The Flow Field Upstream of a Horizontal Axis Wind Turbine

8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine

1/97

University of Massachuses - Amherst

ScholarWorks@UMass Amherst

Wind Energy Center Reports UMass Wind Energy Center

1979

e Flow Field Upstream Of A Horizontal Axis Wind Turbine

K. Modarresi

R. H. Kirchho

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Modarresi, K. and Kirchho, R . H., "e Flow Field Upstream Of A Horizontal Axis Wind Turbine" (1979).Wind Energy Center Reports. Paper 10.hp://scholarworks.umass.edu/windenergy_report/10

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2/97

THE FLOW FIELD UPSTREAM OF

A HORIZONTAL AXIS WIND TURBINE

T echn i ca l Repor t

by

K

Mod arresi and R.H. K i rc hh of f

Energy A1 te rn a t i v e s Program

Un i v e r s i t y o f Massachuset ts

Amherst, Mas sach usetts 01003

June 1979

Prepared f o r t he Un i t ed S ta te s Departmen t o f Energy and Rockwe ll

I n t e rn a t i o na l Rocky F l a t s P l a n t under Co n t rac t PF67025F.

Th is re p o r t was prepared t o document work sponsored by t he Un i ted

S t a t e s Government. N e i t h e r t h e U n i t e d S t a t e s n o r i t s a g e n t t h e

Depar tment o f Energy , no r any Federa l employees, n or any o f t h e i r

co nt r ac tor s , subco nt rac tors , o r t h e i r employees, make any war ran ty ,

exp ress o r i mp l i ed ,

or assume any legal 1 a b i l i t y o r r e s p o n s ib i l t y

fo r the accuracy , completeness, o r use fu ln ess o f any in forma t ion,

a p pa ra tu s, p r o d u ct o r p r oc e ss d is c l o se d , o r r e p r e s e n t t h a t i t s us e

woul

d

n o t i n f r i n g e p r i v a t e l y owned r i g h t s .


3/97

ABSTRACT

A m a t h e m at i c al m odel i s d ev e lo p ed f o r a s t e a d y - s t a t e a x i - s y m m e t r i c

u pst re am f l o w o f a po ro us d is c i n a u n if o r m f l o w f i e l d . The s p ec i a l

c as e o f t h e u ps tr ea m f l o w o f a w i n d m i l l w i t h a nd w i t h o u t a n a c e l l e i s

t r e at e d . F i r s t t h e w i n d m i l l i s c o ns id e re d a s a u n i fo r m d i s t r i b u t i o n

o f s ou rc es a nd th en a s a 1 n e a r d i s t r i b u t i o n o f s ou rc es . S o l u t i o n s f o r

t h e b l a d e d i s c o f t h e w in d f i e l d u pstream a r e o b t ai ne d i n t h e f o r m o f

s t r e a m1 n e s a n d v e lo c i t y v e c to r c o mp o n e n ts .

Sample f l o w p a t t e r n s u p s t re a m o f t h e b l a d e d i s c o f t h e UMass

25

k

w i n d t u r b i n e a r e p r e s e n t e d f o r s e v e r a l p owe r l e v e l s . Docum ented

c o mpu ter p rog ra ms a p p l i c a b le t o a n y w in d t u r b i n e a re a pp en de d.


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TABLE OF COIVTEIVTS

BSTRACT

NTRODUCTION 1

HEORETICAL ANALYSIS 2

I n t r o d u c t i o n t o t h e M o d el in g I d ea

2

o t e n t i a l o f a S ource a t an A r b i t r a r y P o i n t 4

od el f o r t h e Bod y o f a W in d m i l l

o t e n t i a l o f t h e D i s t r i b u t e d Di sc o f S ou rc es

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

5

U n i f or m l y d i s t r i b u t e d d i s c o f s ou rc es

7

L i n e a r l y d i s t r i b u t e d d i s c o f so urc es

8

Sumnary 15

S u p er p os i ti o n o f t h e P o t e n t i a l s 6

V e l o c i t y F i e l d 7

enera l Not e 17

U n i fo r m d i s t r i b u t i o n o f s o ur ce s

8

L i n e a r l y d i s t r i b u t e d c as e 2

U n i f o rm f l o w 3

i n g l e s ou rc e a t t h e p o s i t i o n r

23

Remarks 3

S t r e a m l in e c o n s t r u c t i o n 4

App ly ing the one d imens iona l momentum theory

t o t h e w i n d m i l l s 6

RESULTS 9

REFERENCES 2

PPENDICES 33

IGURES 76


5/97

INTRODUCTION

The p r ob l em of t h e e x a c t s o l u t i o n o f t h e f l o w t h r o u g h a p o ro u s d i s c

has been o f i n t e r e s t s i n ce t h e o r i g i n a l w or ks o f G I T a y l o r [I].

T h i s s o l u t i o n c a n have tw o m a j o r a p p l i c a t i o n s :

a )

F lo w th ro u g h s c re e n s p o w er a b s o rb in g d e v i c e s )

b )

F low th ro u g h p ow er d e v e lo p in g d e v i c e s .

The f i r s t p o i n t o f v i e w h as t o d o w i t h p or ou s b o d ie s w hi ch h ave been

o f p r a c t i c a l i m p o rt a n c e i n t h e p a s t . M os t o f t h es e b o di e s, w h i c h

a r e l i k e pa ra ch ute s, f i s h ne t s, w in d b re ak s, s l o t t e d i n j e c t i o n d i s c s

i n o i l c o m bu st io n cham bers, a n d f i r e d ev el op me nt s i n f o r e s t , c an b e

model ed by a screen.

The s econd p o i n t o f v i e w c o n s id e r ed i n t h i s wo rk d e a l s w i t h

the w ind power deve lop ing mach ines .

These dev ices can be modeled

b y a v e r y p o ro us d i s c i n t h e wi n d f i e 1 d.

The f i r s t c om pr eh en si ve a n a l y s i s o f f l o w t h r o u g h s c re e ns was

c a r r i e d o u t b y T a y l o r a n d B a t c h e l o r 1 94 9) ,

[I] 2 j .

T h e y w e r e i n t e r -

e s t e d i n t h e e f f e c t of t h e s c re en on t h e t ur b u le n c e , a nd t h e r e f o r e

t h e i r s t u d y was o r i e n t e d to w ar d s t h e n o n - u n i fo r m ch a nn e l f l o w , p a s s i n g

t h r o u g h a f l a t s cr ee n. I n 1 959 , E l d e r c o n s i d e r e d t h e m or e g e n e ra l

c as e o f a n i r r e g u l a r - s h a p e d a n d n o n - un i f or m s c re e n

i n a t w o d i m e n s io n a l

c h a n n e l f l o w [ 3 ] .

The p ro b le m of a f i n i t e p l a n e sc re en i n a n

i n f i n i t e f l o w f i e l d

was f i r s t c o n s i d e r e d b y ~ l c h e m a n n n d Weber 1 95 3 ) [ 4 ].

L a te r , i n

1 96 3, T a y l o r c o n s id e re d t h e p ro b le m i n a tw o -dime n s io n a l c a s e [5 ].

T h i s was done b y r e p l a c i n g t h e s c re en w i t h u n i f o r m l y - d i s t r i b u t e d s ou rc es .


6/97

2

F i n a l l y

Koo and James co ns id e re d th e more gen era l case o f the two-

d imens iona l f l o w a round a submerged sc reen [ 2 ] .

F o l l o w i n g t h e i d e a o f m o d e l i n g a ny w in d m a ch in e by t h e c o m b i n a t i o n o f

s ou rc es s i n k s o r v o r t i c l e s t was p ro po s e d t h a t a w i n d m i l 1

can be

modeled by a porous screen.

T h i s wo rk was o r i e n t e d t o w a rd s s o l v i n g

t h e p r o b le m o f a th r e e - d i m e n s i o n a l p o ro u s d i s c i n a s t e a d y - s t a t e

a x i -

s ym m e tr ic u n i f o r m f l o w . The s c re e n was m od ele d b y a d i s t r i b u t i o n o f

s o u rc e s an d t h e p ro b l e m was d i v i d e d i n t o two c a s e s.

F i r s t t h e s im p le

c as e o f m o d e li ng t h e w i n d m i l l b y a u n i f o r m l y - d i s t r i b u t e d d i s c o f s ou rc e s.

Second a m ore r e a l i s t i c m odel was c o n s id e r e d . T a k i ng i n t o c o n s i d e r a t i o n

t h e f a c t t h a t t h e d eve lo pm en t o f p ower i s h i g h e r i n t h e o u t e r r e g i o n

o f t h e w i n d m i l l b l a d es t h e b la d e d i s c was m od ele d by a l i n e a r l y - d i s t r i b u t e d

d i s c o f so urc es . The e f f e c t o f a n a c e l l e and i t s r e l a t i v e o r i e n t a t i o n

t o t h e b l a de d i s c was s t u d i e d i n b o t h c as es .

The v e l o c i t y f i e l d a nd t h e s t r e a m l i n e s we re c o n s t r u c t e d f o r some

n u m e ri ca l exam ples i n a s s o c i a t i o n w i t h t h e

25

kW w i n d m i l l a t t h e U n i v e r s i t y

o f M a s sa c hu s et ts S o l a r H a b i t a t

I


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THEORETICAL ANALYSIS

I n t r o d u c t i o n t o h e M od el in g Id ea

The w e ll -k n ow n i d e a o f m o d e l in g f l o w f i e l d s t h r o u gh a c o m b i n a ti o n

o f i n d i v i d u a l f a c t o r s i s u sed i n o rd e r t o model a p o ro us d i s c a s a

d i s t r i b u t i o n o f s ou rc es i n space w i t h i n a u n i f o rm f l o w f i e l d .

J.K. Koo an d D.F. James

[ 2 ]

deve loped G

I

T a y l o r s [ 5 j i d ea o f

m o d e l i n g a t w o -d i m e ns io n a l s c re e n b y a d i s t r i b u t i o n o f s o u rc e s, b y

m odel i n g t h e s cre e n i n a d u c t . T h i s w ork i s o r i e n t e d t o f i n d a g e n er al

s o l u t i o n f o r a t hr e e- d im e n s io n a l s cr ee n, u s i n g a d i s t r i b u t i o n o f s ou rc es

on a d i s c , i n an a x i -s y m m e t r ic u n i f o r m f lo w .

The d ev elo pm en t o f t h e b a s i c e q u a t i o n s i s b a se d on t h e f o l l o w i n g

p ro ce du re : f i r s t , t h e p o t e n t i a l o f a s ou rc e l o c a te d a t an a r b i t r a r y

p o i n t i n spa ce i s d e te rm in e d , s ec on d, b a se d on t h i s p o t e n t i a l , t h e

c as es of a d i s c w i t h a u ni fo rm o r l i n e a r d i s t r i b u t i o n o f s o urc es a r e

c o n s i d e re d an d t h e p o t e n t i a l on t h e a x i s o f t h e d i s c fo un de d, a nd t h i r d ,

b a se d on t h e h a rm o ni c a nd m ore p a r t i c u l a r l y t h e s y ~ r ~ m e t r i cr o p e r t i e s

of t h e p o t e n t i a l f u n c t i o n a nd t h e s o l u t i o n on t h e d i s c s a x i s by t h e

use o f z on al h arm on ie s, t h e g e n er al s o l u t i o n o f t h e f u n c t i o n i s c o n s t ru c t e d .

The model i s co m p l et e d b y t h e s u p e r p o s i t i o n o f a u n i f o r m f l o w

on t h e p o t e n t i a l o f t h e d i s c .

I n th e ca se o f m od e li ng a w in d m i l l , t h e e f f e c t o f t h e n a c e l l e

on t h e d i s c s a x i s c an be m od ele d b y a s i n g l e s o u rc e, w h ic h ca n r e s u l t

i n d i f f e r e n t bo dy shap es.

The s o l u t i o n i s i n t he fo rm o f an i n f i n i t e s e r i es o f t he

Legendre

a nd A s s o c i a t e d L eg en d re p o l y n o m ia l s . The v e l o c i t y f i e l d a n d s t re a l il l i n e s


8/97

a re con s t ruc ted by a computer p rogram, and w l l be de sc r ib ed l a t e r i n

t h i s r e po r t.

P o t e n t i a l o f a S ource a t an A r b i t r a r y P o i n t

The p o t e n t i a l of a p o i n t so ur ce a t t h e o r i g i n c an be w r i t t e n a s

[ ]:

w here k i s t h e s ou rc e s t r e n g t h .

The p o t e n t i a l a t a p o i n t o f a p o i n t s ource a t S i s : F i g . 1 )

knowing

t h e p o t e n t i a l i s :

By u s i n g a c o o r d i n a t e t r a n s f o r m a t i o n a s shown i n F i g .

2, t h e p o t e n t i a l

c an be t ra n s f o r me d t o s p h e r i c a l

c o o r d i n a t e a s :

then,


9/97

f h e so urc e i s

i n y - z plane, where 8= 1112, then

Model f o r th e Body o f a Windmi l l

The n a c e l l e o f a w i n d m i l l ca n be ap p ro x im a te d a s a p a r a b o l i d o f

r e v o l u t i o n . T h i s i s m od ele d b y a p o i n t s ou rc e i n a u n i f o r m f lo w

[7].

(See Fig.

3).

Tak ing rl and from t he geomet ry o f t he na ce l le , and assuming a

t h e s ou rc e s t re n g t h k and i t s p o s i t i o n r an be found by:

0

T he re fo re , t h e p o t e n t i a l f o r t h e n a c e l l e i n a u n if o r m f l o w can be w r i t t e n

as :

P o t e n t i a l o f t h e D i s t r i b u t e d D is c of S ou rce s

I n t r o d u c t i o n . The t o t a l p o t e n t i a l

a

o f d i s c s s ou rce s i s t h e s o l u t i o n

2

t o t h e L a p la c e E q ua t io n

V

= o ) , w i t h t h e a p p r o p r i a t e bo un da ry c o n d i t i o n s .

S in ce t h e f l o w i s a x i- sy m m et ri c,

t h a t i s , i nd ep en de nt o f a (See


10/97

F i g . 2) t h e s o l u t i o n o f

@ =

0 ca n be w r i t t e n a s [ 8 ] :

~ o

where: Pn(x) = L eg en dre P o ly no mia l o f t h e f i r s t k i n d an d n t h o rd e r.

Knowing tha t on the x -ax is e i s z e ro a nd t h a t Pn ( 0 ) = 1

[8];

t h e

s o l u t i o n o n t h e x - a x i s can b e w r i t t e n a s :

Hence , t o de te rm ine An and Bn,

t h e s o l u t i o n on t h e x - a x i s s h ou ld

be found , expanded i n a power se r ie s o f

r

a n d t h e n e q u a t ed t o e q u a t i o n

5).

C on se qu en tly, t h e f i r s t s t e p i s t o f i n d t h e d i s c s s ou rc e p o t e n t i a l

on t h e x - a x i s .

C o ns id er a d i s c o f d i s t r i b u t e d s ou rc es w i t h t h e r a d i u s A and source

s t r e n g t h p e r u n i t a re a k a s shown i n F i g . 4.

The e lement o f a rea d

i s equal t o

p

d v d p f o r :

o <

p

A and

<

2 ~

s in g e q ua t i on ( 2 ) t h e d i f f e r e n t i a l

p o t e n t i a l a t any f i e l d

p o i n t ( r ,0 ) can be w r i t t e n i n term s o f t h e d i f f e r e n t i a l so urce d i s t r i -

b u t i o n . a s :

T o g e t t h e s o l u t i o n o n t h e x - a x i s ,e q u a t i o n

( 6 )

can be r e s t r i c t e d

t o th e x -a xis , t h a t i s

e =

o. The d i f f e r e n t i a l p o t e n t i a l on t h e x - a x i s

i s :


11/97

and,

I n t e g r a t i o n o f e q u a t i o n 7 ) dep end s o n k . The prob lem can be d i v ided

i n t o tw o c as es : a ) k i s c o n s ta n t o r a u n i f o r m l y - d i s t r ib u t e d s ou rc e d is c

and b ) k i s l i n e a r i n o r a l i n e a r l y - d i s t r i b u t e d sou rce d i s c .

U n i f or m l y d i s t r i b u t e d d l s c o f s o urc es .

k c o n s t )

1f k i s c o n s t a n t, e q u a t io n 7 ) ca n be i n t e g r a t e d a s:

U s i ng a b i onom i a l

expans i on

t

can be shown t h a t , f o r R s m a l l e r

t h a n

A

a n d f o r R g r e a t e r t han A:

n o

T h e r e f o r e x i n e qu at io n 8 ) can be expanded as

[9 ] :

and


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The equating of equations

1 1 )

and 12 ) to equation 5 ) allows n

and n to be determined for the general solution

m

This leads to :

and

Therefore, the general solution becomes:

and for > A

Linear ly d i s t r ibu ted d i s c o f sou rces .

F o r a l i n e a r l y d i s t r i b u t e d d i s c

of sources ,

m where m i s a c o n st a nt .

Hence equation 7 ) can be

w r i t t e n a s


13/97

and i n t e g r a t i o n o f t h i s e qu at io n w i l l

h a v e t h e f o l l o w i n g f o r m

I n o r d e r t o e x pa nd Qx i n a power s e r i e s o f R, t h e s pace i s d i v i d e d

i n t o tw o r e g i o n s : S ee F i g .

5 )

a ) R < A

b) R > A

I n r e g i o n a ) , eac h p o i n t

Q

w h ic h i s i n s i d e a s ph er e o f r a d i u s

A

i s a f f e c t e d b y tw o k in d s o f so ur ce s : F i r s t , t h o s e whose d i s t a n c e f r o m

t he o r i g i n p ) i s le s s th an R R adius o f P o i n t Q ) , t h a t i s p < R, and

second, t hose w i t h

p >

R.

T h e r e f o r e , f o r R

< A,

t h e p o t e n t i a l on t h e x - a x i s c an be w r i t t e n a s :

where Q 1 i s t h e p o t e n t i a l o f t h e sourc es w i t h p <

R

pchanges f rom zero

t o

R

and

m p

i s t h e p o t e n t i a l o f t h e so urc es w i t h

r

>

R

p

changes f rom

R

t o A ).

From equ at ion 15 ) , Q may be wr i t t e n as:


14/97

where = p /R

b i o n o m i a l e x p r e s s i o n c a n b e w r i t t e n a s :

oe

n

=

\ E

n t

o

n

I

6

where:

U s i n g e q u a t i o n

1 6 )

i t

i s o bv io us t h a t :

U s in g t h i s ex pa ns io n i n e q u a ti o n 1 5A ) r e s u l t s i n :

t h e r e f o r e ,

Now s o l v i n g f o r

m2

e q u a t i o n 1 5 ) ca n be w r i t t e n a s:


15/97

where, t =

R P

1 .

U s i n g t h e b i n o m i a l

ex pans ion s hown i n equa t i on 1 6 ) ,

t

can be

shown tha t :

C o n si de r i ng t h i s r e s u l t , e q u a t i o n 1 5 6) ca n be w r i t t e n a s :

t he r e f o r e ;

t

was shown t h a t


16/97

0 = a 1 @

X 2

or

R A

So th e comb ina t ion o f equa t ions 17 ) and 18 )

w i l l

r e s u l t i n

mx

f o r

R < A t h a t i s :

where, tl = R / A ) < 1

R e ar ra ng in g t h i s f o rm ula r e s u l t s i n

The f i r s t t h r e e te rm s o f t h e ex pa ns io n o f I n tl f o r t h e case o f

tl 1 a r e

S u b s t i t u t i o n o f t h i s i n m w i l l g i v e th e f i n a l r e s u l t o f

m x

f o r < A .


17/97

E q u at io n 1 9 ) shows th e p o t e n t i a l o f t h e x - a x i s f o r t h e l i n e a r l y -

d i s t r i b u t e d d i s c o f s ou rc es when R

A.

F o r t h e o t h e r r e g i o n R

>

A),

e q u a t i o n 1 5 ) c an b e w r i t t e n a s

A

f

t = - 51 and t2= e q u a t i o n 1 5C ) c an be r e w r i t t e n a s :

R R

It

was p r e v i o us l y shown t h a t

Hence,

A

s i m p l e i n t e g r a t i o n r e s u l t s i n :

E qu at io ns 1 9) and 2 0) a r e t h e p o t e n t i a l o f a l i n e a r l y - d i s t r i b u t e d

d i s c o f s ou rc es o f r a d i u s

A

o n t h e x - a x is , f o r R s m a l l e r t h a n

A

and R

g r e a t e r t h a n A, r e s p e c t i v e l y .


18/97

To f i n d t h e g e n er a l s o l u t i o n , a s was p o i n t e d o u t p r e v i o u s l y , t h e se

e q u a t i o n s s h o u l d b e e q ua t ed t o e q u a t i o n 5 ) . The r e s u l t o f t h e co m p ar is o n

d e te rm in e s

n

and Bn fo r bo th cases o f

R

>

A, and

R

<

A.

C om pa ris on o f t h e e q u a t i o n 1 9 ) w i t h t h e e q u a t i o n 5 ) shows t h a t :

C omp ar is on of t h e e q u a t i o n 20) w i t h t h e e q u a t io n 5 )

shows the

f o l 1o w i n g r e s u l t s


19/97

where

The gene ra l so lu t i on can be de te rmined by s u b s t i t u t i o n o f e qu at io ns

2 1 ) and 2 2 ) i n e q u a t i o n 4 ) .

The f o l l o w i n g r e s u l t s c an t h u s be d e r i v e d :

For R < A

And, f o r R

> A

where:

=

binomi l

coefficients

n

Summary.

As a c o n c l u s i o n t o p a r t f o u r , i t

i s i m p o r ta n t t o make t h e

f o l l o w i n g summary. The g e ne ra l s o l u t i o n t o t h e p o t e n t i a l o f a d i s c o f

d i s t r i b u t e d s ou rc es of r a d i u s A and s o u rc e d e n s i t y p e r u n i t a r e a k has

been found.

The s o lu t i on has been de te rmined f o r two cases .

The

s o l u t i o n t o t h e u n if o rm d i s t r i b u t i o n o f s ourc es k

=

c o n s t . ) i s shown

i n t h e e q u a t io n s 1 3 ) a nd 1 4 ) .

F or t he l i n e a r d i s t r i b u t i o n o f s ou rc es

k

= m ~ ,

h e p o t e n t i a l i s e s t a b l i s h e d i n e q ua t io n s 2 1 and 2 2 ).


20/97

S u ~ e r ~ o s ii o n o f t h e P o t e n t ia l s

To c o mp le t e t h e f l o w mod el f o r a p o ro u s d i s c and w i n d m i l 1 i n

u n i f o r m f l o w , t h e n e c e s s a ry p o t e n t i a l s s h o u l d be su pe ri mp os ed .

F o r t h e o p e r a t i n g e as e o f t h e p or ou s d i s c o r a w i n d m i l l

w i t h o u t a

n a c e l le , t h e p o t e n t i a l i s :

@ = @ + @

1 2

where:

= p o t e n t i a l o f t h e d i s c

1

@ =

p o t e n t i a l o f t h e u n i fo r m f l o w

2

I n t h e case o f a w ndm i 11, t h e p o t e n t i a l

c a n be w r i t t e n a s:

where :

@

= p o t e n t i a l o f t h e d i s c

1

@

= p o t e n t i a l o f t h e s i n g l e s ou rc e a t t h e p o s i t i o n r o n t h e

2

x - a x i s a s shown i n F i g . 3

@ = p o t e n t i a l o f t h e u n i f o rm f l o w

3

@

was e s t a b l i s h e d i n e q u a t io n s , 1 3 ), 1 4 ) , 2 1 a nd 2 2 ) .

Q 2

can be

d e t er m i ne d f r o m e q u a t i o n s 1 a nd 3 ) a s f o l l o w s :

I n t h i s case, r =

E

= 0. = r 2

so

Q

c an b e w r i t t e n a s


21/97

U sin g r e l a t i o n s I A ) ,

P

c an be t r a n s f o r m e d t o a s p h e r i c a l c o o r d i n a t e .

z

ca s e

=

S h e s ~

and

t L s @ ~

Hence

Ql

i s t h e p o t e n t i a l o f a u n i f o rm f lo w , and

i t

c a n b e re p re s e n te d b y :

V e l o c i t y F i e l d

G en era l N ot e. The v e l o c i t y f i e l d c a n be d e t e rm i n e d b y s u p e r p o s i t i o n

o f t h e v e l o c i t i e s . The t a s k o f t h i s s e c t i o n i s t o f i n d t h e components

o f t h e v e l o c i t y v e c t o r f o r e ach p o t e n t i a l .

The r e l a t i o n between t h e p o t e n t i a l f u n c t i o n and t h e v e l o c i t y v e c t o r

i s known t o b e:

The g r a d i e n t i n s p h e r i c a l c o o r d i n a t e s c a n be shown a s :

I n t h e c as e o f a x i- sy m m et ry t h e g r a d i e n t r e d uc es t o :


22/97

Hence, t h e components o f t h e v e lo c i t y v ec to r can be shown as

i

IJRP . nd U T P .

-

a

b r f

be

where:

uR

= R a d ia l Component o f t h e v e l o c i t y v e c t o r

uT = Tan gen t i a l Com ponent o f t h e v e l o c i t y vec t o r

Now one can supp l y t h i s gene r a t e no t e t o any sp ec i a l case .

U n if or m d i s t r i b u t i o n o f S ou rc es . F o r t h i s c as e t h e p o t e n t i a l was f ou n d

and shown i n equa t i ons 13 ) and 14 ) as f o l l ow s :

a )

i n t h e case o f

R

< A t h e c om pon en ts o f t h e v e l o c i t y v e c t o r c a n

b e d e r i v e d a s f o l l o w s :

T h er efo re , t h e d i f f e r e n t i a t i o n o f e q ua t i on 1 3) y i e l d s :


23/97

Hence:

The tangen t ia l componen t can be wr i t t en as :

a e

U s i n g m f r o m e q u a t i o n 1 3 ) ,

t

c a n b e w r i t t e n t h a t

\ p k ~

r (coso)

b e

U s in g t h e c h a in r u l e , t h e d i f f e r e n t i a l s can b e changed t o

From t h e d e f i n i t i o n o f t h e A s s o c i at e d L eg en dr e p o ly n o m ia l s:

w

irti y rear rangement

where:


24/97

m

Pn x )

=

A s s o c i a te d L eg en dre p ol yn o m i al o f t h e f i r s t k in d , t h e

n

t h

o r de r , and t he

m

degree.

U s i n g e x p r e s s io n 3 1 ) i n t h e e q u a t i o n 3 0 ) i t can eas i l y be shown

t h a t

and

.a n ( G s e )

-

p

b s t 3 )

at

Usin g the n o ta t i o n ASn Co s l ) f o r pnl Cose) , t he above eq uat i on can be

w r i t t e n a s:

aP ( S O )

= A s , 6 s e )

g

.

S u b s t i t u t io n o f t h 2

ec ita

l ? c.:; I. ;?)n t o t h e e q ua t i on 2 9) y i e l d s :

b ) I n t h e c as e o f R > A f o l l o w i n g t h e same p ro c ed u re f o r o f r om

t h e e q u a t i o n 1 4 )

i t

can be w r i t t e n t h a t

and

A c c o r d i n g l y ,

t h e v e l o c i t y co mpo nents o f t h e u n i f o r m l y - d i s t r i b u t e d

d i s c o f s o ur ce s c an b e w r i t t e n a s e q u a t i o n s 2 9 ) an d 3 3 ) f o r th e c as e


25/97

o f R

<

A, a nd a s e q u a t i o n s 3 4 ) a nd 3 5 ) f o r t h e c a s e o f R A.

L i n e a r l y d i s t r i b u t e d c ase.

F o r t h e c as e o f t h e l i n e a r l y - d i s t r i b u t e d d i s c

of sources,

t c a n b e shown t h a t t h e co mpo nen ts o f t h e v e l o c i t y f i e 1d a r e

a s f o l l o w s : See A pp en dix 1 f o r t h e d e t a i l s ) .

F o r R


26/97

And fo r t h e c a se o f R .> A, t c a n b e w r i t t e n :

nd

U n i f o r m f l o w .

t i s q u i t e s i m p le t o show t h a t t h e c omponents of t h e v e l o c i t y

f i e l d f o r t h e u n i f or m f lo w a r e a s f o l lo w s :

and

UT =

U

S c n

S i n g l e s ou rc e a t t h e p o s i t i o n

r

r e p r e s e n t i n g t h e n a c e l l e ).

D i f f e r e n t i a -

t i o n of t h e e q ua t io n 23 ) a c c o r d i n g t o e q u a t io n

2 7 )

y i e l d s t h e v e l o c i t y

c omp on en ts d ue t o t h e b o dy s ha pe o f t h e n a c e l l e a s :

and

Remarks.

I n o rd e r t o u t i l i z e t h e p r e v io u s l y d e r i ve d fo rm ulas f o r t h e

v e l o c i t y f i e l d , t wo c o mp u te r p ro gr am s w er e w r i t t e n .

B o t h of t h e s e p ro -

gram s w ere w r i t t e n f o r t h e g e n er a l

c as e o f t h e p re se nc e of a l l t h r e e

p o t e n t i a l s .


27/97

2

The f i r s t p ro gra m,

P ro gr am P r o j e c t 1 i n Ap pe nd ix 2A was designed

f o r t h e u n i fo r m l y d i s t r i b u t e d d is c o f s ou rc e i n g e ne ra l, and i t s a p p l ic a -

t i o n t o t h e U n i v e r s i t y o f M as sa ch us etts w i n d m i l l i n p a r t i c u l a r .

The second program, Program P r o je c t 2 i n Appendix 2B,

d e a l s w i t h t h e

l i n e a r l y d i s t r i b u t e d d is c o f s ol l rc es i n ge ne ra l, and i t s a p p l i c a t i o n t o

t h e U n i v e r s i t y o f M as sa ch us etts w i n d m il 1 i n p a r t i c u l a r .

The o u t pu t o f b o t h th e p rogram s i s t h e v e l o c i t y f i e l d i n c a r te s i a n

c o o r d i n a t e s a t e ac h p o i n t . The o u t p u t ha s € he f o r m o f : (x , y ) , t h e

c o o r d i n a t e o f t h e p o i n t , (ux , u y ) x , a nd y c om po ne nts o f t h e v e l o c i t y

v e c t o r a t t h e p o i n t ( x ,y ) a nd ( ETA), t h e a n g l e b etwee n u x a nd

uy

Stre am1 n e c o n s t r u c t i o n . The c o n s t r u c t i o n o f s tr ea m1 n e s i s b as ed o n

t h e f a c t t h a t n o f l o w c r o s s es a s p e c i f i c s tr ea m tu be . Once a s l e c t i o n o f

t h e s t r e a m l i n e s s t a r t i n g p o i n t h as been made, o t h e r p o i n t s o f t h e same

s t r e a m l i n e c a n b e f ou n d b y an a p p l i c a t i o n o f t h e c o n s e r v a t i o n o f mass.

I n P efe re nc e t o F i g .

6, t h e v e l o c i t y t h ro u g h t he s t re am t ub e s i s a s

f o l l o w s :

A

i s

u

( o ) , A1 i s u ( l ) ,

A

i s u

Z),

and so on , where u(o) i s

t h e x -c ompo ne nt of t h e v e l o c i t y an a n x - s t a t i o n a nd

yzo

and the same app l ies

f o r

I ) ,

( 2 ) .

.

u ( a ) .

F i g .

7

shows the c ros s s ec t i o n o f t he s t ream tubes .

A c c o r d i n g l y ,

t h e area c or re sp on di ng t o any v e l o c i t y ~ ( n ) an be

f o u n d a s f o l l o w s :


28/97

nd

L

in

=

9 4 no y )

-

An-

H a vi ng t h e p o i n t a ) , t h e s t a r t i n g p o i n t o f t h e s tre am 1 n e , t h e mass

f l o w r a t e o r t h e v o l u m e t r i c f l o w r a t e c an b e fo u n d by summing up t h e f l o w

r a t e t h ro u g h tubes A t o a as shown i n F ig .

8.

When t h e f l o w r a t e t h r o u g h t h e s t re a m tu b e a ) i s d e te rm i ne d a t one

x s t a t i o n , o t h e r p o i n t s o f t h e same s tr e am t u b e c an s u b s e q ~ ~ e n t l y

b e c a l c u l a t e d .

A t ea ch s t a t i o n

x

t h e f lo w r a t e t hr ou g h t h e t u be s w i t h c r o s s - s e c t i o n

A Al Ap . and A shown i n F ig . 7) s h o u l d b e e a s i l y fo un d.

The

de s i r e d s t r eam t ube can be de t e rm i ned by sum il ing up t hese ca l c u l a t ed

f l o w r a t e s , F1, Fp

.

up t o Fn where n i n d i c a t e s t h e p o i n t w here t h i s

s um na tio n i s e qu al t o t h e r e f e r e n c e f l o w r a t e .

I f n i s known, t he y - co -

o r d i na t e can be de t e r m i ned .


29/97

The t a s k o f t h e s t r e am l i n e c o n s t r u c t i o n i s p e rf or m ed b y t wo c o mp ut er

programs.

The f i r s t one c o n s tr u c ts t h e s t r ea m l in e f o r a u n if o r m ly d i s t r i -

b u t e d d i s c o f s o ur ce s i n g e ne r al , a nd f o r t h e U n i v e r s i t y o f M a ss ac hu se tt s

w i n d m i l l , i n p a r t i c u l a r ( se e P r o j e c t S1, A ppe nd ix

2C).

The second program

d e al s w i t h c o n s t r u c t i o n of t h e s t re am l i n e s f o r a 1 n e a r l y d i s t r i b u t e d d i s c

o f s o ur ce s i n g e ne r a l , and f o r t h e U n i v e r s i t y o f M a ss ac hu se tt s w i n d m i l l

i n p a r t i c u l a r , (se e P r o j e c t

S 2

Appendix 20) .

The i n p u t o f t h e s e pr og ra ms c an b e t h e s t a r t i n g p o i n t o f t h e s tre am -

l i n e o r t h e f l o w r a t e t h r o ug h t h e s t re am tu be .

T h e o u t p u t i n d i c a t e s t h e form o f f l o w r a t e , and c o o r d i n a t e i n d i c a t e s

t h e s tr ea m l i n e a t each x s t a t i o n .

Ap p ly in g th e one-d imens iona l momentum theo ry t o t he w i nd mi l l s .

The

s o ur ce s t r e n g t h d e n s i t y p e r u n i t ar ea k c a n be f ou n d i n t h e c as e o f t h e

u n if o rm l y d i s t n i b u t e d s ou rc e d i s c and t h e s lo p e o f k , ( t h a t i s m) f o r t h e

l i n e a r l y d i s t r i b u t e d s o u rc e d i s c , b y t h e one d im e n s io n a l m omentum t h e o r y a nd

L g a l y s theorem.

A c c o r d in g t o t h e L a ga l l y s th eo re m [ 1, t h e f o r c e e x e r t e d u p o n a

p o i n t s ou rc e i n a un if or m fl o w i s 6Xu where i s t h e d e n s i t y o f t h e f l u i d ,

i s t h e s ou rc e s t r e ng t h , and i s t h e f r e e s t re am v e l o c i t y o f t h e u n i fo r m

f l o w .

a ) L i n e a r l y d i s t r i b u t e d d i sc .

I n t h i s case,

t h e s o u r ce - s t re n g t h d e n s i t y i s e qu al t o m r where m

i s th e sl o pe of k a s d e s c r ib e d i n s e c t io n fo u r. A c co rd in g t o L a g a l l y s

th eo re m, t h e f o r c e on t h e d i s c i s :

F

= k (45 )

w he re k i s t h e t o t a l s t r e n g t h of t h e d i s c . How ever, k c an b e d e t e r m in e d

a s f o l l o w s :


30/97

S u b s t i t u t i n g e q u a ti o n ( 4 6) i n t o e q u a ti o n ( 4 5) ;

From one dimensional momentum theory

t

can be shown [11 ] t h a t the

t o t a l f o r c e on t h e d i s c i s ( See F ig .

9

2

F

vA

U U U1)

where u i s t h e v e l o c i t y t h r ou g h

t h e l i s c i l i i d

Sl i s t h e v e l o c i t y down s t r ea m

of t h e d i s c .

2

U sin g t h e B e r n o u l l i s e q ua t i on and t h e f a c t t h a t F

=

T

AP,

t

can be

shown that

[8]:,

and

where S i s t h e a r ea o f t h e d i s c .

F r o m t h e d e f i n i t i o n o f t h e a x i a l i n t e r f e r e n c e c o e f f i c i e n t a,

t

can

be w r i t t e n t h a t

=

1-a)

51

U s i n g e q u a t i o n ( 5 0 )

and

w h i c h c a n b e w r i t t e n a s

M u l t i p l i c a t i o n o f e q ua t io n (52) by (53)

w l l

r e s u l t i n


31/97

s u b s t i t u t i o n i n t o e q u at io n ( 4 9 ) l e a ds t o;

2

=

2 8 ~ A ( l+ a) u 2

The slope m c a n b e d e t e r m i n e d b y e q u a t i n g e q u a t i o n s ( 47 ) a nd ( 5 4 ) , t h a t

b ) U n i fo r m l y d i s t r i b u t e d d i sc .

F o l l o w i n g t h e same pr oc ed ur e, t h e t o t a l d i s c s t re r r g th i n t h i s c as e i s

an d t h e t o t a l f o r ce on t h e d i s c f r o m t h e L a g a l l y s the ore m i s :

E q u at in g t h i s f o r m u la w i t h t h e e q u a ti on ( 54 ) w l l p r o v i d e k hence

; = 2a (l+a ) L I 5 8 )

t c an be shown [ 12 ] t h a t t h e r e l a t i o n be tw een t h e a x i a l i n t e r f e r e n c e

f a c t o r ( a ) a nd t he pow er c o e f f i c i e n t Cp i s a s f o l l o w s :

The niaxi~iiumpower i s developed when a = 1/3 [8 ] .

Thus

i t

has been shown thro ugh L a g a l l y s theorem and th e s im ple one-

d i m e n s io n a l momentuni t h e o r y t h a t t h e r e i s a u n i q u e r e l a t i o n b etw ee n t h e

s t r e n g t h o f t h e d i s t r i b u t e d s ou rc e d is c

k

an d t h e p ow er c o e f f i c i e n t Cp

o f t h e w i n dm i l l .


32/97

RESULTS

The vel oci ty f i e l d and some ch ar ac te ri st ic streaml ine s have been

calculated for the numerical

values appropriate to the 25 kW windmill a t

the University of Massachusetts Sol ar Habitat I See Fig. 10) .

A

ch ar ac te ri st ic fr ee stream veloc ity of 38 ft /s ec 1 1.58 m/sec)

have been considered.

The body of t h i s windmill can be modeled by a

single source of K=647.741bm/sec

294 kg/sec) i n a free stream of

U 8

f t / sec .

A

sample result of the velocity field upstream the windmill

s

shown

a t th e end of P ro ject 1 and Proj ec t 2 [see Appendix A and 2B].

The veloc ity p ro fi le of the uniformly-distributed disc and the lin ear ly-

distributed disc model

s shown

in fi gs . 11) and 12).

The streaml ine s const ructed f o r di ff er en t ca ses a r e shown i n Fig.

13

through 20.

I t

s

obvious from Figs. 13 through 20 t h a t t he ef fe ct iv e change

i n

the f r e e stream veloc ity i s almost negl ig ib le f o r more than two ra di i

upstream of the blade disc.

Fig. 13 repr esen ts t he unifornily di st ri bu te d d isc model without

the body for the case of Cp

Cpma

0.5.

I t s shown that the disc

samples almost 57 percent of the volume of th e f a r upstream wind.

For the

same case,

i f the body i s 1ocated a t the cen ter of the blade disc Fig.

15 ) , 51 percen t of the flow i s sampled.

By moving the si ng le source,

which forms the body to the position

R 2

5.02 f t ) t o model th e University

of Massachusetts windmil 1 , the percentage of th e flow sampled reduces to 35.

The same an al ys is f o r the line ar ly -d is tr ib ut ed disc model Figs. 14,

16, and 18) indicates similar results.

The percentage of the wind sampled


33/97

changes f rom 39 t o 38 and then t o 30.

C om pa ris on o f t h e u n i f o r m m odel a n d l i n e a r mo de l s hows t h a t f o r t h e

same c o n d i t i o n s , t h e l i n e a r m odel s am ple s l e s s f l o w t h a n t h e u n if o rm d i s c .

To see th i s , F i gs . 17 and 18 must be compared.

The u n i f o r m l y - d i s t r i b u t e d d i s c sam ples 35 p e r c e n t o f t h e f lo w , w h i l e

t h e 1 n e a r l y d i s t r i b u t e d d i s c sam ples 3 0 p e rc e nt .

The e f f e c t o f t h e b ody i s n o t r e s t r i c t e d t o t h e s a m pl in g p ro blem .

A lt ho ug h t h i s i s t h e case f o r t h e u n i f o r m l y - d i s t r i b u t e d m odel, i n t h e

c as e o f th e more r e a l i s t i c l i n e a r l y - d i s t r i b u t e d m odel, a s i g n i f i c a n t

d i f f e r e n c e i s o bs erve d

-

t h e p r o b l e m o f l e a k i n g .

F i g s . 16 a nd 1 8 show t h a t due t o t h e h i g h e r r e s i s t a n c e a t t h e o u t e r

reg ion of th e b lades , some o f th e samp led f l o w appears to le ak th rou gh

t h e c e n t r a l r e g i o n n e ar t h e body. T h i s i s q u i t e o b v io u s i n F ig . 12B,

where a t t he s t a t i o n x = l f t ( 0 . 3 ~ ) ~h e y component o f t h e v e l o c i t y i s

down towards t h e c en te r between y=4ft (1.2 ' ) and y - l O f t (3m) .

U sin g th e v e l o c i t y f i e l d t h e p re ss ur e i nc re a se i n f r o n t o f t h e d is c

can q u i t e e a s i l y b e c a l c u la t e d .

I n t h e c ase o f t h e 1 n e a r l y - d i s t r i b u t e d s ou rce d i s c

t h e B e r n o ul

1

equa t ion on th e s tream1 in e

o

s e e F i g .

18)

c an b e w r i t t e n a s

u2 + P

cons t .

1s

A t x

= -

h e v e l o c i t y i s 38 f t / s e c and t h e p re ss ur e i s P w i t h

d e n s i t y & .

A t

x =

1 , a nd y 1 6, t h e v e l o c i t y i s :


34/97

then, i t i s ea sy t o show t h a t

o r

?= P

t i49 .3

6

p = \q. o ps

\ o \ s r

\ e d n L

F o l l o w i n g t h e same p r oc e du re f o r t h e u n i f o r m l y - d i s t r i b u t e d d i s c o f s ou rc es ,

t h e p r essu r e wou l d be :

P p l 6 g s

6 4

Comparing eq ua t io n 60) and 62) ,

i t

i s q u i t e o b vio us t h a t t h e p re s-

s u r e o n t h e s t r e a m l i n e

Q 0

i nc re a se s more i n f r o n t o f t h e l i n e a r d i s t r i b u t e d

sou r ce d i sc.

F i g . 20 shows t h e e f f e c t o f a c ha ng e i n t h e p ow er c o e f f i c i e n t , Cp.

The percen tage of th e f lo w sampled changes f ro m

57

t o 80 p e r c e n t w h i l e

t h e Cp i s cha nged from Cpmax t o Cpmax,2

A ls o, i n t h e 1

m i t

as CP-o, a l l

t h e f l o w w o u ld p as s t h ro u g h t h e b l a d e d i s c u n e f f e c t e d .

The v a r i a t i o n o f t h e v e l o c i t y o n t h e s t a g n a t i o n s tr ea m1 i n e h as be en

r e p r e s e n te d i n F i g . 21.

The d ia gr am shows t h e v e l o c i t y v a r i a t i o n f o r

t h e b ody an d t h e 1 n e a r l y - d i s t r i b u t e d d i s c w i t h t h e bo dy .

t also shows

t h a t t h e s t a g n a t i o n p o i n t has s h i f t e d fo rw ard s, an d t h a t t h e v e l o c i t y

d e cr ea se s mo re r a p i d l y w i t h t h e b l a d e d i s c t h a n f o r t h e b od y a lo n e.


35/97

REFERENCES

T a y l o r ,

G.

I. nd G.K. M at ch el or , Q u a rt . J. ech. Ap pl . Match., 2

1949, pp. 1-28.

Koo, K.J. and F.D. James, J. F l u i d Mech., Vo l. 60, P a r t 3, 1973,

pp. 513-538.

E ld e r, J.W., . F l u i d Mech., Vol. 5, 1959, pp. 355-368.

Kllchemann, D. and J. Weber, Aero dy na mi cs

o

Pr op ul sio n, McGraw-Hi 11,

1953, Chap. 3.

T a y l o r ,

G . I .

I n t h e S c i e n t i f i c Papers o f

G . I .

Tay lo r , Vo l . 3 ,

Cambridge U n iv e rs i t y Press, pp. 383-386.

Es k in az i , S. Vec tor Mechan ics f F lu id s and Magneto f l u id s , Academic

Press, 1967, p. 287.

Ib id . , pp. 308-31 1.

Pi pe s, L.A., an d L.R.

H a r v i l l , A p p l i e d M a th em atic s f o r E n g i ne e rs a nd

P h y s ic is ts , McG raw-Hi l l 1970, pp. 345-348.

Budak, B.M., A.A. Samarski , and A.N. T ikha nov , C o l l e c t i o n f Problems

on Ma them at ica l Ph ys ics , Pergamon Press , 1964, p. 495.

Rober tson, J.M., Hydrodynamics n Theory nd App l i c a t i o n , Pr en t i c e-

H a l l , Inc. , 1965, p. 202.

Wi ls on, R.E. and P.B. Lissaman, A p p l i e d Aerodynam ics o f Wind Power

Machines, NTIS r e p o r t PB-238-595, Chap. 3.

Ibid. , p. 18.


36/97


37/97

APPENDIX 1

1

R


38/97

~ n - I

_L

L

Q

i n r h-1 P , , c s d J

which can be reduced to :

w hi ch i s t h e e q u a t io n

36) .

Equa t ion

3 7 ) c a n b e d e r i v e d a s f o l l o w s , u s i n g t h e same p o t e n t i a l

e q u a t i o n 2 1 ) a n d k n o w i n g t h a t

i t i s e as y t o show t h a t :


39/97

By a 1

i

t l e a l g e b r i c mu1 t i p 1 c a t i o n e q u at io n

37)

c a n e a s i l y b e

de r i ved .

2 )

R

I n t h i s c as e t h e e q u a ti on 22 s h ou ld be d i f f e r e n t i a t e d t o p ro du ce

UR, and UT.


40/97

APPEND IX

: A


41/97

A PPEN D I X

2A

'..,,;I j, J ,

I

s-.,,.;-, ,.. ;-'

';jj

l ? R OJi

1 ;;::

1;

i:' Ll"i.l:im[j.r

~ ~ : # * ~ * $ $ t t : k ; 1 ( ~ $ $

" fi

.,

7

.

.

a,..:c

,..

.

:4

.,,:;,,) ,;:j;; ;C

.\ .\ .. ;:

,,

,,-.

i2

F

(J

f> ,y

[ i, 7: r-- : : :,- ,'

111

2 <

z

{:i

'

-,

r,.

"

,,

';1[=

S (J URCES *

~ 0 i 6 3 ~ L

1

r11 ;.,i

I{ il

f

t

,2

0

i.

71)C

:#

I /

4.

,

2

i

3 u

*

r;;:;yoc

T

H IS

, 7

* ;-

; :-,:-.c, I

7

.

.,:

. i

T

....

- I

~ t'- . . . l . .~t . . . : ' rY'- FIEl.i3 i l X r

UY

7 E T A )

t

*

0 '_

-:)

F 8 A tiN :

f: :

;-

...

>'

r

:i3 r ;::

1

3

i,,

;'

:.:: :i

5

0

R

CE

i>NJJ *

;?. ,a .7 1 s -

'

A

STEJ;',;S

" ' : - ' , b 7 " . - ' I -

%'r ;" - - v'

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s ,?

~ ~ 1 ; -

a . J ,?

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1 ~j> .

I .

4 1%

L

i h r X - A X I S

$

.%.

,

.-.-,,-.

s

2

w - ,L

L

.J~

X .

G$izy3(jz

#

1 3 r ~ 2 4 C [ ; T H I S i R ~ i ~ ~ ; . . A I M

rJ$E " i " " \ " - ' , " '

""'

*

, G 5 ,Jc

.

. ~ i . i . - \ ~ ,~n i : 7 0 L i O W I N r J S *

:[

{

;-

,:-

:.

q

.

'".

-

.?.

-

,322hC)c

.

,

:

2 : . . : i S;< F Sr :

D A T A :

$

O r J 2 7 0 C

t dX y

t s l ' i

y .i z'-TX

IIEL-.'['~i

lq

9

A 1 9

132

;.... \ .r

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*

13028i;C I$ C ; j 2 D: [NS 'rs Ti.. ;: I

-

..

L , . J , L O F d

00290 C

-.

p ,...

-'

.-

'"' -.

I

dl, iZ'Altl..r~c I 1z.1-.S lj V Z N bELOW

:

:.$

(3034CC Z

()?ZZ()C *

, {

3

0 Y-(3x1s X

;5337 Qc

d

;]333i)C + -----------.---.--------------

.-..

( c.4) - 1

) :+I;; .rs

x - n ~ ~ ~ c

-

,,

.-.

t

g 8 ) :31.9 ~ ~ : ~ l : i . j 3 ; x f i 'r;. ~:'i : ~ ~ ~ : . ; p i f ; ? . i " i ~

[N

Y,-~I,;;'Ec+ 0

;., '

.

;

.L

3

C

iii;i- ; 1 ;

-,

-,\jiz ' '

i l : .

I<

T' I N X . . - i l i ' i ? E C T I G N %

%

.,

:.:.-.

]CilZ'L'r'(

: INCKEMiE,< 'T .Il\; Y-.D Ir(;7c

I O N

i.

j I,.:,J i 1

;2354{ji: fit i;hiil[)"S

'ii-li, I l I S C

G0550C x;< :

I J A ~ , I J ~ - 8;: T;..;E_:

.

.*

YY: t.,:,..,L,tjjI G i r Ti.. F

_

)'-I+ h

t.

&

~ , : j 3 7 0 ( ~ I(;?.

:

E+;izNlj'Tt-i OF Tf . . ; , ; : EC)x;'(

S 3 U K C F

t

(

3 3

u C

F:?

: - [~SI'~IQN

TJF

TI.; ;: z.:;j~ y FCIJ ; ; ; ; ~ 8

Ct>S


42/97

00640C UT1: TANGANTInL VELOSITY AT PUIN (RrT) t

00650C DUE TO TWE DISC*

00660C UR1: RfiDIAL VELOSITY AT POINT (RY T)

00670C

DUE

TO TWE DISC

00680C UT2: TANGANTIAL VELOSITY AT POIN (R YT)

* ,

00690C

DUE

TO THE FODY

00700C

UR2: RADIAL VELOSITY AT POIN (RrT) t

00710C DLlE 'TO THE E{OIIY

00720C

UT3: TANGANTIRL VELOSITY AT POINT (RY T) 5

00730C

DUE TO THE FREE STREAM+

00740C UR3: RADIAL- VELOSITY AT POINT (RY T) O

00750C

DUE

THE FREE STREAM,

00760C P(NrX): LEGENDRE FOLYNOMIAL OF THE

t

00770C FIRST KIND AND NTH OKIIER UEFINED-

00780C BY THE FUNCTION P(NrX)*

00790C AS(NYX)

:

ASSOC:IATEII LEGENRRE OF TH E

00800C FIR ST KIN11 FIRST IlEGREE.

00810C AND? TH ORDER IrEFINED

'BY

THE

X

00820C AS(NrX) FUNCTION

t

00830C UX: X COMPONENT OF THE VELOSITY AT X

00840C ANY POINT (RrTIOR ITS EClIVALENT(XX?YY) X

00850C UY: Y COMPONENT OF THE VELOSITY AT

t

OOSbOC ANY POINT(RYT) OR ITS EOlJIVALENT(XXrYY)

00870C ETA: ANGL BETWEEN UX ANDY UY IN

D E G *

t

00880C

00890C

jc

oo900c d HE PROGRAM WAS RUN FOR THE FOl-LOWING

t

00910C VALUES :NX=20 NY=2? I:IELTX=l? :IEl-TY=l U=38

00920C ~A= lb~ Al= 1/3 (F0 R AX* FOWER)rR2-0* X

00930C rK1=647*78

jc -

005'40C

00950C ALSO SEE THE PROJECT REPORT d

00960~*t*lt**~ttt11t*tt*****5**0t*~***** ~* *************** ******

00970 R E A D I N X P N Y Y D E L T X ~ D E L ' ~ Y Y U Y C ; I I A ~ ~ R ~ Y K ~

00980 IN=50

00981 PRINT ~ ~ O ~ N X Y N Y ~ D E L T X Y I ~ E L T Y ~ U Y A ~ A ~ Y R ~ ~ K ~

00982

210 F O R M ~ T ( / / / ~ ~ N X = * Y I ~ Y / ~ ~ N Y = * I I ~ Y / I I ~ I E L T X = * ~ F ~ ~ ~

0 0 9 8 3 + / r ~ U = ~ ~ F l 0 ~ 4 r / ~ t A ~ ~ r F 6 ~ 2 r / ~ ~ A l ~ ~ F ~ O ~ 6

00990

PRIN'T 21

01000 21 F OR HA T( //Y~ ~X Y* VELO SI TY IELIl FOR UNIFORM DISC AND THE BODY

01010.t Y/Y25~Y -----------------------------------------

*

01020 PRINT 10

01030 10 FORMAT(5XrtX IN F T * Y ~ X ~ * YN F T ~ Y ~ X V X U XN F T / S E C * ~ ~ X Y U YN F

010 40 t5X ~tE TA N DEG*tr/r5X?t------- *r9X~t------- *Y6~rt------------ tr5X

01041+~*------------- tr5x7*----------- t

01050 DO 6 III=lrNX

01060 DO 500 II=lrNY

01 070 YY=(II-1)tDELTY

01080 XX=FLUAT(III)*DELTX

01 090 K=2*Al*(l*+Al)*U

01100 R= XX**2)+ YY**2))**0*5)


43/97

01110

NO=O

01120 N l = l

01130

T = - A T A N Y Y / X X )

01140 Tl=COS T)

01150 T2=SLN T)

01160

TO=O*

01170 IF R*GEeA)

O TO S

01180C

01190C

01200C DISC FOR

THE

CASE R A

01210C

01220 :

01230 AK=K/2*

01240 X = R / A

01250 SUM-0,

01260

DO

1 I = l ? I N

01270

N = I - 1

01280 M=N-2

01270 S U M = S U M ~ N ~ X S ~ N - ~ ) ) ~ P N P T O ) + F M P T O ) ) * F N P T ~ ) )

01300 1

CONTINUE

01310

U F < l = A K t P N l

?Tl)-SLJM)

01320 SUM=O*

01330

DO

2

I = l r I N

01340 N-1-1

01350 M=N-2

01360

S U M = S U M + ( ( X ~ ~ ( N - ~ ) ) ~ ( F ( N ~ T O ) + P ( M P T O ) ) : X A S ( N ? T ~ ) )

01370 2

CONTINUE

01380 UTl=AKt SUM-AS NlrT1))

01390

G O

TO 200

01400 5

CONTINUE

01410C

01420C

01430C DISC FOR

THE

CASE

R > A

01440C

01450C

01460

X = A / R

01470 SUM=O+

01480

DO

3

I = l t I N

01490

N = I - 1

01500 M=N+2

01510

S U M = S U M + ( ( N + ~ ) ~ ( X ~ * M ) ~ ( F ( N P T O ) ~ F ( M P T O ) ) * F ( N ~ T ~ ) )

01520

3

CONTINUE

01530 UR?= K/2)tSUM

01540 SUM=O*

01550 DO 4 I = l ? I N

01560

N = I - 1

01570 M=N+2

01580

S U M = S U M ~ X ~ ~ M ) ~ F N I T O ) ~ F M P T O ) ) ~ A S N P T ~ ) )

01590 4

CONTINUE

01600 UTl= K/2) SUM

01610 200 CONTINUE


44/97

01620C

01630C

01640C DODY CALCULATIONS

oibsoc

01660C

01670 D15=(R-(R2$T1))

01680 D16=(((R$Tl)-R2)%*2)

01690 D17=(R1: 2)t(T2%*2)

01700 D18=( (Dlb.tT117)t%(3/2)

01710 AKl=(Kl/(9t3e14))

01720 UR2=AKl$Dl5/D18

01730 Dl9-H2f T2

01740 DllO= RbT1)-H2)tt2)+i Rtt2)t T2tt2)))t~ 3e/2b)

01750 AKJ=K1/(4**3*14)

01760 UT2=AK3tDlY/D110

01770C

01780C

01790C

U N I F O R M FLOW

01800C

01810C

01820 UR3=-UfT1

01830 UT3=UtT2

01840C

01850C

01850C SUFEHFOSITION

01870C

01880C

01890 UR=URl+UR2+UR3

01900 UT=LITl+UT2+UT3

01910C

01920C

01930C

EVALUnTION OF U X

U Y AND

ETA

01940C

01950C

01960 UX=(URtTl)-(UTZT2)

01970 UY=(URtT2)+(UTtTl)

01980 X Y = U Y U X

01990

ET=ATAN XY)t360e/ 2et3e14)

02000

PRINT

~ O O ~ X X F Y Y ~ U X F U Y F E T

02010 300 FORMAT(5(5XrFlOe4))

02020 500 CONTINUE

02030

PRINT

700

02040 700

FORMAT /)

02050 600 CONTINUE

02060 END

02070C

02080C

02090C

02100C


45/97

02130C

02140C

\

02150C

02160C

02170C

02180C

02190~

02200 FUNCTION F (NPX)

02210C

02220C

02230C THIS IS A FUNCTION

TO

CALCULATE LEGENDER POLYNOMIALS

OF

THE

02240C FIRST

K I N D

02250C

02260C

02270 QO=O+

02280 P=QO

02290 IF(N*LT*O) GO TO

1

02300

QO-1.

02310

F=QO

02320 IF(N+ECI+O) O TO

1

02330 Q1=X

02340

F'=Q1

02350 IF(N+EQ+l)

O TO

1

02360 Q2=((3 (Xtt2))-1+)/2.

02370 P=Q2

02380 IF(N+EQ+2) O TO

1

02390 Pl=((Zt(Xt 2))-1*)/2.

02400 P2=X

02410 1=3

02420

2

CONTINUE

02430 P=((((2,*1)-1+)/1)tX Pl)-((I-l,)/I)tP2

02440

IF(I+EO*N)

GO

TO

1

02450 I = I + 1

02460 P2=Pl

02470 P1=P

02480 GO TO 2

02490

1

CONTINUE

02500 RETURN

02510

END

02520C

02530C

02540C

02550C

02560C

02570C

02580C

02590C

02600C

02610C

02620C

02630C

.


46/97

02640C

4

0265OC

02660 FUNCTION hS NvX)

02670 DIMENSION R l000)

02680C

02 5YOC

02700C THIS FUNCTION CALCULhTES ASSOCIATEn LEGENDER FOLYNOMIALS

02710C OF

THE

FIRST K I N D A N D FIRST POWER

02720C

02730C

02740

R O = O +

02750

R A = R O

02760 IF(N*LE*O) GO TO 3

02770 Kl= l- Xlt2))tt0+5

02780 R A z R i

02770 IF N*EQ+.I)GO TO 3

02800 R2=3**Xf((I-(Xtt2))**0*5)

02810 RA=R2

02820 IF N+EQ+2) O

TO 3

02830 Fi=X

02840 F2= 3t XSt2))-1*)/2+

02850 DO iO I=3rN

02860 H(1)=(((2tI)-i)*((i-(Xtt2))ftOt5)*P2)tRl

02070 P=((((2+*1)-1+)/1)tXtP2)-((I-it)/I)*Fl

02880 Pl=P2

02890 P2=P

02900 Ri=R2

02910 R2=R I)

02920 R A = R I )

02930 iO CONTINUE

02940 3 CONTINUE

02950 AS=RA

02960 RETURN

02970 END

02980C

R E A D Y


47/97

RUN

7 8/ 07 /0 4 . 2 2 * 1 5 * 4 8 r

F I L E F ROJ1

V E L O S IT Y F I E L D FOR UNIFORM D I S C AND THE BODY

Y I N F T UX I N FT/SEC UY I N FT / S E C E TA


48/97


49/97

T IME LIMIT

SRU

21.302 UNTS


50/97


51/97

f l l i 3 O I n O h

a 1 * r 9 r b ~ 0 ~ - l @ a - m @ d 9 a

P I O @ U I h @ I n h S M d d 9 C O d O .

b @ m N @ d

D h 9 M D d C - J 0 3 b 7 C . l 0 9 d M O d 9 M d O b M

P 0 3 O d @ V 9 M l J 7 d O d M 9 O C 4 P h h M P C . 4

C T P 6 M M b h

@

r . J q 9 C . J b 9 M

@ b f 4

P O 9 ~ 0 P O Y M W Q 0 9 C ~ J ~ b 7 d 9 b D . d P

I O d C . J M

P

9 9 9 h h a a + + O d C . l C 4 M P U J @ @ @ @ 0 0 0 d d N M M P P b 7 9 d d d d N N

. . . . . . . . .

. . .

L n L l Q d 9 9 9 L7b7b7b7M b 7 L O b 7 b 7 9 9 Q 9 9 9 9

m l J 7 b 7 b 7 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

M M M M n M M

M M E M M M M M M M M M M M M M

M M M M M M M M M M M M M M M M M M M M M M

I I l

t

o o o o c o o

, , , . -


52/97


53/97

E N D

SRU 5 2 . 3 0 9 U N TS ,

RUN COMPLETE


54/97

APPENDIX B

LNH

00100 F'ROGt-ihM F'ROJ2 INPUT OlJTF'UT

00110 IN-50

0 0 1 2 0 ~ * b Z t * t * ~ t t * * * * t t X * * t f j l : * ~ * ~ * ~ ~ * ; j : t ; j : * * * % * ~ * * * * * * * * * * ~ * * * ~ ~ * * * * * * *

OOl3Ot

00140t

OOlSO* PROGRAM FOR LINEAHLY D1STRIE;UTED IlISC 01- SOURCES.

OOllO* WITti THE E{ODY

00170t

00180*

001?0* THIS PROGRAM CAL-CULATES THE VELOSITY FIELD

002002 ( U X ~ U Y P E T A ) OR A LINEAkLY DISTRIBUTED DISC

00210* OF SOURCES AND A STRONG SOURCE AT THE FOSITIUN

00220% R ON .THE X-AXIS*

00230t

00240

002SOt

00260t TO RUN THIS FROGRAM:

00270;k ONE SHOULD INPUT THE FOLLOWINGS

00280

IN RESPONCE TO THE ASK FOR DATA

00290 N X P N Y P D E L T X P K ~ E L T Y ~ U ~ A I A ~ I R ~ I K ~

00300t (ACORDING TO THE DEI-INITION OF

00310*

THE

PARAMETERS GIVEN BELOW:

00320%

00330t

00340X

00350;k PARAMETERS:

00360X SEE PROJl PROGRAM DUCUMENTS*

00370t

ALSO

SEE

THE

PROJECT

REPORT

-

00380t

00390t

00400t

00410*

00420t

00430***U*******t********tt************************* ***~****************

00440 REAL MrK1rL

00450

R E A D ~ N X ~ N Y ~ D E L T X ~ D E L T Y ~ U ~ A ~ A ~ I R ~ ~ K ~

00460 PRINT 210

00470 210 FORMRT ////rlOXrIVELOSITY FIELD FOR LINEARLY DISTRIBUTED DISC+

00480 FRINT 220rNXrNY

IlEL TXTIIELTY

Ur Ar A1 ~ R 2 r K 1

00490 220 FORMAT ~OX,*-------------------------------------------------

O O ~ O ~ + ~ N X = ~ ~ I C J ~ / P ~ N Y = ~ ~ I ~ P / ~ ~ I ~ E L T X = ~ ~

0 0 5 1 0 + ~ A = ~ r F 1 0 ~ 4 r / r t A 1 ~ ~ r F 6 ~ 4 r / r ~ R 2 ~ t r F 1 0 ~ 4 ~ / r ~ ~ l ~ ~ ~

00520 PRINT

200

00530

200 FORMAT(5XrtX IN FT**rlOXrtY IN FT*tr7Xr*UX IN FT/SEC*X

0053ltr7X~SUY N FT*/SEC.*rSXr*ETA IN DEG**r/vSXrX-------- *rlOX,

00532+ --------,7~, -------------,7~, --------------rSX,

00533+ -----------

1

00550

DO 500 III=lrNX

00560 DO

600

II=LrNY

..


55/97

00570 YY-(11-l)*DELTY

00580

XX=FLC)AT 1 DKL-TX

00590 R = ((XXIX2)+(YYS12) *O+S)

00500 6

CONTINUE

00610C R A D I A L COMFONEiNT OF

THE

DISC VELOCITY FIELD

00620

NO=O

00630 Nl=l

00640 N2=2

00650 N3-3

00660 N4=4

00670 N5-5

00680 X==R A

00690 PROD=-O+5

00700

M=3fAi* 1 + A 1 ) tU/A

00710 T2=ATtlN( Y Y / X X ) .-

00720 Tl=CCIS T 2

00730 T3=SIN(T2)

00740

IF(R+GE+ A) GO

TO 5

00750 SUMl=O*

00760

DO 1

I=2rIN

00770 PROD=PKOt~*-0 5-- 1-1 1 I

00780'SUM:L=Sllt~l+((((~1: 1)+1~)/(((2tI)+3~)t((2XI)-2~)))XFROD)

00790 CONTINUE

00800 S1=SUM1

00810 YRODl=-O*S

00820 SUM2=O.

00830 DO 2 I=2r N

00840 N-2II

00850

PRODl=PRODlX(-O+5-(1-1))/1

00860 S U M ~ ~ S U I Y ~ ~ ( ( ( Z ~ ~ F R O I I ~ ) / ( I - ~ ) ) * ( X ~ ~ ( ( ~ X I ) - ~ ) ) X F ( N ~ T ~ ) )

00870 2 CONTINUE

00880 S2=SUM2

00890 AK1=-MtA/2*

00900 DO=2t(S1-(76+/60+))tXfP(N21Ti)

00910

Dl=(9*/2+)t(Xt 2)*F'(NJ~Tl)

00920 D2=3~*(XXt3)tF(N4~Ti)

00930 D3=(5*/6*)t(X8t4)*Y(N5~'rl)

00940 D4=S2

00950

U R l = A K l Z D O + D l - I I 2 + D 3 - D 4 )

00960C R A D I A L

COMPONENT

OF THE

BODY

VELOCITY FIELD

00970 DS=(R-(R2fT1))

00980 Db=(((RfTl)-R2) *2)

00990 D7=(Kt*2)t(T3**2)

01000 D8=((Db+D7)tt(3/2))

01010 AK=(K1/(4*3+14))

01020 UR2=AK*D5/D8

01030 UR3=-UtT1

01040C

T A N G A N T I A L

VELOCITY OF

THE

F R E E STREAH

01050 UT3=U*T3

OlObOC TANGANTIAL VELOCITY OF THE BODY

.

01070 D9=RZtT3


56/97

G: 3D a n

I CnLTjl; lXw

nnnj

r Z Z Z

rl

:

3

W

5

--I ?

r r r 3

t l w C )

I

n

t

z

1 3

O.

u

CL

w


57/97

01590 Sl=SUM1

01600

S 2 = 1 * / 3 * ) t X * t 2 ) t f N O 1 T l )

01610 fiK1=MIfl/2*

01620 URl=AKl (S2tS1)

01630C

TCINGAN

T I A L

COMPONEN TS

01640 UT3=UIT3

01650 D9=R2*T3

01660 D10=((((R*T1)-R2)*%2)+((R**2)t(T3**~?)))**(3*/2~)

01670 AK3=t


58/97

02100 P1=((3*(X**2))-1*)/2.

02110 P2=X

02120

1=3

02130 2 CONTINUE

02140 P~((((2+tI)-I~)/I)*XtF 1)-((1-1t)/I)~f2

02150 I F ( I * E Q * N )

GO

TO 1

02160

I = I + l

02170 P2=f1

02180 F1=P

02190

GO

TO 2

02200

1 CON r I N U E

02210 R E T U R N

02220

END

02230C S ~ ~Y: 5 ~

02240

FUNCTION

AS(NrX)

02250 DIMENSION R(l000)

02260C

THIS

IS

A

FUNCTION

TO

CALCULATE

ASSOCIATE

LEGENDER FOLYNOMIAL

02270C

OF THE

FIRST

K I N D

AN11

F OWER(M)t

02280

RO=O+

02290 RA=RO

02300 IF(NeERt0) GO

TO

3

02310 Rl=(l-(Xtt2))XfO*5

02320

R A = R 1

02330 IF(N*ER+1)

O TO

3

02340 KZ=3+tX*((l-(Xdt2))**0*5)

02350 RA=R2

02360 IF(N+EQ+2)O TO 3

02370

F 1=X

02380 Y2=((3*( **2))-1*)/2,

02390

DO

10 I=3rN

02400 R(I)=(((2tI)-l)*((l-(X*t2))**0~5)*PZ)+Ri

02410 P = ( ( ( ( 2 ~ ~ 1 ) - 1 ~ ) / I ) S X ~ P 2 ~ ~ ~ ~ 1 ~ 1 ~ ) / I ) t P I

02420 Pl=P2

02430 F 2=P

02440 Rl=R2

02450 R2=R(I)

02460 R A = R ( I )

02470 10 CONTINUE

02480 3

CONTINUE

02490 AS=RA

02500 RETURN

02510

END

READY


59/97

RU

VELOSITY F IE LD FOR LINEARLY OSSTRIEUTEO DISCtBODY

UY I N FT./SEC. ETA I N

.

0 0 0 oooo

4 e9155 -10 8261

4 .4470 -1 1 .2858

8 . 8 7 1 9

3 c

J. 4 2 1 2

10.7436

19 .5553


60/97

END

SRU

8 155

LINTS

RUN COMPLETE


61/97

APPEND I X

2C

LNH

00100 PROGRAM F ROJSI(INF1JTvOUTF UT)

00110 DIMENSION U F ( 5 0 v 1 0 0 ) ~ ~ ~ ( 5 0 ) v Y O U T ( 1 0 0 )

00120 REAL KvLvKl

00130 IN=50

00140C

00150 R E A D T N S ~ N ~ ~ N E ~ D E L T X T D E L T Y P U ~ R I A ~ ~ R ~ ~ ~ ~

0 0 1 6 0 t t t f t Z t t d t ~ % f t ~ ~ k t t t d t ~ t ~ ~ d t f ~ ~ t t a C t

00170t

00180%

00190t

00200% PROGRAM FOR UNIFORMLY DISTRIBUTED DISC OF

00210t SOURCES AND THE BODY*

-

00220t

00230t

002408

THIS

PROGRAM

CONSTRUCTS

THE

STREAM

LINES

OF THE

00250t

FLOW FIELD MENTIONED ABOVEr IN TE FORM OFTHE

00260t

COORDINATES OF THE F OINTS ON A

SF CIFIC STREAM

00270% LINE(SF ECIF1ED K{Y ITS STARTING POINT OR BY ITS

002808 MASS FLOW RATE AS WILL BE I IESCRIBED LATEReIAS ITS

00290;3: AND X COORLIINATES; THAT IS FOR EACH ?THE

003008 CORESPONDING

X

VALUE

OF

THE STREAM LINE

S

DETERMINED

00310%

00320f

00330%

00340t

Y

00350

00360t

00370% (NS-1)

LIY \

STREAM LINE: SAI=X

00380t ~ \ ~ ~ ~

00390t

00400t

00410t

00420t

00430%

00440f

00450t

004605

00470%

ORIGIN ---------------------------------------------

00480t NA*DX

NE tDX

00490%

00500*

00510X

00520t TO RUN THIS PROGRAM:

00530t

AIHAVING THE STARTING POINT OF THE

00540t

STREAM LINE; ONE SOULD INFUT THE

005508 FOLLOWINGS IN RESPONCE TO THE ASK

00560t FOR DATA:NSvNAvNEvDELTXvK~ELTYvUvA

05705 - rAl?R2?KlrACORDING TO THE DEFINITION


62/97

0 0 5 8 0 6 O F I H E FARAHETERS GIVEN BEL-OW*

0 0 5 9 0 %

(NOT E: ZN THE F Ktr SEfICE O F TI IE DO-DY

OObOOt

NAXLlX

SOIJL-I

E{E SKEATER THAN

0 0 6 1 0 *

THE STAGN ATION P O IN T OF THE

0 0 6 2 0 t E O U Y

0 0 5 3 0 t

B ) H A V I N G THE MASS FLOW ROTE TtiROUGH

0 0 6 4 0 *

T H E S T R E A H T U E E i U N E S O U L D F I E S T

0 0 6 5 O t

CtiANOt< THE S TfYrEMENT REFH =SUM Z TO

0 0 6 6 0 X

KEFM=MASS FLOW INTER ESTE D I N , (NOTE:

0 0 6 7 0 3

S IN C E FLOW I S I N M IN US

X

D I R E C T I O N P .

O O 6 8 O t T H E MASS FLOW WILL. HAVE A MI NU S

0 0 6 9 0 X

S 1 G N ) T t i E N

THE

F QL-O WIN G I S I N P U T

0 0 7 0 0 t

I N RESFOPICE

T

'THE ASK

F O R

DATA:

G 0 7 1 0 *

N S P N A T N E Y D E L T X ~ D E L T Y I U I A ~ A . ~ . ~ ~ < ~ I K ~ .

0 0 7 2 0 t .

(NOTE: I N T H I S

CASE

NS SuULD RE

0 7 3 0 b S E L E C TE D ON AN E S T I M A T E H A S E v T t I A T

0 0 7 4 0 t I S VALU E THAT ONE

I S

SUREIS GKEATER t

0 0 7 5 0 t T H AN T HE R E A L V A L U E . A LS O . T H E SAM E

0 0 7 6 0 t R E S T R IC ' TI O N I S V A L I EL I FOR HA AS

0 0 7 7 0 * WAS P O I N T E D OUT I N P AR T A 1

0 0 7 8 0 t

0 0 7 9 0 2

0 0 8 0 0 t

0 0 3 1 0 t O U T P U T OF THE PROGRAM:

0 0 3 2 0 t VOLUMETRIC FLOW RATE THROUGH THE

0 0 8 3 0 t

STREAIY

TUEEPAND X-Y CO i IKDINAT E O F

OO84Ot THE rKEAM L I N E *

0 0 8 5 0 $

0 0 8 6 0 $

0 0 8 7 0 8 P A R A M E T E R S :

0 0 8 8 0 f S EE PROJl FROGHAM DUCUMENTS

008901( SEE THE PROJECT REPORT

0 0 9 0 0 t S EE THE ABOVE DIAGRAM?

OO910* NS S T A R T I N G P O I N T O F T H E ST RE AM L I N E ON

0 0 9 2 0 % THE Y - A X I S *

0 0 9 3 0 6 N A: S T A R T I N G P O I N T ON TH E X - A X I S *

0 0 ? 4 0 * N E: E N D I N G P O I N T O F THE STKEAM L I N E ON THE

0 0 9 5 0 t X - A X IS *

O O J 6 O t

0 0 9 7 0 6

:0 09 80 *-- --- --- --- THE PROGRAM WAS

RUN FOR:

0 0 9 9 0 f

NS=12rNA=3rNE=lbrDELTX=l.

0 1 0 0 0 t

D E L T Y = l r ~ U = 3 8 ~ r A = L 6 ~ v A 1 = 0 ~ 3 3

O l O l O t R 2 = O * ~ K 1 = 6 4 7 * 7 8

0 1 0 2 0 *

0 1 0 3 0 *

0 1 0 4 0 * # $ * t t t t * # t * t * * t * f f * 5 * f * * * * * t ** * f * * * * * ** * * $ * * * * * * * * ** * * * * * * t ** % * * * * $ * *

0 1 0 5 0 t

0 1 0 6 0 P R l N T

~ ~ ~ ~ N S I N A T N E I ~ I E L T X P D E L T Y I U T A I A ~ T R ~ I K ~

0 1 0 7 0 2 0 1 F O R M A T ( / / / ~ ~ O X P * S T R E ~ MI N E C U N S r K U C T I O N F O R U N I F O R M L Y D I S T f i I E U

0 1 0 8 0 + ~ / r 3 0 X 1 t D I S C OF SOURCES AND T HE

E f l D Y r / ~ 2 0 X v - - - - - - - - - - - - - - - - - - - - -- - - - - -


63/97

01090+, -------------------------

* ~ / / / ~ ~ N S = ~ ~ I S ~ / ~ * N A = * V I ~ ~ / V * N E = ~ ~ I ~

O l 1 0 0 + F 6 ~ 4 ~ / r * D E L T Y = * r F 6 ~ 4 r / ~ X U = * ~ F 1 O t 5 r / r ~ A = * F - 4 , 1 ~ / r * ~ ~ = ~ ~ ~ 6 ~ ~ ~

0 1 1 1 O ~ t / r * R 2 = ~ r F 1 0 ~ 6 ~ / v * K l = * ~ F 1 0 ~ 4 )

01 120C

01 130C

01 140C

01

isoc

01 l60C

01170 DO 600 III=lrNE

01180 DO 500 II=lrNS

01190 YY=(II-l)*lt

01 200 XX=FLOAl ( I I

01210 K = 2 . * A l * (

1.

tn1 * l J

01220 K = ( ( ( X X : k X 2 ) t ( Y Y t t Z ) ) t J : O t 5 )

01230 NO=O

01240 N1=l \

01250

T=ATAN YY/XX))

01260 Tl=COS(T)

01270 T2=SIN(T)

01280 TO=Oe

01290 IF(R.GE.A) GO TO

5

01300CSttYIttttXttXXXt*XXtfXttt DISC FOR

R < A t X t t t f t t f t d f t t f f t

01310 AK=K/2t

01320 X=R/A

01330 SUH=Ot

01340 DO

1 I = l r I N

01350 N=I-I

01360 H=N-2

01370

SUM=SUM+(Nt(X**(N-i))t(F (NrTO)+F (MiTO))tP(NrTl))

01380

1

CONTINUE

01390 URi=AK*(P(NlrTl)-SUM)

01 400 SUI.I=O,

01410 no I=I,IN

01420 N=I-1

01430 H=N-2

01440 S U M = S U M t ( ( X * f ( N - l ) ) * ( P ( N r T O ) ~ t P ( M I T O ) ) * A S ( N r T l ) )

01450 CONTINUE

01460 UTl=AKt(SUM-AS(NlrT1))

01470 GO TO 200

01480

5

CONTINUE

0 1 4 9 0 C X t t Y Y Y Y t * * t t f X * f t * X t t * t Y * X t *

DISC FOR

R > A **tXt**.fSf tt~tfttt*

01500

X = A / R

01510 SUH=Ot

01520 110 3 I = l r I N

01530

N = I - 1

01540 fl=N+2

01550 S U H = S U M ~ (N + ~ ) * ( X ~ X H ) * ( P ( N ~ T O ) ~ P ( ~ ~ I T O )

X F ( N ~ T ~ ) )

01560 3 CONTINUE

01570 URl=(K/2)*SUtl

01580 sun=o.

01590 DO I = l r I N


64/97

0 1 6 0 0 N = I - 1

0 1 6 1 0 M = N + 2

0 1 6 2 0

SUM=SUM+ X*tf4)t P NrTO)tP MrTO))tAS NrTl))

0 1 6 3 0 4 C O N T IN U E

0 1 6 4 0 U T l = (K / 2 )X S U M

0 1 6 5 0

200

C O N T I N U E

01660Cf tt t t tbttX ~tYtt

THE BODY t t t~tt t t t t f t t t t t t

0 1 6 7 0 D l S = ( R - ( R 2 X T l ) )

0 1 6 8 0 D 1 6 = ( ( ( R t T l ) - R 2 ) * 1 2 )

0 1 6 9 0 D 1 7 = ( R t t 2 ) t ( T 2 t t Z )

0 1 7 0 0 n 1 8 = ( ( D 1 6 + D 1 7 ) t t ( 3 / 2 ) )

0 1 7 1 0 A K l = ( K 1 / ( 4 t 3 * 1 4 ) )

0 1 7 2 0 U R 2 = A K l t D l S / D 1 8

0 1 7 3 0 D 1 9 =K 2 b T2

0 1 7 4 0 DllO= RXTl.)-R2)*t2)t RtX2)t T2Xt2)))tt 3/2)

0 1 7 5 0 A K J = K 1 / ( 4 * 3 * 1 4 )

0 1 7 6 0 U T 2 = A K 3 t D 1 9 /D 1 1 0

0 1 7 7 0 C t ~ X k ~ t l S ~ t ~ t X ~ i ~ t J c t t d X t t ~ d t

N I FO R M

F L O W .

t t b X b t X t t X S b l :

0 1 7 8 0 U R 3 = - U X T 1

0 1 7 9 0 U T 3 = U t T 2

0 1 8 0 O C * t t t t t ~ * t Y S S t I t * t t d t t t t X t X 1 : t S UP ER PO SI T IO N t X X t X t X S * ~ t X S t

01810 U R = U i? l+ U R 2+ U R 3

01820

U T = U T l + U T 2 + U T 3

01830 U X - ( IJ R t T 1 ) - ( U T t T 2 )

0 1 8 4 0 U Y = ( I J R t T 2 ) + ( U T t T l )

0 1 8 5 0 X Y = U Y / U X

0 1 8 6 0

E T = A T A N X Y ) X 3 6 0 * / 2 . * 3 * 1 4 )

0 1 8 7 0 C

0 1 8 8 0 C V E L O C I T Y F I E L D A ND S T R E A H T U B E A RE A S TO R AG E

0 1 8 9 0 C

0 1 9 0 0 l I F ( I I I r I I ) = U X

01910

5 0 0 C O N T I N U E

01920

6 0 0 C O N T I N U E

01930

A T= O *

0 1 9 4 0 DO 6 0 1 I = 2 r N S

01950

AR 1)=3,14t[ DELTY/2)**2)

0 1 9 6 0 I O = I - 1

01970

A T= A T+ A K (

10

01980 AK I)= ~.~~Y ~I-~)XTIELTY)+ DELTY/~))XS~))-AT

01990 6 0 1 C ON T IN U E

02 0 00 C R E FER E N CE M A SS FLOW C O N S TR A U C TI O N

0 2 0 1 0 C

02020 SUMZ=O.

02030 DO 6 I = l r N S

0 2 0 4 0

SUMZ=SUMZ+ AR I)tUF NAII))

0 2 0 5 0 6 C ON T IN U E

0 2 0 6 0 R E F M = S U H Z

0 2 0 7 0 P R IN T S O l rK E F M

0 2 0 8 0

501 FORHAT //r20Xr*VOLUMETRIC

FLO W R A TE THR OU GH TH E S TR E AM T U E E = X p F1 4 . 3 )

0 2 0 9 0 P R I N T 5 0 4

0 2 1 0 0 5 0 4 F O R H A T ( / / / / r Z O X rI X I N F T . t r l 2 X r t Y I N F T .* r/ r2 0X rX -- -- -- -- -- X r l O X r t - - - - -


65/97

02110t------

02120C

02130C STREAM LINES CONS1 \ LJCTION

02140C

02150

AL=Otl

02160

110 I = N A , N E

02170

FL-OW=O.

02180

J=l

02190 10

CONTINUE

02200

F L O W

=FLOW

02210 FLOW=FLC)W A R J)*lJF

P J)

02220 01-LOW= FLOW-REFM

02230 IF AES AL1-OW).LE*AL)

O TO 8

02240

ZF ALLOW.GT.O) GO

r0 9

02250

DELTiY= FLOWl/ FL-0W.I.fFLOW))

02260

YOUT I)= J-2)tDELTY)t IIELTlY~JIELTY)

02270

GO

TO

1 1

02280 9

J=J+l

02290

GO TO

10

02300 8 YOUT I)= J-l)*DELTY

02310

11 CONTINUE

,-

02320 X X X = I L D E L T X

02330 YOUT I)=YOUT I)+ DELTY/2)

02340

PRINT

~OSPXXXFYOUT T)

02350

505 F O R M A T ~ ~ X ~ F ~ O ~ ~ T ~ O X ~ F ~ O ~ ~ )

02360 7

CONTINUE

02370 END

02380C

02390C

02400C

024

OC

.

02420C

02430C

02440C

02450C

02450C

02470C

02480C

02490C

02500C

02510 t

02520

FUNCTION

P N r X )

025301:

02540C

02550C THIS IS A FUNCTION TO CALCULATE LEGENDER POLYNOMIALS OF TEE

02550C FIRST

KIND

02570C

02580C

02590 00=0.

02600 P=QO

. . .

02610

IF N*LT.O)

GO TO

1

.

. .


66/97

02620 170=1

.

02630

P=QO

02640

IF(NeEQ*O) GO

TO

1

02650 Q l = X

;

02660

F=Q1

02670 IF(N*ER*1) GO

TO 1

02680 Q2=((3t(Xtt2))-1*)/2*

02690 ~ = 2

02700 IF(NeERe2) GO TO

1

02710 Fl=((J*(Xt*2))-1*)/2*

-

02720 F 2=X

02730 1=3

02740 2 CONTINUE

02750

P=((((2~tI)-l~)/I)tX*Fl)-((I-1~)/I)tP2

02760 I F I . E Q * N ) GO TO

1

02770 I=ISl

02780 P2=P1

\.

02790 F l=P

02800 G TO 2

02810

I

CONTINUE

02820 RETURN

02830 EN11

02840 FUNCTION AS(N? )

0 2 8 5 0 C t t l t ; t * * t t t ~ t d t : ~ ~ t t * t t * X f t t X t S * S ~ t * * * * * * * ~ * * * t * * * * * * * * * *

02860 DIMENSION R(1000)

02870C

02880C

02890C THIS FUNCTION CAL-CUL-ATES SSOCIATED

LEGENDER

POLYNOMIALS

02900C

OF

THE FIRST KIND NLI FIRST F OWEH

02910C

02920C .

02930

RO=Oe

02940 R A = R O

02950 IF*-(N*LE*O) O TO 3

02960 Rl=(l-(Xt*2))**0*5

02970 R A = R l

02980

IF(N.EQ.1) GO TO

3

02990 R2=3.2Xt((l-(X**2))**0.5)

03000 RA=R2

03010 IF(NsEQ.2)

GO TO 3

03020 Pl=X

03030 P2=((3t(Xtt2))-1,)/2,

03040

DO

10

I=J?N

03050

R(I)=(((2*1)-1)t((l-(X*t2))*~0~5)tP2)tR1

03060

P=((((2~*I)-I~)/I)*X~P2)-((1-1~)/I)*Pl

03070 P1=F2

03080 P2=P

03090 Hl=R2

03100 R2=R(I)

03110 KA=R(I)

. - - .

.

03120 10

CONTINUE


67/97

313 3 ONTINUE

314

AS=RA

315

RETUl N

316 END

R E A D Y


68/97

RUN

7 8 / 0 7 / 0 4 . 2 2 . 5 2 . 1 5 .

F I L E

F KOJS1

S T R E A M

LIN ONSTRU TION FOR

UNIFORMLY

ISTRIRUTE

D I S C

O F SOUR ES

AND THE BODY

T IM E L I M I T S

S RU 25 .2 87 U N TS.

VOLUM ETRIC FLOW RATE THfiOUGH THE STREAM TUBE= -1 5 6 7 5 .7 3 6


69/97

END

SRU 57 256 UNTS.

RUN COMPLETE


70/97

APPEND I X 2D

LIST

78/07/04. 22+56.16.

FILE

F ROJS2

00100 PROGRAM PROJS2(INFUTrOUTPUT)

00110 DIMENSION UF(50r100)rAfi(50)rYOUT(lOO)

00120 IN=50

0 0 1 3 0 C * * * * * t * * ~ * * * * * t t t * * * * * * t * * * * t * t t * t ~ * * * * * * * * * ~ : K * * * * * * * * * ~ * * * ~ * *

00131C

00132C

00133C F R G R A M F O R

L I N E A R L Y DISTRIBUTED

DISC OF

00134C SOURCES

A N D

THE

BODY*

00135C

00136C

001 7C

.

00138C THIS PROGRAM CONSTRUCTS THE STREAM LINES.

00139C FOR TWE LINEARLY DISTRIBUTED DISC OF SOURCES t

001 40C

I N

COMPLETE ACCORKlANCE TO F'ROGFZAM F RUJSl*

00141C

1

00142C X

00143C*****t**************S******************************************

00200 REAL

K1rMrKrL

00220C

00230

H E A D ~ N S ~ N A ~ N E ~ D E L T X ~ I ~ E L T Y Y U I A ~ A ~ ~ R ~ ~ K ~

00231 PRINT

201rNS~NArNE~DELTX~I~ELTYrUrArR1rfi2tK1

00232 201 FORMAT(///r20XttSTREnM L I N E CONSTRUCTION FOR

L I N E A R L Y

DISTRIBUTED

00233tr/r30XrlDISC OF SOURCES

A N D THE

EODYJr/r20Xtd-------------------------

00234.tr

*r///r NS=trISr/rtNA~trI5r/rXb.IE=rkrI5t/t*DELT

0 0 2 3 5 + F 6 t 4 r / r * D E L T Y = * r F 6 ~ 4 r / r * U ~ * r F 1 0 + 5 r / r ~ ~ = * ~ F 4 ~ l r / r ~ A l = * t F 6 ~ 4 r

00236f/r*R2=trF10~6r/rtKl~*tF10.4)

00240

DO

500 I I I = l r N E

00250 DO 600 II=lrNS

00260

YY= II-l)*le

00290

XX=FLOAT(III)

00340

R = X X S Z 2 ) t Y Y * * 2 ) ) * * 0 . 5 )

00360 6 CONTINUE

00370C R A D I A L COMPONENT OF THE DISC

VELOCITY

FIELD

00380 NO=O

00390 N l = l

00400 N2=2

00410 N3=3

00420 N4=4

00430

N5=5

00440 X = R / A

00450 PROD=-0.5

00460 H=3tAlt(

+.+A1 *U /A

00470 T2=ATAN((YY/XX))

00480 Tl=COS(T2)

00490 T3=SIN(T2)

. .

-.

- - . . -


71/97

00500 IF(R*GE*A)

O TO

00510

SUMl=O.

00520

DO 1

I=2rIN

00530 PROD=PROfld(-0.5-(1-1))/I

00540

SUM1=SUEII+((((4tI),f1*)/(((2tI)t3~)~((2*1)-2*)))*PROD)

00550 1 CONTINUE

00560 Sl=SUMl

00570 PRODI=-O*S

00580 SUM2=O+

00590 DO 2 I = ~ P I N

00600

N=2*I

00610

PROPl=PROD1S(-O~5-(1-1))/1

00620

S U M 2 = S l J ~ 2 + ( ( ( I Z P F i O D 1 ) / ( I - l ) ) t ~ X t t ~ ~ 2 ~ I ~ - 1 ~ ~ t P ~ N ~ T

00630

2

CONTINUE

00640 S2=SUM2

00650 AKI== MfA/2

00660

DO=2 (S1-(76./60*))tX*P(t42~Tl)

00670

D1=(9+/2*>%(Xlt2)tF(E431Tl)

00680 D2=3*t(Xif3)tP(N4rTi)

00690 D ~ = ( ~ + / ~ * ) X ( X S * ~ ) ~ P ( N S T T ~ )

00700 IS4=S2

00710

URl=AKlt(DOtDl-D2tD3-D4)

00720C

R A D I A L

COMPONENT OF THE

BODY

VELOCITY FIELD

00730 D5=(R-(R2tTl))

00740 D6=(((RITl)-R2)**2)

00750 D7=(Rt*2)*(T3tt2)

00760 D8=((D6tD7) $(3/2))

00770 AK=(

K / ( 4 3 *

14)

00780 UR2=AKtD5/D8

00790 UR3=-UtT1

00800C T A N G A N T I A L VE~-OCITY F

THE FREE

STREAM

00810 UT3=UtT3

00820C TANGANTIAL VELOCITY OF

THE

B O D Y

00830 D9=R2*T3

00840 DlO=((((RtT1)-R2)**2)t((R**2)*(T3*t2)))t*(3~/2+)

00850 AK3=K/(4*63*14)

00860 UT2=AK3%D9/D10

00870C TANGANTIAL VELOCITY OF THE

DISC

00800 PROD2=-O*S

00890 SUM3=O*

00900

DO

3 I=2rIN

00910

N=2 I

00920

PROD2=FROD2t(-0*5-(1-1))/1

00930

S U M ~ = S U M ~ + ( ( P F < O D ~ / ( ( ~ ~ I ) - ~ ) ) ~ ( X ~ ~ N ) ~

The Flow Field Upstream of a Horizontal Axis Wind Turbine

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Transcript of The Flow Field Upstream of a Horizontal Axis Wind Turbine