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Finite element analysis for the navier - stokes equations

Cheng, Ken Y-K

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Cheng, Ken Y-K (1977) Finite element analysis for the navier - stokes equations, Durham theses, DurhamUniversity. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/9133/

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FINITE ELEMENT ANALYSIS

FOR THE NAVIER - STOKES EQUATIONS

A t h e s i s submitted f o r the degree of

Master of S c i e n c e

i n the

U n i v e r s i t y of Durham

The copyright of this thesis rests with the author.

No quotation from it should be published without

his prior written consent and information derived

from it should be acknowledged.

by

Ken Y-K Cheng

June 1977

ACKNOWLEDGEMENTS

Th i s work was c a r r i e d out under the s u p e r v i s i o n of Mr.J.M.Wilson.

Without h i s constant advice and support the work could not have

s t a r t e d . Without h i s constant guidance and a s s i s t a n c e as w e l l as h i s

personal encouragement i t could not have reached t h i s stage. I am

extremely g r a t e f u l to my s u p e r v i s o r .

I wish to express my thanks to members of s t a f f of the Computer

Unit Department, U n i v e r s i t y of Durham, f o r t h e i r c o n t r i b u t i o n to the

va r i o u s problems r e l e v a n t to the development of the computer programmes.

My g r a t i t u d e must a l s o go to the s t a f f of Durham U n i v e r s i t y

L i b r a r y ,

ABSTRACT

The f i n i t e element method was employed to s o l v e two-dimensional,

unsteady, i n c o m p r e s s i b l e , v i s c o u s f l u i d flow problems. A p r a c t i c a l

computation procedure i s presented. A complete f i n i t e element computer

program has been developed. The numerical technique i s based upon a

general formulation f o r the Navier-Stokes equations making use of a

combined v a r i a t i o n a l p r i n c i p l e f i n i t e element approach. S o l u t i o n to

the system of a l g e b r a i c equations i s approached by the Gaussian

e l i m i n a t i o n scheme. The time-dependent Navier-Stokes equations are

expressed i n terms of a stream f u n c t i o n equation and a t r a n s p o r t

equation. A v a r i a t i o n a l f u n c t i o n a l of the stream f u n c t i o n and a

p s e u d o - v a r i a t i o n a l f u n c t i o n a l of the v o r t i c i t y of the r e s p e c t i v e

boundary value problem i s presented. The p r e s s u r e d i s t r i b u t i o n and

v e l o c i t y p r o f i l e are determined from stream f u n c t i o n . Two numerical

examples are presented and compared w i t h present papers. Some now

i d e a s about the numerical method, obtained through numerical experiments,

are presented and d i s c u s s e d .

- i i i -

CONTENTS

Page

Acknowledgements i

A b s t r a c t i i

Contents i i i

L i s t of F i g u r e s and Tables v

Chapter 1. I n t r o d u c t i o n 1

Chapter 2. V a r i a t i o n a l Formulation of Navier-Stokes Equations 7

2.1 P r i n c i p l e s of V a r i a t i o n a l C a l c u l u s 7

2.2 Navier-Stokes Equations 10

2.3 V a r i a t i o n a l Formulation 15

Chapter 3. F i n i t e Element Model 18

3.1 I n t r o d u c t i o n 18 18

3.2 Matrix Formulation

3.3 I n t e g r a t i o n of the Matrix Equation 26

3.4 E v a l u a t i o n of the Matri c e s of Elements 28

3.5 P r e s s u r e and V e l o c i t y D i s t r i b u t i o n s 31

Chapter 4. Boundary Conditions and Numerical Procedures 39

4.1 Boundary Conditions 39

4.2 Numerical Procedures 43

Chapter 5. Computer Work 46

5.1 I n t r o d u c t i o n 46

5.2 Some D e s c r i p t i o n s 46

5.3 S i m p l i f i e d Flow Diagram f o r the F i n i t e

Element Programmes 49

Chapter 6. T e s t Examples 56

6.1 Example One 56

6.1.1. I n t r o d u c t i o n 56

6.1.2. E n t r y Length 56

6.1.3. I n i t i a l and Boundary Conditions 58 6.1.4. I t e r a t i o n Technique 60 6.1.5. D i s c u s s i o n and Conclusion 60

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Page

6.2 Example Two 69

6.2.1. I n t r o d u c t i o n 69

6.2.2. Boundary Conditions and I t e r a t i o n Technique 69

6.2.3. Conclusion 73

Chapter 7. D i s c u s s i o n 80

7 .1 Convergence Problems 80

7.2 Storage Problems 81

7.2.1. Front S o l u t i o n 81

7.2.2. Banded S o l u t i o n 82

7.3 Computer Time Problems 83

7.4 Boundary Conditions 83

7.5 F i n i t e - E l e m e n t Mesh 83

7.6 Some Observations 85

7.6.1. The Transmission Phenomena of a Mesh L i n e 85

7.6.2. Maximum S t a b l e Time Step 89

7.7 General D i s c u s s i o n 89

Chapter 8 Conclusions 92

References 94

Appendices: A. Matri c e s £Q.U ] and [i.Qv ] 105

B. F i n i t e Element Computer Programs 106

C. Contour Program One 119

D. Stre a m l i n e Contours i n Flow between P a r a l l e l

P l a t e s . 121

E. S t r e a m l i n e Contours 131

F. Sample Data L i s t i n g 132

G. Main R e s u l t s f o r Stream F u n c t i o n and V o r t i c i t y 160

H. Contour Program Two 142

I . Sample Data L i s t i n g 145

- v-

L i s t of F i g u r e s and Tables Page

F i g u r e 3-1 Three-node t r i a n g l e 19

F i g u r e 3-2 Six-node t r i a n g l e 32

F i g u r e 5-1 S i m p l i f i e d flow c h a r t 47

Fi g u r e 5-2 Flow diagram f o r the f i n i t e element programmes 50

F i g u r e 6-1 F i n i t e element mesh f o r f l u i d flow between p a r a l l e l p l a t e s 57

F i g u r e 6-2 Boundary c o n d i t i o n s f o r flow between p a r a l l e l p l a t e s 59

F i g u r e 6-3 V e l o c i t y d i s t r i b u t i o n f o r flow between p a r a l l e l

p l a t e s 63

F i g u r e 6-4 V e l o c i t y d i s t r i b u t i o n 64

F i g u r e 6-5 S t r e a m l i n e contours i n flow between p a r a l l e l p l a t e s 65

F i g u r e 6-6 V e l o c i t y d i s t i r u b t i o n f o r flow between p a r a l l e l p l a t e s . 66

Fi g u r e 6-7 V e l o c i t y d i s t r i b u t i o n f o r flow between p a r a l l e l p l a t e s 67

Fi g u r e 6-8 P r e s s u r e d i s t r i b u t i o n f o r flow between p a r a l l e l p l a t e s 68

F i g u r e 6-9 The geometry and boundary c o n d i t i o n s f o r i n t e r n a l f l u i d flow i n a c o n s t r i c t e d channel 70

F i g u r e 6-10 F i n i t e element mesh f o r f l u i d flow i n a c o n s t r i c t e d

channel 71

Fi g u r e 6-11 Stre a m l i n e contours 74

F i g u r e 6-12 Stre a m l i n e contours 75

F i g u r e 6-13 D i s t r i b u t i o n of the v o r t i c i t y on the w a l l 76

F i g u r e 6-14 V e l o c i t y d i s t r i b u t i o n i n a c o n s t r i c t e d channel 77

F i g u r e 6-15 P r e s s u r e d i s t r i b u t i o n 78

F i g u r e 7-1 Boundary c o n d i t i o n s f o r flow i n a channel of f i n i t e width w i t h some o b s t a c l e s 84

F i g u r e 7-2 Mesh l i n e s f o r a hole on a p l a t e or an o b s t a c l e i n a

f l u i d flow 87

F i g u r e 7-3 Flow round a c y l i n d e r 88

Table 6-1 Streamfunction and w a l l v o r t i c i t y d i s t r i b u t i o n s 62

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Chapter 1 INTRODUCTION

The Navier-Stokes equations governing the f l u i d flow problems, are

known to have a p p l i c a t i o n s to a l a r g e c l a s s of e n g i n e e r i n g problems.

E x a c t s o l u t i o n s of such a v i s c o u s f l u i d flow problem are not c u r r e n t l y

a v a i l a b l e . The n e c e s s i t y of p r o v i d i n g reasonable e s t i m a t e s f o r

complicated flow phenomena l e a v e s r e s e a r c h engineers very l i t t l e c h o i c e .

The numerical approach seems to be one of the very few a c c e p t a b l e t o o l s .

Even the numerical methods fac e d i f f i c u l t i e s a r i s i n g from the non-

l i n e a r i t y and complexity of the boundary i n v o l v e d . The p r e s e n t high­

speed, l a r g e storage d i g i t a l computers have now made i t p o s s i b l e to

s o l v e the Navier-Stokes equations. Numerical s o l u t i o n of the Navier-

Stokes equations u t i l i s i n g modern high-speed computers have been developed

by a number of i n v e s t i g a t o r s .

A f i n i t e - d i f f e r e n c e approach presented by Fromm and Harlow (57) has

had c o n s i d e r a b l e s u c c e s s i n s o l v i n g problems. Lee and Fung (79) used a

method which combines the conforms! mapping and f i n i t e - d i f f e r e n c e

technique to a n a l y s i s v i s c o u s flow problems. M i l l s (91) employed the

f i n i t e - d i f f e r e n c e scheme to s o l v e v i s c o u s flow through a pipe o r i f i c e

a t low Reynolds numbers. Rimon (105) a l s o used t h i s scheme to get

s o l u t i o n s of the i n c o m p r e s s i b l e time-dependent v i s c o u s f l u i d flow past

a t h i n o b l a t e spheroid. Dennis and Chang (35) employed i t to study

problems of steady flow past a c i r c u l a r c y l i n d e r . Using t h i s approach,

Greenspan (51) was doing numerical s t u d i e s of steady, v i s c o u s , i n c o m p r e s s i b l e

flow i n a channel w i t h a s t e p . He a l s o presented i n a second paper (52)

some u s e f u l equations to determine w a l l v o r t i c i t y a t some s p e c i a l s o l i d

w a l l s u r f a c e . L i n , Pepper and Lee (83) employed the f i n i t e - d i f f e r e n c e

techniques to a n a l y z e separated flows around a c i r c u l a r c y l i n d e r . Macagno

and Hung (87) made a study of a c a p t i v e annular eddy u s i n g the f i n i t e -

d i f f e r e n c e method. Roscoe (108) has been u s i n g a new f i n i t e - d i f f e r e n c e

approach to study the three-dimensional Navier-Stokes equations. C a r l s o n

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and Hornbeck (24) analysed the laminar entrance flow of an i n c o m p r e s s i b l e v i s c o u s f l u i d i n a square duct u s i n g the f i n i t e - d i f f e r e n c e procedures.

From these numerous s t u d i e s , i t seems to show th a t the a p p l i c a t i o n s

of the f i n i t e - d i f f e r e n c e method have been l i m i t e d due to c o m p l e x i t i e s

i n the developed computational procedures. I t seems to r e q u i r e l a r g e

amounts of computer time and s t o r a g e . Another important disadvantage of

the f i n i t e - d i f f e r e n c e methods i s the f a c t t h a t these methods r e l y mostly

on meshes of very r e g u l a r and symmetric p a t t e r n s . Great computational

d i f f i c u l t i e s are encountered i f the geometric c o n f i g u r a t i o n of the f l u i d

flow i s complicated and cannot be r e a d i l y transformed i n t o a mesh of

r e c t a n g u l a r p a t t e r n (25,62,63,77).

These d i f f i c u l t i e s can be overcome w i t h the f i n i t e - e l e m e n t method.

The f i n i t e - e l e m e n t has, i n g e n e r a l , c e r t a i n advantages over the f i n i t e -

d i f f e r e n c e approach. These are the ease with which i r r e g u l a r geometries,

non-uniform meshes and i m p o s i t i o n of appropriate boundary c o n d i t i o n s can

be a p p l i e d (4,16,21,37,62,96,97,100,117).

The f i n i t e - e l e m e n t method, developed i n i t i a l l y f o r s t r u c t u r a l and

s o l i d mechanics, has been a p p l i e d to some f l u i d flow problems. S t r u c t u r a l

and n o n - s t r u c t u r a l elements may o f t e n be i d e n t i c a l i n shape and, f u r t h e r ,

be represented by s i m i l a r mathematical e x p r e s s i o n s . The major d i f f e r e n c e

between the e l a s t i c i t y and f l u i d flow problems l i e s i n the boundary

c o n d i t i o n s to be s a t i s f i e d (14,21,22,100).

Oden (95) has presented a t h e o r e t i c a l f i n i t e element analogue f o r the

Navier-Stokes equations, but without a p r a c t i c a l numerical method. Olson

(100) presented a numerical procedure to i n v e s t i g a t e steady i n c o m p r e s s i b l e

flow problems u s i n g stream f u n c t i o n formulation. Some u s e f u l p r a c t i c a l

techniques can be learned from h i s paper. Yamada, Yokouchi and Ohtsubo

(129) used the p r e s s u r e - v e l o c i t y formulation to a n a l y s e steady flow

problems. Tong (124) presented r e s u l t s f o r steady flow u s i n g t h i s method

wi t h p r e s s u r e and v e l o c i t i e s as dependent v a r i a b l e s . Skiba employed

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a v a r i a t i o n a l approach and r e c t a n g u l a r elements to o b t a i n r e s u l t s f o r

steady convection flow i n a r e c t a n g u l a r c a v i t y . R,T-S Cheng (25) suggested

a v e r s a t i l e and widely a p p l i c a b l e q u a s i - v a r i a t i o n a l f o r m u lation to s o l v e

the time-dependent Navier-Stokes equations. Bratanow,Ecer and Kobiske

(16,17) s t u d i e d unsteady i n c o m p r e s s i b l e flow problems u s i n g a p e r t u r b a t i o n

technique f o r treatment of the n o n l i n e a r i t i e s i n the v a r i a t i o n a l formulation

of the v o r t i c i t y t r a n s p o r t equation, and employing hi g h e r - o r d e r f i n i t e

elements f o r a c o n s i s t e n t s o l u t i o n of the governing equations and i n

d e s c r i b i n g the boundary c o n d i t i o n s . Baker (4,5,6) used a G a l e r k i n method

and t r i a n g u l a r elements f o r unsteady flow.

Atkinson, Brocklebank.Card and Smith (2) s t u d i e d c r e e p i n g flow around

a sphere, flow through a converging c o n i c a l s e c t i o n , and developing flow

i n a c i r c u l a r pipe u s i n g the stream f u n c t i o n f o r m u l a t i o n . They employed

three-node t r i a n g u l a r elements w i t h stream f u n c t i o n and i t s f i r s t two

d e r i v a t i v e s s p e c i f i e d a t each node. T h i s kind of f o r m u l a t i o n r e q u i r e s

l e s s computer storage than ve l - o c i t y p r e s s u r e f ormulations, s i n c e there i s

only a s i n g l e equation to be s o l v e d . However, the n e c e s s i t y f o r f i r s t

order continuous ( C ^ 1 ^ ) elements would tend to make e x t e n s i o n to three

dimensional work d i f f i c u l t . Tong and Fung (124) used the stream f u n c t i o n

f o r m u l a t i o n as w e l l to i n v e s t i g a t e s l o w - v i s c o u s flow i n a c a p i l l a r y i n

the presence of moving p a r t i c l e s suspended i n the flow. T h e i r work has

d i r e c t a p p l i c a t i o n to the biomedical problem of determining the i n f l u e n c e

of red blood c e l l s on the flow i n c a p i l l a r y blood v e s s e l s . T a y l o r and

Hood employed the p r e s s u r e - v e l o c i t y formulation to study the problem of shear

induced f l u i d flow past a c a v i t y . Because the same i n t e r p o l a t i o n f u n c t i o n s

were used f o r both p r e s s u r e and v o l o c i t y f o r t h i s problem, the accuracy of

the s o l u t i o n i s open to q u e s t i o n . They have r e c e n t l y presented a

formulation u s i n g higher order shape f u n c t i o n s f o r v e l o c i t i e s than

p r e s s u r e ( 1 1 7 ) . Tay and Davis (116) used v a r i a t i o n a l p r i n c i p l e to

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study the problem of convection heat t r a n s f e r between p a r a l l e l p l a n e s .

Bratanow and E c e r (20) employed a v a r i a t i o n a l approach to a n a l y s e the

three-dimensional unsteady flow around o s c i l l a t i n g wings, and to study

unsteady aerodynamics ( 1 7 ) .

Using q u a d r a t i c polynomials shape f u n c t i o n s f o r v e l o c i t y and l i n e a r

polynomials shape f u n c t i o n s f o r p r e s s u r e , Kawahara and Yoshimura (71)

solved steady flow problems by the Newton-Raphson method and p e r t u r b a t i o n

method, and analysed unsteady flow problems by the p e r t u r b a t i o n method.

L a s k a r i s (77) developed a numerical procedure to study two-dimensional

compressible and i n c o m p r e s s i b l e , steady s t a t e , v i s c o u s f l u i d flow and

heat t r a n s f e r problems. The numerical scheme he presented i s based on

a g e n e r a l formulation f o r the system of hydrodynamic equations, t a k i n g

i n t o f u l l account n o n l i n e a r c o n v e c t i v e terms, v i s c o u s terms, and heat

conduction terms, and u s i n g the method of weighted r e s i d u a l s a p p l i e d

over d i s c r e t e , d i s t o r t e d r e c t a n g u l a r elements of the f l u i d flow r e g i o n s .

Leonard ( 8 0 ) employed the G a l e r k i n ' s method to s o l v e perturbed compressible

flow problems. I s s a c s (69) used a t r a n s f o r m a t i o n s i m i l a r to t h a t used

f o r q u a d r i l a t e r a l i s o p a r a m e t r i c elements to d e r i v e a curved c u b i c

t r i a n g u l a r element which has as nodal parameters the v a l u e of the f u n c t i o n

and i t s two d e r i v a t i v e s , and employed t h i s kind of element to study

p o t e n t i a l flow problems. He compared the t r i a n g u l a r element w i t h a

standard i s o p a r a m e t r i c element, and concluded t h a t t h i s k i n d of t r i a n g u l a r

element w i l l g i v e s i m i l a r a c c u r a c y a t a s i g n i f i c a n t l y lower c o s t . Brebbia

and Smith (110) employed l i n e a r i n t e r p o l a t i o n f u n c t i o n s , a lumped mass

system and a simple ES-ler time i n t e g r a t i o n scheme to a n a l y s e the two-

dimensional, unsteady, i n c o m p r e s s i b l e , v i s c o u s Navier-Stokes equations.

The r e s u l t s a r e not only extremely a c c u r a t e to d e s c r i b e the n a t u r a l

p h y s i c a l phenomena of the problem of v o r t e x s t r e e t development behind

a r e c t a n g u l a r o b s t r u c t i o n but a l s o h i g h l y economic i n computer time.

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There a r e s t i l l many other good papers concerning numerical treatments

of the Navier-Stokes equations. Some of them are chosen and w i l l be

presented i n the r e f e r e n c e s .

I n the present work, the f i n i t e element method was employed to s o l v e

two-dimensional, unsteady i n c o m p r e s s i b l e , v i s c o u s f l u i d flow problems.

A p r a c t i c a l computation procedure i s presented. A complete f i n i t e

element computer program has been developed. The formulation can be

modified to cover a number of d i f f e r e n t s i t u a t i o n s . The same computer

program can be used w i t h only minor m o d i f i c a t i o n to s o l v e other s i m i l a r

problems. The numerical technique i s based upon a g e n e r a l formulation f o r

the Navier-Stokes equations making use of a combined v a r i a t i o n a l p r i n c i p l e -

f i n i t e element approach, a p p l i e d over d i s c r e t e f i n i t e elements of the

f l u i d flow domain where the unknown f l u i d v a r i a b l e s are expressed

continuously i n terms of i n t e r p o l a t i o n f u n c t i o n s and unknown parameters.

S o l u t i o n to the system of a l g e b r a i c equations i s approached by the

Gaussian e l i m i n a t i o n scheme. I t i s b e l i e v e d t h a t t h i s numerical procedure

i s a l s o s u i t a b l e f o r a general three-dimensional problem. The time-

dependent Navier-Stokes equations are expressed i n terms of a stream

f u n c t i o n equation and a v o r t i c i t y t r a n s p o r t equation. A v a r i a t i o n a l

f u n c t i o n a l of the stream f u n c t i o n and a p s e u d o v a r i a t i o n a l f u n c t i o n a l of

the v o r t i c i t y of the r e s p e c t i v e boundary value problem w i l l be presented.

The p r e s s u r e d i s t r i b u t i o n and v e l o c i t y p r o f i l e s are determined from

stream f u n c t i o n .

As i n the c o n v e n t i o n a l procedure of time-dependent f l u i d flow,

a n a l y s i s was o f t e n c a r r i e d out by the incremental method, assuming t h a t

the v a l u e s c a l c u l a t e d i n the preceding s t e p keep constant during the

subsequent s m a l l time increments. T h i s commonly used idea i s a l s o

followed here. To circumvent the n o n l i n e a r i t y i n the Navier-Stokes

equations, the unsteady flow problem i s assumed to be l i n e a r i n the stream

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f u n c t i o n and v o r t i c i t y a t each time s t e p . S t e a d y - s t a t e s o l u t i o n s are

achieved by a l l o w i n g the time-dependent s o l u t i o n s to converge.

To demonstrate the e f f e c t i v e n e s s of t h i s numerical scheme, two

numerical expamples are presented and compared w i t h present papers.

The numerical procedure used seems to be f a i l r y s t a b l e , and flow trends

seem to be w e l l r e p r e s e n t e d . Some new i d e a s about the numerical method,

obtained through numerous numerical experiments, are presented and

d i s c u s s e d . Although t h e i r v a l i d i t y f o r a l l kinds of numerical schemes

has not been a s c e r t a i n e d y e t , i t i s hoped to b r i n g these o b s e r v a t i o n s to

people's a t t e n t i o n .

-7-

Chapter 2 VARIATIONAL FORMULATION OF NAVIER-STOKES EQUATIONS

2.1 P r i n c i p l e s of V a r i a t i o n a l C a l c u l u s

I n t h i s s e c t i o n , some fundamental p r i n c i p l e s of v a r i a t i o n a l c a l c u l u s ,

which w i l l be used i n the subsequent a n a l y s i s are presented.

V a r i a t i o n a l c a l c u l u s i s concerned p r i m a r i l y w i t h theory of maxima and

minima, but the f u n c t i o n s to be minimised or maximised are f u n c t i o n a l s .

The v a r i a t i o n a l c a l c u l u s , i n g e n e r a l , has alwasy been c l o s e l y a s s o c i a t e d

w i t h r e a l i s t i c problems of continuum mechanics. U s u a l l y the f u n c t i o n a l s

whose extreme v a l u e s are sought a r e e x p r e s s i o n s of some form of system

energy. For example, i n f l u i d mechanics, f o r an in c o m p r e s s i b l e , i n v i s c i d

flow, the k i n e t i c energy i s a minimum. And another example i s the

p r i n c i p l e of minimum t o t a l p o t e n t i a l energy f o r e l a s t i c continua.(45,63,127)

L e t us c o n s i d e r a simple f u n c t i o n a l expressed as

4> = f FO,t, f*,tM)d* (2-1 )

where

F i s an a r b i t r a r y f u n c t i o n of one independent v a r i a b l e i X.

^ — 3t ™ ax

Now the v a r i a t i o n of the f u n c t i o n a l i s d e f i n e d i n a manner s i m i l a r to

the c a l c u l u s d e f i n i t i o n of a t o t a l d i f f e r e n t i a l

I t i s obvious t h a t there i s an analogy between f i n d i n g the minimum

or maximum of a f u n c t i o n v i a o r d i n a r y c a l c u l u s and f i n d i n g the minimum

or maximum of a f u n c t i o n a l v i a v a r i a t i o n a l c a l c u l u s . ( 3 7 , 6 3 , I 2 7 )

-8-

So extending the concept of o r d i n a r y c a l c u l u s , the f o l l o w i n g equation i s obtained:

Extending the p r i n c i p l e s of o r d i n a r y c a l c u l u s again, i t can be

l e a r n e d t h a t the f i r s t v a r i a t i o n i s a l s o a commutative operator w i t h

both d i f f e r e n t i a t i o n and i n t e g r a t i o n i f the i n t e g r a t i o n l i m i t s a r e not

to be v a r i e d . (37,63,127)

So the f o l l o w i n g equations may be w r i t t e n .

$( j Fix) = |(JF)«U (2-4)

And equation ( 2-3 ) becomes

I n t e g r a t i n g each item by p a r t s , the f o l l o w i n g equation i s obtained.

= 0 ( 2 - 7 )

B e c a u s e a n d § 1 ^ are a r b i t r a r y a d m i s s i b l e v a r i a t i o n s , the i n t e g r a n d

and remaining terms of equation (. 2-7 ) must v a n i s h . Thus the n e c e s s a r y

c o n d i t i o n s for^&Xx) to minimise 0 ( x ) a r e as f o l l o w s :

-jeL-f 9F )+JL.fdF ) - Q (2-«)

E q u a t i o n ( 2-8 ) i s the governing d i f f e r e n t i a l equation f o r the

problem and i s c a l l e d the Euler-Lagrange equation, or j u s t the E u l e r

equation. The other two c o n d i t i o n s g i v e the n e c e s s a r y boundary c o n d i t i o n s .

From equation (.2 ~- 9 ) the f o l l o w i n g equation may be w r i t t e n . (37,63,127)

X*. I 8F 0

or 0

1 n^ and from equ e i t h e r

9E 0 31? X.

or x z

9 T 0

(2-11)

(2-12)

(2-13)

( 2 - 1 4 )

Equation ( 2 - 11) and (.2 - 13) a r e c a l l e d n a t u r a l boundary c o n d i t i o n s . I f

they are s a t i s f i e d , they a r e c a l l e d f r e e boundary c o n d i t i o n s . Equations

( 2 - 12 ) and ( 2 - 14) a r e c a l l e d geometric boundary condtions or f o r c e d

boundary c o n d i t i o n s . I t may be mentioned here t h a t the Euler-Lagrange

equation e x p r e s s e s only a n e c e s s a r y and not a s u f f i c i e n t c o n d i t i o n f o r a

minimum. So the s o l u t i o n of an E u l e r - Lagrange equation may not y i e l d

a f u n c t i o n t h a t minimises a g i v e n f u n c t i o n a l . (See l a t e r S e c t i o n 2.3

V a r i a t i o n a l F o r m u l a t i o n ) .

(2-9)

(2-10)

-10-

One of the p r i n c i p a l advantages of the f i n i t e element method

employing a s u i t a b l e , v a l i d v a r i a t i o n a l p r i n c i p l e i s . t h a t only the

geometric boundary c o n d i t i o n s need to be s p e c i f i e d . The n a t u r a l boundary

c o n d i t i o n s a r e a u t o m a t i c a l l y i n c o r p o r a t e d i n the formulation'. That i s

why a l l the boundary c o n d i t i o n s . have only been enforced.on r i g i d

boundaries i n t h i s work and why t h e ' n a t u r a l ' boundary c o n d i t i o n s are

always l e f t f o r the program.to approximate when a s u i t a b l e v a r i a t i o n a l

p r i n c i p l e - f i n i t e element method i s employed to d e a l w i t h t e s t problems.

Through numerical experiments i t has been found t h a t when a combined

v a r i a t i o n a l p r i n c i p l e - f i n i t e element method i s employed, the'natural'boundary

c o n d i t i o n s had b e t t e r not be s p e c i f i e d again, otherwise the r e s u l t s

may be i n e r r o r . (21,37,42,63,100,127,131) ( s e e Chapter .4).

A f u n c t i o n a l of two independent v a r i a b l e s has the form

Proceeding i n a s i m i l a r way, i t i s not d i f f i c u l t to d e r i v e the E u l e r -

Lagrange equations and boundary c o n d i t i o n s f o r the above f u n c t i o n a l .

The E u l e r equation f o r equation ( 2 - 1 5 ) i s

S i m i l a r l y , Euler-Lagrange equations and boundary c o n d i t i o n s f o r

other f u n c t i o n a l s may be d e r i v e d . Some more d e t a i l e d d i s c u s s i o n s and

a p p l i c a t i o n s w i l l be presented i n S e c t i o n 2.3.

2.2. Navier-Stokes Equations

The f u l l Navier-Stokes equations r e p r e s e n t i n g a balance of v i s c o u s

f o r c e s , i n e r t i a f o r c e s , and p r e s s u r e f o r c e s a re capable of d e s c r i b i n g

+ ( t ) - [ f F(X,Y, f. %,% t„, tyy -A

( 2 - 15)

3F dF ) 3v l v. a ^ y ( art**) ax ay \ a ^ y

dF 3* V dfv 3* V d% 0

( 2 - 1 6 )

-11-

some of the most i n t e r e s t i n g phenonmena i n f l u i d mechanics. For unsteady,

i n c o m p r e s s i b l e , two-dimensional, v i s c o u s f l u i d flow w i t h i n e r t i a , the

Navier-Stokes equations f o r a n a l y s i n g the motion of the f l u i d s can be

w r i t t e n as

where

u = v e l o c i t y v e c t o r = [ u , v " )

t = time

V = d i f f e r e n t i a l operator = -2— + - 2 — + _3_ ax ay a z

P = d e n s i t y of the f l u i d

F = body f o r c e v e c t o r

^ = kinematic v i s c o s i t y of the f l u i d

P = p r e s s u r e 2 V 1 = L a p l a c i a n operator - ~ #_ + 1? 4 ~ 9 * a 3 V1 5 2 a

u = v e l o c i t y i n X d i r e c t i o n

v = v e l o c i t y i n Y d i r e c t i o n

i n c o n s i s t e n t u n i t s .

The equation of c o n t i n u i t y f o r i n c o m p r e s s i b l e f l u i d i s

V - U. = 0

(2-17)

(2-18)

The v e l o c i t y components u and v may be expressed i n terms of a

stream f u n c t i o n l^r as

9 y (2-19)

The v e c t o r

-12-

may be w r i t t e n as

" = I "ay" > ~ "fx" 3 ( 2 _ 2 0 )

Using equation ( 2 - 1 9 ) , f l u i d r o t a t i o n or v o r t i c i t y i s def i n e d as the

averageangular v e l o c i t y of any two mutually p e r p e n d i c u l a r l i n e elements of

a f l u i d p a r t i c l e .

I n v e c t o r n o t a t i o n , the f o l l o w i n g equation can be w r i t t e n :

0 J = V * U. ( 2 _ 2 i )

A well-known v e c t o r i d e n t i t y shows t h a t f o r any f u n c t i o n P having

continuous f i r s t and second d e r i v a t i v e s ,

V " V P = 0 (2-22)

At the same time, the f o l l o w i n g equation may be w r i t t e n

V * F = 0 (2-23)

Taking the c u r l ( 7 * ) of both s i d e s of equation (2-17) g i v e s

From equations (2-20) and (2-21 ) } the stream f u n c t i o n s are r e l a t e d

to v o r t i c i t y as f o l l o w s :

= - CO (2-25)

Now the p r e s s u r e d i s t r i b u t i o n i s to be c a l c u l a t e d . The p r e s s u r e

f i e l d can be obtained by i n t e g r a t i n g the momentum equation. But, i n

ge n e r a l , i t seems th a t a Poisson type equation y i e l d s more a c c u r a t e numerical

computation r e s u l t s and uses l e s s computer time than the d i r e c t methods

based on the momentum equation (16,21,77). So the Poisson type equation

w i l l be d e r i v e d f i r s t . Equation (2-17) can be w r i t t e n as

-13-

3U t „ 3 U , 3U. ^ I ^ I 9P (2-26) a t ax ^ ay P r* p 3 *

^ L a x a a y 1 J

D i f f e r e n t i a t i n g t h i s equation w i t h r e s p e c t to x g i v e s

v a x a t ' ^ a x ax wJ 'V. ax a y axay >

= _ L ^ E L . _ L J!P_ + ,J (J!2JA_+ 3 2 au > ( 2 _ 2 7 ) P ax p 3x j v k ax'ax ay*ax '

S i m i l a r l y , from equation (2-17)

t t a y P F y T " a y + y L a x 1 a y 1 J ( 2 2 8 )

D i f f e r e n t i a t i n g i t w i t h r e s p e c t to y g i v e s

a t I a y a x "- 3*ay J L a y ay * a y » J a y

p 3y p ay a ^ ax* a y ay* ay J

Combining equations (2-29) and (2-27) g i v e s

a t ^ a x ay J v. a x J + C i ax 2 * ax a y v axay

P v ax ay / P ^ ax ay» > L ax« a *

+ a*au , a 2 a ^ + a z _9J£_] ay'ax 3x* ay ay J a y J

(2-29)

(2-30)

-14-

the f o l l o w i n g can be w r i t t e n (2-18)

V • U = 0

From equation (2-18) i t i s easy to get f o l l o w i n g r e l a t i o n s :

3 7 du d1 a u , a* dv , a* a v _ n ( 2 - 3 1 ) ax* ax a y 1 a * ax* a y a y 2 a y u

v. a x 3 V / u

and

a t ^ ax a y / w ( 2 - 3 2 )

u 3X a V axay * u ax 3 y + ^ a * - 0 ( 2 3 3 )

S u b s t i t u t i n g equations ( 2 - 3 1 ) , (2-32) and (2-33) i n t o equation (2-30)

g i v e s

/ au f , ? av au , / a v y l r c . _ L

I n the absence of body f o r c e s , the f o l l o w i n g equation can be obtained

v p - - p [ ( - » - ) M - i H * a - i 5 r - $ - ]

or

tf2p = r r r a v a n au a i r I v r u ay ax a y J (2-36)

* r L a y 2 a x J V a*ay ; J

- - p a

(2-37)

(2-38)

-15-

where

o = , r a i r au _ 3 u av i * ' i ax a y ax ay J ( 2 - 3 9 )

•> 3 V l a x 1 >• ax ay / J

The a n a l y s i s of unsteady, i n c o m p r e s s i b l e , two-dimensional, v i s c o u s

f l u i d flow i n v o l v e s the simultaneous s o l u t i o n of equations (2-24) and ( 2 - 2 5 ) .

Once the *v|/" and Lj f i e l d a r e known, the p r e s s u r e d i s t r i b u t i o n can be

c a l c u l a t e d from equation ( 2 - 3 7 ) .

2.3 V a r i a t i o n a l Formulation

B a s i c a l l y , the f i n i t e element method r e p r e s e n t s an approximate

procedure f o r s a t i s f y i n g the problem i n terms of i t s v a r i a t i o n a l f o rmulation.

I f the governing d i f f e r e n t i a l equations were a l l l i n e a r the v a r i a t i o n a l

f o r m u l ation would be s t r a i g h t f o r w a r d . The n o n - l i n e a r terms i n the Navier-

Stokes equations seem to have precluded the e x i s t e n c e of an a s s o c i a t e d

v a r i a t i o n a l p r i n c i p l e of the c l a s s i c a l k i n d . I t i s found that the N a v i e r -

Stokes equations cannot be d e r i v e d from a c l a s s i c a l v a r i a t i o n a l p r i n c i p l e

u n l e s s one of the terms t£x (\7 X(/C) or U*( \7t?.j i s zero ( 4 1 ) . However,

i t has been shown t h a t 'pseudo' p r i n c i p l e s can be obtained provided some

terms are not allowed to vary when the f i r s t v a r i a t i o n i s performed. The

p s e u d o - v a r i a t i o n a l f u n c t i o n a l f i n i t e - e l e m e n t method has the advantage of

s i m p l i c i t y and reduced computation. I t i s not a t r u e v a r i a t i o n a l method

s i n c e from another point of view i t can be regarded as a G a l e r k i n method

used w i t h a p a r t i c u l a r approximattion scheme.(21,25,63,100,127)

Equation (2-25) and (2-37) are i n the form of Poisson's equation, f o r

which a d i r e c t v a r i a t i o n a l formulation e x i s t s . From theorems of

v a r i a t i o n a l methods, by i n s p e c t i o n , the v a r i a t i o n a l f u n c t i o n a l s f o r equations

(2-25) and (2-37) can be w r i t t e n as f o l l o w s

-16-

(2-41)

where

* 3 = i j A ( ( ^ - ) 2

+ ( ^ - ) 2 J ^ - J A P P ^

and A i s the r e g i o n of i n t e r e s t . The v a r i a t i o n a l f u n c t i o n a l s 0 and 0„

e x i s t such t h a t the E u l e r - L a g r a n g i a n equations of equations (2-40) and

(2-41) are simply equations (2-25) and ( 2 - 3 7 ) . The f u n c t i o n s l ^ r and P

which s a t i s f y the r e q u i r e d boundary c o n d i t i o n s and which g i v e the f u n c t i o n a l

i n t e g r a l s 0 2 and 0^ are s o l u t i o n s of equations (2-25) and ( 2 - 3 7 ) .

3(A) S i m i l a r l y , by t a k i n g as an i n v a r i a n c e , a v a r i a t i o n a l f u n c t i o n a l 0^

3 "t

(2-42)

e x i s t s (16,19,25,41) such t h a t upon t a k i n g the f i r s t v a r i a t i o n of 0^, the

v o r t i c i t y t r a n s p o r t equation, equation (2-24) w i l l be recovered. The

f u n c t i o n ^ s a t i s f y i n g equation (2-24) and i t s boundary c o n d i t i o n s minimise

0^. Segregating stream f u n c t i o n and v o r t i c i t y s o l u t i o n s a c c o r d i n g to

d i f f e r e n t i n s t a n t s of time reduces the problem to one of c o n s e c u t i v e l y

minimising 0^ and 0g. T h i s can he c o n v e n i e n t l y accomplished by the f i n i t e

element method. (21,25,63,100,127)

A disadvantage of the procedure i s t h a t i t i s not known whether or not

a p a r t i c u l a r pseudo-functional w i l l y i e l d convergence or not u n t i l i t has

been t r i e d , s i n c e a mathematical c r i t e r i o n f o r convergence i s not y e t

a v a i l a b l e . ( 2 1 , 2 5 , 9 3 , 1 0 0 )

N o r r i e and V r i e s (93) suggested t h a t i f a c e r t a i n f u n c t i o n a l does not

converge, one has no choice but to modify i t ; such a change a l t e r s the

-17-

s t i f f n e s s matrix and may r e s u l t i n convergence.

Some experience w i t h the method a s s i s t s i n choosing an a p p r o p r i a t e

f u n c t i o n a l on i n t u i t i v e grounds. N o r r i e and V r i e s p o s t u l a t e tha t : "The

process w i l l converge i f the terms which dominate the p h y s i c a l behaviour

of the system are i n c l u d e d a r e those terms i n the f u n c t i o n a l which are

not i t e r a t e d upon but a r e used only i n the m i n i m i s a t i o n procedure". (93)

One of the disadvantages of the use of stream f u n c t i o n - v o r t i c i t y formulation

i s t h a t u n l e s s the v e l o c i t i e s are e n t i r e l y p r e s c r i b e d on a l l boundaries i t

i s o f t e n i m p o s s i b l e to e s t a b l i s h the v a l u e s of stream f u n c t i o n s on some

p o s i t i o n s of the boundary. T h i s i s p a r t i c u l a r l y s e r i o u s i n m u l t i p l e

connected boundaries, such as are presented by flow around o b s t a c l e s e t c .

To overcome these d i f f i c u l t i e s i t i s n e c e s s a r y to introduce a d d i t i o n a l

c o n s t r a i n t s on the r a t e of boundary work. ( s e e S e c t i o n 6.1, example one)

-18-

Chapter 3 FINITE ELEMENT MODEL

3.1 I n t r o d u c t i o n

The f i n i t e element method i s based on the use of s e r i e s expansions

w i t h i n subdomains or elements, i n t o which the domain of i n t e r e s t i s

d i v i d e d . I t i s a general numerical technique which provides an approximate

piece w i s e continuous r e p r e s e n t a t i o n of the unknown f i e l d v a r i a b l e s i n terms

of polynomials, sometimes c a l l e d i n t e r p o l a t i o n f u n c t i o n s or shape f u n c t i o n s

and model parameters (22,33,37,77,127).

The continuous region i s subdivided i n t o a f i n i t e number of elements

where the nodal v a l u e s and/or the p a r t i a l d e r i v a t i v e s of the dependent

v a r i a b l e s at p r e s c r i b e d p o i n t s , nodes of elements, become the unknown

parameters of the problem. The f i n i t e - e l e m e n t r e p r e s e n t a t i o n of the

dependent f i e l d v a r i a b l e must be able to provide an improved approximation

to the t r u e s o l u t i o n as s u c c e s s i v e s u b d i v i s i o n s of the domain u s i n g s m a l l e r

and s m a l l e r elements i s attempted.

The b a s i c s t e p s of the s o l u t i o n procedure a r e as f o l l o w s (22,33,37,127).

1. D i s c r e t i s a t i o n of the continuum.

2. S e l e c t i o n of i n t e r p o l a t i o n f u n c t i o n s .

3. E v a l u a t i o n of the m a t r i c e s of the elements.

4. Assembly of the element equations.

5. A p p l i c a t i o n of the boundary c o n d i t i o n s .

6. S o l u t i o n of the system equations.

7. C a l c u l a t i o n of any other unknown f i e l d v a r i a b l e s .

3.2 Matrix Formulation

D i f f e r e n t f i n i t e - e l e m e n t models were chosen f o r r e p r e s e n t i n g the

v a r i a t i o n s of s t reamfunctions, v o r t i c i t i e s and p r e s s u r e . Fig.3-1 shows

a t y p i c a l f i n i t e element. The stream f u n c t i o n s and v o r t i c i t i e s were

assumed to vary l i n e a r l y over each f i n i t e element as

-19-

F i g u r e 3-1 T r i a n g u l a r f i n i t e element

f T f U y P (3-1)

aTcx,*; I oTc*,y) J

(3-2)

where the T's and Q's are the t r i a l f u n c t i o n s and the s u p e r s c r i p t ( e )

i n d i c a t e s the >£th element. At the nodal p o i n t s of t h i s element, p o i n t s

i , j and k i n F i g . 3-1,"^i, iOi-.^j, & j, and ^kjG?k a r e

cjt«)= H?>ct)Qpcxi,*)-* Hl^cti&Vcxi^) 0-3)

^ i C t > - H,,e>ct)T,(C)CXK,yo + Hftt)T»w«i.>;

&Vt) = H,ie>£t)€L'°c^yo -*• toaTcx^yo

-20-

I n m a t r i x form, equations (3-1) may be w r i t t e n as

= f c c e , J

(3.4)

and

where

60, 00} it)

and

i n s * )

L T . ^ c x ^ y o

a f t * , * )

T ^ ' C X i , * )

( 3 - 5 )

I n order to express Hn's i n terms o f ^ n ' s orCJn's uniquely, equations

(3-4) can be w r i t t e n as

and

H!e><t)

I H?'(t>

Hf e >(t)

= [ c c e f

HfCt) J

i t ; t t ) J (3-6)

So equations (3-1) and (3-2) g i v e

-21-

iHftt) J t T l

^ ^ v ; , T ^ % y ^ T , t t J ^ y ; ] [ c t e T ,

where

(3-7)

and

H,(e>ct) J

l u j K ( t > j "o*ct>'

( 3 - 8 )

where

For a t r i a n g u l a r element, i f Tn's and Qn's are taken to be l,x and, y

then

Nilw) - l'Pcw) = (at + i>£x+ a y ) / ZA (3-9)

i n which

b £ = yj - j/n C i = XK ~ X,"

-22-

w i t h the other c o e f f i c i e n t s obtained by a c y c l i c permutation of s u b s c r i p t s i n

order i , j , k and where

2 A = 1 * i 4

where A is the area of the element. Equations(3-7) and (3-8) may be expressed by the following matrix

equations.

(3-10)

(3-11)

where T denotes the transpose to the column matrix.

i (e) le) The gradients of 0/r and (m are

(3-15) The finite element models were employed in discretising the variational

functionals in equations (2-40), (2-41) and (2-42). Substituion of equations (3-10), (3-11) and the last equations into equati ons (2—40) and (2—42) gives

(3-12)

(3-13)

(3-14)

-23-

and

+ U ( i f - -If-) (KTH] <A (3-16)

•fe=iU([{^rwf+[{^}>}]>(

(3-17)

The f i n i t e element s o l u t i o n to the problem i n v o l v e s p i c k i n g the nodal

v a l u e s of (ij and ^ so as to make s t a t i o n a r y the f u n c t i o n a l s 0 and 0 . To

make 0. and 0 s t a t i o n a r y w i t h r e s p e c t to the nodal v a l u e s of v o r t i c i t y and 1 ^ stream f u n c t i o n r e s p e c t i v e l y , the f o l l o w i n g c o n d i t i o n s are r e q u i r e d .

= I-fiSf- 6Uk = 0 (3-18)

(3-19)

NN

-24-

where NN i s the t o t a l number of nodes. S i n c e the£fct)i's a n d ^ ^ i ' s a r e

independent, equations (3-18) and (3-19) can hold only i f

= 0 (3-20)

3fi = o (3-21)

Hence from equations (3-21) and ( 3 - 1 7 ) , f o r a t y p i c a l node i the

f o l l o w i n g equation i s requ i r e d

= o U t i l i s i n g equation ( 3 - 1 1 ) , equation (3-22) may be w r i t t e n a s :

• I P ^ W * = 0 ( 3. 2 3 )

I n m a t r i x form, for the e n t i r e element the f o l l o w i n g equation i s obtained

where

)Aie> [ a * ax ay ay

(3-24)

-25-

S i m i l a r l y , from equations (3-20) and ( 3 - 1 6 ) , f o r a t y p i c a l node i the

f o l l o w i n g equation i s r e q u i r e d

2>uG

= o

I n matrix form, f o r the e n t i r e element the f o l l o w i n g equation i s

obtained.

where (3-25)

Sie>

I n s o l v i n g a f l u i d flow problem w i t h the foregoing elements, the u s u a l

assemblage process f o r f i n i t e elements i s followed as w e l l . For the

assembled system the foregoing equations become

(3-26)

(3-27)

where

€ = / (3-28)

-26-

6 • i

(3-30)

(3-31)

(3-32)

where M i s the t o t a l number of elements.

3.3 I n t e g r a t i o n of the Matrix Equation

To i n t e g r a t e equations (2-24) and (2-25) w i t h r e s p e c t to time, the

method suggested by R.T-S Cheng (25) i s used. Cheng considered two

s o l u t i o n s *\ftr\, and OO^at the Nth time s t e p and and a t a time

increment &k l a t e r . Then the governing equations may be expressed as

f o l l o w s :

au)n, vk» 3U). y J u 3<A = JV*CO**» at. ay ax 9 * 3 i i v y

(3-33)

and

And the assembled system equations may be w r i t t e n as f o l l o w s :

(3-35)

(3-36)

-27-

where

K*. = f f (-2hk-3hlL +lhL-M.}Ji*> ( 3 _ 3 7 )

S i f t = ~ ^ L*> UX?* Ni (LA*** (3-38)

(3-39)

€=,JArt,) ' <3-40)

Jw» Zr.JA«)'>«l a y ? x ax a * < 3- 4 1>

The i t e r a t i v e s o l u t i o n procedure s t a r t s by assuming an i n i t i a l values

forCdn ( i . e . , k ) o ) . Then the t i + l ( i . e . , ) i s found from equation (3-35)

and used as the source f u n c t i o n t o determine the&On+l (i.e.,<A) from equation

(3-36). This process i s repeated u n t i l steady s t a t e i s reached. Using

a two-point f i n i t e d i f f e r e n c e formula, the term ^ tO j n+1 can be w r i t t e n

as f o l l o w s ( 25 )

{ * ! « , - ( f L = f 1

so t h a t equation (3-36) becomes

which can be solved at successive time steps f o r the column vector of nodal

values o f v o r t i c i t y . The c o e f f i c i e n t ((Ku>] + [KfcoJ) i s symmetric, banded,

and p o s i t i v e d e f i n i t e . Numerical s o l u t i o n s o f equations (3-35) and (3-36)

were obtained by the Gaussian e l i m i n a t i o n method. Cheng rep o r t s t h a t the

i t e r a t i o n procedure was always s t a b l e f o r s u f f i c i e n t l y small A t . As a

g u i d l i n e , &t should be chosen so t h a t

At < 0./(4O*Re (3-44)

-28-

where A 1 i s the c h a r a c t e r t i s t i c l e n g t h of an element.(25,63,127)

I t has been observed t h a t such a procedure f o r time dependent problems

does not f u l f i l the requirements of the c a l c u l u s of v a r i a t i o n . This i s

because d u r i n g v a r i a t i o n s o f v o r t i c i t y , the term 2uQ 3-t i s t r e a t e d as an

i n v a r i a n c e . This p r i n c i p l e i s , t h e r e f o r e , r e f e r r e d t o as pseudo-

v a r i a t i o n a l p r i n c i p l e .

3.4 Evaluation of the Matrices of the Elements

Equations (3-35) and (3-41) may now be e x p l i c i t l y evaluated using the

d e f i n i t i o n of the i n t e r p o l a t i o n f u n c t i o n s . For l i n e a r t r i a n g u l a r elements

the f o l l o w i n g euqations can be obtained.

where

M' e )(x,y)

2 A =

a, = b. Ci

b,x + c,y 2A

a - i -

cu 1-

(3-45)

1 *

I X, X I Xz %

y 2 - %

= 2« (area of t r i a n g u l e )

(3-46)

The other c o e f f i c i e n t s are obtained by c y c l i c a l l y permuting the

s u b s c r i p t s . From equations (3-45) the f o l l o w i n g equations can be obtained

a y

bi la

26 i = 0 2,3. (3-47)

S u b s t i t u t i o n of equations (3-48) i n t o equation (3-37) gives;

-29-

(3-48) <l-u - w

€ = I

To evaluate M

Some spe c i a l considerations are r e q u i r e d . COn^ 6^ can be t r e a t e d as a

constant w i t h i n the element, and from the i n t e g r a t i o n formula (22,33,61,63)

the f o l l o w i n g equation may be obtained. M

% - - 2 _ - r ^ ^7 (3-49)

(e) .Or the term tdn may be l i n e a r l y i n t e r p o l a t e d i n terms of i t s nodal values as (25,63,127)

(Xe) - (XVMi + uJEa/» * u; f l SA/i (3-50)

i n which case equation (3-38) may be w r i t t e n as f o l l o w s :

Again, employing the i n t e g r a t i o n formula, the f o l l o w i n g equation may be

obtained

-30-

(3-52)

M

= " I A IZ

S i m i l a r l y , from the i n t e g r a t i o n formula, equation (3-40) may be w r i t t e n

as

(3-40)

•e-1 (3-53)

Now the equation (3-41) i s t o be evaluated.

- L L M ( ay 3* ax - 5 y - ; * ^ (3-4D

S u b s t i t u t i o n of equations (3-47) i n t o equations (3-12) - (3-15) gives

a x

(3-54)

20

3 U £ 3^

-31-

S u b s t i t u t i o n of the l a s t equations i n t o equation (3-41) gives

- (-it + + -&v«™>)

M

( b.tUn + k U * + b3U>3n)

e= r ( b , U ) m + b 2Ca«. + b j 6 0 l n )

At present, the governing equations and elements can e a s i l y be

incorporated i n t o the computer programs.

3.5 Pressure and V e l o c i t y D i s t r i b u t i o n s

Now the pressure and v e l o c i t y d i s t r i b u t i o n s are t o be c a l c u l a t e d .

I t may be mentioned here t h a t serious a t t e n t i o n must be given t o the choice

of i n t e r p o l a t i o n f u n c t i o n s f o r pressure and stream f u n c t i o n s . To achieve

the same order of approximation f o r stream f u n c t i o n s and pressure, the

i n t e r p o l a t i o n f u n c t i o n s f o r stream f u n c t i o n s should be higher by one order

than the i n t e r p o l a t i o n f u n c t i o n s f o r pressure. So quadratic t r i a n g u l a r

elements were used f o r the stream f u n c t i o n s , and l i n e a r t r i a n g u l a r elements

f o r the pressure (see Fig.3.2).

-32-

Figure 3-2: Tr i a n g u l a r element w i t h corner nodes 1,2,3 and mid-edge nodes 4,5,6.

N, Lf- L, ( Li + L3)

Ni — Li - U{ U+L,)

N3 = £ - Li(L, + u)

N+ = «• Li Is

Mi U-Lt L2

=

L; =• n a t u r a l coordinates

-33-

3

and

where

N, - tf- L,lU + Li)

AA - ^ - * AO

Mi ~ t-\ =• n a t u r a l coordinates

The gradients o f P w / are

(3-56)

(3.57)

(3-58)

(3-59)

S u b s t i t u t i o n of equations (3-59) and (3-56) i n t o equation (2-41) gives:

(3-60) (e)

Minimisation of the f u n c t i o n a l gives

dft J A t e A L l 3 X J i r t J J j 3 X (3-61)

= 0

-34-

The l a s t equation may be w r i t t e n c oncisely as

where

Q _ fay ay / 3 y f ]

R e f e r r i n g t o equations (3-57) and (3-58), the d e r i v a t i v e s i n equation

(3-65) may be w r i t t e n as

(3-66)

B 1 ^ a y -

• ^ ( c . - c - e . J f !

-35-

+ ~Ttf-(Ci - C * ~ C< )% (3-67)

and

ax ay 1 3X3 y J ' Y I

= 2C,b,-C,b»-c>b,-c,b,- £*fe. ) y ,

«• ( 2C,b*- Crb 3- Csb,- Ct z-C»bJ

+ ( 2C 3b 3- ab, - C, b3 - C 3b 2 - 6*b3)

(3-68)

To s i m p l i f y these c a l c u l a t i o n s as much as p o s s i b l e , the f o l l o w i n g

equations are assumed.

2 For a s o l u t i o n domain of M elements, the system equations are of the

form

[ K P ] { P C t > ] * [ S P ] - fo| (3-70)

-36-

which can be solved f o r the column vector of nodal values o f pressure, f p }

Now the v e l o c i t y d i s t r i b u t i o n i s t o be c a l c u l a t e d . The most usual

procedure o f d e s c r i b i n g incompressible v e l o c i t i e s i s by using the f o l l o w i n g

equations (3-71) i n a d i r e c t manner. I t i s s t r a i g h t f o r w a r d . At present,

an a l t e r n a t i v e way i s presented here. For a two-dimensional f l u i d f low,

the v e l o c i t y components u and v may be expressed i n terms of a stream

f u n c t i o n *Ur (x,y) as

7>V

The v a r i a t i o n a l f u n c t i o n a l s f o r equations (3-71) can be w r i t t e n as

f o l l o w s :

(3-72)

The i n t e r p o l a t i o n f u n c t i o n s f o r stream f u n c t i o n s should also be higher u v

by one order than Ni and Ni , the i n t e r p o l a t i o n f u n c t i o n s f o r u and v.

So a q u a d r a t i c i n t e r p o l a t i o n f u n c t i o n f o r stream f u n c t i o n and a l i n e a r

one f o r f l u i d v e l o c i t y components are adopted.

J- „ (3-73)

t- i 6

where

f\ji*" — A/t,tr ™ Li = n a t u r a l coordinates

a/; = l) - mu - L3)

-37-

Nt = L\ - Lx{ L3+Lt)

N3 = - U( L> + U) (3~74)

Nt " *L,Li

S u b s t i t u t i o n of equations (3-73) and (3-74) i n t o equations (3-72) gives:

= i f A , o ( [ f W ) T f « ] f - 2 ( { # } T W ]

(3-75)

M i n i m i z a t i o n of equations (3-75) gives

= 0 and (3-76)

=• o

From equations (3-76), the element equations may be w r i t t e n as f o l l o w s

-38-

where

The assembled system equations become

LRuHM- [ M f t ] = fo} (3.30

[Rv]\v\ + [ a » H r ] = {<>}

which can be solved f o r the column vector of nodal values of v e l o c i t y

components u and v. Matrices £QU] and [QV] have been evaluated using

equation (3-73), (3-74) and (3-79). Matrices[Qu] and [QV] are given

i n Appendix A.

-39-

CHAPTER 4 BOUNDARY CONDITIONS AND NUMERICAL PROCEDURES

4.1 Boundary Conditions

I n t h i s s e c t i o n , some general ideas about boundary c o n d i t i o n s are

discussed. A f l u i d flow problem, governed by a system of p a r t i a l

d i f f e r e n t i a l equation, i s defined only when a proper set of i n i t i a l and/

or boundary cond i t i o n s i s given. The boundary c o n d i t i o n s are such an

important p a r t o f the d e f i n i t i o n of a problem t h a t the patt e r n s of two

flow f i e l d s can be completely d i f f e r e n t from one another simply due t o

some d i f f e r e n c e s i n the flow boundaries, i n s p i t e o f the f a c t t h a t both

f l o w obey the same system of i n d e f i n i t e d i f f e r e n t i a l equations. One

cannot exaggerate the importance of the e f f e c t s t h a t boundary cond i t i o n s

have on the f l u i d f low a n a l y s i s . I n mixed i n i t i a l - and boundary - value

problems major t r o u b l e s must a r i s e i f the boundary cond i t i o n s are not

pr o p e r l y handled (33,92,128).

Through t h e o r e t i c a l considerations and numerical experiments, i t i s

found t h a t even though the patterns o f two f l u i d flow f i e l d s are the same,

boundary c o n d i t i o n s may s t i l l be d i f f e r e n t . I t seems t h a t boundary

co n d i t i o n s depend not only on f l u i d p atterns but also on the k i n d of

f i n i t e element f o r m u l a t i o n used or on the kind o f f i e l d v a r i a b l e s

employed, and on which ki n d o f i n t e r p o l a t i o n f u n c t i o n adopted. For

example, t o solve the same f l u i d flow problem, i f v e l o c i t y and pressure

are used as f i e l d v a r i a b l e s , then the boundary c o n d i t i o n s f o r t h i s

f o r m u l a t i o n are d i f f e r e n t from those when stream f u n c t i o n and v o r t i c i t y

are used as f i e l d v a r i a b l e s . For another example, when a f l u i d flow

problem i s solved by using a v e l o c i t y - p r e s s u r e f o r m u l a t i o n , unless a

high order element i s used the value of second d e r i v a t i v e ^ has

to be assumed zero which normally would be v a l i d f o r creeping flow only.

This seems t o be equivalent to a high order element being used and "g" T =

boundary c o n d i t i o n being assumed i n t h i s i n t e r p o l a t i o n f u n c t i o n a t the

same time (62,132 ) .

-40-

The boundary c o n d i t i o n s f o r a stream f u n c t i o n - v o r t i c i t y f o r m u l a t i o n problem are of three general types; 1) the s p e c i f i c a t i o n s of the values which stream f u n c t i o n or v o r t i c i t y must assume along the boundary; 2) the s p e c i f i c a t i o n s of the values of the component of the gradie n t of Yr

or GO at the normal t o the boundary; 3) the p r o v i s i o n of some al g e b r a i c r e l a t i o n which connects the value of ijr or 00 t o the values of t h e i r normal components along the boundary (16,55). I n f a c t , i t i s usual f o r values to be s p e c i f i e d a t some parts of a boundary and f o r gradients t o be s p e c i f i e d a t other p a r t s (see example one, Chapter 6 ) . When several d i f f e r e n t i a l equations are to be solved simultaneously, there i s no need f o r the boundary c o n d i t i o n s f o r each equation to be of the same type (55, 63).

I n example one (see Section 6.1), since the flow i s considered

to be f u l l y developed a t the downstream end, the gradients of stream f u n c t i o n

and v o r t i c i t y w i t h respect t o the flow d i r e c t i o n should vanish at the

boundary. These provide the normal boundary con d i t i o n s at the downstream

end. There i s no need t o s p e c i f y the values of stream f u n c t i o n and

v o r t i c i t y a t t h i s boundary. (21,25,55,63,100,110,127)

The mixed type boundary value problem, such as t h a t appearing i n

example one, causes no d i f f i c u l t y i n i t s e l f , provided a scheme can be

found f o r s p e c i f y i n g boundary c o n d i t i o n s associated w i t h f i r s t d e r i v a t i v e s

of stream f u n c t i o n and v o r t i c i t y . For example, to impose the boundary

c o n d i t i o n s f o r the example one, the r e l a t i o n s - ^ r — = = 0 can j u s t

be incorporated i n t o the element s t i f f n e s s m a t r i x . This w i l l give an

a l t e r n a t e form f o r the element s t i f f n e s s m a t r i x which can then be used

f o r elements having node on downstream edge. The disadvantage i s t h a t such

elements must then be incorporated i n t o the computer programe and used as

need a r i s e s . This scheme has been used i n t h i s computer program. An

a l t e r n a t e way i s t o consider the boundary c o n d i t i o n s as c o n s t r a i n t s .

-41-

The set of equations i s expressed as the gross assemblage s t i f f n e s s equation.

The d e t a i l e d procedure can be found from the papers of Bratanow (16,18)

or M artin ( 8 8 ) .

The 'natural' boundary c o n d i t i o n s are somewhat a r b i t r a r y , since there i s

l i t t l e agreement i n the f i e l d of f l u i d mechanics as to what the proper ones

are. The choice i s t h e r e f o r e made on the basis of p r a c t i c a l i t y (33,100).

Usually, the'rigid'boundary c o n d i t i o n s are used. And the programe i s l e f t

t o seek i t s own approximation of the n a t u r a l ' boundary c o n d i t i o n s (see

Chapter 6 ) . (100)

A r i g i d w a l l may be e i t h e r o f two types, n o - s l i p or f r e e - s l i p . The

l a t t e r may be considered t o represent a plane o f symmetry, r a t h e r than a

t r u e w a l l (58,128). I n the examples to be presented, n o - s l i p boundary

c o n d i t i o n would be considered.

To s a t i s f y the c o n d i t i o n of n o - s l i p a t s o l i d w a l l s , the normal and

t a n g e n t i a l gradients of stream f u n c t i o n must vanish at these boundaries.

The t a n g e n t i a l c o n d i t i o n s are s a t i s f i e d by s e t t i n g stream f u n c t i o n constant

along these boundaries. However s p e c i a l a t t e n t i o n i s required t o determine

the boundary formulae f o r the normal c o n d i t i o n s . To determine the

boundary values f o r the w a l l v o r t i c i t y , a p p l i c a t i o n of the n o - s l i p boundary

c o n d i t i o n alone i s not enough. At a p o i n t (Xo.Yo) on the w a l l , the

v o r t i c i t y may be c a l c u l a t e d from the relation.(25,63,121)

where n i s the coordinate normal t o the w a l l . Using a Taylor s e r i e s

expansion, at a p o i n t ( X I , Y l ) along the normal d i r e c t i o n , a small distance

from the w a l l , the f o l l o w i n g equation may be obtained.

e x * , * ; dn1 (4-1)

3"^ a n z. 3 n

(4-2)

-42-

Since the n o - s l i p c o n d i t i o n d i c t a t e s t h a t

0 (4-3) 3TI

then the v o r t i c i t y on the w a l l may be c a l c u l a t e d by

(4-4)

Wall v o r t i c i t y i s then given i n terms of the stream f u n c t i o n

evaluated a t the w a l l and a small distance away from the w a l l . I f

(X1,Y1) i s not a nodal p o i n t , the stream f u n c t i o n "Xj/" ( X I , Y l ) may be obtained

by i n t e r p o l a t i o n between stream f u n c t i o n s of the neighbouring nodal p o i n t s

of ( X I , Y l ) (25,63).

There are several other ways t o compute the surface v o r t i c i t y , which

can e a s i l y be found (4,33,110,128). Equation (4-4) i s c a l l e d the f i r s t -

order one-sided d i f f e r e n c e formula. This formula gave numerical r e s u l t s

i n e x c e l l e n t agreement w i t h the exact s o l u t i o n (128). I t i s found t h a t

the second-order one-sided d i f f e r e n c e formula sometimes led t o unstable

r e s u l t s . The d e t a i l e d discussions may be found from Wu's l a t e s t paper

(128). To play safe, t h i s program employs the f i r s t - o r d e r one-sided

d i f f e r e n c e formula. When using t h i s formula, use of a f i n e r mesh round

the corners i s to be encouraged. This i s not only because there are b i g

v a r i a t i o n s o f values of stream f u n c t i o n round the corners but also because

i t i s hoped t o force the e f f e c t s of the corners t o spread i n t o the f l u i d

i n every d i r e c t i o n , by using a f i n e r mesh i n regions adjacent corners

(see Sec.7.6.1.).

For each system o f equations there are a number of s u f f i c i e n t and

necessary boundary c o n d i t i o n s . For example, f o r a viscous flow the

c o n d i t i o n of n o - s l i p i s s u f f i c i e n t t o determine the flow f i e l d . No

other c o n d i t i o n may be imposed on the r i g i d w a l l (92,100).

-43-

When a problem of v i s c o u s flow i s t r e a t e d by a numerical technique. A c e r t a i n mesh i s used. At the i n t e r i o r mesh-points the governing equations a r e s u b s t i t u t e d by a matrix equation. At each computational s t e p , i n f o r m a t i o n i s tr a n s m i t t e d from each point to i t s neighbouring p o i n t s v i a the numerical computation. I n t h i s way, the boundary mesh-poi n t s i n f l u e n c e t h e i r neighbours and tr a n s m i t the e f f e c t s of the boundary c o n d i t i o n s i n t o the flow f i e l d . Moretti (92) maintained t h a t i f boundary c o n d i t i o n s are not proceeded p r o p e r l y , the r i s k of over-s p e c i f y i n g the boundary c o n d i t i o n s themselves i s faced and, i n a l l p r o b a b i l i t y these o v e r s p e c i f i e d boundary c o n d i t i o n s w i l l not be c o n s i s t e n t w i t h the n a t u r a l of the boundary and the l i m i t i n g forms of the equations of motion ( 6 2 , 9 2 ) .

I t i s not p o s s i b l e to i s o l a t e any p o r t i o n of a f l u i d f i e l d and

obt a i n the s o l u t i o n i n only t h a t p o r t i o n . The d i f f i c u l t y a r i s e s from

the boundary c o n d i t i o n s . I t i s e a s i e r to d e a l w i t h a f l u i d flow

problem on a l a r g e r or e n t i r e flow f i e l d than j u s t on a p a r t of the whole

flow f i e l d .

Some more d e t a i l e d d i s c u s s i o n s about boundary c o n d i t i o n s w i l l be

presented i n example problems l a t e r .

4.2 Numerical Procedures

The p r e s e n t scheme f o r s o l u t i o n of the assembled system equations

(3-35) and (3-43) uses an i t e r a t i v e method to o b t a i n s e l f - c o n s i s t e n t

stream f u n c t i o n and v o r t i c i t y f i e l d s .

The s o l u t i o n to equation (3-35) r e q u i r e s s p e c i f i c a t i o n of stream

f u n c t i o n or i t s normal d e r i v a t i v e s on a l l boundaries. The i n i t i a l

v o r t i c i t y i s not known anywhere. U s u a l l y an i n i t i a l guess of zero

v o r t i c i t y i s o f t e n a p p r o p r i a t e . Equation (3-35) i s then a b l e to be

solved f o r the stream f u n c t i o n . Using the r e s u l t s of stream f u n c t i o n ,

the w a l l v o r t i c i t y can be obtained from equation ( 4 - 4 ) . And then we can

use v a l u e s of w a l l v o r t i c i t i e s and stream f u n c t i o n s to s o l v e equation

-44-

(3-43) f o r the v o r t i c i t y . The v e l o c i t y boundary c o n d i t i o n s provide d e r i v a t i v e boundary c o n d i t i o n s on stream f u n c t i o n . On the boundary, stream f u n c t i o n and the normal and t a n g e n t i a l d e r i v a t i v e s may a l l be s p e c i f i e d . A f t e r the v o r t i c i t y f i e l d has been determined,the v a l u e s of v o r t i c i t y can be t r e a t e d as a source f u n c t i o n , and s o l v e the stream f u n c t i o n f o r the next time i n s t a n t from equation ( 3 - 3 5 ) .

Once the new stream f u n c t i o n f i e l d has been determined the w a l l

v o r t i c i t y f o r the new time i n s t a n t can be obtained by s o l v i n g equation

(4-4) a g a i n . Then equation (3-43) i s r e s o l v e d by a d j u s t i n g the boundary

v o r t i c i t y . T h i s procedure i s repeated u n t i l a s e l f - c o n s i s t e n t stream

f u n c t i o n and v o r t i c i t y f i e l d i s obtained. T h i s procedure not only

circumvents the n o n l i n e a r i t y of the governing equation but a l s o l e a d s

to a l i n e a r a l g e b r a i c system (25,63).

Employing the foregoing procedure, s o l u t i o n s of stream f u n c t i o n and

v o r t i c i t y can be obtained f o r c r e e p i n g flow. Once a convergent Stokes

s o l u t i o n was determined,this s o l u t i o n can be used as the i n i t i a l c o n d i t i o n s

of v o r t i c i t y i n the c a l c u l a t i o n f o r a s o l u t i o n of the governing equations

a t a s m a l l Reynolds number. These s o l u t i o n s of stream f u n c t i o n and

v o r t i c i t y f o r a small Reynolds number are considered as the i n i t i a l

c o n d i t i o n s when the numerical s o l u t i o n of v o r t i c i t y and stream f u n c t i o n

f o r a b i t higher new Reynolds number are s o l v e d . T h i s procedure i s

repeated such t h a t the s o l u t i o n a t a lower Reynolds number i s used as

the i n i t i a l c o n d i t i o n s f o r the higher Reynolds number u n t i l the s o l u t i o n s

of stream f u n c t i o n and v o r t i c i t y f o r a d e s i r e d Reynolds number a r e reached

( 2 5 , 6 3 ) .

To get an a c c u r a t e r e s u l t , the s m a l l e r the increment of Reynolds

number i s , the b e t t e r .

The procedures of the numerical s o l u t i o n a r e summarised as f o l l o w s :

-45-

(1 ) Define s u i t a b l e boundary c o n d i t i o n s f o r both stream f u n c t i o n and v o r t i c i t y .

( 2 ) Assume an i n i t i a l v o r t i c i t y d i s t r i b u t i o n . -i

(3^) Compute stream f u n c t i o n from equation ( 3 - 3 5 ) .

( 4 ) Find the boundary v o r t i c i t y v a lues from equation ( 4-4)

( 5 ) S o l v e the v o r t i c i t y of Stokes flow f o r the next time i n s t a n t .

from equation ( 3 - 4 3 ) . •—Convergence ? I

( 6 ) Read a new sm a l l Reynolds number. *

( 7 ) T r e a t the v o r t i c i t y f i e l d of Stokes flow as a new

^ These boundary c o n d i t i o n s must be kept through t h i s c a l c u l a t i o n .

i n i t i a l v o r t i c i t y d i s t r i b u t i o n .

( 8 ) Compute stream f u n c t i o n from equation ( 3 - 3 5 ) .

( 9 ) F i n d the boundary v o r t i c i t y v a l u e s from equation ( 4 - 4 ) . i

(10) S o l v e the v o r t i c i t y f o r the next time i n s t a n t

a t the small Reynolds number from equation ( 3 - 4 3 ) .

•—Convergence ? i

(11) Read a b i t higher Reynolds number

I

Repeat the foregoing procedure u n t i l d e s i r e d Reynolds

number i s reached.

i (12) Compute pressure d i s t r i b u t i o n , i

(13) Compute v e l o c i t y d i s t r i b u t i o n . 1

(14) S o l u t i o n i s complete.

-46-

Chapter 5 COMPUTER WORK

5.1 I n t r o d u c t i o n

As f a r as a c t u a l a p p l i c a t i o n s of the f i n i t e element method are

concerned, i t seems t h a t computer programs play most important and

p r a c t i c a l r o l e s . The main p o r t i o n of t h i s chapter i s to e x p l a i n how

a d i g i t a l computer can s o l v e f l u i d flow problems by the f i n i t e element

method. Some d e s c r i p t i o n s and a s i m p l i f i e d flow diagram f o r the f i n i t e

element work a re a l s o presented. The a n a l y s i s was programmed i n

'FORTRAN I V computer language.

The flow c h a r t ( F i g u r e 5-1) shows how the system i s used to s o l v e

a f a i r l y s t r a i g h t f o r w a r d problem.

Co n s i d e r i n g the flow c h a r t of F i g u r e 5-1, the boundary c o n d i t i o n s

and other data are read i n at once before any of the c a l c u l a t i o n i s

commenced. I f the t a i l end of the data i s i n c o r r e c t then t h i s e r r o r

w i l l be detected before a s i g n i f i c a n t amount of computer time has been

wasted.

The input data i s d i v i d e d i n t o the f o l l o w i n g main s e c t i o n s .

C o n t r o l data: number of nodes, number of elements, and maximum value of i t e r a t i o n s .

Coordinate data; the coor d i n a t e s of the nodes.

Element data: d e s c r i p t i o n of the topology of the element i n t e r c o n n e c t i o n s .

I n i t i a l and boundary

c o n d i t i o n s : d e s c r i p t i o n of the problems.

The c h i e f purpose of the drawing scheme i s to check the coordinate

data of elements and the element topology. From a drawing of the f i n i t e

element mesh, mistakes i n the data a r e e a s i l y r e c o g n i s a b l e .

5.2 Some D e s c r i p t i o n s

During the e a r l y stage, when the programmes were developing, s e r i o u s

problems of computer storage and the addresses f o r a r r a y s were faced.

To overcome these d i f f i c u l t i e s , a sub-routine c a l l e d DYNMIC sup p l i e d by

-47-

The data i s read from cards or f i l e e t c . and s t o r e d i n some a r r a y s .

No, Stop. Boundary c o n d i t i o n s c o r r e c t ?

Yes

The f i n i t e element mesh can be drawn a t t h i s stage to ensure t h a t the input data i s c o r r e c t .

The elemnts a r e analysed i n d i v i d u a l l y and merged i n t o system equations.

The system equations are solved i n order

to f i n d some primary unknowns.

Some other unknowns are c a l c u l a t e d from the r e s u l t s which have a l r e a d y been determined.

F i g u r e 5.1. S i m p l i f i e d Flow Chart

-48-

the computer u n i t of the U n i v e r s i t y has been used.

T h i s method makes use of the f a c t that FORTRAN a l l o w s the d e f i n i t i o n

of a r r a y s w i t h unknown dimensions w i t h i n sub-programmes. The old main

program i s converted i n t o a subroutine having as arguments the name and

s i z e of each a r r a y r e q u i r i n g dynamic a l l o c a t i o n . T h i s subroutine i s

c a l l e d MAINPR. A small main program i s r e q u i r e d to w r i t e . T h i s s m a l l

main program w i l l read i n the a r r a y dimensions and c a l c u l a t e the space

r e q u i r e d f o r each a r r a y and pass t h i s i n f o r m a t i o n to the subroutine DYNMIC,

the arguments of which a r e i d e n t i c a l to those f o r MAINPR. Subroutine

DYNMIC a c q u i r e s space f o r the a r r a y s and c a l l s MAINPR p a s s i n g the arguments

given to i t , having i n s e r t e d the c o r r e c t addresses f o r the a r r a y s .

The subroutine SOLMIX i s used f o r s o l v i n g matrix equations. Within

SOLMIX a subroutine SOLVE i s c a l l e d . The subroutine SOLVE s o l v e s t h i s

kind of equation.

A ( I , J ) x X ( J ) = C ( I )

f o r X ( J ) by Gaussian e l i m i n a t i o n scheme. There are a l o t of t h i s kind

of present sub-program which can be used. The subroutines SOLMIX and

SOLVE used here were w r i t t e n by P r o f e s s o r J.F.Booker of C o r n e l l U n i v e r s i t y

( 6 3 ) .

The main advantages of the f i n i t e element program are as f o l l o w s :

( 1 ) Complicated boundary c o n d i t i o n s can be in v o l v e d without d i f f i c u l t y .

( 2 ) Changing the type of boundary c o n d i t i o n r e q u i r e s only the change of

input data, and there i s no need to change the computer programmes.

( 3 ) The convergence of the c a l c u l a t i o n can be observed by p r i n t i n g the

va l u e s a t s e l e c t e d p o i n t s a f t e r each i t e r a t i o n .

The main disadvantage i s th a t the achievement of a s u c c e s s f u l s o l u t i o n

depends on the c h o i s e of the c o r r e c t convergence parameters. For some

problems, i t might take a long time by t r i a l and e r r o r before what k i n d s o f

values of parameters a r e the most ap p r o p r i a t e to use are known. So i t

-49-

i s important to w r i t e down and keep the information about these c o n t r o l f a c t o r s . I t may be u s e f u l when t h i s program i s used l a t e r . 5.3 S i m p l i f i e d Flow Diagram f o r the F i n i t e - E l e m e n t Programmes

A flow c h a r t o u t l i n i n g the f i n i t e element procedure i s presented as

f o l l o w s :

-50-F i n i t e Element A n a l y s i s of I n c o m p r e s s i b l e Unsteady Viscous Flow.

S t a r t

( 1 ) Read, echo p r i n t and check data (2) C y c l e f o r each element and form

system m a t r i c e s . ( 3 ) Solve eq.(3-35)

(See F i g . 5 , 3)

i ( 1 ) I n s e r t the n o - s l i p boundary c o n d i t i o n s ( 2 ) Solve eq.(3-43)

Yes onvergence

(See F i g . 5 . 4 )

Solve eq, 'tit I

(See F i g . 5 . 5 )

( 1 ) C a l c u l a t e Q. (2 ) Form matrix Sp ( 3 ) C a l c u l a t e P

[QU] and [QV] (4) Form matr i c e s £RU] , [QU] and [QV]

(See F i g . 5 . 6 )

( 1 ) C a l c u l a t e U,V (2) Write r e s u l t s

Stop

(See F i g . 5 . 7 )

F i g u r e 5.2

F i g u r e 5.3

-51-

Read i n data

i •

P r i n t out data and Draw the f i n i t e e l e ­ment mesh

C y c l e f o r each element and form system m a t r i c e s

CALL SOLMIX

-52-F i g u r e 5.4

I n s e r t the n o - s l i p boundary c o n d i t i o n

From matrix Sw

[SHI] - — U « ) + [KaJ/£t

t

it w ax ax. ay K V

CALL SOLMIX

-53-F i g u r e 5.5

YES

CO*

NO

ICOUNT + 1 ICOUNT

Form matrix S

NO ICOUNT ^ 1 0 0

YES

71+i

[K*] if] CALL SOLMIX

-54-

F i g u r e 5.6

[ W 2Nr ( vH , 3X 1 3 V ^ 9X3V ' J

t Form matrix Sp

S P i =* -[p^GLcCA

- p a

CALL SOLMIX

Form m a t r i c e s [RU] , [Qu] and i[Qv]'

a *

-55-

F i g u r e 5.7

CALL VELSTS

{SM5] [Ga] \f]

(SM6] <

CALL SOLA

[Ru] { u }

AIX

i WRITE RESULTS

-56-

Chapter 6 TEST EXAMPLES

6.1 Example One

6.1.1.Introduction

To t e s t the e f f e c t i v e n e s s of t h i s formulation, and program, and

t h e r e f o r e to a s s e s s s o l u t i o n accuracy, convergence, and s t a b i l i t y , a

f l u i d flow between plane, p a r a l l e l p l a t e s was chosen. The main reason

why i t was s e l e c t e d as the f i r s t example i s that t h i s problem r e t a i n s the

t o t a l n o n - l i n e a r c h a r a c t e r of the Navier-Stokes equations (4.6,25). For

the reason of symmetry, only one-half the problem region was considered.

The c o a r s e s t f i n i t e element mesh which was p l o t t e d by the computer i s

i l l u s t r a t e d i n F i g u r e 6-1.

6.1.2.Entry Length

The e n t r y l e n g t h p l a y s an important r o l e i n t h i s kind of problem.

The f a c t t h a t the duct l e n g t h i s i n s u f f i c i e n t can lead to unstable

r e s u l s t s . So the en t r y length w i l l be d i s c u s s e d f i r s t . The entry l e n g t h

i s defined as the a x i a l p o s i t i o n a t which the c e n t r e - l i n e v e l o c i t y reaches

99% of i t s f u l l y developed value (11,24,55,56). T h i s length can be

determined by experimentation i n which every parameter but the entrance

length i s held f i x e d (14,39,55,56). S c h l i c h t i n g (136) has shown that the

entrance length i s only a l i n e a r f u n c t i o n of the Reynolds number f o r

p a r a l l e l p l a t e channels and p i p e s . T h i s i s true only i f the shape of the

i n l e t v e l o c i t y p r o f i l e i s kept the same ( 5 5 ) . Hai (55) concluded t h a t

the entrance length necessary f o r flow development i s a f u n c t i o n of channel

height, Reynolds number and shape of the entrance v e l o c i t y p r o f i l e f o r the

flow regions considered here. Laughaar (39) suggested that the en t r y

length f o r an i n c o m p r e s s i b l e i s o t h e r m a l laminar of a Newtonian f l u i d flow i n

a c i r c u l a r pipe can be obtained from the f o l l o w i n g simple equation.

Xc = Re x D x K (6-1)

where

in r\j on cn cn CD en

cn ca i n CC ca ca ca i

7

LU LC Kl CC

IT

7

to in IE LCI CO LQ

U1 LC i n 7 C71 ca 1J1 i n i n Lfj

in

IE rn cn m

m a. cn ca rn rvi

i a r\j ru rv

7 ca n

7 ca in

7 /

LT LC cx

IE ca cc ca cc

ca cn

cs

CI rn CO

7 a

a) r-i

to La ca La LC

ca la l a ca

7 in 0)

CO LU

LTJ i n LT

7 —

LU LU _ J ca

UJ r — i —

rvj ID Li­ on

i r rr rr cn

ca LC. i n ru C\J

LC

cr ru ca

-58-

Xc hydrodynamic e n t r y l e n g t h f o r c i r c u l a r p i p e s .

Re Reynolds number = Vav D . P

P Density

dynamic v i s c o s i t y

D pipe diameter

K constant (Laughaar suggested t h a t K = 0.057)

A f t e r Xc i s d e r i v e d , the entrance length f o r a f l u i d flow between plane,

p a r a l l e l p l a t e s can be obtained.

6 . 1 . 3 . I n i t i a l and Boundary Conditions

Now i n i t i a l and boundary c o n d i t i o n s w i l l be d i s c u s s e d .

I n i t i a l Condition:

The duct i s i n i t i a l l y f i l l e d w i t h water of d e n s i t y of 1.0, which i s

i n s t a n t a n e o u s l y a c c e l e r a t e d by a p p l i c a t i o n of a uniform v e l o c i t y of u n i t y

upstream of the duct. The e q u i v a l e n t v o r t i c i t y i n i t i a l c o n d i t i o n i s

v a n i s h i n g everywhere. Of course, t h i s i s by no means the only i n i t i a l

c o n d i t i o n which can be used. But i n t h i s problem, i t i s employed to t e s t

the program.

Boundary Conditions:

R e f e r r i n g to what has been d i s c u s s e d i n chapter 4, the f o l l o w i n g

c o n d i t i o n s are employed ( s e e F i g . 6 - 2 ) .

( i ) To make sure t h a t t h e r e i s no mass i n j e c t i o n c r o s s the upper

where the constant i s determined from the mass f l u x e n t e r i n g the duct.

( i i ) One of the d i f f i c u l t i e s of t h i s problem i s t h a t i t i s not

s u i t a b l e and not even p o s s i b l e to d e f i n e values of v e l o c i t y on the downstream

edge. T h i s does mean i t i s d i f f i c u l t to e s t a b l i s h the v a l u e s of stream

f u n c t i o n s along downstream edge. To overcome t h i s d i f f i c u l t y , some

a d d i t i o n a l c o n s t r a i n t s along t h i s edge were introduced.

w a l l , the f o l l o w i n g equation should be given:

* constant along AB (6-2)

CQ >

•

II

a> to CO CO

L • :

s,

J! i 0

r ! 1 p

•

fi>

I

I i: :

-

TO ro

:

• 1

•

; : : : : i

• ; : ;

: •

ES ;

•

•

J •

i • : :

II

U i i

•

: ; ;

-60-

Because the flow becomes p a r a l l e l along the downstream edge, the normal d e r i v a t i v e s of both v o r t i c i t y and stream f u n c t i o n must be v a n i s h i n g to enforce the flow p a r a l l e l a t the e x i t . So,

M L = o 3TT. along BD (6-3)

a n w

( i i i ) From symmetry, i n the c e n t r e - l i n e of the duct, the v o r t i c i t y and

streamfunction may be defined as f o l l o w s :

f = 0 along CD (6-4)

U) — 0

( i v ) For uniform flow a t the duct entrance, the f o l l o w i n g equations

can be given

t = x y along AC ( 6 - 5 )

UJ = o

where K i s constant.

( v ) The n o - s l i p c o n d i t i o n i s most important ( s e e Chapter 4 ) . The

formula used to c a l c u l a t e w a l l v o r t i c i t y i s as f o l l o w s :

cJi*(x,,y.) = -fc[f(*i,Y.)-iKx.//,)) ( 4 " 4 )

6 . 1 . 4 . I t e r a t i o n Technique

To o b t a i n a s t a r t i n g s o l u t i o n , Stokes' flow was assumed. And stream

f u n c t i o n and v o r t i c i t y f i e l d s were c a l c u l a t e d by g i v i n g Re=0. A f t e r the

Stokes'flow s o l u t i o n was obtained, the i t e r a t i o n process was used to

c a l c u l a t e the flow at s u c c e s s i v e l y l a r g e r Reynolds numbers. The s o l u t i o n s

of stream f u n c t i o n and v o r t i c i t y a t a lower Reynolds number are used as

the i n i t i a l c o n d i t i o n s f o r the s o l u t i o n s a t the next higher Reynolds number ,

I t was found that the numerical scheme was s t a b l e i f a s u f f i c i e n t s mall

A t was chosen.

6.1.5.Discussion and Conclusion

At the e n t r y s e c t i o n where a v e l o c i t y d i s c o n t i n u i t y i s occurred a t

-61-

point A (see F i g u r e 6 - 3 ( a ) ) , the s i n g u l a r i t y i n t r o d u c e s a c o n s i d e r a b l e

d i s t u r b a n c e to the s o l u t i o n . S e r i o u s numerical e r r o r s may be encountered

i n the c a l c u l a t i o n u n l e s s s u f f i c i e n t l y small elements are used.

I n f a c t , because the boundary c o n d i t i o n s are c o n t r a d i c t o r y a t the

p o i n t A, the approximate s o l u t i o n w i l l not be able to s a t i s f y such boundary

c o n d i t i o n s e x a c t l y .

I n a c t u a l computation, two types of entrance conditions a t the

d i s c o n t i n u o u s point have been t e s t e d .

Case 1. U d e c r e a s e s to zero from point E to point A as a parabola

f u n c t i o n . The v e l o c i t y p r o f i l e f o r t h i s kind of entrance c o n d i t i o n was

shown i n F i g u r e 6-3(b).

Case 2. U decreases to zero from point E to point A as a l i n e a r

f u n c t i o n . The v e l o c i t y p r o f i l e f o r the entrance c o n d i t i o n was presented

i n F i g u r e 6 - 3 ( c ) .

The phenomena shown i n F i g u r e s 6-3(b) and 6-3(c) seem to agree w e l l

w i t h Tong and Fung's (124) r e s u l t s (see F i g u r e 6-4).

The input data used i s presented i n Appendix I . I t s main r e s u l t s f o r

streamfunction and v o r t i c i t y from t h i s computer program are given i n Table

6r-l. From t h i s t a b l e , i t i s found t h a t the r e s u l t s seem to be along with

those of Baker and Gasman ( 4 ) . The s t r e a m l i n e contours are presented i n

F i g . 6 - 5 . I t s main contour program i s given i n Appendix C. R e s u l t s f o r

v e l o c i t y and p r e s s u r e are shown i n F i g u r e s 6-6, 6-7 and 6-8. From F i g u r e

6-6 and 6-8, i t i s found t h a t the v e l o c i t y and p r e s s u r e r e s u l t s from t h i s

program seem to be along w i t h those of G o l d s t e i n (49) and Yamada (129) as

w e l l . Some s t r e a m l i n e contours from t h i s program fo r s l i g h t l y h igher

Reynolds numbers are presented i n Appendix D. The s t r e a m l i n e contours of

Baker ( 4 ) f o r Reynolds number of 200 are shown i n Appendix E. I t i s seen

t h a t the contours compare reasonably w e l l thus i n d i c a t i n g t h a t the present

program seems to be a c c u r a t e l y r e p r e s e n t i n g the phenomenon.

-62-

TABLE 6-1

Steamfunction and w a l l v o r t i c i t y d i s t r i b u t i o n

Streamfunction ^jf Wall v o r t i c i t y CO.w

F.E.M.* (Baker)

F.D.M.* (Gosman)

Th i s ** prog.

F.E.M.* F.D.M.* T h i s * * prog.

0.047 0.980 0.976 0.967 4.04 5.05 6.67

0.095 0.982 0.980 0.981 3.55 4.44 3.54

0.156 0.983 0.982 0.982 3.57 4.14 3.43

0.228 0.983 0.983 0.984 3.58 3.71 3.10

0.379 0.983 0.984 0.984 3.60 3.28 3.08

0.521 0.984 0.985 0.984 3 .30 3.12 3.08

1.000 0.983 0.985 0.984 3.51 3.06 3.15

( i )

itt/t/j

( 2 ) * : Re = 2 00 ** : Re = 0.002

-63-

D

( a )

0.0

B 0.5

1.0 Urn

h

0.0

B

J IJ Um 0.0 0.4 0.8 1.2 1.6

( c )

F i g u r e G-3 V e l o c i t y d i s t r i b u t i o n f o r flow between p a r a l l e l p l a t e s .

A: V e l o c i t y d i s t r i b u t i o n at the entrance. C: V e l o c i t y d i s t r i b u t i o n f o r f u l l y developed flow.

-64-

1.0

(a ) Case 1

0.0 r i i

0.5

1.0 (b) Case 2

F i g u r e 6-4 Tong L Fung's R e s u l t s (124) A: V e l o c i t y d i s t r i b u t i o n at. the e n t r a n c e . B: V e l o c i t y d i s t r i b u t i o n f o r f u l l y developed flow,

i

-65-

-66-

1.0

0.5-*-

0.0 i ' i t

0.0 0.4 0.8 1.2 1.6

U Uin

( a ) R e s u l t s from t h i s program

1.0

0.5-.

0.0 ^ H-0.0 0.4 0.8 1.2 1.6

(b> R e s u l t s of G o l d s t e i n

Urn

A; V e l o c i t y d i s t r i b u t i o n a t the e n t r a n c e . B: V e l o c i t y d i s t r i b u t i o n f o r f u l l y developed flow.

F i g u r e 6~o

V e l o c i ty d i s t r i b u t i o n for flow between p a r a l l e l p l a t e s

67

ID

0)

is 0 E

CD

eg OS

CO CM 5; OS 0) I

CN

II CN

CM

II

CM

0)

CM

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to I i

II

CM

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CM I

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-69-

6.2 Example Two 6.2.1.Introduction

The next a p p l i c a t i o n was to the i n t e r n a l f l u i d flow i n a channel of

a r b i t r a r y c r o s s s e c t i o n . The geometry of an example flow passage was

shown i n F i g . 6 - 9 . T h i s type of geometry could provide a good t e s t and

(demonstration f o r t h i s s o l u t i o n procedure because the c o n s t r i c t i o n causes

r a p i d changes i n stream f u n c t i o n and v o r t i c i t y near the c o n s t r i c t i o n r e g i o n .

The c o a r s e s t f i n i t e element d i s c r e t i z a t i o n evaluated were i l l u s t r a t e d i n

F i g u r e 6-10.

6.2.2.Boundary Conditions and I t e r a t i o n Technique

I t has been shown th a t the entrance l e n g t h i s reduced as the shape of

the entrance v e l o c i t y p r o f i l e approaches that of the f u l l y developed p r o f i l e .

And t h i s entrance l e n g t h i s reduced to zero when the entrance v e l o c i t y

p r o f i l e i s i d e n t i c a l w i t h the f u l l y developed p r o f i l e a t which point the

flow i s f u l l y developed a t t h i s entrance to the channel ( 5 5 ) . So

P o i s e u i l l e type flow was used a t the upstream edge i n view of the f a c t t h a t

l a r g e r v alues of X, (see F i g u r e 6-9) would imply a r a p i d growth of computer

time r e q u i r e d . That i s

At the downstream edge (EG) P o i s e u i l l e type flow was a l s o assumed.

I t i s worth emphasizing here t h a t the values of X need be provided l a r g e

enough. Extending the concept of entrance l e n g t h d i s c u s s e d i n example one

to t h i s example, i t i s known t h a t a t higher Reynolds number, a f t e r p a s s i n g

the c o n s t r i c t i o n , the f l u i d flow would have to t r a v e l a much longer d i s t a n c e

Xg before i t was returned to a P o i s e u i l l e type flow. I f the p r o v i s i o n of

Xg was not adequately s p e c i f i e d i n the f i n i t e element mesh, the numerical

procedure may become u n s t a b l e . Even when the c a l c u l a t i o n s are convergent,

wrong r e s u l t s or r e s u l t s which are not expected, f o r example, a r e s u l t from

a d i f f e r e n t boundary c o n d i t i o n , may s t i l l be obtained.

From the governing equation V*f- = - u) and equation ( 6 - 6 ) , the

•

1

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-72-

f o l l o w i n g boundary c o n d i t i o n should be s p e c i f i e d

OJ = 3Y on AT and "EG (6-7)

On the s o l i d w a l l , the n o - s l i p c o n d i t i o n was a p p l i e d as w e l l . And

the f i r s t - o r d e r one-sided d i f f e r e n c e formula (equation ( 4 - 4 ) ) was employed

to c a l c u l a t e w a l l v o r t i c i t i e s . I t i s worth mentioning here that t h i s w a l l

v o r t i c i t y formula i s mainly not s u i t a b l e f o r f i e l d s c o n t a i n i n g c o r n e r s w i t h

high c u r v a t u r e s . Thorn and A p e l t i n t h e i r u s e f u l book " F i e l d Computations

i n E n g i n e e r i n g and P h y s i c s " (121) suggested some h e l p f u l schemes to deal

w i t h t h i s d i f f i c u l t y . Greenspan (51,52) a l s o suggested some other

formulae to s o l v e t h i s problem. However, no matter what kind of scheme

i s being used, i t should be s t r e s s e d t h a t i t i s most important to use a

f i n e r mesh i n the c o n s t r i c t i o n r e g i o n s . I f the elements used are sma l l

enough, then the f i r s t - o r d e r one-sided d i f f e r e n c e formula can s t i l l be

employed. The key point i s th a t the e f f e c t s of the corne r s must be forced

to spread i n t o the f l u i d f i e l d i n every d i r e c t i o n . I t i s worth emphasizing

here t h a t the f i n i t e element method i s simply a man-made method. I t i s

important to help the method by l e t t i n g "him" work as c l o s e to the

phenomenon which occurs p h y s i c a l l y as p o s s i b l e .

R e f e r r i n g to chapter 4 again, i n which some i d e a s about boundary

c o n d i t i o n s have been presented, the f o l l o w i n g boundary c o n d i t i o n s may be

s p e c i f i e d (see F i g u r e 6-9).

ijr = 0 on FG (6-8)

= | on ABODE (6-9)

U) 0 on SfJ (6-10)

At time t=o, val u e s of v o r t i c i t i e s are assumed to be zero everywhere.

And these v a l u e s are considered as the i n i t i a l c o n d i t i o n s when the numerical

s o l u t i o n s of stream f u n c t i o n and v o r t i c i t y f o r creeping flow are s o l v e d .

A f t e r the convergent c r e e p i n g flow s o l u t i o n was obtained, t h i s s o l u t i o n

was used f o r the i n i t i a l c o n d i t i o n s of stream f u n c t i o n and v o r t i c i t y i n

the i t e r a t i o n process to c a l c u l a t e the flow a t a small Reynolds number.

-73-

T h i s process was c a r r i e d out such t h a t the s o l u t i o n a t a low Reynolds number was used as the i n i t i a l c o n d i t i o n s f o r the s o l u t i o n at a higher Reynolds number ( s e e Chapter 4 ) . I t has been found t h a t the i t e r a t i o n procedure was always s t a b l e f o r s u f f i c i e n t l y s mall £ t . Through numerical experiments, i t i s found t h a t a high c u r v a t u r e at point C (see F i g u r e 6-9) has profound adverse e f f e c t s on both numerical s t a b i l i t y and accuracy. Even with the same mesh s i z e , the high-curvature body r e q u i r e s a time step t h a t i s an order of magnitude s m a l l e r than that f o r a low-curvature body i n order to achieve s t a b i l i t y ( 1 0 5 ) . T h i s i s due mainly to the very steep g r a d i e n t s of the v o r t i c i t y i n the f i e l d c r e a t e d by the high c u r v a t u r e . The same g r a d i e n t s a l s o cause severe problems w i t h r e s p e c t to l o s s of accuracy ( 1 0 5 ) . I n order to avoid l a r g e e r r o r s , and a c c e l e r a t e the speed of convergence, i t seems th a t two kinds of time s t e p s , A t , and a s m a l l e r one ^ t g , may be used i n low-curvature regions and h i g h - c u r v a t u r e ones r e s p e c t i v e l y , ( i n t h i s example, t h i s kind of scheme has not been employed y e t . )

6.2.3.Conclusion

The input data used i s presented i n Appendix F. I t s main r e s u l t s f o r

stream f u n c t i o n and v o r t i c i t y from t h i s program are g i v e n i n Appendix G.

The stream l i n e contours are presented i n F i g u r e 6-: 11. The contours seem

to be along with those of Lee and Fung (79) (see F i g u r e 6-12). R e s u l t s

f o r w a l l v o r t i c i t y are shown i n F i g u r e 6-13. From t h i s f i g u r e , i t i s

found t h a t i n the Stokes l i m i t the flow p a t t e r n s are symmetric . before and

a f t e r the c o n s t r i c t i o n , no s e p a r a t i o n occurred a t t h i s l i m i t . T h i s

r e s u l t seems to agree w e l l w i t h both the s o l u t i o n s of Cheng (25) and those

by Lee and Fung ( 7 9 ) . The s l i g h t d i s c r e p a n c i e s among the r e s u l t s f o r the

v o r t i c i t y on the w a l l of Stokes flow from t h i s program and those of Cheng

(25) and Lee and Fung (79) are probably due to the d i f f e r e n c e s i n the

geometries of c o n s t r i c t i o n s and the coarseness of these meshes being used.

The v e l o c i t y r e s u l t s are presented i n F i g u r e 6-14. They seem to be i n

-74-

75

I lllli III III in I III III 11/,I

( a ) The geometry used b\ Lee & Fung

b) S t r e a m l i n e s Contours

FIG.6-12 R e s u l t s or Lee L Fung

76

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good agreement with the s o l u t i o n of Cheng ( 2 5 ) . The pre s s u r e r e s u l t s are given i n F i g u r e 6-15. The s l i g h t d i s c r e p a n c i e s between the s o l u t i o n s f o r pr e s s u r e from t h i s program and those of Lee and Fung (79) are a l s o probably due to the d i f f e r e n c e s i n the geometries of c o n s t r i c t i o n s and the coarseness of the f i n i t e element g r i d .

The r e s u l t s seem to compare reasonably w e l l thus i n d i c a t i n g t h a t t h i s

program seems to be a c c u r a t e l y r e p r e s e n t i n g the p h y s i c a l phenomenon.

-80-

Chapter 7 DISCUSSION

Although the numerical schemes to s o l v e the Navier-Stokes equations have

been d i s c u s s e d , some r e a l i s t i c d i f f i c u l t i e s can s t i l l occur when the schemes

are c a r r i e d out. I n t h i s chapter, some problems w i l l be d i s c u s s e d . And

some o b s e r v a t i o n s about the numerical procedure, obtained through numerous

numerical experiments, w i l l be presented and d i s c u s s e d . These o b s e r v a t i o n s

may be u s e f u l i n the f i n i t e element a n a l y s i s .

7 .1 Convergence problems

The problem of convergence i s one of most important problems i n a

numerical a n a l y s i s . When the foregoing process i s app l i e d to the Navier-

Stokes equations, divergence can take p l a c e i n p a r t s of the f i e l d . Emphasis

has a l r e a d y been put on the need f o r care w i t h the method of determining

the boundary v a l u e s of v o r t i c i t y , but even i f these are known and f i x e d the

f i e l d may s t i l l d i v e r g e i f the mesh s i z e i s too l a r g e (33,42,100,121).

To prevent divergence, Thorn (121) suggested t h a t a t each point the value

of 60 may be a d j u s t e d from the old value CO^, to the newly c a l c u l a t e d value

GJn+1. In other words, CO Its f u l l movement may be given. Thorn c l a i m s

that the movement should be r e s t r i c t e d by combining (rin and&3n+l i n the

pr o p o r t i o n , r:1 where r i s p o s i t i v e . However, i f r i s too l a r g e , i t i s

obvious t h a t the r a t e of advance would be very slow.

In the c a l c u l a t i o n s f o r w a l l v o r t i c i t y , sometimes, i t happens t h a t

to repeat t h i s o p e r a t i o n many times w i l l r e s u l t i n an un s t a b l e o s c i l l a t i o n

of the f i e l d . Thorn and A p e l t (121) suggested t h a t new boundary values

(O = (0)n) + K [(fcin+l) - (^n)w] W W l W }

may be used i n s t e a d of o r i g i n a l ones, where K i s l e s s than u n i t y . The

best value of K can only be estimated by t r i a l and experience. As a s t a r t

they suggested K = 0.5. I n order words the boundary values are only

moved about one-half of the amount i n d i c a t e d by equation ( 4 - 4 ) .

Lee and Fung (79) a l s o employed a s i m i l a r manipulation. They combined

the conformal mapping and f i n i t e - d i f f e r e n c e techniques to i n v e s t i g a t e

-81-

problems of v i s c o u s flow i n a l o c a l l y c o n s t r i c t e d tube. They found t h a t the c a l c u l a t i o n was s t r a i g h t f o r w a r d f o r Re ^ 1 5 . But when Re =i 20, the numerical procedure f a i l e d to converge. To improve the matter, they used an u n d e r - r e l a x a t i o n f a c t o r . They c l a i m t h a t the values of the v o r t i c i t y cOwere g i v e n one h a l f t h e i r t h e o r e t i c a l changes i n each i t e r a t i o n . I n t h i s way they got the r e s u l t s f o r Re=25.

Although i n c o r p o r a t i o n of the u n d e r - r e l a x a t i o n technique seems to be

ab l e to a c c e l e r a t e the speed of convergence, and improve the p o s s i b i l i t y

of convergence, t h e r e i s s t i l l no, to the best knowledge of the author,

apparent t h e o r e t i c a l j u s t i f i e s t L o n f o r such a manipulation of the r e l a x a t i o n

f a c t o r .

Some comparisons between the r e s u l t s of u s i n g a r e l a x a t i o n f a c t o r and

not u s i n g t h i s kind of f a c t o r have been made, and presented i n appendices.

Note that when TURF1=1 (see Appendix D) the r e l a x a t i o n f a c t o r s were

never used, and when TURF1 =0.1, the r e l a x a t i o n f a c t o r s used were 0.1.

7.2 Storage problems

Irs many f i n i t e element problems the amount of core s t o r e r e q u i r e d i s

too g r e a t f o r the computer being used and i t i s n e c e s s a r y to use backing

s t o r e . Sometimes a p e r i p h e r a l such as a magnetic d i s c on a magnetic

tape deck can be used a u t o m a t i c a l l y w i t h i n a program. The n o n - l i n e a r

matrix, because of i t s s i z e , may be s t o r e d out of core, the high-speed

d i s c being the next b e s t p l a c e .

There are l o t s of ways to improve storage problems: such as:

employing the techniques of the f r o n t s o l u t i o n , s u b s t r u c t u r i n g , o v e r l a y i n g ,

e q u i v a l e n c e , or dynamic a l l o c a t i o n , e t c . Some important methods of them

w i l l be b r i e f l y d i s c u s s e d here.

7.2.LFront S o l u t i o n (33,37,67,134)

T h e o r e t i c a l l y the f r o n t s o l u t i o n i s q u i t e simple. There i s a l a r g e

l i n e a r s e t of simultaneous equations w i t h the n stream f u n c t i o n s { l ^ J

of the f l u i d flow f i e l d as unknowns. When a l l the information r e l a t i n g to

-82-

a p a r t i c u l a r v a r i a b l e i s complete then t h a t v a r i a b l e may be e l i m i n a t e d s i n c e an e x p r e s s i o n f o r i t i n terms of the other v a r i a b l e s i n the problem can be obtained. Some unknowns can be e l i m i n a t e d before the complete " s t i f f n e s s matrix" i s formed and t h e r e f o r e the whole of the " s t i f f n e s s matrix" i s never needed i n core a t one time. I r o n s (67) has developed a good f r o n t s o l u t i o n program f o r f i n i t e element a n a l y s i s to s o l v e symmetric p o s i t i v e - d e f i n i t e equations. Hood (134) has a l s o presented another f r o n t s o l u t i o n program which may be used f o r the s o l u t i o n of unsymmetric matrix equations. Using these schemes, core requirements and computer time may be c o n s i d e r a b l y reduced. 7.2.2.Banded S o l u t i o n (33,37,116)

Using the f a c t t h a t the s t i f f n e s s matrix i s square and symmetric and

a l l non-zero terms are concentrated i n a narrow band e i t h e r s i d e of the

l e a d i n g diagonal, g r e a t economies of storage are p o s s i b l e by s t o r i n g only

the band.

To s o l v e equations (3-35) and (3-43) d i r e c t l y by u s i n g , say, the

Gauss-Jordan e l i m i n a t i o n procedure would be very i n e f f i c i e n t i n terms

of computer time and storage, s i n c e that does not take advantage of the

banded nature of [ K y ) and [ [ Kw ] + [Kw] / A t "J .

I t appears that the most e f f i c i e n t procedure i s to s t o r e those elements

of (ity,] a n d ( ( K w ] + [ Kw ] / A t ] those are w i t h i n the band rowwise

i n two v e c t o r s r e s p e c t i v e l y , say [ A ] and" [ B ] , and employ a modified

Gaussian e l i m i n a t i o n scheme w i t h back s u b s t i t u t i o n which takes advantage

of the banded nature of [ Ity] and ( [ K W ] + [ K W ] / A t ] . I n t h i s

procedure, Gaussian e l i m i n a t i o n and back s u b s t i t u t i o n need only be c a r r i e d

out up to the lower and upper edges r e s p e c t i v e l y of the bands. Thus the

zeros of [ K y ] and[(kw] + [Kw] / A t ] o u t s i d e the band a r e not operated

upon and are a c t u a l l y not s t o r e d i n [Ky] a n d [ [ Kw ] + [Kw] / At J .

With t h i s method of s o l u t i o n , i t w i l l be n e c e s s a r y to know the width of the

bands and the l o c a t i o n of the diagonal elements w i t h i n the bands a t every

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row of { K y ] and ( [ Kw] + ( Kw ] / A t ] .

7.3 Computer time problems (6,14,25)

There are a l s o a l o t of methods to save computer time. But i f the

computer program has been prepared, then the f o l l o w i n g suggestion may be

a h e l p f u l way to reduce the computation time used.

I t i s found t h a t the i n i t i a l flow c o n f i g u r a t i o n does not change the

s t e a d y . - s t a t e s o l u t i o n . Regardless of the i n i t i a l c o n d i t i o n s employed,

the procedure does indeed converge to the same s o l u t i o n . Therefore,

provided t h a t there i s no i n t e r e s t i n the t r a n s i e n t s o l u t i o n to the problem,

the a n a l y s i s can be s t a r t e d a t a good i n i t i a l guess w i t h attendant s a v i n g

i n computer time. The s o l u t i o n obtained with a coarse mesh can be

u t i l i z e d as the guess f o r a f i n e r mesh.

7.4 Boundary c o n d i t i o n s

I t has been shown th a t a l l boundary c o n d i t i o n s r e q u i r e d e t a i l e d a t t e n t i o n .

But sometimes, even i f the s i g n i f i c a n c e of boundary c o n d i t i o n s has been n o t i c e d ,

i t i s s t i l l not i m p o s s i b l e to f a c e the problems of how to s p e c i f y s u i t a b l e

boundary c o n d i t i o n s f o r a given problem. A p r a c t i c a l way, but not a good

way from p o i n t s of view of computer storage, to overcome t h i s kind of

problem i s presented here.

I t i s pointed out that the d i f f i c u l t y i n boundary c o n d i t i o n s , sometimes

can be by-passed i f the region to be considered i s changed. U s u a l l y , a

bigger region or the whole region of the problem can be used i n s t e a d of a

s m a l l e r or h a l f the region. For example, to study the problem of vortex

s t r e e t development behind some o b s t r u c t i o n s i n . channel of f i n i t e width, the

boundary c o n d i t i o n s i n the c e n t r e l i n e a re not q u i t e obvious. So i n t h i s

c a s e , the best way to deal w i t h t h i s kind of problem i s j u s t to use the

t o t a l region of the flow f i e l d i n s t e a d of the h a l f symmetric one. (see

F i g u r e 7~1). (110)

7.5 F i n i t e - e l e m e n t mesh

A l i k e l y d i s t r i b u t i o n of contours of f i e l d v a r i a b l e s of i n t e r e s t i s best

be p r e d i c t e d i n advance. Then the mesh i s arranged to be as s i m i l a r to

1

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-85-

the p r e d i c t e d d i s t r i b u t i o n as p o s s i b l e . For example, a f i n e r mesh should

be arranged f o r the regions where the v a r i a t i o n s of streamfunctions or

v o r t i c i t i e s are more pronounced.

I f t h e r e i s no idea a t a l l about the d i s t r i b u t i o n of contours of

f i e l d v a r i a b l e s , an a l t e r n a t i v e way suggested here i s t h a t a coarse mesh

can f i r s t be made to a n a l y s e the problem. I t i s p o s s i b l e to get some

i d e a s of values of r e s u l t s from t h i s a n a l y s i s based on the coarse mesh.

Andthen, these values may be used as a guide to arrange a b e t t e r mesh.

Bearing i n mind that an a p p r o p r i a t e mesh used can save computer storage,

and computation time, but a bad mesh can even make the c a l c u l a t i o n u n s t a b l e .

7.6 Some Obeservations

Through numerous numerical experiments, some obs e r v a t i o n s were made.

They may be u s e f u l i n the f u t u r e a p p l i c a t i o n s of the f i n i t e element method.

Although t h e i r v a l i d i t y f o r a l l k i n d s numerical schemes has not been

a s c e r t a i n e d yet, i t i s hoped to b r i n g these o b s e r v a t i o n s to people's

a t t e n t i o n .

7.6.1.The t r a n s m i s s i o n phenomena of a mesh l i n e

To save computer time and storage, i t i s b e s t to reduce the number

of elements to as few as p o s s i b l e . However to improve and guarantee the

a c c u r a c y and s t a b i l i t y of a c a l c u l a t i o n , i t i s hoped to i n c r e a s e the

number of elements used to as many as p o s s i b l e . T h i s s i t u a t i o n i s l i k e

a famous Chinese proverb which says " I t i s d i f f i c u l t to make a horse f a t

without g i v i n g i t enough food". U s u a l l y , i f the number of elements i s

reduced too much, then i n e v i t a b l y some of the accuracy w i l l be l o s t , and

convergent r e s u l t s can not even be reached. However, even i f t h i s i s

the case, a coarse mesh i s o f t e n forced to be employed i n a complex region,

even though a f i n e r g r i d should have been used, i n view of the l i m i t a t i o n

imposed by the computer. I n t h i s case, the best way to do t h i s i s to

improve the " q u a l i t y " of the mesh being used. I t i s found t h a t mesh

l i n e s seem to have an a b i l i t y of g i v i n g e f f e c t to the c a l c u l a t i o n by

-86-

t r a n s m i t t i n g or spreading newly c a l c u l a t e d values i n t o nodes of neighbouring elements. I t i s known t h a t any numerical method i s simply an instrument to help people do what they want to do, and an a l y s e what they wish to analyse I f the instrument i s hoped to work properly, then i t should be used i n a proper manner. I f the f i n i t e element scheme, the i n s t r u c t m e n t , i s helped by u s i n g a good mesh employing the nature of a mesh l i n e , i t i s not im p o s s i b l e , w h i l e adopting a coarse gridwork i n a c a l c u l a t i o n , to get a s a t i s f a c t o r y and a c c u r a t e and s t a b l e r e s u l t . I t i s found through numerical experiments t h a t i n a region on which a high c u r v a t u r e i s found, more mesh l i n e s should be used t h e r e . The higher a c u r v a t u r e i s , the more mesh l i n e s should be used. And i t i s b e t t e r to do the mesh s y m m e t r i c a l l y u n l e s s the e f f e c t of the mesh on some d i r e c t i o n s i s hoped to i n t e n s i f y through the c a l c u l a t i o n . To i l l u s t r a t e the i d e a , an example i s presented.

F i g u r e 7-2 shows an o b s t a c l e i n a f l u i d flow or a hole on a p l a t e .

The c u r v a t u r e s of points B and D are the same, and so the same number of

mesh l i n e s i s suggested. The c u r v a t u r e a t point A i s the h i g h e s t , then

the numer of mesh l i n e s used should be more than those a t p o i n t s B,C and

D. Note th a t a l l the mesh l i n e s are symmetrical. Furthermore, not only

to save storage but a l s o to reduce computer time, i n a re g i o n where

begger unsymmetric v a r i a t i o n s of r e s u l t s are expected, the number of mesh

l i n e s there should be i n c r e a s e d unsymmetrically. I t i s s i g n i f i c a n t f o r

a u s e r of a f i n i t e element program to l e a r n the e f f e c t of a shape f u n c t i o n

on the choice of a mesh f o r a c a l c u l a t i o n and to have an idea about the

a c t u a l p h y s i c a l phenomena d e s c r i b e d by the problem. For example, from the

foregoing d i s c u s s i o n s , i t i s obvious to conclude that the optimal mesh f o r

a v i s c o u s f l u i d flow depends a l s o on Reynolds number. See F i g u r e 7-3.

A mesh f o r case A should be d i f f e r e n t from t h a t f o r case B. I f the same

mesh i s used to an a l y s e the two c a s e s , then there w i l l be a l o t of computer

time and storage wasted i n a n a l y s i n g f o r case B. Bearing i n mind th a t to

he l p the procedure p r e d i c t the a c t u a l p h y s i c a l phenomena, a s u i t a b l e mesh

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-88-

( 1 ) Case A, Re = 10

( 2 ) Case B, Re = 60.0

Flow round a c y l i n d e r

-89-

d e n s i t y f o r each node must be used to enable the procedure to f o r c e and

spread the e f f e c t of governing equations i n the regions of i n t e r e s t .

7.6.2.Maximum s t a b l e time s t e p A t max.

Through numerical experiments i t i s found t h a t the maximum s t a b l e

time s t e p & t max a l s o depends on Reynolds number, mesh s i z e , s u r f a c e

c u r v a t u r e s of bodies, e t c .

For the example one, i f the input data shown i n Appendix I i s used,

the v a l u e s f o r Atmax are

Re = 0.002 , A t max = 0.00001

Re 1. , A t max = 0.0001

Re = 2. A t max = 0.0001

Re = 3. , A t max = 0.0002

Re 4. , A t max = 0.0002

Re 5. , A t max = 0.0004

Re = 6. , A t max = 0.0004

I t may be mentioned here t h a t i t i s d i f f i c u l t to determine e x a c t l y when

A t max i s reached. Thus, a l l v a l u e s must be considered as approximate.

The parameters which a f f e c t the zones of convergence would a l s o i n c l u d e

maximum s t a b l e time step, Renolds number, mesh s i z e , e t c .

To a c c e l e r a t e the speed of convergence and to make the r e s u l t s s t a b l e , these

parameters should be t r i e d .

7.7. General d i s c u s s i o n s

I t has been found t h a t the f i n i t e element s o l u t i o n algorithm i s

capable of p r e d i c t i n g some n a t u r a l p h y s i c a l phenomena without r e s o r t to s p e c i a l

d e v i c e s . The f i n i t e element method i s ab l e to d e f i n e the nodal p o i n t s and

elements a r b i t r a r i l y to permit f l e x i b i l i t y and easy accommodation of the

complex boundary. Employing the knowledge of f l u i d dynamics concerning

the a n t i c i p a t e d s o l u t i o n d i s t r i b u t i o n , a s m a l l e r element i n regions

c o n t a i n i n g l a r g e r s p a t i a l d e r i v a t i v e s of the dependent v a r i a b l e s can be

used. Some of the other major advantages of the f i n i t e - e l e m e n t method

-90-

over f i n i t e d i f f e r e n c e ones are t h a t d i f f e r e n t shapes may be represented a u t o m a t i c a l l y , v a r i o u s boundary c o n d i t i o n s may be s a t i s i f i e d i n a s t r a i g h t forward manner, d i f f e r e n t element s i z e s may be used to get maximum e f f i c i e n c y . A f i n e r mesh can be used to gain d e t a i l s of the f l u i d flow f i e l d e x a c t l y where d e s i r e d . G e n e r a l l y s i g n i f i c a n t l y fewer equations are r e q u i r e d to provide a given accuracy. (16,62,77,100).

As f a r as computational e f f o r t s a r e concerned, a computer program f o r

the f i n i t e element computation seems to be more complicated than i t s

c o u n t e r p a r t f o r the f i n i t e d i f f e r e n c e method. However, the c o m p l i c a t i o n

stems from the i n t r i n s i c g e n e r a l i t y of the f i n i t e element. The g e n e r a l i t y

of the f i n i t e element method u s u a l l y l e a d s to the computer program to be

a p p l i c a b l e to a c l a s s of s i m i l a r problems (19,25,33).

However, j u s t as a Chinese s a y i n g has i t "There i s nothing i n the world

which i s p e r f e c t 1" The f i n i t e element method a l s o s u f f e r s from some

disadvantages. I t i s known t h a t e r r o r a n a l y s i s i s very important i n

numerical methods. However, up to now, there does not seem to be a method

which can be used to c a l c u l a t e the t r u n c a t i o n e r r o r i n c u r r e d by u s i n g a

p a r t i c u l a r kind of element shape. With the f i n i t e d i f f e r e n c e method, on

the o t h e r hand, the t r u n c a t i o n e r r o r i n v o l v e d i n any f i n i t e d i f f e r e n c e

formula can be analysed using the c a l c u l u s of f i n i t e d i f f e r e n c e . However,

i t i s expected t h a t the t r u n c a t i o n e r r o r i n c u r r e d w i l l be comparable w i t h

th a t of a f i n i t e d i f f e r e n c e mesh of the same s i z e , so i t i s p o s s i b l e to

get some i d e a s about the order of approximation of a p a r t i c u l a r element

shape. When a h i g h e r order of approximation to the unknown f u n c t i o n i s

sought, the s i t u a t i o n may become more complicated w i t h the f i n i t e element

method. With the f i n i t e d i f f e r e n c e method, i n c r e a s i n g the order of

approximation p r e s e n t s no r e a l d i f f i c u l t y (25,116).

I t can be shown th a t the f i n i t e element approach converges to the

exa c t s o l u t i o n as the number of elements i s i n c r e a s e d . S o l u t i o n convergence

w i t h f i n e r f i n i t e element mesh i s very s i g n i f i c a n t f o r numerical s o l u t i o n

of n o n - l i n e a r equations l i k e the Navier-Stokes equations. An i n s u f f i c i e n t

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number of p o i n t s on the s o l i d boundary, a t which n o - s l i p boundary c o n d i t i o n s are s p e c i f i e d may cause u n s a t i s f a c t o r y r e s u l t s . The n o - s l i p c o n d i t i o n determines the amount of v o r t i c i t y c r e a t e d a t the s o l i d s u r f a c e . The h i g h e s t values of the w a l l v o r t i c i t y and the v o r t i c i t y g r a d i e n t , which govern the spreading of v o r t i c i t y i n the f l u i d flow f i e l d a r e found on the s o l i d w a l l (20,25,79,121).

The c o e f f i c i e n t m a t r i x [ [ Kw ] + [ Kw ] / A t ] i s symmetric, banded,

and p o s i t i v e d e f i n i t e . To keep the bandwidth of the c o e f f i c i e n t matrix

to a minimum, the nodal p o i n t s should be ordered i n such a way that the

d i f f e r e n c e i n nodal point numbers f o r any element be a minimum (25,33).

The three-node t r i a n g u l a r element seemed to g i v e q u i t e a c c u r a t e r e s u l t s .

I f the s i z e s of the elements are small enough, the approximation of the

unknown fu n c t i o n w i t h the element i s adequate. Owing to the s i m p l e r

formulation and the a b i l i t y to c a t e r f o r a r b i t r a r y boundary shape, the

three-node t r i a n g u l a r element seems to be adequate f o r most purposes

(63,116).

Although convergence i s expected f o r higher Reynolds number, such

a study was d i s c o n t i n u e d , i n view of the f a c t t h a t f o r higher Reynolds

number f l u i d flow f i e l d , the channel between p a r a l l e l p l a t e s or a

c o n s t r i c t i n g i n t e r n a l passage must be elongated and f i n e r mesh must be

used to get s t a b l e r e s u l t s , thus n e c e s s i t a t i n g t h a t the number of mesh

po i n t s be i n c r e a s e d so mush as to be i m p r a c t i c a b l e f o r t h i s computer.

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Chapter 8 CONCLUSIONS

A g e n e r a l numerical procedure f o r the a n a l y s i s of two-dimensional,

time-dependent,incompressible,viscous f l u i d flow i s presented. A f i n i t e -

element computer program i s developed.

Using a combined v a r i a t i o n a l p r i n c i p l e - f i n i t e element method,

d i f f i c u l t i e s a r i s i n g from the n o n l i n e a r i t y of time-dependent N a v i e r -

Stokes equations have been remedied. The numerical r e s u l t s obtained by

the method have r e v e a l e d very s i m i l a r p r o p e r t i e s to known s o l u t i o n s of

s i m i l a r problems. I n the Stokes l i m i t , the flow p a t t e r n s are symmetric

before and a f t e r the c o n s t r i c t i o n , no s e p a r a t i o n occurred a t the l i m i t .

The h i g h - c u r v a t u r e body r e q u i r e s a time s t e p t h a t i s an order of magnitude

s m a l l e r than t h a t f o r a low-curvature body i n order to achieve s t a b i l i t y .

I n order to avoid l a r g e e r r o r s , an extremely f i n e mesh must be used i n

the regions of l a r g e g r a d i e n t s of the v o r t i c i t y .

The accuracy of the f i n i t e element scheme depends b a s i c a l l y on the

number of nodal p o i n t s i n the f i n i t e element mesh, the time step, and order

of the numerical i n t e g r a t i o n procedure. S t a b l e r e s u l t s can be obtained

f o r a s u f f i c i e n t l y s mall time s t e p . The simple time i n t e g r a t i o n scheme was

found to be s u f f i c i e n t l y a c c u r a t e f o r present t e s t s . To maintain the accuracy

of the c a l c u l a t i o n , the number of i t e r a t i o n s r e q u i r e d i n c r e a s e s s l i g h t l y w i t h

Reynolds number.

F i n a l l y , some important p o i n t s would be s t r e s s e d as f o l l o w s :

( 1 ) The maximum s t a b l e time s t e p &t max a l s o depends on Reynolds number,

g r i d s i z e s and the shape of a body.

( 2 ) Even i f the f i n i t e element formulation used i s the same, the zones of

convergence f o r d i f f e r e n t problems may not be i d e n t i c a l . The zones of

convergence a l s o depend on the nature of the flow problem,

Reynolds number, mesh s i z e s , time step, the way of c o n s t r u c t i n g a mesh, and

the shapes of o b s t a c l e s , e t c .

( 3 ) The boundary c o n d i t i o n s a t the body s u r f a c e play.a d e c i s i v e p a r t i n

the s o l u t i o n procedure.

-93-

( 4 ) The topology and number of elements of a mesh a l s o depend on the n a t u r a l boundary c o n d i t i o n s used. Many elements would be r e q u i r e d to reasonably approximate such n a t u r a l boundary c o n d i t i o n s as occurred i n t e s t problem one.

(5) The f a c t t h a t the numerical procedure w i l l be convergent i f A t

used i s small enough means th a t the i n e r t i a terms play a s t a b i l i z i n g r o l e

i n the scheme.

Using the f i n i t e element method, the region of low p r e s s u r e of a body

i n flow, which accounts f o r most of the drag f o r c e may n u m e r i c a l l y be

c a l c u l a t e d . When the v e l o c i t y of the f l u i d i n c r e a s e s , a symmetric eddies

can be produced behind the body which a r e a l t e r n a t i v e l y shed. For low

Reynolds number c a s e s , t h i s phenomenon can be p r e d i c t e d and the shedding

f r e q u e n c i e s can be found by employing the f i n i t e element method ( 1 3 5 ) .

These problems are i n t e r e s t i n g i n the design of o f f s h o r e s t r u c t u r e s .

I t appears t h a t the f i n i t e element method may be powerful to

p r e d i c t the n a t u r a l p h y s i c a l phenomena of a f l u i d flow f i e l d . The next

major f i e l d s f o r study would i n c l u d e s t a b i l i t y and convergence problems

as w e l l as f u r t h e r r e s e a r c h i n t o a p p l i c a t i o n s of the method to the

design of o f f s h o r e s t r u c t u r e s .

-94-

REFERENCES

1. Adey.R. and Brebbia, C.A., " F i n i t e Element S o l u t i o n f o r E f f l u e n t . D i s p e r s i o n " Numerical Methods i n F l u i d Dynamics, C.Brebbia and J.J.Connor ( E d s . ) , Pentech P r e s s , 1974.

2. Atkinson, B., Brocklebank, M.P., Card, C.C.H. and Smith,J.M. "Low Reynolds Number Developing Flows", Am.Inst.Chem.Eng.J.,Vol.15, J u l y 1969.

3. Atkinson,B.,Card,C.C.H.and Irons,B.M., " A p p l i c a t i o n of the F i n i t e Element Method to Creeping Flow Problems", Trans . I n s t . Chem. Eng. , Vol. 48-, 1970.

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a;- >u - i N o i i ' H ^ o - i'-, o c | ^ j u\ m •—* y rs_ m 4 m o t - - n~ i n *c 1*- UJ r-* l \ J I ' l ^ ^ « 0 r— a ; w (J* y V.J r-1 t \ J [*| J - U i U l ^ • J \ u . l A u i

—* —* i-H r - l r - l r * r - i ,-4 r - l

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o

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r- < r - J . - I -* —* r - l r-< r - l f-H i—I * r - t H ,--{

U - U . O .

—1 o IU U.J V-

APPENDIX G -140-

Nodal points Streamfunction V o r t i c i t y

1 1. 2.934 2 0.995 2.820 3 0.942 2.387 4 0.839 1.953 5 0.695 1.519 6 0,519 1.085 7 0.296 0.560 8 0.0 0.0 9 1.0 2.857

10 0.994 2.679 17 1.0 2.633 18 0.995 2.467 25 1.0 2.177 26 0.996 2.071 33 1.0 1.271 34 0.998 1.368 42 0.999 0.588 49 1.0 0.007 72 1.0 20.785 77 1.0 29.091 78 0.913 14.613 79 0.908 14.665 84 1.0 21.323 111 1.0 0.005 113 0.999 0.585 120 1.0 1.267 121 0.998 1.364 128 1.0 2.173 129 0.996 2.068 136 1.0 2.631 137 0.995 2.465 144 1.0 2.855 145 0.994 2.678 152 1.0 2.934 153 0.995 2.820

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Nodal points Streamfunction V o r t i c i t y

154 0.942 2.387 155 0.839 1.953 156 0.695 1.519 157 0.519 1.085 158 0.296 0.600 159 0.0 0.0

-142-

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