a finite-element analysis of steady, two-dimensional

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1972

A Finite-Element Analysis of Steady, Two-Dimensional, Incompressible, Laminar Flow.Gary Richard WooleyLouisiana State University and Agricultural & Mechanical College

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I 7 3 -1 3 ,6 9 4

f WOOLEY, Gary Richard, 1946- I A FINITE ELEMENT ANALYSIS OF STEADY, TWO| DIMENSIONAL, INCXM>RESSIBLE, LAMINAR FLOW.

The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1972 Engineering, mechanical

University Microfilms, A XEROX Company , A n n Arbor, Michigan


A FINITE ELEMENT ANALYSIS OF STEADY,

TWO DIMENSIONAL, INCOMPRESSIBLE, LAMINAR FLOW

A Dissertation

Submitted to the Graduate Faculty of the Louisiana State University and

Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

inThe Department of Engineering Science

byGary Richard Wooley

B.S., Louisiana State University, 1969 M.S., Louisiana State University, 1970

December, 1972


PLEASE NOTE:

Some pages may have

i n d is t in c t p r in t .

F ilm ed as re c e iv e d .

U n iv e r s ity M ic r o f i lm s , A Xerox Education Company


ACKNOWLEDGEMENT

I wish to express my appreciation to each member of the Engineering Science faculty for their personal and

academic assistance during my years of study. Thanks are

due Dr. D. W. Yannitell for suggesting the problem presented here and for his guidance and assistance in its solution. The assistance of Dr. T. A. Manning in preparing the computer programs is appreciated. Additionally,

Dr. R. L. Thoms deserves my gratitude for his help with

the finite element method. A special acknowledgement is due Dr. D. R. Carver for his academic help, his relentless

assistance in obtaining sufficient financial income for my family, and for affording me the opportunity to teach.

ii


TA B LE OF CONTENTS

Page

ACKNOWLEDGEMENT ............................................ ii

LIST OF F I G U R E S ............................................ vA B S T R A C T ................................................... viChapter

I. DEFINITION OF THE P R O B L E M ...................... 1Governing Equations .......................... 1Non-Dimensional Equations ................... 3Statement of Problem ........................ 4

II. HISTORY OF THE P R O B L E M ......................... 6Analytical Investigations ................... 6Experimental Investigations ................. 9Numerical Investigations ................... 10

III. THE FINITE ELEMENT M E T H O D ...................... 14H i s t o r y ......................................... 14A p p l i c a t i o n .................................... 17Convergence Criteria ........................ 20Interpolating Functions ...................... 21Curved Isoparametric Quadrilaterals . . . . 27Elimination of Internal Nodes .............. 31

IV. METHODS OF S O L U T I O N ............................. 35Method 1 ....................................... 35Method 2 ....................................... 37

V. FINITE ELEMENT FORMULATION .................... 40Method 1 ....................................... 40Method 2 ....................................... 46

VI. DESCRIPTION OF COMPUTER PROGRAMS ............. 53Method 1 ....................................... 53Method 2 ....................................... 62Test P r o g r a m .................................. 69C o m m e n t ......................................... 72

iii


VII. NUMERICAL EXPERIMENTATION AND RESULTS . . . . 73Method 1 ....................................... 75Method 2 ....................................... 80

VIII. CONCLUSIONS AND RECOMMENDATIONS ............... 85Method 1 ....................................... 86Method 2 ....................................... 87

BIBLIOGRAPHY .............................................. 131

APPENDICESA. Interpolating Functions ......................... 137B. Components of the Element Coefficient

Matrix [H]e and Constant Vector {F}e . . . . 138

C. Finite Element Formulation with LinearT r i a n g l e s ......................................... 140

D. Computer ProgramsMethod 1 ..........................................148Method 2 ......................................... 174T e s t .............................................. 209

V I T A .......................................................... 234

iv


L IS T OF F IG U R E S

Figure Page1. Physical Problem, Circular Cylinder .......... 892. Linear Rectangle .................................. 903. Quadratic Rectangle .............................. 91

4. Quadratic Isoparametric Quadrilateral . . . . 92

5. Orientation of Triangular Element ............ 936. Quadrilateral with Twelve Triangles .......... 947. Quadrilateral with Four Triangles;

Quadrilateral with Sixteen Triangles . . . . 95

8. Nine Node Q u a d r i l a t e r a l ......................... 969. Experimental Elements: Thirteen Node Quadri

lateral; Twenty-one Node Quadrilateral . . . 97

10. Circular Cylinder, Surface PressureDistribution .................................... 98

11. Physical Problem, Lens Shaped Cylinder . . . . 9912. Lens Shaped Cylinder, Surface Pressure

Distribution .................................... 10013. Method 1 Results for a Circular Cylinder . . . 10214. Method 2 Results for a Circular Cylinder . . . 11315. Test Program R e s u l t s ................................ 119

16. Method 1 Results for a Lens ShapedC y l i n d e r ............................................ 121

v


ABSTRACT

A solution of the Navier-Stokes and continuity equations for steady, two dimensional, incompressible, laminar

flow employing an iteration technique is presented. The technique begins with an initial guess for the velocity and pressure fields, which is used to compute approximations to

the non-linear terms in the equations of motion. Thus, treating the non-linear terms as known functions, the finite element method is employed to solve the resulting linear equations for new approximations to the velocities and pressure. Using the solution as a new guess, the non-linear terms are approximated again and the procedure is repeated

until two successive solutions are "close."

Presented in the text are two formulations of the governing equations. One consists of the classical form of

two equations of motion and the continuity equation, while the other replaces the continuity equation with a combination of the first derivatives of the equations of motion and the

continuity equation.The agreement with published results for flow past a

circular cylinder is excellent, although the results of the present work are restricted to small Reynolds numbers. Success is achieved with the first formulation, but not the

second.

vi


CHAPTER I

DEFINITION OF THE PROBLEM

A problem of obvious practical importance is the d e scription of the steady motion of a viscous, incompressible

fluid. The present paper analyzes the external flow of such a fluid past a cylindrical body normal to the free stream, resulting in a two dimensional problem. Some simplifying assumptions are made about the body, the governing equations and the boundary conditions are devel

oped, and a method of solution is presented.

Governing Equations

An Eulerian or global formulation of Newton's second

law yields the vector equation of motion of a continuum.If the continuum is assumed to be homogeneous, isotropic,

and if it obeys Stokes' relation that the shear stress is proportional to the velocity gradient, the resulting equation of motion is the Navier-Stokes equation which, for two dimensional, incompressible flow, has the following com

ponents :

uux + vuy * ~(5[ - Px ) + vV2u ,

uv + vv = 1(7 - Pv ) + W 2v ,a y p — /

1


2

where,

p is the fluid density,

v is the kinematic viscosity,

X is the body force per unit volume in the x-direction,

Y is the body force per unit volume in the y-direction,

u is the velocity component in the x-direction, v is the velocity component in the y-direction,

P is the pressure, and subscripts denote partial derivatives. If the pressure is considered as the sum of the hydrostatic pressure, PH , and another term which is denoted the dynamic pressure, P^,

and it is assumed that the only body force acting is

gravity, then with the observation that

3P„ _ 3P„y * « f i y — n- " T F ' - 9y

it becomes clear that,

h + pd>'-^t >

I- ■ I - ^ H + P J - - W ■

Thus, renaming the pressure P as the dynamic pressure, i.e. the difference between the total pressure and the hydrostatic pressure, the Navier-Stokes equations of motion

become,

uu + vu = - I p + vV2u , a y p a


The principle of conservation of mass applied to a

continuum requires that the continuity equation be satis- f ied,

It + 35c(pu) + ^ Cpv)

For steady, incompressible flow the continuity equation simplifies to,

ux + vy ■ 0

Non-Dimensional Equations

To obtain a more useful form of the Navier-Stokes

and continuity equations, the variables may be non-

dimensionalized using a characteristic length, d, and velocity U, which define the new coordinates,

*’ - f . r-l •

velocities,

u , vu = U ’ v = U

and pressure,

p ’ = PU2


4

Substituting these variables into the governing equations,

dropping the primes, and rearranging yields,

+ ra 72“uvx ♦ vvy = - Px + V 2v

ux + vy - 0

Obviously, the only important parameter in this problem is

the well known Reynolds Number,

R = M RE v

Statement of Problem

The problem to be studied is the two-dimensional

steady flow of an incompressible fluid, past a cylinder of arbitrary cross section normal to the flow. To simplify the analysis the body is assumed symmetric about an axis

parallel to the direction of the free stream. The test

problem is the flow past a circular cylinder.Mathematically, the problem is one of solving a sys

tem of three simultaneous, non-linear, elliptic, partial differential equations, subject to a set of forced boundary conditions. The equations are the non-dimensional Navier-

Stokes and continuity equations,

uux + vuy = - Px + 4 v2“ ’


and the boundary conditions are,

y=0 : v=0 , uy =0 , Py=0 ,

r=R(0) : u=0 , v=0 ,

r->°° : u-*l , v-*0 , P-*-0 ,

where r is the radial coordinate from the origin,

r 2 = x 2 + y 2

and R(8) is the cylinder boundary.A sketch of the physical problem including the

boundary conditions is shown in Figure 1 [42].


CHAPTER II

HISTORY OF THE PROBLEM

Flow past a circular cylinder is a problem which has attracted attention for many years. As with most difficult problems, the early stages of development include simplified analytical studies and primitive experimental observa

tion. Subsequently, the early analytical methods are refined and more sophisticated equipment is used in experi

ments to improve results. In recent years, another step of scientific development has been added, that of numerical investigation with a digital computer, which is responsible

for the latest advances in the present problem.

Analytical Investigations

Sir G. G. Stokes obtained the first useful results in

1851 by a simplified analytical investigation [44,45].

Since the inertia forces are quadratic in the velocity, corresponding to the convective terms in the Navier-Stokes equations, he decided the pressure and viscous forces must be the only important forces for small velocities, i.e. at

low Reynolds numbers. This simplification results in a biharmonic equation for the stream function, which must satisfy the usual no slip and free stream boundary

6


7

conditions. An analysis of the boundary conditions led

Stokes to a solution assumption which resulted in,

ifj(r,e) * c |r log(r) “ f + s in (0)>

where is the stream function,

u = , v = -

or in polar coordinates,

vR-dir. = F *e ’ v 0-dir. = - ♦r

r and 0 are the polar coordinates, and C is a constant.The two immediate drawbacks to Stokes' solution are that the velocities become unbounded far from the body, and that

there is no value of C which will satisfy all the boundary

conditions.An improvement on Stokes' solution was made by C. W.

Oseen in 1910 [37]. Stokes' paradox, the non-existence of a solution far from the body, was explained by Oseen to be a result of the singularity of the flow for small Reynolds numbers. He showed that the ratio of the terms neglected by Stokes, the convective terms, to those retained, the viscous terms, is of the order of the product Rr, where R is the body radius, and r is the radial polar coordinate.It is obvious that as r increases the neglected terms are no longer negligible, and Stokes' solution fails. Oseen


suggested a linearization rather than a neglect of the con

vective terms. Therefore, the x-component of the equation

of motion becomes

Uux - - Px * 4 ^ •

where U is the free stream velocity, an obvious improvement on the equation used by Stokes,

Although Oseen presented a solution for the improved Stokes problem for a sphere only, a year later H. Lamb published the solution for a circular cylinder in terms of rectangular cartesian velocity components [32] which are, in non-

dimensional form,

x and y are rectangular coordinates, and r is the radial

coordinate.Many analytical methods have been attempted in an

( i M i * •

where,

1C


effort to improve the results obtained by Stokes and Oseen,

but none have produced a complete solution [5,6,17,18,22,

23,26,31,39,42,51,55,58].

Experimental Investigations

The difficulties of experimental research are obvious, any measuring device causes a perturbation in the flow field. The instability of some flows, particu

larly the pressure distribution, can cause large fluctuations in values due to the perturbation from an experimen

tal instrument.Most of the early experiments were restricted to high

Reynolds number flows, since the accuracy improved with

increasing Reynolds number. Some of the first investigators were able to show good agreement with analytical re

sults as early as 19 21. The preliminary work was done by

Wieselberger [59,60], Relf [40,41], Eisner [14], Thom [48, 49], and Homan [21] in the 1920's and 1930's. Increasing the range of Reynolds numbers for which data were available was the primary purpose of most of their research. After a relatively dormant period in the 1940's, experimental research on viscous flow past a circular cylinder was re

juvenated in the 1950's and 1960's by Taneda [47], Acrivos

[1], Tritton [52], and others. By this time the experimental equipment had improved considerably in quality, providing more accurate results over a larger range of

Reynolds numbers. Additionally, there were more reliable


10

results to compare with at higher Reynolds numbers than the

early analytical work, since the tremendous growth in

numerical solutions occurred during this period as a result

of the introduction of the digital computer. As one might

expect, the improved experimental techniques and equipment produced data which corresponded very closely with the recent numerical solutions, which were restricted to rela

tively low values of Reynolds numbers.

Numerical Investigations

Although numerical methods were used as early as the

1920's to attack the viscous flow problem, they were se

verely restricted until the high speed digital computer was developed sufficiently in the 1950's, when research

with numerical methods realized a tremendous increase. An

early numerical technique, which is the basis for many

current approaches to the problem and is described below, was introduced in 1928 by A. Thom [48]. He developed a finite difference scheme for solving the governing equa

tions in modified form. Since the ideas of Thom required computing facilities, little research was done in the area for over twenty years until the 19 5 0 's, when M. Kawaguti [27] and H. B. Keller [29] made use of the recently developed digital computer to try modified forms of the method introduced by Thom. In the 1960's the power of the finite difference techniques became apparent to many. Thus a

large number of articles appear in the literature for the


11

first time during this period describing numerical solu

tions to the Navier-Stokes equations [2,3,7,8,10,11,19,20,

24, 25, 28, 30, 38, 43, 46, 50, 54].By far, the most popular technique is to convert the

Navier-Stokes and continuity equations to an equivalent but simpler mathematical problem of one order higher and solve

the system by finite differences, which is basically the method introduced by Thom [48]. Introducing the stream

function ip ,

U = lpy , V = - ^ ,

and the vorticity w,

CO = Vx - Uy ,

the governing equations become,

noting that the definition of the stream function automatically satisfies the continuity equation. A transformation

is made to complex polar coordinates where the circular solution domain becomes a rectangle, which is easily and accurately approximated by finite differences. It is rather difficult to choose the location of the external boundary, at which the free stream, boundary conditions are imposed. Therefore an extrapolating procedure, consisting


12

of many solutions with different external boundary distances, is employed. The solutions obtained are extrapo

lated to approximate r -»■ 00 , where r is the radial dis

tance from the body [30].Although the previously discussed method is the most

popular, there have been attempts to analyze the problem in the primitive variables, u, v, P. The most reliable work utilizing this method was published by A. J. Chorin in 1967 [7,8]. He proposed a finite difference scheme for solving the unsteady Navier-Stokes equations in two or

three dimensions in the form,

u t + Px = f(u) , v t + Py = g(v) ,

and the continuity equation in a differentiated form,

T t (ux * V ' °'

The time variable is discretized and at each time step an

iteration technique is employed to calculate new values.His results seem reliable but somewhat restricted to domains

with nice geometry.The finite difference methods have unquestionably

contributed greatly to progress made recently in the investigation of viscous flow, yet a significant restriction

is the necessity of having a domain without irregular boundaries.


13

A considerably more general solution has been a t

tempted recently using the geometrically more powerful

finite element method. The simpler problem of inviscid

fluid flow has been attacked by several authors with reason

able success [4,12,34,35,56], M. D. Olson published the

first useful results of a finite element analysis of v i s cous flow in March, 1972 [36] . He develops a variational principle equivalent to the standard stream function equa

tion, with the pressure eliminated,

+ ^xV ^ y - = 0 *

The finite element method is employed to obtain an approxi

mation to the stationary value of the functional. Since the plate deflection equation is a biharmonic equation, of

the same order as the above governing equation, the author felt an eighteen degree of freedom plate deflection element

might yield good results. The element is a three node

triangle with the stream function, ip, and its five first

and second derivatives, ^v , ij> , treated asa y aa Ay y y

unknowns at each node. Although the analysis by Olson is a significant contribution, some of the results for flow past

a circular cylinder are suspect.


CHAPTER I I I

THE FINITE ELEMENT METHOD

The finite element method is a discretization process

by which a continuous medium, or a system with finite d e grees of freedom, is idealized by a set of finite regions, interconnected at their boundaries by nodal points [61].The values of the desired function at the nodes are the unknowns. Basically a Rayleigh-Ritz technique [9], the method produces a system of simultaneous algebraic equa

tions, whose solution is an approximation to the function which satisfies the governing equation and prescribed

boundary conditions.

History

With the introduction of the high speed digital com

puter, the study of numerical techniques increased greatly in the 1950's. The finite element method is a direct result of two early techniques, structural analysis and

finite differences.Systems with finite degrees of freedom were initially

attacked with structural analysis computer programs. M e m bers, or elements, of the system are connected at joints, or nodes, by imaginary pins to allow only forces to be

14


15

transmitted between members. Forces applied at the joints

of a member, e, are represented as a vector,

{F}6 = <?1

yi

as are the displacements at the nodes,

{ 6 }

Imposing a known stiffness relation for each element be

tween the forces and displacements, a system of simulta

neous algebraic equations is formed

[K]{6} = {F>

where [K] is the "stiffness" coefficient matrix, {F} is the

"force" vector for the n nodes,

and {6} is the vector of unknown displacements at the nodes,

The system of algebraic equations is solved for the d is

placements at the nodes, from which the strains are calculated, and an application of the constitutive relations yields the stresses in a member.

Continuous systems, possessing infinite degrees of freedom, originally were attacked by the finite difference


17

method [15]. The infinite degrees of freedom are approxi

mated by a finite number, and the differential equation to

be satisfied is approximated by a difference equation. A

system of algebraic equations is formed from the difference

equation at each node in the domain, and these equations are solved for the values of the unknown variable at each node. As the number of nodes is increased the approximation to the solution improves, until the error of computa

tion becomes prohibitive.A paper by Turner et al. [53] marked the beginning of

finite element analysis in 1956. The aerospace industry was the first to employ the method on a large scale. When its power became more universally recognized in the 1960's, the finite element method experienced a great increase in

use and analysis by both industrial and academic worlds.

Since the introduction of the method, it has served primarily as a means of stress analysis. However, recently its applicability to other fields has become apparent.

Application

Most mathematical problems in engineering may be represented by one of two equivalent formulations. The more traditional approach is to state a governing differential equation to be satisfied and require certain forced boundary conditions to be met. An alternative formulation is to seek an extremum or at least a stationary value of a

functional, usually defined as an integral over a domain,


18

subject to the same set of forced boundary conditions as

in the first method. An application of variational calcu

lus to find the function which imposes a stationary value

on the functional yields not only the Euler equation, a differential equation in the unknown function, but also "natural" boundary conditions on the function [33,57].

Although the finite difference method directly a p

proximates the differential equation and forced boundary conditions, the finite element method, being a type of Rayleigh-Ritz technique [9], is more suitable to the other

formulation, that of requiring a functional to be stationary. The functional I is defined as an integral of the unknown function {$} and its derivatives over the domain A and its boundary S,

I - f ...) dA + f g({<f>},{<f>}, ...) dS .*A JS X

Let {$} denote the unknown values of the function { } at

the nodes. Within each element the function {<|>}e is

evaluated from the "interpolating" or "shape" functions [N]

and the nodal values {$}e ,

{<f>}6 = [N] {$}e

Treating the nodal values {$} as the "generalized coordinates," the Ritz technique leads to a system of equationsnecessary for I to be stationary,

19

8_I8<t>,

T U T = /

81 J * N J

If the value of the functional over the whole domain is the

sum of the values over each element, i.e.,

I = I I6 , e

then the set of equations mentioned above becomes,

»Lr - T 316 - 0e 9{«}e

For the special case of I quadratic in (<j)} and its

derivatives, it can be shown that the equations take on the

special form,

or simply,

[H]{*> + (F> = 0

where the matrix [H] and the vector {F} are constant such

that,

20

{F±} - I {Pi}® e

This set of simultaneous algebraic equations can be solved

for the unknown values {$}.

Convergence Criteria

As the number of elements is increased, the actual domain is more closely approximated by the finite element representaiton, yet for the solution to converge the func

tional must satisfy certain conditions, which are imposed

on the shape functions [N].The first condition requires the relations f and g in

the functional integral to be single valued:"The element shape functions [N] must be such that

with a suitable choice of {$}e any constant values of {<J>} or its derivatives present in the functional I should be able to be represented in the limit as element size d e

creases to zero" [61] .Another condition is needed to insure that

I - I I6 , e

which is accomplished by requiring the highest derivatives

of {<{>} to be finite:

21

"The element shape functions [N] have to be so chosen

that at element interfaces {<(>} and its derivatives, of one

order less than that occurring in expressions f and g which

define the functional, are continuous" [61].

Interpolating Functions

Much of the recent research in the field of finite

element analysis has involved attempts to develop more sophisticated, and therefore better approximating, interpo

lating functions [N].The first popular two dimensional element was the

three node linear triangle. Although very simple, the linear triangle has proven satisfactory for many problems, and therefore has received much attention [61]. The linear interpolation between nodes, for both geometry and function

variation, is an obvious extension of finite difference methods. In fact, for some cases the same set of algebraic

equations is obtained by both methods, providing the nodes

are in the same positions. As the linear triangle is examined it becomes obvious that the necessary convergence

criteria are satisfied by this element. Not only is the geometry of the element linear, but the variation of a

function within the element is assumed to be linear. Let the function sought be a scalar (j>, then,

<f> = d x + d 2x + d3y ,

is a linear expression for <j>, where d^, d 2 » d^ are

constants to be obtained by imposing the conditions at the

three nodes,

♦i “ dl + V i * V i * 1 ’ 1>2'3 '

Solving the three equations for the constants, and substi

tuting in the linear expression for <}> yields,

<J> = ^ [ ( a 1+b1x+c1y)(t>1+(a2+b2x+c2y)<f>2+(a3+b3x+c3y)<J>3] ,

where A is the area of the element, and,

a i = x 2y 3 - x 3y 2 >

b l = y 2 - y 3 * C 1 = x 3 - x 2 ’

and the other constants are obtained by permuting the indices. In matrix notation the function <J> can be written,

f*'T • ■ < “!• « ! • » ! > < » ! |

where fT, i=l,2,3 are the shape functions,

N 1 = I K Cal + b lx + cly) * etC*’

or more simply,

cj) = <N> {<|)}e

R ep roduced w ith perm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.

One of the major advantages of the formulation using

linear triangles is that, numerical integration is unneces

sary. A big disadvantage is that representing functions

with linear polynomials makes the first derivative in an

element constant, and the second derivative zero. Thus the use of linear shape functions may be too crude an approximation of the actual solution surface, especially if the

number of elements is small.Linear rectangles provide a slight improvement over

linear triangles. Consider the simple linear rectangle in Figure 2. Since the element boundaries are parallel to the coordinate axes, the shape function may include products of

linear terms,

<f> = d x + d 2x + d 3y + d4xy .

Note that along an edge the variation in <j> is linear, since

one variable is constant on any edge. In the manner d e

scribed for linear triangles, the constants d ^ d 2 , d 3 , d 4 can be evaluated, and <J> can be explicitly written as a func

tion of x and y in terms of the nodal values of (|>.A significant improvement on linear elements is ob

tained by non-linear rectangles. To simplify the analysis a scalar unknown is used as before, and to illustrate the

method a quadratic polynomial is employed. But, by no means is the analysis restricted to scalars or quadratic functions! Consider the eight node rectangle shown in

24

Figure 3. Since there are three nodes on an edge, at which the values of {<{>} uniquely define a parabola, the shape

function may contain terms of increasing degree up to the

product of quadratic factors,

<f> = di + d2x + d3y + d4xy + dgx2 + d6y2 + d?x2y + dgxy2 .

Note, however, that the number of terms is limited to eight by the number of constants which can be evaluated from the eight nodal conditions. Imposing these conditions yields,

*1 = di + d2xi + • • • + V i yi •

*8 ■ d l * d 2x8 + • • • - d 8x 8y 8 •

or in matrix notation,

U } e = [c] {d}

If there exists an inverse to [c] then,

{d} = [c] ” 1 (d»>e

and therefore since,

<J> = <1, x, y, . . . , x y 2) {d } ,

<|) can be written,

<t> = <1, x, y, . . . , x y 2) [c] "x {4>}e

25

The interpolating functions N^(x,y) can now be defined,

<j> = <1^, N 2 ........... Ng>{<J)}e ,


<j> = <N>{(j)}e .

Although this formal procedure for obtaining shape func

tions is very direct it has disadvantages. An immediate possibility is that the inverse of the matrix [c] may not exist. Additionally there is always considerable diffi

culty in obtaining the inverse of [c] while maintaining generality with respect to geometry. A close analysis of

the shape functions reveals certain characteristics. Since

the relation,

must hold at any node, it is clear that must vanish at

all nodes except node i, where must be one. Also the variation on an element edge must be uniquely determined

by the nodal values.In view of these conditions the interpolating func

tions can be derived by inspection for linear, quadratic, and cubic variation on an edge, but higher degrees are in

creasingly more difficult. It is convenient to introduce

20

;i sot of normalized coordinates,

where xc , y Q locate the rectangle's centroid, and a and b

are half the lengths of the sides, see Figures 2 and 3.The shape function for the linear rectangle can be expressed

as a product of two factors,

N i = j U - 5Ci)(l + nnj)

which clearly satisfies the condition of being zero at all nodes except i, where = 1. Since interpolation between

nodes is done with a single valued relation, the first convergence criterion is satisfied. Also, the variation on an

element boundary is linear, and there exist two nodes to

uniquely determine the values, so the second criterion is satisfied. In a similar manner the interpolating functions for higher degree polynomials can be constructed. The r e sults for quadratic and cubic variation on an edge are

listed in Appendix A along with the linear variation.Although improved shape functions are usually higher

degree polynomials, there is nothing to prevent the use of

some other functional form, e.g., the trigonometric functions, provided, of course, the function satisfies the con

vergence criteria.


27

Curved Isoparametric Quadrilaterals

To closely approximate a physical domain with non

linear boundaries would require many rectangular elements.

Although the variation of a function within an element can be represented by sophisticated shape functions to reduce

the error, the finite element approximation of the problem is not very effective unless the domain is closely approximated. Thus, it is extremely helpful if the "parent" rectangular elements can be mapped into elements with n o n linear edges. The mapping of rectangles into non-linear quadrilaterals gives the finite element method an obvious advantage over the finite difference method in representing

the solution domain, particularly if the boundaries are

irregular.By far, the most convenient means of writing the

transformation from local £,n curvilinear coordinates to

global x,y coordinates is to employ the already derived shape functions. Considering x and y as functions of £

and n, an expression similar to that for the function 4> is

obtained,

where the functions N^(C,n) are shape functions which


28

determine the transformation. If these functions are re

quired to satisfy the convergence conditions, the trans

formation will be one to one. Although continuity of the

function <J> is evident in the parent element, the values of <j> in the curved elements must be investigated for continuity. Zienkiewicz summarizes the results of the investi

gation in two theorems [61]:Theorem 1. "If two adjacent elements are generated from

'parents' in which the shape functions satisfy continuity requirements then the distorted elements will be contiguous."

Theorem 2. "If the shape functions Ni(£,n) used to define <)> are such that continuity of is preserved in the parent coordinates then continuity requirements will be satisfied in distorted elements."

It should be noted that nodal values of (p may or may not be associated with the same nodes as those used to specify the geometry. If the number of nodes used to de

fine the geometry is greater than the number used to define

the function (j>» then the element is labeled "super- parametric." In the reverse situation the element is called "sub-parametric," which is the more useful type in practice. The third possibility is the more convenient, and is the type used in the present analysis. If the nodes used to define the geometry coincide with those used to

define <t>, and if N i = N! , i.e. the interpolating functions are the same for both the geometry definition and function variation, then the elements are denoted "isoparametric."

The basic non-linear element used in the present work

is the eight node, quadratic, isoparametric quadrilateral

shown in Figure 4, whose shape functions are listed in A p

pendix A. The function <|> is given by,

<j) = d 1 + d JE, + d 3n + d4Cn + d gc 2 + d6n 2 + d?C2n + dg£n 2

which is expressed by the interpolating functions. The

geometry transforms according to the same functions,

x = <N){x}e

y = <N>{y}e

For most problems it is necessary to calculate at

least the first derivatives of some function with respect to x and y. Since the interpolating polynomials, and

therefore all functions defined in an element, depend on

the local coordinates, it is necessary to make a transformation to calculate x and y derivatives. A simple

application of the chain rule results in,

where [J] is the Jacobian matrix,

[J] -

30

For admissible transformations, the derivatives with re

spect to x and y can be calculated,

Some problems require not only first but second d e rivatives with respect to x and y. As with first derivatives, the second derivatives with respect to the global coordinates are obtained from the local derivatives,

' 3 2 3 F -

‘ 2xs

2x5 y5yf

- l 3 2 3?2

3" XS£3x

3y^ 3 y

3 23x3y > = x Px

5 n W V s y?y n <3 2

3£3n3

xCn3x >

j2xnyti y *7 n

3 2

J * ' *

_3_ _ xnn 3x yrm 3y

where the first derivatives are calculated from the Jaco-

bian matrix as discussed earlier,

To calculate each of the matrices above, the derivatives

of x and y with respect to E, and n are obtained from,

x = < N X x } e , y = <N> {y }e

where <N> contains the interpolating functions, which

depend on the local coordinates.

Elimination of Internal Nodes

Nodes within an element need not be included in the overall finite element equations. Equations associated with an internal node may be eliminated from the global equations since they form an independent set of equations. Internal node equations are independent because the finite

element neighborhood for an internal node is simply the element surrounding that node. Consider the element formu

lation for element e,

= [H] e {<f>)e - CF)e ,

where {<(>}e is the unknown variables at the nodes, and [H]e and {F}e are the element matrix and vector. Since {<j>}e can be separated into a part which corresponds to nodes common to other elements {^}e , and one which corresponds to nodes within the element {<|>}e , it becomes clear that one finite

element equation is,

32

- H - = -11— = 0 .9{?}e 9(?}e

Using this relation the element formulation equation can

be partitioned to yield,

919{<}>}e

' 9l e ] f 9I 6 '9{ ? } e 1 J 9{ < j ) } e9I e [ ^ 09{ ? } e J

[H]6 , [H*]€

[H*] , [H] {?}€

Solving the second matrix equation for {<f>}

- x T{?}6 = [H]e ([H* ]e (<}>}e - {F}6) ,

and substituting in the first matrix equation yields,

9 (<£"}[H** ] e { <j>} e - {F*}e

where,

{F* }e = (F}e - [H*]e [H]e {F }6

33

The finite element formulation now contains only the terms

corresponding to the boundary nodes for element e. Employ

ing this procedure to eliminate internal nodes requires

some additional calculations in equation formation, but

reduces significantly the equation solving effort. An obvious simple example is the five node quadrilateral shown in Figure 7. Doherty et a l . have investigated this simple element and many others [13]. More sophisticated elements,

which are discussed in Chapters VI and VII, appear in Figures 6, 7 and 8, The above elimination scheme is applied to each of the internal nodes. It is worth noting that the element matrix is modified each time a node is

eliminated, which is very valuable for some problems.

The generality of the methods described is obvious.An immediate possibility is to extend the analysis to three

dimensions. Additionally, the scalar unknown could be r e placed by a vector function with little complication. Also,

it is apparent that the specification of the interpolating functions is arbitrary, allowing the versatility of changing the functions used without rewriting the program.

Choosing the most efficient element to attack a problem is an art strongly dependent on the engineer's intuition and understanding of the physical problem. However, curved, isoparametric quadrilaterals should be considered in any

problem as a means of reducing input, since fewer elements will be needed, and improving accuracy, as a result of


34

better interpolating functions. The possibilities for element construction and choice of interpolating functions

are innumerable, depending only on the engineer’s

imagination!


CHAPTER IV

METHODS OF SOLUTION

Solutions to the Navier-Stokes and continuity equations are approximated by successively solving a sequence of linear equations by the finite element method. An initial estimate is made of the velocity and pressure fields, from which the non-linear terms are calculated and treated

as known functions. New estimates of the velocities and pressures are computed from the linear system. The pro

cedure is repeated until two successive solutions are

"close." This iteration technique is applied in two forms,

each of which is explained below. Some preliminary in

vestigations using method 1 have been performed by D. L.

Garrett [16].

Method 1

The governing equations from Chapter I may be linear

ized by approximating the non-linear terms,

1 v 2u - P = f(x,y) kE x

- Py = g(x,y) ,

ux + vy * 0 •

35


56

where,

f(x,y) = uux + vvy

g(x,y) = vvx + vvy

Assuming a velocity and pressure field, u°, v°, P°, the

approximating functions can be evaluated,

f°(x,y) = u°u“ + v°u°

g°(x,y) = u°v® + v°v°

The linear equations,

V 2u x - p£ = f°(x,y)

1 V 2v 1 - P * = g°(x,y) ,kE y

ui + V* = 0 ,

and the appropriate forced boundary conditions, are solved by the finite element method for the new velocities and pressures, u 1, v 1 , P 1 . The procedure is repeated until two

successive solutions are close. At the n*"*1 step, the

iteration scheme is,


57

1 V 2u n _ p£ = (uux + VUy )n_1

1 v 2v n - P r e

It is hoped that as n gets large the method will converge to a realistic solution. However, even though two successive solutions may be close, there is no guarantee that the method has converged to a good approximation of the true

solution.

Rewriting the governing equations in a different

form yields,

If the first equation is differentiated with respect to x, the second with respect to y, and the two are added, the result is,

Method 2

V 2u = Re (u u x + vuy + Px )

V 2v = Re (u v x + vvy + Py )


3 8

- vv ) + 37 Cuvx ♦ vvy) ♦ Pxx ♦ Py

Due to the linearity of the Laplacian operator the equation

may be written,

jjL V2(ux ♦ vy) = 4(uux * vuy) * ^Cuvx ♦ vvy) ♦ V2P ,

or, since the continuity equation must be satisfied,

V2P - - ^Cuux ♦ VUy) - ^(uvx ♦ vvy) .

Combining this equation with those for u and v results in a

system of three equations in the velocities and pressure. However, since the new system of equations is one order higher than the original system, an extra boundary condition is necessary. Such a condition does become necessary in the finite element formulation of the pressure equation,

which is discussed in detail in Chapter V. The new system

of governing equations is,

V 2u = f , V 2v = g , V 2P = h ,

where,


39

f = Re (u u x + vuy + Px ) , g - Re (u v x + vvy + Py) ,

h = - ^ T (uux + vuy ) - (uvx + vy •

and the boundary conditions are the original set of physical conditions, plus the additional condition on the pressure needed for the new system to be equivalent to the original. As with method 1, the non-linear functions, f,g, and h, can be calculated from a known velocity and pressure field, which results in a system of three, uncoupled,

Poisson equations in u, v, and P.The iteration technique is essentially the same as

that for method 1, except that the equations are different.

Assuming an initial velocity and pressure field, u°, v°,P°, values of f°, g°, and h°, are computed, and the three

Poisson equations are solved by the finite element method

for new values of the velocities and pressures, u 1, v 1, P 1 . As before, the procedure is repeated until two successive solutions are "close." Again, it is hoped that the itera

tion scheme will converge to a realistic solution.It should be emphasized that no attempt is made to

prove, or disprove, convergence of the iteration schemes.

This task may be not only difficult, but possibly

intractable!


CHAPTER V

FINITE ELEMENT FORMULATION

The general methods of Chapter III will n o w be applied

to the solutions of particular problems. Explicit expressions will be obtained for the simultaneous algebraic equations formed by the finite element method, and techniques of imposing some boundary conditions will be discussed.The two methods described in Chapter IV will be formulated. Additionally, the simple linear triangle will receive some special attention.

Method 1

As stated in Chapter IV the finite element method is

used to solve the system of linear equations,

4 (uxx ♦ u yy> - Px - f - 0

-i-fv + V ) — P — B = 0 ,re 1 xx yy y &

where f and g are functions of x and y, and u, v, and P are subject to forced boundary conditions. An equivalent

40

R ep roduced w ith perm ission of the copyright ow ner. Further reproduction prohibited w ithout perm ission.

41

mathematical formulation is to seek functions u, v, and P,

which will give a stationary value to the functional,

1 - | [~-(u2 + u* + v* + v p + uf + Vg - P(ux + vy)] dxdy .A E

An application of variational calculus to the functional I, yields three Euler equations, which correspond exactly to the above governing equations, and, in addition, the natu

ral boundary conditions,

- - ux — P = 0 for x constant,kE

* r . V yP = 0 for y constant,

Uy = 0 for y constant,

v = 0 for x constant.

Similar expressions are obtained for boundaries which are

not parallel to the coordinate axes. By imposing the

forced boundary conditions,

r = R (0) : u = v = 0 ,

y = 0 : v = Uy = Py = 0

r-*-°° : u - > l , v - > - 0 , P - * - 0 ,

the natural boundary conditions are overridden.The finite element formulation is obtained by


4 2

employing a Ritz technique to find the functions u, v, and

P which render the functional I stationary. Treating the

values of the functions at the M nodes as the unknown

variables, the finite element method requires the equations,

11 TCr 61

eh^" Ie=l Ie=l < 0 , k=l, 2,... M,

to be satisfied. Here u^, v^, are values of the functions at node k, n is the number of elements in the so-

called "finite element neighborhood" of node k, and Ie is the functional I integrated only over the part of the domain occupied by element e. The "finite element neighbor

hood" is the group of elements which contain node k. It

is now necessary to evaluate the derivatives of Ie with respect to the values u^, v^, P^,

3IC3u.

1 f ^ux f 9u _ p ^uxRc x 3uk Uy 3uk' 3uR 3uR dxdy ,

313v.

, 3v 3v ~ 3vi d v x S v T * v y * 8 5v T “ P dxdy ,

' 1 * v ] dxdy ■

where it should be recognized that v and P do not depend on the nodal values of u, and likewise for the other variables.

Recalling the fact that the variables are written,

43

u = (N> {u}e , v = (N>{v}e , P = <N>{P}e ,

the derivatives above can be evaluated,

f3N, ,/3Nii [ v v i \re M 3x \ 3* ’ • • •/ 3y ' 3y ’ ' " ' I

dxdy {u}

' 9N.<N> dxdy {P}6 + fN^dxdy

e J e

31a similar expression for and,

919P,,

-/ 9N1

Nk W • •\ \

- .. / dxdy (u}e + ../ dxdy- ■e - 4

(v}e

In matrix notation these expressions can be written,

313u,

I 31 > = [H] e {F}

313u_

44

where [H]e is the coefficient matrix for element e, and

{F }e is the constant vector for element e, for which ex

plicit expressions are found in Appendix B. Having expres

sions for 31 3 T 3 Tfv * It - * t*ie ^inite element equations

are formed,

Ie=l 319u,

r 91

> , k = l , 2, ... , M,

which can be written in matrix notation as,

[H]{W} - (F> = (0} ,

where [H] is the overall coefficient matrix, {W} is the

vector of unknown velocities and pressures at the nodes, and {F > is the overall constant vector. It is worth noting

that the components of the overall vector {F} are simply sums of the components of the element vectors {F}e corresponding to a particular node, as is true for the overall

matrix [H].The system of linear, simultaneous, algebraic equations

is solved for the components of the vector {W>. Thus, an approximation of the functions u ( x , y ) , v(x,y), P(x,y) is


4 5

given by the values u^, v^, Pk ; k=l, 2, ... , M, where M is

the number of nodes.Having calculated a new estimate of u, v, and P, the

approximating functions f and g can be computed from the

new values. Within an element the values of the functions u and v, and their derivatives, are calculated from the

nodal values,

u = <N){u}e ,

ux = (Nx > {u}6 • uy ■ (Ny ) tu>e *

and similar expressions for v. Recalling the functions f

and g,

f = uux + vuy , g = uvx + vvy

the new values of the functions can be calculated from the new estimates for the velocities at the nodes,

f = « N > { u } e )(<Nx>{u}e) + (<N>{v}e)(<Ny >{u}e)

g - « N ) { u } e)C(Nx>{v}e) + « N > { v } e)(<Ny>{v}e )

From the new values of f and g the governing equations can again be solved for another approximation of u, v, and P.

As described in Chapter IV the procedure is repeated until convergence is achieved.

46

Method 2

The second approach employs the finite element

method to solve the system of Poisson equations,

V 2u = f(x,y) ,V 2v = g(x,y)

V 2P = h(x,y)

subject to the same set of forced boundary conditions

stated for method 1.Since each equation is the same type, and the equa

tions are independent, it is helpful to consider the solu

tion of a general Poisson equation,

v2<f> = iMx,y) ,where <J> and ij> are sufficiently smooth functions of x and y.

The finite element method solves this equation by requiring

a stationary value of the functional,

1 = lA + + dxdy ‘

A stationary value of I is obtained, provided the Euler

equation,

V 2<(> = ip ,

and the natural boundary condition,

47

are satisfied. An analysis of this condition for each

variable follows. The finite element equations are,

I sir - 0 >e=l 3 h

where n is the number of elements in the finite element neighborhood, and k=l, 2, ... , M, where M is the number of nodes. The derivatives of Ie are needed,

Again recalling the functions within an element are written

in terms of the nodal values,

<f> = <N> {<J>}

*x - <Nx > {* ,e ’

the derivative of Ie becomes,

a r

3*k '\c \ 3x ’ ■ • •/

N^dxdy

3N. /3N. \w *tKw ’ ■■■)w dxdy

In matrix notation the equations can be written,

= [H]e{<|>}e - {F }e , k=l, 2, ... , M,0<Pv

48

where [H]e and (F)e are the coefficient matrix and constant

vector for element e, given explicitly in Appendix B. The

finite element equations can be formed,


[H] {<J>} - (F> = 0 ,

where [H] and {F} are the overall coefficient matrix and constant vector, and {<}>} is the vector of unknown values at

the nodes,

The finite element equations are linear, simultaneous, algebraic equations, which can be solved by a standard

method for the values of <f> at the nodes.Having a solution method for a general Poisson equa

tion, the governing equations in u, v, and P can be solved.

A simple substitution of the appropriate variable for <j> in the above analysis yields a system of algebraic equations in the nodal values of that variable, whether it be u, v,

or P .

M

49

It is important to consider the boundary conditions

imposed on a particular variable. As noted above the natural boundary condition on a variable is that the normal

derivative is zero. The forced boundary conditions must be

imposed on each variable separately. For the u equation

the forced boundary conditions are,

r = R(0) : u = 0

* ■ < W 9> : u ■ 1 -

where R is the body radius, and Rmax is the radius to the external boundary. It is clear that allowing the natural boundary condition of zero normal derivative to exist on

the remaining part of the boundary, that being the line of

symmetry, agrees with the physical requirements. The

forced conditions to accompany the v equation are,

r = R(0) : v = 0 ,

r ■ W 9) : v - 0 ’y = 0 : v = 0 ,

which includes the entire boundary, so the natural boundary condition is irrelevant. For the pressure equation there

is only one forced condition,

r - W 9 > : p - 0 •

The zero normal derivative condition agrees with physical

requirements on the axis of symmetry but not on the body.

50

Therefore, it is necessary to impose an additional forced

boundary condition on the pressure on the body. The extra

condition requires the normal derivative of the pressure to

correspond to that prescribed by the flow field. Two suc

cessive solutions for the velocities and the preceding pressure are used to calculate the pressure gradient in the

x and y directions,

Px +1 = <Px + uux + vuy )n ” (uux + vuy)n+1

py +1 = (py + uv* + vvy )n “ (uvx + vvy )n+1 *

After first calculating un+1 and v n+1 in an iteration, the pressure gradient is approximated. From the two components of the pressure gradient, a normal derivative is calculated.

To impose a normal derivative of q(s) on a boundary S, it

is necessary to add a term to the functional,

IBC - - [ qPds Js

where P is the pressure as a function of the location on the boundary. This term requires the specified derivative

as a natural boundary condition. The functions q(s) and

P(s) can be written,

q = (N){q}S , P = (n ){P}S ,

where <N> is the row vector containing interpolating func

tions in one dimensional curvilinear coordinates, and {q}s

51

and (P}S contain the nodal values on the boundary S. The

derivative of the extra term needed for the finite element

equations is,

tBC<N>(q}S Nkds

By adding this term to column K of the element constant

vector {F } e , the normal derivative of P is set equal to q.The imposing of an extra boundary condition on the

pressure is a mathematical necessity, as well as a physical requirement. Recalling that the Poisson equation in P is obtained from a combination of first derivatives of the

motion equations and the continuity equation, it is clear

that an extra boundary condition is needed on P, if the two

systems of differential equations are to be equivalent.Having calculated a new estimate of the velocities

and pressure at each node, the approximating functions f, g, and h can be computed. Noting the definitions of the

functions,

f = Re (u u x * vuy ♦ Px) ,

8 - r e (u v x + vvy * V

h * - K * v y + uuxx + vvyy + 2uyVx + vuxy * uvxy3 •

within each element f, g, and h can be written in terms of

52

the nodal values of u, v, and P,

f = RE[«N>{u}e)«Nx){u}e) + «N>{v}eH < N y>{u}e) + <Nx> (P>e] ,

g = RE[«N>{u}e) « N x){v}e) + «N>{v}e) « N y){v}e) + <Ny>{P}e] ,

h = - [«Nx>{u}6)2 + « N y){v}e)2 + ( ( N X u ^ X ^ X u } 6)

+ «N){v}e) « N >{v}e) + 2 « N ){u}e)((N >{v}e)y y y A

+ «N>{v}e) « N v >{u}e) + «N){u}6) « N >{v}e)] .xy a y

Using the newly calculated functions f, g, and h, the finite element method is employed again to solve the three

Poisson equations for new values of u^, v^, P^,

k=l, 2, ... , M. As described in Chapter IV the procedure

is repeated until convergence is achieved.Since the shape functions for a linear triangle are

very simple, the element formulation deserves special atten

tion. Appendix C contains a detailed discussion of the

formulation with linear triangles.

CHAPTER VI

DESCRIPTION OF COMPUTER PROGRAMS

Computer programs which implement and test the pre

ceding methods are explained in the following discussion. Program listings appear in Appendix D, Two basic programs,

one for each solution method, and a test program are described. Each subroutine is examined separately, and this,

coupled with an abundance of comment cards in the listing, should be sufficient to explain the programming logic.

Method 1

Although many programs are used to attack the problem

with method one, only one is explained here. Others are discussed in the next chapter. The program examined here employs a special variation of the four node quadrilateral, which is formed by twelve linear triangles connected at

nine nodes, five of which are internal nodes, see Figure 6. Using seventy-four elements and ninety-five points requires about 100,000 bytes (hexadecimal) of storage and slightly

over one minute of central processing unit time on an IBM 360/65 model computer. Of course, the time requirement depends directly on the number of iterations performed.Most arrays in the program are dimensioned in and passed

through labeled or unlabeled common blocks.

53


54

Main Program

The main program organizes the calculations, which

are done in the subroutines. Input to the program are the

problem parameters,RE--Reynolds Number,RMAX--Maximum radius on downstream axis,EXPR--Exponent for radial distribution of nodal points,

see subroutine GRID,EXPT--Exponent for transverse distribution of nodal

points, see subroutine GRID,NELRUP--Number of elements in the radial direction on

the upstream axis,NELRDN--Number of elements in the radial direction on

the downstream axis,NELT--Number of elements in the transverse direction

for half the grid, note the total elements in the « transverse direction is 2*NELT,

NPREL--Number of points in the radial direction of an element,

NPTEL--Number of points in the transverse direction of an element,

RO--Body radius at each step in the transverse direction, R(9) in the previous notation.

The finite element grid is generated by calling subroutine GRID, and boundary conditions are set by calling subroutine BOUNDY. An initial guess at the velocity and pressure is calculated in subroutine GUESS, where several guesses are possible. After printing the initial guess and

the problem parameters, the iteration scheme is started.After setting the initial value of the relaxation factor

GAMMA, the new values of velocity and pressure are


5 5

calculated by calling subroutine SOLVE. The values of BIG

and SUM are computed,

BIG = max{|u£ - ujj"11, |v£ - vJJ-11, |p£ - p£_1|, k=l, NP} ,

NPSUM =kH l uk “ u"_1! + lvk “ Vk_1! + - pn_1^

where NP is the number of nodal points and superscripts d e note iteration numbers. The sum is compared to the pre

vious sum to check convergence. Additionally, as the iteration converges the value of the relaxation factor is increased to a final value of unity. If the method has produced a sufficiently small sum, the iteration stops and

the values of u, v, and p at the nodes, which are stored in the vector W, are printed. The Reynolds number is incre

mented and another problem is worked, until the Reynolds number reaches the desired value. The initial guess for each problem after the first is the previous solution.

Subroutine GRID fNELRUP, NELRDN, NELT,NPREL, NPTEL, RMAXDN, EXPR, EXPT)

This subroutine computes the location of the nodal points in terms of the grid parameters, defines the elements

by their nodes, and decides which type point each node is.A polynomial of degree EXPR is used to distribute the points on a radial line. And a polynomial of degree EXPT

distributes in the transverse direction. As EXPR and EXPT increase the points are grouped more densely near the body


and near the axis of symmetry respectively. The upstream

domain is generated by adding elements in the radial direc

tion, until NELRUP is reached. At the beginning of each

radial step the node is marked, in NTYPE, as one on the

body, and at the end the node is marked as an external

boundary node. Also on the axis of symmetry the nodes are marked. Radial lines of nodal points are added until there are NELT elements in the transverse direction, which will correspond to the domain on the upstream side of the y axis. The downstream domain is generated in a similar manner. However, the number of elements in the radial direction is increased by one until NELRDN is reached. The radial and transverse coordinates are generated, and the rectangular coordinates are computed from them. After the coordinates

are calculated, the elements are defined. While stepping out in the radial and transverse directions, the nodes d e

fining each element are stored in the array NODE. Finally

the number of nodal points, NP, number of equations, NEQ, the matrix bandwidth, MBAND, and the number of elements,

NEL, are computed.

Subroutine BOUNDY (NVAR, NBOUND, NCOND)

The boundary conditions are set in this subroutine

according to the flag each node receives in subroutine GRID. NVAR determines the variable to be fixed, either u, v, or P. NBOUND locates the point on the boundary, whether it be on the body, at the external boundary, or on the axis

of symmetry. The code of NCOND determines the value that


57

the variable assumes, which is represented in IB, Setting

the value is discussed in subroutine SOLVE, where IB is

used.

Subroutine GUESS(K)

Here an initial guess for the velocity and pressure field is calculated. The integer K is used to choose the

guess. For test problems special guesses may be desired as in the case for the free stream problem to be discussed in

the next chapter. The initial guess for the real problem

is the Oseen solution of the Navier-Stokes equations, which

is presented in Chapter II.

Subroutine SOLVE

In this subroutine the algebraic equations are

formed, the boundary conditions imposed, and a solution to

the equations computed. The overall matrix and vector are zeroed, and the element matrix and vector are calculated and added to the overall set by calling subroutine QUAD4.

This system of equations is modified to impose the boundary

conditions. By checking the value of each row in the vector IB, the value of the variable corresponding to that

row is chosen, and subroutine MODIFY is called to impose that condition. Finally, subroutine SYMSOL is called to compute the solution to the set of equations.


5 8

Subroutine QUAD4(KK)

This subroutine forms the element matrix, COE, and

vector, CON, for a four node quadrilateral, element KK, and

introduces them into the overall matrix, C OEF, and vector,

CONST. First, the local values of the coordinates and variables are defined by picking the correct value from the overall vectors. The coordinates of the internal nodes are

defined as averages of the corner nodes, likewise for the variables, see Figure 6. The nine node element matrix and

vector are formed by calling subroutine TRI3 twelve times, once for each triangle. In the manner explained in Chapter III, the internal nodes are eliminated. However, before

eliminating these nodes a modification of the equations is necessary. Appendix B shows that the three rows and columns corresponding to a node constitute a matrix of the

form,x o xo x xx x o ,

where x's denote numbers not close to zero, and o ’s denote

zeroes or very small numbers. It is clear that the last row, corresponding to P, cannot be eliminated in the usual

fashion since a zero exists on the diagonal. Therefore, before an internal node can be eliminated, the pressure at that node must be redefined. At the internal nodes the pressures are defined as weighted averages of the other


59

nodes. To illustrate this procedure consider the equations,

lllx l + a 12X 2 + a13X 3

-21X 1 + a 2 2 X 2 + a 23X 3

31X 1 + a 32X 2 + a33X3

Letting x^ = (x^ + x 2)/2 yields,

taU + a 13/2)xl + (a12 * a13/2)x2 + 0 = bl •

Ca21 * a 23/2)xl + (a22 + a23/2)x2 + 0 = b 2 «

Ca31 * a 33/2>x l * (a32 * a33/Z>x 2 * 0 " b 3 '

It is clear that adding fractions of the coefficients of x^ to the other coefficients allows the value of x^ to be defined as a combination of the other variables. Also the equation for x^, the third equation, may now be ignored.In this manner the pressure at the internal nodes is redefined, and the equations corresponding to these nodes are

eliminated by calling ELIM. After the node elimination is completed, the resulting matrix and vector for a four node element is introduced into the overall matrix and vector.

Subroutine TRI3(N1, N2, N3)

The matrix and vector for a linear triangle, with node numbers Nl, N2, N3, are computed and introduced into the matrix and vector for the nine node quadrilateral.


bO

First the element matrix is initialized, The constants a^,

Ik , ci , i=l,2,3, defined in Chapter III, the element area,

the coordinates of the element centroid XBAR, YBAR, and the

interpolating functions evaluated at the centroid, AN1,

AN2, AN3, all are computed for use in the formulation. As expressed in Appendix B, the components of the element matrix H are formed. The second moments of the area and the values of the velocities at the centroid are calculated.

With this information the components of the element vector are formed. Having the element matrix H and vector F, the

components are introduced into the appropriate row and column of the matrix COE and vector CON for the nine node quadrilateral,

Subroutine ELIM (NODE)

By a standard Gaussian elimination procedure the rows associated with node NODE are eliminated from the nine node

quadrilateral matrix COE. The rows to be eliminated are

the rows associated with u and v at that node, i.e. rows

3*N0DE-1, and 3*NODE-2.

Subroutine MODIFY (ROW, VALUE)

This subroutine modifies the overall matrix, COEF, and vector, CONST, to cause the variable corresponding to row ROW to take on the value VALUE after solution of the equations. To illustrate the method consider the system of

linear algebraic equations in x^, X 2 , ,


61

allx l + a 12X 2 + a 13X 3 ii cr

a 21X l + a 22X 2 + a 23X 3 - b 2

a31X l - a32X 2 + a33X 3 ’ b 3

To impose x 2 = U the equations may be written,

all*l + 0 + a 13X 3 = b l “ a12U *

0 + x 2 + 0 = U ,

2u .

The result is two equations in x 1 and x 3 identical to the first and third equations with x 2 = U, and the second equation x 2 = U. Subroutine MODIFY applies this method to

the banded symmetric matrix COEF and vector CONST.

Subroutine SYMSOL

The system of equations,[COEF]{W} = {CONST}

is solved by a standard Gauss elimination method. The method is modified for the special form of storage used to

store the banded symmetric matrix in COEF. The diagonal of the matrix is stored in column one of COEF. The line of components parallel to the diagonal are stored in succes

sive columns,


62

column i

column 2

column 1Therefore the matrix COEF is,

a ll a 12 * * ali

a 22 a23 0

an-l,n

a 0 • • 0nn

The number i is the matrix bandwidth denoted by MBAND, and

n is the number of equations denoted by NEQ.

Method 2

The program in Appendix D for method 2 employs the eight node isoparametric element in Figure 4 to solve the Poisson equations. As a result of the simplicity of the equations, little storage and time is needed for the


6 3

program. Approximately 50,000 bytes (hexadecimal) of

storage and only a few seconds of central processing unit

time are needed.

The following subroutines are the same as in method 1:

MODIFY SYMSOL GRID GUESS BOUNDY .

Main Program

As with method 1, the main program controls input and output operations and organizes calculations. Problem

parameters, which are explained above, are input. A finite element grid is generated by calling subroutine GRID, and

setting the boundary conditions is accomplished by calling

subroutine BOUNDY. An initial guess is made for the v e locity and pressure, and that guess is printed. The

values of the interpolating functions and their derivatives are evaluated in subroutine ONCE. To form the matrix for the Laplacian operator subroutine MATRIX is called. After setting the relaxation factor GAMMA for each variable, the iteration scheme is started. At each iteration step all three Poisson equations are solved. Subroutine SOLVE is used to form and solve the equations. The new solution is compared to the previous solution by calculating SUM and

BIG,


6 4

BIG = max{ | <t>£ - (j)” '1 | , k»l, NP} ,

Nr .SUM = i ,

k=l K K

where 4> may be u, v, or P, and NP is the number of nodal

points. A new guess for the velocity and pressure field is made and a check for convergence is performed. Finally the

results are printed.

Subroutine ONCE (NGP)

Here the interpolating functions and their deriva

tives are evaluated at the gauss points. Three sets of gauss points and weighting coefficients are available. By calling subroutine INTERP, the functions and their deriva

tives are computed, after which they are stored in arrays,

GN-- Interpolating functions,DNA--Derivatives of functions with respect to A (cor

responding to £ in Chapter III),DNB--Derivatives of functions with respect to B (cor

responding to n in Chapter III),DNAA--Second derivatives with respect to A,DNAB--Second derivatives with respect to A and B,

DNBB--Second derivatives with respect to B.The interpolating functions evaluated at the gauss points along the edge B=— 1 are stored in GNB. Finally the derivatives with respect to A and B are evaluated and stored in BDNA and B DNB. The number of gauss points to be used in

65

each direction is passed through NGP.

Subroutine INTERP (A, B)

In this subroutine the interpolating functions and

all of the first and second derivatives with respect to A and B are computed explicitly at the coordinates (A,B).

Subroutine SOLVE

This subroutine forms the equations, imposes the boundary conditions, and computes a solution. Having the matrix components, the equations are formed by calling the element vector subroutine, VECTOR, once for each element. Imposing the boundary conditions and solving the equations

follow the same procedures explained in method 1.

Subroutine MATRIX (KK)

The element matrix H, for element KK, is formed and

is introduced into the overall matrix COE. First, local coordinates of element KK are defined. Forming the components of the matrix listed in Appendix B requires thederivatives of the interpolating functions with respect tox and y, which are computed in subroutine DERIV. Having evaluated the derivatives, the integrands for the matrix components are stored in HI. Also it is clear that the

element area is,

1 1A = / / dxdy = / J |J|d£dn ,

e -1 -1


66

where |J| is the determinant of the Jacobian transformation

matrix from x,y to C,n coordinates, Therefore, the inte

grand for the area is |j|, which is stored in A I , Having

the integrands evaluated at the Gauss points and the weighting coefficients obtained from subroutine ONCE, a

simple Gaussian quadrature integration yields the components of H. These components are introduced into the

matrix COE at the appropriate locations,

Subroutine VECTOR (KK)

The element vector F, for element KK, is formed and

is introduced into the overall vector CONST. Not only are

the local coordinates needed, but also the element veloci

ties and pressures at the nodes, Before each iteration on

the u-equation, the values of the non-linear expressions,

uux , uvx , vuy , vvy

are computed and stored for the elements adjacent to the body. Before each iteration on the P-equation, the pressure gradient is approximated as described in Chapter V.

From this gradient the normal derivative is computed, see Appendix C. As in subroutine MATRIX, the derivatives of

the interpolating functions are obtained from subroutine DERIV. Also needed is the function ip, which is defined in

Chapter V, evaluated at the gauss points. This is accomplished by calling subroutine APPROX. Having the interpolating functions, GN, the value of ip, PSI, and the Jacobian


67

determinant, D ETJ, the integrands may be calculated at the

gauss points and stored in FI. With the weighting coeffi

cients and the integrands evaluated at the gauss points, a gaussian quadrature integration may be performed to obtain

the components of the element vector F. The required normal derivative conditions are imposed on the pressure equation

by adding the appropriate expression to the vector F (see Chapter V ) . The element vector F is introduced into the

proper components of the overall vector CONST.

Subroutine NDERIV (K)

In this subroutine the derivatives of the velocities

and pressures, needed for the normal derivative boundary

conditions on the pressure, are computed. First the d e rivatives of x and y are computed with respect to A and B.

From these derivatives the determinant of the Jacobian matrix is evaluated at one of the nodes, chosen with K, onthe B = -1 element boundary. The derivatives of the inter

polating functions with respect to x and y are computed as explained in Chapter III. Finally the needed derivatives,

u . u , v , v , P , Px* y* x y* x ’ y

are computed.


68

Subroutine DERIV (K)

The purpose of this subroutine is to compute the

Jacobian determinant and the first and second derivatives

of the shape functions with respect to x and y. These quantities are evaluated at the Gauss point defined by K. After determining the derivatives of x and y with respect

to A and B, the Jacobian determinant is computed. The first derivatives are computed as described in Chapter III. For the pressure equation the second derivatives are needed.

Second derivatives of x and y with respect to A and B are computed. The inverse of the transformation matrix, needed

to determine the derivatives with respect to x and y as discussed in Chapter III, is evaluated at gauss point K. Having the transformation matrix, the second derivatives of

the shape functions with respect to x and y are computed.

Subroutine APPROX (K)

Here the function \p, whether it be f, g, or h as discussed in Chapter V, is evaluated at the Gauss point K.First the velocities u and v are evaluated at that point.For each equation the needed factors are computed and the

functions defined in Chapter V are evaluated.


69

Test Program

Since method 2 was unsuccessful, it is necessary to

test the theory and programming for errors. This program

solves a Poisson equation by the finite element method, using linear triangles in one solution and eight node quadratic elements in another solution. Eight node ele

ments are used to solve,

V 2<|> = ip (x,y),

where is assumed to be a constant, C, in one solution,

and a function analytically equivalent to C in another

solution,

*(x,y) * J ? (I xZ) + 575y (§ xy) * T ? (I y!] •

This function is employed as a means of checking the p r o gram’s ability to compute second derivatives with respect to x and y. The quadratic element should be capable of

computing the derivatives of this quadratic function exactly. Also, linear triangles are used for two solutions.

A Gauss elimination scheme is employed to solve the algebraic equations for a solution. Another solution is o b tained by Gauss-Seidel iteration. Additionally, the eigen

values of an amplification matrix are checked.Some of the subroutines have been explained above.

ONCE and INTERP correspond closely to those in method 2.

Subroutine DERIV computes the derivatives exactly as done

70

previously.

Main Program

The main program generates the finite element grid,

computes the analytical solution, and calls the subroutines which do the calculations for the finite element analysis. First, the x and y coordinates of the nodes are generated. Next, the analytical solution is computed at each node and stored in AN. The eight node quadrilaterals

are defined by reading the array NODE, and subroutine ONCE

is called to calculate the interpolating functions and

their derivatives at the Gauss points. Forming the algebraic equations is accomplished by calling subroutine

MATRIX once for each element. Subroutine MODIFY is used to

impose the boundary conditions, after which the equations

are solved by a standard Gauss elimination method in subroutine SOLEL. The equations are formed again using the

alternative formulation for ip described above, which is chosen by the flag II. Next, linear triangles are used to attack the problem. The elements are redefined by reading new values for the array NODE. Subroutine MATR is called once for each element to form the equations. Boundary

conditions are imposed as before by calling subroutine MODIFY. This set of equations is solved once by Gauss elimination, then recalculated and solved by Gauss-Seidel iteration. Finally, the results are printed. Solution values obtained by each method are printed along with the


71

eigenvalues of the amplification matrix.

Subroutine MATRIX (KK)

In this subroutine the element matrix H and vector F

for the eight node quadratic element KK is formed and introduced into the overall matrix COEF and vector CONST. First, the nodal coordinates of the element are defined. Subroutine DERIV is called to compute the derivatives of the shape functions with respect to x and y and the Jacobian determinant, which is the area integrand A I . II is

checked to determine which form of the function PSI to

calculate. Having these quantities, the integrands for the matrix, HI, and vector, FI, are evaluated. A Gaussian

quadrature integration yields the element area A, and the components of the matrix H and vector F. These components

are added to the matrix COEF and vector CONST at the appro

priate location.

Subroutine MATR (KK)

Here the element matrix H and vector F for the three

node linear .triangular element KK is evaluated and added to the global matrix COEF and vector CONST. After the

nodal coordinates are defined, the needed constants a .^ , b^, c^, i=l,2,3, area, and coordinates of the centroid may be computed. The matrix and vector components are calculated as explained in Chapter V and Appendix C and introduced

into the matrix COEF and vector CONST.


72

Subroutine MODIFY (ROW, VALUE)

The procedure explained in method 1 is applied

directly since the matrix is stored in standard order for

this program.

Subroutine SOLEL

This subroutine computes the solution of a set of

linear simultaneous equations by a standard Gauss elimina

tion scheme.

Subroutine SPLIT (ITMAX, EPS)

The solution of the algebraic equations is obtained by Gauss-Seidel iteration. ITMAX is the maximum number of

iterations allowed, and EPS is the measure of closeness

needed for convergence.

Comment

The author recognizes that some inefficiencies exist in the programming. However, since the time and storage

requirements are not excessive for any of the programs,

these inefficiencies are tolerated for the sake of clarity.


73

CHAPTER VII

NUMERICAL EXPERIMENTATION AND RESULTS

Immediate success is not achieved with a direct a p

plication of the previously introduced methods. Some limited success results after many variations of the original programs are tested. Although not all of the numerical experiments are worth mentioning, most of the major modifications, with comments on the results obtained, are ex

plained below.Although each subroutine is tested before being in

corporated into the major program, a check is needed for

the complete program. An obvious test is to assume a known solution and perform the iteration to determine if the solution is a fixed point. A convenient test problem is

obtained by imposing the boundary condition,

r = R(0) : u = 1

The solution to this hypothetical problem is the "free stream" flow,

u = l , v = 0 , P = 0 ,

throughout the domain. To show that the solution is a fixed point, this convenient field is assumed and the iteration method is applied. If the same results are


74

obtained in one step, the solution is a fixed point. To

investigate if the solution is a point of attraction, i.e.

if a "close" guess will converge to the free stream, the

free stream is perturbed, and used as an initial guess. An

example of a possible perturbation is

where r is the radial coordinate, r > l . If the initial guess is corrected by the iteration scheme, the free stream

solution is a point of attraction for this problem.After performing these two simple tests the "real"

problem is attempted. Imposing the realistic boundary

condition,

r = R (0) : u = 0 ,

results in a physically meaningful problem. The Oseen approximation discussed in Chapter II is the initial guess for a very low Reynolds number. Applying the iteration scheme until convergence is achieved results in a corrected

velocity and pressure field, which is the initial guess for another problem at a slightly higher Reynolds number. It is hoped that problems of successively higher Reynolds

numbers can be solved in this fashion.


75

M ethod 1

Linear Elements

After considerable experimentation a computer program

which employs method 1 has been constructed such that the

results agree with known solutions for test problems, but the program's applicability is restricted. D. L. Garrett attempts solutions using simple linear triangles with little success [16]. Although the method successfully

solves the two test problems and does converge for the real problem, the solution obviously is incorrect. Subsequent work reveals a more reasonable solution is obtained with a

five node quadrilateral, which has an internal node that is eliminated, see Figure 7. A simple analysis of the algebraic equations explains the improvement. An inspection of

the components of the coefficient matrix reveals that zeros

are positioned near the diagonal in the equations corre

sponding to u and v, and are located oil the diagonal in every third equation, those corresponding to P. The lack of diagonal dominance in the matrix signifies an unstable

set of equations. As indicated in Chapter III, eliminating the internal node modifies the ill-conditioned matrix. Fortunately this modification includes placing non-zero

numbers on the diagonal of the P equations. Thus, since the zeros no longer exist after node elimination and the solution obtained is an improvement of the linear triangle solution, it seems that the elimination of internal nodes


76

not only reduces the subsequent calculations required, but

also improves a matrix which lacks diagonal dominance. In

fact, this particular property has been investigated at

some length. For at least five types of elements with in

ternal nodes, the diagonal term is compared to the sum ofthe off-diagonal terms in that row before and after internal nodes are eliminated. In each case the diagonal dominance

improves after the elimination of nodes. The ratio of the number of internal nodes to the number of boundary nodes seems to have a direct effect on the improvement of the

matrix. As the ratio increases, the diagonal dominance

becomes stronger.In an effort to benefit from the advantages of node

elimination, solutions of the flow past a circular cylinder

are attempted with many different elements possessing in

ternal nodes and composed of linear triangles. Three that are worthy of discussion are the five node quadrilateral mentioned above and shown in Figure 7, the nine node quadrilateral in Figure 6, for which a program appears in A p pendix D, and the thirteen node quadrilateral in Figure 7.

As expected, the nine node quadrilateral produces thematrix with the strongest diagonal dominance of the three elements, and the five node element matrix has the weakest diagonal dominance. As a result, the program employing the

nine node quadrilateral not only has the fastest rate of convergence, but also produces the smoothest and best ap

proximating flow field as compared to experimental and


77

other numerical work. Also, the program utilizing the five

node element is the slowest and yields the worst approxima

tion. A serious restriction on all of the elements a t

tempted is that the convergence rate decreases as the Rey

nolds number increases, and divergence is encountered before the Reynolds number reaches two. However, for small Reynolds numbers the results compare favorably with previous work. A plot of the pressure on the body versus the angle from the upstream axis is presented in Figure 10. Plotted are the results of the nine node element described

above at Reynolds numbers of 0.5 and 1.0, solutions by Keller and Takami at R£ = 2 [30], and Dennis and Shimshoni at R£ = 1 [10]. These plots show that the finite element

method curves are not only of the correct shape, but the variation with the Reynolds number is in the right direc

tion. Additionally the two curves for R£ = 1.0 are nearly

coincident, indicating excellent agreement with published results. A complete listing of velocities and pressures is

presented in Figure 13.In view of the accurate results obtained for flow

past a circular cylinder at low Reynolds numbers, it is

reasonable to have confidence in the solution obtained for "creeping" flow past a body of arbitrary cross section.The velocity and pressure fields may be computed for any

symmetric body by inputing the proper body radius at each angle to form the surface R(6). The ability to analyze an

arbitrarily shaped symmetric body is felt to be a


78

significant contribution. Employing the same program used for the circular cylinder, the flow past a lens shaped

cylinder, shown in Figure 11, is analyzed. The results of

the computation are the velocities and pressures of Figure

16, and the plot of the pressure distribution on the body

surface in Figure 12.

Non-Linear Elements and Other Experiments

Solutions are attempted with non-linear isoparametric

elements without success. The eight node quadratic element in Figure 4 is the first element employed. Because of the

nature of the matrix produced by this element, division by

zero is attempted in the Gauss elimination solution. This difficulty is easily overcome by utilizing the nine node element in Figure 8. Eliminating the internal node mod i fies the matrix and thus allows the calculations to proceed without attempts to divide by zero. Performing the tests

mentioned above shows the free stream solution to be a fixed point but not a point of attraction. For a close guess at the free stream solution, the iteration corrects

only the velocities. Although the pressure is changed from the initial guess, the final pressure is incorrect. Simi

lar results are obtained with the thirteen and twenty-one node quadrilaterals in Figure 9, which are developed from eight node quadrilaterals with the internal nodes eliminated .


79

After experimenting with many elements, it is clear

that the quadrilaterals composed of linear triangles and

possessing internal nodes produce the best results. In an

attempt to expand the range of Reynolds numbers for which

convergence is achieved, many modifications are investi

gated. A relaxation factor for the new guess at each step of the iteration improves convergence slightly. Feeling that the solution method may "overshoot" the result, each

new guess is computed using some fraction of the calculated correction. Constant relaxation factors, and step func

tions depending on the rate of convergence are attempted. The result is a slight improvement in convergence, but the range of Reynolds numbers is not increased significantly. Various boundary conditions are tested to improve the

answers or increase the region of convergence, neither of

which occurs. As discussed previously, there are certain

natural boundary conditions which result from the finite

element formulation. If these conditions are allowed to

exist on the external boundary of the domain rather than impose the forced conditions the convergence is improved

slightly, as is predicted by Forsythe and Wasow [15] , but the final result is essentially unchanged. The same result is obtained when the forced conditions are imposed on the

velocities while the natural boundary condition on the

pressure at the external boundary is permitted.Additional experiments include another means of

eliminating the zero diagonal term. As a substitute for


80

the continuity equation consider

un+1 + v n+1 + C Pn+1 = C Pn x y

where C is an arbitrarily specified constant. It is clear

that as |Pn - Pn + 1 | -+■ 0, this equation approximates the continuity equation. The finite element formulation using

this equation allows a non-zero coefficient of P in the third equation at each node, which eliminates the zero on the diagonal. However, after several tests it appears that

the optimum value of C is zero, thus the technique is dis

carded.To investigate the iteration properties of the over

all coefficient matrix, the eigenvalues of an amplification matrix are computed. The results seem to indicate that

convergence is not likely. The test is discussed in detail

in the next section.

Method 2

Since many numerical experiments achieve only limited

success, and since node elimination does not sufficiently improve the ill-conditioned coefficient matrix,a new formulation of the equations is sought. The result is method 2 presented in Chapter IV. Although possessing distinct advantages as compared to method 1, this method fails to converge under any conditions tested. The free stream solution is shown to be neither a fixed point nor a point of


81

attraction. Figure 14 shows the divergence from the free

stream solution. Linear triangles, five node quadri

laterals composed of four triangles, and eight node quad

ratic elements are all employed without success.A test program is written to check the important sub

routines. A Poisson equation is solved with linear triangles and eight node isoparametric elements for the values of the unknown at the nodes, which are compared to the values obtained by an analytical solution. Using the same number of nodes for each solution, and therefore more triangles than quadrilaterals, the results indicate the nonlinear quadrilateral element is the more accurate. To in

vestigate the iteration properties of the matrix an ampli

fication matrix is formed and its eigenvalues are computed.

Consider the system of equations,

[A]{x> = (B>

where [A] and {B> contain known constants, and {x} is a vector of unknowns. Writing [A] as the sum of a diagonal matrix [D] and another matrix [C] the system of equations

can be written,

[D ]{x } = {B} - [C]{x} .

After solving this equation for {x}, the Gauss-Seidel

iteration scheme may be introduced,

{xn+1} = - [D] ” 1 [C] {xn }


If the eigenvalues of the amplification matrix [D] [C]

have magnitudes less than one, then convergence is guaran

teed. The eigenvalues of this matrix are computed in the

test program for a rectangular domain. The magnitudes of

the eigenvalues are all less than one with linear triangles, and all but one are less than unity with parabolic quadrilaterals. Shown in Figure 15 are the analytical solution, and two solutions each with triangles and quadrilaterals.

Eight node quadrilaterals are used to solve

V2 <b = C ,

where C is a constant, for which the analytical result is

printed. Additionally, to check the program's ability to compute second derivatives, the quadrilaterals are employed

to solve an analytically identical equation,

The results indicate both solutions are identical and that they agree with analytical results to approximately three digits. The matrix formed with linear triangles is solved by Gauss elimination once and Gauss-Seidel iteration a second time. As expected, since the eigenvalues of the amplification matrix have magnitudes less than one, the

iteration scheme converges to the same values obtained by elimination, which correspond to the analytical solution to

approximately two digits. Although accurate solutions of

83

test problems are only necessary and not sufficient condi

tions of correctness, it is felt that the test program

described here indicates that the subroutines of the pro

gram for method 2 are correct.A test of the iteration properties of the matrix

formed by method 2 is performed in the same manner as that for the test program. Since the equation,

[A]{x} = (B)

formed by the iteration scheme of method 2, has {B> depend

ent on {x}, convergence cannot be guaranteed easily. How

ever, if {x} is close to the final solution, the vector {B} can be approximated as a constant provided its varia

tion with {x} is continuous. Thus, some indication of the

convergence properties can be obtained by computing the

eigenvalues of the amplification matrix. As might be expected, since the method does not converge, the eigenvalues of the amplification matrix for the problems tested do not all have magnitudes less than one. Additionally, the eigen

values of the amplification matrix for method 1 are com

puted with the same result, which likewise is as expected.Another test program for method 2 uses eight node

isoparametric elements to attack the classical Couette and Poiseuille flows. Consider the flow of a viscous fluid

between two plates, one stationary and one moving at a velocity U. The fluid velocity may be calculated analyti

cally as a function of the distance from the bottom plate


84

in terms of the velocity of the moving plate and the pressure gradient, which is assumed constant. By fixing the

pressure at each point in the domain the pressure gradient

is imposed, and the values of the velocity at the moving

plate are fixed as a boundary condition. As might be e x

pected for this linear problem, since no iteration is needed, the program computes the correct velocity field.

It is worth noting, however, that the application of the method to corner flow, employing eight node parabolic elements, and allowing unprescribed pressure and non-linear

velocity terms, is not successful, and that even this slight modification of geometry produces an amplification

matrix with several eigenvalues having magnitudes greater

than unity.


CHAPTER V I I I

CONCLUSIONS AND RECOMMENDATIONS

Although the results of the present work are restricted, several conclusions may be drawn. It is clear that certain viscous flow problems may be analyzed by the

finite element method. The simplified flows which result in linear equations are solved easily. However, attacking non-linear problems with an iteration scheme is not easily

accomplished, and the results may not be accurate. Two iteration formulations are presented in the present paper and the results are discussed below. For each method addi

tional work is needed to further investigate the convergence properties of the iteration. Clearly, alternative formulations are possible for which the convergence proper

ties may be better than those of the methods presented in

this paper. One possibility is the direct formulation of the non-linear governing equations, which requires the solution of a non-linear system of algebraic equations.

This was considered but abandoned due to the lack of truly dependable schemes for solving such systems.

85


86

M ethod 1

Since some success has been achieved for small Rey

nolds numbers, it may be concluded that method 1 is appli

cable to "creeping flow" past a body of arbitrary cross section. From the numerical experiments performed, it is obvious that the pressure is the most difficult of the variables to calculate accurately. Although this fact causes some problems, it does demonstrate a correlation with other methods. Many experimental and numerical investigators claim their techniques are pressure sensitive. The major difficulty with method 1 is the instability of the algebraic equations. This instability is caused by the inherent zero on the diagonal in every third equation,

corresponding to the pressure equation.Some additional work with method 1 might prove worth

while. Of most value is the investigation of possibilities

to improve the diagonal dominance of the algebraic equations. Some methods such as node elimination and approximating the continuity equation are described in the pr e vious chapter; others might exist. Another important problem is the dependence of the rate of convergence on the Reynolds number, a difficulty encountered by other numerical methods also. Further investigation of this dependence

may be of value.


87

M ethod 2

Since little success is achieved with method 2 very

few conclusions are possible. Solving Poisson's equation by the finite element method is nothing new, but the test program for method 2 does illustrate the applicability of

the subroutines. The correct solution of certain test problems further demonstrates the abilities of method 2. However, for the flow past a cylinder divergence is encountered for both the free stream and the realistic boundary conditions. In fact, the iteration scheme does

not converge for any test problems attempted.

In view of these results it is clear that additional work is needed to investigate the cause of the divergence.

Test problems should be devised which would test the spe

cific parts of the method separately. Further analysis of

the eigenvalues of the amplification matrix may reveal use

ful information. To avoid any problems inherent in the finite element method, finite difference schemes may be used to attack the Poisson equation of method 2, which, of

course, restricts the test problems to "nice" domains.


FIGURES


89

M i t t


Figure

1.

Phys

ical

Pr

oble

m,

Circ

ular

Cy

lind

er

9 0


Figure

2.

Linear

Rect

angl

e

91


Figure

3. Qu

adra

tic

Rect

angl

e

92

x


Figure

4. Qu

adra

tic

Isop

aram

etri

c Qu

adri

late

ral

93

x


Figure

5. Or

ient

atio

n of

Tria

ngul

ar

Elem

ent

9 4


Figure

6. Qu

adri

late

ral

with

Twelve

Tr

iang

les

95

/ \ / \/ \ I / \

/ \°\ / \ ^ ---------------- - *

\>o / ‘ \ o /X 1 *

✓ I / x/ \ 1 / \/ \ I / \

" V ---------------^ -------------- *


Figu

re

96


Figure

8.

Nine

Node

Quad

rila

tera

l

9 7

pH

OO


Figu

re

9. Ex

peri

ment

al

Elem

ents

Non-

Dime

nsio

nal

Pres

sure

98

3

Wooley Keller and Takami- — Dennis and

Shimshoni

3

2

1

Angle from Upstream Stagnation, degrees

30 60 90 120 150 180

Re=2

1

Re=l

2

Re=0,5

3

Figure 10. Circular Cylinder, Surface Pressure Distribution


99

I t t I I


Figu

re

11.

Phys

ical

Pr

oble

m,

Lens

Shaped

Cyli

nder

Non-

Dime

nsio

nal

Pres

sure

100

6

0

0

0

Angle from Upstream Stagnation, degrees

0 1801501209030

. 0 R e= l .0

0R e = 0 .5

0

Figure 12. Lens Shaped Cylinder, Surface Pressure Distribution


Figure 13. Method 1 Results for a Circular Cylinder


NO* OF ELEMENTS ON THE UPSTREAM AXIS = 8NO* OF ELEMENTS ON THE DOWNSTREAM AXIS = flNO* OF ELEMENTS IN THE TRANSVERSE DIRECTION = 8EACH ELEMENT HAS 2 POINTS IN THE RADIAL DIRECTION

AND 2 POINTS IN THE TRANSVERSE DIRECTION

TOTAL NUMBER OF ELEMENTS = 64TOTAL NUMBER OF POINTS = 81MATRIX BANDWIDTH = 33MAXIMUM RADIUS ON THE DOWNSTREAM AXIS = 12*00

REYNOLDS NUMBER = 0*50* * 4 * * 4 4 4 4 * « * * * 4 4 4 4 4 4 4 4 4 4 4 4 4 4

ITERATION NUMBER 1 SUM = 1.3E+01 BIG = 1.2E+00

ITERATION NUMBER 2 SUM = 2* 8E + 00 BIG = 6.6E-01

ITERATION NUMBER 3 SUM = 6.4E-01 BIG = 3*1E—C 1

ITERATION NUMBER 4 SUM = 1.4E-01 BIG = 1*3E—01

ITERATION NUMBER 5 SUM = 2* 7E—02 BIG = 5.9E-02

ITERATION NUMBER 6 SUM = 4* 9E—03 BIG = 2* 7E—02

ITERAT ION NUMBER 7 SUM = 3.0E-04 BIG 6.7E-03

ITERATION NUMBER 8 SUM = 2.3E-05 BIG = 1 * 6E—03

ITERATION NUMBER 9 SUM = 2* 6E—06 BIG = 5*1E—04


103

* * * * * * * * * * * * * *

I N I 1 I A L G U E S S

NODE R T U-VELOCITY V-VELOCITY PRESSURE

1 loOG 0*0 0*0 0*0 1*9985222 2* 50 0*0 0*247962 0*0 0*7994093 4*00 0*0 0*458433 0* 0 0 a 4996304 5« 50 0*0 0*610186 0*0 0*3633685 7*00 0*0 0*727518 Oa 0 Go 2855036 8* 50 0*0 0*822884 0*0 0*2351207 10*00 0*0 0*903124 Oo 0 0.1998528 11.50 0*0 0*972344 0*0 0*1737849 1 3m OC 0*0 1*033189 0*0 0*153732

10 1*00 22*50 >0*000000 0.0 1 .8463941 1 2*50 22*50 0*309424 0* 148383 0*73855812 4.00 22*50 0*527028 0*165606 0*46159813 5*50 22*50 0*680937 0. 170806 0.33570814 7* 00 22*50 0*799194 0*173041 Oa 26377115 8* 50 22*50 0*895040 0*174201 0*21722316 10.C0 22*50 0*975561 0*174879 0*18463917 11*50 22*50 1*044960 0*175310 Oo16055618 13*00 22*50 1* 105926 0*175601 C*14203C19 1*00 4 5*00 0*0 0*0 1*41316820 2*50 45*00 0*457806 0*209845 0.56526821 4*00 45*00 0*692634 0*234202 0*35329222 5*50 45*00 0*851743 0*241557 0*25694023 7*00 45*00 0*972235 0*244717 0*20188124 8.50 45*00 1*069242 0.246358 0*16625525 10*00 45*00 1«150441 0*247317 0.14131726 11*50 45*00 1*220270 0*247926 Oo 12288427 13*00 45*00 1*281526 0*248337 0.10870528 1*00 67*50 -OoCOOOOC 0*0 0*76480229 2*50 67*50 C*606189 0* 148383 0*30592130 4*00 67*50 0*858239 O o 165606 0* 19120G31 5*50 67*50 1 * G22549 O o 170807 0.13905532 7*00 67*50 1*145276 0* 173041 0*10925733 8*50 67*50 1*243443 0*174201 0.08997734 10*00 67*50 1*325320 0*174880 0*07648035 1 1.50 67*50 1*395580 0*175310 0*06650436 13*00 67*50 1*457127 0*175601 0.05883137 1*00 90*00 0*C 0*0 0*00000138 2*50 90*00 0*667651 0*000000 0*00000039 4*00 90*00 0*926837 0 3 000000 0*00000040 5*50 90*00 1*093300 0*000000 0 a 00000 ;41 7* JO 90*00 1.216952 o*oooooc 0.00000042 8*50 90* 00 1*315599 0*000000 0*00000043 10.00 90*00 1*397758 OoOOOCCO O.OwOOOG


1 0 4

44 1 1*50 90*00 1*468196 0,00000045 13*00 90*00 1*529862 0*00000046 »CO 112*50 -0*000000 0*047 '2*50 112*50 0*606189 -0*14838248 4*00 112*50 0*858240 -0*16560549 5*50 1 12*50 1*022550 -0* 17080650 7*00 112*50 1*145276 -0*17304151 8*50 112.50 1*243443 -0*17420152 10*00 112*50 1*325320 -0*17487953 11*50 112*50 1*395580 -0*17531054 13*00 112*50 1.457127 -0*17560155 1*0C 135.00 0*0 0*056 2*50 135*00 0*457807 -0*20984557 4*00 135*00 0*692635 -0*23420258 5*50 135*00 0*851744 -0*24155759 7*00 135.00 0*972236 -0*24471760 8*50 135*00 1*069242 -0*24635861 10*00 135*00 lo 150441 -0*24731762 11*50 135*00 1*220270 -0*24792663 13*00 135*00 1*281527 -0*24833764 1*00 157*50 0*0 0*065 2*50 157*50 0*309424 -0*14838366 4*00 157*50 0*527029 -0*16560667 5*50 157*50 0*680937 -0*17080768 7*00 157*50 0*799194 -0*17304169 8*50 157*50 0*895041 -0* 17420170 10.OC 157*50 0*975562 -0*17488071 1 1*50 157*50 1*044960 -0.17531072 13*00 157*50 1*105926 -0*17560173 1*00 180*00 Oo 0 0*074 2*50 180*00 0*247962 -0*00000075 4*00 180*00 0*458433 -0*00000076 5.50 180.00 Os 610186 -0*00000077 7.00 180*00 0*727518 -Oo00000078 8*50 180*00 0*822884 -0*00000079 10*00 180*00 0*903124 -0.00000080 1 1*50 180*00 0*972344 -0*00000081 13*00 180*00 1« 033189 -0*000000

0*000000 0* 000000

-0*764801 -0*305920 -Oo191200 -Oo139055 -0*109257 — Oo 089977 -0*076480 -0*066504 -0*058831 -1*413167 -0*565267 -0* 353292 -0*256940 -0*201881 -0*166255 -0*141317 -0*122884 -0*108705 -1*846393 -0*738557 -0*461598 -0*335708 -0*263771 -0*217223 -0*184639 -0*160556 -0*14203U -1*998522 -0*799409 -0*499630 -0*363368 —Oo 285503 -0*235120 -0*199852 -0*173784 -0*153732


1 0 5

* * * * * * * * * * *

C O R R * * * * * * * * * * *

NODE R T

1 1*0C 0*02 2*50 0*03 4*00 0*04 5*50 0*05 7*00 0*06 8* 5C 0*07 10*00 0*08 1 1*50 0*09 13*00 0*0

10 1*00 22*501 1 2*50 22*5012 4*00 22*5013 5*50 22.5014 7*00 22*5015 8*50 22*5016 10*00 22*5017 1 1*50 22.501 8 13*00 22*5019 1*00 45.0020 2*50 45*0021 4*00 45*0022 5*50 45*0023 7*00 45.0024 8*50 45*0025 10*00 45* CO26 1 1*50 45.0027 13*00 45*0028 1*00 67*5029 2*50 67*5030 4*00 67*5031 5*50 67*5032 7.00 67*5033 8*50 67*5034 10*00 67*5035 1 1*50 67.5036 13.00 67.5037 1*00 90*C038 2*50 90*0039 4*00 90*0040 5*50 90*0041 7*00 90*0042 8*50 90*0043 10*00 90.00

* * * * * * *

E C T E D V * * * * * * *

U-VELOCITY

OoO0*2387150*5783100*6992800*8175250*8561710*9195510*8940741*0000000*00*330065 0*639233 0® 754640 0*855151 0*887490 0*938928 0*912962 1*000000 0*00*541212 0*798829 0*896236 0*960751 0*975066 0*994818 Oo962894 1 *0 0 0 0 0 0 0*00*7267280*9635821*0555421*0918241*0891051*0727331*0290381*000000OoO0*767684 1*028362 1*140931 1* 182537 1*180554 1*145391

A L U E S* * * * 4 * 4 * 4 4 4

V—VELOCITY PRESSURE

0.0 3a 4275370*0 0*7568030*0 I.G48279OaO Oo 644066OoO 0.6217840*0 0*6334650*0 0*4193190*0 0.7985270.0 0.00*0 2*9696140*236029 0*8162330*196858 0*9225480*166030 0*6196370* 121073 0.5667720*095211 0.5966750*058221 0*3845100*053276 0*7510420.0 0.00*0 1.8489390*343145 0.482556Oo 304492 0*6046070*263460 0*4347710*197212 0*4127070*155378 0.4620770*095659 0.3013850*085555 0*6117600*0 0.00*0 0*426651Co 254439 0*1625860*266090 0*1902910*239893 0*2048060*190052 0*1978070*150807 0.2706510*094352 0*1750290*081224 0*4016750*0 0.00*0 -0*8747790*048774 -0.2063390*102461 -0.2272620*099275 -0.0618960*086555 -0*0441100*066251 0.0404880*039209 0.025714


1 0 6

44 1 1.50 90®00 1*087893 0*031847 0*14059545 1 3*00 90®00 1*000000 0*0 0*046 1*00 112*50 0*0 0*0 — 1 o 7875 3947 2® 50 112*50 0*637275 -0*1444 12 —0 #47127946 4.00 112®50 0*929500 -0*094197 -0*54098949 5*50 112.5C 1*074121 -0*092574 —0 u28552550 7® CO 112*50 1*154837 -0*077753 -0*25708751 8® 50 112*50 1*180161 -0*079499 -Oo19106752 10.CC 112*50 1*164422 -0*066174 -0*12667053 11®50 112*50 1*104448 -0*058262 -Oo15772154 13.00 112® 50 1*000000 0*0 OoO55 1®00 135.00 OoO 0*0 —2a 26234056 2.50 135*00 0*411967 -0*219002 -0*70088957 4® 00 135*00 0® 699143 -0*204342 -0*73688258 5® 50 135*00 0*854686 —Oo 216874 -0*46181959 7® CO 135*00 0*976403 -0*202990 —C o 4140 9360 8*50 135*00 1*037135 -0*203819 -0*40440661 1 Oo 00 135* 00 1*076896 -0*167245 -0*26112262 1 1 *50 135*00 1*034815 -0*147553 -0*48873963 1 3®00 135*00 loCOOOOO 0*0 0*064 1®00 157*50 0*0 0*0 -2*44643665 2.50 157*50 0*214047 -0*159014 -0*76802966 4® 0 3 157*50 0*464007 -0*162683 -0*81321167 5® 50 157*50 0*601398 -0*180820 -0*54428968 7®00 157.50 0*738435 —0 a 177693 -0*49272569 8® 50 157*50 0*814904 -0*184113 -0*55331870 10*00 157*50 0*911680 -0*155388 -0*35578371 1 1«5C 157o 50 0*893592 -0*140470 -0*78774972 13®00 157*50 1*000000 0*0 0*073 l.CC 180*00 0*0 0*0 —2* 45725074 2® 50 180*00 0*136777 c.o -0*93118375 4® CO 180*00 0*362364 0.0 -0*83102176 5.50 180*00 0*488066 0*0 -0.60370577 7®00 180*00 0*624075 0*0 -0*51315178 8® 50 180*00 0*701398 0*0 -0*61904279 10®00 180*00 0*821838 0*0 -3*38145580 1 1 • 50 180*00 0*814795 0*0 -0*92120681 13®00 180*00 1*000000 0*0 0*0


1 0 7

NOiu OF ELEMENTS ON THE DOWNSTREAM AXIS = 8

NO. OF ELEMENTS IN THE TRANSVERSE DIRECTION = 8

EACH ELEMENT HAS 2 POINTS IN THE RADIAL DIRECTION AND 2 POINTS IN THE TRANSVERSE DIRECTION

TOTAL NUMBER OF ELEMENTS = 64TOTAL NUMBER OF POINTS = 81MATRIX BANDWIDTH = 33

MAXIMUM RADIUS ON THE DOWNSTREAM AXIS = 12.00

REYNOLDS NUMBER = loOO* * 4 * * 4 4 4 *e 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

ITERAT ION NUMBER 1 SUM = 1.4E+01 BIG = 1 •4E +00ITERAT ION NUMBER 2 SUM = 1.5E+00 BIG = 4.8E-01ITERAT ION NUMBER 3 SUM = 2.2E-01 BIG = 2.CE-01

ITERAT ION NUMBER 4 SUM = 7.4E-02 BIG = 1»1E-C1ITERAT ION NUMBER 5 SUM = 3.0E-D2 BIG = 6.2E-02

ITERAT ION NUMBER 6 SUM = 8.2E-03 BIG = 2.9E-02


ITERAT ION NUMBER 8 SUM = 3.1E—03 BIG = 1.4E-02


ITERAT ION NUMBER 10 SUM = 1* 5E—03 BIG = 7.7E-03

ITERAT ION NUMBER 1 1 SUM = 9.6E-04 BIG = 7.6E-03ITERAT ION NUMBER 12 SUM = 7.1E-04 BIG = 7.5E-03ITERAT ION NUMBER 13 SUM = 4.6E-04 BIG = 5.8E-03ITERAT ION NUMBER 14 SUM = 3.3E-04 BIG = 5.5E-03ITERATION NUMBER 15 SUM = 2.2E-04 BIG = 3.4E-03


1 0 8

I N I T I A L G U E S S* * * * * * * * * * * * * * * * * * * * * * *

NODE R T U-VELOCITY V-VELOCIT

1 1*00 0*0 0*0 0*02 2*50 0*0 0*238715 0*03 4*00 0*3 0*578310 0*04 5*50 0*0 0*699280 0*05 7*00 0*0 0*817525 0*06 8*50 0*0 0*856171 0*07 1C.00 0*0 0*919551 OoO8 11*50 0*0 0*894074 0*09 13*00 0*C loOOOOOO 0*0

10 1*00 22*50 0*0 0*01 1 2*50 22*50 0.330065 0*23602912 4*00 22*50 0*639233 0*19685813 5*50 22*50 0*754640 0*16603014 7.00 22.50 0*855151 0*12107315 8*50 22*50 0*887490 0*09521116 10*00 22*50 0*938928 0*05822117 11*50 22*50 0*912962 0*05327618 13*00 22*50 1*000000 0*019 1*00 45*00 0*0 0*020 2*50 45*00 0*541212 0*34314521 4.00 45*00 0*798829 0*30449222 5*50 45*00 0*896236 0*26346023 7*00 45.00 0*960751 0*19721224 8*50 45*00 0*975066 0*15537825 10*00 45*00 0*994818 0*09565926 11*50 45*00 0*962894 0.08555527 13*00 45.00 1*000000 0*028 1*00 67. 50 0.0 0.029 2*50 67.50 0*726728 0*25443930 4.00 67.50 0*963582 0.26609031 5.50 67.50 1*055542 0*23989332 7*00 67*50 1*091824 0*19005233 8.5C 67.50 1*089105 0* 15080734 10*00 67o 59 1.C72733 0*09435235 1 1*50 67o 50 1*029038 0*08122436 13*00 67.50 1*000000 0*037 1*CC 90*00 C.O 0*038 2*50 90.00 0*767684 G.04877439 4,00 90*00 1*028362 0*10246140 5*50 90o 00 1*140931 0*09927541 7*30 90.00 1*182537 0*08655542 8.50 90.00 1*180554 0*06625143 10*00 90*00 1*145391 0*039209

* * * * *

PRESSURE

3*427537 0*756803 I.C48279 Oo 6440 66 0*621784 Oo 633465 0*419319 0*798527 0*02*969614 0*816233 Oo 922548 0*619637 0*566772 0.596675 0* 3845 1C 0*751042 0*01*848939 0* 402556 0*604607 0*434771 0*412707 0*462077 0*301385 0*611760 0*00*426651 0*162586 0*190291 0*204806 0 o 197807 0*270651 0*175029 0 a 4U1675 0-0

-0*874779 -G.2U6339 -0*227262 -0*061896 -0*0441 1C 0*040488 0.025714


10 9

4445464748495051525354555657585960 61 62636465666768697071727374757677787980 81

11*50 90*00 1*087893 0*03184713*00 90*00 1*000000 0*01*00 112*50 0*0 0*02*50 112*50 0*637275 -0*1444124*00 112*50 0* 9295C0 -0*0941975*50 112*50 1*074121 -0*0925747*00 112*50 1.154837 -0*0777538*50 112*50 1*180161 -0*079499

10*00 112*50 1*164422 -0*06617411*50 112*50 1*104448 -0*05826213*00 112*50 1*000000 0*01*00 135*00 0*0 0*02*50 135*00 0*411967 -0*2190024*00 135*00 0*699143 -0.2043425. 50 135*00 0*854686 -0*2168747*00 135*00 0*976403 -0.2029908* 5C 135*00 1*037135 -0*203819

10*00 135*00 1*076896 -0*16724511*50 135*00 1*034815 -0.14755313*00 135*00 1*000000 0*01*00 157*50 0*0 0*02*50 157o 50 0*214047 -0*1590144*00 157*50 0*464007 -0*1626835*50 157*50 0*601398 -0*1808207*00 157*50 0*738435 -0*1776938*50 157*50 0*814904 -0*1841 1010*00 157*50 0*911680 -0*15538811.50 157*50 0*893592 -0*14047013*00 157*50 1*000000 0*01*00 180*00 0*0 0*02*50 180*00 Q « 136777 0*04*00 180*00 0*362364 0*05*50 180*00 0*488066 0*07*00 180*00 0*624075 0*08*50 180*00 0*701398 OoO10*00 180*00 0*821838 0*011*50 180*90 0*814795 0*013*00 180*00 1*000000 0*0

0*140595 0*0

-1*787539 -0*471279 -0*540989 -0*285525 -0*257087 -0*191067 -0*126670 -0*157721 0*0

-2*262340 —0 *700889 -0*736882 -0*461819 -0*414093 -0*404406 -0*261122 -0*488739 0*0

-2*446436 -0*768029 -0*813211 -0*544289 -0*492725 -0*553318 -0>355783 -0*787749 0*0

-2*457250 -0*931183 -0*831021 -0*603705 -0*513151 -0*619042 -0*381455 -0*921206 0*0


110

* * * * * * * * * * * * * * * * * * * * * * * *

C O R R E C T E D V A L U E S

* * * * * * * * * * * * * * * * * * * * * * * *

NODE R T U-VELOCITY V—VELOCITY

1 1,00 0,0 0,0 0,02 2,50 0,0 Oo 278877 0,03 4,00 0,0 0,646295 0,04 5,50 G,0 0,744028 0,05 7,00 0,0 0,848741 0,06 8, 5C 0,0 0,872392 0,07 10,00 c«c 0,927806 0,08 11,50 0,0 0,897734 0,09 13,00 0,0 1,000000 0,0

10 1,00 22,50 0,0 0,01 1 2,50 22,50 0,379995 0,26947112 4,00 22,50 0,697080 0,18684613 5,50 22,50 0,787780 0,14061714 7,CO 22,50 0,873870 0,09280315 8,50 22,50 0,893797 0,07218816 10,00 22,50 0,940116 0,04209817 11,50 22,50 0,912299 0,04491518 13,00 22,50 1,000000 0,019 1,00 45,00 0,0 0,020 2,50 45,00 0,609804 0,39413421 4,00 45, 00 0,839310 0,30536422 5,50 45,00 Oo903977 0,23529923 7,00 45,00 0,951023 0,16008324 8,50 45,00 0,957173 0,12333725 10,00 45,00 0,978301 0,07263226 11 ,50 4 5,00 0,951178 0,07489127 13.00 45,00 1,000000 0,028 1,00 67,50 0,0 0,029 2,50 67,50 0,800719 0,30002730 4,00 67, 50 0,996977 0,29686931 5,5C 67,50 1,046824 0,23986832 7,C0 67,50 1,058240 0, 17446733 8,50 67,50 1,047517 0,13257434 10,00 67,50 1*037196 0,07940435 11,50 6 7,50 1,006077 0,07759336 1 3,C0 67,50 1,000000 0,037 1,00 90,00 0,0 0,038 2,50 90,00 0,821564 0,08177839 4,00 90,00 1,067266 0,16062440 5,50 90,00 1,143535 0,14375641 7,00 90,00 io 15534 7 0, 1 1751442 8,50 90,00 1,138138 0,08645643 10,00 90,00 lo104453 0,049835

* * * * *

* * * * *

PRESSURE2,065227 0 9 494804 0,642389 0,398218 0,369476 Oo 368589 0 o 240545 0,444579 0,01,728780 0,511581 0,561647 0,380870 0,337713 0,347668 0,221583 0,418852 0,0On 925208 0,260752 0,34865C 0,262383 0,244592 0,270004 0® 174767 0,345420 0,0

-0,028802 0,028135 0,073605 0, 117521 0,115658 C« 162632 0,105216 0,237312 0,0

-0,798446 -0,199662 -0,191848 -0,045585 -0,029213 0,036699 0,023939


I l l

44 11 • 50 90*00 1*061620 0*045006 0* 1100C245 13*00 90*00 1*000000 0*0 0*046 1*00 112*50 OoO 0*0 -1*21612647 2*50 112*50 0*654013 -0* 1154 76 -0*33570148 4*00 112*50 Oo 963830 -0*031451 -0*36730049 5*50 112*50 1*099769 -0*023908 —0*17556450 7*00 112*50 1*164628 -0*010893 -0*15355451 8*50 112*50 1*177155 -0*019817 -0*08423652 10,OC 112*50 1*155375 -0*025115 -0*05729253 11*50 112*50 1*104791 -0*024820 -0*01901354 13*00 112*50 1*000000 0*0 OoO55 1*00 135*00 0*0 0*0 -1*31463156 2*50 135*00 0*397901 -0*187539 -0*43964957 4*00 135*00 0*700616 -0*157705 -0*44847158 5*50 135*00 Oo866734 -0*164569 -0*27002259 7*00 135*00 0*993701 -0*150472 -0*23906860 8*50 135*00 1o 061584 -0*159409 -0*20043761 10*00 135*00 1*105559 -0*145450 -0*13203962 11*50 135*00 1*076474 -0*145861 -0*18264063 1 3*00 135*00 lo000000 0*0 0*064 1*00 157*50 0*0 0*0 -1*26544765 2*50 157*50 0*187030 -0*135462 -0*46387366 4*00 157*50 0*420572 -0*139704 -0*45539367 5*50 157*50 0*554895 -0*161336 —Oo 30992868 7*00 157*50 0*694034 -0*164623 -0*26942469 8*50 1 57* 50 0*776954 -0*188396 -0*29559070 10*00 157*50 0*892476 -0*180943 -0*18558271 11*50 157*50 0*881073 -0*195987 -0*41252772 13*00 157*50 1*000000 0.0 0*073 1*00 180*00 OoO 0*0 -1*20897174 2*50 180*00 0*197814 0*0 -0*54398875 4*00 180*00 0*297813 0*0 -0*45110976 5*50 180*00 0*405146 0*0 -0*33901077 7*00 180*00 0*532865 0*0 -0*27007678 8*50 180*00 0*603553 0*0 -0*34189679 10*00 180*00 0*746158 0*0 -0*19715680 11*50 180*00 0*732682 0*0 -0*54757981 13*00 180* CO 1*000000 0*0 0*0


Figure 14. Method 2 Results for a Circular Cylinder


113

* * * * * * * * * * * * * * * * * * * * * * * * * * * * *

NO* OF ELEMENTS ON THE UPSTREAM AXIS = 4

NO* OF ELEMENTS ON THE DOWNSTREAM AXIS = 4

NO, OF ELEMENTS IN THE TRANSVERSE DIRECTION = 6EACH ELEMENT HAS 3 POINTS IN THE RADIAL DIRECTION


TOTAL NUMBER OF ELEMENTS = 24

TOTAL NUMBER OF POINTS = 93MATRIX BANDWIDTH = 17MAXIMUM RADIUS ON THE DOWNSTREAM AXIS = 19*00

REYNOLDS NUMBER = 1*00 * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

EXACT DOMAIN AREA = 626*75 COMPUTED DOMAIN AREA = 626*65

ITERATION NUMBER IU-VELOCITY NORM = 3*7E-04 V—VELOCITY NORM = 0*0

PRESSURE NORM = l*9E-03

LARGEST TERM LARGEST TERM LARGEST TERM

1*1E—05 OoO3•6E—05

ITERATION NUMBER 2U-VELOCITY NORM = 3*2E-G3 V—VELOCITY NORM = 2.1E-03

PRESSURE NORM = 3»3E-G3


9.5E-C5 7* BE—05 9* 2E—05

ITERATION NUMBER 3U-VELOCITY NORM = 3*7E-04 V—VELOCITY NORM = 4«2E-04

PRESSURE NORM = 4*7E-02


1.8E-05 1*6E—05 1*86-03

ITERATION NUMBER 4U-VELOCITY NORM = 6.3E-02 V—VELOCITY NORM = 5*4E-02

PRESSURE NORM = 1*9E-01


1» 9E—03 1.9E-03 9* 5E—03

ITERATION NUMBER 5U-VELOCITY NORM = l*2E-0l V—VELOCITY NORM = 9*3E-02

PRESSURE NORM s 9.9E+00


6* 3E—03 4* 3E—03 3*9E—01

R eproduced w ith perm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.

1 1 4

* * * * * * ♦ * * * *

I N 1

* * * * * * *

T I A L G U

* * * * * *

1 E S S

* * * * *

♦ * * ♦ * * * * * * * * * * * * * * * * * * * * * * * * *


1 1*CC 0*0 1.000000 0*0 0*02 3*38 0*0 1*000000 0*0 0*03 5*75 0*0 loCOOOOO 0*0 0*04 8. 12 0*0 1*000000 0*0 0*05 10*50 0*0 1*000000 0*0 0*06 12*88 0*0 1*090000 0*0 0*07 15*25 c * o l.COOOOO 0*0 0*0a 17*63 0*0 1*000000 0*0 0*09 20*C0 0*0 1*000000 0*0 0*0

10 1«C0 15*00 1*000000 0*0 0*011 5 * 75 15a 00 1*000000 0.0 0*012 1 0* 5C 15.CO 1*000000 0*0 0*013 15*25 15*00 1*000000 0*0 0*014 20*00 15*00 1*000000 0*0 0*015 1.00 30*00 1*000000 0*0 0*016 3*38 30*00 1*000000 0*0 0*017 5.75 3C* 00 1*000000 0*0 0*018 8* 12 30* CO 1*000000 0*0 0*019 10*50 30*00 1*000 000 0*0 0*020 12*88 30*00 1*000000 0*0 0*021 15*25 30.00 1*000000 0*0 0*022 1 7 * 6 3 30*00 loOOOOOO 0*0 0*023 20*00 30*00 1*000000 0*0 0*024 1*00 45*00 1*000000 9*0 0*025 5 * 7 5 45*00 1*000000 0 * 0 0 * 026 10*50 45*00 1 * 0 0 0 0 0 0 0 * 0 0 * 027 15*25 45*00 1*000000 0 * 0 0 . 028 20*00 45* CO 1 * 0 0 0 0 0 0 0 * 0 0 * 029 1*00 6 0 * 0 0 1 * 0 0 0 0 0 0 0 * 0 0 * 030 3*38 60*00 1 * 0 0 0 0 0 0 0 * 0 0 * 031 5*75 60*00 1 * 0 0 0 0 0 0 0 * 0 0 * 03 2 8* 12 60*00 1 * 0 0 0 0 0 0 0 * 0 0 * 03 3 10*50 60.00 1 * 0 0 0 0 0 0 0 * 0 0 * 034 12*88 60*00 1 * 0 0 0 0 0 0 0 * 0 0 * 035 15*25 60*00 1*000000 0 * 0 0 * 036 17*63 60*00 1 * 0 0 0 0 0 0 0 * 0 0 * 037 20*00 60*00 1*000000 0 * 0 0 * 038 1*00 75*00 1 * 0 0 0 0 0 0 0 * 0 0 * 039 5.75 75.00 1 * 0 0 0 0 0 0 0 * 0 0 . 040 10*50 75.00 1 * 0 0 0 0 0 0 OoO 0 * 041 15*25 75.00 1 * 0 0 0 0 0 0 0 * 0 0 * 042 20*00 75*00 1 * 0 0 0 0 0 0 o * c OoO43 1*00 9 0 * 0 0 1 * 0 0 0 0 0 0 0 * 0 0 * 0


U I V M O O a ! < g 0 > U I » U N H O i D 0 l ^ » U I ^ U M » > O t D a i ^ a > U I » U K > ‘ O « Q : i > l » U I » U M H O l 0 a i N ( M J 1 4 >

S 5 5 » o f u,j.r 8 « S ? r S 5 5 « o ? i . u r S S o y r S J » N = ? ? u r S u ; S o . r S j 5 ;« S a , y 1-gSKSSSSSggKSSggSKSgSSiSggSSSgSSKSSUSiSggSSSSgSSSSSSS

i i i i i i i i P i i i i i P i i i P i i i i i i i i i i P P i P i i i P i i i i i ? ? ? ? r ? f fg g g g s g g g g g g s g g g g g g g g g g g g g g g g g g g g g g g s s s g g g g g g g g g g g g

????????????????? ???????????????????????o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o c c o o o o o o o o o

1 1 6

* * * * * * * * * * * * * * * * * * * * * * * * * * * * *

C O R R E C T E D V A L U E S* * * * * * * * * * * * * * * * * * * * * * * * * * * * *

NODE R T U-VELOCITY V—VELOCITY PRESSURE1 1*00 OoC 1.000000 0.0 0.2401042 3*38 0.0 1.006614 0.0 0.1747513 5® 75 0.0 1.007722 0.0 0.118864A 8® 12 0.0 1® 008110 0.0 0.0845245 10.50 0.0 1.007625 0.0 0.0591756 12*88 0.0 1.0C6475 OoO 0.0392767 1 5® 25 0.0 1.004760 0.0 0.0233508 17.63 0.0 1o002598 0.0 0.0105219 20.00 0.0 1.000000 0.0 0.0

10 1.00 15.00 loGOOOOO 0.0 0.2467181 1 5.75 15.00 1.007374 -0.002239 0.12030812 10.5C 15.0C 1.007220 -0.002052 0.05947313 15.25 15.00 1.004533 -0.001258 0.02367014 20.00 15. 00 1.000000 0.0 0.015 l.CC 30.00 1.000000 0.0 0.26131716 3.38 30. CO 1.005204 -0.003690 0.18006417 5.75 30.00 1.006015 -0.004C29 0.12162518 8. 12 30.00 1.006460 -0.004121 0.08687319 10.50 30.00 1.006112 -0.003812 0.06113820 12.88 30.00 1.005217 -0.003187 0.04074321 15.25 30.00 1.003846 -0.002312 0.02432922 17® 63 30.00 1 a C02099 -0.001259 0.01102823 20.00 30.00 loGOOOOO 0.0 0.024 l.CC 45.00 1*000000 0.0 0.28728225 5.75 45.00 1.004317 -0.005331 0.12652326 10.50 45.00 1*004473 -0.004994 0o06298527 15.25 45.00 1.002859 -0.003055 0.02553128 20.00 45.00 1.000000 0*0 0.029 1.00 60.00 1.000000 O.C 0.32071530 3o 38 60.00 1.001768 —0o005273 0.19129031 5.75 60.00 1« 002298 -0.005614 0.12953832 8. 12 60.00 1.002654 -0.005945 0.09294533 10.50 60.00 1.002593 -0.005540 0.06595834 12.88 60.00 1.002272 -0.004645 0.04437935 15.25 60.00 1.001711 -0.003369 0.02679936 17®63 60.00 1.000943 -0.001820 0.01230337 20.00 60.00 1.000000 0.0 O.C38 1«00 75.00 1.000000 0.0 0.34933239 5.75 75. CO 1.000417 -0.005530 0.13554740 10.5C 75. OC 1.000801 -0.005361 0.06839141 15.25 75.00 1 .000606 -0.003310 0.02830642 20® 00 75. CO 1.000000 0.0 0.043 l.CC 90.00 1.000000 0.0 0.380829


1 1 7

44 3*38 90*00 0*998621 -0*003894 0*20184045 5.75 90*00 0*999063 -0*004635 0*13891146 8.12 90*00 0*999132 -0*004957 0*09977347 10.50 90*00 0*999305 -0*004669 0*07119048 12*88 90*00 0*999512 -0*003956 0*04834849 15*25 90*00 0*999707 -0*002899 0*02952450 17*63 90*00 0,999857 -0*001576 0*01370351 20* CO 90*00 1*000000 0*0 0*052 1*00 105*00 1*000000 0*0 0»38477753 5*75 105* CO 0.997927 -0*003585 0*1430 5054 10*50 105*00 0*998349 -0*003631 0.C7303655 15*25 105.00 0*999089 -0*002294 0*03067456 20*00 105*00 1*000000 0*0 0 o 057 l.CC 120*00 l.CGOOOO 0*0 Oo3890 2658 3*38 120*00 0*997836 -0*001207 0,20975559 5*75 123*00 0*997594 -0*002507 0* 14438160 8* 12 120*00 0*997609 -0*002615 0* 10420661 10*50 120*00 0*997853 -0*002502 0.07478962 12*88 120*00 Qo 998271 -0*002164 0*05108063 15*25 120*00 0*998791 -0*001615 0*03136664 17*63 120*00 0*999361 -0*000878 0.01463865 20*00 120*00 1*000000 0*0 0*066 1*00 135*00 1*000000 0*0 0*36838067 5*75 135*00 0*997510 -0*001343 0*14668268 10*50 135*00 0*997797 -0*001547 0*07542769 15*25 135*00 0*998725 -0*000990 0*03196170 20.00 135.00 1*000000 0*0 0*071 1*00 150*00 1.000000 0*0 0*35214072 3*38 150*00 0*998887 0,000230 0*21447173 5.75 150* CO 0*997726 -0*000710 0*1454 7574 8* 12 150.CO 0*997756 -0*000795 0*10576875 10*50 150*00 0*997956 -0*000786 0*07638776 12*68 150.00 0*998317 -0*000703 0*05228777 15*25 150*00 0*998799 -0*000530 0*03213778 17*63 150* CO 0*999363 -0*000269 0*01502879 20*00 150*00 1*000000 0*0 0*08 0 1.00 165*00 1*000000 0*0 0*33583581 5*75 165*00 0*998093 -0*000198 0*14743082 10*50 165*00 0*998134 -0*000330 0*07614183 15*25 165*00 0*998898 -0*000210 0*03240184 20o CO 165*00 1*000000 0*0 0*085 l.OC 180*00 1*000000 0*0 0*32898486 3*38 180*00 0*999584 0*0 0*21591287 5*75 180*00 0*998138 0*0 0* 14532088 8* 12 180*00 0*998078 0*0 0*10600489 10*50 180*00 0*998234 0*0 0.07677090 12*88 180*00 0*998529 0*0 0.05257791 15*25 180*00 0*998944 0*0 0*03231292 1 7,63 180.CO 0*999445 0*0 0*01511693 20.OC 180*00 1*000000 0*0 0*0


Figure 15. Test Program Results


1 1 9

o « n n o n 4 <o i/t oCM O N 10 o n O <t >O Nn o n ® - •* n o w cvjr n n o v o o<o o> ® o ®H-<fflinoHcooco

o inO N(\J CMH H j j m g n m u n i s u i ' i O K I C O O J -J • » 0 » « 9 O » Q 0 O « * « 4 » O « aH - t o o o - o e o o c o o o o o o o o

q ® n — o n c in in o coinw o n in o n o m ' O cvj -h on o n ® — ■« n o w w o nin n one o ® o a o o v o w w- — c o i n o - c o o c o c o N - t f o i n M ' o o o o— — © O O — a o a O O O O O O O O O O

UJ o _J oO O z o < o

Z UJ UJ 3 O J

o n o N o o o o o o o o o ' O O M ' N i n c ® ® ® c \ i o o o o o o o o o o — — — n r0 O » * * O O * # O ( » * C > O O * * Oo o o o o o o o o o o o o o o o o o ;

v — in 4 n moo c m d w n w• < t n No o n -« s — — — ao —x ® ® ® n <o -h h in oi n o— s - H - H t n -h n -« n — ® n inu. - H - H o m o - H f l o a o o o N - ^ o i n ^ o o o oII— - i - O O O — O O O O O O O O O O O O O

- m <t n in oo <r co «o n n nho n n ® n — n -h - - ® - hd d d N iO - h ( O- Hi nO N On -h -h in -h n - N - i n in in- H 0 > i n o H i s o ( M ) N < i c i n 4 & c Q C• • • • •- - o o o .

• • • • •o o o o o o o o o o o o o

( O C ' m O O ' t f — O O O O O O O O O — 0 4® ® ® n — o o o o o o c o o o o o - n0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 I I I I I I I I

i « o < u o n o o ® o ® i n o c \ i ® o ( \ i — n i n o o ® w o n c u ® o o o o o o o o i n ® o ® — o n — 4 o o ® o o o o o • H i n o ^ N o - N N i n o d d o o o o h ( T ® 0 — ® 0 0 ' ® N 4 0 ® 4 © 0 0 0 » • • • • • • • • • • • • • • • • «- 0 0 0 - 0 0 0 0 0 0 0 0 0 0 0 0 0


-0*0

0004

C*

46 0.0

0*0

0*43

OoO

0*0

-0*C

00C0

1*57

C*C

0.0

0»46

0*0

0*0

Figure 16. Method 1 Results for a Lens Shaped Cylinder


121

* * * * * * * * * * * * * * * * * * * * * * * * * * * * *

NO. OF ELEMENTS ON THE UPSTREAM AXIS = a

NO* OF ELEMENTS ON THE DOWNSTREAM AXIS = 8

NOc OF ELEMENTS IN THE TRANSVERSE DIRECTION = 8EACH ELEMENT HAS 2 POINTS IN THE RADIAL DIRECTION


TOTAL NUMBER OF ELEMENTS = 64TOTAL NUMBER OF POINTS = 81MATRIX BANDWIDTH = 33MAXIMUM RADIUS ON THE DOWNSTREAM AXIS = 12.00

REYNOLDS NUMBER = C.50* 4 4 4 4 * * * * * * * * * * 4 4 4 4 4 4 4 4 4 4 4 4 4 4

ITERAT ION NUMBER 1 SUM = 1.9E+01 BIG = 1.8E+00

ITERATION NUMBER 2 SUM = 3.2E+00 BIG = 7.3E-01

ITERATION NUMBER 3 SUM = 6.3E-01 BIG = 3.4E-01




ITERAT ION NUMBER 7 SUM = 4 .IE-04 BIG = 6.4E-03




122

I N 1: T I A L G U E S S* * * * * * * * * * * * * * * * * * * * * * * * * * * * *

NODE R T U-VELCCITY V-VELQCITY PRESSURE1 1.00 Oo 0 OoO 0.0 1.9985222 2o5C 0. 0 0.247962 OoC 0o7994C93 4.00 0.0 0.458433 0»0 0.4996304 5. 5C c.c 0.610186 0.0 0.3633685 7.00 0*0 0.727518 0.0 0.2855036 8.50 OcC 0.822884 0*0 0.2351207 i o .oc 0* Q 0.903124 0.0 Oo199852a 1 lo5«J c . o 0.972344 O.G 0.1737849 13*00 0*0 1.033189 O.C 0. 153732

1C 0*78 22o 50 -0.010440 -0.113699 2*36717111 2*28 22.50 0.269118 0*142665 0.80982212 3*78 22.50 0.500087 0*164283 G.48846413 5*28 22.50 0.661038 0.170310 0.34969614 6,78 22. 50 0*783478 0. 172803 0.27233015 8*28 22.50 0.882070 0*174069 0.22299516 9*78 22.50 0.964528 0*174799 0.18879317 1 lo28 22.50 1.035361 0.175258 Oo 16368718 12*78 22.50 1*097434 0.175564 C* 14447519 Qo 6 2 4 5.00 -0.238841 -0.400068 2,27930520 2*12 45.00 0*375430 0.194231 0.66659021 3*62 45.00 0.642761 0.230752 0 a 39037822 5a 12 45. CO 0 o 815973 0.240286 0.27601023 6a 62 45. 00 C.944348 0.244115 0.21347024 8* 1 2 45.00 1.046391 0.246026 0. 17403625 9*62 45.CO 1.131084 0.247116 Oo 14689926 11,12 45.00 1.2 03482 0.247795 C. 1270 8327 12*62 45.00 1.266704 0.248247 0.11197828 0.52 67.50 -0.803352 -0.476631 1.47077429 2*02 67.50 0.484643 0*133355 0.37861530 3.52 67.50 0.791154 0. 162389 0.21727331 5a 02 67.50 0.975754 0. 169637 0.15235132 6.52 67.50 1.109234 0.172491 C . 11730133 8*62 67.50 1.214098 0.173900 0.09536234 9.52 67.50 1.300560 0.174697 0.08033635 1 1 a i 2 67.50 1.374160 0.175192 0.06940136 1 2o 52 67.50 1o438248 0*175519 0.06108637 0. 50 90. CO -1.095762 -Oo 000000 0.00000138 2.CC 90.00 0.533679 0.OOGCCC 0.00000039 3,50 90. CO 0.855341 CaCOOOOO C.00000 »40 5.00 90.00 1*043945 0.000003 0.0000004 1 6.50 90.00 1*179111 0.000000 OoOOOOOc42 8.00 90.CC 1.284863 0.OOOCOJ o.ooooo':43 9.5 a 90.CO 1.371861 0.00000c o.oooco;


123

44 1 1*00 9C* 0 0 1*445811 0.00000045 12*50 90*0 3 1.510146 O.OOOOCi.'46 0*52 112*50 -0.803353 0*47663047 2*0 2 112.50 0.484643 -0*13335448 3*52 112* 50 0.791154 -0.16238949 5.02 112* 50 0.975754 -0.16963650 6o 52 112®50 1.109234 -0.17249151 8*02 112*50 1.214099 -0.17389952 9*52 11 2c 50 1*300561 -0. 17469753 1 1*02 112*50 1.374160 -Oo17519154 12.52 112.50 1*438248 -0.17551955 0*62 135.03 -0.238842 0.40006856 2* 12 135.CC 0*375430 -0.19423157 3*62 135® OC C.642761 -0.23075258 5.12 135.00 0.815973 -0.24028659 6*62 135.00 Oo944349 -0.2441156C 8* 1 2 135©00 1.046391 -0.24602661 9.62 135*0 3 1. 131084 -0.24711662 11*12 135o 00 1.203482 -0.24779563 12.62 135*00 1.266704 -0.24824764 Co 7 8 157*50 -0.C1044U 0.11369965 2*28 157*50 0.269118 -Cl. 14266566 3* 78 157*50 0.500087 -0.16428367 5o 28 157*50 0.661038 -0®17031068 6.78 157.50 0*783478 -0.17280369 8*28 157.50 0.882071 -0.17407G70 9*7 8 157.50 0.964528 -0.17479971 1 1.28 157.50 1.035362 -0.17525Q72 12*78 157.50 1*097434 -0*17556573 1*00 180.00 0.0 0.074 2.5C 180* 00 0.247962 -0.00000075 4.00 180.00 0® 458433 -c.oooooo76 5*50 180.00 0*610186 -0.00000077 7* Jt 180.CO 0*727518 -0.00000078 8.50 180.00 0*822884 -OoOOOOOO79 19*00 180.00 Oo 90 3124 -OuOOQGOQ80 1 1.50 180.00 0.972344 -0.00090C81 1 3 © C C 18C.C0 1.C33189 -0.000000

0.000000 OoGuGC'CO

- 1 * 4 7 0 7 7 1 —Oo 3 7 8 6 1 5 - 0 * 2 1 7 2 7 3 —O o 152351 - 0 * 1 1 7 3 0 1 - 0 * 0 9 5 3 6 2 - 0 * 0 8 0 3 3 6 - 0 * 0 6 9 4 0 1 - 0 * 0 6 1 0 8 6— 2o 279302 -0*666589 -0*390378 -0*276009 -0a213469 -0*174035 -0*146899 -0*127083 -0*111978 -2*367170 -0*809822 -0*488464 -Oa 349696 -0*272329 -0*222994 -0*188793 -0®163687 -0*144475 -1*998522 -0.799409 —0* 49963C -0*363368— 0* 285503 -0.235120 -0.199852 -C*173784 -0*153732


1 2 4

* * * * * * * * * * * * * * * * * * * * * * * *

C O R R E C T E D V A L U E S

* * * * * * * * * * * * * * * * * * * * * * * *

NODE R T U-VELOCITY V-VELOCITY

1 laOC 0*0 OoC 0«C2 2*50 0*0 0*380342 0*03 4.C0 OoC 0*668071 0.04 5*5 0 o*c 0*768126 0*05 7*0t C.-C 0*857579 0«C6 8*5C 0*0 0*887136 0.07 luoCO o*c Os 935853 0*08 1 1 * 5 C c*o 0*914092 O.C9 1 3u 06 0*0 loCOOOCO 0* 0

10 C* 78 22*50 0*0 0*01 1 2*28 22* 50 0*437981 0 o 23770012 3.78 22*50 0*692993 0*16402813 5*28 22,50 Oo 799577 0*13325014 6*78 22*50 0,879550 0*09581915 8* 28 22* 50 0*907747 0.0750 1616 9*78 22*50 0.948982 0*04494217 1 1 *28 22*50 Oo928893 0*0441511 8 12*78 22*50 1*000000 0*019 Co 62 45*00 0,0 0*020 2*12 4 5* 30 0*617832 0*28146221 3o 62 45,00 0.812754 0*267 0 3822 So 12 45*00 0*909087 0,21595523 6,6 2 4 5* 00 0.961117 0*16383824 8* 12 45*00 0.977290 0*12658025 9*62 4 5* 00 Oo 993948 0*07786526 11*12 45*00 0*969179 0*07093427 12*62 45, 00 1*000000 0*028 C • 52 67.50 O.C 0*029 2* 02 67* 5C 0*725764 0* 18329930 3o 52 67,50 0*943566 0*23038831 5 ■» C 2 67* 50 1 o 034508 0*20009232 6*52 67*50 1*067972 0* 16176533 8.02 67*50 1 •C 70258 0. 12620834 9o 52 67o 50 1•C 58430 0*07918235 1 1*02 67*50 1*023620 0*06858236 12*52 67*50 1*000000 0*037 ii.SC 90* CC 0.0 0*038 2*00 90*00 0*743410 0*04391539 3 > 50 9Ce CO Co 992657 0*09194440 5 a C 0 90*00 le102 338 0*08892941 6.5C 90*00 1*143519 0*07752242 8»0C 9C* 00 1*146508 0*05918843 9 o 5 0 90*00 1*119805 0*034951

* * * * *

* * * * *

PRESSURE

3.594991 0*607278 0*821862 Oa 523774 Ud495449 0.512465 Oo 340254 0.643523 0,C2*638982 Oo852884 Co744818 0*537515 0*462505 Qo498102 0*315378 0*622122 Oo 00*619995 0o662368 0*436734 Oo 4148 74 0* 330002 Oo 397859 0*248320 C * 518751 OoC

-0 <>270651 0*296028 0.117791 0*220277 0*161965 0*243359 C, 1478 1 1 0* 34 96 50 0«0

— Q, 68 32 40 -0*138082 -C • 17 17 7t -0*030620 -0.025793 Oo 047279 0*027395

1 2 5

44 11.0C 90*00 1*073257 0*028715 0*13192545 12*50 90*00 1.000000 Oo 0 0*046 0.52 I 12*50 0*0 0*0 -1*04003647 2.02 112o5) 0*657628 -0.094212 -0*4651434 8 3.52 112.50 0*915601 -0*072830 -0*39340949 5.C2 112*50 1*049102 -0*066944 -0.24370250 6.52 11 2* 5C 1*120650 -0*058249 -0*19420251 8.C2 112*50 1.146836 -0*059809 -0*15379652 9*52 112*50 1*136064 -0*051343 -0*09867053 1 I *02 112*50 1e 088251 -0*045893 —Oo 12422354 12.52 112*50 1.030000 0.0 OoG55 0*62 135o00 Oo 0 0*0 — 1*50179056 2* 12 135*00 0*500740 -0*18524C -0*74218057 3*62 135.CC Co 733278 - Oo175656 -0*56470850 5*12 135.0C 0.873009 -0*175643 -0*40 119759 6.62 135*00 0.975334 —0 * 164795 -0*3242396C 8*12 135* CO 1*029810 -0.164548 -0.33489C61 9*6 2 135.CO 1 * 064058 -0*135111 -0.20844662 11.12 135*00 1*031053 -0*120798 -0*40778663 12.62 135* CO 1.CJOGOu 0*C OoC64 C* 78 157*50 0*0 0*0 -2*50852065 2*26 157.50 0.313423 -0*174822 -0*68564266 3.78 157*50 0*540819 -C* 142613 -0*69245467 5*28 157*5 0 0*663739 -0*152763 -0*44984468 6*78 157*50 0.78076J -0*146042 -0*40817769 8®2R 157.50 0*846038 -Oo150382 -0*46074170 9*78 157.50 0*926572 -0*124C80 —0 * 2950 3971 11.28 157*50 Oo 912546 -0*115159 -0*66198272 12.78 157*50 loOOOOuO 0*0 0.073 1 a 00 180*00 OoC 0*0 -2.85337974 2.5C 18 0*00 0.247308 0*0 -0*74926575 4 * C C 18C.U0 0*468329 o*c -0*68219576 5,50 180* CO 0*579366 0*0 -0*48944877 7.00 180*00 Oo693939 0*0 —0.4222 3578 8 • 50 180*00 0*757301 0.0 -0*50759479 ICooO 180*CO 0.855728 OoC -0*31704180 1 1.50 180*00 0*847348 0*0 -0*75333281 13.00 180* 00 loCOOOQC 0*0 OoC


NO* OF ELEMENTS ON THE OOWNSTREAM AXIS = 0

NOa OF ELEMENTS IN THE TRANSVERSE DIRECTION = 8EACH ELEMENT HAS 2 POINTS IN THE RADIAL DIRECTION


TOTAL NUMBER OF ELEMENTS = 64TOTAL NUMBER OF POINTS = 01MATRIX BANDWIDTH = 33MAXIMUM RADIUS ON THE DOWNSTREAM AXIS = 12*00

REYNOLDS NUMBER = 1<»C0* * * * * * * * * * * * * * * * * * * * * + * * * * * * *

ITERAT ION NUMBER I SUM = lo1E+01 BIG = 1.5E+0C

ITERAT ION NUMBER 2 SUM = 1« 3E + 0G BIG = 5* 2E—01

ITERAT ION NUMBER 3 SUM = lo 9E-01 B IG = 2»1E-C1

ITERAT ION NUMBER 4 SUM = 6* OE—02 BIG = 1.2E-01

ITERAT ION NUMBER 5 SUM = 2* 4E — 02 BIG = 6.6E-02

I TERATION NUMBER 6 SUM = 6*6E-03 BIG = 2.9E-02

ITERAT ION NUMBER 7 SUM = 3a 5E —0 3 BIG = I.7E-02

ITERATION NUMBER 8 SUM = 2o 5E—03 BIG = 1a5E—U 2

ITERATION NUMBER 9 SUM = 1.6E-03 BIG = loJE-C2

ITERAT ION NUMBER 10 SUM = l*2E-t>3 BIG = 9o IE-03

ITERATION NUMBER 11 SUM = 7e 5E—04 BIG = 6* 5E— 0 3

ITERATION NUMBER 12 SUM = 5o 7E—04 BIG = 7«7E-03

ITERAT ION NUMBER 1 3 SUM = 3o 6E—04 BIG = 5.6E-93

ITERATION NUMBER 14 SUM = 2.SE-04 BIG = SaOE-03

ITERATION NUMBER 15 SUM = 1 # 8E —04 BIG = 3*7E-C3

* * * * * * * * * * *

I N 1

* * * * * *

[ T I A L G

* * * * * * *

u e r* s

* ;* * * *


I 1.» it o* Co 0 CiO 3*5949912 2.50 c.c 0*380342 0.0 0*6072783 4*00 ilt Oo 668071 CaO C o 8 ? 1 8 6 c4 5* SC OoC 0.7 68126 0»0 0.5237745 7 3cn Co - 0.857579 0.>C 0.4954496 3.50 0. 0 0*887136 0*0 0*5124657 ICaCO C b C C j935853 0 0 0 0 a 3 4 C 2 5 48 11*5 0 Oo •: 0.914092 0. 0 0.6435239 13. wr 0* v 1. C u 0 0 0 0 OoC 0 » 0

10 0*76 22o 50 0.0 OoO 2.63898211 2« 2 8 22o 5C 0c 437981 0*2377 00 Oo85288412 3* 78 22. 50 0.692993 I). 164C 28 0.7448181 3 5*28 22o5f. 0*799577 0.133251 0.53751514 6o 78 22.50 0.879550 Oa 095819 0 »4625C515 a.,28 22*50 0 a 907 74 7 Oo 075C16 0.49810216 9. 78 22.50 0.948982 0.044942 0.31537817 1 1*28 22.5 0 0*928893 0.044151 0.62212218 12*78 22*50 I*000000 0.0 O.C19 G>62 4 5o 0-0 0>C 0*0 0 a 5 I999520 2 o 1 2 4 5.00 0.617832 0.281462 Oa66236621 3*62 4 5* C 0 0*812754 0.2670 38 0.43673422 5* 12 45. 0 0 0.90 9087 0.215955 0.41487423 6*62 4 5* CO 0 u961117 C> 1638 38 0.33000224 8 o 12 45.00 0.977290 0.126580 Cl >39785 925 9.62 45* 00 Co 993948 0.077865 0.248 32 026 1 lo 12 45* CO 0*969179 0.0709 34 0>51875127 1 2*62 45a 00 laCOOOOO OoO OoO28 Uo 52 67.50 C.O 0.0 -0.27065129 2 o * 2 67*50 Oo 725764 0.183299 O o 2960 283J 3*52 6 7 o 5 0 0.943566 Oo 230388 0*11779131 5*02 6 7*50 1vC 34508 0* 200 3 92 3.22C27732 6*52 6 7.50 1.C67972 Ca161765 0.1619653 3 8* 02 6 7,» 5 0 1 aC7C 258 Oa1262 38 0*24 335934 9o 52 67*50 1 o C 584 30 0.079182 0.14781135 1 I o 02 67* 50 1oC 2362 0 0.068582 0.34965v36 12*52 67.50 U CC0309 C.C 0 »037 5 90* 'O Co 0 OoC -0 >68324'/38 2d uC 90. CO 0.743410 Oa 043915 -0*138 0 8239 3 > 5 C 90*00 Oc 992657 0* C 91944 —C a 1717724 C 5*00 90. CO 1.102338 0.088929 -0*0300204 1 6 » 5 V 9 0*00 1.143519 0.077522 -C > C2579342 8. CO 90. CO 1.146508 0.059188 0*04727943 9 d SC 90*00 1j 1 I 9805 0.934951 0*027395

R eproduced w ith perm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.

128

44 1 1 ^ 90.00 1o C 7325 7 Oo C28715 0.13192545 12 0 50 9CoC0 luCOO-JCO OoC O.C46 :■ o52 I 12.50 0©C OoO -1.040 03647 2, C2 112© 5" 9o 657628 -0.094212 -0,40514348 3 o 52 112,50 9,915601 -0.072830 -0,3934 3949 5*92 1 1 2® 5v 1,C4911 2 —0,C66944 -0©24 370 25C 6a 52 112.59 1©12065U — 0aC58249 -0.19420251 8 • C 2 • 112.5^ lo146836 - C ,059809 -y,15379652 9,52 112.5? 1©136064 —0 o 05 1.343 -OoC9867053 1 1 j.,2 1 1 2® 5 ) 1,C58251 -0,045893 -0©12422354 12s 52 112.50 i.OoOOCQ o . c OoO55 9*62 1 35,09 T, C OoC — 1 © 5 w179056 2* 12 135,99 0.500740 — G a 185249 — 9a 74218057 3)62 135o 99 0,733278 -0.175656 -0.564 70858 5. 12 135©0 9 0.8730C9 -0©175643 -0.4,119759 6*62 135.0 9 0,975334 -Oa164795 —Oo 32423960 8 o 1 2 135.0? 1,02981. -0.164548 -Oo 3348906 1 9,6 2 135®0 0 1 a 064 J58 -0.135111 — Co 20844662 11.12 135,99 1o C 3 1 353 -Ca120798 —0.4J778663 1 2.62 135,0 0 1j CCOJCv 0,0 OoO64 9,78 157.59 ;J.C OoC — 2.5 j 852C-65 2o 28 157,5C 9,313423 -Oo 174822 —0,68564266 3*78 15 7 o 5 0 C a 540819 -Oa 142613 — 0 o69245467 5*28 157.5^ 0,663739 -Oo 152763 -0,44984466 6© 78 157,59 0,700769 -0,146 v 4 2 —0.4 08 17769 8 a 2 8 15 7 o 5 0 0 « 846038 -0,150382 -0,46974170 9.78 157.59 0.926572 -0,124Q0C -0,2950 3y71 1 1*28 157,50 9.912546 -Oa115159 -0,66198272 1 2o 78 157.50 1.C00000 0,0 OoC73 I 9 0 9 i89oC9 OoO OaC -2,85337974 2 ,50 180.00 C.2473C8 0.0 -Os 74926575 4 o C r' 189,9 0 0,468329 0.0 -0.68219576 5,59 i s o .uo 0,579366 0.0 -C,48944877 7o0 0 1 8>3o 09 0,693939 0 a C -0.) 42223578 8 a 5 C 18C.C0 0.757301 OaC -9.50759479 1 0 r, v o 1 PC.CO C,855728 O.C -Oa31704189 11,5- 180,09 Ca 847348 o . c -0.7533328 1 13*00 18C.C0 loCOCOOj 0,0 OoO


1 2 9

C Cl R r r C T E D V \ L u r S* * * * *

JODE R T U-VELOCITY V-VELOC1TY PRESSURE

1 I jC'J 0a 0 OaC 0 -> 0 2 o 1221422 2,50 O.C Oo435869 Co 0 6 u 39 I 6 563 4u C C c, j Oo 730939 c.o 0,4946254 5, 5C 0o 0 Co 807838 0,0 Co 3171625 7 j C 0 0a : Qje83486 0 » o 9,2695266 8o 5C OoO C,90114* 0,0 U.2926917 1 V lit 0,0 0,942940 o . c Oa 19182Ca ll5b. 0,0 Ca918512 CaC Oa 3516879 13*00 CoO laCOOOOC 0 o f. 0,0

10 )o 78 22o 50 C !> {* c » c 1.495*9011 2,28 22 j 58 0,496264 0,267190 C*51896912 3 > 78 22,50 Ca 746293 C,149229 (jn 4490 1213 5o2e 22a 50 0,830722 0,10 90 91 9,32135114 6e 78 22o 5‘1 0,896543 0,071465 ■j a 2 7 I 0 9015 8 o 2 8 22a 5n Oa 914366 Ca 0559 74 0,28418216 9. 78 22a 50 0,950808 0,032117 0*17845617 1 1 028 22,50 0,929696 0,037272 0,34026718 12*78 22,5: l.COCOOO C*0 e * c19 'Jo 62 4 5, C O O.C 0,0 O o 195 07 12 0 2a 1 2 45, 00 0.68 8545 0.,321978 •J, 36 31742 1 3*62 45, CO ■0, 84993 7 0,2668 39 0,253.3 DC22 5> 12 4 5, JO 0,918 052 0 a 1880 23 0,24618323 6*62 4 5,00 Oo953743 0,130726 0,19525124 8* 1 2 45, >K 0,963156 0*098418 0.22959625 9.62 4 5,00 0,980477 0,058302 Ca 14299726 11*12 4 5o 00 0,959954 3,C61578 0,28898227 12,62 45, or 1,000000 0,0 0.028 0,52 6 7,50 OoC O.C -Oa 39565629 2 , C 2 67, 5C 0,794645 Oo 218 380 0,12474930 3 o 52 67,50 0,976772 0,258969 0 .04054931 5, 02 67, 50 1 a 02 9271 0, 198352 0,12965332 6.52 67,50 1*039783 0,1474 60 0*69528933 8,02 67, 50 1,034880 0,109343 0,14589734 9,52 67,50 1,027888 0,065928 Oo C8865735 H e 02 67,50 1.CO3930 0*064966 Oo 20488636 12, 52 67,5 0 lo COCQOu OoC 0 * C37 0.5C 9 0,00 OoC 0,C -0,62996u38 2a CO 90, 00 0*79583 0 Oe C7 3219 -C.13213639 3 b 5 0 90,,C 0 1,03002 0 O a 143655 -C »1449804 C 5 a C! 9Co JO 1 o 106008 U* 1 2 70 30 -Co 0 178174 1 6, 5 V 90, OC lo119898 0,103861 -0,01527042 b o •:' 90, CO 1,199063 0*0758 56 0 , C 4 I 7 0 143 9, 50 9C» CO 1,C83482 C.04 36 96 G*C24 7 31


1 3 0

44 1 1 iVi 90# 00 1» 049745 0.639987 0 •> 1 >; 2 1 1 ,45 12,50 9 C • 0 ’ t .Cu'OGCw 0*0 O.C46 0.52 112.5^ 0* c 0,0 -0,800 74i.47 2^.2 112*5; 0.683622 -0,068912 -0.3-. 30 1548 3*52 1 12o5r. 0 o 9 5 C 6 6 3 -C.015549 -C•27 012249 3a -2 11 2 o 5 r 1,072892 -0*006122 — Oo 13905550 6 » 52 112,56 1.128498 Oo 0007 53 -Oo I 1233351 8 •» 02 112*50 1 a 142286 -0.007412 -Oo06168452 9i» 52 112« 50 1.125873 -0.015139 -0»v419l65 5 11 o ■; 2 112*59 I® C8650u —C.0158 93 -0*0 J741 154 12*52 112*50 l.-jCCOCOO c.o OaO55 0,62 135*00 0;. C o.c -0.97171256 2a 12 135,00 uo 498864 -0*161416 -0,43415257 3 j 62 135,00 0,743729 -C*133335 -Oo 34484758 5 a 1 2 135*00 0*888062 -Oo128669 — 0.22321159 6,62 135*CO 0.992141 -Co 1 16747 -0* 18273360 a, 12 135*0 1.050483 -Oo123:88 -0.15956761 9,62 135a00 1.087136 -0* 112979 - 9a1022 7262 11,12 135*00 1 a 065437 — Ob 1154 30 -Oo14745163 1 2 o 62 135*6 0 1*COCOOO 0.0 C.C64 0,78 157*50 Co c C.O -1.3461 1765 2 o28 157.50 0.290722 -0.157145 -C j39765166 3.78 157.50 0,508677 -Oo 12 5 •'* 8 9 - j,38346067 So 2 8 157. 5-0 Oo 627664 -0.137239 — C a 24 7 1 9660 6 a 78 157*50 Oo 745926 — 0 » 1348 ) 3 -0* 21802569 8*28 157.50 Oo 815823 -0.153071 -0,24035170 9, 78 157.50 Co 911473 -Oa 1434 ; 1 -Ca13084371 11*28 15 7.50 C a 903431 -0.159178 -9*34 392172 I 2 o 78 157*5<J loCoCCOv OoC 9*073 1 »CC 18 0.CO Oo 0 OoC — 1 a 4 3546174 2,5 0 i s c . c<: 0o 213644 0,0 -2a 4 3034975 4 o c 0 1 8 S o ;»0 G ,411034 0.0 -0* 36926 776 5,5'' 180 a HO Oa 507739 Oo v -Oo 26877677 7 ® 0 C 180.CO )a617339 0,0 — On 22144/78 8,59 l a c . o o Oa 675384 0»C -0-, 2793C679 i c , c : 18C.CC 0*793165 0.0 -C a 16540 *.0 0 1 1 350 18 0*00 0.778479 o . c -C ,448297HI 1 3., 9 0 180,00 i . e o Q o o o 0 a 6 0*0

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Stokes, G. G. "On the Effect of the Internal Friction of Fluids on the Motion of Pendulums," Transactions of Cambridge Philosophical Society, volume 9, part II, 1851, page 8.

Stokes, G. G. "On the Effect of the Internal Friction of Fluids on the Motion of Pendulums," Mathematics and Physics Papers, Cambridge University Press, London and New York, volume 3, 1901, page 1. (Reprinted from 1851.)


1 3 5

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

Takami, II., and H. B. Keller. "Steady Two Dimensional Viscous Flow of an Incompressible Fluid past a Circular Cylinder," The Physics of Fluids, volume 12, number 12, part if, 19fT9, page 51.

Taneda, S. "Experimental Investigation of the Wakes behind Cylinders and Plates at Low Reynolds Numbers," Journal of Physical Society of Japan, volume 11, 1§56, page 3d2.

Thom, A. "An Investigation of Fluid Flow in Two Dimensions," Aeronautical Research Council, Reports and Memoranda, number 11§4, 19 28.

Thom, A. "The Flow past Circular Cylinders at Slow Speed," Proceedings of the Royal Society, series A, volume 141, 153 3, page 651.

Thoman, D. C., and A. A. Szenczyk. "Time Dependent Viscous Flow over a Circular Cylinder," The Physics of Fluids, volume 12, number 12, part II, 1969, page 1 6 .

Tomotika, S., and T. Aoi. "The Steady Flow of Viscous Fluid past a Sphere and Circular Cylinder at Small Reynolds Numbers," Quarterly Journal of Mechanics and Applied Mathematics, volume 3,1950, page 140.

Tritton, D. J. "Experiments on the Flow Past a Circular Cylinder at Low Reynolds Numbers," Journal of Fluid Mechanics, volume 6, 1959, page 547.

Turner, M. J., R. W. Clough, H. C. Martin, and L. J. Topp. "Stiffness and Deflection Analysis of Complex Structures," Journal of Aeronautical Sciences, volume 23, 1956, page 805.

Underwood, R. L. "Viscous Incompressible Flow past a Circular Cylinder at Moderate Reynolds Numbers," Journal of Fluid Mechanics, volume 37, 1969, page 95.

Van Dyke, M. Perturbation Methods in Fluid Mechanics. Academic Press, New York and London, 1964, page 149.

Vries, G. de, and D. H. Norrie. "Application of the Finite Element Technique to Potential Flow Problems," Reports 1_ and Mechanical Engineering Department, University of Calgary, Alberta, Canada, 1969.


1 3 6

57.

58.

59.

60.

61.

Weinstock, R. Calculus of Variations. McGraw-Hill, New York, 1952 .

Whitehead, A. N. "Second Approximations to ViscousFluid Motion," Quarterly Journal of Mathematics, volume 23, 1889, page 143.

Wieselsberger, Von C. "Neuere Feststellungen tiber die Gesetze des Fliissigkeits-und Luftwiderstandes," Physikalische Berichte, volume 22, 1921, page 321.

Wieselsberger, Von C. "Wietere Feststellungen iiber die Gesetze des Fliissigkeits-und Luf twiderstandes," Physikalische Berichte, volume 23, 1922, p. 219.

Zienkiewicz, 0. C. The Finite Element Method in Engineering Science, McGraw-hill, New York, 1971.


APPENDIX A

INTERPOLATING FUNCTIONS

Linear Four Node Element (Figure 3):

N. = J(i + 5 5 ^ ( 1 + nrij)

Quadratic Eight Node Element (Figure 4):

Mid-side Nodes,

? . = < ) , N 4 = |(1 - ? 2) (1 + TiniD

ni = 0 » N i = I (1 + ^ i ) ( l - n 2)

Corner Nodes,

N i = ? (1 + 5Ci)(1 + nni)(eCi + nn± - l)

Cubic Twelve Node Element:

Mid-side Nodes,

Cj = ±1 and Hj ■= ±5

Ni = ^ (1 * « i H l - n 2) (1 * snnj)

(swap variables to get others)

Corner Nodes,

N i = + ^ i ^ 1 + h n ^ t - i o + 9(C2 + n 2)]

137

A P P E N D IX B

COMPONENTS OF THE ELEMENT COEFFICIENT MATRIX [H]e AND CONSTANT VECTOR {F}e

Method 1

Let r = 3i-2 and s = 3j-2 where i = l,n and j = l,n, and n is the number of nodes in an element.

The components of the coefficient matrix [H]e are:'3Ni 3N^ IF 3x 3y

e

He (r, s + 1) = 0

He (r + 1 , s) = 0

H e (r+1, s + 1) = He (r,s)

H e (r+2,s) = He (s,r+2)

H e (r+2,s+1) = He (s+1,r+2)

H e (r + 2 ,s + 2) = 0

138


The com ponents o f th e c o n s t a n t v e c t o r { F >e a r e :

* (r) =

F (r+1) =

F e (r+ 2) = 0

fN^dxdy

gNidxdy

Method 2

Let i = l,n , j = l,n where n is the number of nodes.

The components of the coefficient matrix [H]e are:

f [3N. 3 N . 3 N . 3N.'H C1*j) = L l ~ -5# + - W T y - ,

dxdy

The components of the constant vector {F} are:

F (i) L N ‘ij dxdy


A P P E N D IX C

FINITE ELEMENT FORMULATION WITH LINEAR TRIANGLES

As a result of linear shape functions, much of the formulation with linear triangles may be extended to final

form with little extra effort. Not only may the matrix and vector components be integrated analytically, but imposing the normal derivative boundary conditions may be done alge

braically.The basis for the simplifications is the linear shape

functions,

N i Cx’y) = J K (ai + b ix + c iy ) > i=1>2 »3 >

where A is the element area, and a^, b^, c^ are constants calculated from the nodal coordinates, see Chapter III. To demonstrate the simplifications, the formulation for the

Poisson equation is developed. From Appendix B the com

ponents of the element matrix [H]e are obtained,

f f 3 N . 3 N . 3N. 3N.1H (i,3) “ Je l “53T * ~Sy~ ly-j dxdy ’

A substitution of the interpolating functions yields,

n e (i ,j) = [ (^-)2 (b ibj + cicj) dxdy *

140


1 4 1

But noting that A, Ik , b^ , c^, are all constants the

integral simplifies,

He ( i J ) - & ( b ^ . * c ^ ) .

In a similar manner the element constant vector {F}e com

ponents ,Fe (i) = — [ N ^ d x d y

■'e

may be integrated. Assuming a linear formulation for

iKx,y) = R x + R 2x + R 3y ,

and multiplying inside the integral yields,

pe(i) = " f J A (aiR l + a iR 2x + a iR 3y + b iR lx + b iR 2x2 •'e

+ b iR 3xy + C iR 1y + C.R2xy + C ^ y 2) dxdy .

Recognizing that A, ai , b ± , c ^ Rj, R 2 , R 3 are constants,

and recalling the definitions of centroids and second

moments of areas, the expression becomes,

Fe (i) = - - | ( a ^ ♦ bjRjD - \ (a.R3 ♦ C lRl)

- c iR 3 “S t - “S T (c 1R 2 + b iR3> - b iR 2 ~ W >

where x,y are the coordinates of the centroid, and I^y*Ixy» Iyy ar© the second moments of the element’s area.


1 4 2

Having assumed tj;(x,y) to be linear in the previous

discussion, it is now necessary to compute the coefficients

Rl» R 2* R3 ‘ For the Poisson equation in u the function i|>(x,y) is defined as,

<P(x,y) = R e (u u x + vuy + Px ) .

Substituting the representations of u, ux , and u .,

u = N {u}e ,

ux = {uie • uy ■ <cl'c2>c3>(u)e -

and similar expressions for the other variables, the func

tion iKx,y) becomes,

tCx.y) = \ [(<N>{u>eH ‘u}e) ♦ « N > { v } e) « c i> { u } e) 4A L

+ 2A^bi){P)eJ

Recognizing the coefficients of x and y reveals the con

stants Rj , R 2 , R j ,

R 1 ■ ^ [ « > i > < “ >eH l » ) e) ♦ « 3i> { v n « Cl>lu>e)

+ 2A <bj){P}e]

r 2 ■ J r [ « b 1)(“ )e)«'>i)(u>e) + « b i>{v>e) « c i)(U )e)] ,

£ ♦ (<ci> {v)e) « c i>{u>e)] .

In exactly the same manner expressions for R^, R£, Rg are

calculated for the function iKx,y) in the v and P equations.The normal derivative boundary condition on P is easy

to attack with linear triangles. Recall the components of the pressure gradient are computed from the previous pres

sure and two successive values for the velocities,

p f 1 - CPX - uux ♦ vV " - (uux ♦ vuy )n+1 ,

and a similar equation for P^+1 . By substituting the

product of the interpolating functions, or their deriva

tives, with the nodal values, an algebraic expression isobtained for the components P and P . Locating the x ytriangle at an arbitrary angle 0 to the x axis, see Figure

5, the normal derivative of P is computed,

Pn = Py cos6 “ Px sin6

Writing cosG and sin0 in terms of the nodal coordinates,

the normal derivative becomes,

P . P _ P y 2~y in y L x L >

l = / (x2-x1) 2 + (y2—y i ) 2

Recognizing the constants,

c3 = x 2“x l ’ b3 ’

the final expression is,

c3 ^3p = p + p —*n y L Fx L

To impose this normal derivative, the extra term intro

duced in Chapter V is considered,

IBC = - f qPds Js

If q and P are allowed to vary linearly along the edge,

q (s) = + * p(s:) = pi +

where q^ and P^ are the nodal values and s is measured

along the edge, the integral becomes,

lB C ■ - | [ i i + [ p i * r ( p 2- p i )] d s •


1 4 5

Carrying out the integration yields,

IBC = - Pj |C2q1+q 2) - P 2 ^ ( Z q ^ q j ) •

To form the finite element equations the derivatives of

IBC are needed,

aTBC i aTBC TTJT" = - F (2ql+q2} * = “ 5’(2q2+ql)

By adding these expressions to the element constant vector

{F}e , in row one and two respectively, the normal deriva

tive of P will be set equal to q.


APPENDIX D

COMPUTER PROGRAMS


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V I 6 ) = (V(2) + VI 3) + VI9))/3. GRW 36XC7) = (X{3) ♦ X14) + XI9 ) )/3# GRW 37Y(7> = (Y(3) + Y < 41 ♦ YI9))/3. GRW 38U17 ) = ( U( 3) + U 14 ) ¥ UI9))✓3. GRW 39V I 7 ) = ( V( 3) ♦ V I 4 ) ¥ VI9))/3. GRW 40X ( 8) = ( X{ 4) + XI 1) ¥ XI 9) )73. GRW 41Y (8) = (YC4) + Y(1) ¥ YI9))/3. GRW 42U ( 8) = (U(4) ♦ U( 1) ¥ UI9 ) )/3. GRW 43via) = Iv14) + vi i) ¥ V(9 > )/3. GRW 44

c GRW 45c FORM COEFFIEIENT MATRIX AND CCNSTANT VECTOR FOR FIVE NODE QUADRILATERGRW 46c GRW 47

DO I 1=1.27 GRW 48CON! I )= 0 • GRW 49DO I J=1.27 GRW 50

1 COE(l.J)=0. GRW 51CALL T R 1311*2*5) GRW 52CALL T R 1312* 3*6) GRW 53CALL TR13(3* 4,7) GRW 54CALL TRI3I4,1.8) GRW 55CALL TR 1311.5.8) GRW 56CALL TR 13(2.6.5) GRW 57CALL T R 1313.7.6) GRW 58CALL TR 13(4.8.7) GRW 59CALL TR 1318.5.9) GRW 60CALL TRI3I5.6.9) GRW 61CALL T R 13(6.7.9) GRW 62CALL T R 1317.8.9) GRW 63

c GRW 64c REDEFINE PRESSURE AND ELIMINATE INTERNAL NODES GRW 65c GRW 66c NODE 9 GRW 67c GRW 68

DO 4 1=1.27 GRW 69DO 4 J= 1. 4 GRW 70

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SUBROUTINE MODIFY(ROW.VALUE) GRW 1c GRW 2c THIS SUBROUTINE MAKES CONST(ROW) = VALUE GRW 3c GRW 4

COMMON COEF(250.40).CONST< 250).MBAND.NEQ GRW 5INTEGER ROW GRW 6DO 4 J=2*MBAND GRW 7

c GRW 8c MODIFY COLUMN GRW 9c GRW 10

I = ROW - J + 1 GRW 1 1IF (I .LE. 0) GO TO 2 GRW 12CONST(I ) = CONST!I) - COEF(I.J )*VALUE GRW 13COEF(I.J) = 0. GRW 14

c GRW 15c MODIFY ROW GRW 16c GRW 17

2 1 = ROW + J - 1 GRW 18IF (I .GT. NEQ) GO TO 4 GRW 19CONST!I) = CONST!II - COEF(ROW,JI+VALUE GRW 20COEF!ROW.JI = 0 . GRW 21

4 CONTINUE GRW 22c GRW 23c FIX UNKNOWN AND NORMALIZE DIAGONAL GRW 24c GRW 25

CONST(R CW) = VALUE GRW 26COEF!ROW.1 1 = 1 . GRW 27RETURN GRW 28END GRW 29

170

171

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IF ( IVAR »GT• 0) GO TO 12 GR*22 CONTINUE GR*24 CONTINUE GR*

C GR*C RESULTS GR*C GR*

WRITE (6.28) GR*WRITE (6.38) GR*WRITE (6.40) (J.R(J).T(J).W(3*J-2).W(3*J-1).W(3*J).J=1.43) GR*WRITE (6.50) GRaWRITE (6.40) (J.R(J).T(J).W(3*J-2).W(3*J-1).W(3*J).J=44,NP) GR*

C GRWC FORMAT STATEMENTS GR*C GR*

26 FORMAT (/36K.•ITERATION NUMBER*.13) GR*28 FORMAT (IH1.4(/),36X,29(•* ')//50X,'C O R R E C T E D '. GR*

& • V A L U E S'//36X.29(•* «)/) GR*30 FORMAT (16F5»0) GR*32 FORMAT (1HI•4(/).36X.29(•* •)// GR*

& 36X.*N0« OF ELEMENTS ON THE UPSTREAM AXIS ='.13// GR*& 36X.'NO» OF ELEMENTS ON THE DOWNSTREAM AXIS ='.I3// GR*& 36X.*N0a OF ELEMENTS IN THE TRANSVERSE DIRECTION =',I3// GR*& 36X,'EACH ELEMENT HAS'.13.' POINTS IN THE RADIAL DIRECTION*/ GR*& 4SX*•AND*.13.• POINTS IN THE TRANSVERSE DIRECTION*// GR*& 36X.'TOTAL NUMBER OF ELEMENTS =*.I3// GR*& 36X.'TOTAL NUMBER OF POINTS =*.I3// GR*& 36X.'MATRIX BANDWIDTH =*.I3// GR*& 36X,'MAXIMUM RADIUS ON THE DOWNSTREAM AXIS ='.F&,2// GR*& 36X,'REYNOLDS NUMBER =•,F5.2//36X,29(•* •)/) GR*

34 FORMAT (4F10*0.5I5) GR*36 FORMAT (1H 1.4(/)o36X.29(•* ')>/53X.'I N I T I A L G U E S S'// GR*

& 3 6 X .29C* •)/) GR*38 FORMAT (36X.'NODE•.5X.•R ••8X.•T '•5X.•U—VELOCITY•,2X• GR*

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THREE

POINTS

IN

EACH

DIRE

CTIO

N

Reproduced

with perm

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copyright owner.


prohibited w

ithout permission.

C FOUR POINTS IN EACH DIRECTION C

4 PT(1) = - .8611363115 PT{ 2 ) = - .3399810435 PT(3) = - PT(2)PT{4) = - PT(1). T ( 1 ) = .3478548451 WT ( 2 ) = .6521451548 N T{3) = WT(2).T < 4) = WTC1)

CC STORE INTERPOLATING FUNCTIONS AND DERIVATIVES AT EACH GAUSS POINT C

6 DO 8 1=1.NG A = PT{I)DO 8 J=1•NGB = PT( J JCALL INTERP(A.B)KK = J + NG* Cl-l)DO 8 K=1*8 GN(K*KK ) = N(<)DNAK.KK) = DNDA(K)DNB(K.KK) = DNDB(K)DNAA(K.KK) = DNDAA(K)DNAB(K.KK) = DNDAB(K)

8 DNBBCK.KK) = DNDBB(K)CC STORE INTERPOLATING FUNCTIONS AT GAUSS POINTS ALONG AN EDGE C

DO 10 J = 1.NG A = PT( J )CALL INTERPtA.-l.)DO 10 1=1.3

GR* 36GR. 37GR. 38GR. 39GR. 40GR. 41GR. 42GR. 43GR. 44GR. 45GR. 46GR. 47GR. 48GR. 49GR. 50GR. 51GR. 52GR. 53GR. 54GR. 55GR. 56GR. 57GR. 5BGR. 59GR. 60GR. 61GR. 62GRW 63GR. 64GR. 65GR. 66GR. 67GR. 68GR. 69GR. 70 187

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IF ( I

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Reproduced

with perm

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copyright owner.


prohibited w

ithout permission.

SUBROUTINE MODIFY(ROW.VALUE) GRN 1c GRW 2c THIS SUBROUTINE MAKES CONSTIROW) = VALUE* GRM 3c GRW 4

COMMON COEF< 100.20)•CONST(I 00 I.MBAND.NEQ GRW 5INTEGER ROW GRW 6

c GRO 7c MODIFY COLUMN GRW 8c GRW 9

DO 4 J=2.MBAND GRW 10I = ROW - J ♦ I GRW 11IF (I *LE* 0) GO TO 2 GRW 12CONST!I) = CONST!I) - COEF!I•J )*VALUE GRW 13COEF!I*J) = 0» GRW 14

c GRW 15c MODIFY ROW GRW 16c GRW 17

2 1 = ROW + J - 1 GRW 18IF II •GT• NEQ > GO TO 4 GRW 19CONST!I) = CONST!I) - COEF!ROW*J)*VALUE GRW 20COEF!ROW*J) = 0* GRW 21

4 CONTINUE GRW 22c GRW 23c FIX UNKNOWN AND NORMALIZE DIAGONAL GRW 24c GRW 25

CONSTIROW) = VALUE GRW 26COEF!ROW.1) = 1* GRW 27RETURN GRW 28END GRW 29

205

SYSTEM

OF EQ

UATI

ONS

2 0 6

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6,0.

)

0)

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DNDB

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D8(5

)=.2

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A)

2 1 5

SftSSS3Sf5SSS55SSS5!5S!8SSSSSSS3SSS5SSS* a * a ? a * 5 a a a a " " " * a a a " 5 a * a a a a 5 a a 3 5 5 5 5O O O t f O O O O O O O O O O O O O O O O O O O O O y J O O O O O O O O U )

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k H N n « i n « r > oUJ

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CD 03 0o a e z z : o a i


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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Reproduced

with perm

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copyright owner.


prohibited w

ithout permission.

SUBROUTINE MATRIX(KK) GRM 1c GRH 2c THIS SUBROUTINE FORMS THE MATRIX AND VECTOR FOR AN EIGHT NODE QUADRILGRV 3c AND INTRODUCES THEM INTO THE OVERALL MATRIX AND VECTOR. GRV 4c GRM 5

COMMON COEF!21>22),CONST!2 1 >.NEQ GRM 6£. /BLKi/CC.II GRM 7& /BLK2/N0DE!24*8> GRM 8& /BLK3/XX(21).YY(21) GRM 9& /BLK4/AREA*A I! 16) GRM 106 /BLK5/X(8)*Y(8) GRM 11& /8LK6/DETJ•DNX(8)•DNY! 81*DNXX! 8 1.DNXY18I•DNYYf8) GRM 12& /BLK.7/GNI8*16)*MT!4)*NG*NNG GRM 13DIMENSION HI18*8*16)*H!8*8)*FI(8*16)*F(8) GRM 14

c GRM 15c DEFINE LOCAL COORDINATES GRM 16c GRM 17<0II<\lOo GRM 18

K = NODE!KXoI) GRM 19X!I) = XX!K) GRM 20

2 Y!II = YYIK) GRM 21c GRM 22c COMPUTE INTEGRANDS GRM 23c GRM 24

DO 10 K=1«NNG GRM 25CALL DERIV!KI GRM 26AI!K) = DETJ GRM 27IF 1 I I sEQa 0) GO TO 6 GRM 28PS I = 0* GRM 29DO 4 1=1*8 GRM 30

4 PSI = PSI + DNXXtI)*CC*X!I)*X!I 1/6. ♦ DNYY!I I*CC*Y!11*Y!I I/6. GRM 31& ♦ DNXY!I)*CC*X!I)*Y!I)/3» GRM 32GO TO 8 GRM 33

6 PSI = CC GRM 348 CONTINUE GRM 35

217

DO 10

2 1 8

a a 5 a a a 5 a a a 5 * 5 5 a a « 5 5 3 5 a a a 5 « a 3 5 5 a a ? a ?a (5 U (P o 15 15 O tf D 13 15 O O O 15 tf 1} O 15 tf O O l > 0 0 0 0 0 0 0 1 3 0 0 0

CD II * • I- UJ » • • X •> • • CO H— m a •+ x 3 ** .* -< x z

II II < II « II II 3 V) II O II II 3 II ~#-n*||-o^ju»|)-io«j«»- uO CD CD < CO II * * II 00 II O <0 II “> U^ ,* .* ~ ~ -< • 3II II Ul X X ** X X •* OO O 0 3 O 3 O O D w O 3 O O D « O < Q O < < Q y ) O O o i L Q i / > Q O o i aI*

CD < O CO z

o ~3 Z I- OD (A CD

II Z 0 O O 3 * 0 0


L =

NGDE

(KK.

J)20

COEF

(K,L

> =

COEF

(K.L

) +

HCI.

JI

THIS

SUBR

OUTI

NE

CALC

ULAT

ES

THE

JACO

BI AN

DETE

RMIN

ANT

AND

2 19

S £ S S S S S S S S £ S S S S S S S 3 S S S S S : S S S S S S S S S SD O U O O O O O O O O O O O t f O O O O O U ' J O O U O O O O U U O O O O

X ~Z >0o -

■ co ro 5 5 n X X > > o» * * # uA _ ~ - <X X X X ■>

- aX • 0 S ■) *

N MJ « Z X so >i - II II II II CM II II II II

X T >ii t a


DNX(

I)

= <D

NA =

(DNB

CI«K

)*XA

-

DNA(

I«K)

*XB)

/DET

J

S f t S S ? ; 5 5 : 8 S 5 S ? 8 S S 8 8 S S S 8 S S S S 3 S S 8 5 S S SI S S S S S S S S S S S 5 S S S S 8 S S S 8 S S S 5 S S S S S S S S S

S 2 5 J S S

* § *‘15 _•

l l m iii n H H H H H i» n H n H

S M H l h U H i

= = = 1 1 1 1 1 1 1n <

n w cvi n r< n n « # * < < < < <! r .C (VI (M

< <* I I)) n wI A A

' I I

11 < <# *

n oi in •* rw ~ w -< < < < *i i i i iw w n Z

A n n A

(VI fO (VI l*> .(3 - - (VI •< < < < *

iii: < < <l * * Z

II II II II II II II IIH N n ' i N r i x i g n nzziiiiiiib< < < < < < < < < 0

I + W W » - W W W «

< < < < < < <

AI(3*

2 ) =

(A(1

,2)*

A<3.1>

- A(

1 •1 )*

A(3.

2)l/

DET

AI{3*

3)

= «A

(1,1

>*A{

2.2>

-

A(1,

2)*A

(2.1

))/G

ET

221

p * A i n < l A « M D 0 t O ' < N n 4 l A « M 9SNNNN>NNNNgOOO«0<OCOeOeOOflO

* * ■ » » ■ » » * * * * * ■ * * » »a a a a a a a a a a c r c r a c r a a a c r0 0 0 0 1 5 0 0 0 0 1 3 0 0 0 0 1 5 0 0 0

>• >• > z z z a q o * * *

X X Xz z z a o o # * *

u o o * * *in *> w-« <\j m

44 W 4. 111< co a >< < co **Z Z Z I-O O Q O Q <• >- II II II «n ain

o < < «*4 II II I

O <4 w— X >

X Xo z za a o

-* Z I-44 a ui <>- 3 »- * a>. k - a o oz ui a o h zOE)lLII)lll O N *

OVER

ALL

MATRIX

AND

VECTOR,,

222

* « < \ i t o * i n < o N c o o > o -

B * B Ba a a a o o o o S S S S S S S S S S S S S S g g S g S S g S S S S S S S S S So o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

UI u Z DM Qi - o 3 a o t- a z

< C\J '► UI CM < *. O w . L■> o x al z x < ;, S \ \ !« (\J ro «■ ‘C X X X cj _i _j _i :3 a) is a) t» N N N J

ro -« (M

X X X I I I ~

ro ~ <m >• >■ >•

oi ro •

x x >

) — wX X X >

to -« r

> > 5

II II II II II II < •

* CM —< - < « C X w

c < < 5 m m u o o < < i -


LOCATE

THE

CENT

ROID

(XC1>

♦ X(

2)

+ X(3> >

/3*

2 2 3

t n *- ' 2 n M UJ -* - UJ jII Q * II O •- o - n o « z

z h z w avO W <0 U. D

It Z II UJ I- OO O O O Hi zO t t O Q _ j u a i u

SUBR

OUTI

NE

MODI

FY!R

OW

22 4

h AID«IA>ON(

■ » » » * » * » » * » * » » » * » » » »a S s o s c a o n r a a s a i i c a a B i ro o o o o o o o u o o o o o o o o o o o


THIS

SUBR

OUTI

NE

SOLVES

A

SYSTEM

OF EQ

UATI

ONS

2 2 5

< N K I 4 l A i O N 00 9 O •

S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S c0 0 0 u 0 0 0 0 l 9 0 0 l 9 0 0 0 0 0 0 l 9 l 9 0 0 0 0 0 0 0 l l l 0 0 0 0 l } l ! ^

X < IIO UJ u a _i

-> N O *1

„ UJ •C Z ~ UJ» * • < oI + * W CDn co o < jtt <- < H. « z

CO II «II N II ~ II I-I o z

UJ

• -> * aa • w x -«■

• UJ -> < UJ n :O Z -» I- -i 'UJ • II ~ t• * < II COII *- t *■» II ■» — II« <fIL O

a n • a n ** zw O(0 U


IF <A

8S(A(K,90

) .GT.

I.E-

07)

GO TO

10

WRITE

(6.8)

( I.IC.At I

,K ) .

I = 1 ,

NEQ

)

(40X

*'LA

RGES

T PIVOT

NUMBER

IS AI

2 2 6

I B B B B B B B B ]a a a a a a a a a a a a a a a c e a a a a a a a a ao o o o u a o u o t f t f i j a i j o a o o o i j u o t f o i )

a z o o u. u

* * <• o a* Mw K I< <s oc -»o — O “» otit V I UJ • UJz • z *■* z« M M • w • •*4 w Mi ^ < *4 O* «t ** * VII CQ II II II II•-•II

M- -< X « ^ .« II II • I + II 3 «•— « (/j — z o oUiO CO II z

Z Z II **0 3n o w * - '

<0 II• « I- o> «-> UJ z j co a uj

Reproduced

with perm

ission of the

copyright owner.


prohibited w

ithout permission.

SUBROUTINE SOL IT<ITMAX*EPS) GRW 1c GRW 2c THIS SUBROUTINE SOLVES A SYSTEM OF EQUATIONS GRW 3c BY GAUSS-SEIOEL ITERATION# GRW 4c GRW 5

COMMON A(21•22)•X(21)»NEQ GRW 6c GRW 7c NORMALIZE WITH RESPECT TO THE DIAGONAL GRW 8c GRW 9

N1 = NEQ + 1 GRW 10DO 2 1=1. NEQ GRW 11AA = A( I. I ) GRW 12DO 2 J=1.N1 GRW 13

2 A<I.J> = A(I.J)/AA GRW 14c GRW 15c ITERATE ON THE VECTOR GRW 16c GRW 17

DO 8 IT = l.ITMAX GRW 18II = 1 GRW 19DO 6 1=1.NEQ GRW 20xx = xc i j GRW 21X(I) = A(I.N1 ) GRW 2200 4 J=1,NEQ GRW 23IF (I #EQ# J) GO TO 4 GRW 24X(I> = X(I) - A(I.J)*X(J) GRW 25

4 CONTINUE GRW 26c GRW 27c CHECK FOR CONVERGENCE GRW 28c GRW 29

IF (ABS(XX—X(I)) #LE# EPS) GO TO 6 GRW 30II = 0 GRW 31

6 CONTINUE GRW 32IF 1 I I #EQ# 1) GO TO 10 GRW 33

8 CONTINUE GRW 3410 CONTINUE GRW 35

227

2 28

» *a ao o

t- oui z a ui


Reproduced

with perm

ission of the

copyright owner.


prohibited w

ithout permission.

SUBROUTINE CHECK GRW 1c GRW 2c THIS SUBROUTINE FORMS THE AMPLIFICATION MATRIX GRW 3c AND COMPUTES ITS EIGENVALUES. GRW 4c GRW 5

COMMON COEFC21.22).CONST!21).NEQ GRW 6S /BLKIO/DICI21*21) GRW 7DIMENSION C(21•21)*D(21) GRW 8

c GRW 9c SEPARATE MATRIX INTO A DIAGONAL MATRIX PLUS ANOTHER MATRIX GRW 10c GRW 11

00 2 Is1.NEQ GRW 12DI I) = l./COEFC1*1) GRW 1300 2 J=1« NEQ GRW 14

2 C(I.J ) = COEFII.J) GRW 1500 4 1=1,NEQ GRW 16

4 C< I, I > = 0. GRW 17c GRW 18c FORM AMPLIFICATION MATRIX GRW 19c GRW 20

00 6 1=1.NEQ GRW 2100 6 J = 1•NEQ GRW 22

6 OiC(I.J) = D(I)*C(I,J) GRW 23c GRW 24c COMPUTE THE EIGENVALUES OF QIC GRW 25c GRW 26

CALL HHOLDINEQ1 GRW 27CALL EVAL(NEQ) GRW 28RETURN GRW 29END GRW 30

229

SUBR

OUTI

NE

HHOL

D(N)

2 3 0

- N n « m « N < o » o - « n * | n « S « 2 o - « n s m !o s ® o !o - « | n S|r|

S S S ’ S i S ' S S S S i ' S l ' S S S ’ S S S S ’ i S S S i S S S ’l ] l 9 l ) 0 U I 9 0 0 0 0 U 0 0 0 U 0 l 9 l 9 l 9 U t 9 l 9 l 9 l 9 0 t J 0 0 l 9 0 0 l 9 l 9 l 9 U

X <- ac a h

uj w X 3 X O >• X

♦ a z ~x o a -w w u +< # x* *»> «* ««» (A~ »■* 0» oo x i a (/) • tu i <

a xat a a o

2-1 - <3 O □ Oa <to - 3 O

cm a t

a ca t

Z X 10 CO COO -I z * *2 Q U J J J X V X < <□ M UJ UJu a ct a

w « w * - < « - <Z 2 OJ 2 t- w Z• O O •-‘ • ♦ o i l- « O H — X ♦ • U

N i l « <a — x • n o — — **• i x - • ♦ h — o (0 V) xZ O O © X — — CA — II O •

O C O Z < —• M < —II - 3 < II II CM II + - ~ +

<A II X Z Z XN O 0 — 0 O — O O U . WZ O X l A X O I - C A O W U — <a

o CM

• *■*©- o

VO II <


A<I.J

)

QQ =

QQ +

DBLE

(DUM

>*DB

LE(A

(J,K

)>

Q(I)

= QQ

ACON

IS

= S

♦ DB

LE(Q

CI)>

*DBL

ECA<

I.K)

)

2 3 1

■ * * ■ 2 2 2 3_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ B B B B B B B B B B Ba a a c e a a a a a a a a a a a a a a a a ai j O O C I U O t f O U O O O O U O O U D O U t f

<- wZ wO (/) U 03 # < tn o

H ~

Z ~ - • • <* * II UJ II II 3 *-*->'* Z -J -> 00 CO • H « Z 0 0 * 0 o a < o

• z ►-Z * —w — <8 « II O « II-* ~ - *- 3- Z — h o- - O < UJ zo o o < a uj

COMP

UTES

THE

EI

GENV

ALUE

S OF

B B B B B B B B B B B B B B B B B B B B X B B B B B B B B B B B B X BB B B B B B B B B B a a B B B B B B B B B B B a B B B B B a B a B a aa O O O O O U O O O O O O O O i S O O O O O U U O U O O O U O I J O O U O

o ~ ----—

Z J ~ < c~ < * >t- z j \ ;3 0 10 - (O O S ~ 'B < Z * «CD *-• O J L3 0 X DO UJ .

II II II II I ~ II II z. X <

« uj o a

HI- ~O J ~' • • z «► • COJ — — MI ~ ~ II II

z — W W a - ~ ~ X X J J X O X - I

X _l ~ II II IIII <f w w «0 « COb « i c n «

~ O O l L U . O t t Q . O a Q . X J Q M M Q I O U J O U I U I


ABS(

A<1))

ABS(

MAX-

MIN)

2 3 3

I X X X X X X ]_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ B X X X X X X X X X X Xa a a a a a a a a a a c e a a a a a S a a a i r 0 0 0 1 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

X + b- < O Z h t

a> x o• oH I *■*• X O H

• UJ

z - *» w z aI UJ X X < < • X X H« O H (A • - „ — ■'JO O DD '*>♦ 2 J“ ->V)n»»X«>»X<U.*-V) <* • CO <~ zii ii n it • it z o — w < < a x ii o ii ii a w a

X Z I- t II H “ H II ^ I I --------- Z XU. J I- o< M M M w u. OI L l L U . * - ' 0 < ' - U . o : u i

<\l <0 00

V IT A

Born October 21, 1946, Gary Richard Wooley lived in the New Orleans area until entering college. After being

graduated from East Jefferson High School in May, 1964, he

enrolled at McNeese State College in Lake Charles, Louisiana, in September, 1964, and transferred to Louisiana State

University in Baton Rouge in January, 1965. In January, 1969, he received the Bachelor of Science degree in Mech a n i

cal Engineering from Louisiana State University, after which he entered graduate school in the Department of Engineering Science as a research assistant. He received the Master of Science degree in Engineering Science from

Louisiana State University in May, 1970. While pursuing an education he worked four summers for various companies

in the oil industry, was a National Science Foundation

Trainee for three years, and was a graduate teaching assistant for three and a half years. He is presently a Louisiana State University Dissertation Year Fellow and a candidate for the degree of Doctor of Philosophy in the

Department of Engineering Science.

234


EXA M IN A TIO N AND THESIS R E PO R T

Candidate: G a ry R ic h a rd Wooley

M a jo r Field: E n g in e e r in g M echanics

T it le of Thesis: A F i n i t e E lem en t A n a ly s is o f S te a d y , Two D im e n s io n a l, In c o m p re s s ib leL a m in a r F low

Approved:

D ate of Exam ination:

November 2 2 , 1972

M ajor Profeasonr and Chairman

Dean of the Graduate School

E X A M IN IN G C O M M IT T E E :


a finite-element analysis of steady, two-dimensional

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