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All Theses and Dissertations

1972-8

Application of the Finite Element Method Usingthe Method of Weighted Residuals to TwoDimensional Newtonian Steady Flow withConstant Fluid PropertiesMien Ray ChiBrigham Young University - Provo

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BYU ScholarsArchive CitationChi, Mien Ray, "Application of the Finite Element Method Using the Method of Weighted Residuals to Two Dimensional NewtonianSteady Flow with Constant Fluid Properties" (1972). All Theses and Dissertations. 7085.https://scholarsarchive.byu.edu/etd/7085

L <nS

APPLICATION OF THE FINITE ELEMENT METHOD

0 . o G 'Zr>

~2 6

USING THE METHOD OF WEIGHTED RESIDUALS

TO TWO DIMENSIONAL NEWTONIAN STEADY

FLOW WITH CONSTANT FLUID PROPERTIES

(

A Thesis

Presented to the

Department of Mechanical Engineering

Brigham Young University

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

by

Mien Ray Chi

August 1972

This thesis, ly Mien Ray Chi, is accepted in its present form

by the Department of Mechanical Engineering of Brigham Young

University as satisfying the thesis requirement for the degree of

Master of Science.

'i,-.,? ! ^ . ( < ) 7 >' ! Date'

ii

To My Parents

iil

ACKNOWLEDGEMENTS

Sincere appreciation is expressed to Dr. Howard S. Heaton

for his personal suggestions and encouragement throughout this

thesis. Thanks are also extended to Dr. Henry N. Christiansen

and Dr. Richard A. Hansen for their private instruction.

The author is especially indebted to his parents for their

love and their expressions of confidence and encouragement.

iv

TABLE OF CONTENTS

ACKNOWLEDGEMENTS

LIST OF TABLES .

LIST OF FIGURES.

Page

iv

vii

viii

ChapterI. INTRODUCTION............................................. 1

The Problem Considered The Finite Element Method Summary

II. METHOD OF WEIGHTED RESIDUALS...................... 3

Brief Historical ReviewThe Method of Weighted Residuals

III. GOVERNING EQUATIONS ..................................... 5

Governing Equations Continuity Equation Momentum EquationsGeneral Procedure for Solving the Governing Equations

IV. INTEGRAL EQUATIONS AND DISCRETIZATION .................... 11

Integral EquationsCompatibility Integral Equation Vorticity Transport Integral Equation

Discretization Remarks

V. APPROXIMATIONS AND WEIGHTING VECTOR ..................... 19

Introduction Linear Approximations

Stream Function Approximation Vorticity Approximation

Weighting Vector - Galerkin's Criterion Evaluation of ds Summary

v

vi

Page

VI. FINITE ELEMENT EQUATIONS ................................ 29

OutlineEvaluation of Integral (6.1)Evaluation of Integral (6.2)Finite Boundary Slope Infinite Boundary Slope

Evaluation of Integral (6.3)Evaluation of Integral (6.4)Finite Boundary Slope Infinite Boundary Slope

Summary

VII. ASSEMBLING OF FINITE ELEMENT EQUATIONS .................. 41

The Concept of AssemblingAssembling of the Compactibility Element Equations - The Compactibility Set

Assembling of the Vorticity Transport Element Equations - The Vorticity Transport Set Summary

VIII. SOLUTION METHOD.......................................... 45

OutlineBoundary ConditionsStream Function and Vorticity Solutions by an Iteration

Scheme Velocity Solutions The Use of Digital Computer Solved ProblemsFully Developed Flow Between Two Stationary Parallel

Planes Couette Flow Conclusion

LIST OF REFERENCES................................................. 56

APPENDIX........................................................... 58

Computer Program List

LIST OF TABLES

Table Page

1. Comparison of Exact and Finite Element Solutions forParallel Flow Between Fixed Walls. . . . . . . . . . . . . 52

2. Comparison of Exact and Finite Element Solutions forCouette F l o w ............................................ 54

LIST OF FIGURES

Figure Page

1. Fluid Region Showing the Positive Flow Direction Used inthe Definition of the Stream Function.......... 6

2. Triangular Finite Element.................................. 19

3. Linear Variation of Stream Function of a TriangularFinite Element ........................................... 20

4. A Boundary Element.......................... 26

5. An Interior Node with Neighboring Elements to Show theAssembling Procedures.................... 42

6. An Interior Node with Neighboring Elements to Show theRelation Between the Velocity Components of the Interior Nodes and the Stream Functions of the Neighboring Nodes. . 49

7. Parallel Flow with Parabolic Distribution.................. 50

8. A Typical Finite Element Arrangement ...................... 52

9. Couette Flow Between Two Parallel Flat Walls............... 53

viii

CHAPTER I

INTRODUCTION

The Problem Considered

In this thesis we will be concerned with the two-dimensional

steady laminar Newtonian fluid flow with constant fluid properties in

Cartesian coordinate system. More precisely, we will consider a flow

for which the interesting dependent variables, for example, velocity,

vorticity, stream function, etc., are expressible as functions of two

independent Cartesian coordinates.

The governing equations for the problem considered are the

continuity equation and the momentum equations. We shall first state

these basic equations in terms of the local velocity and then in terms

of the local stream function and vorticity. The use of stream function

and vorticity will enable us to reduce the number of dependent variables

and to facilitate the method of weighted residuals which will be

introduced in the next chapter.

The continuity equation and the momentum equations with the

prescribed boundary conditions constitute a boundary value problem which

is difficult to solve because of the nonlinearity of the momentum

equations. The problem can be solved exactly only for several simple

cases. For complicated cases, numerical methods have to be used. The

finite difference method is an extensively used numerical method. One

disadvantage of the finite difference method is the difficulty in fitting

1

2

irregular boundary shaves to finite difference grids. However, another

numerical method— the finite element method— does not have this

difficulty.

The Finite Element Method

The finite element method was originally developed for the

analysis of stress problems and has been successfully applied to a(13)variety of structure problems. Fluid dynamics and heat transfer

workers have started trying to apply this method to solve the problems in their own £ields . <D , (6) , (9) , (10), (H) , (12)

The development of the finite element method is based on the

fact that a physical problem governed by a set of differential equations

"may" be equivalently expressed as an extremum problem by the

variational principle. An integral which is to be minimized must be

found from the differential equations or directly from basic principles.

For certain types of problems, the integral to be minimized is easy to

find. However, it is generally difficult and may be impossible to

obtain. It is the purpose of this thesis to get rid of this difficulty

by using the method of weighted residuals to accomplish the minimizing

process without actually finding the integral to be minimized. The

concept of the method of weighted residuals will be reviewed in the

next chapter.

Summary

This thesis will solve a general two-dimensional steady fluid

flow with constant fluid properties in Cartesian coordinate system using

the finite element method based on the concept of weighted residuals.

CHAPTER II

METHOD OF WEIGHTED RESIDUALS

Brief Historical Review

The method of weighted residuals was first used by Crandall and (5)Kantorovich to obtain a solution for a differential equation.

(2)Finlayson and Scriven also showed that variational principles applied

to steady-state heat transport problem are applications of the method of

weighted residuals.

The Method of Weighted Residuals

Suppose we are given a mathematical statement,

Ltf^r), f2(r), fn (r)] = 0 (2.1)

~bwhere f^(r) (i = 1, 2, ..., n) are scalar functions of the position

vector r and L stands for a mathematical relation among these scalar

functions. Equation (2.1) holds true if and only if exact solutions for

f^(r) are substituted into it. If approximate solutions are substituted

into Equation (2.1) rather than exact solutions, we will obtain

L[fx(?), f2(r), ..., fn(r)] = R(r)

~bwhere f^(r) (i = 1, 2, ..., n) are approximate solutions and R(r) is the

residual term which is a function of the position vector r. Note that"V ~bR(r) is generally not zero unless exact solutions for f^(r) are

3

substituted into Equation (2.1). Also note that, generally.

4

R(r)dV- + 0

where the triple integral is over the whole region of interest. How­

ever, we might expect to choose an appropriate weighting function W so

that

0

for f^(r) are quite close to their exact solutions.

Various selections of the weighting function have been tried.

In the material that is to follow, we will use Galerkin’s criterion,

because it is closely related to the variational principle as has been

shown in many specific cases. First of all, we will review the basic

governing equations which will be used and correspond to Equation (2.1).

CHAPTER III

GOVERNING EQUATIONS

Governing Equations

The governing equations are the continuity equation and the

momentum equations. As mentioned before, we shall first state these

equations in terms of the local velocity and then in terms of the local

stream function and the local vorticity.

Continuity Equation

The principle of conservation of mass gives the continuity

equation,

for an incompressible flow.

Using Cartesian notations, we shall define the stream function

i(/(x, y) as a point function which has the following relations with the two

velocity components:

(3.1)

or

V * V - 0 (3.2)

3y (3.3)

(3.4)

5

6

ft is easy to show that the assumption of a stream function as a point

function identically satisfies the equation of continuity, since the

order of differentiation of a point function is immaterial. In other

words, the equation of continuity for the steady two-dimensional flow

of an incompressible fluid is mathematically the necessary and

sufficient condition for the existence of a point function called the

stream function.

Following the sign convention in Equations (3.3) and (3.4),

the stream function can be considered as the volume flow rate from left

to right as the observer views a line from A looking toward P as shown

in Figure 1, if we consider a unit depth in the direction perpendicular

to the x-y plane.

Fig. 1.— Fluid region showing the positive flow direction used in the definition of the stream function.

We further define the vorticity £ as

E = _ 9“. (3.5)* Sx 3y

Differentiating Equation (3.3) with respect to y and

Equation (3.4) with respect to x and using Equation (3.5), we obtain

V2^ + 5 = 0 (3.6)

which is called the compatibility equation and will be used to derive

the finite element equations.

Momentum Equations

The principle of conservation of momentum gives the following

momentum equations in x and y directions:

~ = B - I | E - + v V2uDt x p 8x (3.7)

Dv 1 3p . n2— = B ------ + v VzvDt y p 3y (3.8)

for Newtonian flow with constant fluid properties. B^ and are

respectively the body force in x and y directions.

Eliminating pressure p from Equations (3.7) and (3.8) and

using the definition of vorticity Equation (3.5), we obtain

Dt vV2£ = 0

or

— + V • vt; - w*^ = o (3.9)

because

D_ _ 3_ Dt " 3t + v • V

83£For steady flow (— «= 0), Equation (3.9) becomesO t

V • V£ - vV2£ ■ 0 (3.10)

However,

V . V£ - V . (VO - £V • v

Furthermore,

V • V£ = V • (VO (3.11)

for incompressible flow because of Equation (3.2) .

Substituting Equation (3.11) into Equation (3.10), we obtain

V . (VO - vV2£ = 0 (3.12)

which is called the vorticity transport equation and will also be used

to derive the finite element equations.

General Procedure for Solving the Governing Equations

The governing equations to be solved are the compatibility

equation and the vorticity transport equation:

V2i|> + K = 0 (3.6)

V • (VO - W 2£ = 0 (3.12)

The vorticity £ appears in both equations. It is natural to

eliminate the vorticity from these two equations to obtain one equation

involving only the stream function We may solve for £ from

Equation (3.6) to obtain

We also know the relations between the velocity components and the

stream function,

u 3y

_ M3x

(3.3)

(3.4)

or

V =13y3i3x

(3.14)

Substituting Equations (3.13) and (3.14) into Equation (3.12), we

obtain

3y_ it

3x I

(- V2*) + vV1* = 0 (3.15)

Theoretically, we are able to solve Equation (3.15) with prescribed

boundary conditions and get the solution for the stream function.

♦ = Mx, y) (3.16)

With Equation (3.16), we can find the velocity components by Equations

(3.3) and (3.4), the vorticity by either Equation (3.5) or Equation (3.13).

In this thesis, we will formulate two sets of simultaneous linear

equations. One set is formulated from the compatibility equation,

Equation (3.6) and another set from the vorticity transport equation,

Equation (3.12). Instead of eliminating the vorticity, we will use the

iteration scheme to solve for the stream functions. The vorticity

solution will be the intermediate result of the iteration scheme and

10

the velocity components will be calculated with the concept of weighted

averages.

The two sets of simultaneous linear equations will be

formulated in the following chapters. In Chapter IV, we will first

derive the integral equations for both the compatibility equation and

the vorticity transport equation based on the method of weighted

residuals as has been discussed in Chapter II. Then, we will

discretize the integral equations into finite element equations. In

Chapter V, approximate solutions for the stream function and the

vorticity will be derived and the weighting function, introduced into

the finite element equations in Chapter IV, will be chosen so that two

sets of simultaneous equations will be obtained.

CHAPTER IV

INTEGRAL EQUATIONS AND DISCRETIZATION

Integral Equations

Integral equations will be formulated for both the compatibility

equation and the vorticity transport equation.

Compatibility Integral Equation

The compactibility equation from Chapter III is

V2<ji + £ - 0 (3.6)

If approximate solutions of \p and £ are substituted into the above

equation, the left hand side of the equation will not be identically

equal to zero but equal to some residual term which should approach

zero if the approximate solutions approach their exact forms. In other

words

V 2il> + £ = R (4.1)

where

and 5 are approximate solutions and

R is the residual

Based on the method of weighted residuals, we set the weighted average

of the residual to zero over the volume of interest and obtain

11

WRdV- - 0 (4.2)

r r r12

V-

Note that the region of integration of the above integral is the flow

field in which we are interested and has a unit depth in the direction

perpendicular to the x-y plane for our two-dimensional problem.

Substituting Equation (4.1) into Equation (4.2), we obtain

W(V2^ + £)d¥- = 0

or

(WV2ip + W£)dV- = 0 (4.3)

By vector identities,

WV2\|> = WV'Vijj = V.(WVi^) - W 'V i p (4.4)

Substituting Equation (4.4) into Equation (4.3), we have

f[7*(WVi{,) - VW-V\p + W£]d¥- = 0

or

|(W5 - VW*V^)dV- + / 7 • (WV\(;)dV- = 0 (4.5)

By Gauss divergence theorem, the second integral in Equation (4.5)

becomes a surface integral,

V*(WV^)dV- = / / WV<|>*ds (4.6)

Kith Equation (4.6), we can write Equation (4.5) as

(W£ - VWV\f»)dV- + IJ WVij;.ds « 0 (4.7)

13

For our two-dimensional problem, the volume integral and the surface

integral in Equation (4.7) can be written as a surface integral and a

line integral respectively. Thus, Equation (4.7) becomes the

compactibility integral equation

(4.8)

Note that we have used ds to represent both the infinitesional area

element in Equation (4.7) and the infinitesional line segment in

Equation (4.8). Also note that, in Equation (4.8), both £ and ^ are some

kind of approximate solutions and W is a weighting function which has not

been chosen yet. In Equation (4.8), the domain of the surface integral

is the two-dimensional flow region we are interested in and the line

integral is along the boundary of the two-dimensional flow region.

Vorticity Transport Integral Equation

The vorticity transport equation is

7-(V5) - vV2C - 0 (3.12)

If approximate solutions of V and £ are substituted into Equation (3.12),

we will have

V . ( V O - vV25 = R (4.9)

where R is the residual terra. By the method of weighted residuals, we

have

WRdV- = 0 (4.10)

Substituting Equation (4.9) into Equation (4.10), we obtain

W[7-(7C) - v72£]dV- = 0

By vector identities.

WV.(VQ = V.(WVC) - ^*7W

and

W72£ = W7-7£ = 7-(W7S) - 7W*7£

Substituting Equations (4.12) and (4.13) into Equation (4.11)

[7-(WVO - V£ *7W - v7 • (W7£) + v7W-7§]d¥- = 0

or

7W*(75 - v7^)d¥-+ I II [v7• (W7£) - 7-(WV5)]d

By Gauss divergence theorem,

7 * (W7£) dV- = W7£-ds

and

7 • (WV£)d¥- = WV£*ds

(4.11)

(4.12)

(4.13)

, we obtain

V- « 0 (4.14)

(4.15)

(4.16)

Substituting Equations (4.15) and (4.16) into Equation (4.14), we obtain

15

IIIVW-(V£ - vVQdV-+ //'W(vV£ - $ 0 -ds = 0 (4.17)

For our two-dimensional problem, Equation (4.17) becomes

IIVW-(V£ - vV£)dA + W(vV£ - V£)-ds « 0 (4.18)

This integral equation is called the vorticity transport integral

equation. The meanings of the symbols used here are the same as those

used in the compatibility integral equation. Equation (4.8).

Discretization

With the plane region of interest subdivided into a number of

finite elements, Equations (4.8) and (4.18), which are

can be written as summations over the elements,

and

W(vV£ - V O -ds] = 0 (4.20)

Where

16

M = the total number of elements,

A «* the area of m1"*1 element, m= the external boundary of a boundary element.*

As a simplification. Equations (4.19) and (4.20) can be

generated by assembling element equations of the form,

7W.(V? - vV£)dA + W(vV£ - VO-ds - 0 (4.24)

for a boundary element

Note that Equations (4.21), (4.22), (4.23) and (4.24) are not

*A boundary element is a finite element which has at least part of its boundary is the boundary of the whole region of interest.

(4.21)

A

for an interior element

(4.22)

A Sm m

for a boundary element

(4.23)

Am

for an interior element

A Sm

complete by themselves. However, the element equations for all the

finite elements in the region of interest will be assembled in a

certain fashion in Chapter VII to obtain two sets of simultaneous

linear equations which are complete and will be solved.

If we consider a weighting vector consisting of three weighting

functions, we can have

(4.25)

Amfor an interior element

(4.26)

A Sm

for a boundary element

(4.27)

A

for an interior element

VW-(V£ - vV£)dA + W(vV£ - VO*ds = 0 (4.28)

A Sm

for a boundary element

Note again that these equations are not complete by themselves

and should be assembled to obtain complete equations as will be shown

in Chapter VII.

Remarks

18

Element Equations (4.25), (4.26), (4.27) and (4.28) were

derived by using the weighting vector and the approximate solutions for

the stream function \p and the vorticity £. In the next chapter, the

approximate forms of the stream function and the vorticity will be

derived and the weighting vector selected.

CHAPTER V

APPROXIMATIONS AND WEIGHTING VECTOR

Introduction

To formulate the finite element equations, we need the forms of

the approximate solutions for the stream function and the vorticity

and the weighting vector. We also need an expression for ds in the

line integral of Equations (4.26) and (4.28). They will be discussed

in the following sections.

Linear Approximations

First of all, we should select the shape of the finite elements.

We will choose the triangular element and use it in this thesis. A

typical triangular element is shown in Figure 2.

Fig. 2.— Triangular finite element.

Stream Function Approximation

For our two-dimensional steady flow problem, the stream function

19

is a function of x and y, say

>J's=’Kx, y) (5.1)

We assume a linear variation of the stream function in each

triangular element in terms of the discrete stream functions at the

three vertices, say

41 = ax + by + c (5 .2)

or, in matrix form,

= $ C (5.3)

where

4> = ( 1 x y )

C -

Geometrically, this approximation means a plane surface as shown in

Figure 3.

Fig. 3.— Linear variation of stream function in a triangular finite element.

Let the stream functions at the three vertices of a finite

element be i|/ , iji ,

From Equation (5.2), we have

4»i = ai: + byx + c

2 “ ax2 + by2 + c

^3 = ax3 + by3 + C

or, in matrix form

\l> - BC (5.4)

where

\p = ♦ll<J>2*3

I 1 xi y i \B = 1 x2 y2' I1 x3 y3

Solving for C from Equation (5.4), we obtain

C = B"1^ (5.5)

where

2?

V 3- V 2 x3yr xiy3 Xly2'X2yl \

A = y2~y3 y3~yl yr y2

x -x„ x, -x„ x.-x, /\ 3 2 1 3 2 1 /

where

A = the area of the triangular element.

Substituting Equation (5.5) into Equation (5.3), we obtain

* SB"1!})

or

(5.6)

which is the desired approximate form for the stream function within a

triangular finite element.

From Equation (5.6), we can obtain

(5.7)

where

23

We can also evaluate the velocity within a triangular element

by differentiating Equation (5.6). The result is

V = $ = (U) - v (5.8)

where

A , A „ A31 32 33-A-, -A -A21 22 23 /

“w.

Note that the velocity V is uniform throughout the triangular

element based on linear approximation of the stream function in a finite

element.

Vorticity Approximation

Following the same arguments for the linear approximation for the

stream function in the previous discussion, it is obvious that we can

have the following approximations for £ and V£.

(5.9)

(5.10)

where A, A, 4>, 4>' are as defined previously and

is the vorticity vector consisting of the vertex vorticities.

Weighting Vector - Galerkin's Criterion

The choice of the weighting vector is the most difficult part in

the method of weighted residuals. Different choices will lead to

different approximate solutions to a given problem. We will use the

Galerkin method which proves to be simple and powerful in solving two

dimensional fluid flow problems.(4)The Galerkin criterion takes the weighting vector to be the

transpose of the coefficient matrix of the linear approximate form of

unknown physical quantities (stream function or vorticity in our

problem) so that

W - 2A(4>A)T

25

or

(5.11)

where

1xy

By Equation (5.11), we can obtain

T T' VW = 2A A 1 (5.12)

where

26

Evaluation of ds

For our triangular element, ds can be expressed in a simple

form. Recall that d^ is the line segment along the external boundary of

a boundary element. Referring to Figure 4,* we can write ds; in terms of

the slope of the external boundary as follows:

x

Fig. 4.— A boundary element.

The external boundary is a straight line passing through vertices

2 and 3 and can be expressed as

y = mx + b (5 .13)

where

From Equation (5.13),

*Here we limit the external boundary to the 2-3 side of a boundary element. However, we will not lose the generality.

dy = mdx

27

However, ds = (dx, dy)

so ds ** (dx mdx)

or

ds = (l,m)dx for finite m (5.14)

If the slope m is infinite (vertical external boundary), we will

use

ds = (0, dy)

or

ds « (0, l)dy (5.15)

for vertical external boundary.

Summary

For clarity and convenience, the approximations for the stream

function and the vorticity, and the expressions for weight vector and

ds are summarized below. They will be used in the next chapter to

formulate the finite element equations.

* - *A* (5.6)

Viji = G'A'i' (5.7)

28

V

W

VW

Ia !?

— $Ar2a

k !'«

T T 2AA <S>

T T'■ 2AA *

(5.8)

(5.9)

(5.10)

(5.11)

(5.12)

for m = finite

ds = <

v

(1 m)dx

(0, l)dy

for m = 00

(5.14)

(5.15)

yi

CHAPTER VI

FINITE ELEMENT EQUATIONS

Outline

In Chapter IV we obtained element equations (4.25), (4.26)

(4.27), and (4.28), and in Chapter V we obtained the forms of the

approximate solutions for the stream function and the vorticity and

selected the weighting vector. Now we are ready to substitute the

expression obtained in Chapter V into the element equations (4.25),

(4.26), (4.27), and (4.28). It is obvious that we only need to

evaluate the following four integrals:

m

(6.1)

Sm

WVi{) *ds (6.2)

KVW-(VC - vV£)dA (6.3)

m

(6.4)

They will be evaluated in the following four sections.

29

Evaluation of Integral (6.1)

Substituting Equations (5.7), (5.9), (5.11), and (5.12) into

integral (6.1), we obtain

(W? - VW*Vi|i)dA

"m

[ (2AAT$T) (~- *A?)dA (2AAT$T>) •(— * ’A*)]dA

ra

AT [ I I 0TMA]A5 - A V ' n t A

ATnAC - aaW a ^

(6.5)

where A is the area of the mt 1 element.

Qij AkinklAlj

kl

I I dxdy I I xdxdy I S ydxdy ^iu m m

Symmetric

S I x2dxdy I I xydxdy Am

I S y2dxdy I

31

ij

/ a 2 _j_ a 2 21 31 A21A22 + A31A32

A 2 _i_ a 2 22 32

A21A23 + A31A33

A22A23 + A32A33

\Symmetric 23 + A33

Evaluation of Integral (6.2)

Integral (6.2) assumes different forms for finite and infinite

boundary slopes.

Finite Boundary Slope (m = finite)

Substituting Equations (5.7), (5.11) and (5.14) into

Equation (6.2), we obtain

I WVijMds Sm

( 2AA iJ) ) (— 'Ai//) • (1 m)dx

** AT [ I $Tdx](l m)$'A4;8 J ’

■ at s k 4>

(6.6)

Mij - \ i skKj

32

where

\S = (1 / 2 ) ( x ^ - x22)

\ Cm/2Xx32 - x22) + b(x3 - x2) J

m “ (y^ - y2)/(x3 “ x2)

b - (x3y2 - x2y3)/(x3 “ x2>

5 " KJ ■ a2: + -Si

Infinite Boundary Slope (m = <*>)

Substituting Equations (5.7), (5.11) and (5.15) into

Equation (6.2), we obtain

m

(2AAT$T) ( ~ * ' M > ) • (0 l)dy .. _ —

at S* K* ip

(6.7)

where

V = ' S a W

s*’ y3 " y2x2(y3 - y2)

\ (y32 - y22)/2

Kj*= A3j

Evaluation of Integral (6.3)

Integral (6.3) can be split into two integrals.

VW-(V£ - vVOdA

m

rVW'V^dA - v VW•V^dA (6.8)

m m

The second integral is easy to evaluate if compared with the

integral (6.1) and can be written as

V | 1 VW*V£dA = vAP^S.. (6.9)

TOwhere is as defined in Equation (6.5).

The first integral, after substitution of Equations (5.8), (5.9)

and (5 .12), becomes

34

where

J J AVW ♦ V £dA

m

r(2AATiJ>T) • (-- 64-) (— *A?)dA

m

■rr.— AT$T04-[jJ $dA]A£

2A “ 5

V i

m

J. .ij 2A « *

H =

A2iA3i4-i - A31A2i4-. \

A22A3i^i " A32A2i^i

' A23A3i^i “ A33A2i^i /

K - (I.A.. I,A. I.A.,)i ll i i2 i i3

I. = | // dxdy // xdxdy // ydxdy J*• * A A A <

m m m

(6.10)

Substituting Equations (6,9), (6.10) into Equation (6.8), we

35

obtain the integral (6.3) written as

/ /VW • (V£ - vVOdA

Am

(6.11)

Evaluation of Integral (6.A)

As for integral (6.2), Equation (6.4) assumes different forms for

finite and infinite boundary slopes.

Finite Boundary Slope (m = finite)

The first term is easy to evaluate because it is similar to

integral (6.2) and

Integral (6.4) can be split into two integrals.

S

W(vV£ - V5)*ds

(6.12)

m m

(6.13)

The second tern, after substitution of Equations (5.8), (5.9),

(5.11) and (5.14), becomes

36

wve; *dsJ .

m

(2AAT<f.T) 0*) 4>A?) • (1 m)dx

~ [(1 m)64»][AT( | $T $dx)A]5

2A 6Wij^

V i(6.14)

where

6

— 6W. .2A ij

V i

E. = A3i - n,A2i

WiJ ’ AliNlk\j

Ni3 '

Ax Ax 2/2 mAx2/2 + bAx \

Ax 3/3 iuAx 3/3 + bAx2/2\ Symmetric m2Ax3/3 + mbAx2 + b2Ax /

Ax = x3 - x2

Ax2 = x 2 - x 2 X2

Ax3 = x„3 - x„3

* " <y3 ’ y2)/Cx3 x2)

37

b " X3y2 ” X2y3)/(X3 " X2

Substituting Equations (6.13) and (6.14) into Equation (6.12),

we obtain integral (6.4) written as

J W(vV5 - $5) -ds

(6.15)

for a triangular element with finite boundary slope.

Infinite Boundary Slope (m - “)

As for finite boundary slope, we can easily obtain the first

integral in Equation (6.12),

v J WV^’ds = (6.16)

where M ^ * is as defined in Equation (6.7).

The second integral in Equation (6.12), after substitution of

Equations (5.8), (5.9), (5.11) and (5.15), becomes

f - ->WVS-dsJ s

(2A A V ) ( ^ 6 )(— * A O -(0 1) dy

^ [(1 m)6^][AT( I $T$dy)A]|

where

z. * ® — 6*Wij 2A

6* - E±*^i

E * = Ai 2i

w . . = A Nij li lk

N

1 Ay x2Ay

13 ■ x 22iy

, Symmetric

Ay2/2 '

x2Ay2/2

Ay3/3 /

Ay = y3 ■ y2

Substituting Equations (6.16) and (6.17) into Equation (6.12),

we obtain the integral (6.A) written as

r - -W(W£ - V O *dsJ ’

(6.18)

Summary39

The results derived in the last four sections are summarized

below for convenience.

(W£ - VW-V*)dA = Q±j5 - AP (6.5)

m

WVi|> *dsV j

for m = finite (6.6)

for m = » (6.7)m

VW-(V£ - vV£)dA

m

->W(vV£ - V£)*ds

(J., - vAP..)£. (6.11)ij ij ]

(vM^ - Z^) j for m = finite(6.15)

(vM^* - Z ^ * ) ^ for m » « (6.18)m

Substituting these equations into element equations (4.25), (4.26),

(4.27), and (4.28), we obtain the compatibility element equations.

(6.19)

for an interior element

(APij V * J ' V l

for a boundary element with

finite boundary slope

<APu M *)i(i ij J Q <K Vij^j

for a boundary element with

(6.20)

(6.21)

infinite boundary slope

and the vorticity transport element equations.

<JU - vAPiJ,£3 ‘ 0

for an interior element

‘<Jil - '■APy ) - (zi3 " ‘ 0for a boundary element with

finite boundary slope

[ ( J ± . - vAP±j) - (Z±j - - 0

for a boundary element with

infinite boundary slope

These element equations will be used in the next

(6.22)

(6.23)

(6.24)

chapter to

formulate two sets of simultaneous equations.

CHAPTER VII

ASSEMBLING OF FINITE ELEMENT EQUATIONS

The Concept of Assembling

The assembling procedure will be discussed in the following

sections and should be carefully followed to make correct assembling.

The compatibility element equation assumes different forms

depending on the property of the triangular element (interior or

boundary, finite or infinite boundary slope) . We will consider the

boundary element only because it is more general.

Let us look at the compatibility element equation for a

boundary element with finite boundary slope. Equation (6-20).*

where i = 1, 2, 3 and j =1, 2, 3.

Notice that this equation implies three equations corresponding

to three vertices of a triangular element. We assign

Assembling of the Compatibility Element Equations to Obtain

the Compatibility Set

(6.20)

to vertix 1

to vertix 2

*If the boundary slope is infinite, use Equation (6.21).

41

42

(AP^ - M )\Ji = Q3j Cj to vertix 3

Now, let us look at a certain interior node in the region of interest.

For this node there are a number of neighboring triangular elements.

For each of these neighboring elements, there is an element equation

assigned to this certain interior node as we just did. As an example,

refer to Figure 5. Node 1 is the chosen interior node with five

neighboring elements. There are five element equations assigned to

node 1. Each neighboring element assigns one element equation to node 1.

We will use the element equation for a boundary element, i.e. Equation

(6.20) or (6.21), to assign equations to an interior node. In other

words, the edges 23, 34, 45, 56 and 62 are considered as external

boundaries and the region enclosed by these edges is the only region we

are interested in when we are looking at the interior node 1. This con­

cept should be carefully followed as it will be used again. Since we got

five element equations for node 1, we merely add up these five equations to

obtain a complete equation for node 1 which would be in the following form:

Fig. 5.— An interior node with neighboring elements to show the assembling procedure.

A3

‘l*l + *2*2 + *3*3 + V * + *5*5 + *6*6

V l + V 2 + V 3 + V * + b5«5 + b6«6

where ^ and (1 = 1, 2, ...» 6) are respectively the vorticlty and

stream function at node i. and b^ are functions of the positions of

these nodes.

If we apply the same idea to each of the interior nodes in the

whole region of interest, we will obtain a set of simultaneous

equations. Each equation corresponds to an interior node. Thus, we can

have

TY = TC (7.1)

where

T and T are both n x 1 matrix and are functions of the

positions of all nodes in the flow region of interest,

n = the number of interior nodes,

1 = the number of all nodes,

Y * the column vector of the stream functions of all nodes,

£ ** the column vector of the vorticities of all nodes.

This equation is one of the two sets of simultaneous equations

we have been trying to find. It will be called the compatibility set.

Another set of equations will be derived in the next section.

44

Assembling of the Vorticity Transport Element Equations— the Vorticity

Transport Set

We again use the element equation for a boundary element.

Following the same procedures as in the last section and putting all

known and unknown quantities on the left side of the equations, we are

able to obtain the following set of simultaneous equations.

B£ - 0 (7.2)

where

B = a n x 1 matrix and is a function of the positions of all

nodes and the stream functions at all these nodes,

5 = a column vector of the vorticities at all nodes in the

region of interest,

n and 1 = are respectively the number of interior nodes and the

number of total nodes.

This is the second set of simultaneous equations we need and will be

called the vorticity transport set.

Summary

The assembling procedures result in the following two sets of

simultaneous equations:

TV *= T£ (7.1)

BS - 0 (7.2)

Along with suitable boundary conditions, these two sets of equations can

be used to solve a general two-dimensional steady flow problem with

constant fluid properties as will be shown in the next chapter.

CHAPTER VIII

SOLUTION METHOD

Outline

Two sets of equations were derived in the last chapter, i.e.

Equations (7.1) and (7.2). Along with given boundary conditions on

the stream function and vorticity, we can solve these two sets of

equations using as interation scheme. We can obtain the solutions

for the stream functions, vorticities, velocities at all interior nodes.

The procedures will be discussed in the following sections and two

examples are given to show the validity of the method developed in this

thesis.

Boundary Conditions

We need to know the stream functions and the vorticities along

the external boundary of the whole region of interest.

The stream functions at the boundary nodes are obtainable, if the

volume flow rates across the boundaries are determined.

The vorticity boundary conditions are not as easy to obtain as

the stream function boundary conditions, because velocity gradients are

involved. To specify them, we need to distinguish between the solid wall

boundary and the internal fluid boundary. The solid wall vorticities

can be deduced from other information. ^ The internal fluid

vorticities can be measured by hot-wire technique. For the purpose

45

45

of this thesis, we assume the vorticity boundary conditions are known

regardless of the procedures to find them.

Stream Function and Vorticity Solutions by an Iteration Scheme

Refer to Equations (7.1) and (7.2), i.e.

TV *» T £ (7.1)

8£ = 0 (7.2)

The components of the column vector £ in Equation (7.2) are not

all unknowns, because the vorticities on the boundaries are specified.

We can move those terms in each equation of the set (7.2) involving

the boundary stream functions to the other side of the equation. Then,

we can obtain

C = n x n matrix function of the stream functions at all nodes

and the position of each node,

£' * column vector of the unknown vorticities at all interior

nodes,

D = column vector which is a function of the stream functions

at all nodes and the vorticities at all external boundary

nodes.

Now, we can assume the stream functions at all interior nodes.

We also have boundary conditions on the stream functions and vorticities.

C£' = D (8.1)

where

47

So we can solve Equation (8.1) for £', the vorticities at interior

nodes. We then substitute these vorticities into Equation (7.1) to

make the right hand side of Equation (7.1) become constant, or

Ti(j = k (8.2)

where k is a constant column vector.

Note also that the components of <J> are not all unknowns because

the stream functions at the boundary nodes are all specified.

Rearranging the terms in Equation (8.2), we can obtain

eip’ = F (8.3)

where

e « n x n matrix function of the positions of all nodes,

V = column vector of the stream functions at interior nodes,

F = constant column vector.

Then, we can solve for if*’, the stream functions at interior nodes. These

values are compared with the initially assumed stream functions. If

their differences are within some pre-assigned limits. We will accept

the newly calculated solutions or the assumed solutions for the stream

functions at interior nodes. If their difference exceeds the pre­

assigned limit, we use the newly calculated stream functions to calculate

D in Equation (8.1) and solve for 5’• Then, use the new £' to find

F in Equation (8.3) and solve for ip' .

Then, compare this i|j’ with the previous one. If they are close

enough, we have our solution for the stream functions at all interior

nodes. If not, do the same iteration procedures again and again until

the stream functions at the interior nodes converge.

One significant result of the procedures described above is

that, once the stream functions are solved, the vorticities at

interior nodes are obtained. This is obvious because vorticities are

solved during each iteration. The vorticities solved just before the

stream functions converge within a pre-assigned limit are accepted as

our vorticity solutions for the interior nodes.

Velocity Solutions

Recall Equation (5.8), which is

0 - k eJwhere

A31 A32 A33

~A21 "A22 ~A23

or

u 2A A3i^i (8.4)

v -1_ ' 2A A2i*i (8.5)

As we mentioned in Chapter V, Linear Approximations - Stream

Function Approximation, the velocity components u and v are uniform

throughout the triangular element. In essense, it is some kind of

average velocity within a triangular element.

For a certain node, we take the average value of these constant

velocity components for its neighboring elements as the velocity

49

components at this node. The following example shows a general

relation between the velocity components at a certain node and the

stream functions at its neighboring nodes.

Example: With Equations (8.4) and(8.5) some mathematical manipulation shows the following:

2*' <x2- V V ' V ’V V ' V V ' 1'T

+ < v W

2At 1 <y2~y4 >*l+<y3"yl)*2+ < V X2> *

+ (V x3)(.4]

3

3

Where AT = A . + AX][ + + AIV

Fig. 6.— An interior node with neighboring elements to show the relation between velocity components and stream functions.

An alternative way to specify the velocity solutions is to

assume the constant velocity components of each element are the

velocities at the center of the triangular element. Both ways are

acceptable. However, the average velocities over the neighboring

elements will be used in the example problems, because no effort is

intended to find the center of the triangular elements.

The Use of Digital Computer

The complete solution for the problem considered in this thesis

requires numerical calculations. We need to formulate three finite

50

element equations tor each finite element, assemble finite element

equations for each node, and solve two sets of simultaneous equations

by iteration process. A computer program is developed and two

examplar problems with standard solutions are solved using this

computer program. The program solves for the stream functions and the

vorticities. The velocity components are calculated by hand for the

example problems.

Solved Problems

Two problems are solved by the computer program. They are

1. Fully developed flow through a straight channel with fixed

parallel boundaries.

2. Couette flow between two parallel flat walls, one of which

is at rest, the other moving in its own plane with a constant velocity.

They are discussed in the following two sections.

Fully Developed Flow Between Two Stationary Parallel Planes

The solution for this problem is easily obtained by solving the

governing equations with the boundary condition of zero wall velocities.(8)The resulting velocity profile'' , Figure 7, is parabolic and

Fig. 7.— Parallel flow with parabolic velocity distribution.

The stream function is

51

*f '/ u dy 1_ d£

2y dxo

The total volume flow rate through the charnel is

1 dp h3 2y dx 6

Then, the velocity and the stream function at each point in terms

of the total flow rate are

uh 3

(hy - y2) (8.6)

# - ~ (3hy2 - 2y3) (8.7)h 3

The vorticity at each point is

5 = - — (2y - h) (8.8)dy h3

The specific problem considers a 3 ft. long flow channel with

h = 2 ft. and the total volume flow rate ¥ = 100 ft3/hr. The problem

region is divided up into a mesh of triangular finite elements

illustrated in Figure 8.

The exact solutions for the velocity, the stream function and

the vorticity are calculated from Equations (8.6), (8.7) and (8.8).

These exact solutions are compared with the solutions obtained by using

the finite element method in Table 1.

52

9 8 7

Fig. 8.— A typical finite element arrangement.

TABLE 1

COMPARISON OF EXACT AND FINITE ELEMENT SOLUTIONS

NodeVelocity Stream Function Vorticity

Exact Approximate Exact Approximate Exact Approximate

1 56.25 50.16 84.375 83.455 75 73.02

2 56.25 50.16 84.375 83.485 75 73.54

3 75.00 60. 50.00 49.693 0 -.5

4 56.25 50. 15.625 16.218 -75 -75.74

5 56.25 50. 15.625 16.188 -75 -76.26

Couette Flow

53

The general case of Couette flow is a superposition of the

simple Couette flow with a vanishing pressure gradient over the flow(O)

between two stationary flat plates. 1 The velocity solution is

" ■ hD - h t (hy ‘y2)which is shown in Figure 9. Note that U is the constant moving

velocity of the upper wall, and the dimensionless pressure gradient,

u---- ^

Fig. 9.— Couette flow between two parallel flat walls.

^2P = (“ j ) . determines the existence of back-flow. As in the last2pU dxsection, we can write the velocity, the stream function, and the vorticity

in terms of the total mass flow rate, i.e.

u ) y + (—h2 h h2

6V)y2 (8.9)

(8.10). , 3 V U. 2 . ,U 2¥. 3t “ (----- ) y + (------ ) yh3 h h2 h3

„ ,1 2 V 6U. . ,2U 6V.K - (------- )y + (------ )

h3 h2 h h2

54

(8.11)

The specific problem considers a Couette flow channel with

h B 2 ft and the total volume flow rate of 100 ft3/hr. The problem

region is divided up into a mesh of triangular finite elements as

illustrated in Figure 8. The exact solutions are calculated from

Equations (8.9), (8.10) and (8.11) and are compared to the finite

element solutions in Table 2.*

TABLE 2

COMPARISON OF EXACT Aid) FINITE ELEMENT SOLUTIONS FOR COUETTE FLOW

NodeVelocity Stream Function Vorticity

Exact Approximate Exact Approximate Exact Approximate

1 58.125 52.6 81.5625 80.82 62.5 60.772 58.125 52.6 81.5625 80.84 62.5 61.243 72.5 58. 47.5 47.32 -5. -5.4 53.125 47.41 14.6875 15.31 -72.5 -73.5 53.125 47.41 14.6875 15.28 -72.5 -73.5

Conclusion

From the above examples, it is seen that the agreement between

*For Couette flow we have convergence problems. Unless the first guesses are very close to the exact solutions, the iteration solutions keep diverging. The approximate solutions in Figure 10 are the result of the first iteration, when the first guesses are the exact solutions.

numerical and theoretical solutions is very good. The velocity

55

distributions of many physical problems cannot be obtained

analytically. The finite element method can be applied to find the

approximate solutions.

LIST OF REFERENCES

1. Becker, E. B. and Parr, C. H., "Application of the Finite ElementMethod to Heat Conduction in Solids," Rohm and Hass Company Technical Report, S-117, November, 1967.

2. Finlayson, B. A. and Scriven, L. E., "The Method of WeightedResiduals and its Relation to Certain Variational Principles for the Analysis of Transport Processes," Chemical Engineering Science, Vol. 20, 1965, pp. 395-404.

3. Gosman, A. D. and others, Heat and Mass Transfer in RecirculatingFlows, Academic Press, London and New York, 1969.

4. Heaton, H. S., "Improvements in Two-Dimensional Transient HeatConduction Computer Programs," Hercules, Inc. Technical Report, August, 1970.

5. Kantorovich, L. V. and Krylov, V. I., Approximate Methods inHigher Analysis, Interscience, New York, 1958.

6. Myer, G. E., Analytical Methods in Conduction Heat Transfer, McGraw-Hill, New York, 1971.

7. Roache, P. J. and Mueller, T. J., "Numerical Solutions ofCompressible and Incompressible Laminar Separated Flows,"AIAA Fluid and Plasma Dynamics Conference Paper, No. 68-741, June, 1968.

8. Schlichting, H., Boundary-Layer Theory, McGraw-Hill, New York, 1968.

9. Tay, A. G., and G. de Vahl Davis. "Application of the Finite ElementMethod to Convective Heat Transfer Between Parallel Planes," International Journal of Heat and Mass Transfer, Vol. 14, No. 8, August, 1971.

10. Tong, P., and Y. C. Fung. "Slow Particulate Viscous Flow in Channelsand Tubes— Application and Biomechanics." Journal of Applied Mechanics. December, 1971.

11. Vrie3, G., and D. H. Norrie. "The Application of the Finite-ElementTechnique to Potential Flow Problems." Transaction of the ASME. December, 1971.

12. Wilson, E. L., and R. E. Nickell, "Application of the Finite ElementMethod to Heat-Conduction Analysis." Nuclear Engineering and Design, 4. 1966.

56

57

13. Zienkiewicz, 0. C., and Y. K. Cheung. The Finite Element Method— Structural and Continuum Mechanics. McGraw-Hill, Maidenhead, England, 1967.

a ppen d ix

ThS.147 A,T=2.,RAY CUJ eortran DECK «MWR 06/03/72 RACE iC CONGRATULATIONS CONORaTUI AT IONS CONOR aTUl AT IONS CDNORaTLi »T IONSc CONGRATULATION'S CONOR* TulaT ! ONS CONORA Tut A T I (INS CONORA U I * T t ONSC “ NUMRtR Of v*OD£SC NE MJMRER CE El CENTSC w'Jhpeo iHTrojOft NODES FIRSTC CO TuRU EAC« element AND READ NODES COUNTER CLOCKWISEC E0[> INTERIOR ELEMENT STaR f AT ANT OE THREE VERTICESC EOR ROUNDARY ELEMENT START AT THE VERTlX NOT ON BOLNCSrYC NODE Mi NOTE NUMBER0 XM > Y (I) COORDINATE S OE EACH NODEc LAC-n eor Interior elementC LAG = 'l, 2. 3, ETC. FOR BOUNDARY ELEMENTC NRD NUMBER or BOUNDARY MOOES c c

• DIMENSION SA!<5S>.SA!M(S5).v0R(S5),XX(5R).YY(S5>.PD(55T,AnD(5S>.1 1SN0(5S.1C.3),SNE(S5),RAY(SS.55).CATMY<5C.55>.GRACE<55.RS).1 2N0DE(3),x (3).Y(T).S(3),RK(3).RH(3>,1 3a1(13.13),TM(13,13).BB(13.13>.AA<3,3>.R!<3.3>.1 4PLM(3.3)«P<3,3).0<3,3). R.J( 3.3 >. RN (3,3). N< 3,3 >

CC

2 READ<5.9999) NOsET3 9 9 9 9 F O R M A T * 1 1 0 )4 WR!TE(6.e5«R) NOSET9 8883 rOR"AT<lHC, 12(5)

6

ccc

REaO(S,P9) h,NBD7 09 rOR'<AT(2112)e I T I " A. = 19 MINsw-NOn

1 -r MM»NIN*111 REA'.'fE.AV) (xX<T),YYCI),I=l,M>12 <9 FORMAT (■ 2E10.2 I

C READ POUNCaPy STREAM FUNCTIONS1*5 READ!?. 19) (SA!(p.I«N!Ni,M>1 4 19 FCRHAT(cJFlC.T)15 RE AP( 9,2TV ) (VOR(I),I=NIN1,M>16 20? fCrmat(bfu- .3)

C ASSUMING ST' rAM FUNCTIONS for17 REA0(9,2) (SaI<I).I=1*N!n )15 2 FCR''AT(0E1;i.3>: v wH]Te < 6.167 > (SA1U)>IM.K)2: 167 FCR"A*<lH. ,«rlC,3)21 RF AO(N.09) VIS22 99 FORMAT(Elf,3)23 11=194 281 RFAO(S.l) Nf:25 1 FORMAT(Ilf)26 S'.E ( I I ) =nE27 213C DO 3 1=1,M26 DO 3 .1 = 1, M29 Al(I,J)=8.3 7 t u(i,j> =e.•r 4’ X 3 CONTINUE

LnvO

/

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C36 DC t:< 1*1,337 snqi11,wax,I)=NGpt(I)36 1S« CGivTlVir39 20? NE = GNF(I I)4;’ 15* M 71 in.341 J«S'iU( I I .MAX, I )42 X( I >=yx(J>43 V( I > = Y Y tJ)44 71 cont i mjf4 > Nl= M'l 11 ,MiX,l»46 N? = RnI)( I I, “Ax.2)47 N.(rSNn< I 1 « M A X . 3 >46 CALC AMaT(X.Y.AA)49 CALC IMNT(X.Y.RI)

Cc *cr stRca" fu,jcti°n equation

5.?N DO 30 1=1,3SI DC 30 J=l,352 RIM(I,J)e.p

53 38 0(I,J)=L.54 DO 5 1=1,355 DO 5 .1 = 1,3’56 DO 5 K=1,35 7 DO 5 L = 1,356 OC. J)=C(I,J)»AA(K,I).RI<K,L)»AA(L,J>59 5 CONTIIiuF.6 2 CALL ASRCY(Al,Nl,N2,N3,0)

C61 DO A 1=1,262 DC 6 J=I« 363 P(I.J) = (AA(?,I).AA(2,.I)*AA(3,I)»AA(3,44 6 CONTINUE65 P(2,1)=F(1,2)46 P(3,1)=P(1,3>67 P(3,?)=P(2,3)cc68 IF(X(2)-X<3)> 35,A,3569

c75 S(1>=X(2!-X(3>73 WP=X(2)»»2-X(3).»271 S(2) = ,.w/2.72 RM=(Y(2)-Y(3j>/S(l>73 B=(Y(3)*X(2)-Y(3).X(3))/SI1>74 S(3)=»M*S(2)*P*Stl)

Cc

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C1.72 136 5 A X = 1133 DO 111 1=1,u174 DO 111 J=l,«105 111 Bin i, j) = .o

Ccc

176 28 DO 14 1=1,3177 J = SN’D( 11 .BAX, I)178 x (1> = x x < j >179 Y ( I ) = Y Y C J )11C 14 CONTINUE111 N1=6NP( I I , ” A X > 1)112 N? = 9Nr(H,MAY.2)113 NJ = 8ND(I I , y AX■3)ll4 CALL AYATCX.Y/AA)115 CALI P I NT ( X , Y, RI >

CC VOB’ICITy EOtjAT I ONC

116 S(l) = 6Al(M)117 S(2)=SAI(N2)118 S C 3)=?AI(M3)11712C PH2=,C121 DC 16 1=1,3122 RHi=RHl*AA(3,1)*SC1)123 Ph2=rw2»aA(2,I).S(1)

FORTRAN DECK 'KWR 06/0 3 / 7 2 PA«E 3

oxhJ

FORTRAN DECK *MWRTkS.147, CHI124125126 127 126 179ir131132 1 = 3134135136137 136

139143141142143144145146147 143 1 4 9 l5'» l i152153154155156157 156 159163 161 162 143164 146 166 167 163 149 173171172173174

15 CONTINUEDO 14 1*1,3RR< I >*AA<2.l>*RHl = AA<3,l>»RH2

16 CONTINUEL’0 29 1 = 1.3 RK(I)=.0

29 CONTINUEDO 17 1=1.3 DC 17 J=l,3RK{!>sfi.K<I).Rl(i,J)»AA(J. I)

17 cONTI\UEDO 23 1*1,3 DO 23 J=1.3 RJ( T,J)=RH<I >»R*(J)

23 cOn t INlTCCC IF(X(2)-X(3) ) 4.3,41,4cC43 XC*T(2)-X(3)

R"*(Y(2)-Y(3))/XC R=(Y(3)*X(2)-Y(2)4X(3))/XC DO 13 1*1.3

18 RKC I)*RM*AA(2.I)-AA(3. I) o'. (1.1 > = X (3) -X t 7 )RN(l.?>=.5.(x<3)..2-X(2>»o2>95(1.3) = R N * R f. ( 1,2 ) *fi*RN( 1,1) R6(2.?)=(X(!)**3-X(2)*«3)/3. R'«(?,3)=RY»R'i(2.2)»P»“N<i,2)RM3,3) = Rt'«‘2«R9(2,?)*2.»PM»B*R9<i,2)*B**2*l,N(i,i)GO to 45

C41 DO 63 1*1.3 60 R4(!)=AA(2,|>

RN(1, 1 ) =Y(3).Y(?)°N(1,2 ) =X(2)*RN(1,1)RN(l,3)=.5*(y(3)«*2-Y(2)**2)RN(?.2)=X(2>.RN(1,2>R!;(2,3) = x <2).RM(1,3)RN(3,3) = (Y(3)...X-Y(2)*«3)/3.

C<5 cCYTP'uE

RN(2,1 )=R>'(1,2)»N(3.i>sRH(l,3>PN(3,2)=RN(?,3>DO 41 1*1,3 DC 61 J = 1.3

41 wU.J )=.•;. 7 CO 51 1=1,3 DO 51 J = 1»3 DO 51 K =1.3 DO 51 L = 1,3 .

51 U(I,J)* W{I/J)*AA(K»I)*RN(K#L)*Aa(L,J)C

CC= .3DO 22 1*1,3 CC=CC*RK(I)«S(1)

06/03/72 RAGE 4

OXTO

FORTRAN d e c k »m w r147. ,A.T=20,PAY CH! FORTRAN175 22

rcn\'t imjf.

176U

D0 23 1=1,3177 DO 23 J=1,3175 w<I,J)rCC*M< J,J)179 23 CONTINUE1*0 DO 24 1=1,31*1 DO 24 J=1.31*2 RJ<I,J) = .5.(rJ<I,J>*WU,J))/RI<1,1)1*3 24 CONTINUE184 C

122r

CAUL AS(:LY(«R,N1,N2,N3,Rj )1*5 HAX=YAX»11*6 !F(>-4X-Nf.> 24,25 » 27:«7 27 DO 26 1=1,M1 *8 DO 26 J=1,M189 B8(I,J)=bP<I,J)-VIS«TU(I,J)10? 26 C 0 :-i T I • mj r101 DO 13 7 I=1, Mlo2 GBACt<\l,t)»flH<Nl«I>193 137 CONTINUE104 I 1 = 11*1195 IE(I I -NIN> 110,100,135106 120 IE(!T!mF-1> 13,281,147107 147 M A X =1i06 GO TO 2132109

C135 DO 4Ji 1=1,NlN

27 7 431 DP(I>=.U2 .11 DO 128 I=1,NJN2 72 DO 128 Js'JINj ,M2 *5 125

CD'J( I > = dh< D - gRace 11, J)*v o p<J)

2/*CC0

CALL r.LIM(G5A0E,NIN,DD,V0R)

275 DO 33 !=1,N!N2 76 a d d< !> = ,;?277 33 DO(I)=.C2 75 DO 34 1=1,NlN219 DC 34 J=1,M21S DP<I>=DP<!!*RAY(I,J)»VOR<J>211 34 CONTPmt2l2 DO 37 1=1, M n2l3 DO 37 J = u m , M214 ADD!I)=ADD(I)*CATMY(I.J)*S A1(J>215 37 continue216 DO 4j 1=1,n In21 7 nn< 1 !=DD(1)’aOD(1)21 b 43 CONTINUE219 call EL1m <CATHY,NIN,D«.SA!N)22.1

Cdo 42 1=1,NlN

221 !F<aps<Sa IN(I>-8A!(!>>-.1) 50,So,30222 <2 CONTINUE223 3C IE <ITIME-23 1 31,31,777

06/0*5/72 HAGE «s

U)

147, ,A.T«2K#RAY CHI FORTRAN DECK »MWR •22 4 31 ITI “E=ITIHE*1225 DO 4p 1=1.NIN226 A 0 SAI<I> =SAlM<I>227 RPITE<6.44> <SA!<I),I=1,M>22tt A A FORRaT(1HL.AE14.7)229 11=12-0 R A X231 GO TO 2130

cc PRINT vORTICITY AND STRFam FUNCTION232

c777 W°ITF(6.212>

?33 21? FORMAT(1h2»'NOT CONVERGENT')234 cn Tp i3235 52 L'P I T E ( 5, 191)236 191 fcr-at(1r:,'this is the nUrher op iterations’)237 WRITE(6>4766) IT1HE23 5 47A6 forhaT(ih;,Iic)2-39 hRITr(6,9574)24'' 9P76 FORMAT(1 HO.* THIR IS The SOLUTION FOR VORTICITY* )241 wRITF C 6.52) (VOP(I), I = 1.M>242 52 FCRRaT(1H.',5f14.7)243 WRITf(6,117)244 117 FORMAT(1H 0 . 'THIS IS THE SOLUTION FOR stream function245 HR ITE < 6.54) (Sa IN(I)#I=1.MIN)246 54 FORMAT(1HE.5E14.7)247 13 STOP24ti END

NO “ESSAGrS FOR ARCVr COMPILATION,EXTERNAL REFERENCES:

FTIO.RGO FTIO.XMT FT!0.HLT FTIO.KCO aRaTaSBlY EXP2, ELIM

0 6 / 0 3 / 7 2

R I NT

RAGE a

PFTC.ARAT e2f

ON

TRS.H7 A.Wl'.RAY CHI FORTRAN1 SUbPniJT ! NE AYATfX.Y.AA)2 Dt«'AF!ON x <3>. ' ' ( 3>»A«<3, 3)3 AA( 1 , 1 ) = XC2 ) » Y( 3 ) - X! 3 ) * V( 2 )A a«(1.2)=X(3)*Y(1>-X(1)*Y(3)5 a a(I,3)=X(1)oY(?)-X(2>»Y(1>6 AA«2#1) =Y(2>. Y( J>7 AA( 2 . 2 ) = Y< 3 ) - Y( i )8 AA(?,3)=Y(1)-Y(?)9 AA( 7 , 1 ) = X( 3 ) - X( 7 )17 AA(3,2)=X(1>-X<3>11 AA(3,3)=X(2)-X(1)12 RETURN13 END

NO "ESS4G=-S FOR aROVE C0HP IL aTI ON *

#F7C.P!NT

86/23/72 RAr.E T

226

ONUl

ThS, 47, ,A,T=20,»AY CWI FORTRAN DECK fRINT * 06/03/721 S't&SOt'TINE RlMIX.Y.R! >2 DIMENSION X(3),v (3),R[<3,3>,SX(3).SY<3)3 DO 11 1=1,34 DO 11 J=1,35 11 R I ( I . J) =.» . 36 !F(X(?)-X(3)> 3,4,37 4 I D = 18 1 .P01=x(3>-X(l)9 BD2 = X(3)**2-.xQ)**212 R':3 = X (3 > **3-x< 1> **311 pD4 :: X ( 3 > • *4-X < 1 > **412 «M2=(Y(3)-Y(l)>/(X(3)'X(l)>13 PM3 = <v <1)-Y(?))/'<X(i )-X(2))14 Sni = R>-2-RY315 S!'D = R'!?*»2-R"3»«?16 Sr>3 = R'2**3-P“3**317 I F ( ID-4) 63,66,8818 46 I n»?IV .SDi = -S0i2’’ SD2=-0D221 SD3 = -St)322 * 8 RI(l,l>=.S«ani»»2»SDi*RI(i,i0?3 RI (!,’) = (Dd3/3.-.S»x (1) »Rn2) *901 *PU1.2>24 R! (1,3) = Sl2*p.D3/«,.*.5.( Y( 1 )«SDl-X( i) *SD2) »aD2*Xt !)•( ,5.y<i)«SD8-2* 1Y(1'»SD1),R’'1*8T(1,3)25 °I t 2,2) = ( . ?S.RIM-Xt 1) *RD3/'3. )*S01*91(2,2)26 RI(2,3>=SD2*F.D4/(i,*(Y<1).SD1-Xtl)#Sn2).RD3/3,«.5.X<1>*< ,5*X(1>*S26 1D2-Y11 )-sr1)«RD2*BI(2,3)27 R1(1,3)=SD3.p94/l2.*(Y(l).SD2-X(l).SD31*803/3...5»(X(l).«2.SD3-227 1 . *X( 1 )*Y( l).<;D2.Y(l)**2*SDi)*i)D?-X(l )*(X(l)»*2*SD3/3.-V(l)»Y(i)»27 1S02.Y(1>**2*9Dl>.RDl.PI(3,3>28 I F ( ID.tO.l) GO TO 1329 I f < Iq .EO.2) f,0 TO 93-: IM ID.EO.3) GO TO 7731 77 X 11)=SX(2> .32 X(2)=5X(1)33 Y(1)=9Y(2>34 v < 2 > =SY(1>35 ir = 436 GO TO l37 I F (ID.EO.3) GO To 638 9 DO 2.' 1=1,339 x(i> =sx <i>4 3 20 Y(1)=SY(I )4 1 i; PI(2,1)=RI(1,2)42 RI(3,l)=RI(1.3)43 R!(3.?>=R!(2,3>44 RETURN45 3 in = 246 DO 30 1=1,34 7 sx(i>=x(i)43 3£ SY< I)=Y( I )49 I F (<(1) -X ( 2 1 > 1 4 . 7 , 1 45.: 7 X(1> *SX < 3 >■SI X ( 2 )=0 x (1)52 X ( 3 ) = 9 X ( 2)53 Y ( 1 > = G Y ( 3 )54 Y ( 2 ) =5Y(1>

kage: a

ONON

TRS.147 A.T»20,RAY CM I rOPTBAW5556S 7 StJ 59 A "J 61 62 *3 4«4546 67 4 a

y «3)»:,y<z )GC TD 1 '14 Ir{y < t) - x ( 3)) 16,4,156 x<l>«rx<2>X < 2 > = SX(3 >X<3>*5X(1)Y(1>=SY(?)Y(2>r5Y<3>Y(3)*SV(1)ID = ?GO TO 1

15 10 = 3GO TO 1 END

"ESSAGrS rOR Ali°YE C0MPIt-ATInN'EXTERNAL REEERE'ICES:

EXP?.

«F TC, ASRLY

176/03/72 rag: «

e2«

O'>vj

TWS.147 4. T»21.pay Cm 1 rtJPTPAN de ck 'ASBLY

123456 7 6 31 2 11 :2 13

S'JBPO'JTlNE AGf)Lv < A »f<l .M2.N3 DIMENSION a(i3.13).0(3.3) 40.1 ,M) = 4(*'1.N’ 1*0(1.114(‘(1,i,?)eA('Ji ,N?)*()(1,2)A(01.131=A(Ml»M3 1*0(1.3)A (N’,M>=A(M?.M >*0(2.1) 4(n2,'!2)=4(M?.N?)*0(2.2)' A(.M?,'.3)aA(M?,N3)*0(2.3) 4(N5,‘ll)sA(*,3.M ) *Q ( 3,1) A(07,’1?)«4(Mt ,M?)»0(3,2)4 ( N3 , N3 1 = A ( A'3. N3 1 +0 ( 3,3 1 P r T' IBEND

,0)

"tSSAGcS r'OR ABOVE ItaTION,

AFTC.ELI*

26/23/72 page i

221

<7400

TRS.1A7 A.T«2S.RAV C h ! f o r t r a n d e c k 'ELIM .1 SUf-.»0'*TlMF Ff TM(AA#N*RD.7)2 0 t H F f * 3 : C N A*(55.55),B0<55>,A(55,56),2(55)# Tn<55>3 fc s N ♦ 14 d o id: i = i •3 A (I < N*‘-) “BD (!)6 P(I)-I7 DC 1CV J*1.NB ISC A(:,J ) = A A ( I . J )9 **1

1 C 4i CALL r.XCM(A.N.NM,K,!0)1 1 2 I F (A (K .K )) 3 , 9 9 9 , 31 2 3 k k = k * i1 3 D C A J = K K ,N N14 A(K,J ) i A ( K , J ) / a < K . K )1 5 D O A U l , l .1 6 IFtK-n A1,4,A11 7 A1 A ( 1 , J ) = A ( ! , J ) - A ( I » K ) » A ( K , J )IB A C O N T I M j E1 9 K s K K2 C IF 1,2,52 1 5 D O 1 C 1 = 1 , N? 2 D O I D J = 1 ,N2 3 I.F(ID(J)-!)lB,fi,132 < 6 ZM) = A ( J , f-N)2 5 IS coNTp'ur.2 6 RETURN2 7 790 L R I T E I 6,10 D 2 )?e isos FORRA T{19M No UN I CUE S O L U T I O N )?9 RETURN3 C END

N O * £ S S A C rs FOR A B O V E C O H P I L a T I G N .

EXTERNAL refeRe*• C E S :t X C M F T I 0 , W & O F T I O . H L T

«FTC,f XCM

09/03/72 RAGE It

ez<

a \N O

TN$ f147 A.T*2C.RAY CKI fOUTRAN DECK • f XCH

1 St'B°0')T!NE ExCH(A.N,NN,K, ID)2 DIMCNOION A(d5,*;6)»ID(55)3 N R 0 ■*' i K4 NCO'-sm5 B>Af>S(A<K.K))6 DO 2 I=K,N7 DO 2 J=<,N8 ir<Af,SUlt.J)>-f»> 2,2,219 21 NRO'-i* !

in NCOL*J11 M =* A ■' ( S ( A ( J , J » >:2 ? CONTI '..'F13 irc:p'’*-K) 3,3,311« 31 DC- 32 J=K,NN:5 C*4 C r,;io«f» J )16 A(NRO,.,J)=A(K,J)17 22 A(K.J ) =C18 3 CONTJDUE19 I M ’JCOL-K) A,4,41

<1 DO *2 I=1.N?i C«A ( | , dC(h.)22 a (I.NCoL)1A{j,K)23 <2 A ( I,K)«C?4 ! * I P (' COL )25 I 0(DcOL > rID(K)26 I c <>•;> = I27 4 COuTIMJE28 PETIIPN29 END

no N£ssACrs ros arOvt cOmPiIation,PTC VERSION 5 MOD e

016/ 03/ 72 MAGE It

o

APPLICATION OF THE FINITE ELEMENT METHOD

USING THE METHOD OF WEIGHTED RESIDUALS

TO TWO DIMENSIONAL NEWTONIAN STEADY

FLOW WITH CONSTANT FLUID PROPERTIES

Mien Ray Chi

Department of Mechanical Engineering

M.S. Degree, August 1972

ABSTRACT

The finite element method using the method of weighted residuals is applied to solve for the velocity distribution of two- dimensional steady flow with constant fluid properties. The stream function and the vorticity are solved in the intermediate steps.Two examples are included to verify the validity of the method and aspects of the method discussed.

COMMITTEE APPROVAL: