A F D E L I N G DER S C H E E P S B O U W - E N S C H E E P V A A R T K U N D E L A B O R A T O R I U M V O O R S C H E E P S H Y D R O M E C H A N I C A R a p p o r t N o . 4 8 3 - P A C A L C U L A T I O N M E T H O D F O R T H E H E A V E A N D P I T C H M O T I O N S O F A H Y D R O F O I L B O A T I N W A V E S . i r .
J . A . K e u n i n g
j u i y 1979
R e p r i n t I . S . P . 2 6 O c t o b e r 1 9 7 9 n o. 3 0 2 , b l z . 2 1 7 - 2 3 2 D e l f t U n i v e r s i t y of T e c h n o Ship Hydromechanics Laboratory Mekelweg 2DeUt 2 2 0 8 Nelherlands
A C A L C U L A T I O N M E T H O D F O R T H E H E A V E A N D P I T C H M O T I O N S OF A H Y D R O F O I L B O A T I N W A V E S
b y
i r . J . A . K e u n i n g *
1. I n t r o d u c t i o n
D u r i n g the last decades there has been a c o n t i n u i n g interest i n the use o f h y d r o f o i l boats. The f e a s i b i l i t y o f this c r a f t was usually c o n f i n e d t o m i l i t a r y ' pur-poses, as a fast and stable c r a f t and t o passengers-t r a n s p o r passengers-t over m e d i u m range dispassengers-tances.
The c o n v e n t i o n a l h y d r o f o i l boat uses surface piercing dihedral f o i l s f o r n a t u r a l s t a b i l i t y o f the vertical dis-placement. Usually this concept has one surface pier-cing d ü i e d r a l f o i l f o r e and one f u l l y submerged f l a t f o i l a f t . The surface p i e r c i n g f o i l makes this t y p e o f h y d r o f o i l boat rather sensitive t o waves, resulting i n a relatively p o o r seakeeping p e r f o r m a n c e . T h e r e f o r e the use o f these h y d r o f o i l boats is usually restricted to sheltered waters such as rivers, lakes and the M e d i -terranean Sea.
T o i m p r o v e the seakeeping p e r f o r m a n c e n e w types o f h y d r o f o i l boats were developed w i t h o n l y deeply sub-merged f l a t f o i l s . This c r a f t was n o longer stabilised by the variable submerged area o f tlie d i h e d r a l f o i l and as a consequence expensive c o n t r o l systems, i n c o r p o r a -t i n g flaps on -the f o i l s , v/ei-e necessary -t o s-tabilise -the c r a f t . The i m p r o v e m e n t i n seakeeping p e r f o r m a n c e however is great; a safe f l i g h t i n a seastate 5 is possible. To determine, w h e n , i n a specific s i t u a t i o n , the rela-tively inexpensive c o n v e n t i o n a l h y d r o f o i l boat is n o longer feasible, a reliable c o m p u t a t i o n o f the vertical m o t i o n s is needed.
The c a l c u l a t i o n methods most c o m m o n l y used make use o f a description i n the f r e q u e n c y d o m a i n , w h i c h makes a linearisation o f the equations o f m o t i o n ne-cessary. The results o f these methods are p o o r especial-ly i n the s i t u a t i o n w i t h f o l l o w i n g waves. This is largeespecial-ly c o n t r i b u t e d t o the linearisation o f the equations, al-t h o u g h i n al-the descripal-tion o f al-the l i f al-t and drag forces on the foils non-linear e f f e c t s play an i m p o r t a n t r o l e . In this study a description i n the t i m e d o m a i n w i l l be used. T h i s makes i t possible to leave the equations i n their non-linear f o r m . The m o t i o n s are calculated f o r b o t h head- en f o l l o w i n g regular sinusoidal waves. The results o f these calculations are compared w i t h the results o f a n e x p e r i m e n t .
2. The t h e o r e t i c a l m o d e l
2.1. Physical assumptions
In order to f a c i l i t a t e tlie d e r i v a t i o n o f the equations some assumptions concerning tlie geometry o f the
h y d r o f o i l boat and the physics involved had t o be made, these were;
1. the equations are valid f o r a h y d r o f o i l boat o f the c o n v e n t i o n a l t y p e o n l y , i.e. w i t h one surface pier-cing d i h e d r a l f o i l f o r e and one f u l l y submerged f l a t f o i l a f t . N o active c o n t r o l systems are considered. 2. the foils have a constant section over the span and
a rectangular p l a n f o r m . This r e s t r i c t i o n h o w e v e r can easily be r e m o v e d .
3. the c h o r d o f the foils is small, compared w i t h the distance between the f o i l s and the centre o f gravity o f the c r a f t , so v a r i a t i o n o f the centre o f e f f o r t o f the d i f f e r e n t components o f the l i f t f o r c e has n o significant i n f l u e n c e on the m o m e n t s a r o u n d the centre o f gravity o f the c r a f t and the forces due t o the r o t a t i o n o f the f o i l around its o w n centre o f gravity can be neglected.
4. the c h o r d o f the f o i l s is small, c o m p a r e d w i t h the wave length o f the encountered waves, t h e r e f o r e the v e l o c i t y o f the water over the f o i l s may be con-sidered u n i f o r m .
5. the b u o y a n c y o f the f o i l s and struts can be neglec-ted.
6. the t h r u s t o f the propeller(s) acts t h r o u g h the centre o f gravity o f the c r a f t .
7. waves generated b y the f o r e f o i l d o n o t i n f l u e n c e the a f t f o i l .
8. the l i f t and drag c o e f f i c i e n t s o f the f o i l s are con-sidered t o be f u n c t i o n s o f the m o m e n t a r y angle o f incidence, aspect r a t i o and submergence o n l y . The effects o f c a v i t a t i o n , v e n t i l a t i o n and stall w i l l n o t be taken i n t o account, a l t h o u g h these can be incor-p o r a t e d .
1
O -G
1
Figure 1. Geometry of the liydrofoil boat.*) D e l f t U n i v e r s i i y o f T e c h n o l o g y , Ship H y d r o m e c h a n i c s L a b o r a t o r y , Report -ÏSS-P, D e l f t , T h e Netherlands.
e q u i l i b r i u m flight w a t e r l i n e
Figure 2a, Geometry of tlie front f o i l .
Figure 4. Angle of attack and velocities over the front f o i l .
1
Figure 2b. Geometry of the aft f o i l .
Z-X Space fixed coordinate system z-x Translating coordinote syslem z'- x' Body fixed coordinate system
Figure 3. Coordinate systeins.
2.2. Lift 2.2.1. Front foil
T h e expression f o r the h f t o f the f r o n t f o i l can be w r i t t e n as: dc, dc, dc, — A / i + — A / 1 dh dA (1) c o n t a i n i n g the f o l l o w i n g expressions:
Vp — relative velocity of the water over tlie foii
T h e relative water v e l o c i t y over the f o i l is composed o f a p a r t due t o the eciuilibrium v e l o c i t y o f the c r a f t , a part due to the surge m o t i o n and a p a r t due to
the p i t c h m o t i o n . This last c o m p o n e n t varies over t h e span o f the f o i l , because o f the d i h e d r a l shape. T h i s c o m p l i c a t i o n is considered negligible and is avoided b y assuming t h a t the v e l o c i t y at the centre o f e f f o r t is valid f o r the w h o l e f o i l . F o r this ( a n d o t h e r purposes) e l l i p t i c a l l o a d i n g over the f o i l o f the span has been as-sumed. Y i e l d i n g :
{hj - hp+iSpllCp) tgnp{\ -AI3 n))d
( 2 )
Sp - submerged area of the front foil
The h o r i z o n t a l p r o j e c t i o n o f the submerged area o f the f r o n t f o i l depends o n the d i h e d r a l angle and t h e m o m e n t a r y submergance o f the f o i l , y i e l d i n g :
Sp = 2CpCoXgHp{h^p-z ^XpO) ( 3 )
— angle of attack
The m o m e n t a r y angle o f a t t a c k o f the f o i l is c o m posed o f the angle o f a t t a c k in the p o s i t i o n o f e q u i -l i b r i u m , w h i c h is i n c o r p o r a t e d i n the Cj^ c o e f f i c i e n t , and o f a f l u c t u a t i n g part due t o t h e p i t c h - , heave- a n d surge velocities o f the c r a f t . Because o f t h e d i h e d r a l o f the f o r e f o i l a c o i T e c t i o n o f the angle o f a t t a c k is necessaiy. Schuster [ 8 ] f o r m u l a t e d h e r e f o r e :
s i n a * = sina c o s f j ^
w h i c h can be w r i t t e n f o r small angle's o f a t t a c k as a* = a cos jip
T h e m o m e n t a r y angle o f attack then becomes: 1, •
( 4 )
and the f l u c t u a t i n g p a r t t h e n becomes COS/J,
/ ( j - submergence of the front foil Each segment o f the dihedral f o i l has a d i f f e r e n t distance t o the free surface. I n o r d e r to avoid u n -necessary c o m p l i c a t i o n s f o r this purpose t h e d i h e d r a l f o i l has been replaced by a f l a t f o i l w i t h a submergence equal t o the distance o f the centre o f e f f o r t o f the el-l i p t i c a el-l el-l o a d i n g over the span o f the d i h e d r a el-l f o i el-l t o the free surface, y i e l d i n g :
(6)
A ~ effective aspect ratio
The aspect r a t i o o f the f r o n t f o i l depends o n the submergence o f the f o i l . D u e t o end-plate e f f e c t s the g e o m e t r i c a l and e f f e c t i v e aspect r a t i o d i f f e r consider-a b l y . F o r consider-a surfconsider-ace p i e r c i n g f o i l the f o l l o w i n g r e l consider-a t i o n is a d o p t e d :
= 1 , 8 / } ^
The i n t e r a c t i o n o f the foils especially at their con-n e c t i o con-n ucon-nder water is con-n o t w e l l - k con-n o w con-n . T h e r e f o r e i con-n thise case the e f f e c t i v e aspect r a t i o has been sliglitly increased t o account f o r this i n t e r f e r e n c e , y i e l d i n g :
- 2 / 1 = •
e g
2.2.2. A f t foil
2{h eF'
C„ s i n / i . ( 7 )
I n analogy w i t h the f o r m u l a t i o n f o r the f r o n t f o i l the l i f t o f the a f t f o i l has been w r i t t e n as:
, / dCj^ dc^ \
c o n t a i n i n g the f o l l o w i n g c o m p o n e n t s :
— tlie relative velocity of die water over tlie foil
T h i s v e l o c i t y is composed o f a steady p a r t due t o the e c i u i l i b r i u m v e l o c i t y o f the c r a f t and a d y n a m i c p a r t caused b y p i t c h i n g and surging m o r i o n s , y i e l d i n g :
K ^ i - h ^ ^ Ó ( 9 ) — submerged area of tlie aft foil
Since the a f t f o i l is f u l l y submerged the submerged area o f the f l a t f o i l is constant. Situations i n w h i c h the a f t f o i l c o u l d emerge f r o m the water are n o t t a k e n i n t o a c c o u n t . So
'^A " ^A ^A ( 1 0 )
a , — angle of attack
T h e f o r m u l a t i o n f o r the angle o f a t t a c k o f the a f t f o i l is similar to the f o r m u l a t i o n f o r the f r o n t f o i l . T h e angle o f attack o f the a f t f o i l is i n f l u e n c e d h o w ever by tlie d o w n w a s l i o f the f r o n t f o i l . This d o w n -wash b e h i n d the f r o n t f o i l is p r o p o r t i o n a l t o the angle
o f attack o f that f o i l and so the f o l l o w i n g f o r m u l a t i o n f o r the r e d u c t i o n o f the angle o f attack o f the a f t f o i l has been used:
X a v ( 1 1 )
A n a d d i t i o n a l c o m p l i c a t i o n has arisen as a conse-quence o f t h e unsteadiness o f the f l o w : the loading o f the f r o n t f o i l is a f u n c t i o n o f t i m e and so is the d o w n -wash experienced b y the a f t f o i l . T h e d o w n w a s h ex-perienced b y the a f t f o i l at the t i m e t is caused b y the vortices shed o f f b y the f r o n t f o i l o n the t i m e t ~ At. This " d o w n w a s h — t i m e l a g " is equal t o the t r a n s p o r t v e l o c i t y o f the vortices divided by the distance between the t w o f o i l s , i.e.:
A , ( 1 2 )
This poses no special p r o b l e m s t o the s o l u t i o n o f the equations, because these are w r i t t e n i n the t i m e d o -m a i n . The d o w n w a s h n o w beco-mes:
da, da \ dt
and the angle o f a t t a c k :
( a + e + a NL> y y and so the f l u c t u a t i n g p a r t : 1 0 .+ -^A I n w h i c h | a^ ^A da. de^ da dup. At ( 1 3 ) ( 1 4 ) ( 1 5 )
A m = a pit — At) w h i c h value is
dt " 7 --"^
k n o w n i n a t i m e d o m a i n s o l u t i o n .
h^ — submergence of the aft foil
The submersion o f the a f t f o i l is the submersion i n the e q u i l i b r i u m c o n d i t i o n plus a f l u c t u a t i n g part due to the heave and the p i t c h m o t i o n , i.e.:
'^A=KA z + 1 ( 1 6 )
2.3. Drag
2.3.1. Front foil
The drag o f the f o i l is basically composed o f three parts: the p r o f i l e drag, i n c l u d i n g the f r i c t i o n and pres-sure drag, the i n d u c e d drag, caused by the generated l i f t , and the wave drag, caused by the presence o f the free surface. Described i n the usual way the drag reads:
Dp = V2pV],Sp^(C^^+C^.+ Cj,J ( 1 7 )
In this f o r m u l a Vp corresponds to the similar expres-sion i n the l i f t f o r m u l a t i o n , see ( 2 ) .
^FD ^ wetted surface of the foil
This is the m o m e n t a r y submerged area o f the f o i l , i.e.:
SFD (18)
C^^ — profile drag coefficient
Values o f C^^ f o r m a n y f o i l sections are Icnown i n the l i t e r a t u r e . I t appeared t h a t the p r o f i l e drag is larg-ely independent o f the parameters under consideration in this study. T h e r e f o r e C ^ ^ has been considered a constant t h r o u g h o u t the c o m p u t a t i o n .
Cjjj — induced drag coefficient
Wings o f finite span generate t i p vortices w h i c h i n duce a vertical v e l o c i t y over the w i n g , directed d o w n -wards, b y w h i c h the e f f e c t i v e angle o f attack is re-duced and the l i f t gets, a c o m p o n e n t i n the d i r e c t i o n o f the u n d i s t u r b e d f l o w : the induced drag. The mag-n i t u d e o f this drag is d i r e c t l y p r o p o r t i o mag-n a l t o the l i f t generated by the w i n g , leading t o the f o l l o w i n g w e l l k n o w n f o r m u l a t i o n :
(19)
For an e l l i p t i c a l h f t d i s t r i b u t i o n this c o e f f i c i e n t can easily be calculated.
Cjj^^ — wave resistance coefficient
A deeply submerged flat f o i l has the same charac-teristics as an aeroplane w i n g . I n the v i c i n i t y o f a free surface these characteristics however change consider-ably. Due to the pressure field a r o u n d the m o v i n g f o i l waves are generated, i n w h i c h energy is dissipated; so even i n the t w o d i m e n s i o n a l case, i.e. a w i n g o f i n f i n i t e span, a f o i l i n the v i c i n i t y o f the free surface experien-ces a drag f o r c e : the wave resistance. Usually this wave resistance is described as being p r o p o r t i o n a l t o the l i f t squared, see
A C D u -dC Dw
dcl
ACl (19) 2.3.2. A f t foUAnalogous to the drag o f the f r o n t f o i l the drag o f the a f t f o i l c o u l d be w r i t t e n as:
A P A A \ Do ^ ^
( 2 0 ) c o n t a i n i n g the same components as described f o r the f r o n t f o i l . The i n f l u e n c e o f the downwash f r o m the
f r o n t f o i l , on the induced drag-term has Ijeen neglec-ted, because, as f o l l o w s f r o m l i t e r a t u r e , in the range o f angles o f attack here to be expected the v a r i a t i o n i n the i n d u c e d drag c o e f f i c i e n t is very small.
2.3.3. Strut resistance
The strut resistance o f the t w o f o i l c o n f i g u r a t i o n s has been taken as the p r o f i l e resistance o f the m o m e n -tary submerged strut area, y i e l d i n g f o r the s t r u t o f t h e f r o n t f o i l :
Dsj,f, = Vip V] {hp - LSSTR^ * s i n / j , , ) +
+ {hp - LSSTR^ * sinjLi^) C^p ( 2 1 ) F o r the a f t f o i l :
DsTA='/^pVlhCACDo ( 2 2 )
2.3.4. Vertical viscous drag
The drag o f the f o i l s due t o the relative vertical v e l o c i t y between the water and the f o i l s has been acc o u n t e d f o r b y adding an extra " l i f t " f o r acc e . The f o l -l o w i n g expressions have been used f o r the f r o n t f o i -l :
f^VISCF = 'Z^^^^" + I f 0 ) 2 2 C ^ COigllp{h^p-z - \pO)C^y
A f t f o i l :
( 2 3 )
( 2 4 )
2.3.S. Added mass
O n l y the added mass o f the f o i l s f o r the vertical m o t i o n s has been considered.
Usually the dependence on the oscillating f r e q u e n c y o f the added mass poses special problems t o the solu-t i o n mesolu-thods i n solu-the solu-t i m e d o m a i n . I n solu-this case however these d i f f i c u l t i e s have been overcome b y considering regular sinusoidal waves o n l y and neglecting the i n -fluence o f possible non-linear response. M o r e over the dependence o f added mass o f f u l l y submerged flat plates o n oscillation f r e q u e n c y can be neglected and since the f r o n t f o i l w i l l be replaced b y a flat f u l l y submerged f o i l at the equivalent d e p t h this c o u l d be assumed f o r b o t h f o i l s .
The added mass o f flie f o i l s has been assumed t o be equal t o the v o l u m e o f the elipsoid w h i c h has the span and the c h o r d o f the f o i l as its m a i n axes. T h i s v o l u m e equals:
m = y •na'-b
f o r an elliptical p l a n f o r m and high aspect r a f l o ' s . F o r d i f f e r e n t aspect ratio's the c o r r e c t i o n f a c t o r o f M u n k w i l l be used, y i e l d i n g :
' " ' = ^ . P ^ ' ^ F , ^ C ' ^ , M ( 2 5 ) the forces due l o added mass h e r e w i t h become:
^ , „ F = / ^ , 7 | 2 C 2 c o t g p ^ ( / , ^ ^ - z ^ V Ö )
Cz +lpÖ) . ( 2 6 )
f^,nA = K . p \ S ^ C l i ï - \ ^ h (21)
2. 4. Wave excited forces
Consider a wave t r a i n o f regular sinusoidal waves m o v i n g in the d i r e c t i o n o f the negative x-axis. T h e v e l o c i t y p o t e n t i a l o f such a wave o n i n f i n i t e deep water is:
<P = e"''""sin(/cv - .ojt)
The vertical and h o r i z o n t a l o r b i t a l velocities are
s i l l (/cx — wt)
( 2 8 )
( 2 9 )
(30)
( 3 1 ) and the wave p r o f i l e is described b y :
f = = f „ c o s ( / c x - w / )
g dt
Related to the b o d y - f i x e d coordinate system these expressions had to be t r a n s f o r m e d w i t h :
X = x + Vdt (32)
T l i e integral i n the expression is f o r m a l l y necessary because o f the i n c o r p o r a t i o n o f a surge e q u a t i o n . A l -t h o u g h -the s o l u -t i o n i n -the -t i m e d o m a i n made incor-p o r a t i o n o f the integral incor-possible i t has nevertheless been o m i t t e d because o f the f a c t t h a t the i n f l u e n c e b o t h o n the f r e q u e n c y o f encounter and o n the p a t h travelled is small. Expression ( 3 2 ) then becomes:
x ' = x + Vt (33)
and h e r e w i t h the t r a n s f o r m e d expressions ( 2 9 ) , ( 3 0 ) and ( 3 1 ) : K " * f a ' - ' s i n ( c o ^ r + kx) f = f ^ c o s ( c o ^ / + A'x) w i t h : co=kV±oj=k(V± C) ( 3 4 ) (35) ( 3 6 ) ( 3 7 ) I n the equations the upper sign refers to the t i o n w i t h liead waves and the l o w e r sign to the situa- situa-t i o n w i situa-t h f o l l o w i n g waves. This c o n v e n situa-t i o n w i l l be used f u r t h e r on t h r o u g h o u t this r e p o r t . Due to the f a c t
t h a t i n the s i t u a t i o n w i t h f o l l o w i n g waves the speed o f the h y d r o f o i l boat exceeds the speed o f advance o f the waves, at least i n the range investigated, the f r e -quency o f encounter remains positive.
I n the equations ( 3 4 ) t o ( 3 6 ) the value x = 1 ^ refers to the f r o n t f o i l and x = — 1^ t o the a f t f o i l .
The h o r i z o n t a l and vertical o r b i t a l velocities cause relative v e l o c i t y changes and variations i n the angle o f attack over b o t h f o i l s . As apparent f r o m the f o r m u l a ' s ( 3 4 ) and ( 3 5 ) these velocities are a f u n c t i o n o f the d e p t h and so the above m e n t i o n e d variations vary over the span o f the dihedral f o i l . F o r the present s t u d y this was considered t o be an unnecessary c o m -p h c a t i o n . T h e r e f o r e the average value o f t h e e"'^' t e r m over the span o f the dihedral f o i l has been used, i.e.
_ ƒ k(h,f-z-\^e) -kz dz 1 ll^P^Z^lpt h,p~z-\pt ( 3 8 ) T h e i n c o r p o r a t i o n o f the above m e n t i o n e d o r b i t a l velocitites and wave p r o f i l e i n the d i f f e r e n t parts o f the f o r m u l a t i o n s o f the l i f t - and drag forces y i e l d s the wave excited forces o n the f o i l s , i.e.:
Front foil: I n e q u a t i o n ( 3 ) f o r .S*^: + 2Cp coig^p^^ cos{Lo^t + k\p) I n e q u a t i o n ( 5 ) f o r a ^ : ± f ^ c j x s i n ( w ^ r + / c l / , ) I n e q u a t i o n ( 6 ) f o r lip.: + cos{io^t + k\p) I n e q u a t i o n ( 7 ) f o r : + f „ CO%iLO^t + k\p) Cp sm lip I n e q u a t i o n ( 2 ) f o r Vp : + f ^ c o x c o s ( w ^ / - t - f c l ^ ) I n e q u a t i o n ( 2 3 ) ïov Lyj^^p : ± f ^ c o x s i n ( G j / + / ; l ^ , ) Aft foil: I n e q u a t i o n (1 5) f o r : ± r ^ o j c - ^ M s i n ( c o ^ / - A l ^ ) ( 3 9 ) ( 4 0 ) ( 4 1 ) ( 4 2 ) ( 4 3 ) ( 4 4 ) (45)
I n e q u a t i o n ( 1 6 ) f o r : I n e q u a t i o n ( 9 ) f o r K j : T f e - * " c o s ( o j ^ a ^ e A ^ I n e q u a t i o n ( 2 4 ) f o r L yj^f^^ '• ± L o j e - * ' ' sin(Gj / - ,^1 , ) (46) (47) ( 4 8 )
2.5. Corrections for unsteadiness
Tlie c o n t i n u o u s variation o f b o t h relative water speed and angle o f attack due to the waves and the response o f the c r a f t causes the f l o w a r o u n d the f o i l t o be unsteady. U n t i l l n o w the derived equations f o r the l i f t and drag o f the f o i l s refer o n l y to steady o r quasi-steady c o n d i t i o n s and i t is w e l l k n o w n that b o t h forces may d i f f e r considerably due t o unsteadiness. A w i n g o f i n f i n i t e span generates i m m e d i a t e l y a f t e r i t commences t o move, due to b o t h the viscosity o f the f l u i d and the sharp trailing edge o f the w i n g , a vorte.x at the t r a i l i n g edge; the starting v o r t e x . This v o r t e x equals i n strength the c i r c u l a t i o n a r o u n d the w i n g n o w arisen; the b o u n d v o r t e x . The starting vor-tex and the b o u n d vorvor-tex c o m b i n e t o a pair o f vortices w i t h an ever increasing m u t u a l distance; the starting v o r t e x remaining at its place o f o r i g i n a t i o n w h i l e the b o u n d v o r t e x moves away w i t h the w i n g . When the m o t i o n o f the w i n g is constant w i t h respect to its v e l o c i t y and angle o f attack, the f l o w a r o u n d the w i n g becomes steady and the i n f l u e n c e o f the starting v o r t e x is negligible, due t o its large distance t o the w i n g .
I n the case o f an oscillating f o i l however b o t h angle o f attack and v e l o c i t y change c o n t i n u a l l y and so does the c i r c u l a t i o n around the f o i l , causing a sheet o f " s t a r t i n g v o r t i c e s " t o leave the f o i l . The i n f l u e n c e o f
-P
these vortices o n the f l o w around the w i n g can no longer be neglected. I n a three dimensional approach the p r o b l e m is even more c o m p l i c a t e d t h r o u g h the presence o f a pair o f c o n t i n u o u s l y changing t i p vor-tices.
F o r the t w o dimensional p r o b l e m . V o n K a r m a n and Sears [ 1 1 ] f o u n d a c o r r e c t i o n m e t h o d t o be applied o n the steady l i f t t o account f o r the e f f e c t s o f u n -steadiness. T h e i r m e t h o d is basically based on three assumptions, i.e.;
— The f l o w is t w o dimensional, w h i c h implies that atleast the aspect ratio's must be large.
— The vertical oscillatory m o t i o n s are s m a l l ; this i m -plies that the wake o f vortices lies o n the axis t h r o u g h the wings centre.
— T h i n a i r f o i l theory is applicable.
T h e y f o u n d that i f the relative v e l o c i t y between the w i n g and the f l u i d can be expressed i n the f o l l o w i n g f o r m ;
/ " \ • ,
V^^= Bq+2 i: cos nB f / e ' " ' ( 4 9 ) \ 1
t h e n the l i f t per u n i t o f length equals;
: ( 5 w i t h : C coC 2v lOiC
"27
ICOC"2v
+ K , la>c17
(50) (51)Kq,K.^ M o d i f i e d Bessel f u n c t i o n o f the zero and
f i r s t order and o f the second k i n d .
Ogilvie [ 1 6 ] used this c o r r e c t i o n m e t h o d , f o r the c o r r e c t i o n o f the quasi-steady l i f t calculations. His results have been used i n this s t u d y .
First i t is essential t o consider w h i c h c o m p o n e n t s o f the l i f t can be corrected w i t h this m e t h o d . Consider therefore the f o r m u l a t i o n o f the l i f t o f the f r o n t f o i l ;
/ 1 /
Lp = Vip V j X 2CpCotgHp{li^p-z-\p0n,cos{i^J+}c\p)) / - I I - / / I I I / dCp ^«'^ da sin(co / - l - A r l , I V -dC, + - 7 r X { - z - U e + L c o s ( c o / + / : ! „ ) } dlx -V-dCi^ I 2( - z - 1^ 0+ f „ c o s ( c o / + A : l ^ ) dA smM/ V I -^ {2CpCo\giXp(,h-^p-z~\pQ)±l-^zos{u,J-^]i\i-^Cp(zn&)} (52) in w h i c h the d i f f e r e n t c o n t r i b u t i o n s to the l i f t varia-tions are;I ; the v a r i a t i o n i n the submerged area o f the f o i l I I ; stationary l i f t ; compensates the w e i g h t o f the
c r a f t
I I I ; v a r i a t i o n o f the angle o f attack caused by m o -tions and waves
I V ; v a r i a t i o n o f the d e p t h o f the f o i l V ; v a r i a t i o n o f the aspect-ratio o f the f o i l V I ; added mass forces.
The variations i n l i f t described by the c o m p o n e n t s I and V , c.e. the influence o f a rapid change in b o t h
the area and the aspect r a t i o o f a w i n g i n the v i c i n i t y o f a free surface, are u n k n o w n i n the c o m m o n aero-and i i y d r o m e c l i a n i c s . These effects are certainly n o t accounted f o r w i t h the c o r r e c t i o n m e t h o d f o r un-steadiness used here.
A l s o the other t e r m describing the i n f l u e n c e o f the f r e e surface, components I V , w i l l n o t be c o r r e c t e d . K a p l a n showed the c o m p l e x i t y o f the c a l c u l a t i o n o f the l i f t o f a t w o dimensional f u l l y submerged w i n g i n the v i c i n i t y o f the free surface i n an unsteady f l o w . A n ac-curate analysis o f these problems seemed u n p r a c t i c a l i n the scope o f this s t u d y , because i t was n o t l i k e l y to p r o d u c e results w h i c h c o u l d be i n c o r p o r a t e d i n the equations. So o n l y c o m p o n e n t I I I , i.e. the v a r i a t i o n i n l i f t due t o changing angle o f a t t a c k , w i l l be corrected using the m e t h o d derived by Ogilvie [ 1 6 ] . He divided t h i s c o m p o n e n t I I I i n t w o parts:
one independent o f the f r e q u e n c y o f encounter and one dependent, i.e.
dC^ da dC, da 1, 0 y. r ^ c o x s i n C c j ^ r + A ' l ^ ) ( 5 3 ) (54)
and f o u n d as c o r r e c t i o n c o e f f i c i e n t s f o r these parts respectively: D E =
c(—\ + i'^^
, \2vj \Av ( 5 5 ) 2 ^CF 2 C 'CF + / V J 2 ~ (56)The i n f l u e n c e o f the wake behind the f r o n t f o i l i n unsteady f l o w on the l i f t o f the a f t f o i l has also been investigated by Ogilvie using the results o f K a r m a n and Sears. He f o u n d that this i n f l u e n c e is negligible m o s t l y because o f the rapid decrease o f the v o r t i c i t y i n the wake caused by the viscosity o f the f l u i d and that i t certainly is m u c h smaller than the error made by n o t t a k i n g i n t o account the e f f e c t s o f three d i m e n s i o n a l f l o w .
The i n f l u e n c e o f the unsteady f l o w around the f o i l o n the drag has been investigated by Schwanecke [ 9 ] et.al. He calculated and measured the induced drag and the p r o f i l e drag o f a t w o dimensional f o i l in a h a n n o n i c a l -l y osci-l-lating f -l o w . F r o m b o t h the ca-lcu-lations and the experiments he f o u n d :
— that the induced drag decreases as a f u n c t i o n o f t h e reduced f r e q u e n c y , the f o i l section and the charac-ter o f the unsteady f l o w
— t h a t the p r o f i l e drag increased depending o n the va-r i a t i o n o f the angle o f attack and the f o i l section. F r o m this theoretical investigations i t f u r t h e r ap-peared t h a t v a r i a t i o n o f the speed o f advance o f the f o i l caused an a d d i t i o n a l r e d u c t i o n i n the i n d u c e d drag. Overall the e f f e c t o f unsteadiness on the resis-tance consists p a r t l y o f an increase and p a r t l y o f a decrease o f the quasi-steady derived drag. T h e r e f o r e i t has been assumed that the unsteadiness o f the f l o w has no overall e f f e c t on the drag o f the f o i l s .
2.6. Hydrodynamic derivatives
I n the equations describing the l i f t and drag o f the f o i l s as f u n c t i o n o f a n u m b e r o f parameters some dogr 1 ^ D \ , . - - - ' - " " ' ' ' ' ' ' ' \ O 0.6 1 1 ^ head s o e s V 0 4 /--. O.G 10 10 following Boas / dogr 1.0
h y d r o d y n a m i c derivatives appear w h i c l i needed t o be d e t e r m i n e d .
A reliable m e t h o d f o r d e t e r m i n i n g these c o e f f i c i e n t s f o r a given f o i l section and c o n f i g u r a t i o n is b y means o f an e x p e r i m e n t . I n most cases however experimen-tal results w i l l n o t be available and t h e r e f o r e some em-p i r i c a l and some theoretical values f o r these deriva-tives w i l l be used i n this study.
For the l i f t calculation o f a deeply submerged f o i l use has been made o f the l i f t i n g line theory by Prandti and Lancaster:
A f o i l o f finite span is m a t h e m a t i c a l l y described as a n u m b e r o f b o u n d vortices and t r a i l i n g vortices giving the well k n o w n "horse-shoe" t y p e v o r t e x representa-t i o n ; representa-thereby representa-the c i r c u l a representa-t i o n d i s representa-t r i b u representa-t i o n over representa-the span can be described b u t n o t over the c h o r d .
For the foils under consideration v / i t h their large as-pect ratio's this m a t h e m a t i c a l representation o f the f o i l can vep,' well be used.
Tlie t r a i l i n g vortices induce vertical velocities o n tlie f o i l , causmg a decrease o f the e f f e c t i v e angle o f attack and a r o t a t i o n o f the cross force w i t h respect to the d i r e c t i o n o f the u n d i s t u r b e d fiow: w h i c h causes the i n d u c e d resistance.
The w i n g characteristics are d e t e r m i n e d b y calculating the induced velocities and the e f f e c t i v e angle o f attack o f the f o i l and the resulting l i f t and i n d u c e d drag f o r each f o i l section. I n t e g r a t i o n over the span o f the f o i l yields the l i f t en induced drag o f the f o i l .
One necessary assumption f o r f a c i U t a t i n g the c o m p u t a -tions is the e l l i p t i c a l loading o f the f o i l over the span. This assumption is here j u s t i f i e d by the relatively h i g h aspect ratio o f the f o i l s . By d o i n g so the l i f t and i n -duced drag c o e f f i c i e n t o f the deeply submerged f o i l s become: 2i^A
C,
A+3 +2-Cor—
'A (57) (58) When the submergence o f the f o i l gradually de-creases and the f o i l approaches the free surface, the characteristics o f the f o i l change considerably. W i t h e x c e p t i o n o f the lowest velocities waves are generated, caused by the pressure f i e l d a r o u n d the f o i l . I n these waves energy is dissipated w h i c h is derived f r o m the f o i l . So even i n the t w o dimensional s i t u a t i o n the f o i l experiences a drag due to l i f t : i.e. induced drag.As an a p p r o x i m a t i o n the b o u n d a r y c o n d i t i o n on the free surface can be satisfied by assuming a second identical vortex system as the same distance above the free surface. The vortices are o f the same strength a«c/ d i r e c t i o n (this i n c o n t r a d i c t i o n w i t h the g r o u n d e f f e c t o f a w i n g ) .
This s i t u a t i o n can be compared w i t h what is, in the aerodynamics, k n o w n as the unstaggercdbiplane c o n -f i g u r a t i o n . As described here-fore the -free vortices i n the wake o f the f o i l induce velocities over the f o i l , causing d o w n w a s h and induced drag. T h i s k i n d o f i n t e r a c t i o n between the free t r a i l i n g vortices, caused by a n u i n b e r o f b o u n d vortices, and these b o u n d vor-tices, is usually called s e l f - i n d u c t i o n . I n the case o f t h e biplane c o n f i g u r a t i o n there also exists a m u t u a l inter-a c t i o n between the b o u n d vortices o f b o t h f o i l s . This k i n d o f i n t e r a c t i o n is called m u t u a l i n d u c t i o n .
T w o consequences o f this m u t u a l i n t e r a c t i o n are evident:
1. the b o u n d vortices o f the upper w i n g induce a d o w n w a s h c o m p o n e n t over the l o w e r , w h i c h has the opposite d i r e c t i o n to the u n d i s t u r b e d fiow causing: a. a decrease i n l i f t , due t o l o w e r v e l o c i t y
b. a cui-vature o f the fiow at the l o w e r f o i l causing a decrease o f e f f e c t i v e f o i l camber and angle o f a t t a c k .
2. the disturbance o f the fiow around the l o w e r w i n g , i.e. the f o i l , by the vortices o f the upper w i n g , i.e. the image, decreases p r o p o r t i o n a l t o the distance between the t w o wings squared. T h e r e f o r e the l i f t o f the f o i l is dependent o n the submergence o f the f o i l .
W i t h the aid o f the above described m o d e l W a d l i n and Christopher [ 1 7 ] d e t e r m i n e d a c o r r e c t i o n m e t h o d f o r the l i f t o f a f o i l i n the v i c i n i t y o f a f r e e surface. T h e y d e t e r m i n e d a t w o and three d i m e n s i o n a l correc-t i o n c o e f f i c i e n correc-t respeccorrec-tively. W i t h these the l i f t c o e f f i c i e n t becomes:
C,
2K2KjnAa A + 1+2KAI+T) w i t h : „ _ 4 / i ' 2 + 8 / i ' s i n a + 1 4 / i ' 2 + 8//'sina + 2 1 2 \ + w h' = h + /( — sina __4 0 . 0 5 + f / ( 5 9 ) ( 6 0 ) ( 6 1 ) ( 6 2 ) 4 4 cosa + 4 ( ^ + ('/2sina)2 >/2Cosa + ^ 4 4 ( 6 3 )4n 7 ( 2 / ; ' + 'Asina) + (Vicosa)^ + — 4 COSft j 2 ( ' / 2 C o s a ) 2 + ( 2 / i ' + > / 2 s i n a ) ' 1 ' + ( 2 / ) ' + ' / 2 s i n a ) 2
y2C0Sö + y ( 2 / / ' + l/2Sina)2 + (>/2COS«)2 + : ^ ^ 4
I n the case o f a high aspect r a t i o dihedral f o i l the c o r r e c t i o n varies considerably over the span o f t h e f o i l . T o avoid unnecessary c o m p l i c a t i o n s the c o r r e c t i o n f a c t o r at the centre o f the elliptical l i f t d i s t r i b u t i o n over the span has been used f o r the c o m p l e t e f o i l . The equivalent submergence can be w r i t t e n as:
- 2
(' 4
w h i c h f o r m u l a t i o n is valid as l o n g as the f o i l is n o t c o m p l e t e l y emerged o r submerged.
T h i s c o r r e c t i o n m e t h o d is rather cumbersome especial-l y w h e n i t has to be d i f f e r e n t i a t e d f o r instance w i t h respect t o angle o f attack or aspect r a t i o , as i t is ne-cessary to do in the c o r r e c t i o n m e t h o d f o r the e f f e c t s o f unsteadiness used. T h e r e f o r e the e m p i r i c a l correc-t i o n m e correc-t h o d as described b y ICaplan [ 7 ] has also been used. He f o u n d t h a t : (64) CL ^-CL^O - 0 . 4 2 2 e - 1 V C ) (6.5) i n w h i c h : = corrected H f t c o e f f i c i e n t = l i f t c o e f f i c i e n t w i t h o u t free surface e f f e c t s . The i n d u c e d resistance changes also due t o the pre-sence o f the free surface: here is energy dissipated i n the o f f - c o m i n g waves. This part o f the induced re-sistance c o u l d be called wave rere-sistance.
T o avoid c o n f u s i o n , the resistance e.xperienced by a f o i l o f i n f i n i t e span i n the v i c i n i t y o f the free surface w i l l be called wave resistance, the a d d i t i o n a l resistance experienced by a w i n g o f f i n i t e span i n the same situ-a t i o n w i l l be csitu-alled induced resistsitu-ance.
Keldisch Lawrentiei' and K o t c h i n f o u n d f o r the wave resistance the f o l l o w i n g expression:
C
in w h i c h :
F. = / — = m o d i f i e d Froude number
(66)
(67)
Diehl f o u n d l o r the induced resistance in the
v i c i n i t y o f the free surface: 3 3 / / A 3 / 2
2
C Di
33 / ( \ 3/2
H-VA
(68)B o t h f o r m u l a t i o n s have been used i n the c o m p u t a -tions.
The f o r m u l a t i o n derived b y Kaplan and Sutherland f o r the downwash has been used. They d e t e r m i n e d the d o w n w a s h behind a f o i l i n the v i c i n i t y o f the free sur-face related t o this sursur-face f i r s t , and considered there-f o r e this surthere-face t o be r i g i d .
I n analogy w i t h the ground e f f e c t o f a w i n g t h e y f o u n d : ^ / ; \ 3 / 2 - 36 33 3 3 ( ^ V ' ^ l C, (69) F r o m ( 6 9 ) i t can be c o n c l u d e d t h a t the d o w n w a s h b e h i n d a f o i l increases w i t h increasing submergence o f the f o i l . This was c o n f i r m e d b y m o d e l tests. The " d o w n w a s h " i n the o f f c o m i n g wave was then determ i n e d w i t h the aid o f the expression f o u n d b y K e l -disch L a w r e n t i e r and K o t c h i n f o r i n f i n i t e deep water under the assumption that the f l o w closely f o l l o v / s the wave p r o f i l e , y i e l d i n g :
V 2
C Lit
(70)
The c o m b i n a t i o n o f ( 6 9 ) and ( 7 0 ) yields the c o m plete d o w n w a s h : The dihedral o f the f r o n t f o i l c o m -plicates the d e t e r m i n a t i o n o f the downv/ash on the f l a t a f t f o i l . When the dihedral f o i l is replaced by a f l a t f o i l on an equivalent submergence M u t t r a y and Hurst [ 3 ] f o u n d that a c o r r e c t i o n f a c t o r o f 0,65 must be applied due t o the fact that the foils are n o t in the same plane. Due to b o t h the dihedral and the m u t u a l distance the c o r r e c t i o n f a c t o r f o r the c o n f i g u r a t i o n investigated s h o u l d be even considerably higher, b u t an exact value is n o t k n o w n .
2. 7. The equations of motion
W i t h the forestanding f o r m u l a t i o n s f o r the l i f t and drag o f the foils related to t h e i r m o t i o n s and the ex-c i t i n g forex-ces the equations o f m o t i o n s ex-can be derived as f o l l o w s .
2.7.1. Surge
mx + Lpa^p + L^, a.^^^ + / ) , , + - T = 0 ( 7 1 )
w i t h :
2.7.2 Heave
miz -(V+ X )Ó) - Lp -^L^+Dpa pp + D^ a
+ L viscF^ ^ viscA + h ^ ^ p + f^mA + = ° ^^"^ w i t h :
h ' = \v eight o f the h y d r o f o i l boat.
2.7.3. Pitch
f-F-^uF- LfCCppZ^,p + DpappX,,^p - DpZ^^^p ~ f-A •'^mA ~ f-A °^TA ^mA ~ ^^A^ TA ''^mA ~ ^A ^wA
^VISCF^mF^ f' VISCA^mA L(z,ë)^mF
w i t h : A-,„^...x-,„^^ - ' , „ f • ^ „ M
(73) the h o r i z o n t a l and vertical distance between the l i f t or drag forces and the centre o f gravity o f t h e h y d r o f o i l boat respectively, i.e.:
^mF = -^-fcose + {hpp - hp + h^) sine ( 7 4 ) (75) -mF =-^-fSin0 -{hpp~hp + h^)cosd ( 7 6 ) (77) - ^ m ^ = - ^ 4 = 0 5 0 + / ; r ^ s i n ö ^n,A = - i ^ s i n e - / ï ^ ^ C O S Ö 3. S o l u t i o n o f the equations o f m o t i o n
Due to the fact that i n the e x p e r i m e n t described here a f t e r , o n l y measurements w i t h constant speed o f advance were possible, the surge m o t i o n w i l l n o t be taken i n t o account when solving the equations. The i n c o r p o r a t i o n o f the surge m o t i o n however poses no special d i f f i c u l t i e s t o the s o l u t i o n .
I n general the s o l u t i o n o f equations such as the equa-t i o n s o f m o equa-t i o n s here derived, is c o m p l i c a equa-t e d b y equa-the badly n o n linear character o f the equations; describing the l i f t and the drag o f the f o i l s .
The c o m m o n l y used description o f the m o t i o n s i n the f r e q u e n c y d o m a i n asks f o r a linearisation o f the equa-tions around the e q u i l i b r i u m s i t u a t i o n i n the undis-t u r b e d f l i g h undis-t . This m e undis-t h o d was used, among oundis-thers, b y W e i n b l u m [ 1 5 ] and Ogilvie [ 1 6 ] . Due to the linearisation o f the equations they were n o t able t o calculate the s t a t i o n a i y sinkage o f the c r a f t i n waves and their results f o r f o l l o w i n g waves were erroneous. When describing the equations i n the t i m e d o m a i n these may be l e f t i n their non-linear character. When d o i n g so basically t w o methods exist. First the i m -pulse-response technics as first described by C u m m i n s [ 2 3 ] can be used, by w h i c h it is possible to have a r a n d o m exciting-force signal. F o r this study use c o u l d be made o f a simpler m e t h o d , since the e x c i t i n g force signal is f r o m a regular character w i t h k n o w n f r e q u e n -cy and a m p l i t u d e ; so no problems ocCur i n dealing w i t h
the f r e q u e n c y dependent added mass forces. The equations have been solved as a pair o f c o u p l e d d i f f e r -ential equations. The procedure used f o r solving the equations was based on the Runge-Kutta m e t h o d , enabling to solve any n u m b e r o f coupled first order d i f f e r e n t i a l equations. The equations derived had t o be r e w r i t t e n as a n u m b e r o f f i r s t order equations. This has been done by adding t w o equations o f the f i r s t order, so t h a t the equations in their original f o r m :
/ j ( ? ; z, z, é' 9, 0 ) = 0 ƒ2(0, O, ö, z, z, z ) = O were t r a n s f o r m e d t o : (78) q =z q =f fi7:. z, Ó, 0 ) /• = 0 r =f*{Ó,0.z,z) (79)
w h i c h are f o u r first order coupled d i f f e r e n t i a l equa-tions. K n o w i n g the s o l u t i o n o n the equations are simultaneously solved f o r the small time decrement A f , y i e l d i n g the s o l u t i o n o n + At.
A c o m p u t e r program has been developed b y w h i c h the m o t i o n s o f the h y d r o f o i l c r a f t can be calculated w i t h the use o f t h e described m e t h o d .
The p r o g r a m calculates the f o l l o w i n g results:
1. the s t a b i l i t y o f the h y d r o f o i l c o n f i g u r a t i o n dated. Should this check be negative the program ends. 2. the e q u i l i b r i u m s i t u a t i o n f o r the speed o f advance
and the c o n f i g u r a t i o n dated i n the u n d i s t u r b e d f l i g h t .
3. the heave and p i t c h m o t i o n due t o i n c o m i n g regular waves (head o f f o l l o w i n g ) w i t h given f r e q u e n c y and a m p l i t u d e . By means o f a n u m b e r o f switches i t is possible to use the l i f t calculation according to the m e t h o d o f K a p l a n or the m e t h o d o f W a d l i n and Christopher and t o correct f o r the e f f e c t s o f u n -steadiness or n o t .
The o u t p u t o f the program gives f o r each subse-q u e n t time step the heave and p i t c h a m p l i t u d e s and velocities, relative water v e l o c i t y f o r e and a f t , e f f e c t i v e angle o f attack f o r e and a f t , submergence f o r e and a f t and the l i f t c o e f f i c i e n t o f the f o r e and a f t f o i l .
4. E x p e r i m e n t
h i order to check the results o f the calculations i t was decided t o carry out a m o d e l test.
I t has been t h o u g h t to be especially i m p o r t a n t to check the results i n the s i t u a t i o n w i t h f o l l o w i n g waves, because most calculation methods lacked s u f f i c i e n t accuracy i n that s i t u a t i o n .
4.1. Model
The m o d e l used has been derived f r o m the Supra-mar PT 20 h y d r o f o i l boat series. The main particulars o f the m o d e l were: length between t h e f o i l s Lp + 1500 m m dihedral f r o n t f o i l n 3 0 degr. p r o f i l e section f o i l s circular 1/c f o i l s 0.06 chord f o i l s c 80 m m p r o f i l e section struts N A C A 0 0 1 0 design speed K 4 . 0 m/s weight W 25.4 k g l o n g i t u d i n a l m o m e n t o f inertia scale 3.57 kgm^-10
The geometry o f the m o d e l is given i n Figure 6. The m o d e l has been constructed o f a l u m i n i u m alloy ST 5 1 . Great care has been given t o the c o n s t r u c t i o n o f the foils w i t h respect t o section u n i f o n n i t y , con-stant angle o f attack over the span and r i g i d i t y . The angle o f attack o f the f r o n t f o i l c o u l d be adjusted t o compensate f o i ' some t r i m due t o unaccuracy o f the c o n s t r u c t i o n o f t h e m o d e l .
4.2. Expehmental set-up
The tests have been carried o u t i n the t a n k o f the Ship H y d r o m e c h a n i c s L a b o r a t o r y o f the D e l f t Univer-s i t y o f T e c h n o l o g y . The dimenUniver-sionUniver-s o f the tank are: l o n g : 150 m , w i d t h : 4.5 m and d e p t h : 3.5 meters. The m o d e l has been connected t o the t o w i n g carriage by means o f rods i n l o w f r i c t i o n ball-bearings, w h i c h a l l o w e d the m o d e l t o m o v e f r e e l y i n the vertical direc-t i o n , i.e. heave and p i direc-t c h , b u direc-t n o direc-t l o n g i direc-t u d i n a l l y , i.e. surge. The e f f e c t o f the surging m o t i o n o n the ver-t i c a l m o ver-t i o n s ver-t h e r e f o r e has n o ver-t been invesver-tigaver-ted. The m o t i o n s o f the m o d e l have been measured w i t h " w i r e -over p o t e n t i o m e t e r " transducers. These were necessary because o f the large vertical m o t i o n s . T h e waves have been generated at one end o f the t a n k . The waves have been measured at a k n o w n distance i n f r o n t o f the m o d e l .
The e x c i t i n g forces o n b o t h the f r o n t and the a f t f o i l have been measured w i t h strain gauge d y n a m o m e t e r s . For these measurements i t was possible t o fix the m o d e l at any given height corresponding t o the e q u i -l i b r i u m c o n d i t i o n o f the c r a f t at the given speed in s m o o t h water.
A l l signals ha\'e been registered simultaneously o n an U . V . recorder. The f o r c e signals have been f i l t e r e d w i t h a l o w pass f i l t e r t o eliminate the e f f e c t o f high f r e -q u e n c y v i b r a t i o n s o f the carriage.
F r o m these recordings tlie a n i ] 5 l i t u d e s and the phase angles w i t h respect to the i n c o m i n g wave have been d e t e r m i n e d at the centre o f gravity o f the m o d e l .
4.3. Measuring program
T w o series o f tests have been carried o u t : one w i t h head waves and one w i l h f o l l o w i n g waves. Each series consisted o f wave force measurements on the restrain-ed m o d e l and m o t i o n measurements i n 10 d i f f e r e n t wa\e lengths, varying f r o m \JLl\ - 0.72 to \/L/'\ =
1.15 and three d i f f e r e n t wave heights to have a check on the l i n e a r i t y o f the system. The speed o f the m o -del has been 4.0 m/s.
A l i m i t e d n u m b e r o f tests was repeated w i t h a speed o f 3.5 m/s and 4.5 m/s to investigate the i n f l u e n c e o f the speed on the m o t i o n s .
5. Results
5.7. Experime)Ual results
The results o f the e x p e r i m e n t have been p l o t t e d i n Figures 7 to 8 f o r head waves and i n FigLires 9 t o 10
f o r f o l l o w i n g waves. O n l y the results o f the m o t i o n measurements are presented here.
The results are d i r e c t l y derived f r o m the U . V . recordings. The heave a m p l i t u d e has been made d i m e n s i o n -less by d i v i d i n g i t by the wave a m p l i t u d e and the p i t c h angle b y d i v i d i n g it by the m a x i m u m wave slope k'i^. It appeared t o be very d i f f i c u l t to d e t e r m i n e the stationary sinkage o f the c r a f t f r o m the U . V . recor-dings w i t h great accuracy. This is due to the short d u r a t i o n o f the measurement r u n and small deviations in the speed o f advance.
These results should be considered more i n a qualita-tive way than q u a n t i t a t i v e l y .
I n general the measurements have r e p r o d u c e d w e l l and the results show good consistency.
5.2. Computational results
C o m p u t a t i o n s o f the m o t i o n s o f the h y d r o f o i l c r a f t i n waves have been made w i t h the aid o f the f o r e - m e n t i o n e d c o m p u t e r p r o g r a m . T h e wave length range has been chosen slightly larger than the range used d u r i n g the e x p e r i m e n t , i.e. 0.5 < \/L/\ < 1.3.
O e x p e r i m e n t . O e x p e r i m e n t — calculation — calculation
1.5 1.0 0 5 08 1.0 1.2 O e x p e r i m e n t — calculation 90 90 08 1.0 12
\/[7x —
1.5 1.0 0.5 O e x p e r i m e n t — c a l c u l a t i o n Q8 1.0 1.2 Vüx —• 08 1.0 12 \ / l aFigurp 11. Heave iransfer function for different speed of Figure 12. Heave transfer function for different speed of ad-vance. Head waves. vance. Following waves.
w i t h c o r r e c t i o n f o r unsteadiness w i t h o u t c o r r e c t i o n f o r unsteadiness - 0 . 0 2 - O . 0 1 1
II
f O L L O W l l i a S E A S , / -•—. - , / -
"-^^— HFAr> S E A S 1,0 2/1ao
X (»)
6,0Figure 13. Stationaiy sinkage component of the heave mo-tion.
The. calculations have been made b o t h f o r the s i t u a t i o n w i t h head and w i t h f o l l o w i n g waves.
The results are p l o t t e d i n Figures 7 t o 10.
Check has been made on the accuracy o f the results w i t h respect to n u m b e r o f t i m e inten'als per period used and the d u r a t i o n o f the r u n . These results have been made non-dimensional i n the same w a y as the e-x-p e r i m e n t a l results.
6. Discussion o f the results
F r o m the experimental results it can be concluded t h a t the response o f the h y d r o f o i l c r a f t i n liead waves is moderate.
In the f r e q u e n c y range tested there is n o tendency to resonance response f o r b o t h the heave and the p i t c h m o t i o n . The stationary sinkage o f the c r a f t was evident and increased w i t h increasing wave length. I n none o f the tests carried out there v/as a real danger f o r the c r a f t t o crash.
The f r e q u e n c y o f encounter o f the i n c o m i n g waves and consequently o f the m o t i o n s is high. The m o t i o n
A X = 2.25
m H E A D S li A S X C r r •- • •- ' c eo 180 270 3G0B
\ -
1.65
M HEAD SEAS SF. 90 ICO 360OJgxt
X - 3.00
M FOLLOWING SEAS 360Figure 14. Variation in time of the lift of the fronl and aft foU during one wave period.
amplitudes are relatively small but the accelerations at the ships ends are h i g h , o f t e n i n excess o f 0.6 g.
The reaction o f tlie c r a f t to the waves is f a v o u r a b l e : the c r a f t anticipates the disturbance.
In following waves the s i t u a t i o n is q u i t e d i f f e r e n t . The heave response o f the c r a f t is large especially f o r the longer waves, i.e. the lower f r e q u e n c y o f encounter. Due t o the high speed the f r e q u e n c y o f encounter o f the waves remains positive but is m u c h l o w e r than i n head waves. The stationary sinkage o f the c r a f t i n -creases w i t h the wave length and so does the heave
response so there is a real risk o f a crash o f the c r a f t . This actually hapixmed a f e w times d u r i n g the tests. The reaction o f the c r a f t on the waves is i m f a v o u r -able: there is a phase lag w i t h respect to the distur-bance. The speed o f advance has a great i n f l u e n c e on the m o t i o n s : the l o w e r speed o f 3.5 m/s gave m u c h higher response especially f o r the longer waves, i.e. the lower f r e q u e n c y o f encounter.
m o t i o n amplitudes on the wave height is shown de-rived f r o m the e x p e r i m e n t a l results. F r o m these i t is evident that the response o f the h y d r o f o i l boat can be regarded as reasonable linear w i t h respect t o the wave h e i g h t , considering the possible inaccuracy o f the measurements. This could lead t o the assumption t h a t t h e system m a y be considered as a linear system, b u t t l i i s leads to erroneous c o m p u t a t i o n a l results as discus-sed before and shown by K a p l a n , W e i n b h u n and others.
I n the Figures representing the response f u n c t i o n o f t h e c r a f t i n regular waves one a m p l i t u d e f o r b o t h the experiments and the calculations has been used.
The results o f the c o m p u t a t i o n s w i t h the described c o m p u t e r p r o g r a m agi-ee f a i r l y w e l l w i t h the experi-m e n t a l results, as can be concluded f r o experi-m the Figures 7 t o 10. I n a qualitative way the results are i d e n t i c a l , w h i l e q u a n t i t a t i v e l y there still are some discrepancies. T h e calculations p l o t t e d here have been p e r f o n n e d w i t h the p r o g i a m i n c l u d i n g the corrections f o r u n -steadiness and d o w n w a s h timelag.
T h e e f f e c t o f the c o r r e c t i o n f o r the unsteadiness has been investigated by p e r f o r m i n g the calculations f o r t h e same situations w i t h and w i t h o u t this c o r r e c t i o n . F r o m the calculations i t appears that this i n f l u e n c e o n the a m p l i t u d e o f m o t i o n is small. However the ef-f e c t o n the phase o ef-f the m o t i o n s w i t h respect t o the wave is more evident and the c o m p u t a t i o n s w i t h c o r r e c t i o n agree better w i t h . t h e experiments t h a n the calculations v / i t h o u t . The small i n f l u e n c e is remark-able considering a possible r e d u c t i o n i n the forces f r o m about 40%. T h i s observation agrees w i t h the results f o u n d b y Ogilvie [ 1 6 ] w h o applied the correc-t i o n o n his secorrec-t o f lineair equacorrec-tions. He ascribed correc-the small i n f l u e n c e t o the fact that the d o m i n a t i n g forces i n head waves, i n w h i c h s i t u a t i o n the c o r r e c t i o n is m o s t i m p o r t a n t , are a f f e c t e d equally by the c o r r e c t i o n . There is an e f f e c t on the stationai-y sinkage o f the c r a f t , b u t this can be explained by the f a c t t h a t the most i m p o r t a n t non-lineair term o f the equations, b y w h i c h this sinkage is caused, is restricted i n its i n f l u e n -ce o n the t o t a l result.
The stationary' sinkage has an i m p o r t a n t i n f l u e n c e on the behaviour o f the h y d r o f o i l boat i n waves. F r o m b o t h the e x p e r i m e n t a l and the theoretical results i t is evident that the boat oscillates around a new equi-l i b r i u m p o s i t i o n , w h i c h is considerabequi-ly equi-l o w e r i n the water than the e q u i l i b r i u m p o s i t i o n at u n d i s t u r b e d f l i g h t . The magnitude o f this sinkage can easily o u t -range the heave a m p l i t u d e and c o n t r i b u t e d largely t o the risk o f a crash o f the c r a f t , especially i n f o l l o w i n g waves.
A l s o the i n f l u e n c e o f the downwash on the l i f t o f the a f t f o i l has been investigated by a systematical
varia-t i o n o f varia-the downwash r e l a varia-t i o n . There appeared varia-t o be a rather large i n f l u e n c e on b o t h the m o t i o n amplitudes and the phase angles. Considering the rather l i m i t e d knowledge on the e f f e c t o f the d o w n w a s h f r o m a d ü i e d r a l f o i l on a f l a t f o i l this p o i n t certainly needs more a t t e n t i o n . Whereas the phase o f the m o t i o n s on the f r o n t and a f t f o i l are so i m p o r t a n t f o r the response o f the c r a f t the downv/ash-time-lag coupled w i t h the v a r i a t i o n i n the d o w n w a s h must be held responsible f o r these differences.
A n advantage o f the s o l u t i o n m e t h o d used, is f o u n d i n the p o s s i b i l i t y o f c o m p a r i n g the d i f f e r e n t c o m p o -nents o f the system and t l i e i r m u t u a l r e l a t i o n i n the t i m e d o m a i n as quasi-analogue signals. So represents Figure 14 the l i f t on f r o n t and a f t f o i l , the submerged area o f the f r o n t f o i l and the l i f t c o e f f i c i e n t o f the f r o n t f o i l f o r three d i f f e r e n t situations. I n Figure 14A the s i t u a t i o n w i t h m i n i m a l heave a m p l i t u d e is given. F r o m this i t can be c o n c l u d e d t h a t the l i f t f o r e and a f t are almost equal b u t i n anti-phase. N o t i c a b l e is also the character o f the and 5 ^ cuiwe and t h e i r f i n a l result: the l i f t o f the f r o n t f o i l .
Figure 14B represents the s i t u a t i o n i n w h i c h p i t c h i n g is m i n i m a l . Here the h f t o f f r o n t and a f t f o i l are almost i n phase, heaving is m a x i m a l .
I n Figure 14C an e x p l a n a t i o n is souglit f o r the large resonance i n f o l l o w i n g waves. The phase o f the forces w i t h respect t o the waves is u n f a v o u r a b l e : f r o n t minus 7 0 degr. and a f t minus 9 0 degr. The n o n l i n e a r i t y o f the h f t on the f r o n t f o i l is evident. The character o f this l i f t cui-ve is c o n f i r m e d by a force measurement on the f r o n t f o i l i n a p p r o x i m a t e l y the same s i t u a t i o n . I n tlus s i t u a t i o n the f o i l s are actually r u n n i n g i n t o the waves travelling i n the same d i r e c t i o n b u t w i t h slower speed. The du-ection o f the o r b i t a l velocities i n the waves causes the f o i l s t o be actually p o u r e d i n t o the waves crest.
The r e l a t i o n between the crafts-speed, the \ e l o c i t y o f t h é waves i.e. the f r e q u e n c y e n c o i m t e r and the d a m p i n g o f the system determines the response o f the c r a f t . This is c o n f i r m e d b y the tests and c a l c u l a t i o n w i t h the l o w e r speed o f advance o f the c r a f t .
7. Conclusions
The derived equations o f m o t i o n and the s o l u t i o n m e t h o d used appear to give a reliable p r e d i c t i o n o f the m o t i o n o f a h y d r o f o i l c r a f t w i t h a dihedral f o i l fore and a f l a t f o i l a f t i n regular sinusoidal waves. Compared w i t h the usually used description in the f r e -quency d o m a i n , as used b y Kaplan and Ogilvie, the results especially f o r f o l l o w i n g waves are consider-ably better. E v i d e n t l y the linearisation o f the equa-tions o f m o t i o n is n o t advisable.
The stationary sinkage o f the h y d r o f o i l boat is o f p n m e importance when considering the risks o f a crash o f the c r a f t and this sinkage can be calculated w i t h the m e t h o d used.
The superposition o f a separately calculated sinkage and response t o the waves gives m u c h p o o r e r results. The m l l u e n c e o f the d o w n w a s h f r o m the f r o n t f o i l needs f u r t h e r investigation considering the great i m -portance on the overall results.
The a d d i t i o n o f the surge m o t i o n f o r investigation o f the influence o f the surge m o t i o n on the vertical m o -tions should, regarding the i m p o r t a n c e o f relative velocities, give interesting results.
L i s t o f symbols s u f f i x F f r o n t f o i l s u f f i x A a f t f o i l s u f f i x 0 e q u i l i b r i u m p o s i t i o n L l i f t Lyisc viscous l i f t
h , ë added mass forces
D drag W weight r thrust l i f t c o e f f i c i e n t drag c o e f f i c i e n t induced drag c o e f f i c i e n t submergence dependent l i f t c o e f f i c i e n t r wave drag c o e f f i c i e n t c viscous drag c o e f f i c i e n t p r o f i l e drag c o e f f i c i e n t V speed h o r i z o n t a l o r b i t a l v e l o c i t y vertical o r b i t a l v e l o c i t y \\> d o w n w a s h v e l o c i t y X surge z heave 9 p i t c h 171 mass 'e l o n g i t u d i n a l m o m e n t o f i n e r t i a g gravitational acceleration P specific density S submerged area A aspect r a t i o C chord o f the f o i l s span I distance f o i l t o G l o n g i t u d i n a l arm o f the p i t c h m o m e n t
=,„ vertical arm o f the p i t c h m o m e n t a angle o f attack
" . V i ' n o - l i f t ' angle o f attack
adjustable angle o f attack f r o n t f o i l induced angle o f attack
e downwash angle dihedral <f> v e l o c i t y p o t e n t i a l c wave v e l o c i t y OJ wave f r e q u e n c y f r e q u e n c y o f e n c o u n t e r X wave length K = 2nl\ wave n u m b e r G wave p r o f i l e wave a m p l i t u d e r c i r c u l a t i o n 7 v o r t i c i t y Bessel f u n c t i o n s t w o dimensional c o r r e c t i o n f a c t o r three dimensional c o r r e c t i o n f a c t o r c o r r e c t i o n f a c t o r g r o u n d e f f e c t c o r r e c t i o n f a c t o r biplane c o P i f i g u r a t i o n M u n k c o r r e c t i o n f a c t o r r,5 c o r r e c t i o n f a c t o r s p l a n f o r m References
1. Glauert, H., "The elements of aerofoil and airscrew theory", Cambridge University Press, 1948.
2. Hoerner, S.F., "Fluid dynamic drag", 1965. 3. Hoerner, S.F., "Fluid dynamic drag", 1975.
4. Prandti and Tietjens, "Applied hydro- and aeromechanics", Dover Publications Inc., Nev/ York.
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