On the influence of choice of generator lines on the optimum efficiency of screw propellers

Journal of Ship Research, Vol. 34, No. 2, June 1990, pp. 7 9 - 9 1

J o u r n a l of

S h ï D R e ^

Effoeisoey ©f S©rsw Propelters

K. d© Jong^ and J. A. Sparenberg^

The optimum efficiency of a screw propeller, with or without end plates, depends on the geometry of the generator lines of blades and end plates. Quality numbers of propellers have been calculated for a variety of generator lines. The shape of the planform of end plates is discussed in order to minimize their viscous resistance.

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

I N T H I S PAPER we generalize i n some respect t l i e subject discussed i n [1].^ Tliere the i n f l u e n c e o f a n end plate applied to a propeller w i t h o u t r a k e , a c t i n g i n a homogeneous and incompressible p a r a l l e l f l o w , is considered and theoretical results are compared w i t h e x p e r i m e n t a l ones. The obtained results were favorable and the question arose i f there w o u l d exist s t i l l more e f f i c i e n t shapes o f propellers, w i t h r a k e , dif-f e r e n t types o dif-f end plates or curved generator lines.

The o p t i m i z a t i o n t h e o r y we used for t h e c a l c u l a t i o n of the p o t e n t i a l t h e o r e t i c a l c i r c u l a t i o n d i s t r i b u t i o n along blade and end p l a t e is, as i n [1], a l i n e a r theory. Hence i n t h e f i r s t instance the results are v a l i d for l i g h t l y or moderately loaded propellers. However, end plates or r a k e become more favor-able i n the case of h e a v i l y loaded propellers. The reason is t h a t , w h e n T is the t h r u s t of t h e propeller, t h e k i n e t i c en-ergy losses are p r o p o r t i o n a l to T^, w h i l e the viscous losses as we w i l l discuss (Section 9) can be t a k e n a p p r o x i m a t e l y p r o p o r t i o n a l to T. T h i s means t h a t w h e n T is s u f f i c i e n t l y large, a n increase i n the q u a l i t y number g of a propeller w h i c h affects the k i n e t i c energy losses can raise the efficiency i n spite o f an increase o f the viscous resistance caused b y rake, by b r o a d e n i n g o f t h e blade t i p a n d b y t h e end plate w h e n the propeller diameter is k e p t the same. W e hope t h a t l i n e a r theory w i l l indicate directions i n w h i c h t h e search f o r a bet-t e r propeller i n bet-the h e a v i l y loaded case can be c a r r i e d oubet-t. T h a t t h i s is n o t w i t h o u t prospect follows f r o m the experi-ments i n [1], w h i c h showed a n increase i n efficiency by end plates, calculated by a l i n e a r theory, of about 1.5 percent i n model scale. Because the i n f l u e n c e of the l a r g e r Reynolds n u m b e r i n the case of a full-scale propeller is favorable, i t can be expected t h a t a h i g h e r increase i n efficiency w i l l be possible i n t h a t case.

The m e t h o d of c a l c u l a t i o n has to be s u f f i c i e n t l y f l e x i b l e to cope w i t h the v a r i o u s propeller geometries we w a n t to i n

-^Mathematics Department, State University of Groningen, Gronin-gen, The Netherlands.

^Numbers in brackets designate References at end of paper. Manuscript received at SNAME headquarters May 16, 1988; revised manuscript received January 13, 1989.

vestigate. Moreover, t h e calculation of the p o t e n t i a l theo-r e t i c a l q u a l i t y n u m b e theo-r q has to be theo-r a t h e theo-r accutheo-rate, because a s m a l l increase i n q o f some percent can be already r e l e v a n t for t h e design of a propeller. T h i s is the reason t h a t we checked t h e accuracy of our computer progi'am f o r some spe-cial propellers by c o m p a r i n g i t s results w i t h those obtained f r o m Bessel f u n c t i o n expansions analogous to the w o r k of Goldstein [2], or w i t h the results f r o m a f i n i t e - d i f f e r e n c e cal-c u l a t i o n .

I t w i l l be shown t h a t the i n f l u e n c e of end plates alone i s , i n a c e r t a i n sense, stronger t h a n the i n f l u e n c e alone of r a k e or c u r v a t u r e of t h e generator l i n e o f the blade. . Besides, i t t u r n s out t h a t w h e n end plates are applied, a n a d d i t i o n a l r a k e or c u r v a t u r e does not c o n t r i b u t e s u b s t a n t i a l l y to t h e efficiency anymore. F u r t h e r , a d e v i a t i o n of the p l a n f o r m of the end p l a t e f r o m the cylinder t h r o u g h the blade t i p s seems to worsen propeller efficiency.

F i n a l l y , t h e shape o f end plates is discussed globally, w i t h respect to three i m p o r t a n t items. F i r s t , i n order to reduce t h e i r viscous resistance, t h e i r area has to be as s m a l l as pos-sible w i t h o u t increasing the danger of cavitation. Second, the bound v o r t i c i t y o f a blade has to be conveyed smoothly to i t s end plate. T h i r d , t h e f r e e v o r t i c i t y has to be smooth. I t t u r n s out t h a t t h e best we probably can do is to place t h e end p l a t e such t h a t t h e blade t i p is i n the m i d d l e of the t o t a l span o f the end plate. However the p l a n f o r m o f a n end p l a t e w i l l have r e l a t i v e l y s h i f t e d halves, each w i t h a chord l e n g t h at its r o o t w h i c h is equal to or somewhat l a r g e r t h a n h a l f of the chord l e n g t h o f the t i p of the blade such t h a t t h e w h o l e blade t i p is covered. The leading edges and the t r a i l i n g edges of t h e end p l a t e halves have to be chosen such t h a t the free v o r t i c i t y does n o t induce large velocities o n the blade-end p l a t e c o n f i g u r a t i o n .

2. B a s i c n o t a t i o n s

For our geometrical considerations we use t w o types of coordinate systems, the cartesian system {x,y,z) and t h e c y l i n -d r i c a l system (a;,7-,e) ( F i g . 1). We consi-der a screw propeller, r o t a t i n g w i t h constant a n g u l a r velocity w i n the negative 9d i r e c t i o n a r o u n 9d t h e xaxis, i n a n incompressible an9d i n v i s -cid p a r a l l e l f l o w \J i n the positive x - d i r e c t i o n . I t has to

l i v e r a prescribed t h r u s t T i n t h e negative x-direction. A t the t r a i l i n g edges of the blades free vortex sheets are foi-med. The l i n e a r i z a t i o n assumption is t h a t t h e vortex lines of t h e free v o r t e x sheets are helices, l e a v i n g t h e considered p o i n t of the t r a i l i n g edge i n t h e positive x-direction. T h e shape of such a h e l i x is g i v e n b y

— X = const. , r = r*

₍₁₎

where const, and r* are appropriate constants. So no r o l l - u p or b r e a k d o w n of v o r t e x sheets is i n c l u d e d i n o u r model. The h u b is represented b y a two-sided i n f i n i t e l y l o n g c i r c u l a r c y l i n d e r o f radius r^, w h i c h is t h e i n n e r r a d i u s of t h e pro-peller.

The surfaces a t w h i c h t h e v o r t i c i t y i s located a n d w h i c h f l o a t i n t h e l i n e a r theory, w i t h o u t change of shape, i n t h e positive x-direction, we c a l l reference surfaces. I n a l i n e a r t h e o r y t h e blades a n d e n d plates have t o move i n a close neighborhood of these surfaces. F i r s t w e discuss the refer-ence surfaces a t some m o m e n t o f t i m e t, say t = 0. For later use t h e y are two-sided i n f i n i t e l y long. Consider i n t h e xy-plane (Pig. 1) t h e t w o generator lines a n d g2 of a blade a n d a n e n d plate, respectively:

^ i : X = /iCy), 0 < X s Xi, s 3/ < r^, f(rh) = 0 (2)

gz.

y = Ux),

X2 < X < X3, ^2 < < rs

(3)

where fi and /a are s u f f i c i e n t l y smooth functions and the points

( x i , r i ) , (X2,r2), a n d (xsj^g) i n t h e x3'-plane are denoted i n F i g . 1. We demand t h a t g2 pass t h r o u g h t h e p o i n t (x,^) = ( x i , r i ) o f ^ i .

The lines gi and g2 are now r o t a t e d w i t h constant a n g u l a r velocity w around t h e x-axis i n t h e positive B-direction a n d a t t h e same t i m e t r a n s l a t e d w i t h constant velocity U i n t h e positive x-direction. B y d o i n g t h i s , gi and gz w i l l describe t h e reference surfaces denoted by Hi and H2, respectively, of t h e screw blade and t h e end plate under consideration. I f t h e r o t a t i o n a n d t r a n s l a t i o n process "started a t x = - w " , Hi a n d H2 w i l l stretch f r o m x = -00 t o w a r d x = +00. T h e surface of the h u b i s denoted b y H3: Hi:Q + - ( f i { r ) - x ) = 0,rh^r: H2.Q--X U

X*, r = fiix*), X2 s X*

;x3

(4)

(5) (6) The outer r a d i u s i-p of the propeller is d e f m e d as t h e gi-eatest r-value a t t a i n e d b y points (x,r,e) s i t u a t e d on t h e reference surfaces.

E n d plates can be "one-sided" or "two-sided." A one-sided end p l a t e has a generator l i n e gi [equation (3)] w i t h X2 = Xj or X3 = X l , w h i l e for t h e two-sided end p l a t e X2 < Xi < X3.

On t h e surfaces Hi, u n i t normals are introduced ( i = 1,2,3) i n t h e local c y l i n d r i c a l coordinate directions (4, le) a t t h e p o i n t {x,r,%) under consideration

U7- wr

- ^ , - m i

111 1 +

CÜ?-TJ

il) «2 U f i l x - - e

u \

V

u - f i \ x - - ^ ,1, CO / w

1.,

iL

-"-,)'

UU^

\ CO / I Vcor (8)

4 = (0,1,0) (9)

The d i r e c t i o n o f u n i t n o r m a l Hi is t a k e n positive w h e n i t is p o i n t i n g i n t h e d i r e c t i o n of increasing 0, w h i l e t h e positive d i r e c t i o n o f «2 a n d « 3 is t h a t o f increasing r. W e r e q u i r e o f gi, a n d t h e velocities co a n d U to be such t h a t H2 w i l l n o t c u t i t s e l f T h e n i n a l l points of i ï g a u n i q u e exists. W e define the " + " side Ht of Hi to be oriented i n t h e d i r e c t i o n o f t h e u n i t n o r m a l n,- (i = 1,2,3). T h e " - " side H^ is t h e opposite side.

I f t h e propeller has N blades, we sometimes a d d to s y m -bols b e l o n g i n g t o blades a n d e n d plates a second index j = 1 , N . Symbols belonging to t h e h u b keep t h e single index 3. F o r instance, f o r t h e reference surfaces w i t h t h e i r u n i t n o r m a l s we use: Hij, Hzj, Hg a n d d i j , f i i j , ( j = 1. N), respectively. Reference surfaces coincide w i t h each other a f t e r r o t a t i o n over a n angle i2iT/N (i = 1, ...,N) around t h e X-axis. I f there is no danger o f confusion, t h e second i n d e x w i l l be o m i t t e d .

Now we discuss the l i f t i n g - l i n e system w h i c h , i n t h e l i n e a r theory, can represent t h e screw propeller. W e t a k e t h e l i f t -i n g l-ines l -i j , hj -i j = 1, N) a n d 4 as t h e -i n t e r s e c t -i o n o f the reference surfaces Hij, H2J a n d H3 w i t h t h e yz-plane (Fig. 2). I n t h i s w a y t h e h u b is represented b y a c i r c u l a r l i f t i n g l i n e I3 a n d can be treated i n t h e same w a y as t h e screw blades a n d t h e end plates. So, v o r t i c i t y t h a t is f o r m e d on t h e h u b i n o u r model i s l e a v i n g I3 a l o n g helices [equation (1)] l y i n g a t t h e surface H3. We could have t a k e n t h e l i f t i n g lines i n m a n y other ways, f o r instance, a l o n g t h e generator l i n e s gl a n d g2 a n d (y = r^) i n t h e xy-plane. However, t h e choice we made here seems to be more palpable.

For t h e iV-bladed screw, along a r e l e v a n t 1/Nth p a r t (for w h i c h we t a k e j = 1) of t h e l i f t i n g - l i n e system, l e n g t h pa-r a m e t e pa-r s apa-re intpa-roduced. A l o n g t h e l i f t i n g l i n e I3 o f the h u b , the l e n g t h p a r a m e t e r CTJ is increasing f r o m t h e p o i n t (r,e) = (/•A,0) t o (r,e) = (rf,, - 2'TT/JV). A l o n g t h e l i f t i n g l i n e o f t h e blade, we t a k e t h e l e n g t h p a r a m e t e r a i i n c r e a s i n g f r o m (rfi) = (rft,0) t o (7-,9) = iri,

U Xl). I f end plates are present t h e l e n g t h p a r a m e t e r o-g is used, increasing along 1^,1 f r o m t h e

Fig. 2 Lifting lines /,,,, / j , , , I3 and length parameters o-j of a 3-bladed screw propeller rotating witfi angular velocity w in the negative e-direction

end values of CT,(Ï = 1,2,3) are denoted b y o-,- s and a,_e, re-spectively. The 0-2 value corresponding to the intersection of li^i and ^2,1 we denote by

(J2,bi-N e x t we define t h e sign o f t h e bound v o r t i c i t y along ^2,1,13 and o f t h e fi-ee v o r t i c i t y along the helices (1). The bound v o r t i c i t y F/a,) ( i = 1,2,3), w h i c h is the c i r c u l a t i o n a r o u n d It a t the place o;, is t a k e n positive w i t h a " r i g h t - h a n d screw" i n the positive d i r e c t i o n o f the p a r a m e t e r CT;. The f r e e vor-t i c i vor-t y yiidi) sivor-tuavor-ted avor-t Hi (i = 1,2,3) along helices is reck-oned positive w h e n , w i t h a r i g h t - h a n d screw, i t has a posi-t i v e componenposi-t i n posi-the posiposi-tive x-direcposi-tion. For reasons o f s i m p l i c i t y we take t h e f r e e v o r t i c i t y yiiu,) per u n i t o f l e n g t h i n the o-j-direction ( i = 1,2,3), w h i c h however is not pei-pen-dicular to the helicoidal vortex lines. T h e n between the boimd v o r t i c i t y r,(o-,) and the f r e e v o r t i c i t y 7,(0-;) we have the re-l a t i o n s

7i(o-/) = - ^ ( c T i ) , ( i = 1,2,3) a<Ji

(10)

F i n a l l y , we w a n t to have a t our disposal a measure w h i c h gives us some i n f o r m a t i o n about the extent to w h i c h the f l u i d "outside" the end plate reference s u r f a c e s H ^ j U = 1, ...,N) is b e i n g separated f r o m t h e f l u i d "inside" these surfaces. Therefore, we introduce the r a t i o of covering k of the end plates. Suppose the h e l i x t h r o u g h the p o i n t (x,r,e) = (x2,/-2,0) cuts the yz-plane a t the p o i n t (.ï,r,e) = (0,r2,e2), w h i l e t h e h e l i x t h r o u g h (x3,7-3,0) cuts t h a t plane at (0,7-3,63). For t h e Af-bladed screw propeller w i t h end plates, t h e r a t i o o f cov-e r i n g k is dcov-efincov-ed as

k = 2)

2-17 (11)

A screw propeller w i t h o u t end plates has = 0. I n t h i s pa-per, o n l y screws w i t h 0 s ^ £ 1 are considered.

3. O p t i m i z a t i o n a n d e f f i c i e n c y

B y o p t i m i z i n g , i n t h i s paper, w e m e a n m i n i m i z i n g the k i -netic energy of the t r a i l i n g v o r t e x system, w h i c h is f o r m e d at the l i f t i n g lines, representing the screw propeller. A shght extension of the classical o p t i m i z a t i o n procedure of Betz [3] yields t h e boundary-value p r o b l e m f o r the p o t e n t i a l of t h e o p t i m u m t r a i l i n g v o r t e x system f a r b e h i n d the propeller. I n other words, the procedure can be f o r m u l a t e d as follows.

I n a f l u i d w h i c h is at rest, t r a n s l a t e the reference surfaces Hi j a = 1,2)7 = 1, . •., A^) a n d H3, considered to be r i g i d a n d impermeable, w i t h a v e l o c i t y \ i n the positive x - d i r e c t i o n .

T h e f l u i d is unbounded and i n v i s c i d w h i l e the velocity f i e l d is i r r o t a t i o n a l . For the reference surfaces to become stream s u r f a c e s , t h e y n e e d t o h a v e v o r t i c i t y \yi^j ( i = 1,2; j = 1, N) and X73 along hehcoidal lines (1). T h e n t h i s

v o r t i c i t y is the free v o r t i c i t y shed by t h e o p t i m u m propeller. The t o t a l c i r c u l a t i o n around the free v o r t i c i t y sheets has to be zero because the v o r t i c i t y of propeller blades a n d free vortex sheets is free of divergence. T h e n f a r b e h i n d t h e o p t i -m u -m propeller, a u n i q u e l y v a l u e d and ti-me-independent po-t e n po-t i a l ^ ^ ( x , ^ ^ ) of po-the f l u i d velocipo-ties exispo-ts oupo-tside po-the vorpo-tex sheets. The p o t e n t i a l $ belongs t o a u n i t t r a n s l a t i o n a l ve-l o c i t y \ o f t h e f r e e vortex sheets. Because the N bve-lades w i t h end plates of the screw are a l i k e and e q u a l l y spaced, t h i s o p t i m u m p o t e n t i a l w i l l be such t h a t 7i,p = 7i,g (i = 1,2; p,q = 1 , N ) . For 73 there is a n analogous property, and there-f o r e o n l y along the r e l e v a n t 1 / i V t h p a r t othere-f t h e l i there-f t i n g l i n e system (Section 2) has the o p t i m u m p o t e n t i a l to be f o u n d . A t t h a t r e l e v a n t p a r t of the l i f t i n g lines the o p t i m u m cir-c u l a t i o n d i s t r i b u t i o n s \Ti(<Ji) (i = 1,2,3) s a t i s f y

r;(cT,) = mtia,) over Hi (i = 1,2,3) (12) The constant X w i l l be d e t e r m i n e d d i r e c t l y b y t h e c o n d i t i o n

t h a t a t h r u s t T has to be delivered.

So the boundary-value p r o b l e m f o r the o p t i m u m p o t e n t i a l $(x,y,2) becomes (see also reference [4])

A $ = 0 —r- = (riij-h) on Hij (i = 1,2; j = 1, driij dfi: , .,N) 0onH3 (13) (14) (15)

where A is t h e Laplace operator.

I t can be proved [4] t h a t t h e disturbance velocities f o l l o w -i n g f-i-om the potent-ial f u n c t -i o n <-i> have the property t h a t they are perpendicular to the h e l i c o i d a l lines (1).

B y the l a w o f J o u k o w s k i we f i n d f o r t h e t h r u s t T 2

T = iVptoX 2

'•ri.i

dfTi

ATpcüX^) m - r - t d f j i (16)

where f((Ti) is t h e vector p o i n t i n g f r o m t h e o r i g i n

0

t o w a r d t h e considered p o i n t CT; of Z; 1. The vector t((Ji) i n t h e yz-plane is the u n i t t a n g e n t to Z,-,! at CT = CT,- and p is the f l u i d density. F r o m (16) follows the u n k n o w n v e l o c i t y X.

W h e n the o p t i m u m propeller is m o v i n g f o r w a r d w i t h a vel o c i t y U i n f vel u i d a t rest, t h e u n k n o w n v e vel o c i t y X, i n the vel i n -ear theoi-y, is t h e velocity b y w h i c h t h e o r e t i c a l l y the f r e e v o r t i c i t y sheets i n f i n i t e l y f a r b e h i n d t h e p r o p e l l e r t r a n s l a t e backwards by t h e i r o w n induced velocities.

I n our l i n e a r theory, the k i n e t i c energy E o f the p a r t o f t h e t r a i l i n g v o r t e x sheets t h a t is f o r m e d per u n i t of t i m e equals t h e k i n e t i c energy i n f i n i t e l y f a r b e h i n d t h e screw i n between t w o planes x = const, and x = const. + U. Hence i n the o p t i m u m case we have

\grad^\^dxdydz ZVpX' 2 r r i=l J J OsxsU 34> — mn ds (17) J U N E 1990 81

where dS is an e l e m e n t a l surface area o f the considered p a r t of i ï ; , i . U s i n g (14), t h e last e q u a t i o n can be reduced to

.2 2 E = Npu>\- C 1=1 J / , ,

ml f-tddi

E l i m i n a t i n g \ f r o m (16) a n d (18) gives y 2 (18) (19) E = —r 2 p a ) i V 2 mlr-tdGi '•=1 •'',•,1

N e x t we introduce t h e q u a l i t y n u m b e r of a propeller. Con-sider a r o t a t i o n a l s y m m e t r i c actuator disk w i t h t h r u s t T , constant loading T/'nirl - r\), velocity of advance U, and t h e same i n n e r and outer r a d i u s as t h e considered propeller. I t s k i n e t i c energy loss per u n i t of t i m e E^c has the v a l u e

E„, (20)

T h e n the q u a l i t y number q, w h i c h is independent o f t h e t h r u s t T, is defined by

. ^ < 1 E

(21)

The l a r g e r q is, t h e better t h e propeller c o n f i g u r a t i o n w i t h respect to k i n e t i c energy losses. The q u a l i t y n u m b e r has to be s m a l l e r or equal to one because t h e actuator d i s k w i t h constant l o a d i n g is a propeller w i t h t h e highest e f f i c i e n c y i n the l i n e a r theory [ 4 ] . For o u r screw propeller we t h u s have

Uirirl

2 f rh) i=i JL,

m l r ' t d G i (22)

B y (22) we f i n d t h a t g is a f u n c t i o n o f rp, r^, w, U, N a n d t h e shape of t h e generator lines. T h e n , f r o m simple d i m e n -sion analysis, i t f o l l o w s

q = qiwrp/U, rJrp,N, h ) , k = 1,2, . . . (23) where t h e dimensionless 8^ represent t h e shape o f t h e gen-erator lines. A l t h o u g h we could have i n c l u d e d r^/rp a n d N i n t h e 8^, we m e n t i o n e d these e x p l i c i t l y because i t evokes a better picture of the propeller under consideration. U s i n g the q u a l i t y n u m b e r , t h e e f f i c i e n c y t] o f t h e propeller can be w r i t -ten as TU TU + E 1 + 2pU\(4 - ii)qj (24) 4. S y m m e t r y p r o p e r t y

B y the o p t i m i z a t i o n procedure of Section 3 we can o b t a i n a u s e f u l s y m m e t r y p r o p e r t y of o p t i m u m propellers. We con-sider t w o screw propellers A a n d B w i t h the same rotational velocity u around t h e x-axis, w h i l e each is a c t i n g i n a par-a l l e l f l o w U i n t h e positive x-direction ( F i g . 3). A par-a n d B par-are r e l a t e d to each other i n t h a t the l i f t i n g lines of B equal those of A a f t e r r o t a t i n g t h e l a t t e r ones over ir radians a r o u n d t h e y-axis. L e n g t h parameters CT; (i = 1,2,3) a l o n g t h e r e l e v a n t l i f t i n g lines of A are i n t r o d u c e d as was done i n Section 2. The same parameter values occur at con-esponding points o f propeller B.

The o p t i m i z a t i o n procedure is applied to b o t h propellers. So, t h e i r reference surfaces have to be t r a n s l a t e d w i t h ve-l o c i t y \ i n the positive x - d i r e c t i o n . Suppose t h e cave-lcuve-lated o p t i m u m free v o r t i c i t i e s f o r A a n d B are

A B

Fig. 3 Lifting lines of screw propellers A a n d 8 , 3 blades

A-"^""* ,B^-^'"\(i=l,2,3) (25) respectively. N o w we r o t a t e the reference surfaces, the

op-t i m u m v o r op-t i c i op-t y a n d op-t h e op-t r a n s l a op-t i o n a l v e l o c i op-t y \ of screw A over IT radians a r o u n d t h e y-axis. A f t e r t h e r o t a t i o n the r e f erence surfaces o f A coincide w i t h those o f B . T h e n , chang-i n g t h e schang-ign of t h e t r a n s l a t chang-i o n a l velocchang-ity a n d o f t h e v o r t chang-i c chang-i t y of t h e r o t a t e d screw A , t h i s v o r t i c i t y has become i d e n t i c a l to t h e o p t i m u m v o r t i c i t y of B, or

^•Y,(-i) = 57,(cr,o _ ( j = I 2,3) (26) Hence b o t h propellers have the same o p t i m u m c i r c u l a t i o n

d i s t r i b u t i o n s a n d as a consequence t h e same q u a l i t y n u m -ber.

I t is not d i f f i c u l t to show t h a t w h e n we have t w o generator line configurations w h i c h are symmetric i n the xy-plane w i t h respect to t h e yaxis, t h e corresponding l i f t i n g l i n e c o n f i g u -r a t i o n s i n t h e yz-plane a-re also s y m m e t -r i c w i t h -respect to the y-axis.

As a special case of t h e f o r e g o i n g we consider a screw pro-peller w i t h generator lines ( F i g . 1)

g i : X = 0, TA < y < r^

g2- y = fzix), Xa < X < X3

(27) (28) where Xg = x^ a n d fzix) = fzix). F r o m (27) a n d (28) i t f o l -lows t h a t t h e l i f t i n g l i n e li^i r e p r e s e n t i n g a blade coincides w i t h p a r t o f t h e y-axis, and t h e l i f t i n g l i n e Za.i r e p r e s e n t i n g a n end p l a t e is s y m m e t r i c i n t h e yz-plane w i t h respect to t h e y-axis. T h e n (26) i m p l i e s t h a t t h e o p t i m u m fi-ee v o r t i c i t y o f end plates a n d h u b is s y m m e t r i c a l i n t h e sense

liici) = 7i(o-,> + cTie - o-,), ( i = 2,3) (29) A l s o , we can o b t a i n b y our s y m m e t r y p r o p e r t y some i n -s i g h t i n t o t h e i n f l u e n c e of a r a k e . Con-sider a -screw w i t h a "rake angle" a, a n d end plates w h i c h are p a r t o f t h e c y l i n d e r ?- = r i = rp. (30) ^ i : X = y t a n a , < y < r i ^ 2 : y where we i n t r o d u c e d r i A I = (X3 - x i ) / ( x i - X a ) , 0 < ^ < cc (31) (32) w h i c h w i l l be called t h e end plate r a t i o . T h e n b y our s y m -m e t r y p r o p e r t y such a propeller has t h e sa-me efficiency as one w i t h r a k e angle a . T h i s means t h a t t h e q u a l i t y n u m -ber q or t h e efficiency iri is a s y m m e t r i c f u n c t i o n of a w i t h respect to a = 0 deg. Hence a s m a l l r a k e angle a w i l l cause only v e r y s m a l l changes of O(a^) i n the efficiency. O f course t h i s r e s u l t also holds f o r a propeller w i t h o u t e n d plates.

follows t h a t the q u a l i t y n u m b e r is independent of the direc-t i o n of direc-the r o direc-t a direc-t i o n a l velocidirec-ty co.

Applications of the s y m m e t r y property w i l l be used l a t e r

5. S o m e a s p e c t s o f t h e o p t i m u m f r e e v o r t i c i t y I n Section 3 we stated t h a t t h e o p t i m i z a t i o n procedure yielded a Neumann boundai-y-value problem. I n realizing t h a t t h e o p t i m i z a t i o n prob le m can be considered as a p o t e n t i a l -flow problem around t h e impermeable reference surfaces w i t h o u t thickness, we s h a l l derive some a p r i o r i knowledge about the o p t i m u m vorticity distributions we are looking for. The r e s u l t i n g o p t i m u m potential flow we w i l l denote by OFF.

I n t h i s flow pr obl e m we use f o r the disturbance velocity at each side of a reference surface Hid = 1,2,3) the n o t a t i o n

vti<Ji), vTi'rd (33)

reckoned positive w h e n h a v i n g a positive component i n the positive a,-direction. A s we r e m a r k e d below (15), these ve-locities are perpendicular to t h e h e l i c o i d a l lines (1) i n gen-eral, hence also to the h e l i c o i d a l lines w h i c h f o r m a surface

Hi. T h e n f o r the o p t i m u m free v o r t i c i t y 7i(a,), w h i c h i n

Sec-t i o n 2 we defined per u n i Sec-t of l e n g Sec-t h i n Sec-the CT,- d i r e c Sec-t i o n , we have

yiiui) = (viiGi) - vtia,)) cos(ai,s,) (34)

where S ; is the d i r e c t i o n o n Hi pei-pendicular to the helices. Because of t h e i r o r t h o g o n a l i t y w i t h the helices, the veloc-ities vt and vt i n the neighborhood of the intersection of t w o reference surfaces can be a p p r o x i m a t e d by the correspond-i n g veloccorrespond-ity components of a t w o - d correspond-i m e n scorrespond-i o n a l p o t e n t correspond-i a l flow at an angle p. The plane of t h i s two-dimensional flow is the flat plane W perpendicular to the common h e l i x of the t w o considered reference surfaces. T h e angle p is the d i h e d r a l angle, w h i c h is the angle enclosed at the angular p o i n t by the intersection lines of t h e plane W w i t h the reference sur-faces. The nearer to t h e common h e l i x , t h e better t h e ap-p r o x i m a t i o n for vt and vt.

For instance, i n F i g . 4 a representation is given of the i n -tersecting reference surfaces H^ (of blade) a n d ffg (of h u b ) , h a v i n g an angle a (0 < a < -rr) between t h e l i f t i n g lines k and Is i n the yz-plane. The d i h e d r a l angle p = p(a) is r e l a t e d to a by

tan p = V l + M-l t a n c t , 0 < a < TT where we made use of t h e n o t a t i o n

where the indices are connected to li a n d I3, respectively, and R is the distance of a p o i n t of W to the a n g u l a r p o i n t [Fig. 4(right)].

A p p r o x i m a t i n g vt(R) and vtiR) by WiiR) and WaiR), re-spectively, i t follows f r o m t h i s t wo -dimens io nal flow p r o b l e m t h a t (35) U (36)

T

'I

0

Fig. 4 {left) Angle a In yz-plane; (right) ditiedral angle (3 in plane W

\vï{R)\ = \wtiR)\ - \vUR)\ " \wUR)\

const. R" ,R-^0 (38) where p = pi^) depends on p i n the w e l l - k n o w n w a y (for instance [5])

L a t e r on we w i l l use analogously fXp = |x(rp). I n t h e plane W we denote the velocities of the flow w i t h i n the angle p and at i t s sides by

WiiR), WsiR) (37)

p = - - l,(0<p<<»)

P

(39)

and const, is u n k n o w n . N o w i t is clear t h a t a t t h e intersec-t i o n of intersec-t h e h u b and a reference surface of intersec-the blade, intersec-the ve-locities (33) t e n d to zero, w h i c h gives b y (34)

7 i ( c j i ) ^ 0 , O l i ffi,. (40)

The same reasoning is v a l i d f o r the v o r t i c i t y at Hi of a screw blade e n d i n g a t a two-sided end plate

- y i ( o i ) ^ 0 , CTi t <Ti,e (41)

I n t h e O F F the fluid remains a t rest i n t h e inside r e g i o n of H3. Therefore we have

VaicJs) = •• CI3

^ 0-3,,

According to (38) we have

1^3 (0-3) 0 , a g t 0-3.e O r CT3 \ ^3,0

I t is easy to derive t h a t c o s ( a 3 , S 3 ) i n (34) equals 1

COS(0-3,S3) = , O3,, : 0-3 ^ <^3,e

(42)

(43)

(44)

S u b s t i t u t i n g (42), (43) a n d (44) i n expression (34), i t follows t h a t at the i n t e r s e c t i o n o f t h e h u b and the reference surface of the blade 73(03), the f r e e v o r t i c i t y a t the h u b w ü l have the behavior

73(03) • c^3 t o'a.e or (T3 i CT3,; (45)

W h e n the r a t i o of covering equals one (k = 1) a n d the reference surfaces H2 of the end plates coincide w i t h t h e c y l -i n d e r r = rp, the O F F behaves l -i k e an u n d -i s t u r b e d p a r a l l e l flow i n the r e g i o n outside H2. T h e n we find, analogous to (45):

72(02)

-IJ-p

a2-^

o'2,w

(46)

W i t h t h e considerations given above we have f o u n d l i m i t values of the o p t i m u m free v o r t i c i t y 7, at t h e helices of i n -tersection of the reference surfaces. I t is n o t d i f f i c u l t to de-r i v e f de-r o m (38) the w a y i n w h i c h the f de-r e e v o de-r t i c i t y tends to these values. As a n example, t h i s behavior of t h e o p t i m u m f r e e v o r t i c i t y i n the neighborhood o f t h e common h e l i x of i ï i a n d H3 is g i v e n i n F i g . 5, where h cuts ^3 u n d e r a n angle oi

7^ 17/2.

I t is seen t h a t 73 has a zero d e r i v a t i v e at 03,^ a n d an i n finite d e r i v a t i v e a t 03,^. T h i s is caused by t h e r e l e v a n t d i -h e d r a l angle being larger or smaller t -h a n T T / 2 . T-he v o r t i c i t y

7 i has an i n f i n i t e d e r i v a t i v e at the i n t e r s e c t i o n of Hi and

H3 [ F i g . 5(c)]. N o t e t h a t t h i s is a consequence of the

as-s u m p t i o n t h a t t h e h u b ias-s a two-as-sided i n f i n i t e l y l o n g cylinder.

( 0 t'l.s '

Fig. 5 Betiavior of optimum free vorticities in neighbortiood of connection of screw biade to hub

To find w h a t happens i n t h e t h e o r y w h e n the h u b down-stream of a n o p t i m u m p r o p e l l e r has a finite l e n g t h , we r e f e r for instance to [6]. However, t h e more realistic behavior given there w i l l not n e a r l y i n f l u e n c e t h e f l o w at t h e r e l e v a n t w o r k i n g parts of the propeller.

The o p t i m u m f r e e v o r t i c i t y a t t h e connection of the ref-erence surfaces o f a two-sided end plate a n d a blade e x h i b i t s a s i m i l a r behavior.

A t the t i p of a reference surface o f a blade w h e n t h e end plate is absent, a square-root s i n g u l a r i t y o f the free v o r t i c i t y w i l l occur

const.

7i(ai) - , , o-i t Ol,, (47)

Ol

Square-root s i n g u l a r i t i e s w i l l also arise a t the t i p s o f t h e reference surfaces of two-sided end plates w h e n these sur-faces have no h e l i x i n c o m m o n

72(02)

72(02)

const. V o 2 , e - 0'2 const.

V.

, 02 i

O-2,s (48) (49) ~ 0-2,s

S t i l l , a s i n g u l a r i t y of the f r e e v o r t i c i t y o f another type occurs w h e n a onesided end p l a t e is applied. Suppose f o r i n -stance we have a one-sided end p l a t e

[(xi,ri)

=

(a;2,r2)]

w i t h generator l i n e (3)

g2- y = fzix), X2 S X < aCg

7-2 < y < 7-3, (Xl,ri) = (X2,7-2) . (50) I n t h e yz-plane a t Q we t a k e a as t h e smallest angle enclosed by t h e l i f t i n g lines h a n d kiO <OL< 77), ( F i g . 6). U s i n g (34) a n d (38) we find

7I(CTI) ~ const. (CTI,<, - dif , CT f CTI_<, (51)

72(02)

~ const. (CT2 - a-2fiif , CT2 i CT2,S = CT2,6; (52) i n w h i c h f o r b o t h expressions t h e u n k n o w n constants are t h e same. The order p now has a negative v a l u e depending on p = p(a), (35): - T T

+ P

2TT - p , ( - V 2 < P < 0 ) (53)

0

h z

F i g . 6 One-sided end plate wilh angle a between lifting lines at corner Q

Hence the f r e e v o r t i c i t y density becomes i n f i n i t e at t h e j u n c -t i o n o f -the one-sided end pla-te a n d -t h e blade. T h i s is caused by t h e i n f i n i t e velocities of the O F F w h i c h occur a t the cor-ner Q w h e n t h e f l u i d f l o w s ( F i g . 6) a l o n g Ht a n d Jfg •

F r o m (40), (41), (45)-(49), (51) and (52) i n combination w i t h (10), we o b t a i n i n f o r m a t i o n about dTi/dat at t h e values i n question o f CT,- (for instance see F i g . 14 i n Section 9).

6 . N u m e r i c a l m e t h o d

T h e geometry o f the reference surfaces t h a t can be con-stmcted w i t h the generator hnes of Section 2 can vary r a t h e r strongly. To cope w i t h t h e m a n y k i n d s of possible bound-aries, t h e n u m e r i c a l method f o r the o p t i m i z a t i o n p r o b l e m needs to be f l e x i b l e . O u r n u m e r i c a l m e t h o d approximates, at t h e r e l e v a n t p a r t s of the reference surfaces Hi, the o p t i -m u -m free v o r t i c i t y 7; ii = 1,2,3) b y -means o f piecewise l i n e a r f u n c t i o n s .

For t h e v o r t i c i t y 7i(CTi,J, CT,>,S < CT,-,„ < CTi,„,e (t = 1,2,3; n = 1, N), belonging to the boundary-value p r o b l e m (13)-(15), we can w r i t e d o w n three coupled s i n g u l a r i n t e g r a l equa-tions. As t h e relevant parts o f t h e reference surfaces we choose t h e p a r t w i t h parameters CT,-I, ( i = 1,2,3) (see F i g . 7). A t a p o i n t CT;,i = CT;,i of t h e l i n e Z;,i, t h e component, l o c a l l y n o r m a l to t h e reference surface i ï , - i , o f t h e v e l o c i t y t h a t is induced by a concentrated h e l i c o i d a l v o r t e x o f u n i t s t r e n g t h c u t t i n g t h e yz-plane at t h e p o i n t CTJ-,„ w i t h ij,n) ( j , l ) is denoted b y

isr,j>a,o-.>), Vi, j = 1.2,3; ij,n) ^ ( i , l ) ] (54) T h e a f o r e m e n t i o n e d velocity component, b u t n o w w i t h ij,n) = ({,1), is denoted by

Ki,i,liVi_l,<Ti^t}

a = 1,2,3) (55)

(Oi,l - CTi.i)

U s i n g these notations, we find t h a t t h e t h r e e coupled i n t e -g r a l equations become °gi,,-,l(CT,-,i,g,-,i) Ó/,l) (o-i,i 3 N

7i(CTi,i) do,-,

3 « rtJj,„,e

+ 2 ) 2

-K^ij>(*i,i,o>>)

yM J daj,„ = (71,-1

•

r;,)(CT,-,i),

CTi,i,, < CT,-,i < CTi,i,,, a = 1,2,3) (56) T h e integi-als i n equations (56) are t h e c o n t r i b u t i o n s of two-sided i n f i n i t e l y l o n g v o r t e x sheets induced a t t h e p o i n t o;,i = CT,-,i of the l i f t i n g l i n e h^i i n t h e l o c a l l y n o r m a l d i r e c t i o n to' t h e v o r t e x sheet at t h e p o i n t CT,- 1. T h e k e r n e l f u n c t i o n s /fi,,-! ( i = 1,2,3) a n d = 1,2,3; n= 1, ...,N; ij,n) ¥^ ( i , i ) ] c o n t a i n , i n connection w i t h t h e B i o t a n d S a v a r t l a w .

F i g . 7 View in the yz-piane of iength parameters CT/,„ and free vorticity 7,, [; = 1,2,3; n = 1 (W = 3)], used in integral equations (56)

i n f o r m a t i o n about the helicoidal shape o f the vortex sheets, the choice of the generator lines, and the n u m b e r o f blades N. I n order to avoid the w r i t i n g of v e r y l e n g t h y f o r m u l a s , we leave the s t r a i g h t f o r w a r d t a s k of finding expressions f o r the k e r n e l f u n c t i o n s to the reader. The first t e r m o f each of the three equations ( 5 6 ) is a Cauchy p r i n c i p a l v a l u e i n t e -gi-al, due to the fact t h a t t h i s t e r m involves the velocity con-t r i b u con-t i o n of a vorcon-tex sheecon-t acon-t poincon-ts on con-the sheecon-t icon-tself. I f the considered screw propeller has no end plates or a h u b w i t h zero radius (r^ = 0 ) , values of i and j equal to 2 or 3 or b o t h do n o t t a k e p a r t i n expressions ( 5 4 ) - ( 5 6 ) .

I f 7i is expected to have no square-root s i n g u l a r i t y o f the type (47), ( 4 8 ) or ( 4 9 ) and no s i n g u l a r i t y o f the type ( 5 1 ) and ( 5 2 ) , we approximate i t b y a piecewise l i n e a r f u n c t i o n of CT, along the l i f t i n g l i n e Z,-. I f , however, yi does have such a sin-g u l a r i t y for one or more values of the index i, we m a k e use of appropriate coordinate t r a n s f o r m a t i o n s

Oi,„ = CTi,„(Ti), CT,-,„,5 < CTi,n < (Ti^n.e .

f^.Xs =S < CT,-„,„ {n=l,...,N) ( 5 7 ) W i t h the use of (57) the l e f t - h a n d side o f (56) becomes

'iï'i,,-l[CT,-i,CT,-i(T,)] sn^^O'a, , , • 7i[oi.i(Ti)] —r~ (T,) dTi cTjj, [O;,I(T,) - CT,-,i] d^i

+ 2 2 KijM.i.o-jJrj)^yj[aaTj)]-^{T,)dTj ( 5 8 )

j = l n = lJo-. ^ "T;'

The t r a n s f o r m a t i o n CT,_„ = CTi,„(T,), ( 5 7 ) , i s chosen such t h a t •Vi(Ti), defined by

7;(T,) = 7,[f^.-,r.(T,)] ^ (T,) , i n = l , . . . N ) ( 5 9 ) (ZT;

r e m a i n s finite f o r CT,>,S T; < CTi,„,e. T h e n instead o f approx-i m a t approx-i n g 7approx-i(CT,) by a papprox-iecewapprox-ise l approx-i n e a r f u n c t approx-i o n of CT;, we ap-p r o x i m a t e 7i(T,) by a ap-piecewise l i n e a r f u n c t i o n of T,-.

For instance, consider a screw propeller w i t h o u t end plates. The o p t i m u m free v o r t i c i t y of t h e blade 7I(CTI) has at the t i p the behavior ( 4 7 ) . To deal w i t h t h i s s i n g u l a r behavior a n appropriate coordinate t r a n s f o r m a t i o n CTJ = CTI(TI) is giv-en b y

•n- (TI - CTi,) CTidi) = CTi,s + (CTi,e - CTi,J COS

Oi,,=£

2 (CTI,, - CTi,e)_

Ol £ CTi,e , CTi,s < Tl < CTi,e

T h e n t h e f u n c t i o n 7I(TI) ( 5 9 ) w i l l have a finite v a l u e f o r TI I CTi,(,. A n analogous t r a n s f o r m a t i o n can be made to handle the square-root s i n g u l a r i t i e s at t h e tips o f t h e two-sided end plates.

W h e n a one-sided end plate is applied, t h e character of the s i n g u l a r i t i e s at the j u n c t i o n o f blade and end plate d i f f e r s f r o m t h e square-root s i n g u l a r i t y . For instance, i n t h e case where we have generator l i n e ( 5 0 ) , we have f o u n d ( 5 1 ) and ( 5 2 ) . T h e n we have applied

CTI(TI) = CTi,, - (CTI,, - CTi,,) \ 1 Tl - CTi,,

- 1 "

CT2(T2) = CT2,e +

_Ol,e - CTi,;

0-l,s < CTi < CTi_„ CTi,^ Tl < CTi_<, ( 6 1 ) (o'2,e - Ta)^ ( 2 p + 3) (CT2,. -CT2,.) V2(o-2,s + 0-2,e) =S < CTg,, - 0-2,s + (02,e - CT2,s) 2p + 2 2p + 3 j 2(T2 - CT2,J .(02,e - 02,s). 1 CT2,,<T2< V2(CT2,s + 0-2,e) ( 6 2 ) N e x t we introduce the parameters (i = 1,2,3). W h e n no s i n g u l a r i t y of 7,- occurs, we p u t i);,- = CT;; w h e n a s i n g u l a r i t y of 7i occurs, we p u t = T,;. For t h e parameters i];,, we i n t r o -duce t h e meshes

O'i.s = 'I'i-.O < ll'i,! < • • . < <\li,m, = 0-,>, (ï = 1,2,3) (63)

w h i c h need n o t be equidistant. For the base f u n c t i o n s ( F i g . 8) of t h e piecewise l i n e a r a p p r o x i m a t i o n of 7i(CT,) or o f 7j(T,), we choose

B,j (4-,) = i<\ii - - '^u-O, -l-u-i < < <\'u = i<i>u+i - >!'i)/(^^iJ+l - 'I'uX iiij < '\>i < 'I'ij+i

= 0, elsewhere iorj = 1 , m , - - 1

and

BiM) = («l».-,! - 4'.)/(>l'i,i - •l'.,o), «l-i.O <<\ii< vl»;,!

= 0, elsewhere Bi.„f<\ii) = (4-; - «l'.>,-i)/(>|J,>, - 4'.,™;-!), •V,-,».,-! < 4<i < il-r,».,.

=0, elsewhere a = 1,2,3) (64) The a p p r o x i m a t i o n of 7i(CT,) or 7i(T,) (i = 1,2,3) along t h e l i f t

-i n g l-ines t h e n becomes

m,-[7i(CT,) or 7i(Ti)] = Ci,JBiJ(^\>.), [i = 1,2,3) ( 6 5 ) j=o

w i t h appropriate real constants C,^ (t = 1 , 2 , 3 ; j = 0, mt). Now we have t o t a l l y

2 i^i + 1) (66)

(60) Fig. 8 Base functions

u n k n o w n s Cij. These u n k n o w n s have to be determined by t h e i m p e r m e a b i h t y conditions (14) and (15) and by the zero t o t a l c i r c u l a t i o n condition o f our o p t i m i z a t i o n problem.

As we have seen i n Section 5, we also have to cope w i t h the local behavior o f t h e o p t i m u m f r e e v o r t i c i t y i n the neigh-borhood of the junctions o f t h e reference smfaces o f t h e blades a n d h u b ( F i g . 5), or of blades and two-sided end plates. A t those places the free v o r t i c i t y has, locally, a large or a n i n -finite derivative. I n order to obtain accurate results, we make use of appropriate local r e f i n e m e n t s of t h e i|)-meshes (63) where necessary. O f course i t is possible to use a more ac-curate m e t h o d to handle t h e last-mentioned behavior. T h i s is l e f t undone because the s i n g u l a r behavior of the deriva-t i v e appears deriva-to have a n e x deriva-t r e m e l y local characderiva-ter and has no i n f l u e n c e on the first t h r e e d i g i t s o f t h e q u a l i t y n u m -ber q.

The imperaieability condition is satisfied at the l i f t i n g lines k a = 1,2,3) i n the m i d d l e of the meshpoints (JJ,. I n F i g . 9, d i s t r i b u t i o n s of collocation points over the l i f t i n g lines are depicted f o r some screws, w h e n coordinate t r a n s f o r m a t i o n s and r e f i n e m e n t s are applied as explained. Each collocation p o i n t yields a l i n e a r equation f o r t h e u n k n o w n s C , j . I n each such p o i n t for every base f u n c t i o n the velocity induced by the v o r t i c i t y on N h e l i c o i d a l strips s t r e t c h i n g f r o m x = t o w a r d x = +<» is d e t e r m i n e d u s i n g the l a w of B i o t and Sa-v a r t . I n d e t e r m i n i n g t h i s i n f l u e n c e the integi-ations a l o n g the parameters ijf; over a mesh o f t h e l i f t i n g lines are c a r r i e d out by Gaussian rules. For t h e integi'ations along the helices a s p l i t has been made. A l o n g the first M w i n d i n g s o f a h e l i x (1) i n the positive and negative .t-direction, t h a t is, f o r

-M-nU/oi < X < M-nU/iü (67) diverse n u m e r i c a l i n t e g r a t i o n r o u t i n e s are used t h a t are

ad-equate f o r the integi'and's behavior. For t h e c o n t r i b u t i o n of t h e r e m a i n i n g w i n d i n g s s t r e t c h i n g t o w a r d x = +<» and x =

—00, asymptotic expansions have been used. The n u m b e r of w i n d i n g s M t h a t is n u m e r i c a l l y integi-ated, we determine i n r e l a t i o n to the requested accuracy.

I t is easy to see t h a t the c o n d i t i o n of v a n i s h i n g t o t a l cir-c u l a t i o n yields another l i n e a r equation f o r the cir-constants C,-^ (65).

The a p r i o r i knowledge (40) at t h e h u b a n d (41) i n the case of the presence of twosided end plates is p u t i n t o the n u m e r i c a l procedure b y o m i t t i n g i n t h a t case the base f u n c -t i o n s Bi,o(&g-t;|'i) or 5i,„,j (&g-t;|/i) or b o -t h [see (64)] f r o m represen-t a represen-t i o n (65). T h e n w i represen-t h our choice of represen-the paramerepresen-ters, base f u n c t i o n s and collocation points, the i m p e r m e a b i l i t y condi-t i o n and condi-the zero condi-t o condi-t a l c i r c u l a condi-t i o n c o n d i condi-t i o n y i e l d as m a n y l i n e a r equations as there are u n k n o w n s C y . T h i s is also t r u e i f no end plates are applied.

For the case of a one-sided end plate, the above paragi-aph does not hold. T h e n the base f u n c t i o n Bi^mj (^l^i) cannot be o m i t t e d because 7 i ( o i ) now becomes i n f i n i t e at the j u n c t i o n

W (b) (c)

F i g . 9 Distribution of 50 coliocation points over lifting lines; three blades ( N = 3 ) ; (a) no end plates; (b) one-sided end plates; (c) two-sided end plates

{k= 1)

of blade and end plate. T h i s w o u l d mean t h a t we have one u n k n o w n C , j more t h a n there are l i n e a r equations for the Cij (i = 1,2,3; J = 0, . . . , m,). For t h i s case, however, we can use the e x t r a l i n e a r e q u a t i o n for the u n k n o w n s derived by the knowledge t h a t (51) and (52) have t h e same constant factor.

A f t e r s o l v i n g the system o f l i n e a r equations, we o b t a i n an a p p r o x i m a t i o n for the o p t i m u m free v o r t i c i t y l e a v i n g the l i f t i n g lines. The o p t i m u m circulation distributions along the l i f t i n g lines can be obtained by p r o p e r l y i n t e g i - a t i n g the free v o r t i c i t i e s . U s i n g (22), the q u a l i t y n u m b e r q can be ob-t a i n e d .

7. C h e c k o n a c c u r a c y of n u m e r i c a l m e t h o d As we s h a l l see i n Section 8, t h e difference i n q u a l i t y n u m b e r q of screw propellers w i t h d i f f e r e n t generator lines w i l l sometimes be s m a l l . However, i t seems n o t impossible a p r i o r i t h a t geometrical shapes w h i c h cause s m a l l favorable increments i n q can be included i n the propeller design. Therefore, i t is i m p o r t a n t to have a n idea how accurately t h e o p t i m u m c i r c u l a t i o n d i s t r i b u t i o n s and t h e q u a l i t y n u m -ber q can be approximated u s i n g t h e n u m e r i c a l m e t h o d de-scribed i n the previous section. We repeat t h a t the a i m of t h i s paper is n o t to calculate the q u a l i t y n u m b e r f o r a r e a l propeller w i t h such h i g h accuracy, b u t to find t r e n d s w h i c h we hope w i l l occur also i n r e a l i t y .

As a first check we compare the o p t i m u m c i r c u l a t i o n dis-t r i b u dis-t i o n s obdis-tained by our n u m e r i c a l m e dis-t h o d w i dis-t h dis-the series expansion of Goldstein [2]. H e considered a n A''-bladed screw (Fig. 10) w i t h o u t end plates, h a v i n g a straight generator hne

g,: X = 0, 0 = /-A < y < r-p (68) w i t h a zero h u b r a d i u s = 0. I n t h i s special case we can

easily p r e d i c t w i t h t h e considerations of Section 5 the be-h a v i o r o f tbe-he o p t i m u m free v o r t i c i t y f o r CTI i o i ^. T be-h a t is, because oi^^ here corresponds to the o r i g i n O, t h e plane W of Section 5 is perpendicular to the x-axis and therefore co-incides w i t h t h e yz-plane. T h e CTI d i r e c t i o n is perpendicular to the h e l i c o i d a l lines, and the d i h e d r a l angle (3 enclosed by two screw blades equals {2T\/N). U s i n g (34) we conclude

7i(o-i) = const, (o-i - CTi,J^2 / , a i J, CTI,^ (69) Hence a t 0 the free v o r t i c i t y o f t h e blade of a n o p t i m u m one-bladed propeller has a square-root s i n g u l a r i t y . T h e free vor-t i c i vor-t y of a n o p vor-t i m u m vor-two-bladed screw vor-tends vor-to a finivor-te v a l u e at O and i n case of an o p t i m u m screw w i t h more blades 7i(o-i) tends to zero at O. I t is evident t h a t at the blade t i p a square-root s i n g u l a r i t y of 7i(o-i) occurs.

O u r n u m e r i c a l m e t h o d is applied to t h i s propeller u s i n g coordinate t r a n s f o r m a t i o n s a n d r e f i n e m e n t s suited to t h e expected behavior, as e x p l a i n e d i n Section 6. Accurate i n t e -gi-ations along M = 1 2 w i n d i n g s o f t h e helices are used to-gether w i t h asymptotic expansions for the contribution of the r e m a i n i n g parts of the helices.

Goldstein's method to o b t a i n a series expansion f o r t h e opt i m u m c i r c u l a opt i o n d i s opt r i b u opt i o n along opt h e blade has been i m -plemented i n a computer progi-am i n order to y i e l d a more accurate solution t h a n could be g i v e n i n [ 2 ] a t t h a t t i m e . I n F i g . 1 1 t h e results o f t h e comparison are given f o r increasing numbers r i j of the collocation points on the r e l e v a n t l i f t i n g l i n e o f t h e blades. I t is seen t h a t good convergence is achieved. U s i n g n-h = 3 0 collocation points, the o p t i m u m c i r c u l a t i o n d i s t r i b u t i o n s of the series expansions and of our n u m e r i c a l m e t h o d e x h i b i t a difference of less t h a n 0 . 1 percent over t h e e n t i r e range 0 < p, < |Xp. T h e q u a l i t y n u m b e r q is approxi-m a t e d i n f o u r d i g i t s f o r nt = 2 0 , as follows f r o approxi-m t h e table i n F i g . 1 1 .

As a second check we consider another special case, t h a t is, a propeller w i t h a nonzero h u b r a d i u s r^, s t r a i g h t gen-erator lines, zero r a k e angle, a n d end plate reference sur-faces Hij ( j = 1, N) c o i n c i d i n g w i t h the outer c y l i n d e r , 7- = rp. The r a t i o of covering equals one, k = 1. We adapt Goldstein's method of s o l u t i o n to t h i s case.

C h a n g i n g to h e l i c o i d a l coordinates ((ji,0 w i t h \x defined by

( 3 6 ) a n d 5 by 5 = 8 tü we consider $ (5,|x) to - • f C L j J t ) ( 7 0 ) ( 7 1 )

where <t> _{C„[x.) is the p o t e n t i a l of t h e O F F . W e have to de-t e r m i n e &lde-t;i&gde-t; only i n de-t h e r e g i o n

G : jjL, < ^Jl < fJip, 0 < 5 < TT/TV (72)

because o f the s y m m e t r y p r o p e r t y of Section 4 , and because <ï> is a constant f o r |x a |J.p and 0 < (Jt < |X/,. So, 4 has to be d e t e r m i n e d f r o m (for instance, [ 4 ] ) ' $ + ( 1 + , . ^ ) — = 0 ( 7 3 ) n u m e r i c a l method; n b 1 5 0.73650 2 0 0.73949 5 0 0.73954 s e r i e s e x p a n s i o n ; q ^0.73954

F i g . 11 Comparison of numericai metfiod with series expansion; (ip = 5; W = 4; series expansion; . . . n,, = 5; for = 20 and = 50 resuits of

both methods coincide

w i t h b o u n d a r y conditions ( F i g . 1 2 ) 5 = i r / i V : é = 0 t = 0 1 + p." 9 $ dp. ( 7 4 ) ( 7 5 ) ( 7 6 )

F o l l o w i n g Goldstein we t a k e a cosine expansion f o r $ i n the ^-direction. I n d e v i a t i o n f r o m h i s expansion f o r the open propeller of F i g . 1 0 , m o d i f i e d Bessel f u n c t i o n s o f t h e second k i n d cannot he o m i t t e d f o r t h e propeller considered now, be-cause of the nonzero h u b radius. S k i p p i n g t h e analysis f o r reasons of b r e v i t y , we give d i r e c t l y t h e r e s u l t i n g series so-l u t i o n : $ ( L f i ) = 4 ; E VTi/vp.) + aJC^ivp.) cos(vO + bMvtx)] - T ^ , ( 7 7 ) ( m + 1/2)' w i t h V = ( m + V2)ZV ( 7 8 )

I n t h i s expression K„ and 1^ are t h e m o d i f i e d Bessel f u n c -tions as d e f i n e d i n [ 7 ] , w h i l e t h e Tj^^ f u n c t i o n s are t h e same as those used i n [ 2 ] . For every m [m = 0 , 1 , 2 , . . . ) t h e a „ a n d 6m are d e t e r m i n e d by t h e b o u n d a r y conditions ( 7 6 )

aJ<:'Xvp.h) + bjlivp^u) = -Tl^Wh) ( 7 9 ) a,„K;(v|jtp) + hj',{vp.p) = - Tl,{vv.p) ( 8 0 ) where the p r i m e s denote d i f f e r e n t i a t i o n s w i t h respect to t h e

a r g u m e n t VIJL.

O n the r e l e v a n t l i f t i n g lines, t h a t is, o n those boundaries of G where N e u m a n n conditions are demanded, t h e t e r m s of expansion ( 7 7 ) f o r values o f m w h i c h are n o t too large (for example, m ^ 1 0 ) can be calculated d i r e c t l y u s i n g ( 7 9 ) a n d ( 8 0 ) . I t is possible to o b t a i n as m a n y t e r m s as desired i n series ( 7 7 ) . N a m e l y , b y u s i n g asymptotic expansions (for i n -stance [ 8 ] ) , w e f m d f r o m ( 7 9 ) and ( 8 0 ) f o r l a r g e r values o f m

aji^ivp.) = r i . v ( v | x , ) - T I > | J t p ) - ^ KlivtXf,) /p(vp-A) Klivix^p) 7;(v|Xp). KiviXp) G ( 8 1 ) d f i d f i + ft'

F i g . 12 Boundary conditions o n BG for <t> = 4>(^,M.)

Analogously we have Table 2 Quality number q, straight generator lines, e n d plate ratio £ = 1

-TUviXp)

/ > | j j

Tv(vp.p) (82)

The last quotients i n (81) and (82) can again be approxi-mated by asymptotic expansions.

We compared the above-mentioned accurate series expansion for problem (73)(76) w i t h t h e results f r o m the n u m e r -ical method. I t is f o u n d t h a t by increasing t h e n u m b e r o f collocation points, good convergence of the o p t i m u m circu-l a t i o n d i s t r i b u t i o n occurs on h u b , bcircu-lade and end pcircu-late to t h e values f o u n d by the series expansion. I n Table 1 the com-parison of the q u a l i t y n u m b e r q is g i v e n . The n u m b e r of col-location points on the l i f t i n g lines of hub, blade and end plate are denoted by ri/,, a n d / i , , respectively.

A s a double check f o r t h e last-mentioned case, a finite-difference a p p r o x i m a t i o n has been made, w h i c h also con-verges to the series expansion i n r e f i n i n g the meshes of t h e finite difference gi-id. The n u m e r i c a l method of discrete he-licoidal vortices was also tested f o r t h i s case. We f o u n d t h a t t h i s method d i d not give results w h i c h we considered to be accurate enough f o r our purpose.

8. N u m e r i c a l r e s u l t s

I n t h i s section the q u a l i t y n u m b e r q is g i v e n f o r a n u m b e r of propeller geometries. The s y m m e t r y property of Section 4 implies, as has been stated there, t h a t the q u a l i t y n u m b e r

q is independent of the d i r e c t i o n o f r o t a t i o n . Hence a l l

gen-erator l i n e c o n f i g u r a t i o n s i n t h i s section can be i n t e r p r e t e d to generate propellers t h a t rotate i n either of the two direc-tions. Calculations are presented f o r screws w i t h

r j r p = 0.2 (83) I n the numerical method the t o t a l number of collocation points for a l l screws equals 50.

Influence o f rake angle a and r a t i o of covering k

Consider the propeller w i t h s t r a i g h t generator lines (30) and (31). Hence t h e screw blade has a r a k e angle a, and the end plate reference surface is p a r t of t h e c i r c u l a r c y l i n d e r t h r o u g h the blade t i p and is s y m m e t r i c w i t h respect to the blade t i p = 1, (32)]. I n T a b l e 2 the q u a l i t y n u m b e r q as a f u n c t i o n of the r a k e angle a and the r a t i o o f covering o f the end plates k is g i v e n f o r d i f f e r e n t numbers o f blades

N and d i f f e r e n t values of the dimensionless q u a n t i t y |jLp = ix>rp fU. F i g u r e 13 shows a gi-aph o f q.

F r o m Section 4 we k n o w t h a t the g r a p h o f q is s y m m e t r i c w i t h respect to the plane a = 0 deg, f o r k = const. I t is seen f r o m the n u m e r i c a l results t h a t g as a f u n c t i o n of a indeed has a zero d e r i v a t i v e a t a = 0 deg. F r o m t h i s we conclude t h a t i n order to increase g, the a p p l i c a t i o n of s y m m e t r i c a l end plates is more effective t h a n g i v i n g the screw a r a k e angle. F u r t h e r m o r e f o r a = const., the q u a l i t y n u m b e r g as a f u n c t i o n o f k is seen to have n u m e r i c a l l y a zero d e r i v a t i v e at ^ = 1.

Influence o f asymmetry o f end plate reference surface A g a i n , the propeller has s t r a i g h t generator lines and the end plate reference surface is p a r t of t h e c i r c u l a r cylinder t h r o u g h the blade t i p , as i n t h e preceding case. The d i f f e r -ence is t h a t now the end plate refer-ence surface can be

Table 1 C o m p a r i s o n of quality number JJ; W = 4, n,, = 1, (ip = 5

0° 15° 30° 45° O .450 .532 .582 .453 .535 .584 .463 .644 .592 .482 .561 .608 3 .566 .646 .691 .569 .649 .694 .580 .668 .702 .602 .676 .717 4 0.00 .648 .722 .762 .652 .726 .764 .663 .733 .771 .685 .750 .784 5 .708 .774 .809 .711 .776 .811 .722 .784 .817 .742 .799 .828 6 .752 .811 .841 .755 .813 .844 .765 .820 .849 .783 .833 .859 7 .587 .636 .665 .588 .636 .665 ,588 .638 .667 .592 .643 .673 3 .690 .736 .762 .691 .737 .763 .693 .739 .765 .700 .745 .770 4 0.25 .758 .800 .822 .759 .800 .822 .762 .803 .825 .769 .808 .829 5 .804 .841 .861 .805 .842 .861 .808 .844 .863 .815 .849 .867 6 .837 .870 .887 .838 .871 .888 .841 .873 .889 .847 .877 .893 7 .679 .704 .719 .679 .704 .719 .679 .705 .720 .679 .706 .721 3 .774 .796 .809 .774 .796 .809 .774 .797 .809 .775 .798 .811 4 0.50 .832 .851 .861 .832 .851 .861 .832 .862 .862 .834 .853 .863 5 .869 .886 .895 .869 .886 .895 .870 .887 .895 .871 .888 .896 6 .896 .909 .917 .895 .910 .917 .895 .910 .917 .897

1

.911 .918 7 .732 .744 .750 .732 .744 .751 .733 .746 .751 .734 .746 .752 3 .823 .831 .835 .823 .831 .835 .823 .831 .836 .824 .832 .836 4 C.7S .875 .881 .884 .875 .881 .884 .875 .881 .884 .876 .882 .885 5 .907 .912 .914 .907 .912 .914 .907 .912 .914 .906 .912 .914 6 .928 .932 .934 .928 .932 .934 .928 .932 .934 .928 .932 .934 7 .750 .757 .761 .750 .757 .761 .751 .758 .762 .753 .760 .763 3 .839 .842 .844 .839 .842 .844 .840 .843 .845 .841 .844 .845 4 1.00 .889 .891 .891 .889 .891 .891 .889 .891 .892 .890 .891 .892 5 .919 .920 .920 .919 .920 .920 .919 .920 .921 .920 .920 .921 6 .939 .939 .939 .939 .939 .939 .939 .939 .940 .939 ,939 .940 7 N 2 3 4 2 3 4 2 3 4 2 3 4 _I'p

a s y m m e t r i c w i t h respect to the blade t i p , 5^ 1, (32)]. I n Table 3, f o r screws w i t h zero rake angle, the results are given for k = 0.25, k = 0.5, and k = 0.75, respectively. I t follows t h a t the a s y m m e t r y has a positive i n f l u e n c e o n g; however, the l a r g e r k, t h e less g is i n f l u e n c e d by ^. T r i v i a l l y , f o r k = 1 t h e q u a l i t y n u m b e r g is unaffected by ^, hence f o r t h i s case the v a l u e o f g can be f o u n d i n Table 2.

Because of t h e s y m m e t r y property (Section 4) i n t h i s case a screw w i t h end plate r a t i o ^ (a = 0 deg) has t h e same

qual-0.3

2

1

3 1 4

4I

6 1 8 8 12! 16 I6I24 I32 s e r i e s , expansion i .89243 .89149 .89141 .89145 .89149

F i g . 13 Quality number q as a function of ral<e angle a a n d ratio of covering k; p.p = o>rp/f = 4 ; W = 3

Table 3 Quality number q, straight generator lines, rake angle a = 0 deg, W = 3 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.6 0.75 3 .636 .704 .744 .636 .704 .744 .643 .709 .748 4 .736 .796 .831 .737 .796 .831 .744 .802 .834 5 .800 .851 .881 .800 .851 .881 .808 .857 .884 1 2 00

i t y n u m b e r as one w i t h end plate r a t i o T h i s rises t h e expectation t h a t the a s y m m e t r y of the end plate reference surface cannot cause, i n the neighborhood of ^ = 1, substan-t i a l changes i n q, f o r substan-the same reason as a s m a l l r a k e angle a could not n e a r l y i m p r o v e q i n t h e neighborhood o f a = 0 deg. The n u m e r i c a l results of Table 3 c o n f i r m t h i s expec-t a expec-t i o n . E v e n i n case o f a one-sided end plaexpec-te expec-the change is s m a l l .

To i l l u s t r a t e the effect of ^ 5^ 1 w h e n the screw has a n o n -zero r a k e angle, analogous results are g i v e n f o r a propeller w i t h r a k e angle a = 30 deg (Table 4). Here the m i n i m u m of q appears f o r a case w i t h a s y m m e t r i c a l end plate refer-ence surfaces (^ 7^ 1). The screw w i t h the one-sided end p l a t e and the gi-eatest angle enclosed between screw blade and end plate has the largest v a l u e o f q. However, differences a g a i n are s m a l l . Also here, f o r k = 1, the q u a l i t y n u m b e r q is u n -affected b y ^, and its value can be f o u n d i n Table 2.

We w i l l discuss the i n f l u e n c e of the a s y m m e t r y of the end plate reference surface (^ 1), i n connection w i t h the v i s -cosity o f water, i n Section 9.

Influence o f c u r v a t u r e o f generator line ffi o f blade

The end plate reference surfaces are p a r t of the c i r c u l a r c y l i n d e r t h r o u g h the blade t i p s w i t h ^ = 1. The first screw of Table 5 is the first one of Table 3. The generator l i n e gi of the blade o f the second screw is the one obtained by i n -t e r p o l a -t i n g w i -t h a cubic spline -the poin-ts

(x,y) = {0,rh), (O.lrpfiArp), (0.157p,0.6rp), (O.lr^.O.Szp), (0,rp)

(84)

Table 5 Quality number q, end plate ratio ^ = 1, W = 3

0 0 0

30 . 3 0 3 0 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 Jt 3 .632 .704 .546 .704 .644 .705 .533 .706 .568 .705 4 .645 .796 .660 .797 .658 .797 .646 .797 .684 .798 5 .721 .851 .735 .852 .733 .852 .722 .852 .757 .853

B y our s y m m e t r y property the q u a l i t y n u m b e r o f t h i s screw is the same as f o r the propeller obtained b y u s i n g a s i m i l a r spline t h r o u g h the points

{x,y) = (0,/v,), (-0.1rp,0.4rp), (-0.15rp,0.6rp),

(-0.1rp,0.8/-p), (0,rp) (85) The t h i r d screw has a 30-deg rake. The generator lines gi of the blades o f the f o u r t h and fifth screw are obtained by u s i n g splines t h r o u g h the points obtained a f t e r a skew t r a n s f o r m a t i o n w i t h angle 30 deg has been applied to the points (84) and (85), respectively.

W h e n the screw has no end plates {k = 0), the c u r v a t u r e of gl does have i n f l u e n c e on q, t h e last screw h a v i n g the highest q. W h e n end plates w i t h ^ = 0.5 are applied, the i n f l u e n c e of the c u r v a t u r e is negligible.

I n f l u e n c e o f curvature of generator line ffa o f end plate I n Table 6 examples are g i v e n f o r screws w h e n the ref-erence surfaces of the end plates are not p a r t of the c i r c u l a r cylinder w i t h r a d i u s rp. The generator lines gi of t h e blades are straight, and a l l have zero rake except the last one, w h i c h has a 15-deg r a k e .

I n T a b l e 6 a l l examples are such t h a t t h e r a d i a l range of t h e end plates is rp/20. A g a i n the first screw of t h i s table is the first one of Table 3. The t h i r d and f o u r t h screw i n Table 6 have generator lines g2 w i t h p a r a b o l i c - l i k e shapes, w h i l e the second screw has a g2 w h i c h is h a l f s t r a i g h t and h a l f parabolic. The generator lines g2 o f t h e end plates o f t h e fifth and s i x t h screw are s t r a i g h t and i n c l i n e d ; i n t h e i r m i d p o i n t s t h e y are m e e t i n g gi.

We conclude t h a t the q u a l i t y n u m b e r q decreases w h e n g2 is l e a v i n g the outer circular cylinder.

T a b l e 4 Quality number q, straight generator lines, rake angle a = 30 d e g , W = 3

0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0,5 0.75 0.25 0.5 0.75 Jt 3 .639 .707 .745 .637 .705 .745 .638 .705 .745 .640 .705 .744 .661 .712 .748 4 .739 .799 .832 .738 .797 .832 .739 .797 .831 .741 .797 .831 .752 .805 .835 5 .803 .853 .882 .802 .851 .881 .803 .852 .881 ,805 .852 .881 .814 .859 .884 0, 0.5 1 2 CO J U N E 1990 89