C o n t e n t s l i s t s a v a i l a b l e a t S c i e i i o e D i r e c l
Applied Ocean Research
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a p o r
O C t A N
R E S E A R C H
Generalised Wagner model of water impact by numerical
conformal mapping
T.I. Khabakhpasheva*'*, Yonghwan Kim'', A.A. Korobkin'^
^r/ieorerica/ Department, Lavrentyev Institute of Hydrodynamics, Novosibirsk, Russia^Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul, Republic of Korea '^School of Mathematics, University of East Anglia, Nonvich, UK
Cro.ssMark
A R T I C L E I N F O
Article history: Received 17June 2013
Received in revised form 15 October 2013 Accepted 19 October 2013 Keywords: Water impact Wagner tlieory Conformal mapping A B S T R A C T A n u m e r i c a l m e t h o d t o solve t b e p r o b l e m o f s y m m e t r i c r i g i d c o n t o u r e n t e r i n g w a t e r v e r t i c a l l y at a g i v e n t i m e -d e p e n -d e n t spee-d is p r e s e n t e -d . T h e m e t h o -d is base-d u p o n t h e so-calle-d g e n e r a l i s e -d W a g n e r m o -d e l . W i t h i n t h i s m o d e l t h e b o d y b o u n d a r y c o n d i t i o n is i m p o s e d o n t h e a c t u a l p o s i t i o n o f t h e e n t e r i n g surface, t h e f r e e -s u r f a c e b o u n d a r y c o n d i t i o n -s are l i n e a r i -s e d a n d i m p o -s e d o n t h e p i l e - u p h e i g h t , w h i c h i-s d e t e r m i n e d a-s p a r t o f t h e s o l u d o n . T h e h y d r o d y n a m i c pressure is g i v e n b y t h e n o n - l i n e a r B e r n o u l l i e q u a d o n . T h e h y d r o d y n a m i c pressures w h i c h are b e l o w t h e a t m o s p h e r i c v a l u e are d i s r e g a r d e d . The c o n f o r m a l m a p p i n g o f t h e f l o w r e g i o n o n t o the l o w e r h a l f - p l a n e is used. The v e l o c i t y p o t e n U a l o f t h e f l o w is g i v e n i n a n a l y d c a l f o r m once t h i s m a p p i n g is k n o w n . The c o n f o r m a l m a p p i n g is c a l c u l a t e d n u m e r i c a l l y . The o b t a i n e d r e s u l t s are v a l i d a t e d w i t h respect to t h e k n o w n s o l u t i o n s f o r w e d g e a n d c i r c u l a r c y l i n d e r . The n o v e l t y a n d p r a c t i c a l i m p o r t a n c e o f t h e p r e s e n t a p p r o a c h are d u e to a special a c c u r a t e t r e a t m e n t o f t h e f l o w s a n d t h e pressures close t o t h e c o n t a c t p o i n t s b e t w e e n t h e e n t e r i n g b o d y a n d w a t e r f r e e s u r f a c e . This special t r e a t m e n t is r e q u i r e d f o r r e l i a b l e p r e d i c d o n o f t h e h y d r o d y n a m i c p r e s s u r e a l o n g the w e t t e d p a r t o f t h e c o n t o u r d u r i n g its i m p a c t o n t o t h e w a t e r surface a n d t h e s u b s e q u e n t e n t r y .
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1. Introduction
This paper deals w i t h the t w o d i m e n s i o n a l p r o b l e m o f the u n -steady l i q u i d f l o w caused by an i m p a c t on its f r e e surface. The s t u d y is m o t i v a t e d by the s l a m m i n g p r o b l e m f o r ships In severe sea, w h e n the b o w part o f the ship exits the w a t e r and t h e n subsequently en-ters It again. The r e s u l t i n g i m p a c t loads m a y cause h i g h - f r e q u e n c y v i b r a t i o n o f the ship h u l l . This p h e n o m e n o n is k n o w n as w h i p p i n g [ 1 ] . W h i p p i n g may c o n t r i b u t e to the w a v e - I n d u c e d v l b r a d o n o f the h u l l increasing the w a v e - i n d u c e d stresses In the h u l l by up to 20% [ 2 ] . The i m p a c t loads can be c o m p u t e d b y using the s t r i p t h e o r y f o r e l o n -gated ships such as container ships. Computations by the strip t h e o r y are questionable f o r the b o w part o f the ship, w h e r e the s l a m m i n g pressures are essentially three-dimensional. However, the s l a m m i n g loads p r e d i c t e d by t h e strip t h e o r y can be corrected (see H e r m u n d -stad and M o a n [ 3 ] ) b y using a f a c t o r w h i c h relates t w o - d i m e n s i o n a l and t h r e e - d i m e n s i o n a l s l a m m i n g loads. This factor Is p r o v i d e d by the t h r e e - d i m e n s i o n a l W a g n e r t h e o r y o f w a t e r i m p a c t (see Scolan and K o r o b k l n [4], Fig. 17). In the present paper w e restrict ourselves to the t w o - d i m e n s i o n a l p r o b l e m o n v e r t i c a l e n t r y o f a s y m m e t r i c s m o o t h section. This case corresponds to heave and p i t c h ship m o t i o n s In head seas.
I n i t i a l l y the l i q u i d Is at rest and occupies the l o w h a l f - p l a n e y < 0 (see Fig. 1 a). The l i n e y = 0 corresponds to the u n d i s t u r b e d p o s i t i o n o f the free surface. The shape o f a s y m m e t r i c t w o - d i m e n s i o n a l b o d y is described b y a f u n c t i o n / ( x ) , w h e r e / ( - x ) = / ( x ) , / ( O ) = 0 , / ( O ) = 0 and
f[x) > 0 f o r x / 0. Here the p r i m e stands f o r t h e x - d e r i v a t i v e . The b o d y
touches the f r e e surface y = 0 at a single p o i n t taken as the o r i g i n o f the Cartesian coordinate system Oxy. A t some Instant o f time, taken as i n i t i a l ( f = 0), the body starts to penetrate the l i q u i d v e r t i c a l l y w i t h the body v e l o c i t y h{t) being prescribed (see Fig. 1 b). Here the o v e r d o t stands f o r the t i m e d e r i v a t i v e and h{t) is the vertical d i s p l a c e m e n t o f the body. The p o s i t i o n o f t h e body at t i m e Instant t is g i v e n b y t h e e q u a t i o n y = f{x) - h{t). The region occupied by the l i q u i d changes In time. A t any time Instant t > 0 the l i q u i d b o u n d a r y consists o f t h e f r e e surface a n d the w e t t e d p a r t o f the m o v i n g body, w h i c h Is k n o w n as the contact region b e t w e e n the l i q u i d and the body surface. N o t e t h a t the jets at the p e r i p h e r y o f the contact region are n o t s h o w n i n F i g . l . The contact region starts to g r o w f r o m a single p o i n t I n the w a t e r Impact p r o b l e m , w e need to d e t e r m i n e the f l o w caused by the m o v i n g body, the pressure d i s t r i b u t i o n i n the contact r e g i o n and the contact region Itself. The p r o b l e m is c o m p l i c a t e d since the c o n t a c t region and the p o s i t i o n o f f r e e surface are u n k n o w n I n advance and s h o u l d be d e t e r m i n e d together w i t h the l i q u i d f l o w .
To s i m p l i f y the f o r m u l a t i o n o f the I m p a c t p r o b l e m , g r a v i t y a n d surface tension are neglected, and the l i q u i d is m o d e l l e d as i n v i s c i d • Corresponding auttior. Tel.: + 7 9607947219.
E-mail address: tana@tiydro.nsc.ru (T.I. Khabal<hpasheva).
0141-1187/S - see front matter © 2013 Elsevier Ltd. All rigtits reserved. Iittp://dx.doi.org/10.1016/j.apor.2013.10.007
30 T.I. Khabakhpasheva et al. /Applied Ocean Research 44 (2014) 29-38
Fig. 1. Slcetclies of (a) initial configuration and (b) the flow caused by the impact.
a n d incompressible. The flow caused by I m p a c t is I r r o t a t i o n a l , t w o -d i m e n s i o n a l an-d s y m m e t r i c w i t h respect to the s y m m e t r y line x = 0. Presence o f the air above the w a t e r surface Is n o t taken into account. These assumptions are discussed b y Khabakhpasheva and K o r o b k l n [ 5 ] . These assumptions w e r e j u s t i f i e d and proved t o be reasonable b y c o m p a r i n g the obtained results w i t h e x p e r i m e n t a l data and the results predicted by CFD. Even i n such s i m p l i f i e d f o r m u l a d o n , the p r o b l e m o f w a t e r I m p a c t Is still c o m p l i c a t e d and its s o l u t i o n can be o n l y achieved by n u m e r i c a l methods [ 6 - 8 ] .
The p r o b l e m can be a d d i t i o n a l l y s i m p l i f i e d i f w e restrict ourselves to the i n i t i a l stage o f Impact w h e n the p e n e t r a t i o n o f the body Is small, a n d to b l u n t bodies w i t h [ f (x)| « 1 i n the contact region. The r e s u l t i n g a p p r o x i m a t e m o d e l o f w a t e r Impact Is k n o w n as the Wagner m o d e l [9 ]. W i t h i n this model, the w e t t e d part o f t h e body Is a p p r o x l m a t e d by an e q u i v a l e n t fiat disc, the b o u n d a r y conditions, b o t h i n the contact region and o n the free surface, are linearised and imposed o n the I n i t i a l l y u n d i s t u r b e d l i q u i d level, y = 0. The a p p r o x i m a t e b o u n d a r y value p r o b l e m w i t h respect to the c o m p l e x v e l o c i t y p o t e n t i a l Wo(z, t)
= (fix,y, t) + i X x , y , t ) , z = x + iy, has the s o l u t i o n
Wo(z, t) = / / ! ( f ) ( z - y z 2 - c 2 ( t ) ) , ( 1 ) w h e r e <p(x, y, t ) Is the v e l o c i t y p o t e n t i a l o f the Induced flow and fix,
y, t ) Is the stream f u n c t i o n (see Oliver [ 1 0 ] ) . Here the f u n c t i o n c(t)
describes the w i d t h o f the contact region, - c ( t ) < x < c(t). This f u n c t i o n Is d e t e r m i n e d by the so-called W a g n e r c o n d i t i o n , w h i c h requires that the elevation o f the free surface at the contact points x = ± c ( t ) Is equal to the vertical coordinate o f the b o d y s u r f a c e / ( c ) - h{t) at these points. This c o n d i t i o n can be reduced to t h e e q u a t i o n (see K o r o b k l n [111)
J"^ / [ c ( t ) s l n 0 ] d ( 9 = | f t ( f ) (2)
w h i c h does n o t require any k n o w l e d g e about the flow. In the con¬ tact region, t h e s o l u t i o n (1) provides <pix. 0, t ) = -h{t)y/c^ -x^. The pressure p(x, 0, f ) o n the w e t t e d part o f the b o d y is given b y the linearised B e r n o u l l i equation p = -p(pt(.x, 0, f ) , w h e r e p is the l i q u i d density. The h y d r o d y n a m i c force Fo(t) acting o n the entering body Is g i v e n b y f ( f ) = tl{ma(.t)h{t))/dt w i t h i n the W a g n e r theory, w h e r e 'na(t) = {K/2)pc'^{t) is the added mass o f the flat disk a p p r o x i m a t i n g the w e t t e d area o f the body. I t Is w e l l k n o w n that the Wagner theory overpredlcts the h y d r o d y n a m i c force f o r moderate deadrise angles o f the body b u t provides a reasonable a p p r o x i m a t i o n f o r small deadrise angles, arctan(/'(x)) < 2 0 ° , w h e n the loads are very h i g h .
To I m p r o v e the performance o f the W a g n e r model, i t was m o d i f i e d by I n c l u d i n g nonlinear terms b u t o n l y i n the B e r n o u l l i equation for the pressure (see K o r o b k l n [12]). The c o r r e s p o n d i n g m o d e l is k n o w n as the m o d i f i e d Logvlnovlch m o d e l ( M L M ) . This m o d e l was validated against n u m e r i c a l and e x p e r i m e n t a l results on w a t e r e n t r y o f b o t h s i m p l e shapes (wedge, circular c y l i n d e r ) and ship sections. It was no-ticed i n [ 1 3 ] that the M L M can be used i n calculations o f s l a m m i n g loads f o r ship sections w h i c h are n o t t h i n ( b o w sections). For m o r e general shapes, such as bulbous ship sections, the so-called gener-alised W a g n e r m o d e l was r e c o m m e n d e d .
The generalised Wagner m o d e l ( G W M ) was i n t r o d u c e d by Zhao et al. [ 6 ] . W i t h i n this m o d e l the b o d y b o u n d a r y c o n d i t i o n Is Imposed on the actual p o s i t i o n o f the e n t e r i n g surface, the free-surface boundary c o n d i t i o n s are linearised and Imposed o n the p i l e - u p height, w h i c h
is d e t e r m i n e d as p a r t o f the solution. The h y d r o d y n a m i c pressure Is given b y the nonlinear Bernoulli equation. The m o d e l was I n -vestigated by M e l , L i u and Yue [14] using the c o n f o r m a l m a p p i n g technique. The c o n f o r m a l m a p p i n g was used to t r a n s f o r m the flow region I n t o the l o w e r half-plane. The w e t t e d p a r t o f a s y m m e t r i c section was d e t e r m i n e d b y the Wagner c o n d i t i o n . The c o r r e s p o n d -ing Integral e q u a t i o n was solved by us-ing Chebyshev p o l y n o m i a l s . This m e t h o d was f u r t h e r developed by M a l l e r o n et al. [ 1 5 | , w h o ac-counted f o r singularities o f the c o n f o r m a l m a p p i n g at the i n t e r s e c t i o n points b e t w e e n t h e free surface and the m o v i n g body. Nevertheless, they n o t i c e d that the predictions o f the h y d r o d y n a m i c forces acting on e n t e r i n g bodies by G W M are not as good as those b y M L M , w h e n w e compare the theoretical forces w i t h the measured ones. This re-sult is n o t logical because f o r m a l l y G W M Is superior w i t h respect to M L M . However, i n b o t h the original Wagner m o d e l ( G W M ) and in M L M the integrations o f the calculated pressures are p e r f o r m e d rather accurately i n contrast to the G W M , in w h i c h calculations are done n u m e r i c a l l y w i t h o u t account for b o t h the flow and pressure singularities at the contact points. G W M predicts the pressures m u c h better t h a t M L M w h e n compared w i t h the f u l l y n o n - l i n e a r p o t e n t i a l s o l u t i o n but, strangely, the total h y d r o d y n a m i c force b y M L M bet-ter fits available e x p e r i m e n t a l data. A possible e x p l a n a t i o n o f this strange result was g i v e n by Malenica and K o r o b k l n [ 1 6 ] w h o h i g h -l i g h t e d t h a t G W M m a t h e m a t i c a -l -l y Is m o r e c o m p -l e x t h a n b o t h O W M , w h i c h a l l o w s analytical solutions and has no d i f f i c u l t i e s w i t h n u m e r i -cal treatments because computers actually are n o t i n v o l v e d , and f u l l y n o n - l i n e a r p o t e n t i a l m o d e l , i n w h i c h there are no contact points due to j e t flows i n a p r o x i m i t y o f the body surface and the v e l o c i t y field and the pressure d i s t r i b u t i o n are s m o o t h over the w e t t e d p a r t o f the e n t e r i n g body. G W M by Zhao et al. [6] j u s t looks s i m p l e b u t It is n o t so. Intensive theoretical t r e a t m e n t of this m o d e l Is needed to increase the p o t e n t i a l o f G W M to be used i n practical n u m e r i c a l calculations. I n the present study, analytical calculations o f the flow a n d pres-sure d i s t r i b u t i o n s are e m p l o y e d m u c h deeper t h a n i t has been done so far. Singular components o f the flow v e l o c i t y a n d the pressure are distinguished and treated carefully. The original p r o b l e m is reduced to t w o n o n - l i n e a r i n t e g r a l equations, w h e r e one o f t h e m serves to evaluate the c o n f o r m a l m a p p i n g o f the flow region and the o t h e r one to c o m p u t e the p o s i t i o n o f the contact p o i n t B o t h equations are s i n -gular b u t t h e i r solutions can be obtained w i t h g o o d accuracy. W e use the c o n f o r m a l m a p p i n g technique as in Mel et al. 114] a n d M a l l e r o n et al. [ 15|. Singularities o f the corresponding c o n f o r m a l m a p p i n g w e r e studied b y K o r o b k l n and i a f r a t i [ 1 7 ] . The o b t a i n e d s o l u t i o n s p r e d i c t accurately the h y d r o d y n a m i c forces for b l u n t sections s i m i l a r to M L M but a d d i t i o n a l l y t h e y give access to the pressure d i s t r i b u t i o n s w h i c h is n o t available w i t h i n M L M and can be used for bulbous b o w sections. Calculations o f the forces and the pressures by the present v e r s i o n o f the G W M are as q u i c k as i n M L M but require precalculations w h i c h depend o n l y on the body shape b u t not o n the b o d y m o t i o n .
2. Formulation of the problem
W i t h i n the generalised W a g n e r m o d e l the l i q u i d flow Is g o v e r n e d by the equations v 2 ^ = 0 ( y < ƒ ( X ) - / 1 ( f ) , |x| < c ( t ) ) u ( y < H ( t ) , | x | > c ( t ) ) , ( 3 ) V = 0 ( y = H ( t ) , | x | > c ( t ) ) . ( 4 ) S , ( x , t ) = v > y ( x , H ( t ) , 0 ( | x | > c ( f ) ) , ( 5 ) <Py = f'W'P.-hiC) ( y = / ( x ) - / i ( t ) , | x | < c ( t ) ) , ( 6 ) V ^ O ( x 2 + y 2 _ o o ) , (7) H ( t ) = / [ c ( f ) ] - / i ( t ) . (8)
S{X,0) = 0. c ( 0 ) = 0 . (9)
Position o f the section e n t e r i n g w a t e r is given b y the equation y =J{x)
- hit), w h e r e the s m o o t h even f u n c t i o n ; ( x ) describes the shape o f the
secdon. Position o f t h e l i q u i d free surface after the impact, y = S{x, f), S{-x, t) = S{x, t), S(x, 0 ) = 0, a n d the h o r i z o n t a l size o f the contact region, - c ( f ) <x< c(t), b e t w e e n t h e l i q u i d and the body are u n k n o w n in advance and have t o be d e t e r m i n e d as part o f the solution. The f r e e -surface b o u n d a r y conditions (4) and (5) are linearised and imposed on the line y = Hit), \x\ > c(t), w h e r e H ( t ) = S[c(t), t ] . The body boundary c o n d i t i o n (6) is taken i n its o r i g i n a l f o r m and is imposed on the actual position o f the body surface.
If the f u n c t i o n c(f) is k n o w n , t h e n one can calculate H( t ) by using (8) and solve the boundary-value p r o b l e m (3), (4), ( 6 ) and (7). Then one integrates the k i n e m a t i c c o n d i t i o n ( 5 ) subject to the i n i t i a l conditions (9), evaluates S[c(t), f ] = Hit) and checks Eq. (8). If the f u n c t i o n c(t) was g i v e n correctly, Eq. (6) is I d e n t i c a l l y satisfied at any time instant f.
Once the boundary-value p r o b l e m ( 3 ) - ( 9 ) has been solved, the h y d r o d y n a m i c pressure p(x, y, f ) i n the f l o w region is c o m p u t e d by using the non-linear and unsteady Bernoulli equation
pix,y.t) = ^p(^Vt+l\^'P\^y (10)
Gravity a n d surface tension effects are n o t taken i n t o account w i t h i n this approach, as w e l l as the presence o f the air.
Note that the conditions (4) and (6) do not m a t c h each o t h e r at the contact points, x = ± c(t). As a result, the f l o w velocity is singular at these points and the pressure ( 1 0 ) tends to - o o w h e n the contact points are approached. In the present version o f G W M , o n l y positive pressures matter.
The p r o b l e m ( 3 ) - ( 1 0 ) Is m u c h more complicated than b o t h the O W M and M L M . This is due to the fact that the flow region changes in time w i t h i n G W M b u t n o t w i t h i n O W M and M L M . As a result, the c o m p l e x velocity p o t e n t i a l (1) cannot be used w i t h i n G W M a n d the flow should be c o m p u t e d at each time instant. Eq. (2) for t h e coor-dinates o f the contact points x = ± c ( f ) also cannot be used i n the G W M a n d the f u n c t i o n c(f) is c o m p u t e d by using the original W a g n e r c o n d i t i o n t h r o u g h i n t e g r a t i o n o f the kinematic boundary c o n d i t i o n (5). Note that this m o d e l cannot be derived f r o m the f u l l y n o n l i n e a r potential model, w h i c h is resembled by G W M b u t has no f o r m a l rela-tion w i t h i t . Nevertheless, the pressure d i s t r i b u t i o n s p r o v i d e d by the G W M (3 ) - ( l 0) are v e r y close to those by the f u l l y nonlinear p o t e n t i a l m o d e l f o r all shapes studied so far.
Despite o f a significant d i f f e r e n c e between G W M and the O W M , the s o l u t i o n o f w h i c h Is given by ( 1 ) , some Ideas and finding f r o m the o r i g i n a l Wagner m o d e l can be applied to the p r o b l e m ( 3 ) - ( 9 ) i f a c o n f o r m a l m a p p i n g o f the flow region onto the l o w e r h a l f - p l a n e is k n o w n .
3. Conformal mapping
The v e l o c i t y p o t e n t i a l <pix, y, t) is d e f i n e d i n the region i2(c) s h o w n in Eq. (4). Note that this region is d i f f e r e n t f r o m the flow region, w h i c h is bounded by the f r e e surface y = S(x, t ) . The region fi(c) depends o n the coordinates o f t h e contact points x = ± c(t) and the shape o f the section, b u t not on the t i m e f d i r e c t l y . I t is convenient to I n t r o d u c e a u x i l i a r y complex plane ( = ^ + it} and the c o n f o r m a l m a p p i n g z = z ( f ) o f the l o w e r half-plane ;/ < 0 o n t o the region i2(c) i n t h e z - p l a n e as (see Fig. 2 f o r b o u n d a r y correspondence)
z = i H ( t ) + F ( f , c ) , (11) w h e r e F ( f , c) is an analytic f u n c t i o n i n < 0, such that F( ± 1, c) = ±c
a n d F ( f , c ) ~ F o c ( c ) i : as \(\ c o . A r e a l f u n c t i o n F:,c(c) is the c o e f f i c i e n t in the f a r - f i e l d asymptotics o f the c o n f o r m a l m a p p i n g (11). The real and i m a g i n a r y parts o f the analytic f u n c t i o n s F ( f , c) o n the b o u n d a r y Ï; = - 0 , - (X)< f < -F oo are denoted a s X ( f , c) and y(?, c). The intervals
©
Fig. 2. Tlie boundaiii correspondence for the conformal mapping z = z(i;, c).
^ < - 1 and f > 1 o f the boundary = 0 correspond t o the lines y =
Hit), \x\ > c(t) i n the z-plane. Therefore, y ( § , c) = 0, w h e r e |$| > 1. The
i n t e r v a l - 1 < ^ < 1 , = 0 corresponds to the w e t t e d p a r t o f the body surface, w h e r e y =Jix) - hit), w h i c h gives
y ( ? . c ) = / [ X ( t c ) ] - / ( c ) ( l f l < l ) . ( 1 2 ) Eq. (12) indicates that the boundary-value p r o b l e m w i t h respect to the
analytic F(<;, c) is nonlinear and can be solved f o r d i f f e r e n t c separately. I t is seen that the c o n f o r m a l m a p p i n g is k n o w n i f the f u n c t i o n X ( f , c), w h e r e |?| < 1, i n (12) is k n o w n . W e shall derive an i n t e g r a l equation w i t h respect to the d e r i v a t i v e X f ( f , c).
Let us consider the f u n c t i o n G ( c , c) = i ( F ^ ( f . c) - Fy,ic))y/^^ - 1, w h i c h is analytic i n the l o w e r half-plane <; < 0 e n d decays as f oo. Here ^ _ i ~ < as |cl ^ oo and ^i;^ - 1 = w h e r e f = ^ - iO and t > 1, ^ f ^ - 1 = - - / ^ ^ - 1, w h e r e ^ = § - (0 and f < - 1 ,
- 1 = - i V l --§2, w h e r e f = ^ - iO a n d |?| < 1. The real part o f the f u n c t i o n G ( f , c) on the boundary f = ^ - lO is
[ 0 ( I ^ I > 1 ) .
9 A [ G ( ^ - I O , C ) ] =
[ ( X f ( ? , c ) - F ^ ( c ) ) y i - f 2 im<\) a n d i m a g i n a r y part is
J [ C ( ? - ! 0 . c)] = y f ( ? . c ) y i - f 2 ,
w h e r e |^| < 1. The real and imaginary parts o f the analytic i n ;/ < 0 f u n c t i o n are related by the H i l b e r t f o r m u l a [18], w h i c h gives
n ( f . c ) y w 2 = i P . v . | \ x f ( r , c ) - F , , ( c ) ) ^ ^ Ï ^ J ^ . ( 1 3 ) w h e r e the integral Is understood as the Cauchy p r i n c i p a l value i n t e -gral a n d - 1 < f < T i n f o r m u l a (13), the d e r i v a t i v e y f ( f , c) is calculated b y using (12), Yf (^, c) = ƒ [ X ( f , c)|X{($, c), and
P.v.f ^ ï ^ ^ = -.^ ( R K l ) .
I t is seen t h a t i t is convenient to i n t r o d u c e a n e w u n k n o w n f u n c t i o n
Uil;, c) = X t ( f . c j ^ l - ?2/F^(c), I f l < 1, w i t h respect to w h i c h Eq.
(13) takes the f o r m
f [ X ( f . c ) ] U ( f , c ) - l Uir,c)dr ( | f | < l ) . (14)
This is a singular nonlinear integral e q u a t i o n w i t h respect to the f u n c -tion U ( f , c). The f u n c t i o n U ( f , c) is an even f u n c t i o n o f f f o r s y m m e t r i c shapes. The s o l u t i o n o f Eq. (14) should satisfy the e q u a l i t y
1 / ' ^ ^ ^ = 1. ( 1 5 )
w h i c h f o l l o w s f r o m (14) and the c o n d i t i o n t h a t X ^ ( f , c ) \ / l - f ^ _ i . Qas If I ^ 1. The latter c o n d i t i o n comes f r o m the local analysis o f the c o n f o r m a l m a p p i n g near the contact points [ 1 7 ] . The local analysis provides i n the present notations
F ; « . C ) = c , ( i : - 1 ) - ' - " " ' / ' - F C , ( ? _ I ) ' - 2 > * ) / - T + q |-(^ _ l y - K o / . T j ( i g )
as f - i - 1, w h e r e Ci(c) and C2(c) are real u n d e t e r m i n e d f u n c t i o n s a n d
yic) is the deadrise angle at the contact p o i n t , yic] = t a n ^ ' [ f ( c ) ] . The
a s y m p t o t i c f o r m u l a (16) shows that U ( f , c) = 0 [ ( 1 - f / " ' ^ " ' ' f ^ " ' " ] as f 1 w h i c h j u s t i f i e s t h e c o n d i t i o n ( 1 5 ) . The local analysis also shows
34 T.I. Khabakhpasheva et al./Applied Ocean Research 44 (2014) 29-38
and the s o l u d o n o f the i n t e g r a l equation ( 1 4 ) is g i v e n by
-1/2 ( / ' ^ + l ) c o s ) / - f 2 / t S ( f , c ) = 4;^2(c) ( l + f ) » *
-[1 + +2iL cos y f
(53)
S ( l , c ) = 4;<2(c) 2 - ' ' o - ' / 2 c o s K , M = ( | ^ )
The f u n c t i o n m(c, CQ) i n Eq. (38) is g i v e n f o r the circular c y l i n d e r by the f o r m u l a
2 c g ( l + . ) ( c ^ c g ) - ' - ' - ' ° '
ko^c^>W(c + C o ) ' * ' ^ « ) ( l - X ) ^ ' C + Co
J
(54)Both Eq. ( 5 4 ) and Eqs. (39) and (53) p r o v i d e 2 i - i / / f o ( c )
m(c, c) =
6.2. Numerical integration of the integral equation (14)
The i n t e g r a l e q u a t i o n ( 1 4 ) is solved b y iterations f o r a g i v e n value o f c. As a first guess w e t a k e X < ° ' ( f , c) = cf and c o m p u t e the s o l u t i o n [ / ( f , c). Then w e calculate F ^ ( c ) and the next a p p r o x i m a t i o n o f X ( i ) ( f , c) b y ( 1 9 ) . The o b t a i n e d f u n c t i o n X ( i ' ( f , c) Is s u b s t i t u t e d In ( 1 4 ) and the e q u a t i o n is solved again. A t each step o f i t e r a t i o n s j, j > 1, w e c o m p u t e
s = m a x X t J J ^ . c )
-o<«<i ' X " - " ( f , c )
Iterations stop w h e n S Is smaller than a prescribed value S>. The p r o -cedure consists o f t w o steps: (1) n u m e r i c a l s o l u t i o n o f Eq. ( 1 4 ) f o r a g i v e n X ( f , c); ( i i ) i n t e g r a t i o n s i n (19). The results o f calculations are v a l i d a t e d w i t h respect to the k n o w n solutions f o r w e d g e and c i r c u l a r cylinder.
For a g i v e n f u n c t i o n X ( f , c) l e t us d e n o t e / [ X ( t , c)] i n (14) b y y ( f ) . The dependence o f the s o l u t i o n o n c Is d r o p p e d i n this subsection. W e use the f a c t t h a t U ( f ) is an even f u n c t i o n . Then Eq. (14) can be w r i t t e n as
Jr J 0
r-F?
U ( r ) d r = f ( 0 < f < l ) . ( 5 5 ) The f u n c t i o n U ( f ) is sought i n the f o r m (17) w i t h S ( f ) being a n e w u n k n o w n s m o o t h f u n c t i o n w h i c h has a c e r t a i n value S ( l ) at f = 1.The i n t e r v a l 0 < f < 1 is d i v i d e d I n t o N subintervals [ f j , f j + , ], w h e r e f j = i i - \ )/N, 1 < J < N . The f u n c t i o n S ( t ) is a p p r o x i m a t e d by a constant Sj w i t h i n the i n t e r v a l f j < f < f j + , . The constants Sj are u n k n o w n . There are N such u n k n o w n constants. They are d e t e r m i n e d by c o l l o c a t i o n m e t h o d . Collocation points Xj are placed In the Intervals [ f j , f j + 11 i n such a w a y t h a t Xj = f j -i- aj, w h e r e aj, 0 < a ; < 1 / N are parameters o f the c o l l o c a t i o n m e t h o d . I n the present calculations aj = 1/(2N). This is, the c o l l o c a t i o n points are placed at the m i d d l e o f the intervals [ f j , f_,- + , ] .
Eq. ( 5 5 ) leads to N algebraic equations
w h e r e
/
tnin +l r - X i r + Xj ( l - r j ^ d r .(56)
(57)
The i n t e g r a l i n (57) has been reduced to a p o w e r series, coefficients o f w h i c h ffi decay as w h e n i ^ oo. In calculations 100 terms i n the series w e r e retained.
Fig. 3. The function S(i;, c) calculated for c = 0.125, 0.25 and 0.5 (lines 1-3) as the solution of the integral equation (14) for the unit circular cylinder (solid line) and by formula (53) (dashed line).
S u b s t i t u t i n g (17) i n t o (19), w e a r r i v e at the f o r m u l a e f o r F^{c) and Xixj. c) X[Xj.c)=F^{c) J - 1 J2S"l''"-Ln+^] + S j { L j - L { X j ) ) (58) (59) w h e r e Ln = L (?n) L{xj) =
j
(1 - r)''dr
72o~xj)^j2
m=0 am m+l<othe coefficients am are calculated b y the recurrence r e l a t i o n
2 m -
12 m ao = 1.
N e x t w e calculate y{Xj) = f'[X{xj, c ) l and solve the system (56) again. The iterations stop w h e n S < c/lOON. Usually less t h a n five steps o f iterations are needed.
Validation for unit circular cylinder. The n u m e r i c a l a l g o r i t h m o f
this subsection is validated against the k n o w n s o l u t i o n (51), (53) f o r the circular cylinder. I n the present case, 0 < c < 1. The f u n c t i o n S(f, c) f o r several fixed values o f c is s h o w n i n Fig. 3 f o r N = 40 i n (56). I t is seen t h a t the n u m e r i c a l s o l u t i o n oscillates near the centre o f the contact region. However, close to the c o n t a c t points, w h e r e the pressures are expected to be m u c h h i g h e r t h a n at the centre, the agreement is v e r y good. For finer mesh i n f w i t h N = 100 these oscillations disappear. However, calculations w i t h N = 100 do n o t p r o v i d e s i g n i f i c a n t i m p r o v e m e n t In terms o f S ( l , c) and the c o e f f i c i e n t
Focic) s h o w n i n Fig. 4.
Validation for the wedge of 45'. I t was s h o w n t h a t the proposed
a l g o r i t h m w e l l reproduces the a n a l y t i c a l s o l u t i o n ( 5 0 ) f o r the wedge. The analytical f o r m u l a e (50) p r e d i c t t h a t the f u n c t i o n S(f, c) is I n -d e p e n -d e n t o f c. W e fin-d exact value S ( l , c) = 0.8409 an-d n u m e r i c a l values S ( l , c) ^ 0.8483 f o r N = 40 ( r e l a t i v e e r r o r is 0.9%) and 0.8440 f o r N = 100 (relative e r r o r Is 0.4%). The f u n c t i o n S(f, 0.5) is s h o w n i n
\
[
"—' i J L . , ^ f.(0 /
\
. L _ _ _Fig. 4. Tlie function S( 1, c) calculated as tlie solution of the integral equation (14) w i t h N = 100 for the unit circular cylinder (solid line) and by formula (53) (dashed line) and the coefficient F^[c) calculated by formula (19) and the solution of integral equation ( 1 4 ) w i t h N = 100 forttie unit circular cylinder (solid line) and by formula (51) (dashed line).
S(£.0.5)
^^^^^^
ü
0i
iC b )
Fig. 5. (a) The function S(f, 0.5) predicted by the integi-al equation (14) for the wedge of 45 degrees w i t h N = 100 (solid line) and by formula (50) (dashed line), (b) The coefficient f ^ ( c ) calculated by formula (19) w i t h N = 40 (solid line) and by formula (49) (dashed line).
Fig. 5a w i t h JV = 100 and the c o e f f i c i e n t F x ( c ) i n Fig. 5b w i t h N = 40. It is seen, t h a t the agreements are v e r y good.
6.3. Integral equation (38) and its solution
Accurate calculation o f the size o f the c o n t a c t region, w h i c h is described b y the f u n c t i o n c(f), is crucial f o r the success o f the pressure p r e d i c t i o n . I n equation (38), the shape f u n c t i o n / ( c ) and the power/£-(c) are k n o w n , and the f u n c t i o n m(c, CQ) depends o n l y on the c o n f o r m a l m a p p i n g .
The f u n c t i o n ni(c, CQ) is given by f o r m u l a ( 5 4 ) f o r the u n i t circular c y l i n d e r a n d calculated by the m e t h o d described i n Section 5. The a n a l y t i c a l and n u m e r i c a l results are c o m p a r e d I n Fig. 6. I t is seen t h a t the a g r e e m e n t is very good.
The I n t e g r a l equation (38) is solved by the c o l l o c a t i o n m e t h o d w i t h CniB.v b e i n g the m a x i m u m value o f the f u n c t i o n c(t) and Nc the n u m b e r o f the collocation points q = iAc, w h e r e 1 < i < Nc. Ac =
Fig. 6. The function m(c, co) computed by the piesent numerical method (solid lines) and by the analytic formula (54) for c = 0.25,0.5 and 0 7 5 (lines 1-3).
Cmax/Nc. Eq. (38) provides
f(Ci) =
m ( C j , C o ) d c o
2 if.2
o/'-i
, » ( Q ) ' ( 6 0 )
w h e r e C j _ i = 0 at j = 0 and 1 < / < Nc. The p r o d u c t s N{cQ)m{cu CQ), w h e r e Cj _ i < CQ < c,-, are a p p r o x i m a t e d by linear f u n c t i o n s
N(r,i)ra(r,.r,i)=N(rj i ) m ( r , . r j , ) + | N ( C j ) m ( r i . r , ) - N (r, i ) m ( r , . r , i ) | ( r „ ^ c , , ) / A r . (61)
S u b s t i t u t i n g ( 6 1 ) i n (60) and rearranging terms, w e o b t a i n A c / ( Q ) - E > ' i N j m i . j |cj+,Si.j+, - Cj_,S,.j + P(j - Pi.j+i 1 Hi = N,- = N ( C i ) . ' I.J ƒ -m,., |Pi,, -c,_,S,., ] mi,j =m{Ci.Cj).
codco
(62)f i
dCn ^ - ' ( l - c 2 / c ? ] The coefficients S j j and Pij are calculated by f o r m u l a e: C i 2 ' - 2 ' ' i 2 ( 1 - / ^ i ) • C j - i 2Ci
V
2ci .2\ ' - ' ' i w h e r e the f u n c t i o n i((z) is ( 6 3 ) ( 6 4 ) (65)For the u n i t c i r c u l a r c y l i n d e r the calculated f u n c t i o n N(c) Is s h o w n i n Flg.7 by the s o l i d line. I n the calculations c,„ax = 1 and Nc = 50.
The dashed line is f o r this f u n c t i o n predicted by the W a g n e r t h e o r y f o r equivalent parabolic c o n t o u r y = ( 1 / 2 ) x 2 . I n the W a g n e r t h e o r y
N{c) = ( l / 2 ) c . It is seen t h a t the W a g n e r theory can be used i n c o m b i
-n a t i o -n w i t h the a p p r o x i m a t i o -n o f the circular c y l i -n d e r by a parabolic c o n t o u r f o r 0 < c < 0.4.
6.4. Pressure distribution and the total force
The h y d r o d y n a m i c pressure i n the w e t t e d area, 0 < x < c o f t h e body is g i v e n by Eqs. ( 2 5 ) - ( 2 7 ) w h i c h yield
p = />/i2 / l , ( f , c ) ( l ~ f ) -k{c) _ ' 4 2 ( t . C ) ( l - f ) -2k{c:
+
( 6 6 )36 T.I. Khabakhpasheva et al. /Applied Ocean Research 44 (2014)29-38
0 0,1 0 2 0.3 0.7 0.9 O.ït
Fig. 7. Tlie function N(c) for the unit circular cylinder predicted by Eq. (38) (solid line) and by the Wagner theory (dashed line).
1
!
1 " " 5 0,1 0.15 0 2 0,25 0.1 !1
!_ J
]——
r
1 " ]r
1 "1
—
1
—
i
j 0 0 1 0 2 Ö.3 0 4 0 5 0 6 0.7 0 3 0.0Fig. 8. The functions P,{x, c) (solid line) and P„(x, c) (dashed line) f o r the unit circular cylinder at c = 0.3 (a) and c = 0.6 (b). Calculations are performed w i t h N = 200 and Nc = 100.
W h e r e t h e f u n c t i o n s / l i ( f , c), / l 2 ( f . c), P „ ( f , c) and /13(c), 744(c) are s m o o t h , d e p e n d e n t o n l y o n the shape o f the body a n d can be p r e -c o m p u t e d . The values o f the f u n -c t i o n s 7 4 , ( f , -c), / l 2 ( f , -c), P,v(f, -c) are calculated at the c o l l o c a t i o n points f = Xj o f Eq. (14). The derivadves W i t h respect to c In / 4 , ( f , c) and 743(c) are calculated b y the finite d i f f e r e n c e m e t h o d . A t points d i f f e r e n t f r o m the c o l l o c a t i o n points, the f u n c t i o n s are calculated by linear i n t e r p o l a t i o n . The f u n c t i o n s Pv(f, c) a n d Pvv(f, c), w h e r e x =X{l, c), d e p e n d o n l y o n the b o d y shape. These f u n c t i o n s o f x are d e p i c t e d f o r the u n i t circular c y l i n d e r In Figs. 8 and 9 f o r c = 0.3 and c = 0.6. Details o f the f u n c t i o n s Pvix, c) close to the c o n t a c t p o i n t x = c, w h e r e Pv - c o , are n o t s h o w n i n these figures. It is seen t h a t P^ix, c) is rather small d u r i n g t h e i n i t i a l stage w h e n c « 1.
P
1 T \ / !y r
1 1 . - - ' 1 L•
/ / / / / f ^ / i ii
•
/ / / \—
—
1
1-—-T
ii
1 1 ii
Fig. 9. The added mass of the unit circular cylinder M„(c)/p. Solid line is for N = 150, Nc = 100, crosses are for N = 50, Nc = 40, and dashed line is f o r the added mass of "equivalent flat plate", Mo/p = ( j r / 2 ) c ' .
Fig. 10. The funcdon £(c) f o r the unit circular cylinder: Solid line is f o r N = 150, Nc •¬ 100, crosses are for N = 50, Nc = 40.
The h y d r o d y n a m i c force F(t) is g i v e n b y the f o r m u l a (28), w h e r e o n l y p o s i t i v e pressures are integrated. W i t h i n the a p p r o a c h b y M e l et al. [14], w i t h f . ( t ) = 1 i n (28), the force Is g i v e n b y (29) and does n o t r e q u i r e calculations o f the pressure d i s t r i b u t i o n . I n (29), Ma(c) is the added mass o f the b o d y and £ ( c ) Is a f u n c t i o n o f the size o f the w e t t e d region c. The f u n c t i o n s Ma(c)//) and £ ( c ) f o r the u n i t c i r c u l a r c y l i n d e r are s h o w n i n Figs. 9 and 10. It Is i m p o r t a n t to n o t i c e t h a t E(c) is negative and tends to a non-zero value as c ^ 0. Therefore, the second t e r m In (29) provides an i m p o r t a n t c o n t r i b u t i o n to t h e total force F„,/j,(t) f o r s m a l l times. W i t h i n the W a g n e r t h e o r y o f " f l a t -disk a p p r o x i m a t i o n " the h y d r o d y n a m i c force Fw(t) is g i v e n by the first t e r m In (29), w h e r e Ma{c) = {jr/2)pc\t) and c ( t ) = 2 ^ / m . For constant speed V o f the cylinder, the W a g n e r m o d e l provides F,v(f) = 2;7-p\/2R.
A better a p p r o x i m a t i o n o f the force is given by t h e second-order W a g n e r t h e o r y [19], w h e r e , F it) pV^R By using Eqs. (51 £ ( c ) ^ - ^ R 2;r - 2 (TT -F 2) y V t / R .
(53) f o r a circular cylinder, one can s h o w t h a t
as c 0. Thus, the m e t h o d by M e l et al. [14] u n d e r p r e d i c t s the force a t t = 0 b y 2 5 % .
The forces calculated b y the positive pressure i n t e g r a t i o n a n d b y the m e t h o d o f M e i et a l . [ 1 4 ] are s h o w n i n Fig. 1 1 . For c o m p a r i s o n , e x p e r i m e n t a l results by A r m a n d and Cointe [19] and M L M p r e d i c t i o n [ 1 2 ] are also s h o w n . I t is seen t h a t the forces by G W M and M L M are v e r y close to each o t h e r f o r 0 < f < 0.2. The f o r m u l a ( 2 9 ) u n d e r p r e d i c t s
pV-R ,
1
1i .1
: ! - . .i i
I I I i ; i1
i i . 1 ' : I'l/R Fig. 1 1 . Tlie force acting on tlie circular cylinder of radius R entenng water at constant speed V. The non-dimentional force given by the generalised Wagner model (28) w i t h integration of the positive pressures is shown by the solid line, by the method of Ivlei et al. | 1 4 | by the dashed line. Crosses are for experimental results by Cointe and Armand [20|, and pluses are for MLM 112].the force o n this i n t e r v a l . However, f o r 0.2 < f < 0.45 the p r e d i c t i o n by G W M is very close to t h a t by ( 2 9 ) b u t M L M u n d e r p r e d i c t s t h e force o n this i n t e r v a l . The deadrise angle at the contact points at Vt/R = 0.2 is estimated as 70". One m a y conclude that M L M can be used for shapes w i t h deadrise angles smaller than 70=.
6.5. Summary of the numerical algorithm
In this subsection, w e s u m m a n s e the steps o f the described algor i t h m to h e l p a algoreadealgor w i t h its i m p l e m e n t a t i o n . The a l g o algor i t h m p algor o -vides the pressure d i s t r i b u t i o n a l o n g the w e t t e d part o f an s y m m e t r i c section e n t e r i n g w a t e r v e r t i c a l l y , and the total h y d r o d y n a m i c force F(t) a c t i n g o n this section. The n u m e r i c a l procedure consists o f f o u r steps. First t w o steps do n o t require the m o t i o n o f t h e b o d y b u t o n l y its s h a p e / ( x ) .
Step 1. Numerical conformal mapping.
A t this step w e specify the h o r i z o n t a l p o s i t i o n x = c o f the c o n t a c t point, 0 < c < Cmax, and solve n u m e r i c a l l y the integral e q u a t i o n ( 1 4 ) w i t h respect to the regular f u n c t i o n S(?, c) i n t r o d u c e d b y (17). Note t h a t i n t h e present a l g o r i t h m w e proceed b y steps i n c b u t n o t i n time t. The dependence c = c ( f ) w i l l be recovered at the end o f the n u m e r i c a l calculations f o r a prescribed m o t i o n o f the body.
The integral e q u a t i o n ( 1 4 ) is discretised leading to the system o f equations (56), w h e r e k = ( 1 / 2 ) - ( y ( c ) / j r ) and y ( c ) is the dead-rise angle o f the body at x = c, tan / ( c ) = ƒ ( c ) . The coefficients Aj„ i n the sys-t e m are g i v e n by insys-tegrals ( 5 7 ) w h i c h are evaluasys-ted n u m e r i c a l l y . The coefficients / ( x j ) i n ( 5 6 ) depend o n the s o l u t i o n , y{Xj) = f'[X{xj.c)],
this is w h y the system ( 5 6 ) is solved by Iterations. As the i n i t i a l guess w e t a k e X ( 0 ) ( f , c) = c f , w h i c h gives Y^°\xj) = f'{cxj). The linear sys-tem (56), w h e r e y ( X j ) is a p p r o x i m a t e d by ƒ ( c X j ) , is solved by the Gauss m e t h o d w i t h respect to Sj = S ( f , c), f j < f < f j + i .
N e x t w e calculate F ^ ( c ) by the f o r m u l a ( 5 8 ) and the first I t e r a t i o n X ( i ' ( X j , c) o f the c o n f o r m a l m a p p i n g by ( 5 9 ) . If the i n i t i a l guess X<°'(Xj, c) = cXj a n d the result o f the first i t e r a t i o n X(''(x,-, c) are close to each other, iterations are t e r m i n a t e d and w e proceed to the n e x t value o f the c o o r d i n a t e o f the contact p o i n t c + Ac, w h e r e Ac is a prescribed step In c. I f the result o f the first i t e r a t i o n is n o t satisfactory, w e c o n t i n u e w i t h iterations s e t t i n g y{Xj) = / ' [ X ' i ) ( X j . c)] a n d s o l v i n g the linear system (56) once again. Then w e c o m p u t e n e w a p p r o x i m a t i o n s of Foc(c) a n d X^^1{Xj, c) b y (58), ( 5 9 ) and c o n t i n u e w i t h iterations i f X ( ' ' ( X j , c) and X<^'(Xj, c) are n o t close e n o u g h to each other.
At the e n d o f step 1 w e c o m p u t e d the m a p p i n g X ( f , c) f o r a certain mesh i n 0 < f < 1 and f o r d i f f e r e n t values o f c. W e c o m p u t e d also the regular f u n c t i o n S ( f , c) w h i c h is p r o p o r t i o n a l to the d e r i v a t i v e X f ( f . c) = S ( f , c ) . F ^ ( c ) • (1 - f ) ^ - ( ' / 2 ) ( i + 5 ) - ( V 2 ) ^ 0 < f < 1. Note
t h a t negative values o f f and X ( f , c) are n o t considered due to the s y m m e t r y o f the flow d o m a i n . In a d d i t i o n , the d e r i v a t i v e X c ( f , c) is c o m p u t e d by n u m e r i c a l d i f f e r e n t i a t i o n .
Step 2. The Wagner condition by integral equation.
At this step w e c o m p u t e the ratio N(c) = h{t)/c{t) o f the v e r t i c a l speed /7(f) o f the body and the h o r i z o n t a l speed c(C) o f the c o n t a c t p o i n t b e t w e e n the f r e e surface and the m o v i n g b o d y surface. It is i m p o r t a n t t h a t this r a t i o depends on the shape o f the b o d y b u t n o t on its m o t i o n w i t h i n the generalised W a g n e r m o d e l (see Section 5), and I t is governed by the integral e q u a t i o n ( 3 8 ) . This e q u a t i o n is solved n u m e r i c a l l y by f o r m u l a e ( 6 2 ) - ( 6 5 ) . These f o r m u l a e r e q u i r e the f u n c t i o n m(c, CQ), w h i c h is d e f i n e d by ( 4 0 ) t h r o u g h the c o n f o r m a l m a p p i n g c o m p u t e d at step 1. C o m p u t a t i o n o f the f u n c t i o n m(c, CQ) has been reduced to the i n t e g r a t i o n o f the o r d i n a r y d i f f e r e n t i a l e q u a t i o n (47) w i t h the I n i t i a l c o n d i t i o n f = 1 at c = CQ. The d i f f e r e n t i a l e q u a t i o n (47) was integrated by the f o u r t h - o r d e r R u n g e - K u t t a m e t h o d f o r the values o f CQ w i t h step A c The step o f the i n t e g r a t i o n was also equal to A C Note t h a t this is the f u n c t i o n JW(f(c CQ), CQ) o n the right-hand side o f Eq. (47), w h i c h w e need to c o m p u t e m ( c CQ) by ( 4 6 ) b u t n o t the f u n c t i o n f ( c CQ) i t s e l f The f u n c t i o n /W(f, CQ) is c o m p u t e d by ( 4 5 ) n u m e r i c a l l y at each step o f the i n t e g r a t i o n .
Step 3. Elements of tlie hydrodynamic pressure.
The pressure d i s t r i b u t i o n i n the contact region o f the e n t e r i n g body, 0 < X < c, Is g i v e n i n the p a r a m e t r i c f o r m ( 2 5 ) - ( 2 7 ) together w i t h the c o n f o r m a l m a p p i n g x = X ( f , c), w h i c h was c o m p u t e d at step 1. The c o m p u t a t i o n s o f step 2 enter these f o r m u l a e o n l y t h r o u g h the f u n c t i o n N(c) i n (26). Regular elements i n ( 2 5 ) - ( 2 7 ) , w h i c h are n o t singular, are d i s t i n g u i s h e d in f o r m u l a (66). These e l e m e n t s are p r e -c o m p u t e d as f u n -c t i o n s o f f and -c
The m o t i o n o f the b o d y is described by the p e n e t r a t i o n d e p t h h{t) w h i c h m u s t be prescribed i n the present a l g o r i t h m . Note t h a t the f u n c t i o n N(c) provides the r e l a t i o n b e t w e e n the p e n e t r a t i o n d e p t h h and coordinate o f the contact p o i n t c b y
c
h = j N{c)dc.
0
C o m p a r i n g the latter r e l a t i o n w i t h the g i v e n f u n c t i o n h = /i(t), w e c o m p u t e the f u n c t i o n c = c ( f ) and finally the pressure d i s t r i b u t i o n p(x, t ) at any p o i n t o f the w e t t e d area u s i n g i n t e r p o l a t i o n b e t w e e n the nodes Xj and the c o n f o r m a l m a p p i n g f r o m step 1.
Step 4. Hydrodynamic force.
The h y d r o d y n a m i c force F(t) is c o m p u t e d n u m e r i c a l l y by u s i n g the Integral (28). The u p p e r l i m i t f > ( t ) In this i n t e g r a l is i m p o r t a n t f o r accurate p r e d i c t i o n o f the force by the generalised W a g n e r m o d e l . The f u n c t i o n f ' ( f ) is the r o o t o f the pressure (25), w h i c h is the closest to f = 1. To c o m p u t e the root, w e need the regular e l e m e n t s o f t h e pressure and the f u n c t i o n c = c(t) f r o m step 3. The e q u a t i o n p ( f , ( t ) , t ) = ph^ ( f ) Pv ( f , ( f ) . c ( f ) ) + ph"{t) P „ ( f , ( t ) . c ( f ) ) = 0 is solved by the b i s e c t i o n m e t h o d w i t h respect to f - ( f ) at each time i n s t a n t t. Finally the force F(t) is evaluated n u m e r i c a l l y using (28).
7. Conclusion
The generalised W a g n e r m o d e l o f w a t e r i m p a c t w a s i n v e s t i g a t e d by m a t h e m a t i c a l means. Both the v e l o c i t y field and pressure d i s t r i b u -t i o n p r e d i c -t e d by -this m o d e l are singular a-t -the c o n -t a c -t poin-ts. These singularities w e r e d i s t i n g u i s h e d and separated. The regular parts o f the pressure d i s t r i b u t i o n w e r e c o m p u t e d . The w a t e r i m p a c t p r o b l e m was reduced to t w o singular i n t e g r a l equations, n u m e r i c a l solutions of w h i c h w e r e c o m p a r e d w i t h available analytical results f o r wedges and c i r c u l a r cylinders. The n u m e r i c a l a l g o r i t h m s o f s o l v i n g these t w o integral equations w e r e described i n d e t a i l .
I t was s h o w n that, once the n u m e r i c a l confornnal m a p p i n g is k n o w n , b o t h the v e l o c i t y field and the pressure d i s t r i b u t i o n are g i v e n
38 T.I. Khabakhpasheva et al. /Applied Ocean Research 44 (2014) 29-38
b y a n a l y t i c a l f o r m u l a e w i t h singular and regular elements b e i n g separated. The l a t t e r helps to i n t e g r a t e the pressure accurately, to d e t e r -m i n e t h e t o t a l h y d r o d y n a -m i c force.
I t was s h o w n t h a t the m e t h o d o f [ 1 4 ] , i n w h i c h b o t h negative a n d p o s i t i v e pressures are i n t e g r a t e d , u n d e r p r e d i c t s the f o r c e d u r i n g t h e e a r l y stage b u t is r a t h e r accurate a t the l a t e r stages. W i t h i n this m e t h o d t h e c a l c u l a t i o n o f t h e force is r e d u c e d t o evaluations o f t w o integrals I n f o r m u l a ( 2 9 ) . I t is I n t e r e s t i n g to n o t i c e t h a t the p r e d i c t i o n o f the f o r c e b y M L M is v e r y close to t h a t b y t h e o r i g i n a l G W M f o r deadrise angles s m a l l e r t h a n 7 0 ° .
The a p p r o a c h o f this paper can be generalised to a s y m m e t r i c c o n tours e n t e r i n g c a l m w a t e r w i t h b o t h v e r t i c a l a n d h o r i z o n t a l c o m p o -nents o f t h e b o d y v e l o c i t y . The approach is a p p l i c a b l e o n l y t o m o t i o n s w i t h p r o v i d e m o n o t o n i e e x p a n s i o n o f t h e c o n t a c t r e g i o n I n b o t h d i -rections.
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