Applied Ocean Research 42 (2013) 70-7S
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Applied Ocean Research
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Hydrodynamics of a submerged hydrofoil advancing in waves
G.D.Xu3•^ G.X. Wub.'
'College of Shipbuilding Engineering, Harbin Engineering University. Harbin 150001, China
^Department of Mechanical Engineering, University College London, Torrington Place, London WC1E7]E. UK
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A R T I C L E I N F O
Article history: Received 21 May 2012
Received in revised form 1 May 2013 Accepted 2 May 2013
Keywords: Hydrofoil
Boundary element method Free surface effect
Wave radiation and diffraction
A B S T R A C T
The h y d r o d y n a m i c problem of a hydrofoil travelling at constant speed in w a t e r w a v e s has been investigated through velocity potential theory. T h e boundary conditions on the free surface have been linearized, and the effects are accounted for through the Green function. T h e overall problem is decomposed into the steady forward speed problem and periodic w a v e radiation and diffraction problems. E a c h of these problems is solved using the boundary integral equation over the hydrofoil surface together w i t h a vortex sheet behind the trailing edge. The body surface boundary condition is imposed on its m e a n position. As a result the steady potential w i l l contribute a w e l l - k n o w n irij term to the body surface boundary condition on the radiation problem. T h e n u m e r i c a l difficulty in dealing w i t h this t e r m is effectively resolved through a difference method. T h e effects of the thickness on the w a v e radiation and diffraction are investigated. T h e applicability of various reciprocity relationships in this problem is discussed.
© 2013 Elsevier Ltd. A l l rights reserved.
1. Introduction
The h y d r o d y n a m i c force and m o m e n t o n a h y d r o f o i l can be v e r y d i f f e r e n t w h e n i t moves near the free surface. This is one o f t h e m a j o r concerns o f high speed vehicles such as h y d r o f o i l s . As i t encounters i n c o m i n g waves, t h e h y d r o f o i l m o v i n g along w i t h the ship w o u l d experience o s c i l l a t i o n induced b y the waves, leading to a c o m b i n e d w a v e r a d i a t i o n and d i f f r a c t i o n p r o b l e m .
Giesing and S m i t h [1 ] developed a m e t h o d to calculate t h e l i f t i n g force o f a single or m u l t i l i f t i n g bodies m o v i n g near w a t e r surface. The sources and sinks w e r e d i s t r i b u t e d over the b o d y surface to solve the b o u n d a r y integral equation, together w i t h a v o r t i c i t y to satisfy the K u t t a c o n d i t i o n at t h e t r a i l i n g edge. The Green f u n c t i o n used i n the i n t e g r a l equation satisfies the linear f r e e surface b o u n d a r y c o n d i t i o n and the r a d i a t i o n c o n d i t i o n . Similar w o r k based on the panel m e t h o d for a h y d r o f o i l at steady f o r w a r d speed includes those by Y e u n g and Bouger [ 2 ] , Bal [3] f o r t w o d i m e n s i o n a l f o i l and b y Xie and Vassalos [4] for three d i m e n s i o n a l problems. Bal [5] also studied a surface p i e c i n g h y d r o f o i l .
I n earlier w o r k o n unsteady m o t i o n o f f i s h or f o i l , the b o d y sur-face c o n d i t i o n is satisfied on the centre plane o f the body, i n c l u d i n g those b y W u [ 6 - 8 ] a n d L i g h t h i l l [ 9 ] i n the u n b o u n d e d f l u i d d o m a i n . W u [ 1 0 ] considered the i n c o m i n g free surface w a v e . Its e f f e c t was h o w e v e r i n c l u d e d o n l y i n the b o u n d a r y c o n d i t i o n on the f o i l surface and the b o u n d a r y value p r o b l e m was still solved i n the u n b o u n d e d f l u i d d o m a i n . Grue et al. [11 ] considered the f u l l effects o f t h e linear surface w a v e o n an oscillatory plate w i t h o u t thickness t h r o u g h the
' Corresponding author. Fax; +44 20 7388 0180.
£-mai7 addresses: g.wu@iucl.ac.uk, gx.wu@meng.ucl.ac.uk (G.X. Wu). 0141-1187/$ - see front matter ® 2013 Elsevier Ltd. All rights reserved. http://dx.d0i.0rg/l 0.1016/j.apor2013.05.001
a p p r o p r i a t e Green f u n c t i o n , i n the c o n t e x t o f p r o p u l s i o n o f a ship, i n the sense t h a t energy can be extracted f r o m waves as the h y d r o f o i l m o v i n g f o r w a r d w i t h heave and p i t c h m o t i o n s . Z h u et al. [ 12 ] s t u d i e d the p r o p u l s i o n o f a t h r e e - d i m e n s i o n a l f o i l near the f r e e surface.
The present w o r k is concerned w i t h a h y d r o f o i l o f thicl<ness. This has been considered e x t e n s i v e l y i n the l o w speed aerodynamics w i t h n o w a v y f r e e surface [ 1 3 ] . Here the focus is o n the h y d r o d y n a m i c b e h a v i o u r o f a h y d r o f o i l a d v a n c i n g i n waves. The b o d y surface c o n -d i t i o n is no longer satisfie-d on the axis line o f t h e f o i l , b u t o n t h e m e a n p o s i t i o n o f the surface. I n this sense, f o r the f o r w a r d steady p r o b l e m and w a v e d i f f r a c t i o n p r o b l e m w i t h o u t b o d y oscillation, t h e b o u n d a r y c o n d i t i o n o n the b o d y surface is satisfied o n its exact l o -c a t i o n . W h e n there is a b o d y os-cillation, the -c o n d i t i o n is satisfied o n its mean p o s i t i o n . The Green f u n c t i o n w h i c h satisfies the linear f r e e surface b o u n d a r y c o n d i t i o n is adopted. This means t h a t the i n t e g r a l e q u a t i o n used involves o n l y the body surface.
W h e n there is the nonzero d i s t u r b e d steady p o t e n t i a l , the w e l l -l<nown nij t e r m w i l l appear i n the b o u n d a r y c o n d i t i o n o n the b o d y surface f o r the unsteady p o t e n t i a l [ 1 4 ] . The accuracy o f this t e r m is a m a j o r challenge i n this type o f n u m e r i c a l s o l u t i o n because i t contains the second o r d e r d e r i v a t i v e . Here a scheme based o n the finite d i f f e r e n c e m e t h o d is adopted. N u m e r i c a l test has s h o w n t h a t t h e m e t h o d gives v e r y accurate result. The mj t e r m is t h e n i n c l u d e d i n t h e b o d y surface b o u n d a r y c o n d i t i o n o n the r a d i a t i o n p r o b l e m , w h i c h seems to have been absent i n the previous w o r k o n a submerged h y d r o f o i l .
V e r i f i c a t i o n o f the m e t h o d o l o g y is first p e r f o r m e d t h r o u g h convergence study and t h r o u g h c o m p a r i s o n w i t h the p u b l i s h e d e x p e r i m e n -tal data. The e f f e c t o f the thickness is investigated. This is f o l l o w e d b y the detailed results f o r h y d r o d y n a m i c related to w a v e r a d i a t i o n a n d
CD, Xu. G.X. Wu /Applied Ocean Research 42 (2013) 70-78 71 Z w a v e X h heave S o s ^^^^^— :^p[tch Surge ^ , y \ U J=ar-'- - - - " V j j • r *" Surge ^ , y \ U 11
-i '
— iFig. 1. Sl<etch of a submerged hydrofoil advancing in waves.
d i f f r a c t i o n . The a p p l i c a b i l i t y o f various r e c i p r o c i t y relationships f o r the p r o b l e m is investigated.
2. Mathematic equations
The h y d r o d y n a m i c p r o b l e m o f a h y d r o f o i l m o v i n g i n waves near the f r e e surface at constant f o r w a r d speed U, as s h o w n i n Fig. 1, is considered. W e d e f i n e a Cartesian c o o r d i n a t e s y s t e m oxyz m o v i n g w i t h the h y d r o f o i l at the same speed U. oxy is a l o n g the still w a t e r plane, x is i n the d i r e c t i o n o f f o r w a r d speed U and z p o i n t s u p w a r d . The c h o r d o f the h y d r o f o i l is C = 21, and the distance o f the t r a i l i n g edge to the m e a n w a t e r line is h. W e d e f i n e the Froude n u m b e r as Fn = U/VgC, and denote a as the attack angle. The c h o r d centre [Xc, Zc) is a t equal distance to the t r a i l i n g and l e a d i n g edges. The i n c o m i n g w a v e is assumed sinusoidal b o t h t e m p o r a l l y and spatially and the h y d r o f o i l is i n h a r m o n i c surge, heave and p i t c h m o t i o n s . W h e n the attack angle a and m o t i o n a m p l i t u d e are s m a l l , the p r o b l e m can be described by the linearized v e l o c i t y p o t e n t i a l t h e o r y w i t h a v o r t e x w a k e b e h i n d the t r a i l i n g edge. The t o t a l p o t e n t i a l 0 can be w r i t t e n as
U4>{x,z)-.,2 ,
R e [ ^ ( x , z ) e ' ' - f ] ( 1 ) : wo±(.(üQ/g)U is the encounter f r e q u e n c y , + and c o r r e -* (X, z, t)
w h e r e w =
spond to w a v e f r o m the r i g h t and l e f t h a n d sides respectively, wo is the w a v e f r e q u e n c y . 0 i n the e q u a t i o n is due to t h e steady f o r w a r d m o t i o n o f the h y d r o f o i l . The p o t e n t i a l (t> related to the periodic m o t i o n can be w r i t t e n as
4> = Ao(pQ + A\(p\ + Asrps + A^rps + Ayipj (2)
w h e r e <po is the p o t e n t i a l due to i n c i d e n t w a v e a n d (pj is due to its d i f f r a c t i o n by the h y d r o f o i l ; AQ is the i n c o m i n g w a v e a m p l i t u d e , and
AJ^AQ; (pj (J = 1 , 3 , 5 ) are the p o t e n t i a l s due to surge, heave and p i t c h
m o t i o n s respectively; A, ( i = 1 , 3 , 5 ) are the c o r r e s p o n d i n g a m p l i t u d e s o f these m o r i o n s .
The p o t e n d a l due to the steady m o r i o n sarisfies t h e Laplace equa-rion
i n the fluid d o m a i n . Its b o u n d a r y c o n d i t i o n s can be w r i t t e n as
3 n tlx
(3)
(4) on the f o i l surface, w h e r e n= (ux, Hz) is the i n w a r d n o r m a l o f the b o d y surface, and
K<I>Z + Ipxx = 0 (5) on the f r e e surface, w h e r e K = g/iP- and g is the acceleration due to g r a v i t y . I n the far field, w e have
V ^ = 0 , x ^ + o o and V j = f i n i t e , x - c o . (6) (7) -1.0
Ü
-N=100 N=200 Exp. Ausman (1954) - Giesing&Smith (1967) u p p e r s u r f a c e / -0.8-^ -N=100 N=200 Exp. AusiTian(1954) •Giesing&Smith (1967) u p p e r s u r f a c eFig. 2 . Pressure distribution on the NACA4412 foil at Fn = 1.03 and a = 5 . (a) h/C 0.94 and (b)h/C = 0.6.
The p o t e n t i a l s due to r a d i a t i o n a n d , d i f f r a c t i o n s a t i s f y Laplace e q u a t i o n
V > j = 0 , j = l , 3 , 5 , 7 (8)
i n the fluid d o m a i n . On the b o d y surface the b o u n d a r y c o n d i t i o n can be w r i t t e n as [ 1 4 ] dn ^ icüUj + Utrij, ( j = 1, 3, 5 ) d<P7 _ d<po an ~ an w h e r e ( n , , n3, ns) = (Hx, Hz, Z H X - X H Z ) , X = X~XC, Z = Z - Z C a [ z ( 0 , - l ) - x 0 z ] ' an (9) (10) ( ' " " ' " 3 . m s ) = - | ^ , - f ,
Its b o u n d a r y c o n d i r i o n o n the free surface can be w r i t t e n as [15,16]
<Pjz + —fjxx - '2iT<Pjx - vpj = 0 ( 1 1 )
w h e r e T = iUm/g) and v = {tó^/g).
The p o t e n t i a l due to t h e i n c i d e n t w a v e can be w r i t t e n as
- S_^z±i(i(QX+e)
72 C D . Xu, GX Wu /Applied Ocean Researcli 42 (2013) 70-78 0.04
0:03-^
Ö-02-0.01-^ 0.00 O 0;8n 0.64 D ; 40.2
0.30 0.250 ^
0.20
0.1:5 H 0.10 0•t/C=0.10
t/C=0.15
t/G=0.20
Fn
(a)
—'t/G=0.10
— t/C=0.15
• -:t/C=0.20
Fn
(b)• -1/0=0.10
---t/G=Q.i5
t/C=Ö.20
2
4Fn
:(c) 6Fig. 3. Thickness effect of a symmetric Joukowsky foil with a = 5° and h/C = 0.6. (a) Resistance, (b) lift and (c) pitch moment.
w h e r e /<b = {^Q/g) is the w a v e n u m b e r and e is the phase o f the i n -c o m i n g w a v e .
Once the b o u n d a r y value p r o b l e m is solved, the force and m o m e n t on the f o i l can be o b t a i n e d f r o m the i n t e g r a t i o n o f pressure p. W e have [14,16]
(13) + R e [(/loF„jo + F „ j , + As Fuji + AsFujs + AoF^p) e'"']
w h e r e Fsj a n d F^jj are the steady force and the c o m p l e x a m p l i t u d e o f the unsteady force, respectively.
30A
30A
0
-30•-t/C=0.10
- --t/C=0.15
- t/G=0.20
0.80 0.85 0.90 0.95 1.002x/C
(a)
- " - « 0 = 0 . 1 0
-^--t/C=0.15
4Ö4 •t/G=0.20
20-1 1.05 -20 0 . 8 0 0.85 0.90 Ö.95 1.Q0 K.052x/G
Fig. 4. «ij terms for the symmetric Joukowsky foil of various thickness with h/C= 1.0 at Fn = 0.6 and a = 5°. (a) mi and (b) ma.
The f o r m e r d u e to f o r w a r d speed is obtained f r o m the f u l l B e r n o u l l i e q u a t i o n
Fg = pnjdS = -pU^f^ ( - ^ x + ^ V ^ V ^ ) rijdS, j = 1,3, 5 ( 1 4 )
The later due to w a v e r a d i a t i o n a n d d i f f r a c t i o n is obtained f r o m the B e r n o u l l i e q u a t i o n linearized based o n the m o t i o n a m p l i t u d e . W e have the r a d i a t i o n w a v e force
•^Uji =-P
J So
+ W-y<Pi)njdS, i , j = l , 3 , 5 ( 1 5 )
w h e r e W = U V ( 0 - x ) .
^ 2 I n E q . (15), and the t e r m , w h i c h involves the g r a d i e n t of I W | [ 1 4 , Eq. (8.7)1, is n o t i n c l u d e d . This is because the t e r m is n o t f r e q u e n c y d e p e n d e n t and w i l l c o n t r i b u t e to the force as a generalised stiffness, or the restoring force. The i n c l u s i o n o f such a t e r m w o u l d n o t have caused any n u m e r i c a l d i f f i c u l t i e s here. I n fact, once m i and rns are obtained, Eqs. ( 1 7 ) and (18) can be used to f i n d <l>xx (or <f>zz) and <pxz,
^ 2 w h i c h can be used to f i n d the g r a d i e n t o f | W I easily.
The c o m b i n e d i n c i d e n t w a v e f o r c e and d i f f r a c t i o n w a v e force can be w r i t t e n as
^(F„,„ + F „ ^ J =
C D . Xu, GX. Wu /Applied Ocean Researcli 42 (2013) 70-78 73 2-1 -2 A t/C=0.1.0 - > - t / . G - 0 . 1 5 - - - 1 / 0 = 0 , 2 0 0.0 0,2 o.<( 0.6 a.(
vC
(a)
1.0 ••-tfC=0.10 •*-t/C=0.15 - - t / C = 0 . 2 0 0.2 O.A 0,6 0.8vG
(c)
1.0 0.0-1 E 0.00/
i f
•-t/C=0.10 '-t/C=0.15 --f/C=0.20 0.0 0.2 0.4 0.6 0.£ v C 1.0 (b) S -1.0i -1.5 •2,5-.-t/c=o.to
- - - t / C = 0 . 1 5 - • • - t / C = 0 . 2 0 0.0 0.2 0.4 0.6 O.e 1.0 v C (d) •3 23
"5' K o j -1 -2-1s.
- • - t / C = 0 , 1 0 • ' - t / C = 0 . 1 5 - - - 1 / 0 = 0 . 2 0 0.0 0.2 0,4 0:6vC
(e) o:8 1.Ö i f f • —t/C=0.1Ö ---tyc=0;i5 • - - t / C = 0 , 2 öo:o o;2 0.4 o.e o.e 1,0
vC
(f) 1 0.0 0.2 t/C=0.10. VC=0.15 -•••-t/C=0.2Ö 0,4 0.6 0,8vC
(g) 1.0 - • - t / c = o , - i o - • ! - t / C = 0 , 1 5 - - t / C = 0 , 2 b 0,0 0,2 0.4 0.6 O.E v C (h) 1,0Fig. 5. Hydrodynamic force on a symmetric Joul<owsl<y foil with various thicltness ratio, at Fn = 0.6 and h/C = 1.0.
One m a y notice t h a t the body surface b o u n d a r y c o n d i t i o n o f the t h e b o d y surface. W e have unsteady m o t i o n contains the second order derivatives o f the steady
potential. This is usually p r o b l e m a t i c i n n u m e r i c a l calculation. One scheme is to solve the mj t h r o u g h a b o u n d a r y i n t e g r a l equation, w h i c h was f i r s t proposed b y W u [17]. Here w e calculate mj terms d i r e c t l y based o n a f i n i t e difference m e t h o d f o r derivatives o f a n d along
nil ' an = ~<t>xx^x — iPxi^z al (713 an Pzx"x - Vxxi'z • al ( 1 7 ) ( 1 8 )
74 CD. Xu, GX Wu /Appiied Ocean Research 42 (2013) 70-78 01 2^
-1
(p = i c o n + U m .
— ( p „ = U m 3
„..--•"1
0.0
/ y \ y 0;2 0.4 Ö.6v G
(a)
0.8 1.0 0.08 0.04 0 . 0 0 4 -0.04 -0.08• — <p„=icon„
0.0 0.2 0 . 4 0.6 0-8 1.0v G
(b) C D DC1-<p^=i(ong+Umg
0.0 0.2 0.4 0.6 0.8 1.0vG
(c)
4^
3
2 1' 0-j• —<p^=i(0ng+Umg
• • - ( P ^ = i ( ö n ^
Ö.Q0.2
Ö.4
0-60.8
1,0
v G (d) Fig. 6. The contribution of mj terms on the radiation force with atFn = 0.6, / i / C = 1.0, ff = 5°, and t/C = 0.15.o f the b o d y surface. F r o m m i , ma, w e have ms = Hz (0x - 1) - hxfz + Z m i - Xm3 Green's t h i r d i d e n t i t y gives 3C{p,q) 3 tin ( q ) - G ( p , < / ) (19) dSn ( 2 0 ) SQ+SW++SW
w h e r e A{p) is the solid angle at p o i n t p(x, z) o n the b o u n d a r y , and G is the Green f u n c t i o n due to a u n i t source at q(^, ri), s a t i s f y i n g the same free surface and r a d i a t i o n b o u n d a r y c o n d i t i o n s as the p o t e n t i a l ,
Sw + , Siv - are respectively the upper and l o w e r sides o f the w a k e
surface S,v s h o w n i n Fig. 1. As w e use the linear t h e o r y , the w a k e is parallel to f o r w a r d speed U and extends to i n f i n i t y f r o m the t r a i l i n g edge. W e d e f i n e the d i p o l e d i s t r i b u t i o n as
The c o n t i n u i t y o f the n o r m a l v e l o c i t y across the w a k e gives
S<pw+
90W-(21)
(22) w h e r e the negative sign is due to d i f f e r e n t d i r e c t i o n s o f the n o r m a l . S u b s t i t u t i n g Eqs. ( 2 1 ) and ( 2 2 ) i n t o Eq. (20), w e have
A{p)<l>iP) = J ' 9 G ( p , q ) " 9 G ( P . q ) a Tin 0 ( 9 ) - G ( p , q ) 3 0 ( g ) a tin dSa (23) anw+ lj.(q)dSw
w h e r e X j is the x coordinate o f t h e t r a i l i n g edge.
To solve the above p r o b l e m , the b o u n d a r y o f t h e f o i l surface is d i v i d e d i n t o N small segments. I t is assumed t h a t the p a r a m e t e r over each segment is constant. Thus Eq. ( 2 3 ) can be w r i t t e n as
N E j = i ^<Pj - Yij 30) an
r
J -cc 3G 9nw+ dSw = 0 (24) w h e r e aC{p,q)aua dSa Yij J Sj [ G{p,q)dSg
and I, j are the n u m b e r s o f the segments c o r r e s p o n d i n g to p and q respectively. The dipole i n the w a k e is related to the p o t e n t i a l j u m p at the t r a i l i n g edge, w h i c h w i l l be discussed i n d e t a i l b e l o w .
Before p r o v i d i n g n u m e r i c a l results and s t a r t i n g discussions, i t w o u l d be a p p r o p r i a t e to c o m p a r e the present w o r k w i t h t h a t o f Grue et al. [ 1 1 ] . Both are based o n a s i m i l a r m a t h e m a t i c a l m o d e l , or the linearized v e l o c i t y p o t e n t i a l t h e o r y . H o w e v e r there are some m a j o r differences. First, because the m e a n p o s i t i o n o f the h e a v i n g a n d p i t c h -i n g plate -i n [11 ] -is a l o n g the d -i r e c t -i o n o f f o r w a r d speed, o r the m e a n attack angle is zero, there is no steady p o t e n t i a l and t h e r e f o r e no mj t e r m . W h e n a f o i l has thickness nij t e r m w i l l appear e v e n at zero attack angle. The second d i f f e r e n c e is the s o l u t i o n t e c h n i q u e . For a plate, they w e r e able to use the inverse m e t h o d to solve t h e i n t e g r a l e q u a r i o n [18]. Such a m e t h o d is n o t applicable f o r the f o i l w i t h t h i c k -ness and the BEM described above is adopted i n this w o r k . The t h i r d
CD, Xu, CX Wu /Applied Ocean Researcli 42 (2013) 70-78 75 -10 -20 -30 4 6 F n
(a)
10 80-^ 40 -40 -80 0 2 4 6 8 10 F n (b)Fig- 7. /„35 and/u53 of a symmetric Joukowsky foil with t / C = 0,15, (ï = 5 , and vC = 0.1,
d i f f e r e n c e is t h a t they focused o n s e l f - p r o p u l s i o n w h i l e the present w o r k is o n w a v e d i f f r a c t i o n and r a d i a t i o n i n the c o n t e x t o f m o t i o n o f a boat s u p p o r t e d b y the h y d r o f o i l s .
3. Numerical results and discussions
3.1. Verification and ttie thicl<ness effect
The steady m o t i o n is used f o r convergence study and c o m p a r i -son. The c o n t i n u i t y o f the pressure o b t a i n e d f r o m the linear B e r n o u l l i e q u a t i o n across the w a k e requires the d i p o l e ii i n Eq. ( 2 4 ) to be c o n -stant. This is t h e n obtained by the d i f f e r e n c e o f the potentials o n the t w o segments attached t o the t r a i l i n g edge o f the f o i l . Eq. ( 2 4 ) t h e n has N u n k n o w n s w h i c h are o b t a i n e d f r o m the N c o n d i t i o n s i m p o s e d at the centres o f N segments. Convergence s t u d y and the v e r i f i c a t i o n o f the present m e t h o d have been c a r r i e d o u t on NACA4412 f o i l . Meshes w i t h N = 100, 200 are used and the pressure d i s t r i b u t i o n over the b o d y surface is given i n Fig. 2, w h i c h is n o n - d i m e n s i o n a l i z e d as
( l / 2 ) p U 2 (25)
The f i g u r e shows that results f r o m t w o sets o f meshes are a l m o s t g r a p h i c a l l y indistinguishable, w h i c h means t h e pressure d i s t r i b u t i o n has converged w i t h respect to the mesh. The results are c o m p a r e d w i t h the e x p e r i m e n t a l data o f A u s m a n [19], t a k e n m a n u a l l y f r o m the paper o f Giesing and S m i t h [ 1 ] , and a very good agreement can be
f o u n d . The n u m e r i c a l results o f [1] are also i n c l u d e d i n the figure, w h i c h are slightiy d i f f e r e n t o n the upper surface.
W e n o w consider the thickness e f f e c t o f the f o i l . Fig. 3 presents the resistance, l i f t i n g force and m o m e n t o n s y m m e t r i c J o u k o w s k y foils o f d i f f e r e n t thickness against Froude n u m b e r . The n o n - d i m e n s i o n a l i z e d resistance, l i f t and p i t c h m o m e n t are d e f i n e d as
Fs\ Fs3 ^ Fs5
CR
( l / 2 ) p f / 2 C ' Ci ( l / 2 ) p U 2 C ' CM ( l / 4 ) p U 2 C 2 (26)
The f i g u r e shows that w h e n there is n o surface wave, or w h e n Fn = 0 or Fn = 00, the e f f e c t o f t h e thickness o n the force is m a r g i n a l . C o m b i n i n g w i t h the free surface, the e f f e c t o f the thickness is v e r y significant, especially near the Froude n u m b e r s w h e r e the resistance or l i f t is peaked. This suggests that a t h i n f o i l t h e o r y t h r o u g h a plate m a y become less accurate w h e n there is surface w a v e .
W e t h e n consider the thickness e f f e c t o n the mj t e r m s o f these foils. To calculate the mj terms based o n Eqs. ( 1 7 ) and ( I S ) , the first order d e r i v a t i v e at the m i d d l e o f the element is first calculated t h r o u g h fi-n i t e d i f f e r e fi-n c e a f t e r the p o t e fi-n t i a l o fi-n the fi-nodes is o b t a i fi-n e d t h r o u g h i n t e r p o l a t i o n . The second order d e r i v a t i v e is calculated i n a s i m i l a r m a n n e r w h e n the finite d i f f e r e n c e and i n t e r p o l a t i o n are applied to first order derivatives. N u m e r i c a l test has been c o n d u c t e d and con-vergence o f the scheme is achieved. Finer m e s h is needed to give converged results, especially at places w h e r e the curvature is h i g h . Fig. 4 shows that the results f o r these second o r d e r derivatives near the f o r e body. W e notice that rrij terms are q u i t e large near t h e lead-i n g edge. C o m p a r lead-i n g the curves o f mj t e r m s o f these folead-ils o f d lead-i f f e r e n t thickness, the value is m u c h higher near the l e a d i n g edge f o r a t h i n -ner f o i l . W e f u r t h e r notice t h a t nj i n Eq. (9) are t h e components o f the n o r m a l and thus t h e i r magnitudes are always less t h a n one. As a result, the second t e r m could p l a y an i m p o r t a n t role, w h i c h reflects the significance o f the e f f e c t o f the steady p o t e n t i a l o n the unsteady p o t e n t i a l and t h e r e f o r e the unsteady forces.
3.2. Wave radiation and diffraction
For the periodic m o t i o n , t h e pressure at the w a k e is o b t a i n e d t h r o u g h the linear B e r n o u l l i e q u a t i o n
p = ia)(pi -U(pix (27)
w h i c h ignores the e f f e c t o f the steady d i s t u r b e d p o t e n t i a l . As i n N e w -m a n [18], its c o n t i n u i t y across the w a k e -means t h a t /* i n Eq. (21) satisfies
io>ii-Uiix = 0 (28)
This gives
l z ( x ) = tiQe'''w' (29)
w h e r e l<w = w/U, the value of II{XT) can be o b t a i n e d f r o m the d i f f e r e n c e b e t w e e n the potentials on the t w o elements attached to the t r a i l i n g edge, as i n t h e steady p o t e n t i a l .
We consider the h y d r o f o i l advancing i n a regular w a v e and w i t h small a m p l i t u d e h a r m o n i c heave and p i t c h m o t i o n s . Fig. 5 gives these forces o n t h e s y m m e t r i c J o u k o w s k y f o i l o f t / C = 0.1,0.15,0.2, w i t h Fn = 0.6, / i / C = 1.0 and w = 5 ° . The result is presented i n t h e f o r m o f real and i m a g i n a r y p a r t and is n o n - d i m e n s i o n a l i z e d as
I'u3j
pgl ' ƒ, u5j
Pu5j
(30)
The figure shows that the thickness has s i g n i f i c a n t e f f e c t o n the forces and r o t a f l o n a l m o m e n t , especially w h e n vC > 0.17. This is consistent to w h a t is observed i n Fig. 3. I n a d d i t i o n , because o f e f f e c t o f the thickness o n the mj t e r m , this w i l l f e e d i n t o t h e r a d i a t i o n force t h r o u g h Eq. (9). As i t has been seen i n Fig. 4, t h e rrij t e r m can be v e r y large. W h e n i t varies, its e f f e c t o n t h e r a d i a r i o n p o t e n t i a l can be v e r y s i g n i f i c a n t at l o w e r frequency, as discussed i n last section. W e
76 CD. Xu, CX. Wu /Applied Ocean Research 42 (2013) 70-78
notice i n Fig. 5 thiat there is a sharp v a r i a t i o n at vC 0.17. This i n f a c t corresponds to r 0.25. The data closest to this p o i n t used to p l o t these curves are at i ; C = 0.168 c o r r e s p o n d i n g to r = 0.246 < 0.25 and y C = 0.2 c o r r e s p o n d i n g to r = 0.268 > 0.25, respectively. The reason f o r the sharp v a r i a t i o n at this p o i n t can be e x p l a i n e d b y the w a v e s t r u c t u r e discussed i n the a p p e n d i x and the w o r k o f [ 1 1 , 1 6 , 2 0 ] ,
W e f u r t h e r s t u d y the c o n t r i b u t i o n o f the nij terms to the unsteady force o n the f o i l w i t h t / C = 0.15 at Fn = 0.6, h/C = 1.0, a = 5 ° . Fig. 6 gives the r a d i a t i o n f o r c e and m o m e n t w h e n the body surface c o n d i -tions d(pj/an = iwnj and Sipj/dn = Umj are used separately. As s h o w n i n the figure the force due to dcpj/dn = U rrxj is m u c h larger t h a n t h a t due to d(pj/dn = imj at l o w e r f r e q u e n c y ; the c o n t r i b u t i o n o f nij terms to fu33<fu55 is d o m i n a n t . As the f r e q u e n c y increase, the c o n t r i b u t i o n o f dipj/dn = icoUj to /u33 becomes i m p o r t a n t , as s h o w n i n Fig. 6(a). H o w e v e r , Fig. 6(c) a n d (d) shows t h a t drpj/Bn = icorij has m i n o r c o n -t r i b u -t i o n rela-tive -to -t h a -t o f dipj/dn = Umj. A l l -these s h o w -t h a -t -the mj terms have m a j o r i n f l u e n c e to the unsteady r a d i a t i o n forces, espe-cially at l o w e r frequency.
For a submerged body, T i m m a n and N e w m a n [21 ] have s h o w n t h a t the h y d r o d y n a m i c forces associated w i t h the w a v e r a d i a t i o n / „ | j ( i , i = 1,3, 5) satisfy
/ m j ( U ) = ( - Ü ) (31)
w h i c h was f u r t h e r c o n f i r m e d b y W u and Eatock Taylor [ 2 2 ] at l o w f o r w a r d speed. For a t w o d i m e n s i o n a l b o d y o f f o r e / a f t s y m m e t r y , Eq. (31) rneans t h a t
fuij = - f u j i (32) except
/ u l 5 = /u51
I t was, however, f o u n d b y W u a n d Eatock Taylor [ 2 3 ] t h a t this r e l a -t i o n s h i p is v a l i d o n l y a-t l o w f o r w a r d speed. A -t large f o r w a r d speed, W u a n d Eatcok Taylor [ 2 3 ] have s h o w n t h a t
R e ( L , j ) = Re{f,ji), l m ( / „ j ) = - l m ( / , j , ) (33) w h i c h do n o t r e q u i r e the b o d y to have f o r e / a f t s y m m e t r y .
W e consider an e x a m p l e at vC = 0.1 and p l o t / u 3 5 a n d / u 5 3 against Froude n u m b e r i n Fig. 7. The data p l o t i n Fig. 7 start f r o m Fn = 0.05 w i t h a n i n c r e m e n t o f 0.05. The results do n o t satisfy the T i m m a n and N e w m a n r e l a t i o n at l o w Froude n u m b e r . This is i n f a c t n o t a surprise, because the h y d r o f o i l does n o t have f o r e / a f t s y m m e t r y a n d because o f the presence o f the w a k e e f f e c t here. One can also see i n the figure t h a t Eq. (33) is n o t satisfied e i t h e r i n m o s t cases. This is again due to the e f f e c t o f the wake, as t h e r e l a t i o n s h i p does n o t r e q u i r e f o r e / a f t s y m m e t r y o f the body. There are i n f a c t f u r t h e r relationships to l i n k these h y d r o d y n a m i c forces w i t h the r a d i a t e d w a v e at i n f i n i t y f o r a n o n - l i f t i n g b o d y [ 2 4 ] . H o w e v e r because o f the w a k e effect, such r e l a t i o n s h i p m a y n o t be relevant here.
Fig. 8 presents the w a v e e x c i t a t i o n force and m o m e n t due to i n -c i d e n t p o t e n t i a l and d i f f r a -c t e d p o t e n t i a l w h e n the w a v e is f r o m the r i g h t h a n d side, or the head sea. For the d i f f r a c t i o n p r o b l e m s , there is also a r e l a t i o n s h i p l i n k i n g the force w i t h the a m p l i t u d e o f the d i f f r a c t e d w a v e at i n f i n i t y [24] and a r e l a t i o n s h i p l i n k i n g the a m -p l i t u d e s o f the r e f l e c t e d and t r a n s m i t t e d w a v e s themselves [ 2 5 ] , The w a k e here h o w e v e r has changed the v a l i d i t y o f these r e l a t i o n s h i p s . I n particular, f o r a n o n - l i f t i n g body the w a v e s t r u c t u r e is t h a t discussed
CD. Xu, CX. Wu /Applied Ocean Research 42 (2013) 70-78 77
i n A p p e n d i x A. For a h y d r o f o i l i n waves, however, because the v o r -tex.sheet i n Eq. (29) extends to the i n f i n i t y , the waves w i l l become m o r e c o m p l i c a t e d . The curves s h o w the force, i n c l u d i n g c o n t r i b u t i o n f r o m b o t h i n c o m i n g w a v e and d i f f r a c t e d w a v e . The d i f f e r e n c e s due to the thickness become s i g n i f i c a n t w h e n vC increases. These curves increase w h e n vC becomes larger and t h e n decrease a f t e r t h e y reach t h e i r peaks. I t is expected t h a t these curves approaching zero w h e n vC be v e r y large because o f the e"**'' t e r m i n the i n c i d e n t w a v e .
4. Conclusions
The p r o b l e m of an oscillatory h y d r o f o i l a d v a n c i n g i n waves is s t u d -ied t h r o u g h the linearized v e l o c i t y p o t e n t i a l t h e o r y i n the f r e q u e n c y d o m a i n , t h r o u g h the b o u n d a r y i n t e g r a l m e t h o d together w i t h a v o r -tex sheet b e h i n d the t r a i l i n g edge. The e f f e c t o f the thickness is h i g h l y significant, especially near the Froude n u m b e r s w h e r e the resistance or l i f t is peaked. A n e f f i c i e n t scheme f o r the rrij t e r m s is proposed, w h i c h is f o u n d to be s u f f i c i e n t l y accurate. Its e f f e c t on the oscillatory m o r i o n o f the f o i l is i n c l u d e d a n d is f o u n d to be v e r y s i g n i f i c a n t . The a p p l i c a b i l i t y o f r e c i p r o c i t y relationships f o r the w a v e radiation and d i f f r a c t i o n p r o b l e m s is i n v e s t i g a t e d . It is f o u n d that these r e l a t i o n -ships are generally n o t v a l i d , p r i n c i p a l l y because o f the v o r t e x sheet.
The present w o r k can be e x t e n d e d to the f u l l y n o n l i n e a r p r o b -lems. In fact X u and W u [ 2 6 ] have considered a f o i l i n large a m p l i t u d e m o t i o n w i t h o u t free surface i n the c o n t e x t o f p o p u l a t i o n , e n e r g y ex-t r a c ex-t i o n and flying. The b o u n d a r y c o n d i ex-t i o n is i m p o s e d on ex-the exacex-t p o s i r i o n o f the body surface, a n d the m o r i o n and d e f o r m a t i o n and the v o r t e x sheet are tracked n u m e r i c a l l y . W h e n there is a free surface w i t h n o n l i n e a r b o u n d a r y conditions, the Green f u n c t i o n s used i n the present paper are no l o n g e r applicable. Rankine source technique has be used i n the b o u n d a r y e l e m e n t m e t h o d , a n d c o m p l e x i n t e r a c t i o n s of the free surface and r o l l i n g up o f the v o r t e x sheet have to tackled.
Aclaiowledgements
The first a u t h o r w i s h e s to state t h a t the research was conducted at the D e p a r t m e n t o f Mechanical Engineering, U n i v e r s i t y College L o n -d o n as p a r t o f his PhD stu-dy. This w o r k is p a r t i y s u p p o r t e -d by Lloy-d's Register Foundation (LRF) t h r o u g h the j o i n t centre i n v o l v i n g U n i v e r -sity College London, Shanghai Jiaotong U n i v e r s i t y a n d H a r b i n Engin e e r i Engin g UEnginiversity, to w h i c h the a u t h o r s are m o s t g r a t e f u l . LRF s u p -ports the a d v a n c e m e n t o f e n g i n e e r i n g - r e l a t e d education, and f u n d s research and d e v e l o p m e n t t h a t enhances safety o f life a t sea, on l a n d and i n the air.
Appendix A
For the steady p o t e n t i a l </>, the Green f u n c t i o n can be w r i t t e n as [15] r gKz+'i) G = \nr + lnr' + 2p.v.j — c o s f c ( x - f )
dk
( A l ) + 2 7 r e ^ ( ^ + " ) s i n * c - ( x - f ) m -w h e r e r = ^{x - + {z - r' = ^ { x - ^ f + ( z + 7 , f , and p. v. dicates the Cauchy p r i n c i p a l i n t e g r a l .For the p o t e n t i a l r e l a t e d to the h a r m o n i c m o t i o n , the Green f u n c -tion can be w r i t t e n as [15,16]
G = I n r - In r
' +
p.v. /'°^e''(^+") \A (fe)e'''(''-^) + B(fe)e-"'(''-^)l
dfey 0 L J (A2) w h e n T < ( 1 / 4 ) , w h e r e B{k) = 1 1 / 1 1 V l + 4 r V'<-'<3 l<~l<4, l < l = ^ { ^ - 2 r + V ^ ^ ) . fe2 =
2 ^ ( l - 2 r - y i ^ )
'^3 = ^ ( l + 2 r - F V l - F 4 r ) , = ^ (l + 2 r - v T T 4 7 )
Ö1 V T T 4 f 'VTT47
a n d the ± sign i n last t e r m is taken positive w h e n j = 1,2, and negative w h e n j = 3 , 4 ; and
G = In r - I n r
' 4- p.v. pe''(z+'i) \c (fc)e*(''-^) + B (k)e~<'"-^-^^dk
w h e n r > ( 1 / 4 ) , w h e r e
^ ^'^^" r^k^-v{\-2x)k +
v^This indicates t h a t there w o u l d be f o u r waves o f d i f f e r e n t w a v e n u m -ber w h e n T < ( 1 / 4 ) , t w o t r a v e l l i n g b e f o r e the f o i l a n d t w o b e h i n d ; t h e r e are t w o waves p r o p a g a t i n g b e h i n d the f o i l , w h e n r > (1 / 4 ) .
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