on numerical studies of explosion and implosion in air · 2.4 test 4: exact solution (solid line)...

On Numerical Studies of

Explosion and Implosion in Air

FU Sau-chung

A Thesis Submitted in Partial Fulfilment

of the Requirements for the Degree of

Master of Philosophy

in

Mathematics

© T h e Chinese University of Hong Kong

June 2006

The Chinese University of Hong Kong holds the copyright of this thesis. Any per-

son(s) intending to use a part or whole of the materials in the thesis in a proposed

publication must seek copyright release from the Dean of the Graduate School.

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Thesis/Assessment Committee

Professor ZOU Jun (Chair)

Professor XIN Zhou Ping (Thesis Supervisor)

Professor AU Kwok Keung Thomas (Committee Member)

Professor WANG Xiao Ping (External Examiner)

Abstract of thesis entitled:

On Numerical Studies of Explosion and Implosion in Air

Submitted by FU Sau-chung

for the degree of Master of Philosophy

at The Chinese University of Hong Kong in June 2006

Explosion and implosion in air are rigorous fluid dynamics phenomena and

hence, those studies are highly restricted in a laboratory. Computational fluid

dynamics becomes an important tool to investigate such problems. Numerical

calculations of hyperbolic conservation laws play an important role in such

problems. The resolution of such numerical schemes is highly improved, since

the introduction of a class of non-oscillatory explicit second-order accurate

finite difference schemes by using the concept of total variation diminishing

(TVD). Recently, the development of a class of numerical schemes so called

relaxation schemes contributed the field on its simplicity, without using either

Riemann solvers spatially or a nonlinear system of algebraic equation solvers

temporally. The leading order approximation of the relaxation schemes, so

called the relaxed scheme when applying to the one-dimensional scalar conser-

vation law was proved to be TVD.

In this research, the problem of explosion and implosion in air with spher-

ically or cylindrically symmetric flow is studied numerically by using the re-

laxed scheme. The governing equations are basically the Euler equations with

a geometric source term. A splitting method is employed to deal with the inho-

mogeneous equation. The numerical dissipation of the relaxation scheme is to

a large extent determined by a matrix A which is constructed when applying

the scheme to the problem. The appropriate choice of matrix A is critical in

i

the accuracy and practicality of the scheme. An algorithm to choose the ma-

trix and technique in applying the relaxed scheme is developed in addition to

maintaining the good features of the relaxed scheme. Some numerical tests are

carried out to test the reliability of the algorithm to the Euler system. Numer-

ical simulation of four cases namely spherical explosion, cylindrical explosion,

spherical implosion and cylindrical implosion are discussed. The numerical

results are presented and compared with some previous numerical, analytical

and experimental results.

ii

摘要

空中向外爆炸及向內爆炸是很劇烈的流體動力學現象，所以，在實驗室內做有

關的硏究是有一定程度的困難，於是，計算流體動力學便成爲了硏究有關課題

的一個重要的工具，而雙曲守恆律的數値計算在此類問題中擔任了重要的角

色。自從引進了全變差下降(TVD)的槪念而發展的一類非振還、顯式、二階精

度差分格式後，數値計算格式的分解能力大大地提升。最近，一類名爲鬆弛格

式(Relaxation Scheme)，它免去運用空間上的黎曼解算子或是時間上的非線性代

數方程組解算子，這簡單性給予數値計算這範疇上很大的意義。當應用在一維

單守恆律時，鬆弛格式的首階近似，又名放鬆格式(Relaxed Scheme)，已證明是

TVD 的。

在本硏究中，放鬆格式將用來探討球面對稱及圓柱對稱的空中向外及向內爆炸

的問題。問題的控制方程爲歐拉方程帶有一個幾何有關的源項，這非齊次方程

將用分裂法來處理。鬆弛格式的數値耗散很大程度基於運用這格式時所構造的

一個矩陣‘A’，格式的準確度及實用性取決於能否適當地構造它。在保持放鬆格

式的優點這前提下，本論文提供了構造‘A’的算法及應用放鬆格式的技巧，並使

用實際的數値測驗來證實了本算法的可靠性，並作了四個數値模擬：球面對稱

的向外爆炸、圓柱對稱的向外爆炸、球面對稱的向內爆炸及圓柱對稱的向內爆

炸。計算結果將會跟以前數値計算的、分析的及實驗的結果作一比較。

Acknowledgement

I would like to take this opportunity to express the deepest gratitude to my

research supervisor Prof. Z. P. Xin for his guidance and generous support

to me. Many thanks are given to him for giving me the chance to study

in this field. Also, thanks must go to my colleagues in the office for their

encouragement and inspiring discussion.

This research has been supported by The Institute of Mathematical Sci-

ences (IMS), The Chinese University of Hong Kong. I am much grateful to

IMS for the award of a postgraduate studentship to me.

iii

“The heavens declare the glory of God;

the skies proclaim the work of his hands. ”

Psalms 19.1 (NIV)

iv

Contents

Abstract i

Acknowledgement iii

1 Introduction 1

1.1 Background of Explosion and Implosion Problems 1

1.2 Background of the Development of Numerical Schemes 2

1.3 Organization of the Thesis 5

2 Governing Equations and Numerical Schemes 6

2.1 Governing Equations G

2.2 Numerical Schemes 8

2.2.1 Splitting Scheme for Partial Differential Equations with

Source Terms 8

2.2.2 Boundary Conditions 9

2.2.3 Numerical Solvers for the ODEs - The Second-Order,

Two-Stage Runge-Kutta Method 10

2.2.4 Numerical Solvers for the Pure Advection Hyperbolic

Problem - The Second-Order Relaxed Scheme 11

3 Numerical Results 29

3.1 Spherical Explosion Problem 30

3.1.1 Physical Description 32

V

3.1.2 Comparison with Previous Analytical and Experimental

Results 33

3.2 Cylindrical Explosion Problem 46


3.2.2 Two-Dimensional Model 49

3.3 Spherical Implosion Problem 52


3.4 Cylindrical Implosion Problem 53


3.4.2 Two-Dimensional Model 53

4 Conclusion 65

Bibliography 68

vi

List of Figures

2.1 Test 1: Exact solution (solid line) and numerical solution (square

symbol) for pressure, density, velocity and temperature at time

t = 0.2 22



t = 0.15 23



t = 0.012 24

2.4 Test 4: Exact solution (solid line) and niimeriral solution (square^


t = 0.035 25



t = 0.8 26

3.1 Test 6: Comparison of the smearing effect at the contact discon-

tinuity among exact solution (solid line) and numerical solutions

(rela:xed scheme - square symbol, Harten's scheme - dashed line,

modified Harten's scheme - dash-dot line) at time t 二 1.0 31

3.2 Numerical simulation of the explosion problem for (a) pressure

(b) velocity (c) density and (d) temperature from time t = 0.2

to t = 0.4 35

vii


(b) velocity (c) density and (d) temperature from time 力= 0 . 6

to t 二 1.0 37


(b) velocity (c) density and (d) temperature from time t 二 1.15

to t = 1.30 39



to t 二 1.90 41



to t = 2.4 43



to t 二 2.9 45

3.8 Comparison between the relaxed scheme and the (a) analyti-

cal (Erode), (b) experimental (Boyer) results of the spherical

explosion 47

3.9 Comparison between the spherical and the cylindrical explosion. 48

3.10 Comparison between the 2D model and the quasi-ID model of

the cylindrical explosion at time (a) t 二 1.65 and (b) t = 2.2. . . 50

3.11 Three-dimensional plot of the pressure contour of the cylindrical

explosion at time t = 2.5 51

3.12 Numerical Simulation of Implosion for (a) Pressure (b) Velocity

(c) Density and (d) Temperature from time t = 0.05 to t = 0.15. 55

3.13 Numerical simulation of the implosion problem for (a) pressure


to t 二 0.27 57

viii



to 力二 0.35 59



to It 二 1.0 61

3.16 Comparison between the spherical and the cylindrical implosion. 62

3.17 Comparison between the 2D model and the quasi-ID model of

the cylindrical implosion at time t 二 0.4 63

3.18 Three-dimensional plot of the pressure contour of the cylindrical

implosion at time t = 0.5 64

ix

List of Tables

2.1 Data for Five Riemann Problem Tests 21

3.1 Data for the 6th Riemann Problem Test 29

3.2 Initial Data for Explosion Problem 32

3.3 Initial Data for Implosion Problem 52

X

Chapter 1

Introduction

1.1 Background of Explosion and Implosion Problems

Studies of spherical blast waves have long been interested by many researchers

([1], [2], [3], [4], [5]). An experimental investigation of the explosions of a glass

spheres under high internal pressure was carried out by Boyer [2] in 1960. A

moving outward shock wave and a moving inward rarefaction wave generates

at the time the glass sphere broken instantaneously. The former is called

the main shock. Due to the dimensionality effect, a shock, also called tlie

second shock, follows from the tail of the rarefaction wave and increases with

strength. The second shock reflects from the origin of the sphere and starts

propagating outward. Theoretical investigation predicts that once the second

shock interacts with the contact discontinuity, similar to the formation of the

second shock, a third shock generates and moves inward. Similarly, fourth,

fifth, sixth shocks would be formed provided that the time is long enough and

the strength of the explosion is large enough. However, hardly can the third

shock be observed in experiments carried out in a laboratory ([2], [5]). Some

analytical results were studied in order to find the explicit formulae for the

location of the shocks and the contact discontinuity ([1], [3]).

For the implosion problem, since it is very difficult to break a spherical

diaphragm instantly ([2], [5]). The resulting flow of the experiment was com-

1

CHAPTER 1. INTRODUCTION 2

pletely nonuniform and the experimental results were not satisfactory. Because

of the difficulty in carrying out an experiment in the laboratory, computa-

tional fluid dynamics becomes important to investigate the problem. Sod [6

carried out a numerical experiment on a converging cylindrical shock by using

Glimm's method and operator splitting. Sod successfully captured the second

shock produced by the interaction between the main shock and the contact

discontinuity.

Due to the limitation of the experimental study, numerical simulations pro-

vide a new approach to investigate the fluid dynamics problem. For example,

Liu et al. [7] simulated the explosion and implosion problems by using a mod-

ified Harten's scheme. They successfully captured the main features of the

problems. Recently, high resolution and robust numerical methods and nu-

merical technique are developed quickly. It would be valuable to carry out

some numerical simulations by using different classes of numerical methods to

test the practical use of the recently developed methods. Also, the physics of

the explosion and implosion problems would understand more thoroughly by

the simulations of different numerical schemes.

1.2 Background of the Development of Numerical Schemes

Cylindrical and spherically symmetric wave motion arises naturally in the the-

ory of explosion in air. In these situations, the multidimensional equations may

be reduced to essentially one-dimensional equations with a geometric source

term to account for the second and third spatial dimensions. The govern-

ing equations become a nonlinear system of hyperbolic conservation laws with

source terms. One of the approach to tackle such inhomogeneous system is

to split for a time step, into the 'advection problem' which is a homogeneous

hyperbolic problem, and the 'source problem' which is a system of ordinary

differential equations (ODEs). The solvers for the two split problems can be

chosen independently and the numerical methods for both fields are highly


developed. There is a vast literature on ODEs and its numerical methods (see

8] for reference). For the pure advection (homogeneous) hyperbolic problem,

two requirements on numerical methods, namely high-order (at least second-

order) of accuracy and absence of spurious oscillations have been competing

in the history of computational fluid dynamics. A prominent class of nonlin-

ear method, total variation diminishing (TVD) method, is one of the most

significant achievements in the development of numerical methods for partial

differential equations in the last two decades.

A class of explicit second-order finite difference schemes for computation

of weak solutions of hyperbolic conservation laws was developed by Hart en [9

in 1983. These highly nonlinear schemes, which were proved to be TVD, are

obtained by applying a non-oscillatory first-order accurate scheme to an ap-

propriately modified flux function. Following his work, other robust numerical

methods or techniques were developed, e.g., essentially non-oscillatory scheme,

subcell resolution, etc ([10], [11], [12], [13]). In simulating the explosion and

implosion problems, Liu et al. [7] improved the Marten's TVD scheme to obtain

a high resolution on a contact discontinuity by modifying the scheme with the

technique of the artifical compression method (ACM) ([14], [15]). The above

schemes are one-dimensional in nature and they can be used to calculate a

higher dimensional problem with the help of a dimensional splitting method.

However, they are not easy to extend to the multi-dimensional version. So, it

would be useful if a class of numerical schemes can be developed, for which

can generalize to a higher dimensional version easily and keep its simplicity

and explicit nature.

Later, a class of numerical schemes called relaxation schemes for systems

of conservation laws in several space dimensions was introduced by Jin & Xin

16] in 1995. The second-order schemes were shown to be TVD in the zero

relaxation limit for scalar equations. The advantage of the relaxation scheme

is that neither spatially Riemann solvers nor temporally nonlinear system of

algebraic equation solvers are required. Also, it keeps its simplicity for general-


ization to a higher dimensional space. Such advantages are interested by many

researchers to further investigate and develop the scheme recently ([17], [18],

19], [20], [21], [22], [23], [24], [25], [26], [27], [28]). The idea of the relaxation

scheme is to use a local approximation by constructing a linear hyperbolic sys-

tem with a stiff lower order term that approximates the original system with a

small dissipative correction. The leading order approximation of the relaxation

scheme in the small dissipative correction limit is called the relaxed scheme.

During the construction process, a constant matrix ‘A’ is to be constructed.

In order to ensure the dissipative nature of the leading order approximation

of the relaxation system, a dissipative condition is required. Although the

dissipative condition can always be satisfied by choosing a sufficiently large

A. However, larger A introduces more numerical viscosity, so, it is desirable

to obtain the smallest A to meet the numerical stability condition. Evje and

Fjelde [24] showed that the relaxation scheme produces a poor approximation

for a typical mass transport example which involves transition from two-phase

flow to single-phase flow. This is due to the 'over-estimate' in choosing the

matrix A causing the excessive 'smearing out'. They solved the problem by a

flux splitting method so that the choice of the matrix A was possible to become

'reasonably' small. So, the numerical dissipation of the relaxation scheme is

to a large extent determined by the matrix A and the choice of A becomes

critical in the application of the scheme.

In applying the relaxation scheme to the explosion and implosion problems,

we encounter a similar difficulty in choosing the matrix A. As suggested by Jin

& Xin [16], the matrix A can be roughly estimated by the characteristics of the

original Euler equations such that the characteristics of the relajcation system

interlace with those of the Euler equations, or simply take the largest eigenvalue

of the Euler equations over the whole space-time domain. However, when the

shocks reflect at the origin, the physical quantities will inflate dramatically

and causing a 'sudden inflate' of the eigenvalue at this specific time. Surely,

the matrix A estimated by the above method will be too large for most of


the calculation time. It would be a main contribution for this research to find

an algorithm or at least, some guidelines to estimate the matrix A for the

explosion and implosion problems.

1.3 Organization of the Thesis

The objective of this research is to understand the physics of the explosion and

implosion problems through numerical simulations; meanwhile, the technique

of the numerical scheme for the problem is developed. More specifically, an

algorithm of using the relaxed scheme on the problems is developed. In chapter

2, the mathematical model are stated. The governing equations of a spherically

or cylindrical symmetric flow are basically the Euler equations with a geometric

source term. A splitting method is employed to deal with the inhomogeneous

equation and the Euler system is mainly solved by the relaxed scheme. The

appropriate choice of the matrix 'A' is critical in the scheme. An algorithm

to choose the matrix is also developed in chapter 2. The numerical results

are presented in chapter 3. Four cases namely spherical explosion, cylindrical

explosion, spherical implosion and cylindrical implosion are discussed. The

numerical results are compared with some previous numerical, analytical and

experimental results. Also, two-dimensional models are calculated in order to

compare with the approximated cylindrical symmetric models. Finally, some

concluding remarks are made in the last chapter.

Chapter 2

Governing Equations and Numerical Schemes

The main interest in this research is to simulate the explosion and implosion

problems numerically, and to analyze the numerical method involved. In this

chapter, we develop the mathematical model for the problems. After choosing

a suitable numerical method for the problems, the details and techniques of

the method will be discussed.

2.1 Governing Equations

In the theory of explosion or implosion in air, the problem is normally regarded

as an inviscid flow (Sod [6], Liu et al. [7]). By dropping the effects of the

body force and the thermal conduction, the equations for an unsteady, two-

dimensional, compressible inviscid flow, also called the Euler equations are as

follows:

f + T + T =。’ （2.1)

6

CHAPTER 2. GOVERNING EQUATIONS AND NUMERICAL SCHEMES 7

where

( \ I \ ( \ p pu pv

pu + p puv U 二 , F肌二， G ( U ) = . (2.2)

pv puv pv"^ + p

V E ) \ {E-Vp)u ) \ {E^j))v )

Here p is the density, p is the pressure. E is the total energy, u and v are

the velocity components with respect to the x and y-direction respectively.

Similarly, the one-dimensional (ID) Euler equations can be written as,

f + T =。’ （2.3) where

( \ I \ P pu

U= pu , F { U ) = 叫 h p . (2.4)

V E ； V {E^p)u

The equation of state for a perfect gas is required for the closure of the

problem,

E 二 + Ifm'. (2.5) 7 —丄 2

where 7 is the ratio of specific heats, which is a constant for a calorically

perfect gas and dependent on the temperature for a thermally perfect gas. In

our model, we assume the gas is calorically perfect and having 7 = 1.4 (i.e.

for air).

Wave motion in the explosion and implosion problems is normally cylin-

drical or spherically symmetric in nature. The multi-dimensional Euler equa-

tions may be reduced to essentially one-dimensional equations with a geometric

source term vector, S{U), to account for the second and third spatial dimen-

sions. It is usually written in the form dU dF{U) ^ 广

1 + （2.6)


where

( \ ( \ ( \ p fm pu

U 二 pu，刚二 fm'+p ，聊二 - d pu^ . (2.7)

\ E ) \ {E + p)u y \ {E+p)u

Here now x is the radial distance from the origin and u is the radial velocity.

For a. 二 1, (2.6) returns to our one-dimensional Euler equations (2.3) and

it represents a plane one-dimensional flow. For a 二 2, (2.6) represents the

cylindrical symmetric flow which approximates a two-dimensional flow. For

a = 3, (2.6) represents the spherically symmetric flow, an approximation to a

three-dimensional flow.

A difficulty occurs in solving (2.6) due to the arising of the singularity at

the origin, x = 0. In order to remove the singularity, the following form of the

conservative equations is adopted to ensure that the resulting finite difference

numerical scheme employed is compatible with the original equations at the

origin (Liu et al. [7]). At the origin，

dp I dfjii 瓦小二。，

dE di^E + p)u ^ ( �

i + = 0，（2.幻

u = 0.

2.2 Numerical Schemes

2.2.1 Splitting Scheme for Partial Differential Equations with Source

Terms

The governing equations (2.6) are non-linear system of hyperbolic conserva-

tion laws with source terms. Such inhomogeneous system can be calculated

numerically by a splitting method. We split the governing equations (2.6) into

the pure advection hyperbolic system, which is a homogenous system, same as


(2.3)，

dU dFiU) , �

i + ~ ^ : 。，（2.9)

and, a system of ordinary differential equations (ODEs)，

I " 二风 [ / ) . (2.10)

By using the data at 力二 n M , where A力 is the step size of time, and n is

an integer, as the initial condition and solving the pure advection hyperbolic

problem to find the state at 力二 (n+1) A力,we get an intermediate state. Then,

using this intermediate state as the initial condition and solving the ODEs. We

get the result at 力=(n + I)At, at last.

By re-written the procedure above as follows:

U{t = {n+l)At) 二二 (2.11)

Here, 幻 and 龙）can be any numerical solvers with the time step At for

the ODEs and the pure advection hyperbolic system respectively. If S 脚 and

are at least second-order accurate solution operators, Strang [29] showed

that the splitting scheme (2.11) is second-order accurate in time.

2.2.2 Boundary Conditions

At the origin, the equations (2.8) are discretized as follows:

, � A 力 - 购 )

仇二仇 ^ h ‘

pn+l — p n 力 ( 五？ + P S 胸 - 必鍾风 - 税 - a ^ ，（2.12)

PS+1 = (7-1)丑？+1,

= 0.

Here and after, the subscript 'j' indicates the spatial positions at x = jAx*,

where Ax is the step size for space. So subscript '0', T and '2' indicate the

points at the origin, x = Ax and x 二 respectively. Also, the superscript

'n' indicates the temporal positions at 力二 nAt.


Reflective boundaries are employed at the origin. This boundary conditions

can account for the coming and interaction of waves from the axi-symmetric

direction. A fictitious state ITlj is defined from the known state U言 inside the

computational domain,

二 P], 二-u�and (2.13)

For the end side of the domain, open-end boundary conditions, or so called,

transmissive boundaries are used so as to allow the passage of wave without

any effect,

p]+i - P], = u] and (2.14)

2.2.3 Numerical Solvers for the ODEs - The Second-Order, Two-

Stage Runge-Kutta Method

In (2.11), the numerical solvers S � � and 龙） can be chosen independently.

We first concentrate on solving the ODEs (2.10). For simplicity, it is preferred

to choose an explicit method. However, if the system of ODEs is stiff, nu-

merical difficulties arise especially for explicit methods. Even using implicit

method. Leveque [30] observed that large overshoots and non-physical wave

speeds occurred for hyperbolic conservation laws with stiff source terms. So

checking the stiffness of the ODEs is upmost important. As stated by Lambert

8], a nonlinear system of the form (2.10) is said to be stiff if

(i) Re{\i) < 0, i = 1 ,2 , . . . , m, and,

(ii) � � m i n j \ R e [ X i ) .

Here Re{Xi) denotes the real part of the complex number Ai, which is the

eigenvalue of the Jacobian A{U)三藉.In our ODEs, the Jacobian is found to

be

f 0 0 ^

A{U) = 2u 0 . (2.15)

^ -jEu/p + (7 - 1)仏3 jE/p - 3 ( 7 - 1)1/72 -fu j


So, the eigenvalues are u and ya, and then the stiffness ratio is

maXj\Re{Xi)\ : minj\Re{Xi)\ 二 7,

which is, for air, only 1.4 in our case. Hence, our system of ODEs is not stiff.

The second-order, two-stage Runge-Kutta explicit method is chosen to use

due to its simplicity and high accuracy. For time step At, when the system

evolutes from time t介 to f^+i，where f^ — nAt, the Runge-Kutta method is

described as

u � = I T +

U � = [ / � + A:i5，(f +ALL/�）， (2.16)

[T+i 二 - + [/⑵-. 2 L

Here and after U几 and [/时丄 represent the solutions at time f^ and t 几 r e -

spectively.

2.2.4 Numerical Solvers for the Pure Advection Hyperbolic Prob-

lem - T h e Second-Order Relaxed Scheme

A relaxation scheme based on the corresponding relaxation systems was devel-

oped by Jin & Xin [16] a decade ago. The advantage of the relaxation scheme

is that neither spatially Riemann solvers nor temporally nonlinear system of

algebraic equation solvers are required. Also, it keeps its simplicity for gen-

eralization to a higher dimensional space. Higher dimensional systems can be

treated in the same way as in the one dimension because of the constant linear

characteristic fields. Such advantages are interested by many researchers to

further investigate and develop the scheme recently.

Our pure advection hyperbolic problem (2.9) is a system of conservation

laws in one space variable. A corresponding relaxation system can be intro-

duced, dU dV ^ r.3

1 二 0 V g K a t 十彻， ‘

dV dU 1 i + A - = e〉0’ (2.17)


where e is a positive constant, calling the relaxation rate. The 3 x 3 matrix A,

A = diag{ai,a2,a3), ai,a2, as > 0, (2.18)

will be determined later.

The idea of the relaxation scheme is to solve the relaxation system (2.17)

instead of the original hyperbolic system (2.9). The special feature of the linear

characteristic fields and localized lower order terms of the relEDcation system

results in the advantages of the scheme mentioned before. The discretization

of the relaxation system is called the relaxing scheme. The leading order

approximation of the relaxing scheme in the small e limit is called the relaxed

scheme.

For simplicity, we may choose

A = al, a>0, (2.19)

where I is the identity matrix. The reduced system, i.e. the zero e limit of the

relaxation system is dissipative provided that

— < 1. (2.20) a

Here A = maXi\Xi{U)\ where A are the genuine eigenvalues of F'(U). For ID

scalar conservation laws,

专t等“鳴 (2.21)

where / is a scalar valued function. We have a corresponding relaxation sys-

tem,

du dv n ‘ + T 二 0, u,v G R , at ox

芸+ 二 - - { v - f { u ) ) ^ e > 0 . (2.22)

ot ox €

The dissipative condition now is referred to as the subcharacteristic condition

by Liu [31], < f{u) < v ^ for all u. (2.23)


It will be shown later that this condition plays an important role in the numer-

ical stability. Also, the dissipative structure of the first-order correction to the

original system implies that the numerical solution to the relaxation system

converges to the entropy solution of the original system [16 .

As observed in the numerical tests by Jin k Xin, for the case of compressible

Euler equations, the relaxing scheme and the relaxed scheme produce essen-

tially the same result. So, in our problem, we employ the second-order relaxed

scheme, instead of the relaxing one, for the pure advection hyperbolic part.

We describe below the one-dimensional version of the relaxed scheme we are

going to employ. The following second-order explicit Runge-Kutta is applied

in the time discretization.

"⑵ 二 " ⑴ — 巧 ( " ⑴ ） — ⑴ ) ） ’ （2.24)

� + 1 = 1 [[/- + � - . Z ‘- �

Here Fj^i is the numerical flux detailed below, and we use uniform girds for

both spatial and temporal space with step sizes Ax and At respectively.

In order to obtain a second-order accuracy in spatial discretization, van

Leer's MUSCL scheme [32] are used. The MUSCL scheme uses the piecewise

linear interpolation to achieve the second-order accuracy. For the p-component

of the flux, it is written as

+ & < —。、 )， (2.25)

where

cjf 二 (i^⑷O/JVi) 土的 ⑷ ( W ) 干「严n)树疗)，(2.26)

没士 = 士树、 R - 伸干 y/^ryff广

] — 土 v / ^ y j ? 广— F ( P 卿干 v^yj办". “


(/)(0f) is called the slope limiter function. Sweby [33] gave out a general con-

dition of the slope-limiter function for the scheme to be TVD,

0 < ^ < 2 , and —e

0 < _ < 2 (2.28)

In particular, we use the one introduced by van Leer [32] in this research,

m 二 (2-29)

We want to remark here that due to the explicit and the special structure of

the linear characteristic field of the relaxation scheme, the components of the

solution vector can be obtained in parallel computing without any modification

of the scheme.

Theorem:

For the slope-limiters satisfying (2.28), by choosing the matrix A as (2.19),

the corresponding second-order relaxed schemes (2.24) to (2.27) for the ID

scalar conservation law (2.21) are TVD provided that the subcharacteristic

conditions (2.23) and the following CFL condition are satisfied,

辟:^ \ (2.3。）

Proof of the Second-Order Relaxed Scheme to be T V D for ID scalar

Conservation Law

Our proof essentially follows the same as Jin & Xin's [16] analysis with only a

slight modification.

A scheme is called TVD if its total variation TV{u) ：二 — Uj

decreases in time,

< T V �

The relaxed scheme with the method of line takes the form

At )


：二 - ‘ (,(略 — ^ K - i ) ) + 盖 ( � 1 - +

+ 4 ( 7+1 — + - < ) • (2.31)

By the subchar act eristic condition (2.23), we have

C] — ( ^ + 权 4 ) ^ 0, (2.32)

At ( / ( 略 1) - m)\ D] ：二 — ( • 一： j - 0. (2.33)

By substituting Cj and Dj, the relaxed scheme (2.31) becomes

二 u] - - uU) + - U-)

+ 4 (^7+1 - 斤 + - 〜+) • (2.34)

Then we have

- (1 - Q - - u]) + ~ u]^,)

( 以 ? + 巧 ( 2 . 3 5 )

where

^i+l 二去[( 7+2 - - (S-+1 - ^7)

- O + « - < i ) ] . (2.36)

By rewriting cr, using C] and D”

= — & 2 ) ^ + 2 ( 4 3 — 4 2 )

(2.37)

and observing that

= C〕_等 (2.38)

GJ D八 u ; � u ] � ) = (1)， (2.39)


we get

切 - “ 广 = [ i - ( i - m ) + c ^ i 一

+ ( 1 - 嘲 C M 抖 1) \ 乙 }

+ f l + 没州） — — 略 1). (2.40)

By using the CFL condition (2.30) and the condition for slope limiter (2.28),

we can prove that the three coefficients on the right-hand side of (2.40) are

non-negative,

1 + 崎 + 1 ) - 榮 (2.41)

We then obtain

- <

1 — (1 - m ) + c ] - ( 1 + I ) . ] I略1—以?I

+ + _ 辦、 - 略 2 -略 i l . (2-42) V �"j+2 /

By (2.39), the slope limiter condition (2.28) and the fact that

_ 二 0, V6> S 0,

we have

等 C h W ] - 二树灯 )略 1 -

= (2.43)


Now we obtain

K i i -

1 - ( 1 — I m ) ) c] - ( i + \ < m ) ) 巧 ] —

+ ( 1 - 臺州 ; i K - �

+ ( 1 + 臺術 + 1 ) ) 1 ^ 州 1 < + 2 — 略 1 |

- l m + 2 ) D j + 2 \ u ] ^ 3 — (2.44)

By summing up over all j yields,

= ^%-ooKtl - y^iI < S告—⑴I略1 = TV{u-) (2.45)

Up to now, we have proved that the MUSCL relaxed scheme with one-step

time discretization is TVD.

In order to get second-order accuracy in time, the second-order Runge-

Kutta splitting scheme is used: ⑴ 7/几 (1)

We have already proved that

释 ⑴ ）二

= - + ^^ (^K+i) - ^ K ) ) I

By using the result of Shu k Osher [12], we can show that the second-order

Runge-Kutta splitting scheme is also TVD,

T 寧 + 1 ) =


= 頌 — u]\ + 全 - t^i) + At — I

< 全卿， + 全歸 ⑴ ）

= T V { u ^ ) ,

and we complete the proof of the theorem.

We remark that no rigorous proofs are available for the TVD property

for either the multi-dimensional scalar case or ID system conservation laws.

However, we are going to develop an algorithm to choose the matrix A and

setup some criteria for somewhat similar to the CFL condition (2.30) in the

next section. Then, a number of numerical test will be carried out to give

confidence to our algorithm, before applying to the explosion and implosion

problems in the next chapter.

Algorithm for the Relaxed Scheme

We want to solve the ID Euler system (2.9) by using the relaxed scheme

ineiiLioiied before. For siiiipliciLy, we choose the inauix ds in (2.19) for the

system. In order to ensure the relaxed system to be dissipative, the condition

(2.20) must be satisfied. In fact, this can always be satisfied by choosing a

sufficiently large A. However, larger A introduces more numerical viscosity,

so, it is desirable to obtain the smallest A to meet the numerical stability

condition.

Although we cannot obtain any CFL condition other than the case of ID

scalar for the scheme to be TVD, we use the condition (2.30) as a reference.

We define the CFL number as

CFL number := (2.46) Ax

For returning to the scalar case, the CFL condition (2.30) is expressed as

CFL number < (2.47)


Since the eigenvalues of the ID Euler equations are u and w 土 c where

c = (2.48)

is the sound speed. In order to obtain the minimum A, and fulfill the dissipative

condition (2.20), one can take

a = max{sup\u - sup\u\^, sup\u + (2.49)

here the supremum is taken over all t and x in the whole space-time domain.

However, as evidenced by the previous researches ([2], [6], [7]), we know that

the quantities of the eigenvalues would have a large variation along time, espe-

cially when the shocks reflect at the origin and the physical quantities would

inflate dramatically. If we simply take the supremum over all time and space,

the resulting A would become too large relative to some time intervals. Exces-

sive numerical viscosity would be injected during such time intervals arid the

accuracy of the calculation would be lower in general. So, we propose to choose

a 'dynamic' matrix A such that it is tailor-made for each time step interval.

Let lis consider a system evoliitins: from time f^ to 力 I n s t e a d of taking

the supremum of eigenvalues over all time, Lhe idea is to lake I lie supremum

over this time step interval only. Now, we want to estimate the maximum

eigenvalues inside the time interval The estimate follows the spirit

of the Godunov scheme. We assume the state at 力“as a piecewise constant.

The state keeps constant along every spatial interval i, x^^ i]. Evoluting

the system to the time t奸i is equivalent to solve a Riemann problem with the

initial condition of the state at f . The state at the position x] will remain

unchanged inside the time interval At provided that maXi | A | < 盡 where Xi

is the eigenvalue of the system. Then the constant ''a' is chosen as the same

maimer as (2.49), but now, the supremum is taken over t G and for

all X in the spatial domain. By using this method of estimate, the constant

'a will be different for each time step. For each time step interval, we suggest


the CFL number to be,

CFL number == v ^ ^ ^ = maXi \ —^ < (2.50) A x A r c — 2

which matches with the CFL condition (2.47) when returning to the scalar

case. By using the CFL condition in the scalar case as a reference, we impose

no additional constraint on the CFL number for the method proposed in this

section.

We remark that the above method provides only a rough estimate since

the Godunov scheme is only first-order accurate. So, this method in no way

guarantees that the estimate always satisfies the dissipative condition, however,

a close estimate to the minimum A is expected to obtain. In practical use, we

set the time step in the computer program by

At = (CFL number) x A x 剛

J(5 X max{\u + c|, \u\, \u — c\)

By changing the adjusting factor j3 as long as the numerical stability is still

maintained, we believe that the solution is reliable.

Numerical Test for the Algorithm

As mentioned in the previous section that we can only prove the T V D property

of the ID scalar conservation law. However, in this research, what we want

to calculate instead, is the system of equations. In order to apply the relaxed

scheme on the Euler system with the algorithm developed in the previous

section, we have done a series of numerical tests to support it.

It should be noted that we are not going to test the accuracy of the numer-

ical scheme in this section, instead, we want to give evidence that the scheme

do converge to the 'correct' solution and it is numerically stable. So in the

following parts, instead of comparing with other numerical methods, we com-

pare our numerical solution with the exact solution. Once the result shows

that the scheme converge to the exact solution, we conclude that the scheme

is numerically stable and hence reliable.


Test PL UL PL PR UR PR Xo

1 1.0 0.75 1.0 0.125 0.0 0.1 0.5

2 1.0 -2.0 0.4 1.0 2.0 0.4 0.5

3 1.0 0.0 1000.0 1.0 0.0 0.01 0.5

4 5.99924 19.5975 460.894 5.99242 -6.19633 46.0950 0.4

5 1.0 2.0 0.1 1.0 -2.0 0.1 0.5

Table 2.1: Data for Five Riemann Problem Tests.

Five tests are selected to test the performance of the numerical scheme.

They have been used by Toro [34]. Relaxed scheme with the algorithm devel-

oped in the previous section are tested. In all the tests, the spatial domain is

X £ [0，1]. The ratio of specific heats is 7 二 1.4. The step size is Ax 二 0.005

and the CFL number: 0.3. For each test, different step size, CFL number

and adjusting factor (3 in (2.51) have already been tested repeatedly to ensure

that the result is numerically stable. The tests are Riemann problems with

the initial data summarized in table 2.1, in terms of the primitive variables.

All quantities are dimensionless. Subscript 'L' and ‘R, represents the left and

right constant values of the initial data discontinuous at the position x = xq.

The first test is similar to the 'Sod test problem' [35]. It is, in fact, a

modification of the Sod test and the solution consists of a right shock wave, a

right traveling contact wave and a left sonic rarefaction wave. This test is very

useful in assessing the entropy satisfaction property of numerical methods.

The second test is called '123 problem' and its solution consists of two

strong rarefactions and a trivial stationary contact discontinuity. The solution

region between the non-linear waves is close to vacuum, so this test is useful

in assessing the performance of numerical methods for low density flows [36].

The Richtmyer (or two-step Lax-Wendroff) method fails to give a solution to

this problem.


1 \ i \ 0J9 r \ 0.9 i- \ OS i- \ 0.8 — \ 0.7 r \ 0.7 ；• \

\ ^ ^ i o 5 ^ \ io.6 L • i 1 o

0.4 — I CW 二 _• Li

03 - 0.3 02 r 0.2 -0.1 r ^ 0.1 -

0 r I I I I I—I I I—I—I—I—I—i—I—\ \ i I \ ~ F I I 1 t I t I • I I I I I • > , , I , I 0 0.25 0.5 0.75 1 0.25 0.5 0 75 1

K X 1J6 r- 3S r-1.4 L

/ i ^ / … r

： / - r • 、‘「 / = • 奋： / 专 ^ - / S2S ^ 5 I

06 fc 2J8 -： 2.4 L X

D.4 7 X

。” " l \ J 一。 r . . , . I _ , _ , _ I _ I _ _ I _ , _ , _ , _ _ I _ I _ I _ I _ _ 1 _ _ 1 _ _ I 1 8 _ . _ _ . I _ _ , _ , _ , _ , I _ , I , 1 0.25 0.5 0.75 1 0 D.25 0.5 075 1

* X

Figure 2.1: Test 1: Exact solution (solid line) and numerical solution (square symbol)

for pressure, density, velocity and temperature at time t = 0.2.


LZ i i I _ I I I I I I I I I I I I I I 1 I 1 I I 1 c I I I I I I . I I I . I . . I I I , I I

0 0 . 2 5 0 . 5 0 . 7 5 1 0 0 . 2 5 0 . 5 0 7 5 1 X X

I:丨 / 1 \ y -2 r I < . . • . . _ I 0 h • . • . I . • • • I . . . . I . . , . I

0 0 . 2 5 0 .5 0 .75 1 0 D 2 5 0 5 0 . 7 5 1 X X


for pressure，density, velocity and temperature at time t = 0.15.


t 8 r 门誦 r \

9 0 0 - \ - ‘

; \ . 800 7 \

[ •. I 迎 r X i 3 ：

a 'TOO ^ ^ 3 0 0 — 2 - -

200 — } - • 仰 p ^

0 一 L — — ：__I I I I I—I—I—I—J—I—I—I—I—I I I 1 1 I I pj r . « . . I . . . . I • . • . t . . . . I

0 0 . 2 5 0 . 5 0 . 7 5 1 0 0 . 2 5 0 . 5 0 . 7 5 1 X X

： [ r — t 16 1. / 2000 -

，4 ^ / • -

, 1 2 - / = 1 5 0 0 -

t ； / ^ • I '0 r / I • ^ 3 r / i 1000 -

- / ^ ： :

” Z 5 0 0 - • :l/ t__ s - . . • • _ I _ _ I _ _ I _ _ I _ 1 _ I _ 1 _ _ I _ 1 _ _ 1 _ I _ _ I _ I _ _ 1 _ _ I _ _ I 0 h . . . . I . . , I ' • , , . . I

0 : 2 5 0 5 0 7 5 1 0 0 25 0 .5 D.75 1 Jf A

Figure 2.3： Test 3: Exact solution (solid line) and numerical solution (square symbol)

for pressure, density, velocity and temperature at time t 二 0.012.

Test 3 is the left half of the 'blast wave problem' by Woodward and Colella

37], which is a very severe test problem. The solution consists of a left rarefac-

tion, a contact discontinuity and a strong right shock with Mach number^ 198.

The Richtmyer (or two-step Lax-Wendroff) method fails to give a solution to

this problem.

Test 4 is a very severe test, too. Its solution represents the collision of two

strong shocks produced by the above test (Test 3) and consists of a left facing

shock traveling very slowly to the right, a right traveling contact discontinuity

and a right traveling shock wave. So, all three discontinuities are traveling to


1800 r- � 30 ： r r ^

1600 - , - • •

1鄉 7 . 25 - ‘ I I • • 1200 - ： •

- 2 0 - I w t= 10C0 - ^ i ； : S 800 - 2 � i CL , O . I'

鄉 r i L ： 5 -20D 7 , -

�0 0.25 0.5 0.75 1 °0 0.25 ' ' ' ' 05 ' ' ' ' 075 ' ' ' ' 1 X X

25� 3DD r- •� _

如： 250 i 15 - ‘

： 200 -

2 ： •t - ^ I ： •_ g : si 150 -

• li : • 10D -D - ；

( ' � � • .10 r . • • • I__1 I_1__I I__1 I_1_I 1__1_I n ~• • • • I . . . . I . 薩. . . I

0 025 D5 075 1 "o 0 25 CI .5 075 1 * -A



the right.

The fifth test simulates the collision of two uniform streams. Its solution

consists of two strong symmetric shock waves and a trivial contact discontinu-

ity. In this test, the adjusting factor [3 should be at. least 6.5 for stable solution,

while P ^ I is large enough for all other tests.

In figures 2.1 to 2.5, it is shown that the relaxed scheme with the algorithm

in the previous section passes all the five tests above. Although we cannot

prove the TVD property of the scheme when applying to the Euler system,

the numerical experiments above show the applicability of our scheme to the


6 J- 6F : 5: ^ 5 r

- _ ‘ 4 . 5 i- I

41 , 4 i-

I 丨 r . ” I 3 - 讓 3 —

I ' = 2 . 5 ；.

2 - 2 ^

: 1.5 i-1 - 1 ： ^ 丨、

； 0 . 5 r

Ci ^^^^^^ L I I I I I I I I I I I I I I ^^^^^^ n t I 1 I I I I I I I—I—I—I—1—1—I—I—L—I I I 0 0 . 2 5 0 . 5 0 . 7 5 1 0 0 . 2 5 0 . 5 0 . 7 5 1

>C X

, 2 . 5 厂

-

2 ^ f ] 2 -

I -S 1.6 -

•f , • ： 1 ® 0 - g

- ； I 1 - I •1 -

D.5 --2 一 I' ‘ J 丨塞

- I I • i l l I I I I I I 1 i l l • • I I 11 • • • f 1 . . . . I • . . . I . • . . I 0 0-25 0.5 0.75 1 "o D.25 05 0.75 1 X X




Euler system.

Two-Dimensioned Case

One of the advantage of the relaxation scheme is its simplicity for generaliza-

tion to a higher dimensional space. Since the governing equation (2.6) is an

approximation to the multi-dimensional case. So, it is worthwhile and con-

venience to calculate the original multi-dimensional system by the relaxation

scheme in order to analyze the effectiveness of the approximated equation. We

choose to calculate the two dimensional case for comparison with the cylindri-

cal symmetric approximated model.

Considering the two-dimensional Euler System (2.1) and (2.2), similar to

the ID case, a corresponding relaxation system can be introduced,

dU dV dW ^ TT, ot ox oy

dV . dU 1 T /rTW + 乂石二 ⑴），

BW dlJ 1 ^ + = —{W-G{U)), e > 0 . (2.52) at oy e

For simplicity, as in (2.19), we consider the 3 x 3 matrices /I 二 a I and B : hi,

where a, 6 > 0 are to be chosen. Jin & Xin [16] showed that the reduced system

(2.52) is dissipative provided that \2 " 2

- + ^ < 1 . (2.53) a 0

Here A 二 ma:Ci|Ai((7)| and ji = max^\|2i{U)\ where 入 and yu are the gen-

uine eigenvalues of F'{U) and G'{U) respectively. The relaxed scheme can be

expressed as follows:

U⑴二『-f加、U, - F]事))

U � = u⑴ — 芸 ( F j + “ U � ) — F ) _ “ U � ) )


『+1 二； + (2.54)

where At, Ax and Ay are the step size for time, for spatial x-direction and

spatial y-direction respectively. Indices 'n', and ‘j, are corresponded to 't',

‘X，and ‘y，respectively.

For the p-component of the flux，it is written as

+ 全 - 《力， (2.55)

绍!(『）二 I (作。)+ G(PnU�办 I麵焚—(4，)

+ 去《+1), (2.56)

where the slope limiters are

一二 F�P�m) 土 • 增 " - F � ( 化 1 , � 干讽恕说

‘‘一沙 ) ( " r + i，》士沙)(喝）干 v/^ 吻)"， )

二 ⑷(「；士錢 ( 恋 - 干 v [yg)，n) (2.59)

一士坊 - G � 干彻愁 _

We define the CFL number as

CFL number max v ^ ^ ^ ] , (2.61) V k Ay；、乂

so that it would naturally return to (2.47) for degenerating to the case of ID

scalar conservation law.

Similar algorithm as the previous section in order to obtain a rough esti-

mation of matrix A and B are used. However, numerical stability should be

taken more carefully by the adjusting factor P because of the more uncertainty

in this 2D case now.

For the next chapter, we would use the numerical scheme stated and the

algorithm developed in this chapter to calculate the explosion and implosion

problems.

Chapter 3

Numerical Results

In this chapter, results of numerical simulations of explosion and implosion

problems are presented. The numerical method developed in the previous

chapter is used. The results are shown to be reliable by comparing with the

data calculated by the modified Harten's scheme [7]. The modified Harten's

scheme attempts to have a high resolution for contact discontinuities in addi-

tion to maintaining the good features of Harten's TVD scheme. A technique

in the same spirit of the artifical compression method (ACM) is used in the

modified Harten's scheme. The original ACM [14] is capable to maximize

the resolution of a steady progressing profile. In addition to the five tests in

the previous chapter, in order to test the smearing problem of our numerical

scheme at contact discontinuities, we carry out the sixth test as shown in table

3.1.

This Test 6 corresponds to an isolated contact discontinuity moving slowly

to the left. In figure 3.1, the relaxed scheme (square symbol) is compared with

the Harten's scheme [9] (dashed line) and the modified Harten's scheme [7

Test PL ul PL PR Ur PR Xo

6 1.4 -0.1 1.0 1.0 -0.1 1.0 0.5

Table 3.1: Data for the 6th Riemann Problem Test.

29

CHAPTER 3. NUMERICAL RESULTS 30

(dash-dot line). It is seen that the resolution for contact discontinuities of the

relaxed scheme is not as high as the modified Harten's but better than the

Harten's scheme. In the following presentation of the explosion and implosion

problems, the results of the relazxed scheme and the modified Harten's scheme

would be shown on the same graph. This comparison is significant because

these two numerical schemes are based on completely different ideas.

3.1 Spherical Explosion Problem

For the spherical explosion problem, for purpose of comparison, we choose

the same model as such experimentally investigated by Boyer [2]，analytically

analyzed by Erode [1] and numerically discussed by Liu et al. [7]. A glass sphere

with radius Xq = 1 in. initially filled with air at pressure p' j = 326 p.s.i. and

temperature T' = 299 K is exploded in air at the atmospheric pressure and the

same temperature inside the sphere. The superscript 'prime' {') denotes the

dimensional quantity. The dependent variables are non-dimensionalized via

P 二丨fL P 二 PV(PoCO^), u - (3.1)

Here Cg is the initial sound speed calculated by (2.48),in which the correspond-

ing density is Pq 二 p'L and the pressure is determined by the ideal gas law.

The independent variables are non-dimensionalized via

力=力7 ( 4 4 / 4 ) ’ x = x'l{Ax',). (3.2)

As mentioned in chapter 2, in our model, we assume the gas is calorically

perfect and having 7 : 1.4 (i.e. for air). The initial values corresponding

to the non-dimensionalized quantities are calculated and summarized in table

3.2.

At time ^ = 0, the glass sphere with radius x 二 0.25 is broken and it is an

analogue to a Riemann problem. In the following numerical results, A x = 0.05,

CFL number二 0.3，the adjusting factor (3—1.


间 1 广 • Rela>«<i Scl-»em«

“ ———-Marten's Scheme - Modified Marten's Scheme

i^l ^ x-i

- C, 1.3 - r,

a> - 〜： •

I 4 f … ^ OS ^

OS I 11 - p 1 L

Ojb -O � G .25 0.75 1 。.岛丄• • • • ^ ^ • • • • ols • • • • 0 .75 ' ' ' ' 1 X K

D � 2J0 -；- pT；

： 2 .4 L >f 。叶 23 L

^ : ' •f 1 2.1 ^ t i I - i

1 JO - r. E*act

n 1 s ^ Harten'sTclieme ⑴ _ Mocified Marten's Scheme • I 7 —

1 j6

J" • • • • 0i5 • _ I _ • • • '0.l6 • • 0 “ • • • • 0.5 0 I _ _ _ ；

X K

( b ) Exact Solution

• R e l a x e d S c h e m e P~ — 一一 — Marten's S c h e m e f —- — fVIoclified H arte n s S c h e m e

1 4 • • • • • m-m — 辑 m — _ _ • _ •

- — � � . - 、、-\

1 . 3 - \

• f J

1 . 1 - \ •\ ： \

一 \ _ \ - • •

1 _ ^

O Q I I 1 I I I I I 1 I I 1 i 1 1 0 . 3 0 . 4 0 . 5

X

Figure 3.1: Test 6: Comparison of the smearing effect at the contact discontinuity

among exact solution (solid line) and numerical solutions (relaxed scheme - square

symbol, Hapten's scheme - dashed line, modified Harten's scheme - dash-dot line) at

time t = 1.0.


Internal Density p'H 二 26.1886 kgm—3 pfj 二 21.733

Internal Velocity u丨h = 0 m/s uh = ^

Internal Pressure p'fj = 326 p.s.i. pn 二 15.514

External Density p' = 1.205 kgm—3 p乙二 1.0

External Velocity u'l = 0 m/s ul = 0

External Pressure p'L = 15 p.s.i. pi 二 0.715

Unit Time t' = 293 /is 力二 1

Sphere Radius j/q 二 1 in. xo = 0.25

Table 3.2： Initial Data for Explosion Problem.

3.1.1 Physical Description

The present numerical results are in fairly good agreement with the results

obtained by Liu et al. [7]. Immediately after the broken of the glass sphere at

力二 0, a spherical shock (main shock) is generated and moves outward in the

air. The compressed gas is expari(ieci through a spherical rarefaction wave and

a contact discontinuity separates the expanded gas from the air compressed

by the shock. Different from an one-dimensional shock tube problem, due to

the three-dimensionality of the flow, the region between the tail of the rar-

efaction wave and the main shock wave is not a steady-state region. The high

pressure gas upon passing through a spherical rarefaction wave, due to the

increase of volume from the dimensionality, must expand to a lower pressure

than that reached through an equivalent one-dimensional expansion. This

'over-expansion' must be compensated by a compression wave or shock. This

is the physical explanation of the formation of the second shock in a spherical

explosion (see [2]，[3] for reference). This second shock is rather weak and

propagates outward initially. Then its strength increases with time and the

second shock stops moving outward, but starts reversing to the inward direc-


tion at about t = 0.6. The contact discontinuity propagates outward slowly

from X 二 0.25 to 0.7 and starts reversing the direction at about t = 0.8 (see

figures 3.2 and 3.3). It is seen from the figures that the results calculated from

the relaxed scheme is highly agreed with those calculated by the modified

Harten's scheme.

The second shock reflects from the origin at about t = 1.2 and interacts with

the contact discontinuity at about t = 1.8. It propagates through the contact

surface and continues moving outward. The interaction causes the contact

discontinuity propagates slightly outward again and; meanwhile, ari inward

rarefaction wave is generated during the interaction. Similar to the formation

of the second shock, a rather weak, inward moving third shock follows (see

figures 3.4 and 3.5).

Similarly, the third shock reflects from the origin at about t = 2.7 (see

figures 3.6 and 3.7). It can be expected that a similar situation as stated above

would repeatedly occur resulting in the formation of the fourth, fifth and sixth

shocks, but they are too weak to be observed or formed in our simulation. The

third shock seems a bit more sharp when calculated by the modified Harten's

scheme than by the relaxed sclieine. Any the main features of the explosion

problem are captured by the relaxed scheme successfully.

3.1.2 Comparison with Previous Analytical and Experimental Re-

sults

Since the shocks and contact discontinuity especially the second and third

shock waves, are rather weak, the positions are obtained based on a rather

objective observation. However, for the purpose of comparison of the main

features of this problem, this rough observation is enough. In figure 3.8(a),

the analytical result predicted by Erode [1] are compared. The relaxed scheme

shows a good agreement with Brode's result. Figure 3.8(b) shows the compar-

ison between the result of the present numerical scheme and the experimental

result obtained by Boyer [2]. There are two main deviations for the results.


(a) 1 6 �

I u 容 t> t—0.2. Re laxed S c h e m e

X • t=0 .3 Re laxed S c h e m e 二i • t=0 .4 Re laxed S c h e m e

12 —f t=0 .2 Modrf ied Har ten 's S c h e m e 幻 - t t=0 .3 Modi f ied Har ten 's S c h e m e

I f — — _ _ 1=0.4 Modrf ied Har ten 's S c h e m e .2 10 - t 2 ： 1 岩 ^ - ^ Rarefact ion W a v e

t “ \ in i £ ： I £ 4 - \

— % S e c o n d Ma in S h o c k ： \ S h o c k

n I I I I I I I I I — I — I — I — I 1—I 0 0 . 5 1 1 . 5 2

Radial Distance (Dimensionless)

(b) 2 r~

— t> t= 0.2 Re laxed S c h e m e — D t=0.3 Re laxed S c h e m e — o t=0.4 Re laxed S c h e m e _ t=0.2 Modi f ied Harten 's S c h e m e — t=0.3 Modi f ied Harten 's S c h e m e

^ 5 _ — — — — t=0.4 Modi f ied Harten 's S c h e m e a> -

I : 遍

I 1 - i h r c i r

Li/ -邏 1 ^ ^ ^ Main Shock

M 丨 t 0 ^ 1 ‘ 1 1 ~ ~ L u l l丨 " I I I i i • III 丨丨 1 i J ^ i i i w i J ! — 細 _ l i i i .•丨 _ j r _ . _ . i i . " _ . » _ . “ . _ i _ i i i i J

0 0 . 5 1 1 . 5 2



(c)

\ t> t=0.2 Relaxed S c h e m e 冗 V D t=0.3 Relaxed S c h e m e ^ ^ T O t=0.4 Relaxed S c h e m e

_ \ t=0.2 Modi f ied Marten's S c h e m e _ r t=0.3 Modified Marten's S c h e m e

t 一一 — — t=0.4 Modif ied Harten 's S c h e m e

^ 1 5 - 1

g - t Rarefact ion wave

•i _ \ ^ > Contact Discontinuity

r � \ X 0> _ \ S e c o n d /

: ^ ^ f e l ^ Main Shock

f) 1 I I I I I I I I I I I I I 1 1 1 1 1 1 0 0 . 5 1 1 . 5 2

Radial Distance <Dimensionless�

(d)

2 . 5 —

- > ^ l ^wip®*®*^ Main S h o c k

0 Sfe I ^ ^ ^ '1 1 ^ 、二

$ ~ \ ^ J ^ G o n t a c t Discontinuity

•色 - \ ? OJ I \ 1 ' i t C> t=0.2 Relaxed S c h e m e 3 . ' \ i> J • t=0.3 Relaxed S c h e m e 2 J I • t=0.4 Relaxed S c h e m e ^ — ^ N a - ^ j ^ ^ t=0.2 Modif ied Marten's Scheme g" - t=0.3 Modrfied Marten's Scheme g . — — — - t=0.4 Modif ied Marten's Scheme

: S e c o n d Shock

n 1 1 1 1 1 1 1 1 I I I I I I 1 1 1 1 1 1 0 0 . 5 1 1.5 2


Figure 3.4: Numerical simulation of the explosion problem for (a) pressure (b) ve-

locity (c) density and (d) temperature from time t 二 1.15 to t 二 1.30.


(a)

1 . 2 5 -

一丨 / I 瓦 1 r y r t ^ ^ ^ M a i n S h o c k

H i / y ^ ^ / > t = 0 . 6 R e l a x e d S c h e m e

J^ / m a t = 0 . 8 R e l a x e d S c h e m e i W o t = 1 . 0 R e l a x e d S c h e m e

^ - 1=0.6 M o d i f i e d M a r t e n ' s S c h e m e 0 . 2 5 — S e c o n d ^ 1 = 0 . 8 M o d i f i e d H a i t e n ' s S c h e m e

f) J luiijiMmiiiwi nur “ • I 1 I ‘ • I I I I • 1 I I I I 0 0 . 5 1 1 . 5 2

R a d i a l D i s tance ( D i m e n s i o n l e s s )

(b) •> t = 0 . 6 R e l a x e d S c h e m e n t=0. 8 R e l a x e d So h e m e

一 • t= 1 .0 R e l a x e d S c h e m e 1 _ t = 0 .6 M o d i f i e d Mar ten ' s S c h e m e

— t = 0 . 8 M o d i f i e d Hapten 's S c h e m e _ Jf — — — — t = 1 . 0 M o d i f i e d Mar ten ' s S c h e m e

— : / _ ^ ^ ^ S e c o n d S h o c k

r \ Shock

I " ^ '' 0 5 — I y ^

_ 1 I I I I I I I—I—I—I—I—1—；—I—I—I—1—i—I—I 0 0.5 1 1.5 2

R a d i a l D i s t a n c e ( D i m e n s i o n l e s s )


(c) t> t=0 .6 Re laxed S c h e m e • t=O.S Re laxed S c h e m e

: A o t=1 .0 Re laxed S c h e m e _ » t=0 .6 Modrf ied Har ten 's S c h e m e _ P t=0.8 Modrf ied Har ten 's S c h e m e

2 J — — — — t=1 .0 Modrf ied Har ten 's S c h e m e

f 轉 Contact Discontinuity

I：： 7 £ f l ^ S e c o n d S h o c k

— — _ L

I ••••••••叫•»«••，•••••• Hn r\ OPJpHHyMBfflg I • I I ' I I I I ' I 丨 ‘ ‘ I I

0 0 . 5 1 1 .5 2 R a d i a l D i s t a n c e ( D i m e n s i o n l e s s )

( d )

M a in Shock

I ： ^ ^ t/> 匿 _ 丨_ — __ Contact Discontinuity

i : I a> 1 — > t=0.6 Re laxed S c h e m e 3 Ir a t=0.8 Relaxed S c h e m e % 孤 J ^ • t=1.0 Relaxed S c h e m e ^ — ^V.a^T'^^f i r t=0.6 Modif ied Harten's S c h e m e g - I ^^iSafeBfi^ t=0.8 Modrfied Harten's S c h e m e = 0 5 _ - - - - t=1.0 Modrfied Harten's S c h e m e

卜 . ： ^ I I ‘ r “ S e c o n d Shock

(MWHIlWMIWWMMWhMBWit � I I • I I 1 J I I I I I I I I 1 1 1 1 1

0 0 . 5 1 1 . 5 2





(a)

Y -, ^ t=1.15 Relaxed S c h e m e -丨 ° t=1.20 Relaxed S c h e m e

2 ， • t=1.30 Relaxed S c h e m e

'trt" ‘ p t=1.15 Modif ied Marten's S c h e m e t=1.20 Modif ied Haiten's Scheme

- g 2 ^ — — — — t=1.30 Modif ied Marten's S c h e m e

•i ^ g T I c 1 ^ —L

Q ^ Reflected Second Shock

一 - , f rom Origin Main Shock

1 I ； u

0.5 1 I",

Shock n Li_r I • I I I I I 1 1 1 1 I I I I I I I I

0 0 .5 1 1.5 2 Radial Distance (Dimensionless)

(b)

1 厂

- Main Shock 0 . 5 ^

o Second Shock ^ ^ "uT I Reflected from Origin „

！9L 丨 > t=1.15 Relaxed Scheme 公口 t=1.20 Relaxed Scheme

l i V M • t=1.30 Relaxed Scheme - / t=1.15 Modified Haiien's S c h e m e

^ t=1.20 Modified Marten's Scheme _ — — — - t=1.30 Modified Harten's Scheme

-1.5 - f \

- Second Shock

1 1 1 1 1 I I ‘ I I I � I I I I I I I 1 1 “ 0 0 .5 1 1.5 2



(c)

10 IP

^ > t=1.15 Relaxed S c h e m e 卜 • t=1.20 Relaxed S c h e m e r o t= 1.30 Relaxed S c h e m e

^ [“ t = 1 . 1 5 M o d i f i e d M a r t e n ' s S c h e m e > t=1.20 Modified Marten's S c h e m e

§3 r - — - - t=1.30 Modified Marten's S c h e m e

OJ P E I 0 [

4 U S e c o n d S h o c k Its |- Ref lected from Origin g Ji C o n t a c t D i s c o n t i n u r t y

O Main Shock

•丨丨丨丨I_丨丨"•‘丨旧丨川•賺丨"II丨1丨II

- e CO nd S h o c k r \ l n m n i ^ * ^ 1 I t I • ' I • I I I i I I I I 1 1 1

0 0 .5 1 1.5 2 R a d i a l D i s t a n c e ( D i m e n s i o n l e s s )

(cl)

6 r^

5.5 — » t二1.1 5 Relaxed Schem e 二 • t=1.20 Relaxed S c h e m e

_ — - 5 • t= 1 . 3 O R e l a x e d S c h e m e ^ S t=1.15 Modified Marten's S c h e m e _aj 4.5 fr t=1.20 Modified Marten's S c h e m e "E ^ — — — - t=1.30 Modified Marten's S c h e m e .2 4 0-绘日 1 3.5 Ef 己 3 婆 Second Shock Reflected from Origin ^ Main Shock 3 2 . 5 ^ ^ C o n t a c t D i s c o n t i n u r t y ^^^

t•； o.s.L、^：^^^^^^^'.™

二 Second Shock n I I I I I I I I 1 1 1 1 1 1 1 1 1 1 1 1

0 0 .5 1 1.5 2 Radial Distance (Dimens丨on丨ess)




2 ~

> t=1 .65 Re laxed S c h e m e ^ ^ ° t=1 .80 Re laxed S c h e m e ^ ^ ^ o t=1 .90 Re laxed S c h e m e - t=1 .65 Modi f ied Har ten 's S c h e m e

t=1 .80 Modi f ied Harten 's S c h e m e ITT 1 5 _ ^ ^ — — _ _ t = 1 . 9 0 Modi f ied Har ten 's S c h e m e 0 - 丨 nter€i.ction between

'5) — S e c o n d S h o c k and Contact Discontinuity g _ F o r m i n g Third S h o c k M o v i n g T o w a r d Or ig in

1 — 、丨知

1 ：芝 0 . 5 一 M a i n S h o c k

- S e c o n d S h o c k

f) I I I I I I I I I I t I I I I I I I I I_I I 1 I 1__I I I I I_I 0 0 . 5 1 1.5 2 2 . 5 3


(b)

Q 5 _ S e c o n d S h o c k Interacting with

- Contact Discontinuity Ma in Shock forming Third Shock

一 — ^

I ：

- ci . c . , > t= 1.65 Re laxed S c h e m e u_o .5 - S e c o n d S h o c k • t=1.80 Re laxed S c h e m e

• t=1.90 Re laxed S c h e m e > — t=1.65 Modi f ied Harten's S c h e m e

- t=1.80 Modi f ied Harten's S c h e m e — — — _ t=1.90 Modi f ied Harten's S c h e m e

—I—I—I—I~I—I—I—1—I_I—I—I—I I_I I I_I I_I I I I I_I—I—I—I—I_I 0 0 . 5 1 1.5 2 2 . 5 3



(c)

6 一

- > t=1 .65 Re laxed S c h e m e L 口 t=1 .80 Re laxed S c h e m e

5 H o t=1 .90 Re laxed S c h e m e -一._ I t=1 .65 Modi f ied Harten's S c h e m e W I t= 1.S0 Modi f ied Harten's S c h e m e 2 [ # — — t=1 .90 Modi f ied Harten's S c h e m e

M " f m

£ f S e c o n d S h o c k

I 已 p B W S e c o n d S h o c k Interacting with >% F & W Contact Discontinuity

2 I f % fo rming Third Shock

S p Ma in S h o c k

1 -

n I r I I I I I I I I I I I__I I__I_I 1__I 1__I__I__1_I_I_I__I_I 1 1_I 0 0 . 5 1 1 .5 2 2 . 6 3


(d)

> t=1 .65 Relaxed S c h e m e - Q t=1 .80 Relaxed S c h e m e J • t=1.90 Relaxed S c h e m e

^ ]_ t=1 .65 Modif ied Harten's S c h e m e r t=1 .80 Modif ied Haiten 's S c h e m e

_ _ _ _ t=1 .90 Modif ied Harten's S c h e m e K I § ^ S e c o n d Shock Interacting with Contact Discontinuity

4 forming Third S h o c k

i . 4 h ^ = � \ Ma in Shock

I 1 r M \ a> I te^ Contact Discontinuity

卜 L ——^ 0 5 ?" S e c o n d Shock

n r I I I I I I I I I I I I I I I I I 1—I—I—I—I—I—I—I—I—I I I I 0 0 . 5 1 1 . 5 2 2 . 5 3





(a)

2 r— ‘ t=2.0 Relaxed S c h e m e

- � t=2.2 Relaxed S c h e m e - o t=2.4 Relaxed S c h e m e - t=2.0 Modif ied Marten's S c h e m e - — ~ ^ t=2.2 Modif ied Marten's S c h e m e

1 5 一 — _ — _ t=2.4 Modif ied Marten's S c h e m e

'vi - ^

I , - K I X ~ 1 — V l ^ T _ S e c o n d S h o c k

I 0 5 Main S hock

Q- ) Third Shock — ^ — ^ Moving Toward Origin

n • I '丨 I_I I I I I_I I I I—I_1—I—I—I—I_I—I—I—I—I_I—1—I—I—I_I 0 0.5 1 1.5 2 2.5 3


<b)

> t=2.0 Relaxed Soli erne 0.5 厂口 t=2.2 Relaxed Scheme

_ • t=2.4 Relaxed Scheme t=2.0 Modified Marten's S c h e m e

- t=2.2 Modified Marten's S c h e m e _ — — — _ t=2.4 Modrfied Marten's Scheme

l/T _ Third Shock o> 售 0.25 -

^ ^ Main Shock

_n OR I I—I—I_I—I—I—I—I__I—I—I—I—I_I—I I I I_I I I I I_I—I—I—1—I_I 0 0.5 1 1.5 2 2.5 3



<c)

> t=2 .0 Re laxed S c h e m e ® r " o t=2 .2 Re laxed S c h e m e

�： • t=2 .4 Re laxed S c h e m e L t=2.0 Modi f ied Marten's S c h e m e r t=2.2 Modified Haiten's S c h e m e

^ r — — — _ t=2.4 Modi f ied Marten's S c h e m e

f ；

•H 4 r T h i r d S h o c k

I - ^ ^ E 3 ， C o n t a c t D i s c o n t i n u i t y

警 2 斷 Z 0 S e c o n d S h o c k Ma in S h o c k

爆 ^ ^ “ f) • I I I I I I I • I _ I I I I • I I I I I I I I I I I i I I I

0 0.5 1 1.5 2 2.5 3 Radial Distance (Dimensionless)

(d)

> t=2.0 Re laxed S c h e m e - • t=2.2 Relaxed S c h e m e

3 “ • t=2.4 Relaxed S c h e m e t=2.0 Modif ied Hai ten 's S c h e m e t=2.2 Modi f ied Marten's S c h e m e

- . _ __ — — t=2.4 Modified Haiten's S c h e m e iS 2.5 it

^ I ___ 囊 S e c o n d Shock

1 2 | 一啣二

Oi ‘ ， I Ma in S h o c k

I \ O) S. Contact Discontinuity

n L_i—I I—I I I 1 I I I I I I I I I I I I 1—I—I—I——I—I——i——I—I—I—I 0 0.5 1 1.5 2 2.5 3





<a)

1 �

I ^ Second Shock

^ y ^ Refleotion of Third Shock .E 0 . 5 from Origin

3 — j n ' t=2.5 Relaxed Scheme ijy r w - t=2.7 Relaxed Scheme

n � “ t=2.9 Relaxed Scheme Q- 一 ^ t=2.5 Modified Marten's Scheme

_ t=2.7 Modified Marten's Scheme 一 — — — — t = 2 . 9 Modified Marten's Scheme

n I I I • I I I 丨 I I I I I 1 1 1 1 1 1 1 1 0.5 1 1.5 2

Radial Distance (D imens ion less�

( b )

o t=2.5 Relaxed Scheme _ a t=2.7 Relaxed Scheme

• t=2.9 Relaxed Scheme - t=2.5 Modified Haiten's Scheme

t=2.7 Modified Marten's Scheme 0.25 - — — — — t=2.9 Modified Marten's Scheme

<D -

/ Second Shock

Third Shock

n OR ‘ I I I I I I I I I 1 1 1 1 1 1 1 1 1 fi _。」。0 0.5 1 1.5 2



(c )

6 I — - > t=2.5 Re laxed S c h e m e - • t=2.7 Re laxed S c h e m e - • t=2.9 Re laxed S c h e m e

5 广 t=2.5 Modi f ied Marten's S c h e m e � t=2.7 Modi f ied Marten's S c h e m e

^ ^ “ — — — — t=2.9 Modi f ied Marten's S c h e m e K �

I 寸 cu -£ 3 -

己 [p Contact Discontinuity

I 2 � ^ ^ ^ ^ ^ ^ ^ S e c o n d S h o c k

1 ^ ^ ^ 細麵丨丨 " 丨丨

n � I I • I 睡 _ _ ‘ • • 丨 I I 1 1 1 1 1 1 1

0.5 1 1.5 2 Radial Distance {D imens ion less�

(d) t> t=2.5 Relaxed S c h e m e

p Q t=2.7 Relaxed S c h e m e 2.5 — • t=2.9 Relaxed S c h e m e

J- t=2.5 Modif ied Ha i ten 's S c h e m e 1 t=2.7 Modrfied Marten's S c h e m e

^ J \ _ _ _ _ t=2.9 Modified Harten's Scheme

% 2 1\ _ _ _ _

i l r — ^ ― c -|<fc. % fe^ S e c o n d S h o c k

I I V \ S 广 Contact Discontinuity E >

门 I I t I I I I I I I I I I I I I I ‘ ‘ I I 0 0.5 1 1.5 2





First, the time for the second shock to reflect at the origin is later in the ex-

periment. Second, before the interaction with the second shock, the contact

discontinuity in the experiment do not indicate any inward motion. Our de-

viations obtained are the same as those calculated by the modified Harten's

scheme. Liu et al. [7] suggested three reasons for the deviations. First, the glass

sphere may not have broken instantaneously and it results in a not completely

spherically symmetric flow. Second, the moving fragments of the broken glass

diaphragm causes the energy loss in the flow system. The adjustment by in-

creasing the pressure may partly compensate the loss of the kinetic energy

residing in each fragment, but the loss of the thermal energy is not considered.

Third, the effect of the slower moving fragments, which have much greater

inertia due to their higher density than air，in the flow and across the contact

discontinuity may greatly contribute to the asymmetry of the flow.

3.2 Cylindrical Explosion Problem

For the cylindrical explosion problem, same numerical settings are used as in

the spherical case except that a = 2.


The physics of the cylindrical flow is similar to the spherical flow. Figure

3.9 compares the movement of shocks and contact discontinuities for these

two flows. The main shock is moving faster in the cylindrical flow than in

the spherical flow. Both second shocks propagate outward at first and then

reverse backward to the origin. The cylindrical second shock reverses at a

later time (about t = I) than the spherical case (t = 0.6). So, the reflection

of the second shocks from the origin is later for the cylindrical case (t : 2.05)

than the spherical case(^ = 1.2). Although it is not shown in the figure, it is

expected that the third shock of the cylindrical case would be formed at about

t 二 3.2 when the second shock interacts with the contact discontinuity.


(a)

5 p - > Ma in S h o c k - Re laxed S c h e m e I D S e c o n d S h o c k - Relaxed S c h e m e

4 <3 Third S h o c k - Re laxed S c h e m e 二 • Contact Discontinuity - Re laxed S c h e m e - Ma in S h o c k - Brode Analytical

g 4 — > S e c o n d S h o c k - Brode Analytical • ^ 三 • Contact Discontinuity - Brode Analyt ical .2 3.5 二

E 3 二 Z S E . ^ 2 . 5 - Z 0 = M a i n S h o c k C • r>> ra 2 - -17? 二；口

••口口口

1 = ’、>、斤口 z �

« 1 厂一 C o n t a c t Discontinuity •口•口口口口 Third S h o c k

r S e c o n d S hock • •二•口口f" , � � < 〈工…了 f) • I I I I ^ I I I 1__*L，UniU • I_I I I I I__I I I I I_T iHr rtl__I I I 0 0.5 1 1.5 2 2.5 3

T i m e (D imens ion less )

(b)

5 r— > fAain Shock - Relaxed S c h e m o 二 u S e c o n d S h o c k - R e l a x e d S c h e m e

4 ^ _ c T h i r d S h o c k - R e l a x e d S c h e m e 二 o Contact Discontinuity - Relaxed S c h e m e 3 » Ma in S h o c k - Boyer Exper iment

沿 4 — • S e c o n d Shock - Boyer Exper iment • ^ 三 • Contact Discontinuity - Boyer Exper iment >>

.S 3.5 -= = z E 3 二 Z

— r>> Q I g 三 Ma in Shock

•它 = z ••口

•2 : - Contact Discontinuity n•口口

"CJ . - • 4 4 • • •门 6口 9 • • 2 ^ ~ • • 一 • ••口 - Third Shock

S e c o ” d S h o c k • / o - , /-I 1~ I I I I r I 1 I I I I 口印口 I I I I 产 fa I I I I I I __I I__I

O 0 . 5 1 1 .5 2 2 . 5 3 Time (D imension less)

Figure 3.8: Comparison between the relaxed scheme and the (a) analytical (Brode),

(b) experimental (Boyer) results of the spherical explosion.


5 > Main Shock - Spherical ： • Second Shock - Spherical ： < Third Shock - Spherical

4.5 - o Contact Discontinuity - Spherical 一 -_ V Main Shock - Cylindrical i? 4 L • Second Shock - Cylindrical a> - • Contact Discontinuity - Cylindrical

I 3 . 5 L

E 3 二 Q ： Main Shock

I . 少 Z Z

一 - •口

夸 = ••口•• Third Shock

n ~ I I I [ I I I I I I I Ofn^ 1 I I I I I I t , ^ • T I I I I I 0 0.5 1 1.5 2 2.5 3

Time (Dimensionless)

Figure 3.9: Comparison between the spherical and the cylindrical explosion.


3.2.2 Two-Dimensional Model

As mentioned before, the convenience of the generalization to a higher dimen-

sional space of the relaxation scheme is one of its advantages. We take this

advantage to calculate the two-dimensional model of the explosion problem in

order to compare with the cylindrical approximation model (quasi-ID model).

The 2D scheme has been described in the last section of the previous chapter.

As a 2D 'Riemann problem，，the two sets of initial data are separated by a circle

with radius 0.25. For the points adjacent to the quadrilateral cells cutting the

initial discontinuity, the initial data is modified by assigning the area-weighted

value of the two sets of data. This 'smoothing' process avoids the formation of

small amplitude waves created at early times by the staircase configuration of

the data. In the following numerical results, CFL number二 0.3, the adjusting

factor j3 = 1, Ax = Ay = 0.1

In figure 3.10(a), the quantities of pressure, density, velocity and temper-

ature between the quasi-lD and 2D models are compared at 力=1.65. The

radial distance in the 2D model is extracted from the positive x-a:x:is. At this

particular time position, the second shock has been reflected from d i e � n g m

but has not interacted with the contact discontinuity yet. Figure 3.10(b) shows

the situation at time t 二 2.2, where the third shock has just been formed and

propagating inward. Agreement is good between the two models except for

the position of the contact discontinuity and the position very near the ori-

gin. We also observe that there is a strange 'trough' for the velocity profile

at about the position of the contact discontinuity. Since the diversities stated

above are much worse if we don't carry out the 'smoothing' process of the

area-weighted for the initial values, or using a coarser gird. So, it is believed

that the rectangular grid of our finite difference scheme which results in a

non-circular geometry of the initial data causes those diversities stated above.

Although the density and temperature profiles at the position very near the

origin are different for the two models, however, the highly agreement of the


(a) 2厂

2 D M o d e l ： 2 D M o d e l I . ” o Quasi 1D Model 1.9 二 o Model

i i r y 0.2 ^__:

ofc=f , • . , I I I • , . • . I . • • i I ' ' -I • ' ' ' j ' ' ' ' 3 ' ' ' ' i Distance (nimeitsinnless Ratlml D istancc < Dimension less)

2.5 r-2Q Model • 20 Model OJ3 L o Quasi-1 • Modal o QuasMD Model

-0.4 - •

： I . _ . _ ~ « — • ~ I — ' ~ • • , I ~ ~ — ' ~ I ^ 2 3 A 0 1 2 3 4 R dUil Distance (D linenslunlesst Radial D»tarice mirrieris'iciriles )

(b) 1 " r on … � . 2D Mendel

： - M o d e . ；。 Q u a m D Model

3.5 -

l i I ” t o e S 2 ]

of • • I • j I • I I ” I , I j I • •,丄 � 4 ' ' ' ' -i • ' ' ' j ' ' ' ' i ' ' ' ' A R.Klial Disrance 4 Dimension less) Rartkil O isrtancR <r>inu?i»s«>nless>

0-4 r 2D Model F 2D Model

o Quasi-1D Modal ^ ^ ^ = Quas-iD Model

I � : A X t ~ . _ � f

t � � X i ：

- o . e ： . . . . . , . . • • • • • . . • : . • • _ . . ‘ . I . _ . _ . _ . _ I _ . . _ . . _ I -OB^‘ ‘ ‘ ‘ ：： ‘ 2 3 4 Q 1 2 3 ^ RaOMI Distcirice |D imensioriless) Rodiol D istance (Diinen ioi l'evs)

Figure 3.10: Comparison between the 2D model and the quasi-ID model of the

cylindrical explosion at t ime (a) t 二 1.65 and (b) t = 2.2.

CHAPTER 3. NUMERICAL RESULTS ,5]

Figure 3.11: Three-dimensional plot of the pressure contour of the cylindrical ex-

plosion at time i — 2.5.

shock position away from the origin shows that the quasi-ID model is a good

approximation to the cylindrical symmetric problem. Our reflective boundaries

in (2.13) and the compatible equations at the origin (2.8) are acceptable for

the simulation of the reflection of shock at the origin. Also, negligible or even

none dimensional effect are observed in affecting the waves interaction and re-

flection on the numerical scheme. Figure 3.11 shows the three-dimensional plot

of the pressure contour at time t 二 2.5. The roughly circular shape indicates

the cylindrical symmetric of the flow.


Internal Density 二 1.0

Internal Velocity ul = 0

Internal Pressure pl — 1-0

External Density = 4.0

External Velocity uh 二 Q

External Pressure pH = 4.0

Table 3.3: Initial Data for Implosion Problem.

3.3 Spherical Implosion Problem

The initial condition of the implosion problem are summarized in table 3.3.

The radius of the sphere is 0.25. In the implosion problem, the numerical

settings are Ax = 0.05, CFL number= 0.3，the adjusting factor (3=1.


At the time of the glass sphere broken (t 二 0): a moving inward shock wave

(main shock) and a moving outward rarefaction wave form. A contact discon-

tinuity, which is moving inward separates the shock and the rarefaction waves.

Different from the explosion problem, there is no second shock follows at this

moment (see figure 3.12). The main shock then reflects from the origin and

interacts with the contact discontinuity at about t = 0.2. Due to the inter-

action, the contact surface stops moving inward. The contact discontinuity

propagates slightly outward for a short time and then stays stationary at the

position X 二 0.175. A second shock which is moving inward is generated due to

the interaction. The main features of the implosion problem are captured by

the relaxed scheme, as effectively as the modified Harten's scheme. Pressure

is imploded at about t = 0.3 that indicates the reflection of the second shock

at the origin (figure 3.14). We don't observe the formation of the third shock


when the second shock encounters the contact discontinuity. It may because

it is physically too weak to observe.

3.4 Cylindrical Implosion Problem

For the cylindrical implosion problem, same numerical settings are used as in

the spherical case except that a = 2. For the initial setting stated above, it is

exactly the same model as Sod [6] and Liu et al. [7 .


As in the explosion case, the physics of the cylindrical implosion is similar

to the spherical implosion. Figure 3.16 compares the movement of shocks and

contact discontinuities for these two implosion flows. The main shock interacts

with the contact discontinuity at about the same position in both cases. Unlike

the case of the explosion problem, the main and second shocks are moving a

bit slower in the cylindrical flow than in the spherical flow. No third shock is

observed in both cases and it may because the third shock is too weak to be

observed

3.4.2 Two-Dimensional Model

Similar to the analysis in the explosion problem, a comparison between 2D

model and the quasi-ID model is made. The numerical settings of the 2D

model are, CFL number= 0.3, the adjusting factor 二 1 and Ax = Ay = 0.05.

Good agreement is shown in figure 3.17 between the 2D model and the quasi-

ID model except at the position near the origin. Due to the reasons stated

before in the explosion section, same conclusion can be drawn in this implosion

case. The quasi-ID model as an approximation model to the cylindrical flow

is acceptable. Figure 3.18 shows the three-dimensional plot of the pressure

contour at time t 二 0.5. The roughly circular shape indicates the cylindrical

symmetric of the flow.


(a)

OQ > t = 0 . 0 5 R e l a x e d S c h e m e ‘ r � ° t = 0 . 1 0 R e l a x e d S c h e m e

i io o t =0 .13 Re laxed S c h e m e 1 3 J- I <3 t = 0 . 1 5 R e l a x e d S c h e m e

4 ； j\ t=0 .05 Modi f ied Ha i ten ' s S c h e m e 1 a L. {' t = 0 . 1 0 Modif ied Marten's S c h e m e

"vT r ； I t=0 .13 Modi f ied Ha i ten ' s S c h e m e trt 4 /中< t=0 .15 Modi f ied Marten's S c h e m e

1 14 . 2 \ c 12 V" Ref lected M a in S h o c k f rom Or ig in <U “ : �

I 8 二 cn - <3> •ij t/t -6 - \ \ Q- ： M a i n S h o c k

I riiiii'iiiiiiii imin iin ih 11 ill 11 r IM r 1 1 1 — i — i — i — i — I — i — i — i — i — I — i — i — i — i — I O 0.1 0 . 2 0 . 3 0 . 4 0 . 5


(b) o t=0 .05 Relaxed S c h e m e

� a t=0 .10 Relaxed S c h e m e ： o t=0 .13 Relaxed S c h e m e

. _ - t = 0 . 1 5 R e l a x e d S c h e m e 1 . o 一

: t=0 .05 Modi f ied Ha i ten 's S c h e m e = t=0 .10 Modi f ied Marten's S c h e m e

- H 1 二 — — — — t=0 .13 Modi f ied Marten's S c h e m e ^ - — "1=0.15 M o d i f i e d H a r t e r V s S c h e m e

. 1 0.5 - 一C工 Reflected Ma in S h o c k f r o m Origin

I I ' V Eo.5 > -1 ri： M a i n S h o c k

-

二 • -1.5 - o

r I I I I I I I I I I I I I I I I I I I I—I—I—I—I—I 0 0.1 0 . 2 0 . 3 0 . 4 0 . 5



(c) 1Q > t = 0 . 0 5 R e l a x e d S c h e m e

p o t=0 .10 Re laxed S c h e m e 1 \ o t=0 .13 Re laxed S c h e m e

'"j ^ <1 t=0 .15 Re laxed S c h e m e g t t=0 .05 Modi f ied Ha i ten 's S c h e m e

o 丨， t =0 .10 Modi f ied Harten 's S c h e m e 一 S t=0 .13 Modi f ied Ha i ten 's S c h e m e Jj) ^ — t=0 .15 Modi f ied Harten 's S c h e m e

考： j \ A .O < 经 ^ \ Ref lected M ain S h o c k f rom Orig in ^ J / V � �

i — o ：

“ y S i — Contact Discontinuity 2 一

~ p Main Shock I r V “ r , L . i i U r > “ _ , , - M < f * ^ I I I 1 1 1 ~ I I I — — I — — I ~ I ~ ~ I 1——I 1——J

O 0.1 0 . 2 0 . 3 0 . 4 0 . 5 Radial Distance (Dimensionless)

<d)

p > t = 0 . 0 5 R e l a x e d S c h e m e � D t=0 .10 Relaxed S c h e m e

o t=0.1 3 Relaxed S c h e m e � ^ t = 0 . 1 5 R e l a x e d S c h e m e _ ^ t=0 .05 Modi f ied Harten's S c h e m e

t=0 .10 Modi f ied Hai ten 's S c h e m e - \> - — —— — t=0 .13 Modi f ied Harten's S c h e m e

-S fi _ t=0 .15 Modi f ied Hai ten 's S c h e m e

§ k \ 巧 ^^^^^ Reflected Ma in Shock from Origin ��

a -4 K x ^ z Contact Discontinuity

f 2 - Ma in S h o c k ^ ^ ^ ^ ^ ^ ^

I I 1 t I I I I I I I t I I I I t I I I I 1 1 1 1

O 0.1 0 .2 0 . 3 0 . 4 0 . 5 Radial Distance (Dimensionless)

Figure 3.12： Numerical Simulation of Implosion for (a) Pressure (b) Velocity (c)

Density and (d) Temperature from time t — 0.05 to t = 0.15.


(a)

� > t=0 .18 Re laxed S c h e m e : ° t =0 .20 Re laxed S c h e m e

o t=0 .24 Re laxed S c h e m e : < t =0 .27 Re laxed S c h e m e “ t =0 .18 Modi f ied Har ten 's S c h e m e

.^^_ _ t =0 .20 Modi f ied Har ten 's S c h e m e 办 — — — - t = 0 . 2 4 Modi f ied Har ten 's S c h e m e

‘ ^ ^ t=0 .27 Modi f ied Har ten 's S c h e m e

： M o v ' i n g TowaJd O

O • I I I • I I I I 窗丨 I I I I I I I I I O 0.1 0.2 0.3 0.4 0.5

Radial Distance (D imensionless)

(b) 1 厂 > t=0 .18 Relaxed S c h e m e

- ° t=0 .20 Re laxed S c h e m e - o t=0 .24 Re laxed S c h e m e _ < t=0 .27 Re laxed S c h e m e _ t=0 .18 Modi f ied Harten 's S c h e m e _ t=0 .20 Modi f ied Harten 's S c h e m e

"vT 0.5 — — — — - t=0 .24 Modi f ied Harten 's S c h e m e ^ _ ^ ^ ^ t=0 .27 Modi f ied Harten 's S c h e m e

"cu > - 0 . 5 - S e c o n d Shock

- Mov ing Toward Origin

一 1 I 1 I I I 睡 I f • I I 丨 I I I I I I I I I I I I I I O 0.1 0.2 0.3 0.4 0.5



(C)

[T > t=0 .1S Re laxed S c h e m e _ 口 t =0 .20 Re laxed S c h e m e - • t=0 .24 Re laxed S c h e m e — < t=0 .27 Re laxed S c h e m e _ ！ — t=0 .18 Modi f ied Har ten 's S c h e m e ~ t=0 .20 Modi f ied Har ten 's S c h e m e

— 一一一 t=0 .24 Modi f ied Harten 's S c h e m e gj - M a i n S h o c k t=Q.27 Modi f ied Har ten 's S c h e m e •Si 丨 n teractwith g - Contact Discontinu'rty •JJ5 - to fo rm S e c o n d S h o c k g - M o v i n g T o w a r d Or ig in

^ � M • S h k

Ihock*^ Contact Discontinuity o I I I I I I I I 1 1—I 1 1 I • I I I I I I I I I I I

O 0.1 0.2 0.3 0.4 0.5 Radial Distance (Dimensionless)

(d)

> t = 0.1 S Relaxed S c h e m e _ ° t=0.20 Relaxed S c h e m e

• t=0 .24 Relaxed S c h e m e _ < t=0 .27 Relaxed S c h e m e

. . 广 - t=0 .18 Modi f ied Harten 's S c h e m e g ^ - t=0 .20 Modi f ied Harten 's S c h e m e h — 一一一 t=0 .24 Modi f ied Harten 's S c h e m e "c - t=0 .27 Modif ied Harten 's S c h e m e .O -c 5 — Main Shock 丨irteract with 髮 _ tp p. ^ ^ ^ ^ Contact Discontinuity .E _ V i ^ t o form S e c o n d S h o c k & - Mov ing Toward Origin

I 4 -

oJ - r \ I 鬼货 _ 一一一一――…Contact Discontinuity

# ^ | i Shock Main S h o c k

9 ll I I I I I I I I I I I I I I I I ' I ' I 1 1 1 1 J 0.1 0.2 0.3 0.4 0.5 Radial Distance (Dimensionless)




(a)

^ r~ Imp los ion of > t=0 .2S Re laxed S c h e m e 二 P r e s s u r e due to ° t =0 .30 Re laxed S c h e m e - Ref lect ion • t =0 .32 Re laxed S c h e m e : o f S e c o n d S h o c k < t = 0 . 3 5 Re laxed S c h e m e

_ “ at O r ig in t=0 .2S Modi f ied Har ten 's S c h e m e _一“ ' — t=0 .30 Modi f ied Har ten 's S c h e m e Jg / — — — - t = 0 . 3 2 Modi f ied Har ten 's S c h e m e Qi ' ^ t=Q.35 Modi f ied Har ten 's S c h e m e

i 發， M a i n S h o c k

^ - Ref lected ^ ^ " ‘ 4 ^ ^ ~ —

r j I I I I I I I I I I I I I I I I I I 1 1 I O 0 . 2 5 0 . 5 0.75 1

Radial Distance (D imension less)

(b)

t=0. 28 Re laxed S c h e m e � t二0.30 Re laxed S c h e m e

- o t=0 .32 Re laxed S c h e m e 0.2 一民弹 < t=0 .35 Re laxed S c h e m e

- r p ^ t=0 .28 Modi f ied Harten 's S c h e m e - r S ^ iJv t=0 .30 Modi f ied Ha i ten 's S c h e m e - a 7 — _ t=0 .32 Modi f ied Harten 's S c h e m e

Jg - f / / ] / 1 t=0 .35 Modi f ied Harten 's S c h e m e

‘ ：旧 ^ ^ I 1 I—I—I I I I I I i ' • I ' 1 ' I ' ' I

0 0.25 0.5 0.75 1 Radial Distance (Dimensionless)


(c)

7 n-^ t=0 .2S Re laxed S c h e m e 口 t=0 .30 Re laxed S c h e m e • t=0 .32 Re laxed S c h e m e < t=0 .35 Re laxed S c h e m e

^ - t=0 .28 Modi f ied Marten 's S c h e m e — b t=0 .30 Modi f ied Marten 's S c h e m e "t/T M — — — - t=0.32 Modified Marten's Scheme 2 I t=0 .35 Modi f ied Marten 's S c h e m e

•1 ,,丨 Ma in S h o c k

11 K ^ ^ o 3 \

Reflected Contact Discontinuity _ S e c o n d _ S h o c k

O I I I I I I I 1 I I I I I I I I I I 1 1


(d)

5 厂 Reflected > t=0.2S Relaxed S c h e m e - S e c o n d 口 t = 0 . 3 0 R e l a x e d S c h e m e

Shock • t=0 .32 Relaxed S c h e m e - < t=0 .35 Relaxed S c h e m e

一急 ^ ^ t=Q.28 Modi f ied Marten's S c h e m e ^ t=0 .30 Modif ied Marten's S c h e m e a> — — — - t=Q.32 Modif ied Marten's S c h e m e

"c ^ t=0 .35 Modif ied Marten's S c h e m e 0 4 -f g ^ 1 1 :: 1 Q ‘> ^ ^ ^ Contact Discontinuity

¥ B -<’ f

I � 卜 . ： ^ ^ 一脚

I I I 1 I I I I ) I I I I I I I I I 1 I


Figure 3 . 4 : Numerical simulation of the explosion problem for (a) pressure (b) ve-



(a )

6 � t = 0 . 4 R e l a x e d S c h e m e _ ° t = 0 . 6 R e l a x e d S c h e m e - • t = 0 . 8 R e l a x e d S c h e m e - < t = 1 . 0 R e l a x e d S c h e m e

5 .5 — t = 0 . 4 M o d i f i e d H a r t e n ' s S c h e m e _ --.. “ t = 0 . 6 M o d i f i e d H a r t e n ' s S c h e m e JJ5 I — — — - t = 0 . 8 M o d i f i e d H a r t e n ' s S c h e m e at fe t = 1 . 0 M o d i f i e d H a r t e n ' s S c h e m e

.O - S e c o n d 爹、 g - S h o c k 量

.i 4.5 二 S h o c k

I ~ o r . I [ I I I I I I I • I t I I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I 0 0.25 0.5 0.75 1 1 . 2 5 1.5 1.75

Rad ia l D is tance ( D i m e n s i o n l e s s )

( b )

- > t=0 .4 R e l a x e d S c h e m e _ ° t = 0 . 6 R e l a x e d S c h e m e

Q 2 _ • t = 0 . 8 R e l a x e d S c h e m e - < t=1 .0 R e l a x e d S c h e m e - 1=0-4 M o d r f i e d Ha r ten ' s S c h e m e _ t = 0 . 6 M o d i f i e d Har ten ' s S c h e m e - S e c o n d 铁 — — — - t = 0 . 8 M o d i f i e d Ha r ten ' s S c h e m e

g - S h o c k k \ t = 1 . 0 M o d i f i e d Ha r ten ' s S c h e m e

I I I I I I I I I I I I I I_I__I__1_I__I_I__I__I_1_I_I_I__I_I_I~I~I~I~I~I~I 0 0 . 2 5 0 . 5 0 . 7 5 1 1.25 1 . 5 1 . 7 5

Radia l D istance (D imens ion less )


<c)

6 � > t=0 .4 Re laxed S c h e m e ° t=0 .6 Re laxed S c h e m e o t=0 .8 Re laxed S c h e m e < t=1 .0 Re laxed S c h e m e

t=0.4 Modrf ied Harten 's S c h e m e t=0 .6 Modrf ied Harten 's S c h e m e

5 - t=0 .8 Modi f ied Harten 's S c h e m e </> ^ t=1 .0 Modi f ied Harten 's S c h e m e

•i ^ .Q 1 ^ ^ ^ ^ M a i n S h o c k I S "> u:' ^ ^ ~ Contact Discontinuity

。！J � 1 1 1 1 1 1 1 1 i I 塞 1 1 1 1 1 1 1 _ I _ _ I _ I _ I _ I _ I _ I _ I _ I _ I _ I _ _ I _ I _ I ~ I ~ I ~ I ~ I

0 0.25 0.5 0.75 1 1.25 1.5 1-75 R a d i a l D i s t a n c e ( D i m e n s i o n l e s s )

( d )

4 5 � ^ t=0.4 Relaxed Sc ! ‘ eme _ ° t = 0.6 Relaxed S c h e m e - • t=O.S Relaxed S c h e m e -爽 < t=1.0 Relaxed S c h e m e

^ . t=0.4 Modi f ied Harten's S c h e m e ^ 4 - a t=0.6 Modi f ied Harten's S c h e m e S 容^ f e — — — - t=O.S M o d m e d Harten's S c h e m e

"E , ^ ^ t=1.0 Modrf ied Harten's S c h e m e I

S 3.5 - I £ ] C o n t a c t D i s c o n t i n u i t y

a i：

3 ^ ：7 g “ ， Main Shock

I 2.5 ；- ^^Tl^^S^^；[；

� • • I I I _ ， I I I I I 1 丨_I_I_I_I 1_I_i_1 1_I_I_I——I_I 1_I~I~i 1~I~I 0 0.25 0.5 0.75 1 1.25 1.5 1.75

R a d i a l D i s t a n c e ( D i m e n s i o n l e s s )

F i g u r e 3.4: Numer i ca l s imulat ion of the explos ion p r o b l e m for (a ) pressure (b ) ve-

l o c i ty ( c ) density and (d ) t e m p e r a t u r e f r o m t i m e t 二 1.15 t o t 二 1.30.


p > Main Shock - Spherical

- • Second Shock - Spherical 1> _ o Contact Discontinuity - Spherical _ Main Shock - Cylindrical &

2 1 一 • Second Shock - Cylindrical / 广 S • Contact Discontinuity-Cylindric^il X • 0) j> Z —薩 0 - 7 m 0.75 - tx X ^ / •i — y z <U Main Shock > / Z 门 � 1 0 . 5 - Z / Z Second Shock

I : 1 -符 ^ ^ i jn Contact Discontinuity

：V八 0 I I ^ I I—I—I—I—I—I—I—I—I—I—I—I—I—I

0 0.25 0.5 0.75 1 Time (Dimenstonless)

Figure 3.16: Comparison between the spherical and the cylindrical implosion.


5 r- m 5 -2D Wtottel - 2D Model

a QLHSi-ID Model a ° Quasi-1D MoOei

I ； I

r| \f~ W • • _ _ I , • • • • . . , • I . , , , I 2 5 I I •‘ III_I I I_I_I—I—I—t—I_I——I——I——I

2 5� R d D rstancefDunensionilss) ‘ � Rad 'l fDimensionllss, ‘

5 p

^ I r I : "|��l : I / 竺 - : \

1.2: 1 / 1 : : \ • 善 � . 3 — V y I 3 : \

-0.5J I I I I • ‘ I ‘ ‘ ‘ I ' ‘ ‘ ‘ ‘ J 0.25 0.5 0.75 1 � Rmllll Distancefoimensionl!..) R miial D bt. nce (Dimensionless)

Figure 3.17: Comparison between the 2D model and the quasi-ID model of the

cylindrical implosion at time t 二 0.4.


Figure 3.18: Three-dimensional plot of the pressure contour of the cylindrical im-

plosion at time t 二 0.5.

Chapter 4

Conclusion

The present development provides a demonstration on the numerical simula-

tion of explosion and implosion problems by using the relaxed scheme.

Cylindrical and spherically symmetric explosion and implosion in air are

numerically simulated. The governing equation is a nonlinear systems of hy-

perbolic conservation laws with source terms. The inhomogeneous system is

split for a time step, into the advection problem which is a homogeneous hy-

perbolic problem: and the source problem which is cl system of ordinary differ-

ential equations (ODEs). The ODEs is solved by the second-order, two-stage

Runge-Kutta method. The pure advection hyperbolic problem is calculated

by the second-order relaxed scheme developed by Jin & Xin [16]. A constant

matrix is constructed in the application of the relaxed scheme. The nu-

merical dissipation of the relaxation scheme is to a large extent determined by

the matrix A and the choice of A becomes critical in applying the relaxation

scheme. An algorithm to choose the matrix is developed. The matrix is cho-

sen 'dynamically' such that it is tailor-made for each time step interval. The

matrix constructed as A = al where the constant a defined as the square of

the maximum eigenvalue inside the time interval for all spatial do-

main. We impose no additional criteria on the CFL number for the algorithm

proposed in practical use. Though we cannot prove the TVD property of the

scheme in system of conservation laws, a series of numerical tests are carried

65

CHAPTER 4. CONCLUSION 66

out to exam the applicability of the algorithm on the Euler system. The test

results are satisfactory.

By applying the relaxed scheme in the explosion and implosion problem,

four cases namely spherical explosion, cylindrical explosion, spherical implo-

sion and cylindrical implosion are presented. For the explosion problem, at the

time immediately after the broken of the spherical or cylindrical diaphragm,

the main shock is generated and moves outward in the air. The compressed

gas is expanded through a rarefaction wave and a contact discontinuity sepa-

rates the expanded gas from the air compressed by the main shock. Different

from an one-dimensional shock tube problem, due to the three-dimensionality

of the flow, the region between the tail of the rarefaction wave and the main

shock wave is not a steady-state region. The high pressure gas upon passing

through a spherical rarefaction wave, due to the increase of volume from di-

mensionality, must expand to a lower pressure than that reached through an

equivalent one-dimensional expansion. This 'over-expansion' is compensated

by a second shock. This second shock is rather weak and propagates outward

initially. Its strength increases with time and the second shock stops moving

outward, but starts reversing to the inward direction and reflects from the

origin. Also, the contact discontinuity propagates outward slowly and starts

reversing the direction until it interacts with the reflected second shock. The

contact discontinuity propagates slightly outward again and a third shock gen-

erated similarly. The numerical simulation has successfully captured all the

main features of the explosion problem including the propagation of the main

shock, the reflection of the second shock, the interaction with the contact dis-

continuity and the formation of the third shock, as well. Comparison is made

between previous numerical results [7], analytical results [1] and experimental

results [2]. Also, thanks to the advantage of the relaxation scheme to gen-

eralize to a higher dimensional space easily, a 2D model with circular initial

condition is calculated to check the cylindrical symmetric model. The results

of our quasi-ID model is found acceptable.

CHAPTER 4. CONCLUSION 67

For the implosion problem, at the time of the spherical or cylindrical di-

aphragm broken, an inward moving shock wave (main shock) and an outward

moving rarefaction wave form. A contact discontinuity, which is moving inward

separates the shock and the rarefaction waves. Unlike the explosion problem,

there is no second shock follows at this moment. The main shock then re-

flects from the origin and interacts with the contact discontinuity causing the

formation of the second shock (similar to the formation of the third shock in

the case of the explosion problem). Due to the interaction, the contact surface

stops moving inward but propagates slightly outward for a short time and then

stays stationary. The main features of the implosion problem, the reflection of

the main shock, the movement of the contact discontinuity and the formation

of the second shock are captured by the relaxed scheme, as effectively as the

modified Harten's scheme. We don't observe the formation of the third shock

when the second one encounters the contact discontinuity. It may because it

is physically too weak to observe. A 2D model has been presented so as to

compare to our cylindrical symmetric quasi-ID model.

The present research contribnteR a picture to understand the phvsics of the

explosion and implosion problems through the numerical siniulaLion by the

relaxed scheme. Meanwhile, the technique of the application of the relaxed

scheme is developed especially on the choosing of the important matrix A,

which plays the critical role on the accuracy and practicality of the scheme.

Although the matrix A chosen by the algorithm developed in this research

has shown to be practically capable for most Euler problems by the numeri-

cal tests, it is still a rough estimate and in no way guarantee the dissipative

condition to be always satisfied. Based on the present finding, it would be

valuable for the future to find a procedure for choosing A that will rigorously

be proved to fulfill the dissipative condition. Moreover, noted the smearing

effect on the contact discontinuity (figure 3.1(b))，equipping the relaxation

scheme with ACM technique can also be considered as a future development

so as to eliminate the smearing effect.

Bibliography

1] H. L. Erode, "Theoretical Solutions of Spherical Shock Tube Blasts," The

RAND Corporation, RM-1974, 1957.

2] D. W. Boyer, "An Experimental Study of the Explosion Generated by a

Pressurized Sphere,” J. Fluid Medi., vol. 9，pp. 401-429, 1960.

3] M. P. Friedman, "A Simplified Analysis of Spherical and Cylindrical Blast

Waves," J. Fluid Meek., vol. 11, pp. 1-15, 1961.

4] W. E. Baker, Explosions in Air^ Wilfred Baker Engineering, San Antonio,

1973.

5J I. I. Glass, Nonstationary Flows and Shock Waves^ Oxford Science Publi-

cations, Oxford, 1994.

6] G. A. Sod, "A numerical study of a converging cylindrical shock," J. Fluid

Mech., vol. 83, pp. 785-794, 1977.

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