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This is to certify that the

thesis entitled

REDUCTION OF THE NUMBER OF INDEPENDENT
VARIABLES AND OPTIMIZATION IN SWIRLING FLUID FLOW

presented by

Michael Jerome Brink

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Mechanical Engineering

H zv, h/WK/épé

Major Agrofessor

Date May 12, 1971

 

0-169

 

 

 

T JUN 031995

W
/ :17 1‘ Q

 

 

 

 

 

ABSTRACT

REDUCTION OF THE NUMBER OF INDEPENDENT VARIABLES
AND OPTIMIZATION IN SWIRLING FLUID FLOW

By

Michael Jerome Brink

 

The major objective of this investigation is to determine so—
lutions to the system of fundamental equations governing inviscid, in—
compressible swirling flow in (R, 6, Z) space by using group theory to
reduce the number of independent variables and obtaining the solutions
for the resulting system of ordinary differential equations.

Transformations obtained by the application of continuous
one-parameter group theory are applied to the fundamental system of
non-dimensional equations for conservation of mass, conservation of mo-

mentum, and conservation of energy. The number of independent varia—

 

bles is reduced from three to two to one using the absolute invariants
determined for each transformation group. The resulting system of non—
linear ordinary differential equations is then solved numerically by
Hamming's modified predictor—corrector technique. The one—dimensional
solutions are transformed back to the three—dimensional space using the
transformations obtained from the application of the continuous one-
parameter group theory.

Computer curves are presented for both the one—dimensional
and the transformed three—dimensional solutions to the conservation of
mass and conservation of momentum equations. Two continuous one—
parameter transformation groups and their invariants which were discov-

ered and then utilized to accomplish the reduction in the number of

Michael Jerome Brink
independent variables for cylindrical polar coordinates are also pre—
sented.

A Mayer type optimization problem is solved using the one—
dimensional representations for conservation of energy and the swirl
parameter to optimize the one—dimensional representation for pressure.
The one-dimensional optimization results are transformed back to the
three-dimensional space using the transformations obtained from the ap—
plication of the continuous one—parameter group theory.

Computer curves are presented for the optimization results in

both the one—dimensional and the three-dimensional space.

 

 

 

REDUCTION OF THE NUMBER OF INDEPENDENT VARIABLES

AND OPTIMIZATION IN SWIRLING FLUID FLOW

By

Michael Jerome Brink

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

197 l

 

 

 

 

4M2

PLEASE NOTE:

Some Pages have indistinct
print. Filmed as received.

UNIVERSITY MICROFILMS

 

i mm/

To Marylyn, David, Bridjette, and Cynthia —

for infinite patience and understanding

ii

ACKNOWLEDGMENTS

The author wishes to express his sincere appreciation to his
guidance committee chairman, Professor M. Z. v. Krzywoblocki, for his
valuable guidance during the period of graduate study and research.

He also wishes to thank Professors C. R. St. Clair, Jr.,
Terry Triffet, Robert Wasserman, and C. P. Wells for serving on his
guidance committee.

Thankful acknowledgment is extended to Mr. W. Arntson for his
assistance in computer programming and Mrs. J. Harris for her excellent

typing.

 

'5

TABLE OF CONTENTS

Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . . . . . . vii
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
I. FUNDAMENTALS . . . . . . . . . . . . . . . . l
l. I Introduction. . . . . . . . . . . . l
l. 2 Scope of This Investigation . . . . . . . . . . . . . 2
1-3 Group Theory. . . . . . . . . . . . . . . . . . 3
l.h Optimization Techniques . . . . . . . . . . . . . . . ll
II. MATHEMATICAL SWIRLING FLOW MODEL . . . . . . . . . 15
2. 1 Type of Fluid Flow and Coordinate System. . . . . . . 15
2. 2 Fundamental Equations . . . . . . . . . l6
2 3 Boundary Conditions and Swirl Parameter . . . . . . . l9
2. h Non—Dimensional Equations . . . . . . . . . . . . . . 21
III. REDUCTION OF THE NUMBER OF INDEPENDENT VARIABLES . . . . . . 2h
3.1 Three to Two Reduction in the
Number of Independent Variables . . . . . . . . . . 2A
3.2 Two to One Reduction in the
Number of Independent Variables . . . . . . . . . . MA
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . 60
IV. SOLUTIONS FOR THE CASE .5 = -1. . . . . . 62
A. l One—Dimensional Solutions of
the Ordinary Differential Equations . . . . 62
h 2 Three—Dimensional Pressure and Velocity Curves. . . . 68
h.3 Summary and Recommendations for Further Research. . . 70
V. OPTIMAL SOLUTIONS. . . - 76
5 1 Application of Optimization Techniques
in X Space. . - 76
5 2 Three—Dimensional OptimaI Pressure
and Velocity Curves . . . - . - - - - - - - - 82
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . 85
. 88

BIBLIOGRAPHY . .

 

iv

 

it...

 

APPENDICES

APPENDIX A:

APPENDIX B:

APPENDIX C:

APPENDIX D:

APPENDIX E:

TABLE OF CONTENTS

NUMERICAL DATA FOR THE SOLUTIONS TO THE

SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS.

CALCOMP PLOTTER ROUTINE.

LEAST SQUARES POLYNOMIAL CURVE
FITTING PROGRAM. . . . . .

PRESSURE AND VELOCITY PROGRAMS
AND NUMERICAL DATA . .

PROGRAMS AND NUMERICAL DATA
FOR OPTIMUM FUNCTIONS. . . . . . . .

Page

92

113

1&1

lbs

Table

Tl Numeric:
of Ordii

C.1 Curve F‘
C-2 Curve F

13.1 Sample

L-rJ
H

Numeric

tfil
n)

Numeri<
‘ and Ta]
,
T

 

|I
T't.

LIST OF TABLES

Page

Numerical Data for Solutions to the System
of Ordinary Differential Equations. . . . . . . . . . . . . 108

Curve Fitting Data for the Interval O.lOOOOSXSO.A653l . . . 129

Curve Fitting Data for the Interval O.h653lSXSl.OOOO. . . . 135
Sample Data Pressure and Velocity Functions . . . . . . . . lhh
Numerical Data for Optimum One-Dimensional Functions. . . . 1&8

Numerical Data for Optimum Pressure
and Tangential Velocity . . . . . . . . . . . . . . . . . . 15h

vi

2.1 Cylir

2-? Velm

Iva

V1

A
:U

 

igure

L7
§.8

;.10

1.11

LIST OF FIGURES

Page
Cylindrical Polar Coordinates .7. . . . . . . . . . . . . . 17
Velocity Components . . . . . . . . . . . . . . . . . . . . 18
(R, 8, Z)+(nl, n2) Reduction R—Momentum Equation (2.h.7). . 39
(R, 6, ZT-an, n2) Reduction e-Momentum Equation (2.h.8). . A0
1’ n2) Reduction Z—Momentum Equation (2.h.9). . Al

(R, 6: Z)—*(n ,
Mass Equation (

(R, e, z)+(n

n2) Reduction Conservation of
2.h.10). . . . . . . . . . . . . . . . . . . M2

(R. e, Z)+(n , n ) Reduction
Bernoulli Equation (2.h.ll) . . . . . . . . . . . . . . . . A3

(R, 9, Z)+(n , n ) Reduction Swirl
Parameter Equation (2.h.ll) . . . . . . . . . . . . . . . . M3

(n1, n2)+(X) Reduction Equation (3.1.66). . . . . . . . . . 5h

 

(n1, n2)a(X) Reduction Equation (3.1.72). . . . . . . . . . 55
(n1. n2)+(X) Reduction Equation (3.1.77). . . . . . . . . . 57
(n1, n2)+(X) Reduction Equation (3.1.82). . . . . . . . . . 58
(n , n )+(X) Reduction Equation (3.1.83)

T2“ 59
an Equation (3.1.8h) . . . . . . . . . . .
One—Dimensional Functions . . . . . . . . . . . . 67
Pressure Curves . . . . . . . . . . . . . T2
Radial Velocity Curves. . 73
Tangential Velocity Curves. . . . . . . . . . . 7h
Axial Velocity Curves . . . . . . . 75

76

Real Space Functions. . . . . . . . . .

G2 Curves S Parameter . . .
GA Curves S Parameter . . .
Optimum Pressure Curves .

Velocity Curves for Optimum Pressure.

viii

Page
83
8h
86
87

 

p-Pr
rieiz'Ph
u,V,W—Ve

D-De

 

NOMENCLATURE

Pressure

Physical coordinates, independent variables

Velocity components

Density

Velocity

Internal energy and base of the natural logarithm system
Total energy change

Bernoulli constant

 

Lagrange multipliers

Augmented function

Functional forms of optimal constraint equations
Swirl aspect ratio = r2/z2

Tangential energy fraction = v2/(h/p)

Swirl parameter = n1 N2

Axial swirl length

Tangential velocity at the outer swirl radius
Dimensionless independent variable = r/zO
Dimensionless independent variable = z/ZO
Dimensionless radial velocity component = u/vo
Dimensionless tangential velocity component = v/vO

Dimensionless axial velocity component = w/vo

2
Dimensionless static pressure = p/(DVO)

ix

H — Dimensionless Bernoulli constant = h/(pvi)

Ti — Constants

a - Group parameter

1 — One—parameter group

Q — Differential form of k—th order in m independent variables
5 - Arguments of the k—th order differential form

1. — Independent variables, absolute invariants of Al

i — Absolute invariants of Al

E - Group constant = t2/tl

d - Group constant = tA/t

2

P — Two—dimensional representation for the radial velocity
component

- Two-dimensional representation for the tangential velocity
component

'3 - Two—dimensional representation for the axial velocity
component

IA — Two—dimensional representation for static pressure
k - Constant

1 — Constants

c — Group parameter

L — One—parameter group

i — Independent variable, absolute invariant of El

7 - Group constant = Yl/k

 

Absolute invariants of B1

One-dimensional representation for the radial velocity
component

One-dimensional representation for the tangential velocity
component

One—dimensional representation for the axial velocity
component

One-dimensional representation for static pressure

_ 2 2 = 2
Constant - G1 + G3 G

Denotes differentiation with respect to the independent
variable

xi

 

1.1 Introduc
Sw.‘
engineers Si}
cently, it 5
ed in Wheele
"black hole"
Ge

which the t:
that the £11
00W in H1
and rotatin
’I

in the gene
ferential .
Tyne“ tET
Were chPe
“mponents
ble [6, 11

to the flu

are Obtai;

I. FUNDAMENTALS

Introduction

Swirling fluid flows have been of interest to scientists and
gineers since 1797 when Venturi studied the free vortex. Most re—
itly, it seems that swirling flow may appear in the motion encounter-
in Wheeler's theory of gravitational collapse [51] and the so—called
Lack hole" in space.

Generally, swirling flow may be categorized as any flow in
Lch the tangential velocity component has a finite magnitude such
it the fluid has a macroscopic rotary motion. Typical flow examples
:ur in Hilsch tubes, vortex amplifiers, tornados, large fuel tanks,

i rotating fluid machinery.

The system of fundamental equations describing swirling flow
the general case includes three—dimensional, nonlinear partial dif-
'ential equations which are not amenable to solution by standard ana—
iical techniques. Therefore, investigators of swirling type flows
'e compelled to make various simplifying assumptions. All velocity
ponents were assumed to be functions of only one independent varia-

[6, 11, 1h, 15, 16, 30, 36, A7] or particular velocity components
e assumed to have only certain special forms [13, 29, 35].

In this work, the major objective is to determine solutions
:he fundamental equations governing swirling flow. The solutions
obtained by accomplishing a reduction in the number of independent

1

 

 

variables uti
ing ordinary
The
one-parametei
differential
transformed
dependent va

differential

Ti
dimensional
of momentum

C
reduce the
c°ml>lish t1
groups are
fTuition by
tion. The
linear. H

temine ti

dimenSiOm

Stredried '

dimenSion

Tilhensi0r

The aTM:

 

 

2

bles utilizing one-parameter group theory and solving the result-
rdinary differential equations.

The minor objective of this work is to discover two sets of
arameter transformation groups such that the syStem of partial
rential equations for the model in three independent variables is
formed into a system of partial differential equations in two in-
dent variables which is then transformed into a system of ordinary

ential equations.

rchpe of this Investigation

The fundamental fluid flow model is posed using the three-
asional system of equations for conservation of mass, conservation
amentum, and conservation of energy.

One—parameter group theory is applied twice in succession to
:e the number of independent variables from three to one. To ac—
Lish this reduction, two sets of one—parameter transformation
is are discovered since the one—parameter group theory allows a re~
Lon by one in the number of independent variables per transforma—
. The resulting system of ordinary differential equations is non—
nu Hamming's predictor—corrector numerical method is used to de—
Lne the solutions to this system of equations.

Calculus of variations techniques are applied to the one—
sional system of equations to determine an Optimum solution con—
ned by a transformed swirl parameter.

The one-dimensional solutions are then transformed from one-
sional space to two-dimensional space and next to three-
sional space using the two sets of transformations discovered by

application of the continuous one-parameter group theory.

 

 

1.3 Group Th1
The

a particular

particular 0;

1. Closure.

n)

. Associat:

w

. Left Ide

4;-

V1

‘ COIHmute

Not all grl

fifth aKid

[21]. The

Trans forms

 

 

The axioms [2, 18] which a set of elements must follow under

ticular operation are stated below. A star (u) symbol denotes the

cular operation between two elements.

losure.

ssociativity.

.eft Identity.

.eft Inverse.

Tommutativity.

If a and b are two elements of the set, then

a*b = c is also a unique element of the set.

When three or more elements are formally manipula—
ted using the defined operation, the order of mani-
pulation makes no difference. Hence,

(a*b)*c = a*(b*c).

Among the elements there is an identity element,
denoted by I, with the property of leaving the ele—
ments unchanged after formal manipulation with the

defined operation. Therefore, I*a = a.

Each element, a, in the group has a left inverse

a“1 such that a'l*a = I.

When two or more elements are formally manipulated
using the defined operation, the order of manipula—

tion may be reversed. Hence, a*b = b*a.

‘11 groups satisfy the fifth axiom. Those groups that satisfy the

axiom are knOWn as Abelian or commutative.

Continuous one—parameter groups are multiplicative groups

The groups of interest in this work are continuous one-parameter

formation groups. The defined operation for multiplicative

 

 

transformatio

tions. Hence

Aset of tra
tion of tran
values a1 at
such that A:

and

Having defi

Cept is i101

1' Closure
2- T5soci

3- Left 1

 

 

h

sformation groups is the successive application of the transforma—

3. Hence a point x is taken into a point X as follows:

X=A X=A (A x)EA A x
a2 a1 a2 a1 ,

 

2

l
where a and a are parameters.

of transformations is said to be closed under the defined opera—

of transformation multiplication if, given any set of parametric

l 2

s a and a , a set of parametric values a3 can always be found

that A 3 is a unique member of the set of given transformations
a

A A = A .
al a2 a3
1g defined the operation for closure, the transformation group con—

is now formulated in terms of the group axioms.

Ilosure. A A = A l. .l

)x=(A A )A x (1.3.2)

A a1 a2 a3

lssociativit . A A
y .2 .3

al(

left Identity. A transformation I exists such that

IA x = A I x = A x 1.3.
a1 a1 ( 3)

Any transformation which leaves a point unaltered

is defined as an identity transformation.

 

 

T. Left lnve:

@
Q is unaltere
E.

Tr: i.e.
QTXK...

then the fun

De

Twit y).
following (it

knowing T
To“ restm

of the for]

 

5
eft Inverse. An inverse transformation A_i belonging to the set
a

exists such that
A_]l_Alx= Ix=x. (1.3.11)
a a
Definition 1.3.1 - If a numerically valued (scalar) function

unaltered by an r—parameter continuous group of transformations,

.e.
(xl,..., xm, yl,..., y“) = Q(il,..., Em, yl,..., in), (1.3-5)
the function Q(x, y) is called an absolute invariant of Ar'

Definition 1.3.2 - The m+n functions, Ql(x, y),...,
K, y), are said to be functionally independent if and only if the

ring determinant is non-vanishing.

39.1... .aQrfl
3X1 Bxi

. #O(i=l,...,m;j=1,...,n) (1.3.6)
E. ”391131

ring the approach developed by A.J.A. Morgan [Ah], attention is
astricted to a one-parameter continuous group of transformations

2 form:

A : (1.3.7)

F3 = fJ(yl,..., yn; a) (j = l,..., n; n21)

 

 

where a is a

form a subg:
and the dep‘
tions. The
up to any r
atives of 1
(1.3.7), ti
one-perms

known as e

SOT-Ute inv

and

whue R

6

'e a is a numerical parameter. The transformations
i1 = fl(xl,..., xm; a) (1.3.8)

1 a subgroup SA of Al. The xi and the yJ denote the independent
the dependent iariables of a system of partial differential equa—
lS. The yj are considered to be differentiable functions of the xi
,0 any required order. If the transformations of the partial deriv-
'es of the yj with respect to the xi are appended to those of

i-7), the resulting set of transformations also forms a continuous

-parameter group, Ak New groups constructed in this manner are

l'
n as enlargements of the group Al.
The group Al (1.3.7) has m+n-l functionally independent ab—

te invariants

n(xl,...,xm),...,n (x1 ..,xm) (1.3.9)

1
l n l m
g1(yl""= yn, xiv”, xm),..-, gn(y y , x x ) (1.3.10)
it is possible to choose them such that the Jacobian

8(gl,..., gn)

 

 

0 (1.3.11)
3(yla"'9 yn) aé
R 8(nl,..., nm—l) ] = m—l (l 3.12)
3(xl,..., Km)

e R denotes the rank of the indicated Jacobian matrix.

 

Th

tions of the

where the g

in (1.3.10

yj) lmpliC
to be exae
I

defined as

The condi

lent com?

The ml"
the Subg
(1.315)

absolute

ally im

 

7

The yj and i3 are considered to be implicitly defined as func-

ons of the x1 and i1 by the equations

zj(xl,. , xm) = gj(yl,. , y , xl,. , xm) (1.3.13)
235:1, ,.em) = gj(§rl,...,yn, 21,...)21‘1) (1.3.111)

ere the gj are the absolute invariants of the group (1.3.7) indicated

(1.3.10).

Theorem 1.3.1 — A necessary and sufficient condition for the
, implicitly defined as functions of xl,..., xm by equation (1.3.13),
- l m '3 ' 1' 't1
be exactly the same functions of x ,..., x as the y , imp 1C1 y
l

fined as functions of i ,..., Em by equation (1.3.lh), are of the

,..., 2m is that

z (x1 ... xm) = 23(il,..., 'm) = 2 (il,..., Em). (1.3.15)

e condition given by equation (1.3.15) can be replaced by the equiva—

nt condition

). (1.3.16)

pendent absolute invariants of

e n -, nm—l are functionally inde

1"'
e Subgroup SA of the group (1.3.7). The equivalence of equations
.3.l5) and (1.3.16) is seen from the group theory statement that any

solute invariant of a group is expressible in terms of the function—

1y independent absolute invariants.

 

 

Pg
ential equa’t
same fimcti<
solutions 0:

W
differentia

essary to c

form,

BY equatiom
Properties
expressed

point (x1,

m indepem

WhOSe arg
are Consi

ments of

8

Definition 1.3.3 — Solutions of a system of partial differ-
;ial equations which have the property that the y3 are exactly the
1e functions of the x1 as the y3 are of the ii are called invariant
.utions of a system of partial differential equations.

When considering invariant solutions of systems of partial
?ferential equations, Theorem 1.3.1 demonstrates that it is only nec—
;ary to consider equations (1.3.13) or (1.3.1h) to be given in the

.m,

). (1.3.17)

gj(yla"" yIl, xlgn': xm) = Fj(nl,--~, nm—l

equation (1.3.11) and the assumed continuity and differentiability
>perties of the functions F3 and gJ, it follows that the yj can be
>ressed as functions of the F3 and xi in some neighborhood of the

.nt (xl,..., xm).

Y3 = yJ(F -. F . xl,..., xm) (1.3.18)

1" ’ n

Definition 1.3.u — A differential form of the k-th order in

ndependent variables is a function of the form

1 n
¢(xl,..., xm, yl,..., yn,...,8E1 ,...,aEx k) (1.3.19)
8(xl)k mm)

J
se arguments include derivatives of the yJ up to order k. The y

l - _
considered to be functions of the x . For convenience, the p argu

zp.

. 1
ts of the differential form will be de51gnated by z ,...,

 

2i

variant unde

of the group

where h is l

M is some f

i

ally invari

nation of 1

where o is

MdMisg

Variant u

of the gr

Where (1) 3

tial equ
grWP or

f0I‘Ins o

 

9
Definition 1.3.5 - A differential form ¢ is conformally in—
nt under a one—parameter group Al’ if, under the transformations

1e group, it satisfies the relation
@(Zl,..., 2P) = M(zl,..., zp; a) ©(zl,..., zp) (1.3.20)

: ¢ is exactly the same function of the 2's as it is of the 2's and

some function of the 2's and the parameter a.

Definition 1.3.6 — A differential form ¢ is constant conform—
invariant under a one—parameter group Al, if, under the transfor—

>n of the group, it satisfies the relation

3

<1>(§1,..., 2P) = M(a) <1>(zl ..., 21’) (1.3.21)

2 s is exactly the same function of the 2's as it is of the 2's

1 is some function of the parameter a.

Definition 1.3.7 - A differential form Q is absolutely in-

int under a one-parameter group Al, if, under the transformations

18 gTOUp, it satisfies the relation

¢(El,..., 2P) = ¢(z ,..., zp) (1.3.22)

3 ¢ is exactly the same function of the 2's as it is of the z'S.

Definition 1.3.8 - A system of k—th order partial differen-

 

= O is invariant under a continuous one-parameter

J

p of transformations A1

equations d

if each of the k—th order differential

. k
3 Q is conformally invariant under the transformations of Al. The

 

k .
group A1 15 t

cussed new

be a system
11(22) indepe
yli' ' I , YD V

at under tl

T
in the Syste
invariant 1x
Variant Sol

0f a system

33%

which is a
independem

the ”1‘1 ab:

those of e‘

independen

be reamed

10
up A: is the k-th enlargement of the group Al which has been dis—
sed previously.

Let

1 1 3k 1 3k n
a (x ,..., xm, y ,..., yn,...,——X———,...;—JL——J = 0 (1.3.23)
J 1 k k
3(x ) 3(Xm)
a system of partial differential equations of the k-th order in
3) independent variables xl,..., xIn and n(21) dependent variables
..., yn where each of the differential forms is conformally invari—

under the k—th enlargement of the group A1.

 

Theorem 1.3.2 - If each of the differential forms $1,..., on
he system of partial differential equations (1.3.23) is conformally
riant under the k-th enlargement of the group (1.3.7), then the in—

ant solutions of (1.3.23) can be expressed in terms of the solutions

system of the form

B (

J ”1""’ nm-l’ F

l,..., n,.. , k ,...,—k—

h is a system of k—th order partial differential equations in m—l

pendent variables ”1"" In equation (1.3.2h), the n's are

9 nm_l '
n—l absolute invariants of the group SA and the functions FJ are
1
a of equation (1.3.17).
Theorem 1.3.2 gives the conditions under which the number of

pendent variables in a system of partial differential equations can

educed by one in the process of obtaining invariant solutions.

1.11 Optimiz

Th
dard techniq
[ii]. The c
minima of f1
mined. Hem
the assembl;

or a surfac

In equatim
i
d:
fimetions

331d ’E
yk

for which

(Mi-2) s:

whieh inv
frereciom o
Ween the
It is as:

ing end (

 

ll
Optimization Techniques
The optimization techniques used in this work are the stan—

techniques from the calculus of variations as presented by Miele

The calculus of variations is concerned with the maxima and
a of functional expressions where entire functions must be deter—
. Hence, the unknown of the calculus of variations problem is
ssembly of an infinite set of points which identify either a curve
surface, depending upon the nature of the problem.

A functional form to be optimized may be defined as shown

9 E [T(X, yk)]: + iff 0(x, yk,‘y£) dx . (1.h.1)
l

nation (l.h.l), subscripts i and f denote initial and final points

d u
g E aék. The form 9 is optimized by finding that spec1a1 set of

ions
(x) k = 1,..., n ' (i.h.2)

1ich it is either a maximum or a minimum. The class of functions

3) satisfy the constraints

¢J(X.yk.yk’) =0 J=1,...,p<n (1.17.3)

involve n = n-p degrees of freedom. The number of degrees of
>m of a differential system is defined to be the difference be—
the number of dependent variables and the number of equationS.
x) are consistent with the follow—

assumed that the functions yk(

1dcmflhflmw.

 

 

The problem
Bolza proble
Tl

troducing a

and by form

Motion.

It is know
SatiSfy nor

Whid} are 1

The differ
and the E);
knowns. .1
““8 wiei

Sim-tans

12

wfixis Yki) = 0 u = 1..... q (1.11.11)
wu(xf. ykf) = O u = q + 1,..., s s 2n+2 (i.h.5)

blem as formulated in the preceding paragraph is known as the
roblem.
The Bolza problem can be treated simply and elegantly by in~

1g a set of variable Lagrange multipliers
A (x) j = 1,..., p (1.h.6)

?orming the following expression which is known as the augmented

1.

p
K=o+2 Ada (1.11.7)
J

(own that the optimal arc (the special curve optimizing 9) must
not only equations (1.h.3) but also the following equations
'e known as the Euler—Lagrange equations.
8

g—x(%K—.—) "3&10 k=1,... n (1.h.8)

yk yk
erential system composed of the constraining equations (l.h.3)’
Euler—Lagrange equations (1.h.8) includes n+p equations and un-
Therefore, the solution of this sytem of differential equa—
elds the n dependent variables and the p Lagrange multipliers

eously.

There are 2n+2 boundary conditions to which equations (1.3.3)

 

and (11.8
the bounda‘
plied by t‘

condit ion .

The trans
systems 0:

and (1.1;:

investiga
The Legen

The Leger

F0? 8. re.

first ca
2%; th
Ullmber c
second (
integra;
known a:

are val

13
id (1.h.8) are subjected. Equations (1.h.h) and (l.h.5) supply 5 of
ie boundary conditions and 2n+2—s of the boundary conditions are sup—

Lied by the following condition which is known as the transversality

)ndition.
n K n K f
[dT+ (K- 2 Ly‘)dx+z Lay] =0 (1.14.9)

1e transversality condition is to be satisfied identically for all

stems of infinitesimal displacements consistent with equations (1.h.h)

.d (1.11.5).

After an optimal arc has been determined, it is necessary to
vestigate whether the functional 9 attains a maximum or minimum value.
6 Legendre-Clebsch necessary condition is used to determine this.

e Legendre—Clebsch condition for a relative minimum is

n n 2
z )3 flf___;&y’5y’ Zo. (1.1+.10)

r a relative maximum, the inequality (1.h.10) is reversed.

There are two particular cases of the Bolza problem. The
rst case is when the integrand of equation (1.h.l) is identically
r0; that is, when 0 E O. This case is known as a Mayer problem. The
fiber of end conditions 5 is less than 2n+2 for a Mayer problem. The
:ond case is when the functional to be optimized appears entirely in
3egra1 form; that is, when T E O in equation (l.h.l). This case is
an as a Lagrange problem. Equations (l.h.8), (1.h.9), and (l.h.10)

: valid for both of these particular caseS-

Ti

to this worl

the reader :

excellent r

 

1h
This section reviews only the optimization relations relevant
this work. For a comprehensive treatment on calculus of variations,
e reader is referred to the literature on calculus of variations. One

cellent reference on calculus of variations is Akhiezer [1].

 

2.1 Type c
1
since the w

The follow:

 

Therefore

. °°mPressiT

Cm“! of
coordinat

tions Shc

 

 

II. MATHEMATICAL SWIRLING FLOW MODEL

Type of Fluid Flow and Coordinate System

In swirling flow, the fluid has a macroscopic rotary motion

e the velocity has a tangential component of finite magnitude.

following assumptions are made for this type of flow:

Viscous effects are negligible.

The dependent variables are not functions of time
(3? = o).

The fluid is incompressible (p = const.).
Gravitational and electromagnetic body force effects
are negligible.

The flow is isentropic.

afore, the class of flows of interest are inviscid, isentropic, in-

ressible flows.

Since the fluid has a macroscopic rotary motion, the natural

:e of coordinates is a cylindrical polar coordinate system. These

iinates are related to cartesian coordinates by the familiar rela-

: shown as equations (2.1.1), (2.1.2). and (2-1-3)-

r2 = x2 + y2 (2.1.1)
tane = y/x (2'1'2)
Z = 2 (2.1.5)

15

 

 

 

In Figure 2.
ahasic syst
has a unique
W211.

T

will be wri

The welocit

Figure 2.2

2-9 Fonda

in Section

Serwetion

in fluid e

tion of 11'

l6
gure 2.1, the cylindrical polar coordinates are shown referred to
ic system of cartesian coordinates. Any point not on the z—axis
unique representation (r,9,z) with the restrictions r20 and
7T.
The vector form for velocity in cylindrical polar coordinates

be written as shown by equation (2.1.h).

E = u é (2.1.h)

+ v ée + w e

I" Z

elocity components at a point in (r,e,z) space are shown in

a 2.2.

?undamenta1 Equations

The equations governing the types of swirling flow described
:tion 2.1 are based on the basic laws of conservation of mass, con—
;ion of momentum, and conservation of energy.

The conservation of mass is given by the continuity equation

lid dynamics.

'-'+ ——'= 0 Conservation of Mass (2.2.1)
(Continuity Equation)

For inviscid, isentropic, incompressible flows, the conserva—

tf momentum is given by the three Euler equations.

311 W 311 12 = _ .1. 22 r—Momentum (2.2.2)
r p

 

 

l7

b:(r,6,z)

 

 

r r cos 6

 

 

r sin 6

FIGURE 2.1 CYLINDRICAL POLAR COORDINATES

 

IL

 

18

 

 

/

FIGURE 2.2 VELOCITY COMPONENTS

 

 

The Eule‘

equation

thermody

may he 1

Q is th
to the
work, i
tationa

ESSUmPt

u 31. 1.31. W 2X. Ez__ 1 .gR

3r + r 36 Bz r ' ' pr 39 e-Momentum (2.2.3)
u3_w iLw WEE..- 122

3r r Be + z — ' p 32 Z-Momentum (2.2.h)

e Euler equations are non—linear, first order, partial differential
nations.

The conservation of energy is expressed by the first law of
ermodynamics. For the assumed model, the first law of thermodynamics

y be written as the familiar steady flow energy equation.

Q = A(e + p/p + éqz + gz) (2.2.5)

is the total energy change or difference between the heat energy added
the system and the shaft work done by the system. Assuming no shaft
7k, isentr0pic flow, no temperature variations, and negligible gravi—
3ional body forces, the quantities Q, e, and gz all vanish. For the
sumptions mentioned, equation (2.2.5) reduces to Bernoulli's equation:

h = p + %pq2 = constant. (2.2.6)

The fundamental governing equations for this investigation
a (2.2.1) — conservation of mass; (2.2.2), (2.2.3), and (2.2.h) _

iservation of momentum; and (2.2.6) — conservation of energy.

3 Boundary Conditions and Swirl Parameter

I;

The boundary conditions are all Specified at the outside swirl

lius, r0, for any a and any 2. The Specific values of the dependent

riables will be denoted as shown in the following equations. At

four independent variables

20

u = uo
v = vO
w = wO
P = P0-

(2.3.1)

(2.3.2)

(2.3.3)

(2.3.11)

have specific values at the outside swirl

as and these values are the boundary conditions for this investiga—

The swirl parameter is defined to be the product of a geomet—

similarity ratio known as the swirl aspect ratio and a kinematic

Larity ratio known as the tangential energy fraction.

Swirl Parameter

Swirl Aspect Ratio

Tangential Energy Fraction

(2.3.1)

(2.3.2)

(2.3.3.)

ituting equations (2.3.2) and (2.3.3) into equation (2.3.1), the

.parameter equation becomes

(2.3.11)

uantity h in equation (2.3.h) is determined from Bernoulli's equa-

(2.2.6).

 

 

21

rNon—Dimensional_Equations

 

For greater generality and convenience, all variables are
dimensionless. A reference length, 20, which is the axial length
1e swirl, reduces the radial and axial lengths into dimensionless
linates. Similarly, the velocity compOnents and static pressure
1ade dimensionless by comparing them to the tangential velocity,
and the tangential dynamic pressure, ovg. The dimensionless vari-

; are defined as follows:

R 2 r/zO (2.h.i)
z 2 z/zO (2.h.2)
U s u/vO (2.h.3)
V E V/Vo (2.h.h)
w E w/vO (2.h.5)
P ; p/pvg. (2.11.6)

Application of the dimensionless variables to equation (2.2.2)

is the following intermediate result.

 

 

 

2 2
Uvg 8U VVE 3U WVE 8U vo v2 _ 1 pvo 3P
+ ____ _____—-—__—
203R R20 88 ZOBZ 20 R p Z03R

result reduces to

van 17.311 was V2. if: R—Momentum (21.7)

GB + R 56'+ az ’ ET' ‘ as“

22
Similar applications of the dimensionless variables to equations
(2.2.3), (2.2.h), (2.2.1), and (2.2.6) yield the following set of non—

dimensional equations:

U 8V V 3V W 3V UV 1 SP

3R+ R e+ S—Z—+-§_ -§a e—Momentum (2.)408)
U 3W V 8W W 3W 3P

3 + R 33.+ 52-- _ BZ Z-Momentum (2-h-9)
3U U 1 8V 8W .
—- —- -* _= Di Oh. 0
3R + R + R 3 + BZ 0 Conservation of ass (2 l )
H = p +1/2(U2 + V2 + W2) Bernoulli Equation. (2.h.ll)

The quantity H in equation (2.h.11) is defined to be

H E h/pvg. . (2.h.12)

The dimensionless boundary conditions at R = R0 are:

U = U0 (2.u.13)
V.= V0 = 1 (2.h.1h)
w = W0 (2.h.15)
P = Po' (2.h.16)

When converted to dimensionless variables, the SWirl para—

meter (2.3.h) becomes

R2 V2. (2.h.17)

(. ix»)

 

i

 

23
Equations (2.h.7), (2.h.8), (2.u.9), (2.h.1o), and (2.h.ii)
are the non-dimensional forms of the fundamental governing equations

for this work.

(1‘ lat .(lll. (

 

 

III. REDUCTION OF THE

NUMBER OF INDEPENDENT VARIABLES

In this chapter, the ideas outlined in section 1.3 are ap-
plied to the non-dimensional model equations given in section 2.h. The
number of independent variables is reduced from three to two and then

from.two to one.

3.1 Three to Two Reduction in the Number of Independent Variables

The three independent variables (R,6,Z) are reduced to the
two independent variables (nl,n2). (The new independent variables
(nl,n2) are functionally independent absolute invariants of a subgroup
of the one-parameter continuous group of transformations, A1. The one-
parameter continuous group of transformations, Al, will subsequently be
referred to as the group A1.

The group Al is chosen as shown below:

R = eaii R U = eath U E = eatY P
Al: 6 = e + 9,132 7 = eat5 V (3.1.1)
2 = eat3 Z W = eat6 W

‘ the rou
where t1, t2, t3, th’ t5, t6, and t7 are constants and a is g p
Parameter. The group labeled as (3.1.1) has the general form specified
by equation (1.3.7). The transformed independent variables are func—
tions of the parameter a and the previous independent variables.

2h

 

 

25
ilarly, the transformed dependent variables are functions of the
ameter a and the previous dependent variables. There is no unique
nod for specifying groups such as (3.1.1) and the form for 6 is
t proposed by Hall [2b, pg. 93]. By specifying a as shown, the au-
r discovered that the group Al would satisfy all the required con-
ions to affect a reduction by one in the number of independent vari-
es for the system of equations given in section 2.h.

The group A (3.1.1) must satisfy the four group axioms given

1
equations (1.3.1), (1.3.2), (1.3.3), and (1.3.h). To demonstrate
satisfaction of the group axioms, only the transformations on R and
re considered. All the remaining transformations are similar to the
nsformation on R and if the R transformation satisfies the group ax-
s, so will the remaining transformations.

The closure axiom, equation (1.3.1), is applied using the

001 A i to denote a particular transformation utilizing the parameter
a

altl

2 _
a t - _ _
A A R = A e l R) = A R - e R —
al 3.2 al( a1
(3.1.2)
1 2 1 2 a3t _
1 1 1 2 l ‘ = ' + 1t -
A A e = A (()+ a t ) A 6 9 a 2
a1 a2 a1 2 a1
(3.1.3)
— 1
6+ 321:2 + 8.1132 = e + (a1 + a2 )t2 = e + 8.3t2 : Aa3 6
— . .h
Where a1 + a2 = 8-3. (3 l )

26
closure axiom is satisfied since Aa3 and A:3 have been shown to be
que members of the set of given transformations.
Equation (1.3.2), the associativity axiom, is applied where

denotes a particular transformation utilizing the parameter ai.

2 1 2 3
al(Aa2 A 3) R = A 1 e(& + a3)tl R = ea tl e(a + a )tl R =
a a

(3.1.5)
e[al + (a2 + a3)]tl R

3t (a1 + a2)t a3t
(A A A R = A A ea 1 R = e l e l R =
31 a2) a3 a1 a2

(3.1.6)
e[(al + a2) + a3]tl R

ce equation (3.1.5) equals (3.1.6), the associativity axiom is satis-

d for the type of transformation employed on R.

= e + [a1 + (a2 + a3)]t2 (3.1-7)

1 1 _ 1 2 3
(Aa2 Aa3) 6 — Aal(6 + (a + a )t2)

_ l 2 3
1 11:2) 11:3 6 = 11:, A; (e + 8.3132) _ e + [(a + a ) + a 14.2 (3.1.8)

ition (3.1.7) equals equation (3.1.8) which demonstrates that the
Dciativity axiom is satisfied also for the type of transformation

Loyed on 9.

Any transformation which leaves a point unaltered is by defi—
Lon an identity transformation. For the group Al, it is seen by in—
:tion that this definition is satisfied when the parameter a equals

>. The left identity axiom, equation (1.3-3). is satiSfied When

a = 0. (3.1.9)

 

27

R = eatl R = R (3.1.10)
9 = e .+ at2 = 9, (3.1.11)

For the types of transformations employed on R and 6, it is
by inspection that the inverse transformation is obtained by allow—
the parameter a to be negative. Therefore, the left inverse axiom,

:ion (1.3.h), is satisfied when

a = —a. (3.1.12)
Example,
A'1 A R = e'atl eatl R = IR = R (3.1.13)
a a
‘a e = A;l(e + at2) = A: I; .= 5 — at2 = e + at2 — at2 = e. (3111+)

;roup Al (3.1.1) satisfies the four group axioms for closure, asso—
.vity, left identity, and left inverse.

Theorem 1.3.2 gives the conditions under which the number of
>endent variables in a system of partial differential equations can
‘duced by one in the process of obtaining invariant solutions. The
,imensional equations given in section 2.h must be conformally in—
,nt under the k—th enlargement of the group Al (3.1.1). Conformal
iance is defined by Definition 1.3.5 and given in equation form by
ion (1.3.20). For convenience, equation (1.3.20) for conformal in—

nce is displayed in this section.

- l P
442%.”, 21’) = M(z%..., 2?; a) <I>(z,..., z ) (1.3.20)

 

I;

 

 

28

By conformal invariance, the group Al (3.1.1) is formally sub—

1 into an equation and the mathematical manipulations indicated
equation are performed. The resulting form must then be the

equation (1.3.20) for the equation to be conformally invariant.

The group Al (3.1.1) is formally substituted into equation
the R—Momentum equation.

ea<2th—tl) U .e—U

E 36 17 33 v at? 17? 31?
T. + : -: + —_ — : + *2 =
3R R 39 3Z R 3B 3B
(th+t5"tl) X3_U ea(t6+th—t3) W E ea(2t5—tl) K2
R as + BZ ' R
a(t -t ) i
+ e 7 l 2%. (3.1.15)

lential terms in equation (3.1.15) must equal each other to ob-
necessary form for conformal invariance as indicated by equa—

3.20). Equating the exponential terms, the following results

ned:

1 = tu + t5 - t1 = t6 + tu — t3 — 2t5 — t1 = t7 — t1, (3.1.16)
(3.1.17)

t

th = t5, t11L = t6 + t1 - t3, 7 = 2th'

dent from equation (3.1.15) that the function M of equation

is only a function of the parameter a. By Definition 1.3.6,

rential form represented by equation (3.1.15) is constant con—

invariant.
The group A (3.1.1) is formally substituted into equation
1

:he e—Momentum equation.

 

 

 

 

 

a? ea(th+t5‘t1) U 3_V

Ea? Va? Wav UV 1
:+:—_+ —_+—_+:—._=
6R R 39 32 R R a . 3R
ea<2t5_tl) V 8V ea(t6+t5_t3) w 3V ea(tu+t5_tl) UV
R 36'+ 3Z'+ ‘3?
a(t -t )
e 7 l 1 BP
+ .—
R 36 (3.1.18)

>onential terms in equation 8.1.18) must equal each other to ob—
1e necessary form for conformal invariance as indicated by equa—

..3.20). Equating the exponential terms, the following results

;ained:

th + t5 — t1 = 2t5 — t1 = t6 + t5 — t3 — t7 — t1, (3.1.19)

th = t5, t5‘= t6 + t1 — t3, t7 = 2th. (3.1.20)

son of equations (1.3.20) and (3.1.18) indicates that the func—
of equation (1.3.20) is only a function of the parameter a. By
ion 1.3.6, the differential form represented by equation (3.1.18)
tant conformally invariant.

The group Al (3.1.1) is formally substituted into equation

, the Z—Momentum equation.

ea. (th+t6_tl) U E

Ufl+iflflfl.£=
afi' R'ae 32 a 3R
ea(t5+t6—tl) V aw ea(2t6—t3) w 21
+ R'_6'+ 3Z
t —t )
+ ea( 7 3 g; (3.1.21)

 

 

3O

nobtain the necessary form for conformal invariance as indicated by

nation (1.3.20), the exponential terms in equation (3.1.21) must

ual each other. The following results are obtained by equating the

:ponential terms:
th + t6 — t1 = t5 + t6 — tl = 2t6 — t3 (3.1.22)
(3.1.23)

t1+ = t5, 2t6 = t7.

imparison of equations (3.1.17), (3.1.20), and (3.1.23) yields the

(sult,
(3.1.211)

“om equations (3.1.22) and (3.1.2h), it follows that
(3.1.25)

tl = "t3.
(mparison of equations (1.3.20) and (3.1.21) indicates that the func—
on M in equation (1.3.20) is only a function of the parameter a.
erefore, the differential form represented by equation (3.1.21) is

nstant conformally invariant according to Definition 1.3.6.
The group A1 (3.1.1) is formally substituted into equation

~h-10), the Conservation of Mass equation.

_ _ .. — t —t ) a(t -t )
3_U+H+1§1+§.W==ea(hlw ‘5 ”1‘1
3R" E R a 32 3B R
8L(135451) 1 av ea<t6't3) fl (3.1.26)
'R‘a-+ az

 

31
e exponential terms in equation (3.1.26) must equal each other to ob-
Lin the necessary form for conformal invariance as indicated by equa—
.on (1.3.20). Equating the exponential terms, the following results

'e obtained:
th = t5, t5 = t6 + tl — t3, th = t5 = t6. (3.1.27)

Luations (3.1.27) agree with the results obtained from the previous
>nformal invariance verifications. Again, by comparison of equations
..3.20) and (3.1.26), the function M in equation (1.3.20) is seen to
: only a function of the parameter a. Therefore, the differential
)rm represented by-equation (3.1.26) is constant conformally invariant
:cording to Definition 1.3.6.

The group A1 (3.1.1) is formally substituted into equation

!.h.ll), the Bernoulli equation:
—- 2 2
H - P — 1/2 (E2 + V? + we) = H _ eatY P — 1/2 (ea tn U

+ ea2t5 V2 + ea2t6 W2)- (3.1.28)

r conformal invariance, the exponential terms in equation (3.1.28)
st equal each other as indicated by equation (1.3.20). The results

equating the exponential terms are shown below:

— — .1.2
t7 = 2th, t, - t5 - t6. (3 9)

iations (3.1.29) agree with the results obtained previously. Compari—

1 of equations (3.1.28) and (1.3.20) indicates that the function M in

lation (1.3.20) is a function only of the parameter a. By Definition

3-6, the differential form represented by equation (3.1.28) 15

 

 

 

32

constant conformally invariant.

By inspection, it is seen that equation (2.h.17) for the
Swirl Parameter is constant conformally invariant under the group A1
(3.1.1).

Since the equations (2.1.7), (2.h.8), (2.h.9), (2.h.1o),
(2.h.1l), and (2.h.17) are all constant conformally invariant under the
K—th enlargement of the group Al (3.1.1), Theorem 1.3.2 states that the
number of independent variables for the system of equations mentioned
above can be reduced by one in the process of obtaining invariant solu—
tions.

Using the results (3.1.17), (3.1.2h), and (3.1.25) obtained
from the conformal invariance verifications, the group A1 (3-1-1) is

nodified as shown below:

§.= eatl R fi-= eath U 2‘: e2ath P
A - 6': e + at 7': eath v (3.1.30)
1' 2

2.: eatl Z W-= eath W

mere t1, t2, and th are constants and a is the group parameter.
The group A1 (3.1.30) has m+n—l = 6 functionally independent

msolute invariants. Equation (1.3.9) gives the form for m-l = 2 of

he absolute invariants and equation (1.3.10) gives the form for the

emaining n = h absolute invariants. Definition 1.3.1 and equation

1-3-5) give the definition and general form for an absolutely invari—

nt form. The manner in which absolute invariants are selected is not

ell defined. The functions n1 and n2 are chosen as fOllOWS‘

 

 

 

 

33
= 6—6 RP (3.1.31)

n1

-6 Zb.

e (3.1.32)

n2

For the functions in equations (3.1.31) and (3.1.32) to be absolute in—

variants, the following relations must hold:

nlfif) = nl(9,R) (3.1.33)

n2(6,2) = n2(6,Z). (3.1.3h)

The values of the exponents p and b of equations (3.1.31) and (3.1.32)
must be determined such that equations (3.1.33) and (3.1.3h) are true.
The required tranformations from the group Al (3.1.30) are utilized in

the following equations to determine p and b:

n1 = 8—9 Bp = e—(9+at2) epatl Rp = ea(ptl_t2) e”e rp

P = tQ/tl (3.1.35)

e_(e+at2) ebatl Zb = ea(bt1—t2) e-G Zb

_ . . 6
b — t2/tl (3 1 3 )

Equations (3.1.35) and (3.1.36) are substituted into equations (3.1.31)
ind (3.1.32):

= 8‘9 Rt2/t1 (3.1.37)

“2 = 6—9 Zt2/t1. (3.1.38)

 

 

 

31.

functions fl, f2, f3, and fh are chosen to resemble n1 and n2:

-3 e -5 e -5 _
fl = Ue l , f2 = Ve 2 , f3 = We 39, f, = Pe one. (3.1.39)

the functions in equations (3.1.39) to be absolute invariants, the

.owing relations must hold:

flfi. 5) = fl(U. 6). rem. 6) = f (V. 6). (3.1.1.0)

f3(W, 6) = f3(w, e), rh(§, 6) = fh(P, e). (3.1.h1)

values of the exponents 6 6 63, and 5h of equations (3.1.39)

1’ 2’
. be determined such that equations (3.1.h0) and (3.1.h1) are true.

required transformations from the group A1 (3.1.30) are utilized in

following equations to determine the exponents 61, 52, 63, and 6h:
— - ‘ -6 9+ t a t -5 t -5 e
= Ue 616 = oath Ue l( a 2) = e ( h l 2) Us l

51 = th/t2 (3.1.h2)

— -6 5 ati -62(6+at2) a(th-52t2) -529
- - e Ve = e Ve

2
62 = th/tg (3.1.h3)
f3 = We‘635 = eati We—63(9+at2) = ea(tu-63t2) We—53e
53 = tu/t2 (3.1.hh)
ft = Fe_6h§ = e2ath Pe-6u(e+at2) = ea(2th—6ut2) Pe—ohe
(3.1.h5)

6h = 2th/t2.

 

 

 

35
e values of the 61's are substituted into equations (3.1.39) to ob-

in the following results:
_ 6 —t t 6 —t t 6 -2t t 6
=Uetu/t2,f2=Ve u/2,f3=weu/2,fh=1.e M2,

(3.1.h6)

‘The condition given by equation (1.3.11) must be satisfied by

a chosen functions fl, f2, f3, and fa.

J = a U, v, w,, P # O (3‘l'u7)

M1 M1 M1 M1
3U av aw aP

erg ar2 ar2 erg
3U av aw 8P
J = (3.1.148)
ar3 ar3 8f3 af3
3U av aw 8P

afu th arh afu
3U 8V 8W, 3?

 

 

ng equations (3.1.h6), equation (3.1.h8) becomes:

 

 

e_t2+/t28 O O 0
J =
o O 64mk26,0
-2t t 6
o o o e h/ 2

 

 

 

36
The Jacobian determinant J is non—vanishing for the chosen functions
fl, f2, f3, and ti. Therefore, the condition (3.1.h7) is satisfied.
The condition given by equation (1.3.12) must be satisfied

by the functions ml and n2.

3(n1. n2)
R m = m—1 = 2 (34-19)

The rank of the indicated matrix must equal 2. Expanding the matrix

given in equation (3.1.h9) gives the following result:

 

Bnl Bnl Bnl
SR 36 BZ

s M. (3.1.50)

3n2 3n2 3n2
QB 39 az

Using equations (3.1.37) and (3.1.38), equation (3.1.50) becomes:

3. — 0
R n1 n1
= M, (3-1051)
0 ”n2 E'n2
where E a t2/tl. (3.1.52)

Since M is a 2x3 matrix and the row rank equals the column rank for a
matrix, 2 is the maximum rank that M in (3.1.51) can have. Both rows
>f M in (3.1.51) are linearly independent and hence, M has rank of 2.
The condition indicated by equation (3.1.h9) is satisfied for the cho-

Len functions nl and n2-

 

37
It has been demonstrated that equations (3.1.37), (3.1.38),
(3.1.h6) represent the functionally independent absolute invariants

the group A (3.1.30).

1
Application of equation (1.3.17) to the invariants of the

up Al (3.1.30) yields the following result:

fj(U, v, w, P, R, e, z) = F3011, n2) J: 1, 2, 3, 1.. (3.1.53)

ations (3.1.h6) are rearranged using the symbolic equation (3.1.53):

d6

W = eme F3011, n2), P = e209 F1.(nl. n2). (3.1.55)
where a E th/t2' (3-1-56)

For convenience, equations (3.1.hh) and (3.1.h5) and the par—

1 derivatives of these equations are listed together.

nl = e_6 R5 (3.1.57)

n2 = e_9 Zg (33-1-58)
a? = 0 W ' 0

a

an1 = _n ..“2 = -.. (3.1.59)
39 1 ea 2
3'11 5 33.12. = E n
3R =§“1 az z 2

 

 

1111‘- !)4‘))

 

 

38
The relations (3.1.5h) through (3.1.59) are now used to re-
:e the number of independent variables in the system of equations

ren in section 2.h. The three independent variables (R,e, Z) are

Luced to the two independent variables (n1, n2) in equations (2.h.7),
.h.8), (2.h.9), (2.h.10), (2.h.11), and (2.h.17). Figures 3.1
rough 3.6 illustrate the transformation details and give the final

:ult for each of these equations. Note that the final result dis-

wed in each figure is obtained by summing all the transformed terms

ren in the figure.

 

 

 

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ht
A three—to-two reduction in the number of independent varia-
has been accomplished for the system of equations given in section
The new system of equations for the fluid flow model in (nl, n2)
: consists of equations (3.1.66), (3.1.72), (3.1.77), (3.1.82),

83), and (3.1.8h)

Two to One Reduction in the Number of Independent Variables

The two independent variables (n1, me) are reduced to the one
xendent variable (X). The new independent variable (X) is a func—
.1ly independent absolute invariant of a subgroup of the one—

Leter continuous group of transformations, B The one—parameter

1'
nuous group of transformations, Bl’ will subsequently be referred
the group Bl'
Taking advantage of the experience gained from choosing the

> A (3.1.1), the group B is chosen as shown:

1 1
_ = k — = Yl F
nl c nl F2 0 2
- _ k - _ Y1 (
- _ F — c F 3.2.1)

Bl‘ n2 0 n2 3 3
- _ Y ' = 2)(1 F
Fl -- C 1 Fl Fu C 14

k and Y1 are constants and c is the group parameter. The group
ed as (3.2.1) has the general form specified by equation (1.3.7).
ransformed independent variables are functions of the parameter e
1e previous independent variables. Similarly, the transformed de-
nt variables are functions of the parameter c and the previous de—

1t variables .

 

 

 

 

 

MS

The group Bl (3.2.1) must satisfy the four group axioms given
.ations (1.3.1), (1.3.2), (1.3.3), and (1.3.h). The form of the
brmations in group Bl (3.2.1) is the same as the form for the R
'ormation in group A1 (3.1.1). Since the R transformation in
A1 (3.1.1) satisfies all the group axioms, it follows that the
'ormations in the group Bl (3.2.1) satisfy all the group axioms.

The conditions under which the number of independent varia—
n a system of partial differential equations can be reduced by
, the process of obtaining invariant solutions are given by
‘m 1.3.2. The two-dimensional equations (3.1.66), (3.1.72),
‘7), (3.1.82), (3.1.83), and (3.1.8h) must be conformally invari-
.der the k-th enlargement of the group Bl (3.2.1). Conformal in-
.ce is defined by Definition 1.3.5 and given in equation form by
on (1.3.20). To verify conformal invariance, the group B1
) is formally substituted into an equation and the mathematical
lations indicated by the equation are performed. The resulting
ust then be the same as equation (1.3.20) for the equation to be
mally invariant.

The group Bl (3.2.1) is formally substituted into equation

the two—dimensional representation of the R-Momentum equation.

U‘
V
v

' - ’ - ‘ 1/5 - - a? _ -2 - -
L ' F2) “1 3=l-+ (5F3(2l) ' F ) n2 3%; F2 + “F1F2
an]. ”2 2

‘ 3F 2Y n 1/5 _ 3F

fit = c2Y1(€Fl — F2) n1 1 + c 1(5F3(_lJ F2) "2.35l

)fil anl U2 2

_ C2Y1F2 + c2YldF F — c2Yl£n th (3.2.2)
‘ 2 l 2 1 3n—

1

 

ii lll.|| I II

 

 

M6
; evident from equation (3.2.2) that the function M of equation
.20) is only a function of the parameter c. By Definition 1.3.6,
lifferential form represented by equation (3.2.2) is constant con-
illy invariant.
The group Bl (3.2.1) is formally substituted into equation

.72), the two—dimensional representation of the e-Momentum equation.

'-‘ 'a'+ "l/€_- '3F2+FF+QF2
.Fl F2)1nl 8—?)— (€F3(fi_1_) 13‘2)n2 as 1 2
1 2 2
- fi 3 — H ”h + 2a? — c2Yl(EF - F ) n 8F2
1 5i. 2 5:— h 2 1 3n
1 2 1

_ C2Yln th _ C2Yln2 th + C2Y12th (3.2.3)

anl 3n2

equation (3.2.3), it is evident that the function M of equation
20) is only a function of the parameter c. By definition 1.3.6,
.ifferential form represented by equation (3.2.3) is constant con—
lly invariant.

The group Bl (3.2.1) is formally substituted into equation

77), the two—dimensional representation of the Z-Momentum equation.

 

 

 

 

 

_ _ _ _ a" _ fi l/E _ _ 3‘ _ _
(EFl F2) n1 §fi§'+ (5F3(:l- ' 2) 2 3:; + o”T2F3

1 n2 n2
- l _
(Ll) M fig 3 u = c2Yl(€Fl — F2) nl 3F3 + czYl EF3 ”1 1/5 F2) n2 3F3
— 8— — .—
n2 n2 ”1 n2 an2

2y 2v n l/E 3F
1 _ 1
+ c 0LF2F3 c E(_£) 02 __£ (3.2.h)
n2 . 3n2

parison of equations (1.3.20) and (3.2.h) indicates that the func—
1 M in equation (1.3.20) is only a function of the parameter c.
refore, the differential form represented by equation (3.2.h) is
stant conformally invariant according to Definition 1.3.6.

The group Bl (3.2.1) is formally substituted into equation

-.82), the two—dimensional representation of the Conservation of

1 equation.
_ ai - 3F _ 3F E 1/5 _ 3F
Enl___1_+Fl+0LF2u-nl__2_—n2__2+£(_i) n2—
3711 301 3n2 n2 3712
Y 3F Y Y Y 8F Y 8F
= c lénl l + c 1Fl + c laF2 - c 1n1 2 — c lng 2
Bnl Snl 3n2
l/E
Y n 8F
+ C lE(—l) ”2 —e§ (3.2.5)
n2 3n2

‘ evident from equation (3.2.5) that the function M of equation

.20) is only a function of the parameter c. By Definition 1.3.6,

lifferential form represented by equation (3.2.5) is constant con—

Llly invariant.

 

 

 

 

 

 

M8
The group Bl (3.2.1) is formally substituted into equation

(3.1.83), the two-dimensional representation of the Bernoulli equation.

2ae _ _2 -2 -2 2ae 2y
H - e (Fu + 1/2(Fl + F2 + F3)) = H — e (c th
+ l/2(c2YlFi + c2YlF: + can-Fin (3.2.6)

Comparison of equations (1.3.20) and (3.2.6) indicates that the func—
tion M of equation (1.3.20) is only a function of the parameter c. The
iifferential form represented by equation (3.2.6) is constant conform-
ally invariant according to Definition 1.3.6.

The group Bl (3.2.1) is formally substituted into equation

(3.1.8h), the two—dimensional representation of the Swirl Parameter

:quation.
s — (Sioz/g e2aeF2/H = s — c2Yl(j;J2/€ e2aeFS/H (3.2.7)
E2 2 n2

Tom equation (3.2.7), it is seen that the function M of equation
1.3.20) is a function of only the parameter c. By Definition 1.3.6,
quation (3.2.7) is constant conformally invariant.

The group B1 (3.2.1) has m+n-l = 5 functionally independent
bsolute invariants. Equation (1.3.9) gives the form for m—l = l of
ie absolute invariants and equation (1.3.10) gives the form for the
amaining n = h absolute invariants. Definition 1.3.1 and equation
L-3.5) give the definition and general form for an absolutely invari—

1t form. As previously mentioned, the manner in which absolute

 

 

 

M9
invariants are selected is not well defined. The function X is chosen

as shown below:

X ='- ' (3.2.8)

This choice is suggested by the appearance of this ratio in equations
(3.1.66), (3.1.72). (3.1.77), (3.1.82), and (3.1.8h). For the function

X to be an absolute invariant, the following relation must hold:
X(filg fig) = X(nl, 02)- (3.2.9)

Phe specific condition for the absolute invariance of X is determined

my substituting equation (3.2.8) into relation (3.2.9):

3| 3|
l—l
J

. (3.2.10)

[\J
J
NIH

[sing relations from group Bl (3.2.1) for n1 and n2, it is evident that
tquation (3.2.10) is satisfied and that X is an absolute invariant.

Fhe functions g1, g2, g3, and 5h are chosen as shOWn:

0'

_ -o _ F n—c —2o

gl4 = th1 . (3.2.11)

or the functions in equations (3.2.11) to be absolute invariants, the

ollowing relations must hold:

g16112 51) = s1(F1. n1). 32(52, fil) = s2(F2. n1) (3.2.12)

v

83(F3: fil) = 83(F3, 01), gh(ihs 7]1 = gh(Fh’ n1)' (3'2'13)

 

 

 

 

SO
3 value of the exponent o in equations (3.2.11) must be determined
:h that symbolic equations (3.2.12) and (3.2.13) are true. The re-
Lred transformations from the group B1 (3.2.1) are utilized in the

Llowing equations to determine a:

= — _—o = Yl k -0 = Yl—okF —0
$1 Flnl c Fl(c nl) C 1n ,
- -— Y k —0 Y —ok —0
= F O = l = 1 F
g2 2n1 c F2(c ”1) ° 2n1 ’
(3.2.lh)
— _-0 Y1 k -0 Yl-ok —o
= = F = F ,
a3 F3nl c 3(c n1) c 3n

_-2o _ 2Y1 k —20
— c Fu(c n1)

_ - 2(Yl-Ok) '20
sh ' thl C thl .

)m the set of equations (3.2.1h), it is evident that the gi will be

solutely invariant if

0 = Yl/k. (3.2.15)

, The condition given by equation (1.3.11) must be satisfied by

a chosen functions g1, g2, g3, and gh.

= a(gl, 829 33, 5h) # 0 (3.2.16)

3 F1, F2, F3, Fh

agl agl Bgl 381
BFl 3F2 3F3 3Fh
3&2 , 3g2 age 3s2
3F are 3F aFu
J = l 3 (3.2.17)
383 £3. 3%; E;
aFl 3F2 3F3 th

3g, asu 33h 38h
BFl 3F2 8F3 ”"1.

 

 

 

 

 

 

 

51

Using equations (3.2.11), equation (3.2.17) becomes:

 

 

—0

n1 0 O 0

—o

0 n1 0 0

J:
—o
0 0 n1 0
—2o
0 O 0 nl

Phe Jacobian determinant J is non—vanishing for the chosen functions
g1, g2, g3, and gh. Therefore, the condition (3.2.16) is satisfied.
The function X must satisfy the condition given by equation

(1.3.12).

R 27%;; = m—1 = 1 (3.2.18)
nl, n2

Fhe rank of the indicated matrix must equal 1. Expanding the matrix

;iven in equation (3.2.18) gives the following result:

2.31 E s M. (3.2.19)

sing equation (3.2.8), equation (3.2.19) becomes,

—n
.1. _l = M. (3.2.20)
n2 n2

 

 

 

 

52
.nce M is a 1x2 matrix and the row rank equals the column rank for a
Ltrix, 1 is the maximum rank that M in (3.2.20) can have. The columns
? M in (3.2.20) are not linearly independent (one is a multiple of the
:her) and hence, M has rank of 1. The condition indicated by equation
3.2.18) is satisfied for the chosen function X.

It has been demonstrated that equations (3.2.8) and (3.2.11)
apresent the functionally independent absolute invariants of the group
_ (3.2.1).

Application of equation (1.3.17) to the invariants of the

roup Bl (3.2.1) yields the following result:
gJ(Fl, F2, F3, Fh, n1, n2) = GJ(X) 3 = 1, 2, 3, h. (3.2.21)
luations (3.2.11) are rearranged using the symbolic equation (3.2.21)

> obtain the following general form for the Fi:

F = nKOG,(X)

i 1 1 (3.2.22)

For convenience, equation (3.2.8) and the partial derivatives

'the function X are listed together.

= .2.23

x nl/n2 (3 )

.831 = .1. it = ‘11 (3.2.2.)
2

8"1 n2 an2 n2

The relations (3.2.22) through (3.2.2u) are now used to reduce

= number of independent variables in the system of equations given in

 

 

 

53
tion 3.1. The two independent variables (n1, n2) are
independent variable (X) in equations (3.1.66), (3.1
1.82), (3.1.83), and (3.1.8h). Figures 3.7 through 3.
transformation details and give the final result for
ations. Note that the final result displayed in each

ned by summing all the transformed terms given in the

reduced to the

.72), (3.1.77),

11 illustrate
each of these
figure is ob-

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Amm.H.mV ZOHBHDGH onBopomm AxVIAmc .Hcv oH.m mmseHa

m H m H
o u .ow\H+wa I .oxm + o Ao I av + 0 Ash + HV

m H .mmm H m .mmm m we
a c wa n m c c xw u m c Il.m
. o w\H+H on o w\H am EHAHCv
m H m
once I ma
N H OWWW H VN .Wmm N
C H C CI H CI
(was New o mam
m m H H m Hmm H H .mmm H
.0 + Cl H C .0 + C Cl H CI
A a .wao AHIo a mom o v mam
H H H
woc n a
Hem Hem

 

 

 

 

59

 

Amm.m.mv

Asm.N.mv

Asm.H.mV 20Ha<pem 92¢ Amm.H.mV ZQHBHDGH 20Heoenmm Axv+Amc .Hcv

m H
C H
m\mo om memo ”\mx m

 

m m H owceemo H
Ame + me + meVm\H I m u a

HH .m "$5on

60
To obtain a complete reduction to one independent variable in
equations (3.2.57) and (3.2.58), the constants a and 0 must vanish.
Let a = 0 = 0, equations (3.2.32), (3.2.h2), (3.2.H9), (3.2.56),

(3.2.57), and (3.2.58) then become:

5x010; - €Xl+l/EG3Gi - G: = -£XG£ (3.2.59)
EXGlGé + GlG2 — £Xl+l/£G3Gé = 0 (3.2.60)
EXGlG; — €Xl+l/€G3G; = gxl+l/§GL (3.2.61)
G1 + axe; — gxl+l(50; = 0 (3.2.62)

01‘ = H - l/2(Gi + G: + G?) (3.2.63)

3 = Xa/gGg/H. (3.2.6h)

A two to one reduction in the number of independent variables
has been accomplished for the system of equations given in section 3.1.
Equations (3.2.59) through (3.2.6h) comprise the new system of equa—

tions for the fluid flow model in (X) space.

3.3 Summary

The partial differential equations of the fluid flow model in
(R, 6, Z) space have been transformed into partial differential equa—
tions in (n1, n2) space by the application of the continuous one—
Subsequently, the partial

parameter transformation group A1 (3.1.30)~

differential equations of the fluid flow model in (n1, n2) space have

 

 

 

61
men transformed into ordinary differential equations in (X) space by
the application of the continuous one-parameter transformation group

31 (3.2.1).

 

 

 

'1’: HHHHHH

 

 

 

 

 

IV. SOLUTIONS FOR THE CASE 5 = —l

+.l One-Dimensional Solutions of the Ordinary Diffgrential Equations

The group constant E which is defined by equation (3.1.52)
ippears as an exponent in the transformation equations (3.1.57) and
:3.1.58). 5 may be chosen to be positive, negative, or zero. The ef-
?ect of each of these choices is readily determined after substitution
)f equations (3.1.57) and (3.1.58) into equation (3.2.8). For each

:hoice, the variable X is influenced as shown:
g>0, x = (R/Z)E; . g<0, x = (Z/R)l€|; a = 0, X = 1.

‘or the choice E = O, X becomes a constant and is not a variable and
Ihe system of ordinary differential equations (3.2.59) through (3.2.62)
ecomes meaningless.
For an example in this investigation, 5 is chosen to be nega—
ive. By inspection, it is noted that for E = —1:
1. The system of ordinary differential equations
is simplified.
2. The velocity components will be proportional
to l/R in (R, 0, Z) space and the pressure
will be proportional to 1/R2 in (R, e, Z) space.
3. The system of ordinary differential equations on
page 60 is greatly simplified as the exponential

term 1 + 1/5 vanishes.

62

 

63
The system of ordinary differential equations consisting of equations
(3.2.59), (3.2.60), (3.2.61), and (3.2.62) simplifies to the following

system of ordinary differential equations:

—XGiG£ + G3G£ — GS = XGfi (h.1.1)
—XGlGé + Gng + G3Gé = o (h.1.2)
—XGlGé + G3Gé = —G£ (u.1.3)
-XG£ + Gly+ G; = o (h.1.h)

where G; E dGi/dX. Since this system of ordinary differential equa—
tions is non—linear, a numerical analysis approach is required to de—
termine its solutions.

Hamming's modified predictor—corrector numerical method is
utilized to obtain solutions for the system consisting of equations
(h.l.l), (h.1.2), (u.1.3), and (u.1.h). The method employs the follow—

ing numerical calculations:

Predictor: Pj+l = Gj_3 + %§(203 — G3_l + 2G5_2)
MOdifier: MJ+l = Pj+1 - %%%(Pj - C3)

Corrector: Cj+1 = $(9GJ — G3“2 + 3h(M3+1 + 2G: - G3-1))
Final Value: Gj+l = Cj+1 + I%I(Pj+l - Cj+l>

 

6h
If the results are known at the equidistant points XJ 3, XJ 2, XJ 1'

and X g the results at point X = XJ + h (where h is the step size)

J

can be computed using the numerical equations given above. Hamming's

3+1

predictor—corrector method is not self starting; that is, the function—
al values at a single previous point are not enough to get the func—
tional values ahead. To obtain the starting values, a special Bunge—
Kutta procedure followed by one iteration step is added to the
predictor—corrector method.

It may be demonstrated that the convergence criterion given
by Wilf [52, pg. 98] for Hamming’s modified predictor-corrector method
requires that the step size be chosen such that h<8/(3l8F/3NB+1|) Where
) =‘M’ . For an extensive discussion on Hamming's modi—

Xj+1’ M3+1 3+1

fied predictor—corrector method, the reader is referred to Carnahan

F‘(

[9, pp. 381—392] or Wilf [52, pp. 96-108].

Appendix A contains the computer printout of the IBM scienti—
fic subroutine DHCPG, a double precision arithmetic routine utilizing
Hamming's method, which was used in this work to solve the system of
non-linear ordinary differential equations. The numerical computations
were performed by an IBM 360 computer and the numerical results are
displayed in Table A.1.

After rearrangement, equations (h.l.1), (h.1.2), (h.1.3), and

(h.l.h) become:

G’ = C/E (h.l.5)
h

- ’ - ’ h.l.6

G3 — Gh/A ( )

 

 

III. III III

 

 

65
G,” = (GlG2)/A

G1 = (G1 + G3)/X

 

where A s X01 — G3, 0 s —(AGl)/X, and E s X + 1/X.

The boundary conditions (2.h.13), (2.h.1u), (2.u.l5), and
(2.h.16) are constants and transform to X space unaltered. The values

for the boundary conditions are selected as shown:

U6+Fl*+Gl* = 1 o, V6+F2*+G2* = 1.0,
(8.1.9)
W6>F3*+G3* = 1.0, P6+Fh¥+Gu* = 0.5.

In X space, Gj* E Gj(X*) where J = l, 2, 3, h.
The boundary point X* is the left limit of the X interval
(X*5XSX2). Using the result obtained from substituting equations

(3.1.57) and (3.1.58) into equation (3.2.8), X* is defined to transform

L

U)

indicated by the following equation:

 

—e —1
n e R
X* a .53.: e O = %-. (u.1.10)
n2 e' z"1 0

For the selected boundary conditions, the numerical solutions
2 equations (u.1.1), (u.1.2), (8.1.3), and (u.1.t) are plotted for the
Iterval 0.13XSl.O in Figure 8.1. The output from the computer program
Ling the DHCPG subroutine was stored as the input to a subroutine
ich controls a Calcomp plotter from which the plots were produced.

pendix B contains a printout for the Calcomp plotter routine.

 

 

 

 

66

At the point x = 0.h6531 in Figure n.1, the solution func-
:ions G1, G2, and G3 all have a discontinuous derivative. Gh (the one-
limensional representation for pressure) reaches its maximum value and
% (the one—dimensional representation for the tangential velocity com—
»onent) reaches its minimum value. Therefore, the point X = 0.h6531 is
esignated as the "tangential quasi—stagnation point" for this flow.

The solutions G1, G2, G3, and GM were determined by simul-
aneously solving the one-dimensional representations for conservation
f momentum and conservation of mass. To verify these results, the
ne-dimensional representation for conservation of energy, equation

3.2.63), is used to determine the Bernoulli constant H at each point X.

From equation (3.2.63), it is seen that the equation for H is:

3
H = on + 1/2 2 G2. (u.1.11)
i=1 1

.e value of H is fixed by the selected boundary conditions and for
is case, H = 2. Data values for H were computed at each point X
ing equation (h.l.ll) and the solutions G1, G2, G3, and Gh' From the
data values listed in Table A.1, the maximum percent deviation of H
am the value H = 2 is determined to be 0.28%. This maximum percent
viation occurs at only one point. Most of the percent deviations for
are in the range from 0.00% to 0.09% and hence, equation (3.2.63) is
:isfied.

Polynomial expressions for the solutions G1, G2, G3, and GM
2 determined using a Gaussian least squares method known as Polfit.
’olfit computer printout is included in Appendix C. The polynomial

>ressions for the interval 0.15X50.h6531 are:

 

 

 

 

 

67

ONE DIMENSIUM‘IL FUNCTIONS
, Cl SOUQRE.02 CIRCLE.L‘.3 FBIRNGLE.GLI PLUS /
3

 

 

 

 

 

FIGURE lI».l ONE-DIMENSIONAL FUNCTIONS

 

 

 

 

68

G1 = 0.88531 + 1.1ohox + o.hu657x2 (h.1.l2)

G2 = 1.h366 — 9.5897x + 82.897'x2 - 383 95X3

 

+ 829.38X“ — 691.71IX5 (h.l.l3)
G3 = 1.0909 — 0.91970X + 0.13199x2 (u.1.1u)
Gh = 0.h07h0 + 1.031hx — 1.09h1x2. (u.l.15)

The polynomial expressions for the interval 0.h653lSXSl.0 are:

G1 = 1.5338 - 0.29709X + 0.1I3202x2 (h.1.16)

02 = —36.18u + 2h1.ohx — 636.39X2 + 838.23x3

— 5h6.9ux” + 1h1.25x5 (u.l.17)
G3 = 1.h35o — 1.7h66x + 0.30862x2 (u.1.18)
Gt = 0.31091 + 1.55ohx — 1.7556x2 (h.l.19)

8.2 Three—Dimensional Pressure and Velocity Curves

Application of equations (3.1.5h), (3.1.55), (3.1.57),
(3.1.58), (3 2.8), and (3.2 22) to this case results in the following

transformation equations:

x = nl/n2 = Z/R (u.2.1)

 

 

 

 

G1(X) = Fl(nl, ”2) = U(R> 6: Z):
(u.2.2)

2(X) = F3(nl’ n2) = V(R§ 9,1Z),

G3(X) = F3(nl, n2) = W(R: ea'Z),

(8.2.3)
Gt(X) = Fu(nl, n2) = P(R, e, z).

The three—dimensional equations for pressure and velocity
components are obtained by substituting the transformation relations
(H24),Uua2hzmd(k23)imoemmfimm(ulu2)nnmgh(klum.

For the interval 0.15Z/R50.h653l, the polynomial expressions become:
U = 0.88531 + 1.10h0(z/R) + 0.1I1I657(z/R)2 (h.2.h)

v = 1.h366 - 9.5h97(Z/R) + 82.897(z/R)2 - 383.95(Z/R)3

+ 829.38(z/R)h - 691.71I(z/R)5 (h.2.5)
w = 1.0909 — 0.91970(Z/R) + 0.13199(z/R)2 (h.2.6)
P = 0.h07ho + 1.031h(z/R) - 1.09u1(z/R)2. (h.2.7)

For the interval O.h653lSZ/R51.0, the polynomial expressions become:
U = 1.5338 — 0.29709(Z/R) + o.lI32.o2(z/R)2 (h.2.8)
v = —36.18h + 2u1.0h(z/R) — 636.39(z/R)2 + 838.23(Z/R)3

_ gratin/II)h + iui.2s(z/R)5 (h.2.9)

 

 

 

 

70

w = 1.8350 — 1.7866(Z/R) + 0.30862(z/R)2 (8.2.10)
P = 0.31091 + 1.5508(Z/R) — 1.7556(Z/R)2. (8.2.11)

The absence of the 6 coordinate in the transformed equations

 

Tor U, V, W, and P signifies that for this case an axially symmetric

‘1ow is obtained.

The U, V, W, and P curves are plotted as functions of R using

 

as a parameter. For Z as a parameter, the plot intervals become:
.OZSRS2.189lZ and 2.1891ZSR510Z. The Calcomp plotter routine is given
nput data from the author's subroutines PC05El and PC05E2 which calcu—
ate data points from the polynomial expressions for pressure and the
elocity components. Appendix D contains the PC05El and PC05E2 subrou—
The programs and a sample of the computed data for these curves.
Figures 8.2, 8.3, 8.8, and 8.5 display the curves for pres—
lre, radial velocity, tangential velocity, and axial velocity, respec—

.vely. For the parametric values Z = 0.1, Z = 0.8, and Z = 0.7, the

 

Irresponding tangential quasi—stagnation point locations are

= 0.21891, R = 0.85968, and R = 1.5088. In each set of curves except

8 set for axial velocity, the location of the tangential quasi—stagnation
int is pronounced by an abrupt slope change for a particular curve.

gure 8.6 displays the real space functions U, V, W, and P on a single

agram for the parametric value Z = 0.7.

3 Summary and Recommendations for Further Research

Numerical solutions to the non—linear ordinary system of dif—
‘ential equations have been obtained for selected boundary conditions

an the group constant E = -1. These solutions are plotted in Figure

 

 

71

8.1. Polynomial expressions were determined for each of these solu—
tions and the polynomial expressions were transformed from the one-
iimensional (X) space back to the three-dimensional (R, 6, Z) space.
The three—dimensional polynomial solutions for pressure and the velo—
city components are plotted in Figures 8.2, 8.3, 8.8 and 8.5.

Solutions to the case E = —1 for boundary conditions other
than those reported on in this chapter were investigated by the author.
iowever, the investigations were not completed as the computer expense
vas prohibitive. The author's bill for computer time on this investi-
gation is $2,765.87. During this investigation, funds for computer
time were generously provided by the Process Engineering Department of
leneral Motors Institute, Flint, Michigan.

Many interesting areas for further research are exposed by
:his work. The author suggests that profitable results could be ob-
;ained from investigation into the case 5 = —l in this work, transfor-
1ations for a system of equations in spherical coordinates, and trans—
formations for a system of equations including derivative boundary

:onditions.

 

 

 

U. QB

 

‘-' I

72

PRESSUREtP]
Z=.1 SGURRE.Z;.8 LIHCLE.Z;.7 TRIRNELE

x

\\
\
\\ \
\

I A N

 

08890 1. 70 2. 50 31. 30 8'. 10

FIGURE 8.2 PRESSURE CURVES

I
4.

(.14

.70

fil‘
8.5L1

 

 

73

RRDIRL VELOCITTIUJ
Z—.1 SOURRE.Z=.8 CIRCLE,Z=.7 TRIFINBLE

 

 

 

FIGURE 8.3 RADIAL VELOCITY CURVES

 

 

 

 

 

 

 

 

 

78

TRNGENTIPL VELUCITYIV]
22.1 SOURRE.ZL.8 CIRCLE.ZL.7 TRIQNCLE

 

 

 

 

”TM ____,_¢l.————A
48*" H
// ’/K_/a/
/ ,
_,/
/
I I l I _r I
2.5L‘ 3,30 8.10 ~:.BU r1.70 5,50

 

FIGURE 8.8 TANGENTIAL VELOCITY CURVES

 

 

 

7S

,QXIRL VELOCITY (N)
Z=.1 SOURRF.Z=.8 CIRCLE.Z:.7 TRIQNCLE

 

lax—— «L
M77— MIA/A

7r

FIGURE 8.5 AXIAL VELOCITY CURVES

 

 

IIIIIJIIIIIIII
II

 

 

UNLIIUNO
70

&

Iu.rl|_ dl I'ILC r
ESQ

030

l

10
I

&

 

fflJU

76

RHE.V—CIRCLE.H—TRIRNGLE.P-PLUS
7

 

REAL SPACE FUNCTIONS

 

 

 

 

 

 

V. OPTIMAL SOLUTIONS

1 Application of Optimiggpion Techniques in X Spacey

The mathematical optimization model employs the one—
mensional representations for the conservation of energy (equation
.2.63)) and the swirl parameter (equation (3.2.68)). For the case

= -1, these equations become:

_ 2
GM = H — 1/2(G§ + G2) where 02 = 0% + G3 (5.1.1)

S = GS/(HX2). (5.1.2)

8 optimization problem is posed to discover those functions G2 and G
at will optimize the function G8 subject to the restriction imposed
the swirl parameter S.

Let

t1 = G8 - H + 1/2(G§ + 02) = 0 (5.1.3)
and ¢2 = 03 — SHX2 = 0. (5.1.8)

r augmented function for this case is written as:

2

A
1 2 2
: = Al¢l + i2¢2 = AlGu _ AlH + 57(02 + G ) + X2G2 — AQSHX . (5.1.5)

76 A

 

 

 

 

 

77
Comparison of equation (5.1.5) to equation (1.8.7) demonstrates that
this problem is formulated as a classical Mayer type problem (refer to
section 1.8).
Following the proposition given in reference [81, pg. 33] for
solving problems not involving derivatives of the dependent variables,

a change of dependent variables is introduced.

Let o’ = 07,, 8’ = 0.2, and Y’ = G. (5.1.6)

Substitution of equations (5.1.6) into equations (5.1.3), (5.1.8), and
(5.1.5) yields the following expressions for the constraint equations

and the augmented function:

7. = of — H + 1/2(s’2 + V2) = 0 (5.1.7)
1
452 = 6’2 - SEX2 = 0 (5.1.8)
! Al ’2 ’2 ’2 2 (5 l 9)
K=Alo—A1H+§~(B +7 )+>\2B —)\28HX. ..

A single degree of freedom exists for this problem as there
are three dependent variables (a, B,Y) and two constraint equations
(5.1.7) and (5.1.8). Hence, one optimum requirement can be imposed on
a. The following end conditions are specified for this problem: xi,
xf, df, Yi, and Yf. Now the optimum problem is specifically formulated
as follows: In the class of functions a(X), 8(X), and Y(X) which are
consistent with the constraint equations (5.1.7) and (5.1.8) and the
specified end conditions, find that special set which minimizes the
difference Ar = Tf — Ti, where T = a. Note that for df specified, min—

imizing the difference AT corresponds to max1lelng ai.

 

 

 

78

Applying equation (l.8.8), one Euler—Lagrange equation is ob—

tained for each dependent variable.

d. 3K 8K _
ETIEE’) — 33.. 0 (5.1.10)
d 3K 3K _

32435,) _ 38 _ 0 (5.1.11)
iirié.) _ 25.: 0 (5.1.12)

dX 37’ av

Substitution of equation (5.1.9) into equations (5.1.10), (5.1.11), and

and (5.1.12) leads to the following results:

A1 = 0 or A1 = constant . (5.1.13)
(Al + 2x2) 8” + (xi + 215) s” = 0 (5.1.18)
(17” + iiY’ = 0. (5.1.15)

Substitution of equation (5.1.13) into equation (5.1.15) leads to the

following result:

Y — L where L is an integration constant. (5.1.16)

Since Al is a constant, an obvious solution to equation (5.1.18) is ob—
tained by choosing 12 to be a constant. Let

-A
_ _ 1. (5.1.17)
12 — constant — 5—-

 

 

79
The value of Al is determined by applying equation (1.8.9),
:he transversality condition. The specific transversality condition
?or this problem becomes:

2 :2
- B Y ’2 2
d _ _. _. B
. a (Al(H + 2 + 2 ) + A ( + SHX )) dX + Aldo + (Al + 2A

2 ) 8’88

2

. f
+ A Y dY] = 0. (5 1.18)
l 1

{quation (5.1.18) is to be satisfied identically and for the specified

Ind conditions this then requires that:

81' = -1, (5.1.19)
1
rom equation (5.1.13),
f
d = .
if Al O
lerefore,
= = = — , .l.2O
Alf Ali Al and Al 1 (5 )

lus, from the application of the Euler—Lagrange equations and the
'ansversality condition; Al, A2, and Y’ have been determined.

If expression (1.8.10), the Legendre—Clebsch necessary condi—
on, is negative, the optimum obtained is a maximum. For this case,

.pression (1.8.10) becomes:

* ’ 5 ’<0 (5.1.21)
Kyiyg 58: VJ

where repeated indicial subscripts indicate summation.

 

 

 

 

 

   

By inspection of the form of the augmented function, equation (5.1.9),

it is seen that

K . = o for j ¢ k. (5.1.22)

Utilizing equation (5.1.22), the specific expression for expression

(5.1.21) becomes:

22 A2 "2
Ka,a,(5a ) + KB,B,(6B ) + KY,Y,(6Y ) <0. (5.1.23)

From equation (5.1.9), the following relations are determined:

K , , = o, K828, = 11 + 2A2, K , = A . (5.1.2h)

Upon substitution of equations (5.1.2h), expression (5.1.23) becomes:

(Al + 2A2) (68’)2 + Al(6Y’)2<O. (5.1.25)

From equations (5.1.17) and (5.1.20) it is indeed evident that expres-
sion (5.1.25) holds and hence a maximum is obtained for this case.

The optimum expression for GM is obtained by substituting
equations (5.1.6), (5.1.8), (5.1.16), and (5.1.20) into equation
(5.1.7).

Gh = H — 1/2(SHx2 + L2) (5.1.26)
Equation (5.1.26) represents the optimum Gh when the sum G2 = G5 + G; =

42 = a constant and G is governed by the swirl parameter S as given by

2

equation (5.1.2). Gh attains its maximum value at X = Xi.

 

 

 

_'.
_

 

 

81
Since GM is the one—dimensional representation for pressure,
it is restricted to positive values. This restriction on GM places

the following restriction on the swirl parameter:

s<2 — L2/H. (5.1.27)

From Table A.l, the values H = 2.0000 and L2 = 2.6951 are selected
since they occur at the point where the function GM is a maximum as de—
termined by the solution of the system of ordinary differential equa—
tions in Chapter IV. For the selected values of H and L2, equation
(5.1.27) requires that s<o.6525.

Computing the maximum Gh at X = Xi = 0.1 using equation
(5.1.26) with the above—mentioned values for H and L2, the computed
maximum GM for S = 0.25 is found to be within 0.02% of the maximum
ralue given in Table A.l. For 8 = 0.65, the computed maximum value of
}h is found to be within 0.63% of the maximum value given in Table A.1.
Therefore, the maximum GM value determined by equation (5.1.26) com-
>ares favorably with the maximum Gh value determined by the solution
rf the system of ordinary differential equations in Chapter IV.

Figures 5.1 and 5.2 illustrate the G2 and GM curves generated
rom equations (5.1.2) and (5.1.26) using S as a parameter. The S val-
es chosen are S = 0.25, S = O.h5, S = 0.55, and S = 0.65. The maximum
alue of GM occurs at the left end of the X interval where G2 is at a
inimum.

The curves were plotted by a Calcomp plotter which is con—
rolled by the output data from an IBM 360 computer- The computer fa—

.lity used during this investigation is located at General Motors

 

 

 

 

82
Institute, Flint, Michigan. Appendix E contains the program for compu—

ting the data for these curves and the actual curve data.

5.2 Three-Dimensional Optimal Pressure and Velocity Curves

The three—dimensional equations for optimum pressure and tan—
gential velocity are obtained by substituting the transformation rela—
tions (h.2.1), (h.2.2), and (h.2.3) into equations (5.1.26) and

(5.1.2):
P = H — 1/2(SH(Z/R)2 + L2), (5.2.1)
V = [SH]l/2 Z/R. (5.2.2)

Equation (5.2.1) represents the optimum pressure and equation (5.2.2)
represents the tangential velocity for optimum pressure.

The optimum P and V curves are plotted as functions of R
using Z as a parameter and S = 0.65. For Z as a parameter, the plot
interval becomes: ZSRSIOZ. The maximum value of P occurs at the right
end of the R interval where R = lOZ. At R = lOZ, V is a minimum.
Therefore, the maximum pressure occurs at the point where the tangen-
tial velocity is a minimum. This same result is obtained in Chapter IV
for the transformed equations of the solutions to the system of ordi—
nary differential equations.

These curves were also plotted by a Calcomp plotter. The
:omputing program and the actual computed data are contained in
lppendix E.

Figures 5.3 and 5.h display the curves for optimum pressure

Lnd tangential velocity, respectively. For the parametric values

 

 

 

 

 

62 CURVES S—PRRRMETEH
S=.25 SUURHE.S=.U5 CIRCLE.S=.55 THIRNGLE.S=.BS PLUS

FIGURE 5.1 G2 CURVES S PARAMETER

 

 

    

 

 

I

LLZH
J

 

GU-HXIS
.16 '

O
._.___. .L-

0.08

CL). GU

 

 

8h

04 CURVES S‘PRRQMETER
52.25 SUUQRE,SL.QS CIRCLE,SL.SS T9IRNGLE.S:.65 PLUS

0150 ciau 0170 0150 0190 1100

FIGURE 5.2 Ga CURVES S PARAMETER

 

 

 

85
Z = 0.1, Z = O.h, and Z = 0.7, the corresponding maximum pressure loca—
tions are R = 1, R = h, and R = 7. Hence, the maximum pressure value
occurs at the rightwend point for each pressure curve in Figure 5.3.
Similarly, the minimum value of the tangential velocity occurs at the

right—end point for each velocity curve in Figure 5.h.

 

5.3 Summary

A one—dimensional optimal expression for GM has been deter—
mined for the case where the G2 expression is governed by the swirl
parameter and the expression G2 is required to be a constant. Optimal

curves for G2 and GM are plotted in Figures 5.1 and 5.2. The expres—

 

sions for G2 and GM were transformed from the one-dimensional (X) space
back to the three-dimensional (R, 6, Z) space omitting the passage to
the two—dimensional (n1, n2) space. The three—dimensional expressions
for optimal pressure and tangential velocity are plotted in Figures

5.3 and 5.h.

 

 

86

FRMILY 0F UPFIMUM PRESSURE CURVES 52.65
Z=.1 SOURRE.Z=.Q CIRCLE.Z=.7 TRIRNGLE

32
I

SSUU‘RE (P)

.2”
I

UPIIM%M PRE

16

0.

0.08

 

I
00.10 0190 1 . 70 ausu 3'. 3E1 u'.10 4. 90

FIGURE 5.3 OPTIMUM PRESSURE CURVES

 

 

 

 

87

TRNGENTIRL VELOCITY FUR UFTIMUM PRESSURE 52.65
D Z=.1 SUURRE.Z=.Q CIRCLE.Z;.7 TRIRNGLE
ID

I
J

L00

 

UCITYlV)

030

l

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060

0. 4'0"

U.8U

‘u UU

1, 10 0'. an 1'. 70 2150 3’. 30 u'. 1:: 4190 5. 70 6.51‘
R-RXIS

d‘

FIGURE 5.h VELOCITY CURVES FOR OPTIMUM PRESSURE

 

li.‘ Ill l".\l 1" El l: llll.

 

 

BIBLIOGRAPHY

10.

ll.

l2.

l3.

 

BIBLIOGRAPHY

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Batchelor, G. K., "On Steady Laminar Flow with Closed Streamlines
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Beckett, Royce and Hurt, James, Numerical Calculations and
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Benjamin, T. B., "Significance of the Vortex Breakdown Phenomenon",
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Bichara, R. T., and Orner, P. A., "Analysis and Modeling of the
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Birkhoff, Garrett, H drod amics, Second Edition, Princeton
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Bookston, J. M. and McKeachie, D. D., "Notes on the Numerical
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Carnahan, Brice, Luther, H. A., and Wilkes, James 0., A lied
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Deissler, Robert G. and Perlmutter, Morris, ”An Analysis of the
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Hildebrand, Francis B., Finite Difference E uations and Simulations,
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APPENDICES

 

 

APPENDIX A

APPENDIX A

NUMERICAL DATA FOR THE SOLUTIONS

TO THE SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

A computer program utilizing Hamming's modified predictor—
corrector method for numerically solving a general system of ordinary
differential equations is presented in this appendix. This program is
written in Fortran IV language and includes a file save such that the
computed output data can be stored and used as input data to the sub-
routine which controls a Calcomp plotter. The IBM scientific subrou—
tine DHCPG mentioned in Chapter IV is included in this particular pro-
gram.

Table A.l presents the numerical data for the solutions to
the system of non-linear ordinary differential equations given in
Chapter IV. In addition to values for G , G2, G3, and GA’ values for
H, S, L2, and we are listed. The values of these functions at each
value of X are displayed in the order shown in the following 2X5

matrix:

X value G1 G2 G3 Gu H

92

 

 

 

 

 

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APPENDIX B

APPENDIX B

CALCOMP PLOTTER ROUTINE

A special library subroutine named L9MIKE was used to control
the Calcomp plotter which plotted all the curves displayed in this the-
sis. This subroutine allows the user to title curves, title axes, se—
lect scale lengths, plot parametric curves where the plot interval is a
function on the parameter, specify the order and number (up to 20) of
variables to be plotted per diagram, and identify each curve by at
least 10 identification marks (squares, circles, triangles, etc.). The
input data for the L9MIKE subroutine is taken from a stored file and
all data points in the file are plotted, but only 10 or 11 identifica—
tion points are marked on each curve. The L9MIKE subroutine is written
in Fortran IV language.

This appendix presents the computer printout listing of the

L9MIKE subroutine used for controlling the Calcomp plotter.

113

 

 

 

 

 

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APPENDIX C

 

 

APPENDIX C

LEAST SQUARES POLYNOMIAL CURVE FITTING PROGRAM

A computer program utilizing the method of least squares for
determining a polynomial approximation of tabular data is presented in
this appendix. The program is written in Fortran IV language and the
least squares portion is a part of a library of standard scientific
subroutines. The program is named PCOSMB and it will fit 2 through 100
points with a polynomial of degree 1 through 11. For this program, the

standard error of estimate is defined as:

1/2

1 i 2
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1=l

Standard error of estimate E -——j;':'7313j"“‘

In the equation for the standard error of estimate, m represents the
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fitting data for the functions G1, G2, G3, and Gh in the interval

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APPENDIX D

PRESSURE AND VELOCITY PROGRAMS AND NUMERICAL DATA

This appendix presents the computer program utilizing the
subroutines PC05El and PC05E2 to calculate the pressure and velocity
component data for the polynomial equations given in Chapter IV.

Table D.l displays some sample data for these functions at

each R value in the order shown in the following 3X5 matrix:

R value U V W P H

The subscripts indicate the value of the Z parameter used to calculate
each of the functions U, V, W, P, and H. Only a single page sample of
the data is included in this appendix as it would take 76 pages to dis-

play it all.

1H1

 

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APPENDIX E

 

 

 

APPENDIX E

PROGRAMS AND NUMERICAL DATA FOR OPTIMUM FUNCTIONS

 

This appendix presents the computer programs utilized to gen-

erate the data for the functions G Gh’ V, and P as given in Chapter V.

23

Table E.l displays the data for the functions G and GM at

2
each X value for different values of the swirl parameter S in the order

shown in the following 2X5 matrix:

X value G'25 g-25 g-h5 g-h5 g-55
, ‘ 2 u h 2

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for these functions is displayed in the order shown in the following

2X5 matrix:

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each of the functions V and P.

115

 

 

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