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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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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 .
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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
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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 .
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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
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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,
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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
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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 :
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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
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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:
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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:
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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
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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 .
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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 .
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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
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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 ).
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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
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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 :
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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 :
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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:
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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
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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
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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 .
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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 :
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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 .
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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 :
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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
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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 .
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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
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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 :
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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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.
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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.
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8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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APPENDIX 1
1
R
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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~ 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 :
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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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.
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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APPEND IX
: A
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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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'
. .:'.:,?
s ,?
~ ~ 1 ; -
a . J ,?
.-, .-;
;..
L
..I
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
-
*
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
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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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)
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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
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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
~ O O ~ X X F Y Y ~ U X F U Y F E T
02010 300 FORMAT(5(5XrFlOe4))
02020 500 CONTINUE
02030
700
02040 700
FORMAT /)
02050 600 CONTINUE
02060 END
02070C
02080C
02090C
02100C
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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
.
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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
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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T IME LIMIT
SRU
21.302 UNTS
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
50/97
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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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
, , , . -
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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E N D
SRU 5 2 . 3 0 9 U N TS ,
RUN COMPLETE
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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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
..
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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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
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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END
SRU
8 155
LINTS
RUN COMPLETE
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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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
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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 - - - - - - - - - - - - - - - - - - - - -- - - - - -
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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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
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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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 - - - - -
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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
~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
.
. .
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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
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313 3 ONTINUE
314
AS=RA
315
RETUl N
316 END
R E A D Y
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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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
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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END
SRU 57 256 UNTS.
RUN COMPLETE
8/18/2019 The Flow Field Upstream of a Horizontal Axis Wind Turbine
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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)
. .
-.
- - . . -
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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 ) ~
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