mach uniform algorithms. For instance

van der Heul et al [56] carried out a stability analysis based on Fourier analysis of the coupled

system, that gives a prediction for the maximum allowable time step for both (semi)-implicit

and explicit methods as a function of the Mach number. Similar analysis could be applied

here to show the stability of our scheme in terms of the Mach and CFL numbers.

3.3 Numerical results

The numerical examples are presented in two different subsections. The rst part is

dedicated to single-uid ow problems ranging from low to high Mach numbers. In the

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second part, we provide numerical results of all speed multi-phase ow computations. For

calculations in Section 3.3.1.1 through Section 3.3.1.7 we used k = 3 (the number of

iterations) and for all other calculations, we used k = 2. We used = 1 for all computations

in this thesis.

3.3.1 Application of the second order primitive preconditionertechnique to single-uid ow problems

For the single-uid ow case, we solve similar test problems as in [40, 7] in order to explore

the resolution gained by our proposed preconditioner.

3.3.1.1 Sods shock tube problem

The shock tube problem considers a long, thin, cylindrical tube containing a gas separated

by a thin membrane. The gas is assumed to be at rest on both sides of the membrane, but

it has different constant pressures and densities on each side. At time t = 0, the membrane

is ruptured, and the problem is to determine the ensuing motion of the gas. This problem

was rst studied by Riemann, and known as the Riemann problem. The solution to this

problem consists of a shock wave moving into the low pressure region, a rarefaction wave

that expands into the high pressure region, and a contact discontinuity which represents the

interface.

Table 3.1: Error analysis for Sods shock tube problem based on three levels of gridrenement.

k = 2 iterationsNumber of mesh points 100 200 400Error in total mass 5 .99 10 4 3.54 10 4 2.30 10 4Error in total energy 2 .47 10 3 1.75 10 3 1.38 10 3

k = 5 iterationsNumber of mesh points 100 200 400Error in total mass 2 .85 10 5 2.46 10 5 2.42 10 5Error in total energy 1 .99 10 4 1.88 10 4 1.87 10 4

The shock tube problem of Sod [57] considers the Riemann problem with the following

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initial data on the interval 0 x 1,

((x, 0), u(x, 0), p(x, 0)) = (1, 0, 1) if x 0.5(0.125, 0, 0.1) if x > 0.5. (3.42)

When producing computational results for this test problem, we used 400 grid cells and

the CFL ( |u|max t < CFL x) number is 0.3. We used the conservative explicit building block

given in Section 3.2.1. Our numerical results indicate higher resolved solutions for a given

time step and given mesh size than the numerical results reported in [ 40, 7]. Our results,

shown in Figure 3.1, have no spurious oscillations at any shock or contact discontinuities.

A conservation study is done for the mass and total energy with different mesh resolutions.

Table 3.1 shows that the conservation errors are within acceptable ranges. Table 3.1 also

shows that the conservation property of our method improves when more iterations are

performed.

3.3.1.2 Laxs shock tube problem

We present the calculations for the Riemann problem,

((x, 0), u(x, 0), p(x, 0)) = (0.445, 0.698, 3.528) if x 0.5(0.5, 0, 0.571) if x > 0.5, (3.43)

used by Lax in [58]. The computational domain is taken to be [0 , 1]. The grid resolution is 400

cells and the time stepping condition is CFL = 0 .3 (|u|max t < CFL x). The conservative

explicit building block (Section 3.2.1) is used for this problem. Again as in the previous test

case, our results indicate higher resolution than [ 7] and comparable resolution to [ 40] for a

given mesh size and time step. Figure 3.2 displays well resolved shock and contact solutions.

3.3.1.3 Strong shock tube problem

Here we present a strong shock tube problem with the following initial condition,

((x, 0), u(x, 0), p(x, 0)) = (1, 0, 1010) if x 0.5

(0.125, 0, 0.1) if x > 0.5. (3.44)

The computational domain is taken as the interval [0 , 1]. For this problem, we use the

conservative explicit building block (Section 3.2.1). As noted in [40], the initial condition( 3.3)

creates a supersonic shock associated with extreme jumps in velocity and pressure. It is well

known that purely non-conservative schemes fail to compute strong shocks due to their

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0 0.5 10

0.2

0.4

0.6

0.8

1DENSITY

0 0.5 10

0.5

1VELOCITY

0 0.5 10

0.2

0.4

0.6

0.8

1PRESSURE

0 0.5 10

0.2

0.4

0.6

0.8

1MACH NUMBER

0 0.5 12.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4ENTHALPY

0 0.5 10.8

1

1.2

1.4

1.6

1.8

2ENTROPY

Figure 3.1: Numerical results of Sods shock tube problem at t = 0.15. Solid lines representthe reference solutions and dashed lines represent the Mach uniform results.

intrinsic inability to calculate correct shock speeds. This is also discussed in [ 40]. Xiao

[40] computed correct shocks for this problem, yet there are still density overshoots in his

results. Figure 3.3 shows the numerical results at t = 2.5 10 6. We used 400 grid cells

during the computation while keeping the CFL ( |u|max t < CFL x) at 0.3 as in [40]. As

can be seen from Figure 3.3, there are no overshoots at any place while obtaining correct

shock calculations.

3.3.1.4 Mach 3 shock test

The Mach 3 shock tube experiment uses the following initial conditions,

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0 0.5 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6DENSITY

0 0.5 10

1

2VELOCITY

0 0.5 10.5

1

1.5

2

2.5

3

3.5

4PRESSURE

0 0.5 10

0.2

0.4

0.6

0.8

1MACH NUMBER

0 0.5 10

5

10

15

20

25

30ENTHALPY

0 0.5 10

2

4

6

8

10

12ENTROPY

Figure 3.2: Numerical results of Laxs shock tube problem at t = 0.12. Solid lines representthe reference solutions and dashed lines represent the Mach uniform results.

((x, 0), u(x, 0), p(x, 0)) = (3.857, 0.92, 10.333) if x 0.5(1, 3.55, 1) if x > 0.5. (3.45)

Here we use the conservative explicit building block (Section 3.2.1) with 400 grid cells and

the CFL ( |u|max t < CFL x) number 0 .5 on the domain of [0, 1] to produce the numerical

results. Figure 3.4 indicates that our numerical results fairly compare to [ 40].

3.3.1.5 High Mach ow test

The High Mach number shock tube problem uses the following initial data on the interval

0 x 1,

((x, 0), u(x, 0), p(x, 0)) = (10, 2000, 500) if x 0.5(20, 0, 500) if x > 0.5. (3.46)

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0 0.5 10

0.2

0.4

0.6

0.8

1DENSITY

0 0.5 12

0

2

4

6

8

10

12x 10

4 VELOCITY

0 0.5 10

2

4

6

8

10x 10

9 PRESSURE

0 0.5 10

0.5

1

1.5

2MACH NUMBER

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5x 10

10 ENTHALPY

0 0.5 10

2

4

6

8

10

12x 10

9 ENTROPY

Figure 3.3: Numerical results of a strong shock tube problem at t = 2.5 10 6. Solidlines represent the reference solutions and dashed lines represent the results of the all speedprimitive preconditioner.

The number of grid cells are 400 and the CFL ( |u|max t < CFL x) number is 0.4 for

this problem. The conservative explicit building block (Section 3.2.1) is used here. Figure

3.5 demonstrates well resolved shocks with correct shock locations and less smeared contact

discontinuities than [ 40]. Here the Mach number can reach 240. This problem illustrates

the robustness and stability of our method, even in the case of extremely high Mach number

(high speed) ows.

3.3.1.6 Interaction of blast waves

In this subsection we present the numerical results for the problem of two interacting

blast waves. This problem, introduced by Woodward and Colella [ 59], involves multiple

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0 0.5 10.5

1

1.5

2

2.5

3

3.5

4DENSITY

0 0.5 10.5

1

1.5

2

2.5

3

3.5

4VELOCITY

0 0.5 10

2

4

6

8

10

12PRESSURE

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5MACH NUMBER

0 0.5 13

4

5

6

7

8

9

10ENTHALPY

0 0.5 12

3

4

5

6

7INTERNAL ENERGY

Figure 3.4: Numerical results of a Mach 3 shock tube problem at t = 0.09. Solid linesrepresent the reference solutions and dashed lines represent the results of the all speedprimitive preconditioner.

interactions of strong shock waves and other discontinuities. Initial conditions are

((x, 0), u(x, 0), p(x, 0)) =(1, 0, 103) if 0 x < 0.1(1, 0, 10 2) if 0.1 x < 0.9(1, 0, 102) if 0.9 x < 1.

(3.47)

The boundary conditions at x = 0 and x = 1 are reective solid wall conditions , i.e, we

dene auxiliary states vn0 ,...,v

n r +1 for the left wall and v

nM +1 ,...,v

nM + r for the right wall by

n i+1 = ni , u

n i+1 = u

ni , p

n i+1 = p

ni , for i = 1,...,r (3.48)

nM + i = nM i+1 , u

nM + i = u

nM i+1 , p

nM + i = p

nM i+1 , for i = 1,...,r (3.49)

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0 0.5 10

20

40

60

80

100

120DENSITY

0 0.5 1500

0

500

1000

1500

2000VELOCITY

0 0.5 10

0.5

1

1.5

2x 10

7 PRESSURE

0 0.5 10

50

100

150

200

250MACH NUMBER

0 0.5 10

2

4

6

8

10x 10

5 ENTHALPY

0 0.5 10

1

2

3

4

5

6x 10

4 ENTROPY

Figure 3.5: Numerical results of a high Mach number shock tube problem at t = 1.75 10 4.Solid lines represent the reference solutions and dashed lines represent the results of the allspeed primitive preconditioner.

To make a consistent comparison with Xiaos [ 40] paper, we used 400 grid cells. 1600

cells were used to produce a numerical reference solution. We used the conservative explicit

building block with CFL ( |u|max t < CFL x) number 0 .3 for this test problem. Figure 3.6

shows that we achieved more accurate results compared to [ 40] and observed no spurious

oscillations which were noted in [40]. Here we note that there have been many reported higher

resolution results for this problem, e.g, using ENO [ 50, 51, 52], but these previous results

were based on using purely explicit non-preconditioned methods. We have observed with

our new preconditioner that shock resolution is not adversely effected by the preconditioner

itself. For instance if one uses high order ENO/WENO schemes [ 50, 51, 52, 53, 54] as the

explicit building block, then we expect that our primitive preconditioning technique would

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produce similar shock resolution as explicit ENO/WENO methods [ 50, 51, 52, 53, 54].

0 0.5 10

1

2

3

4

5

6

7DENSITY

0 0.5 15

0

5

10

15VELOCITY

0 0.5 10

100

200

300

400

500PRESSURE

0 0.5 10

0.5

1

1.5

2

2.5MACH NUMBER

0 0.5 10

500

1000

1500

2000ENTHALPY

0 0.5 10

200

400

600

800

1000

1200ENTROPY

Figure 3.6: Numerical results of two interacting blast waves problem at t = 0.038. Solid

lines represent the reference solutions and dashed lines represent the results of the all speedprimitive preconditioner.

A justication for why the primitive preconditioner does not adversely effect the resolu-

tion of the explicit building block can be seen by considering the implicit correction equations

in Section 3.2, i.e, consider the pressure correction equation,

pn +1 ,k pexp,k

t = c2 U

n +1 ,k(3.50)

where pexp is produced by an explicit building block. We observed that the difference term

U gets very small with an increasing number of iterations. This means that pn +1 gets

closer to pexp (i.e, U 0 as k , then pn +1 pexp ). This tells us that the implicit

correction step spatially preserves the accuracy of the explicit building block. In other words,

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if one provides a high order explicit building block, then our method will spatially maintain

the high order of accuracy of the given explicit building block.

3.3.1.7 Two symmetric rarefaction waves

0 0.5 10

0.2

0.4

0.6

0.8

1DENSITY

0 0.5 12

1.5

1

0.5

0

0.5

1

1.5

2VELOCITY

0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4PRESSURE

0 0.5 10

0.5

1

1.5

2

2.5

3MACH NUMBER

0 0.5 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1INTERNAL ENERGY

0 0.5 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6ENTHALPY

Figure 3.7: Numerical results of two symmetric rarefaction waves problem at t = 0 .15. Solidlines represent the reference solutions and dashed lines represent the results of the all speedprimitive preconditioner.

Here we present a test problem that has two symmetric rarefaction waves. The initial

conditions, on the 0 x 1 interval, are

((x, 0), u(x, 0), p(x, 0)) = (1, 2, 0.4) if x 0.5(1, 2, 0.4) if x > 0.5. (3.51)

Our results are comparable to Xiaos [ 40] results. Figure 3.7 shows that there are some

computational inaccuracies observed in the ow elds as in [ 40], when using the conservative

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0 0.5 10

0.2

0.4

0.6

0.8

1DENSITY

0 0.5 12

1.5

1

0.5

0

0.5

1

1.5

2VELOCITY

0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4PRESSURE

0 0.5 10

0.5

1

1.5

2

2.5

3MACH NUMBER

0 0.5 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1INTERNAL ENERGY

0 0.5 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6ENTHALPY

Figure 3.8: Numerical results of the two symmetric rarefaction waves problem computed bythe second version (preconditioning an explicit building block based on the primitive Eulerequations) of the algorithm at t = 0.15. Solid lines represent the reference solutions anddashed lines represent the results of the all speed primitive preconditioner.

explicit building block. For instance, we have an overestimation in the pressure prole near

the expansion center, an unphysical pulse in the internal energy eld near the center of

the low pressure region, and inaccurate velocity representation. These inaccuracies are

improved signicantly (Figure 3.8) by using the second version of our algorithm. For

instance, we precondition the non-conservative Euler equations (using the non-conservative

explicit building block, Section 3.2.2) during the implicit correction step. We note that theimprovements when using the non-conservative explicit building block are due to the fact

that there doesnt exist shock discontinuities in the ow.

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3.3.1.8 Smooth ow test (Mach zero limit)

In this section, we show the results of the zero Mach limit ows which are dominated by

acoustic waves. We use the following smooth functions as initial conditions,

u(x, 0) = 0 (3.52)

p(x, 0) = p0 + p 1(x),

where p0 = 106, = 1.0, and

p1(x) = 60 cos(2x ) + 100 sin(4x ). (3.53)

We initialize the density eld by the following isentropic relation,

(x, 0) = p(x, 0)A

1

, (3.54)

where A is a constant which is determined by letting p0 = A0 with 0 = 10 3.

Here we use periodic boundary conditions on 0 x 1. The numerical results are

produced by applying the second version of our algorithm where we precondition the non-

conservative explicit building block (Section 3.2.2). Preconditioning the conservative explicit

building block works ne as shown in Section 3.3.2.4, but the computation would be less

efficient due to the explicit CFL restriction. Preconditioning the non-conservative explicit

building block enables us to use larger time steps, i.e, CFL= 3 (( |u| + c)max t < CFL x).

Figure 3.9 and 3.10 together with Table 3.2 clearly demonstrate the second order convergence

of our newly proposed scheme.

Table 3.2: Error analysis for the zero Mach test problem based on three levels of gridrenement.

Error in acoustic pressure Error in acoustic pressure Order of accuracy

from 200 to 400 mesh from 400 to 800 meshl1 norm 48.70 8.93 2.45l norm 27.94 5.12 2.45

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0 0.2 0.4 0.6 0.8 10.9999

0.9999

0.9999

1

1

1

1

1

1.0001

1.0001

1.0001x 10

6 PRESSURE

800 grid cells400 grid cells200 grid cells

Figure 3.9: Numerical results of a smooth ow test with CFL=3 at zero Mach limit.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2x 10 4 MACH NUMBER

800 grid cells400 grid cells200 grid cells

Figure 3.10: Numerical results of a smooth ow test with CFL=3 at zero Mach limit.

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3.3.2 Application of the second order primitive preconditionertechnique to multi-phase ow problems

In this section, we present the application of the second order primitive preconditioner

technique to multi-phase ow problems. The major part of this section is dedicated to the

underwater explosion problem. In the rst three subsections, the problem description of

an underwater explosion, its numerical procedure, and some computational results will be

given. In the last subsection, the oscillating water column problem in a 1-D tube and some

numerical results will be presented.

3.3.2.1 Problem statement and mathematical description (an underwater ex-plosion)

We are interested in studying non-linear bubble dynamics, which plays an important role

in many areas of contemporary science and technology such as: oil industry, in which bubbles

are essential for lifting the heavy oil to the surface; chemical reactors, in which bubbles are

used to increase the contact surface between the gas-liquid phases; ship hydrodynamics, in

which collapsing bubbles are the main cause of the propeller damage and limiting factors for

the speed of vessels; some medical and many other technological applications (including our

current subject underwater explosions) can also be added [ 47, 60, 61, 62].

Here we pay special attention to underwater explosions. An underwater explosiongenerates a very strong shock wave and a high pressure gas bubble. Therefore an underwater

explosion can be modeled as the growth and collapse of a gas bubble. For this problem, we

make the following assumptions: bubble expansions are assumed to be spherically symmetric,

so variation only occurs in the radial direction. The system is assumed to be adiabatic, i.e, we

ignore heat convection. We assume that surface tension and viscous effects are negligible (See

Appendix B). We also assume that the gas contents of the bubble are spatially uniform (but

vary with time); i.e, we dont solve the gas-dynamics equations inside the bubble. Finally we

assume that water is compressible and obeys the Tait equation of state (see equation ( 3.66)

below), which is independent of internal energy [ 63].

With the above assumptions, the governing equations become the inviscid compressible

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Euler equations in spherical coordinates,

t

+ 1r 2

(r 2m)r

= 0, (3.55)

mt

+ 1r 2

(r 2um)r

= pr

, (3.56)

where , u, m, and p represent density, velocity, momentum, and pressure of water

respectively.

We enforce a Dirichlet type moving boundary condition for water pressure, i.e, we specify

water pressure as the bubble pressure at the bubble wall.

The time dependent bubble pressure is determined by using the adiabatic form of the

JWL (Jones-Wilkins-Lee) equation of state for explosive materials,

pbubble (t) = Ae R 1 (

V ( t )V I

) + Be R 2 (

V ( t )V I

) + C (V (t)V I

) (+1) , (3.57)

where V I is the volume of the bubble at the radius RI , and A, B,R 1, R2,R I , and are the

standard constants for TNT, and C is an arbitrary constant which is determined by letting

V (t) = V 0, pbubble (t) = pbubble 0 at t = 0.

The position of the bubble interface (the material surface) is captured by updating the

following level set equation,t + u

r = 0, (3.58)

where is the level set function and u is the external velocity eld (water velocity). The

level set equation ( 3.58) states that is constant along particle paths. This means that if

the zero level set of is initialized as a material surface (bubble interface) between water

and gas, then it will always be a material surface at later times [ 64, 65]. We take < 0 in

the gas region and > 0 in the water region (Figure 3.11). Hence, we have

(r, t )> 0 if r water= 0 if r < 0 if r gas,

(3.59)

where represents the bubble interface, and it is dened as the zero level set of ,

= {r | (r, t ) = 0 }. (3.60)

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> 0

water region

< 0

= 0 (bubble interface)

gas regionr

Figure 3.11: Bubble interface (the material surface) represented by the level set function.

3.3.2.2 Numerical procedure

For an adiabatic underwater explosion, the governing equations of water are the conti-

nuity and momentum equations, ( 3.55) and (3.56). Thus the same second-order primitive

preconditioner algorithm given in Section 3.2 applies here (except for the energy part, due

to the Tait equation of state). Here we shall only concentrate on the details of the moving

boundary conditions; i.e., the extension procedure (extrapolation) of the water eld variables

into the gas region at each time step, along with a short description of the level set algorithm.

We impose two different types of Dirichlet boundary conditions when solving the elliptic

equation for the new pressure eld ( 3.5). At the left end of the computational domain (gas

region), we specify the water pressure as the bubble pressure which is calculated via thebubble volume. At the right end, we specify the hydrostatic water pressure pinf as a pressure

boundary condition.

We call one particular cell, in front of the bubble interface, the critical cell (Figure

3.12). This critical cell plays an important role during the computation, because the moving

boundary condition at the left will be determined by this cell. At each time step, we will

have to nd a new critical cell due to motion of the bubble interface.

When i equals the critical cell (i.e, i = critical ) we impose the following boundary

conditions when discretizing ( 3.5),

pn +1critical 1 = r

pn +1bubble +

r

pn +1critical , (3.61)

where is the distance between the critical cell and the bubble interface.

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P Pinf

L

PM

water region

Bubble interface

gas region

M+1/2

bubbleP

critical+1/2PcriticalPcritical-1/2P

r

Critical cell r

M-1/2P

Figure 3.12: A moving boundary condition representation for pressure.

For i equals the last cell (i.e, i = M ) we have,

pn +1M +1 = r

L pinf + L r

L pn +1M , (3.62)

where pinf is the water pressure at some distance L.

We also apply the following boundary condition for the gradient of the pressure; i.e., for

the pn +1 ,k term in ( 3.5),

pn +1r,critical 12 = pn +1critical p

n +1bubble

, (3.63)

and

pn +1r,M + 12 = pinf pn +1

M L . (3.64)

Now we shall discuss the extension procedure for the ow variables from water region into

the gas region. We must use this extension since we dont solve the gas dynamics equation

inside the bubble for this particular problem, and we need the values of the ow variables in

the gas region because we have to solve the level set equation in the whole domain including

the gas and water. Also one has to solve (3.1) using values in the ghost regions. Here we

remark that the discretization of the explicit step does not use the ghost uid treatment

to dene the ghost values. The extension procedure to dene the ghost values is done via

linear extrapolation of the ow variables from water into the gas, irrespective of the jump

condition that might exist at the moving boundary.

After nishing the extension procedure at each time step, we are ready to solve the level

set equation in the whole domain. The level set equation ( 3.58) is solved by way of the

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following second order differencing,

n +1 ,ki n +1 ,0i

t = u

( n +1 ,k 1)+( n +1 , 0)2

i

( n +1 ,k 1)+( n +1 , 0)2

i+1 ( n +1 ,k 1)+( n +1 , 0)

2i 1

2 r . (3.65)

3.3.2.3 Numerical results of the underwater explosion

Here we present the numerical results of the underwater explosion test problem, to show

the performance of our second order non-oscillatory all speed multi-phase ow algorithm

developed in Section 3.2. We recall that underwater explosions can be modeled as the growth

and collapse of spherically symmetric gas bubbles, and considering the other assumptions

we made in Section 3.3.2.1 we note that the numerical results come from 1-D calculations

of the adiabatic inviscid Euler equations together with the level set equation in the radial

direction.The method is tested by using the following initial data:

The initial water density is taken to be 1.00037984 g/cm 3.

The water pressure is calculated by the Tait equation of state [ 63],

p = B[(/ ) 1] + A (3.66)

where B = 3 .31E + 09d/cm 2, A = 1.0E + 06d/cm 2, = 1 .0g/cm 3, and = 7.15.

The initial water velocity is taken to be 0.0, and initial water pressure is 1 .0E +07 d/cm2

.Finally the initial bubble radius is taken to be 16 cm, and the initial bubble pressure is

P 0 = 7.8039E + 10d/cm 2.

The conservative explicit building block (Section 3.2.1) is used here with CFL= 0 .25

(( |u| + c) t < CFL x). In Figure 3.13 and Figure 3.14, we demonstrate the second order

accuracy of the proposed scheme away from the discontinuities. The code is run for four

different grid settings, 250 , 500, 1000, and 2000. The enlarged part of the pressure prole,

Figure 3.14, clearly indicates convergence of our method at discontinuities. Figure 3.15 shows

the bubble radius as a function of time during the growth and collapse of the explosionbubble. It can be seen that our solution is in very good agreement with the benchmark data

[63].

We note that this problem is a difficult problem to compute due to the high impedance

mismatch at the explosive-water boundary. Nevertheless, we capture this discontinuity

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waterair airU

11 0x fs fsx x

Figure 3.16: A closed tube in which a column of water oscillates back and forth, watercolumn initially moves from left to right with speed U .

a pressure difference is built up across the water column resulting that the column of water

decelerates to the right, makes a stop, and then accelerates to the left. This time a reverse

pressure difference is built up across the water column redirecting the ow from left to right

again. As a result of this continuous process, the water column starts to oscillate. The

governing equations and the numerical procedure are the same as in Section 3.3.2.1 and

3.3.2.2 except all formulations have to be written in cartesian coordinates. The following

initial data are used as in [ 66],

p + Bp ref (1 + B) pref

= ( ref

) (3.67)

where the same equation of state ( 3.67) can be used for both water and air with pref =

ref = 1, w = 7, a = 1.4, Bw = 3000, B a = 0, pw0 = pa 0 = 1, u = 1, and xf s = 0.1. To be

consistent with Koren et al [66], we plot the time evolution of the pressure coefficients,

P (x = 1, t) = p(x = 1, t) pa (0) pa (0)

, P (x = 1, t) = p(x = 1, t ) pa (0) pa (0)

, (3.68)

and the time evolution of the relative mass error,

M (t) ma (t) ma (0)

ma (0) , (3.69)

where ma (t) is the total mass of air in the tube at time t. Figure 3.17 and Figure 3.18

are produced by using both conservative and non-conservative explicit building blocks. We

used CFL= 0 .8 ((|u| + c)max t < CFL x) for the conservative explicit building block, and

CFL= 3 (( |u| + c)max t < CFL x) for the non-conservative explicit building block. The

results from both building blocks are comparable, yet we observe that the non-conservative

explicit building block is more efficient (it uses large CFL numbers). With this problem,

we demonstrate the ability of our method for computing low Mach (weakly compressible,

i.e, it is evident by the Figure 3.19) that divergence of the velocity e

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