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Journal of Wind Engineering and Industrial Aerodynamics 39 (1992) 139-149

Elsevier Science Publishers B.V., Amsterdam - - Print ed in The Netherlands

139

ns tea dy actuator d i sc m od el for hor izonta l ax i s

w i n d t u rb i n es

Jens N S~ rensen and A sger M yken

Department of Fluid Mechanics Building 404 Technical University of Denmark

DK-2800 Lyngby Denmark

S u m m a r y

A non-linear and unsteady actuator disc model for horizontal axis wind turbines is presented.

The model consists of a finite-differencesolution of the axisymmetric Euler equations in a vortic-

ity-streamfunc tion formulation. We here show some results, steady as well as unsteady, for an

actuator disc with a prescribed elliptic load distribut ion and for the 20 m radius Nibe turbine.

Generally, the results are found to be in good agreement with measurements.

1.

Introduct ion

T r a d i t i o n a l l y p e r f o r m a n c e c a l c u l a ti o n s o f w i n d t u r b i n e s a r e b a s e d o n e i t h e r

b l a d e - e l e m e n t m o m e n t u m t h e o r y o r v o r t e x m o d e ls . H o w e v e r , i n re c e n t y e a rs

g e n e r a li z e d a c t u a t o r m o d e l s h a v e b e c o m e p o p u l a r . T h e r e a s o n f o r t h i s i s p ro b -

a b ly t h a t , a l t h o u g h b e i n g c h e a p t o r u n o n a c o m p u t e r t h e b l a d e - e le m e n t t h e o r y

i s base d on assu m pt ion s th a t have never been jus t if i ed . Espec ia l ly in o f f-des ign

c o n d i t i o n s t h e b l a d e - e l e m e n t m o d e l i s i n p o o r a g r e e m e n t w i t h m e a s u r e m e n t s .

V o r t e x th e o r ie s , o n t h e o t h e r h a n d , r e p r e s e n t t h e p h y s i c s a d e q u a t e ly b u t a r e

e x p e n s iv e t o e m p l o y a n d s u ff e rs o f t e n f r o m c o n v e rg e n c e p r o b l e m s w h e n f re e

w a k e s a r e c o n s i de r ed . A s a s u p p l e m e n t t o t h e b l a d e - e l e m e n t a n d v o r t e x th e o -

r ie s the ac tu a to r d i sc m ode l desc r ibes the f low f ie ld adequ a te ly wi th ou t be ing

too exp ens ive to u t il i ze on a com pute r .

S i n c e t h e c o n c e p t o f t h e a c t u a t o r d is c f i rs t w a s f o r m u l a t e d b y F r o u d e [ 1 ] i t

h a s b e e n c l os el y r e l a te d t o t h e o n e - d im e n s i o n a l m o m e n t u m t h e o r y a n d m u c h

confusion about i t s appl icabi l i ty in descr ib ing complex f low f ie lds s t i l l exis ts .

A l t h o u g h t h e a c t u a t o r d i s c c o n c e p t is a m a i n i n g r e d i e n t i n t h e b l a d e - e l e m e n t

t h e o r y , a s f o r m u l a t e d f o r e x a m p l e b y G l a u e r t [ 2 ] , i t s h a ll b e e m p h a s i z e d t h a t

t h e a c t u a t o r i s a p h y s i ca l m o d e l t h a t e n a b l e s o n e to p u t d i s c o n t i n u it ie s i n t o

the gov ern ing flow equa t ions . In the case o f a ro to r the ac tu a to r d i sc i s de f ined

a s a p e r m e a b l e s u r fa c e n o r m a l t o t h e f r e e s t r e a m v e lo c it y o n w h i c h a n e v e n l y

d i s t r ibu t ion o f b lade fo rces ac t s upo n th e f low. In i t s genera l fo rm the f low f ie ld

0167-6105/92/ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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i s d e t e r m i n e d b y t h e u n s t e a d y , a x i s y m m e t r i c E u l e r o r N a v i e r - S t o k e s e q u a -

t i o n s , wh ic h me a n s t h a t n o p h y s i c a l r e s t r i c t i o n s h a v e t o b e imp o s e d o n th e

k in e m a t i c s o f t h e f lo w .

Th e f i r s t n o n - l i n e a r a c tu a to r d i s c mo d e l f o r h e a v i ly l o a d e d p ro p e l l e r s wa s

fo rm u la t e d b y W u [3 ] . A l th o u g h n o a c tu a l c a l c u l a t i o n s w e re c a r r ie d o u t , t h i s

w o r k d e m o n s t r a t e d t h e o p p o r t u n i t ie s f o r e m p l o y i n g t h e a c t u a t o r d i sc o n c o m -

p l ica ted conf igura t ions as e.g . duc ted p rope l le r s and p rope l le r s wi th f in i te hubs .

L a t e r i m p r o v e m e n t s , e sp e c ia ll y o n t h e n u m e r i c a l t r e a t m e n t o f t h e e q u a t i o n s

a re d u e t o e . g . Gre e n b e rg [4 ] , S c h mid t a n d S p a re n b e rg [5 ] , a n d Le e a n d

Gre e n b e rg [6 ] .

I n th e a p p l i c a t io n o f t h e a c tu a to r c o n c e p t f o r w in d tu rb in e a e ro d y n a m ic s ,

t h e f i r s t n o n - l i n e a r mo d e l wa s s u g g e s t e d b y M a d s e n [ 7 ] . H e d e v e lo p e d a n a c -

tu a to r c y l in d e r mo d e l t o d e s c r ib e t h e f lo w f ie ld a b o u t a v e r ti c a l a x i s w in d tu r -

b i n e, t h e V o i g h t - S c h n e i d e r o r G y r o m i ll . T h i s m o d e l h a s l a t e r b e e n a d a p t e d t o

t r e a t h o r i z o n ta l a x i s w in d tu rb in e s . A th o ro u g h r e v i e w o f a c tu a to r d i sc mo d e l s

fo r r o to r s i n g e n e ra l a n d win d tu rb in e s i n p a r t i c u l a r h a s r e c e n t ly b e e n g iv e n

b y v a n K u ik [8 ] .

Th e me th o d p ro p o s e d h e re i s a n a c tu a to r d i s c mo d e l b a s e d o n a f i n i t e -d i f -

f e re n c e so lu t i o n o f t h e u n s t e a d y , a x i s y m m e t r i c Eu le r e q u a t io n s. T h e k in e m a t -

i cs o f t h e f lo w i s d e s cr ibe d b y t r a n s p o r t e q u a t io n s fo r v o r ti c i ty co a n d s wi r l

v e lo ci ty w , a n d a P o i s s o n e q u a t io n fo r t h e s t r e a m fu n c t io n ¥ . Th e i n f lu e n c e o f

th e ro to r o r , i h e f l o w f i e ld i s t a k e n in to a c c o u n t b y r e p l a c in g th e b l a d e s b y

v o lu me fo r c e s wh ic h a r e t r e a t e d a s s o u rc e t e rms in t h e e q u a t io n s . S in c e t h e

in t e r a c t i o n b e twe e n th e b l a d e fo r c e s a n d th e f lo w fi e ld i s mu tu a l , i t i s n e c e s s a ry

to seek the so lu t ion i t era t ive ly . Th is i s done by a t ime-s tep p ing p rocedu re w hich

a t e v e ry in s t a n t a s s u re s a t ime - t ru e s o lu ti o n . Th e v o lu m e for c es a re e s t im a te d

f r o m m e a s u r e d , t w o - d i m e n s i o n a l a ir fo i l a t a .

A s o p p o s e d t o m o s t o t h e r a c t u a t o r d i s c m o d e l s , t h e p r e s e n t m o d e l i s t h r e e -

d i m e n s i o n a l i n t h e s e n c e t h a t t r a n s p o r t e q u a t i o n s f o r b o t h m o m e n t u m a n d

m o m e n t o f m o m m t u m a r e t a k e n i n t o a c c o t m t . F u r t h e r m o r e , a s t h e m o d e l i s

b a s e d o n d i r e c t t i m e - t r u e s i m u l a t i o n , i t i s c a p a b l e o f h a n d l i n g r o t o r s w o r k i n g

i n u n s t e a d y c o n d i t i o n s . T h u s , f o r e x a m p l e d y n a m i c i n f l o w a n d / o r u n s t e a d y

p i t c h i n g o p e r a t i o n s c a n b e d i r e c t l y i m u l a t e d .

2 . F o r m u l a t i o n o f t h e m o d e l

As s u m in g a x i a l s y m m e t ry a n d in v is c id , i n c o mp re s s ib l e f l o w c o n d i t i o n s , i n

t e rm s o f c y l in d r ic a l c o o rd in a t e s (x ,r ,0 ) w i th c o r r e s p o n d in g v e lo c i ty v e c to r

u,v,w) t h e E u le r e q u a t io n m a y b e wr i t t e n a s fol lo ws :

Co n t in u i ty :

~ ur ) t O vr) O,

Ox ~ r = (1 )

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x - m o m e n t u m :

0 u + Ou Ou 1 O p l

~

at U ax+ V~= pox p (2 )

r - m o m e n t u m :

av Ov av w 2 1 0p i

~ + U ~ x + t ~ r - r = p Or p ' (3 )

~-m o m e n tu m :

~ w w a w v w f t

o t + u ~ x + V -- ~r t - - r - - p 4 )

where p deno tes the p ressure , p the a i r dens i ty , t the t im e va r iab le and ?=

( f ~ , f r , f t )

i s the vo lum e fo rce ac t ing on the ro to r decom posed in x -, r- and 0 -d i rec t ions ,

respect ively .

P r ef er ri ng a f o r m u l a t i o n i n v o r t ic i t y a n d s t r e a m f u n c t i o n w e i n t ro d u c e t h e

fol lowing def in i t ions:

Vort ic i ty :

Ov i~u

c°=~x-

, ~ r

5 )

s t ream:funct ion:

o~,_ o~_

O x - r v ' O r - - r u .

(6 )

T a k i n g th e c u r l o f th e m o m e n tu m e q u a t i o n s a n d i n tro d u c in g th e d e f i n i ti o n s

5 ) a n d 6 ) , w e g e t tw o m o m e n tu m e q u a t i o n s fo r t r a n s p o rt o f v o r t ic i ty a n d

swir l ve loc i ty and a Po i sso n equat ion for the s tream funct ion ,

ox . i

0 ~ - ~ x \ V j j = \ 0 x - - ~ -r Jp ' (7 )

O w O u w ) O v w ) 2 ~ ) = ~

- - ~ 4 O x t- O r

8 )

Ox2

r O r ~ C - O - ~ = r ° ~

(9 )

T h e r e s u l t in g s y s t e m o f g o v e rn i n g e q u a t i o n s a r e n o w g i v e n b y E q s. ( 6 ) - ( 9 ) .

T h e a d v a n t a g e o f t h e p r e s e n t f o r m u l a t i o n is t h a t t h e t h r e e t r a n s p o r t e q u a t i o n s

f o r t h e v e lo c it ie s n o w a r e r e p la c e d b y t w o t r a n s p o r t e q u a t i o n s f o r co a n d w , a n d

t h a t t h e p r e s s u r e i s e l i m i n a t e d f ro m t h e e q u a t i o n s . F u r t h e r m o r e , b y i n tr o d u c -

i n g t h e s t r e a m f u n c t i o n ~ , e v e r y w h e r e i n t h e f l o w d o m a i n t h e c o n t i n u i t y i s

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a u to m a t i c a l l y s a ti s fi e d . Th e e q u a t io n s a r e h e r e p u t i n t h e s o - c al l ed c o n s e rv a -

t i v e fo rmu la t i o n w h ic h a s s u re s e n e rg y to b e p r e s e rv e d .

F o r fu n d a m e n ta l i n v e s t ig a t i o n s , t h e f ie ld o f v o lu m e fo r ce s m a y b e s p e c if ie d

a p rio r i a n d th e k in e m a t i c a l p ro p e r t i e s c a l c u la t e d . Ap p l i e d o n a c tu a l r o to r c o n -

f ig u ra t i o n s, h o we v e r , t h e fo rc e f ie ld a n d th e k in e m a t i c s a r e c o u p le d a n d h a s t o

b e d e t e rm in e d i t e ra t i v e ly .

To c lo s e ~he s y s t e m o f g o v e rn in g e q u a t io n s c a l c u l a t i o n d o m a in , a n d b o u n d -

a ry - a n d in i t i a l - c o n d i t i o n s h a v e t o b e d e t e rmin e d . A t t ime t=0 we l e t t h e

f lo w fi eld b e g iv e n b y a p a r a l l e l fl o w wh e re u - Uo - - c o n s t a n t a n d v = w - - 0 . Th e

c a l c u l a ti o n d o m a in i s d e f in e d b y a n i n flo w a n d a n o u t f l o w p l a n e , a n d a n a x i s

o f : , y mm e t ry a n d a l a t e r a l b o u n d a ry . F o r n u m e r i c a l e f f ic i e n cy i t is imp o r t a n t

to r e d u c e t h e c a l c u l a t i o n d o ma in a s mu c h a s p o s s ib l e w i th o u t d i s tu rb in g th e

f lo w to o mu c h . G e n e ra l l y s p e a k in g , o n ly t h e s y m m e t ry c o n d i t i o n i s o f n a tu r a l

o r ig in wh e re a s t h e o th e r b o u n d a r i e s a r e im p le m e n te d o n ly t o lim i t t h e c a l c u-

l a t i o n d o m a in .

Th e b o u n d a ry c o n d i t i o n s e m p lo y e d in t h e p r e s e n t w o rk a r e a s f o ll o ws:

Inf low:

~V

u = U o , - ~ x = O , C O = O , w = O . (10)

Outf low:

0v 0CO 0u 0w

o -- ;= o o - 7 = - C O o = o .

11)

La te r a l b o u n d a ry :

u f U o O r v)

0---7-=0, COl0, w =0 . (12)

S y m m e t r y a x is :

0U

~ r = 0 , v = 0 , C O l0 , w = 0 . ( 13 )

F r o m t h e se f o r m u l a s b o u n d a r y c o n d i t io n s f o r t h e s t r e a m f u n c t i o n a r e r e a d il y

d e riv ed . I t m a y b e n o t e d t h a t t h e r a d i a l v e lo c ity is a ll o we d to v a ry a lo n g th e

~ateral boundary thu s th is bound ary cond it ion i s m ore general and less restr ic-

t ive t h a n a s s u m in g i t t o be a s t r e a ml in e .

W i t h t h e r o t o r l o c a t e d a t X--Xd a n d l im i t e d b y r e [0 , r a ] t h e v o lu me fo r c e s

a r e d i s t r ib u t e d a s

f - ( fx , fr , ft ) J(X = Xd ,r~r d) ,

(14)

wh e re J d e n o te s t h e D i r a c fu n c t io n .

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3 . N u m e r i c a l t e c h n i q u e

F r o m a m a t h e m a t i c a l p o i n t o f v ie w t h e m o m e n t u m e q u a t i o n s a re h y p e rb o li c

an d th e Po i s so n eq u a t io n e ll ip ti c. Th es e p ro p er t ie s h av e to b e re f lec ted in th e

n u m er ica l so lu tio n p ro cedu re . W e h ere emp lo y a f ini t e-d i ffe rence t ech n iq u e

c o m b i n e d w i t h t h e a l t e rn a t in g - d i re c t i o n -i m p l i c it ( A D I ) m e t h o d o f P e a c e m a n

an d Rach fo rd [9 ] wh ich i s seco n d -o rd er accu ra te in time a n d u n co n d i t io n a l ly

s tab le .

T h e so lu tio n of th e d e f in it io n eq u a t io n s (6 ) a n d th e Po i s so n eq u a t io n (9 )

is accomp l ished by em ploy ing second-order acc ura te cen t ra l -d i f ference d iscre-

t iza t ion . At each t im e s tep Eq . (9 ) n eeds to be fu lly sa t i s f ied and i t therefore

r e p r e s e n t s t h e m o s t t i m e - c o n s u m i n g p a r t o f t h e c o m p u t a t io n . T o s p ee d u p t h e

convergen ce ra te the re laxat ion tec hn iq ue o f W achpre ss [ 10] i s u t il ized . For

th e cases ca lcu la ted u p to n ow, th i s t ech n iq u e a ssu red co n v erg en ce wi th in 8

i t e ra t io n s p e r t im e-s t ep .

Ow in g to th e h y p erb o l ic i ty o f th e m o m en tu m eq u a t io n s , f i r s t o rde r accu ra te

u p win d in g fo rmu las a re emp lo yed . Th u s , ex emp li fi ed b y th e t o - eq u a t io n , th e

convect ive term s are d iscre t ized as fo llows:

x-derivat ive:

~ ( u t o ) ~ [ ( u t o ) i . y - ( u t o ) i _ l j ] / A x , f o r u i j > ~ O ,

(15)

Ox - [ [ ( u t o ) i + , j - ( u t o ) i . j ] / A x , for

U i

< 0 ,

r-derivat ive:

8 ( v t o ) ~ [ ( v t o ) i . ~ - - ( v t o ) i .~ _ ~ l / A r , forvi.~>~0,

~ ) r = ( [ ( v t o ) i j + l - ( v t o ) i j l / A r , for v i ~

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ro to r s a n d ro to r s wo rk in g in u n s t e a d y c o n d i t i o n s , t h e fo c u s w i l l m a in ly b e o n

bas ic f low cases wi th p resc r ibed load ing .

4. I . R otor w ith prescribed loading

T o c o m p a r e t h e p r e s e n t m o d e l w i t h t h e s i m p l e m o m e n t u m t h e o r y , i t is c o n -

v e n ie n t t o i n t ro d u c e s o me g e n e ra l d e f in i t i o n s . Th e t h ru s t T , d e f in e d a s t h e

to t a l a x i a l p r e s s u re fo rc e a c t i n g o n th e d is c, ma y b e d e f in e d a s

T - - y F x d A , (17)

Ad

wh ere Fx deno tes the ax ia l su r face fo rce (Fx = f f , d x a n d Ad i s t h e a r e a o f t h e

a c tu a to r d is c. Th e c o r r e s p o n d in g p o w e r c o n v e r t e d t o o r f ro m th e f l o w is d e t e r -

m i n e d b y

P = f F xu dA. (18)

t/

Ad

I n t h e c a s e o f a w i n d t u r b i n e t h e s t r e a m s u rf ac e p a s s i n g t h r o u g h t h e a c t u a t o r

d i sc h a s a c ro s s - s e c t io n a l a r e a , Ao , a t t h e i n flo w b o u n d a ry wh ic h i s s m a l l e r t h a n

A ~, t h e a r e a a t o u t f l o w d o w n s t r e a m th e d is c. Ac c o rd in g to t h e l a w o f c o n s e r -

v a t io n o f ma s s , we g e t

A o U o = f u d A f y u d A

19)

Ad A~

F r o m t h e k i n e m a t i c s t h r u s t a n d p o w e r m a y a i s o b e o b t a i n e d a s

T - p U ~ A d - f pu 2 dA, (20)

A~

P ~ ½pU ZAd- I ½pu3dA ( 2 1 )

Aoo

w h e r e t h e c o n t r ib u t i o n o f t h e p r e s s u r e o n t h e b o u n d a r y o f t h e s t r e a m s u rf a ce

has been ignored ( see Glauer t [2 ] ) . T he ax ia l f low induc t ion fac to r , a , i s g iven

a s

Ud Ao

a = 1 - V o o = 1 - - A d , 2 2 )

w h e r e U d d e n o t e s t h e a v e ra g e a x i a l v e l o c i ty p a s s i n g t h r o u g h t h e a c t u a t o r d i s c .

F i n a l ly t h e t h r u s t c o e f f ic i e n t C T a n d t h e p o w e r c o e f f i c i e n t C e ar e d e f i n e d a s

T P

C r = ½ P U ~ A d , C p = a • ( 23 )

~pUoAd

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F r o m o n e - d i m e n s i o n a l m o m e n t u m t h e o r y , w e re c a ll t h e c l as s ic a l r e s u lt s

C Tf4a 1- -a) , C p= 4a 1-a) 2,

( 2 4 )

f r o m w h i c h t h e w e l l - k n o w n B e t z l i m i t s t a t e s t h a t t h e m a x i m u m p o w e r t h a t

c a n b e e x t r a c te d f ro m t h e w i n d is g iv e n b y C p ~ = 1 6 / 2 7 a t a = 1 /3 .

E q . ( 2 4 ) w a s d e ri v e d b y a s s u m i n g a c o n s t a n t n o r m a l l o ad o n t h e a c t u a t o r

s u r f a c e . B y i n s p e c t i o n o f E q s . ( 7 ) a n d ( 8 ) i t i s r e a d i ly se e n t h a t t h i s c a se do e s

n o t p r o d u c e a n y s w i r l v e lo c i ty , i.e . w = 0 . F u r t h e r m o r e , a s t h e v o l u m e f o r ce i n

E q . ( 7 ) i s g i v en a s t h e d e r i v a t i v e o f t h e n o r m a l l o a d , a v o r t e x s i n g u l a r i ty o c c u r s

o n t h e e d g e o f t h e a c t u a t o r d i sc . A t t i m e t = 0 t h i s c o n c e n t r a t e d v o r t e x s t a r t s

t o d r i v e t h e fl ow , a f t e r w h i c h a d i s t r i b u t i o n o f v o r t i c e s i s c o n v e c t e d d o w n -

s t r e a m t h e d is c . B e c a u s e o f t h e d i f fi c u lt i e s i n i n t r o d u c i n g a s i n g u l a r v o l u m e

f o r c e i n t h e e q u a t i o n s , i t w a s d e c i d e d to e m p l o y a n e q u i v a l e n t e l li p ti c d i s t r i-

b u t i o n o f v o l u m e f o r ce s

d T 23[ r ) ] 1/2

Fx-- -½pU o 1 - -R CT (25)

w h e r e R d e n o t e s th e r a d i u s o f t h e r o t o r a n d CT s d e f i n e d b y E q . ( 2 3 ) .

B y a d j u s t in g CT a n d c a l c u l a t i n g t h e r e s u l t i n g a v e r a g e a x i a l v e lo c i t y a t t h e

p l a n e o f t h e a c t u a t o r , i t is p o s s ib l e t o c o m p a r e t h e p r e s e n t m o d e l w i th t h e o n e -

d i m e n s i o n a l m o m e n t u m t h e o r y . T h i s i s s h o w n i n Fi g. I w h e r e t h e p o w e r co e f-

f i c i e n t is s h o w n a s f u n c t i o n o f t h e a x i a l i n d u c t i o n f a c t o r . T h e t w o c u r v e s s h o w

t h e s a m e t r e n d s , b u t t h e p r e s e n t m o d e l a l w a y s g iv e C p - v a l u es le s s t h a n t h o s e

o b t a i n e d b y th e m o m e n t u m t h eo r y. T h u s , c o m p a r e d w i th t h e t h e or e ti c al m a x -

i m u m v a l u e o f 0 .5 9 , w e c a l c u l a t e a m a x i m u m v a l ~ e o f 0. 53 . T h e d i f fe r e n c e i n

t h e t w o c u r v e s m a y b e e x p l a i n e d b y t h e d i f f e r e n t lo a d d i s t ri b u t i o n s .

I n t h e c a l c u l a t i o n s , a d o m a i n o f x e [ 0, 6 R ] a n d r e [ 0 ,2 R ] w a s e m p l o y e d , w i t h

0

tO

d

o

d

o

d ~

0

/ • M 0 m e n t u m - t h e 0 r y

_ _ P r e s e n t

_ ~ ,~ .o ~ . . . ~ : ~ E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 2 0 0 . 3 0 0 . 4 0 0 . 5 0 0 . 6 0

O

F i g . 1 . P o w e r c o e f f i c i e n t v e r s u s f l o w i n d u c t i o n f a c t o r .

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120 nodepoints in the x-direction and 40 in the r-direction. The actuator disc

was located at x = 2R. As shown in Fig. i solutions were obtained for a-values

up to a = 0.40. When increasing fur ther the load, it became impossible to obtain

convergence. This is attributed to the limits of the calculation domain, as the

expansion of the wake here exceeds the lateral boundary. This is illustrated in

Fig. 2 where the resulting streamlines for a= 0.35 and a = 0.40 are shown. The

lower figure, corresponding to a = 0.40, has yet not converged, and it is to be

expected th at only an increase in the calculation domain will make a converged

solution possible. In future studies, the influence of the size of the calculation

domain on the solution will be investigated further.

To study the model's ability of calculating unsteady flows, a calculation with

momentary shift of thrust coefficient, from CT O.1to CT--0.3, was carried

out. In Fig. 3, the resulting flow induction factor is shown as function of time.

. _ _

_ ~ _ ~ . j ~ ~ _ . . ~ ~ _ ~

F i g . 2 . S t r e a m l i n e s ; u p p e r : a = 0 . 3 5 l o w e r : a = 0 . 4 0 .

0 10

0.08

0.06

0

0.04

0.02

° ~ oo ,o o . . . . . ~ o o o . . . . . ~ o . o b . . . . . ~ do , oo

T ime sec )

F i g . 3 . C h a n g e o f b l a d e l o a d ; a x i a l f l o w i n d u c t i o n f a c t o r v e r s u s t i m e .

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W e h e r e s e e t h a t i m m e d i a t e l y a f te r c h a n g i n g t h e lo a d t h e i n d u c t i o n f a ct o r

i n c r e a s e s d r a m a t i c a ll y a f te r w h i c h i t o n l y s l o w l y c o n v e r g e s t o w a r d s i ts fi n a l

s t a t e .

4 2 C a l cu l a t i o n o f N i b e 2 0 m r a d i ~ r o t or

T o t e s t t h e m o d e l o n a p r a c t ic a l r o to r c o n f i g u r a t i o n c a l c u l a t i o n s w e r e c o m -

p a r e d w i t h m e a s u r e m e n t s o f t h e 2 0 m . ra d iu s N i b e r o t o r [ 1 1 ] . I n F ig . 4 t h e

m e a s u r e d C p -c u rv e is c o m p a r e d w i t h r e s u lt s f r o m t h e p r e s e n t m o d e l a n d t h e

0 . 5 0

0 . 4 0 / /

/ " .

0 . 3 0 \

\

\

Q

0 . 2 0

• •

• * •

M e a s u r e d d a t a

0 . 1 0 ~ P r e . ~ e n t

B l a d e e l e m e n t t h e o r y

0 . 0 0 . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . ~ ~ . . . . . ~

0 . 0 0 2 . 0 0 4 , 0 0 6 . 0 0 8 . 0 0 I 0 . 0 0 1 2 . 0 0 1 4 . 0 0 1 6 . 0 0

i p ra i io

F i g . 4 . C o m p a r i s o n

o f

m e a s u r e d a n d c a l c u l a t e d p o w e r c o e f i~ c i e n t f o r N i b e - B r o t or .

5 0 , ~

0 L

i

t o t 6 2 o ~ 4 0 5 q o ~ 5 0 5 5 6 0 , 5 l o

F i g . 5 . M e a s u r e d f la p w i s e b e n d i n g m o m e n t t i m e r e p r o d u c e d f r o m [ 1 2 ] ) .

u • i a i v • , , a ~, • u v i v i , i , , . ~ . , , , , . | ~ T I | , ~ i n vnn l l l l u u , , u , u u • i , uu w , v u W v i , , u n v u . , I

o . . o o o . o o 3 o . o o oo s o . o o 6 0 . o 0 7 0 . o o

T i m e sec )

F i g 6 C a l c u l a t e d p o w e r c o e f f ic i e n t t i m e

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b la d e -e l e m e n t t h e o ry . C a l c u l a t io n s w e re h e re c a r r i e d o u t f o r t i p - r a t io s , A=

~R

Uo, rang in g f rom 4 .0 to 9 .5. For h igher k -va lues , owing to the l im i t s o f the

c a l c u l a ti o n d o m a in , c o n v e rg ed s o lu t i o n s w e re n o t o b t a in a b l e . Ho w e v e r , f o r t h e

c a s e s c a l c u l a t e d , t h e p r e s e n t mo d e l i s e v e ry wh e re i n b e t t e r a g re e me n t w i th

e x p e r ime n t s t h a n th e b l a d e -e l e m e n t t h e o ry , wh ic h a lwa y s p re d i c ts v a lu e s h ig h e r

t h a n m e a s u re d .

M e a s u r e m e n t s p e r f o rm e d o n t h e N i b e -B w i n d tu r b i n e h a v e d e m o n s t r a t e d

the e f fec t o f p i tc h ang le cha nges [ 12 ] . I f the p i tc h ang le i s dec reased , fo r ex -

a mp le , t h e a n g le o f a t t a c k i n c r e as e s a n d m o re t h r u s t i s p ro d u c e d . Th i s r e s u l t s

fu r th e r in an inc rease o f the in duce d ve loc it ies . T he induced ve loc i t ies , how-

e v e r, a r e c r e a t e d b y th e v o r t i c i ty i n t h e w a k e , wh ic h o n ly s lo wly a r e c o n v e c t e d

d o wn s t r e a m. Th e re fo re t r a n s i e n t u n s t e a d y e f f e c t s w i l l o c c u r . I n t h e N ib e -e x -

p e r im e n t a s q u a re v a r i a t i o n o f t h e b l a d e p i t c h a n g le wa s a p p l i e d a n d th e r e-

s p o n s e o n th e f l a p wis e mo me n t wa s a n a ly s e d . Th i s i s s h o wn in F ig . 5 wh e re

th e a v e ra g e d f la p wis e mo m e n t i s s h o w n a s fu n c t io n o f t ime .

To t e s t t h e d e v e lo p e d a c tu a to r m o d e l , a s im i l a r e x p e r im e n t w a s c a r r ie d o u t

n u me r i c a l l y . W e h e re c h a n g e d th e p i t c h a n g le w i th a s q u a re fu n c t io n o f l e n g th

3 5 s , c o r r e s p o n d in g to t h e p e r io d e m p lo y e d in t h e N ib e -e x p e r im e n t . T h e o u t -

c o me is s h o wn in F ig . 6 wh e re t h e r e s p o n s e o f t h e p o we r c o e f fi c ie n t i s s h o wn

a s fu n c t io n o f t ime . C o m p a r in g th e tw o f ig u re s t h e q u a l i t a t i v e ly b e h a v io u r o f

th e t im e - r e s p o n s e s a r e s e e n to b e i n v e ry g oo d a g re e m e n t . T h u s , i t i s b e l i ev e d

th a t t h e d e v e lo p e d m o d e l i n p a r t i c u l a r w i l l b e a v a lu a b l e t o o l f o r a n a ly s in g

uns teady e f fec t s .

e f e r e n c e s

1 R . E . F r o u d e , O n t h e p a r t p l a y e d i n p r o p u l s i o n b y d i f fe r e n c e s o f f l ui d r e s s ur e , T r a n s . I n s t .

N a v a l A r c h i t e c t s 3 0 ( 1 8 8 9 ) 3 9 0 .

2 H . G l a u e r t , A i r p l a n e p ro pe ll er s, n: D u r a n d , W . F . , A e r o d y n a m i c T h e o r y ( D o v e r , N e w Y o r k ,

1 9 6 3 ) .

3 T . Y . W u , F l o w t h r o u g h a h e a v i l y l o a d e d a ct :~ P t e r i s c , S c b i f f s t e c h n i k 9 ( 1 9 6 2 ) 1 3 4 .

4 M . D . G r e e n b e r g , N o n - l i n e a r a c t u a t o r d i s c t h e o r y , Z . F l u g w i s s e n s c h . 2 0 ( 3 ) ( 1 9 7 2 ) 9 0 .

5 G . H . S c h m i d t a n d J . A . S p a r e n b o r g , O n t h e e d g e s i n g u l a r i t y o l u t i o n o f a n a c t u a t o r d i s c w i t h

l a rg e c o n s t a n t n o r m a l l o ad , . S h i p R e s . 2 1 ( 2 ) ( 1 9 7 7 ) 1 2 5 .

6 J . H . W . L e e a n d M . D . G r e e n b e r g , L i n e m o m e n t u m s o u r c e i n s h a l l o w i n v i s ci d f lu id , . F l u i d

M e c h . 1 4 5 ( 1 9 8 4 ) 2 8 7 .

7 H . A . M a d s e n , T h e a c t u a t o r c y l i n d e r a f l o w m o d e l f o r v e r t i c a l a x i s w i n d t u r b i n e s , A a l b o r g

U n i v e r s i t y C e n t r e ( 1 9 8 2 ) , D e n m a r k .

8 G . A . M . v a n K u i k , O n t h e l i m i t a t io n s f F r o u d e s a c t u a t o r d i s c c o n c e p t , T e c h n i s c h e U n i v e r -

s it ei t i n d h o v e n ( 1 9 9 1 ) .

9 D . W . P e a c e m a n a n d H . H . R a c h f o r d , T h e n u m e r i c a l s o l u t i o n o f p a r a b o l i c a n d el li pt ic i ff er -

e n ti a l e q u a t i o n s , J . S o c . I n d . A p p l . M a t h . 3 ( 1 9 5 5 ) 2 8 .

1 0 E . L . a c h s p r e s s , I t e ra t i ve o lu t i o n s f e ll ip ti c y s t e m s ( P r e n t i c e - H a l l , n g l e w o o d C li ff s, J ,

1966}.

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11 Nibe maleprogram, Ene rgim inisteriets of Elvmrkernes Vindkraftprogram (1987), DEFU ,

Lundtoftevej 100, DK -2800, Lyngby.

12 S. Oye, Un steady wake effects caused by pi tch-angle changes, 1st . Syrup. on the A erodyn-

amics of Wind Tu rbines (1986).