Application of variational method to the vertical hydrodynamic impact of axisymmetric bodies

Applied Ocean Research 39 (2012) 75-82

Contents lists a v a i l a b l e at SciVerse ScienceDirect

Applied Ocean Research

j o u r n a l h o n n e p a g e : w w w . e l s e v i e r . c o m / l o c a 1 e / a p o r

Application of a variational method to the vertical hydrodynamic impact of

axisym-metric bodies

Flavia M. Santos, Leonardo Casetta, Celso P. Pesce'

Offshore Mechanics Laboratory. Department of Mechanical Engineering, Escola Politécnlca, University of Sao Paulo, Brazil

A R T I C L E I N F O A B S T R A C T

Article history:

Received 18 November 2010

Received in revised form 15 August 2012 Accepted 10 October 2012

Keywords:

Desingularized variational numerical method Weak formulation

Hydrodynamic impact problem Axisymmetric bodies

The application of a desingularized variational n u m e r i c a l method to the vertical h y d r o d y n a m i c impact problem of a x i s y m m e t r i c bodies is addressed here w i t h i n the so-called G e n e r a l i z e d von K a r m a n Model (GvKM). A w e a k formularion is used and the velocity potential is numerically a p p r o x i m a t e d in a Sobolev space. Trial functions are c o n v e n i e n t l y w r i t t e n as finite summations of elementary potentials. A m a i n advantage of the proposed technique is the fact that a first-order error in the velocity potential computation implies a second-order error in the added mass value. Good agreement in added mass calcularions is verified for a sphere and for a n oblate spheroid in comparison w i t h results obtained from W A M I T ®

) 2 0 1 2 Elsevier Ltd. All rights reserved.

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

First studies of the l i y d r o d y n a m i c i m p a c t p r o b l e m go bacl< to 1930s and are associated to t h e classical w o r k s o f v o n K a r m a n [ 1 ] and W a g -ner [ 2 ] . Such authors w e r e i n i t i a l l y m o t i v a t e d w i t h the i m p a c t o f seaplane floats d u r i n g ' l a n d i n g ' . B o t h v o n K a r m a n and W a g n e r ap-proaches f o l l o w f r o m t h e i n t e r p r e t a t i o n o f the i m p a c t i n g body as an equivalent f l a t plate t h a t is supposed to be collapsed o n t o the h o r i z o n -tal plane. Wagner approach takes i n t o account the p i l e d - u p w a t e r ef-fects at the v i c i n i t y o f the i m p a c t i n g body. Recent w o r k s o n the t h e m e are devoted to t w o - d i m e n s i o n a l (see, e.g. [ 3 - 5 ] ) and a x i s y m m e t r i c a l (see, e.g. [ 6 - 8 ] ) cases. I n the latter, cones and spheres are considered. Instead o f the usual f l a t plate a p p r o x i m a t i o n , S h i f f m a n and Spencer [9,10] show that, w h e n c o n s i d e r i n g cones and spheres, lens and e l l i p -soids can be c o n v e n i e n t l y used. W i t h i n the t h r e e - d i m e n s i o n a l cases, m e n t i o n m u s t be done to the inverse W a g n e r m e t h o d w h i c h was d i s -cussed b y Scolan and K o r o b k i n [ 1 1 ] i n the c o n t e x t o f e l l i p t i c contact lines.

N u m e r i c a l technique is one o f the c u r r e n t research fields o n the t h e m e . The w o r k s o f Zhao and Faltinsen [12,13|, Takagi [ 1 4 | , Peseux et al. [ 1 5 ] , Sun and Falrinsen [ 1 6 ] , W u [ 1 7 ] , and X u et al. [ 1 8 ] can be cited i n this sense. H o w e v e r , despite the advances i n this k i n d o f approach, arbitrariness i n the g e o m e t r y o f the i m p a c t i n g b o d y and f u l l satisfaction of the b o u n d a r y c o n d i t i o n on the o r i g i n a l body can be handled o n l y w i t h i n f e w m e t h o d s .

The purpose of this paper is to discuss a n u m e r i c a l technique w h i c h is designed to deal w i t h t h e p o t e n t i a l flow due to the i m p a c t i n g body. A v a r i a t i o n a l m e t h o d ( V M ) is used to evaluate the added mass value

* Corresponding author. Tel.: +55 11 30915424; fax:+55 11 38131886. E-mail address: ceppesce@usp.br (CP. Pesce).

0141-1187/$-see front matter © 2 0 1 2 Elsevier Ltd. All rights reserved. http://dx.doi.Org/10.101S/J.apor.2012.10.002

o f a x i s y m m e t r i c bodies d u r i n g the v e r t i c a l entrance i n t o w a t e r . The m e t h o d is based on a v a r i a t i o n a l approach, o r i g i n a l l y proposed by A r a n h a and Pesce [ 1 9 ] and constructed t h r o u g h a w e a k f o r r h u l a t i o n to t r e a t the linear w a v e - b o d y i n t e r a c t i o n p r o b l e m . I t can be considered t o be a m e t h o d o f f u n d a m e n t a l solutions or desingularized m e t h o d (see, e.g. [ 2 0 | ) . Recently, the desingularized t e c h n i q u e was also used b y Scolan [21,22] to simulate a b r e a k i n g w a v e i m p a c t o n t o a vertical w a l l .

2. M a t h e m a t i c a l f o r m u l a t i o n of the h y d r o d y n a m i c i m p a c t p r o b l e m

The p o t e n t i a l p r o b l e m w h i c h f o l l o w s f r o m the f o r m u l a t i o n o f h y -d r o -d y n a m i c i m p a c t p r o b l e m is consi-dere-d to be w e l l characterize-d b y the d o m i n a n c e o f i n e r t i a l forces. The l i q u i d surface is assumed t o be an e q u i p o t e n t i a l surface, w h i c h t h e n a l l o w s the usual analogy w i t h the i n f i n i t y f r e q u e n c y l i m i t i n the f r e e surface o s c i l l a t i n g floating b o d y p r o b l e m . The v e r t i c a l i m p a c t force can be calculated i n t e r m s o f a d d e d mass v a r i a t i o n w i t h the p e n e t r a t i o n d e p t h .

Consider t h a t the l i q u i d is i n i t i a l l y at rest. A t t h e i n i t i a l instant o f time t = 0, the body touches the f r e e surface at a single point, w h i c h is t a k e n as the o r i g i n o f a Cartesian c o o r d i n a t e s y s t e m Oxyz. The l i q u i d is assumed to be i n v i s c i d and the flow i r r o t a t i o n a l . A p o t e n t i a l scalar f u n c t i o n , (pix,y,z,t), defines the v e l o c i t y field. B o d y forces are assumed to be negligible. Hence, the n o n l i n e a r i m p a c t p r o b l e m can be f o r m u l a t e d as f o l l o w s :

= 0, i n t h e l i q u i d d o m a i n V , ( 1 )

76 Flavia M. Santos et al./Applied Ocean Research 39 (2012) 75-82

Fig. 1. The bulk ofthe liquid, the description of the free surface elevation, and wetted surface of the body.

Fig.2. Sketch ofthe different models of water impact; see Table 1.

a ^ a , o n the free surface S , , dt ax dx dy ay az

dó 1

+ - v</) • = 0, o n the free surface SF ,

<p, 1] Q as

( 3 )

( 4 )

( 5 )

w h e r e is a single-valued f u n c t i o n t h a t describes the f r e e surface e l e v a d o n z = ri{x,y,t) (see Fig. 1),

( x , y , f = 0) = 0, ?;(x,y, t = 0) = 0. (6)

The v e r t i c a l c o m p o n e n t o f the h y d r o d y n a m i c i m p a c t force can be calculated b y the pressure field i n t e g r a t i o n over t h e w e t t e d surface o f t h e b o d y SB (see [ 2 3 ] ) :

F = J ^^pn,dS = -pJ ^^(^^+^-Vct,.Vcl>y,dS, (7)

w h e r e p is the pressure field and p is the density o f the l i q u i d .

3. Generalized v o n K a r m a n Model

I n the present paper, the n u m e r i c a l s o l u t i o n o f the v e r t i c a l h y d r o -d y n a m i c i m p a c t p r o b l e m is solve-d w i t h i n the so-calle-d G v K M . ' In such approach, the exact b o d y boundary c o n d i t i o n is f u l f i l l e d and the w e t c o r r e c t i o n is n o t taken i n t o account (see Fig. 3).

I n t e r m e d i a t e models as the Generalized v o n K a r m a n M o d e l ( G v K M ) , t h e Generalized W a g n e r M o d e l ( G W M ) (see [13,24,25]), the M o d i f i e d Logvinovich M o d e l ( M L M ) (see [ 2 6 , 2 7 ] ) can be considered to be located b e t w e e n the original v o n K a r m a n m o d e l and the f u l l n o n l i n e a r m o d e l .

Table 1 presents basic characteristics o f each one.

W i t h i n G v K M , Eqs. (1 ) - { 7 ) are s i m p l i f i e d as f o l l o w s (see Fig. 3 f o r d e f i n i t i o n s o f V, SB and Sf)^

A(j) = 0, i n the l i q u i d d o m a i n V,

(V(/) - U ) • n = 0, o n the w e t t e d surface S B ,

(8)

(9)

1 This model has been recently used by Malenica and Korobkin [26| for ship hulls during water impact.

Except at the intersection line of body with the free surface

Original problem

Generalized v o n Karman M o d e l

Fig. 3. Sketch ofthe three-dimensional contact area in the Generalized von Karman Model (GvKM).

on the free surface SF, ai] _ atp

at ~ 3 z '

^,<t> = 0, on the free surface S f ,

dt

0 , 0

w h e r e convective terms are neglected i n Eqs. ( 1 0 ) and (11). The vertical h y d r o d y n a m i c i m p a c t force is g i v e n b y

d^

b e i n g also here convective t e r m s so accordingly neglected.

f pnzdS=-pf ^nzdS = - p ^ [ (ptt^dS,

J SB J SB •^t atj SB

( 1 4 )

4. T h e equation of m o t i o n

The h y d r o d y n a m i c i m p a c t p r o b l e m can be f o r m u l a t e d f r o m the v i e w p o i n t o f the Lagrangian f o r m a l i s m . In this sense, the v e r t i c a l i m p a c t force is d e t e r m i n e d f r o m the added mass f u n c t i o n b y means o f the usual expression:

F = 4 ( M , f ) , (15)

w h e r e Mj, is the added mass w h i c h is d e f i n e d w i t h i n t h e b u l k o f the l i q u i d , i.e. e x c l u d i n g the jets (see 128]).

A c c o r d i n g to N e w t o n ' s law, the force a p p l i e d to t h e b o d y is g i v e n b y

(16)

w h e r e m is the mass of the body. Eqs-(15) and ( 1 6 ) t h e n lead to the f o l l o w i n g e q u a t i o n o f m o t i o n :

( 1 7 )

( 1 8 )

w h e r e M'b = dMb/d^.

Let dimensionless time, position, a n d added mass be, respectively, d e f i n e d as

Flavia M. Santoset al. / Applied Ocean Researcli 39 (2012) 75-82 77 Table 1

Models of water i m p a c f

Aspects/model vKM WM GvKM GWM MLM Full nonlinear model

Free-surface (<p = 0) a t 2 = 0 atz = 0 atz = 0 a t z = )f atz = 0 a t z = ) ;

Wet-correction No Yes No Yes Yes Yes

Exact body shape No No Yes Yes Approximately Yes

' See Fig. 2 for geometrical details.

w h e r e a is the characteristic radius o f the i m p a c t i n g b o d y (the radius o f a sphere, or the s e m i - d i a m e t e r o f an e l l i p s o i d ) , fi = m Jmu is the specific mass w i t h mo, the mass o f l i q u i d t h a t is displaced b y the t o t a l l y i m m e r s e d body, and Uq is the v e r t i c a l v e l o c i t y at the v e r y f i r s t i n s t a n t o f impact, f = Q~. Eq. (18) t h e n takes the dimensionless f o r m o f

r + : 0 . (20)

Eq. ( 2 0 ) , " integrated w i t h i n i t i a l c o n d i t i o n s ?*(0) = 0; f ' ( O ) = 1, leads to the d e t e r m i n a t i o n o f the i m p a c t force i n Eq. (15).

The f u n c t i o n s M^{i;*) and M ' b ( f *) are calculated at each t i m e step, or i n t e r p o l a t e d f r o m pre-calculated values, f o r a range o f p r e d e f i n e d positions f *.

5. T h e v a r i a t i o n a l method and the h y d r o d y n a m i c i m p a c t p r o b l e m

Fundamentals o f the present n u m e r i c a l t e c h n i q u e f o l l o w f r o m the v a r i a t i o n a l m e t h o d w h i c h was considered by Pesce and Simos [ 2 9 ] i n the c o n t e x t o f p o t e n t i a l f l o w s p r o b l e m s due to the m o t i o n o f t h r e e -d i m e n s i o n a l bo-dies i n u n b o u n -d e -d l i q u i -d . The m e t h o -d was o r i g i n a l l y conceived to address the linear 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 o f free surface waves by f l o a t i n g bodies (see [ 3 0 ] and [ 1 9 ] ) .

A l t h o u g h the h y d r o d y n a m i c i m p a c t p r o b l e m is n o t associated w i t h a t o t a l l y submerged body, a consistent l i n e a r i z a t i o n o f t h e f r e e sur-face c o n d i t i o n leads to an e q u i v a l e n t p o t e n t i a l f l o w p r o b l e m a r o u n d a d o u b l e b o d y i n an u n b o u n d e d f l u i d at each i n s t a n t o f time. The free surface b o u n d a r y c o n d i t i o n , 0 = 0, can be then recovered b y c o n s i d -e r i n g t h -e i n f i n i t -e f r -e q u -e n c y l i m i t i n th-e usual o s c i l l a t i n g f l o a t i n g body p r o b l e m (see, e.g. [23] and 131]), Thus, at each i n s t a n t o f time, the a d d e d mass value, w h i c h is p r o p e r l y associated to b o t h the penetra-t i o n d e p penetra-t h and w e penetra-t penetra-t e d area o f penetra-the i m p a c penetra-t body, is r e q u i r e d (see Fig. 4). This is solved b y recalling the classical result t h a t the added mass of the i m p a c t i n g body can be w r i t t e n i n t e r m s o f the added mass o f the c o r r e s p o n d i n g double b o d y (see [ 2 3 | ) .

The m e t h o d presented here is based o n a v a r i a t i o n a l w e a k f o r -m u l a t i o n o f the associated p o t e n t i a l p r o b l e -m . The s o l u t i o n o f the c o r r e s p o n d i n g weak e q u a t i o n is o b t a i n e d i n a f i n i t e d i m e n s i o n space, so spanned by a set o f t n a l f u n c t i o n s - w h i c h satisfy the f i e l d equa-tion (Laplace equaequa-tion) - and is e n f o r c e d b y the w e l l posed b o u n d a r y c o n d i t i o n s on the p e n e t r a t i n g double b o d y surface. The set o f t r i a l f u n c t i o n s is constructed f r o m e l e m e n t a r y p o t e n t i a l singular f u n c t i o n s , such as poles, dipoles and v o r t e x rings. Singularities are placed 'inside' the body, w h i c h thus avoids n u m e r i c a l p r o b l e m s i n surface integrals. In t h i s sense, the m e t h o d m a y be classified as a 'desingularized' one. The m a i n advantage is t h a t the added mass values are d e t e r m i n e d w i t h a secondorder error i n the energy n o r m , since they are b i l i n -ear q u a n t i t i e s o f the w e a k s o l u t i o n f o r t h e c o r r e s p o n d i n g potentials. In o t h e r w o r d s , a rough a p p r o x i m a t e s o l u t i o n w h i c h is o b t a i n e d f o r the p o t e n t i a l leads to a second-order e r r o r a p p r o x i m a t i o n f o r the added mass coefficients. This does t u r n c o m p u t a t i o n fast. A c t u a l l y , the added mass values are m a t h e m a t i c a l l y analogous to the usual Rayleigh q u o t i e n t i n c o n t i n u u m mechanics (see [ 1 9 ] ) . Moreover, the

Fig. 4. Geometric parameters ofthe double body.

b o d y b o u n d a r y c o n d i t i o n s are exactly satisfied, instead o f a p p r o x i -m a t i n g the i -m p a c t b o d y by an e q u i v a l e n t flat plate, w h i c h i -m p l i e s t h a t the o r i g i n a l t h r e e - d i m e n s i o n a l shape o f the b o d y is taken i n t o account (see Fig. 3). H o w e v e r , the effects o f the local f r e e surface e l -e v a t i o n ar-e n o t consid-er-ed i n th-e pr-es-ent w o r k and shall b-e i n c l u d -e d i n a f u r t h e r paper. I n t h i s sense, the present m e t h o d m a y be classified as a GvKM m e t h o d (see [ 2 6 ] ) . A s u m m a r y o f t h e v a r i a t i o n a l m e t h o d presented b y Pesce and Simos [ 2 9 ] is presented i n the f o l l o w i n g sec-tion.2

5.1. Formulation of tlie variational metitod

Let ^ ( r ) be a n u m e r i c a l a p p r o x i m a t i o n o f cp{T ) a n d { T / r ) ; j = 1, . . . , N} a l i n e a r i y i n d e p e n d e n t set o f t r i a l f u n c t i o n s w h i c h satisfy the Laplace e q u a t i o n and the proper evanescence c o n d i t i o n . W r i t i n g

0 ( r ) = ^ q j T ; ( r ) ,

a n d f r o m Eqs. ( 8 ) - ( 1 3 ) , the w e a k equation is presented as

w h e r e

V{^^f)= [ i/fUndS

J SB

C(0,lA)= / v0-ni/rds'

The w e a k e q u a t i o n corresponds to a linear algebraic s y s t e m i n the u n k n o w n c o e f f i c i e n t s {qj;j= 1 N], i.e. G q = V G = {G{Ti.Tj)] q={</;•} V = { U ( T , ) 1 ( 2 1 ) (22) (23) (24)

78 Flavia M. Santos etal./ Applied Ocean Research 39(2012) 75-82

Dipole, rings o f dipoles and v o r t e x rings are e m p l o y e d as t r i a l f u n c t i o n s , see A p p e n d i x B.

5.2. A measuremetit ofthe error of the weak solution

The added mass is a h y d r o d y n a m i c parameter w h i c h is related to integral p r o p e r t i e s o f the f l o w f i e l d (see [ 2 3 ] ) . In fact, as p r e v i o u s l y discussed i n earlier sections, secondorder errors i n added mass v a l -ues are associated to first-order errors i n the p o t e n t i a l flow field (see [ 2 9 ] ) .

I n this w o r k , the e r r o r o f the n u m e r i c a l a p p r o x i m a t i o n ^ ( r ) is e s t i m a t e d t h r o u g h the f o l l o w i n g proposed parameter:

fiftc ( 2 5 )

w h i c h can be v i e w e d as a n e r r o r f o r the w e a k s o l u t i o n r e g a r d i n g t h e satisfaction o f the b o u n d a r y c o n d i t i o n on the b o d y surface.

6. Results

This section aims at i l l u s t r a t i n g the a p p l i c a t i o n o f the v a r i a t i o n a l m e t h o d to the i m p a c t p r o b l e m o f a x i s y m m e t r i c bodies. In order to demonstrate t h e efficacy o f the m e t h o d to calculate the added mass, the p r e - c a l c u l a t i o n and f o r m e r i n t e r p o l a t i o n approach was adopted, instead o f s o l v i n g the added mass p r o b l e m at each i n s t a n t o f i n t e g r a -tion. The results w e r e separated i n t o t w o sections: one section f o r a sphere, and one section f o r a n oblate spheroid. N u m e n c a l results w e r e p e r f o r m e d b y M a t i a b ® r o u t i n e s . W A M I T ® ' ' s o f t w a r e was a d o p t e d as the n u m e r i c a l p a r a d i g m f o r the added mass coefficients.

6.1. Sphere

The t h r e e - d i m e n s i o n a l i m p a c t o f a sphere o f radius a is taken. A l t h o u g h the i m p a c t force reaches its m a x i m u m value a t the i n i t i a l stage, i.e. at s m a l l p e n e t r a t i o n depths, larger values f o r f * are also considered (0 < f * < 1) f o r added mass calculations.

The set o f t r i a l f u n c t i o n s is considered to be composed b y a single v e r t i c a l d i p o l e and b y ( N j p - 1) concentric rings o f discrete dipoles, all placed at the z = 0 s y m m e t r y plane, w h e r e Njp is the n u m b e r o f t r i a l f u n c t i o n s . The v e r t i c a l d i p o l e is p o s i t i o n e d at the o r i g i n and the rings o f discrete dipoles placed a r o u n d the s y m m e t r y axis. The r a d i i o f the rings o f discrete dipoles, Rj, 1 < j < {Njp - 1) is g i v e n as Rj = [0.045 + ( j - 1 )AR|rc, w h e r e AR {Njf - 2) = 0.93 and Vc is the d o u b l e b o d y radius w h i c h is associated w i t h t h e considered p e n e t r a t i o n d e p t h .

Since the e l e m e n t a r y s o l u t i o n o f a d i p o l e leads to the flow due to a sphere, the systematic used i n this w o r k is to i n c l u d e a single v e r t i c a l d i p o l e at the o r i g i n i n o r d e r to emulate the flow a r o u n d the 'inscribed sphere' w i t h radius equal to the p e n e t r a t i o n d e p t h , f , a n d dipoles placed along circular rings - or, s i m p l y , rings o f dipoles^ - to repre-sent the w h o l e d o u b l e b o d y surface. Obviously, a systematic i n c l u s i o n of rings o f discrete dipoles can i m p r o v e the results because the e m -ulated flow a p p r o x i m a t e s to the flow around the exact d o u b l e body surface. I n fact, this can be seen i n Fig. 5(a) and (b), w h i c h presents the dimensionless added mass results, = Mb/{4/3 pna^), a n d the

Sbc parameter (see Eq. ( 2 5 ) ) as a f u n c t i o n o f N^f, respectively. I t can

also be checked i n Fig. 6(c) t h a t second-order errors i n added mass values are associated to firstorder errors i n the n u m e r i c a l a p p r o x i -m a t i o n o f the v e l o c i t y p o t e n t i a l . I n Fig. 6(c), ËM>^ is a -m e a s u r e -m e n t o f the 'error' i n the dimensionless added mass w h e n c o m p a r i s o n is made w i t h W A M I T ® .

" WAMIT®: 'Wave Analysis Massachusetts Institute of Technology'. It is a computer program based on the linear and second-order potential theory for analyzing floating or submerged bodies in the presence of ocean waves.

^ Not to be confused with (continuous) rings of dipole densities, which could be employed, as well. X r • « » • & « * < > --^>Q-a---o B ti:!:^iz--±":tziz±z----^--• = 0.10 I Ü 1 ^

Fig. 5. Sphere; numerical results and convergence as a function of Njr: (a) dimension-less added mass; the dotted line represents the results obtained through WAMIT®; (b) weak solution boundary condition error; (c) dimensionless added mass error. The legend in (a) is also used in figures (b) and (c).

O-fti at; o.u SIS

Fig. 6. Dimensionless added mass for an impacting sphere: (a) full penetration range and (b) small penetration range and power fitting.

Fig. 5(a) shows t h a t the agreement a m o n g n u m e r i c a l results is f a i r l y good. This is p a r t i c u l a r i y v a l i d f o r small p e n e t r a t i o n depths, w h e r e the h y d r o d y n a m i c i m p a c t force reaches its m a x i m u m value. M o r e o v e r , a systematic i n c l u s i o n o f i n n e r equi-spaced rings o f dis-crete dipoles i m p r o v e s the result b y r e d u c i n g the errors etc and (see Fig. 5(b) a n d (c)).

A n o t h e r set o f t r i a l f u n c t i o n s composed b y dipoles a n d v o r t e x rings is also used to calculate the added mass, see A p p e n d i x C f o r details. O p t i m a l values f o r Sbc are o b t a i n e d (see Fig. 5 ( b ) ) and the c o r r e s p o n d i n g added mass results are c o m p a r e d w i t h the n u m e r i c a l p a r a d i g m as a f u n c t i o n o f f * as s h o w n i n Fig. 6(a).

In Fig. 6(a), i t can be n o t i c e d t h a t f o r the p a r t i c l e value o f f * = 1, i.e. a t o t a l l y i m m e r s e d hemisphere, the exact a n a l y t i c a l value is p r o m p t l y recovered. This is done w i t h a single v e r t i c a l d i p o l e as expected.

A p o w e r fitting f o r MJ( f *) is o b t a i n e d f o r s m a l l p e n e t r a t i o n depths, as s h o w n i n Fig. 6(b), and so the d e r i v a t i v e M ' J ( ^ * ) can be p r o m p t i y o b t a i n e d . The fitted p o w e r f u n c t i o n s M ^ ( f *) and M ' b ( f ' ) are t h e n used d u r i n g the i n t e g r a t i o n o f the e q u a t i o n o f m o t i o n , Eq. (20), at each rime step.

N u m e r i c a l results f o r the added mass are also c o m p a r e d w i t h a s y m p t o t i c a n a l y t i c a l results (see Fig. 7 ) . T h r o u g h w e l l k n o w n a s y m p -totic techniques and s i m i l a r i t y theory, i t can be s h o w n t h a t the added mass o f a p e n e t r a t i n g sphere m a y be a p p r o x i m a t e d b y e i -ther = ( 3 V 3 / ; r ) f i f Wagner's a p p r o a c h is considered, or

Flavia M. Santoset al./Applied Ocean Research 39 (2012) 75-82 79 0.30 0.25 0.20 0.15 O.IO 0.05 0.00 \Vagni:r — - - v o n Karniiiii V M

Fig. 7. Comparisons of added mass of a spfiere: Wagner and von Karman approaches and variational method (VM).^

Fig. 9. Geometric parameters of the oblate spheroid.

„

%-ti-i---t.-.'11 A) ?li

Fig. 10. Oblate spheroid (b/a = 0.6); numerical results and convergence as a function of Wrf: (a) dimensionless added mass; the dotted line represents the results obtained through W A M I T ® ; (b) weak solution boundary condition error; (c) dimensionless added mass error. The legend in (a) is also used in figures (b) and (c).

6.2. Oblate spheroM

Fig. 8. Dimensionless penetration, velocity and acceleration (impact force) of an impacting sphere vs. dimensionless time, for three values of specific mass.

= ( 2 ^ / 2 / 7 ^ ) f i f vQp, Karman's approach is tal<en instead; see, e.g. Casetta [ 3 2 ] .

i t can be seen i n Fig. 7 t h a t the n u m e r i c a l values are close to those o b t a i n e d f r o m the o r i g i n a l v o n K a r m a n a p p r o x i m a t i o n . As a m a t t e r o f fact, n u m e r i c a l values are smaller due to the t h r e e - d i m e n s i o n a l i t y o f the e n t e r i n g body, accounted f o r i n the v a r i a t i o n a l m e t h o d as w e l l as i n t h e c o m p u t a t i o n s w i t h W A M I T ® . These effects are disregarded i n v o n K a r m a n m o d e l , w h e r e the w e t t e d surface is a p p r o x i m a t e d b y an e q u i v a l e n t f l a t disc.

On the o t h e r hand, the n u m e r i c a l results are q u i t e d i f f e r e n t f r o m the results o b t a i n e d w i t h i n Wagner's a p p r o x i m a t i o n , w h i c h essen-t i a l l y correcessen-ts v o n Karmans' m e essen-t h o d by essen-t a k i n g essen-the ' p i l e d - u p ' w a essen-t e r e f f e c t i n t o account. Such an e f f e c t is t o be f u r t h e r i n c o r p o r a t e d i n the v a r i a t i o n a l m e t h o d technique.

A f t e r the calculation o f M | ( f *) a n d M ' J ( f *), Eq. ( 2 0 ) can be i n t e -g r a t e d under i n i d a l c o n d i d o n s f *(0) = 0 and f *(0) = 1. Fi-g. 8 shows p o s i t i o n , v e l o c i t y and acceleration f o r d i f f e r e n t values o f t h e specific mass, p.

The same procedure w h i c h is presented to o b t a i n the added mass o f a sphere, Section 6.1, is a p p l i e d to the case o f an oblate s p h e r o i d {b/a = 0.6), a b e i n g the h o r i z o n t a l and b the v e r t i c a l s e m i - d i a m e t e r s of t h e s p h e r o i d {b is o n the r e v o l u t i o n axis); see Fig. 9. Large values f o r w e r e also considered (0 < f * < 0 . 6 ) . Fig. 10 presents t h e added mass results and the parameters e^* and as a f u n c t i o n ofNjp.

As i n the case o f t h e sphere, the agreement b e t w e e n the n u m e r i c a l results is q u i t e g o o d . This is p a r t i c u l a r l y v a l i d f o r t h e systematic i n -c l u s i o n o f t r i a l f u n -c t i o n s (see Fig. 1 0 ) . O p t i m i z e d values f o r -can be o b t a i n e d f r o m Fig. 6 ( b ) and its c o r r e s p o n d i n g added mass values are c o m p a r e d w i t h the n u m e r i c a l p a r a d i g m as a f u n c t i o n o f f * , as s h o w n i n Fig. 11(a). A p o w e r fitting f o r M J ( f *) is t h e n d e t e r m i n e d , f o r small p e n e t r a t i o n depths, as s h o w n i n Fig. 11(b).

In t h e same w a y , a f t e r M J ( f ' ) and M ' b ( f *), Eq. ( 2 0 ) can be i n t e -grated u n d e r i n i t i a l c o n d i t i o n s f *(0) = 0 and f ' ( 0 ) = l . F i g . 12 shows p o s i t i o n , v e l o c i t y and acceleration, f o r d i f f e r e n t values o f t h e specific mass, p.

7. C o n c l u s i o n s

Based o n the w e l l - k n o w n analogy w i t h the i n f i n i t y f r e q u e n c y l i m i t i n the usual o s c i l l a t i n g floating body p r o b l e m , a v a r i a t i o n a l m e t h o d

80 Flavia M. Santoset al./Applied Ocean Research 39 (2012) 75-82

Fig. 11. Dimensionless added mass for an impacting oblate spheroid (b/a = 0.6): (a) full penetration range and (b) small penetration range and power fitting.

Fig. 12. Dimensionless penetration, velocity and acceleration (impact force) of an impacting spheroid vs. dimensionless time, for three values of specific mass.

was a p p l i e d to the e v a l u a d o n o f t h e added mass o f a x i s y m m e t r i c b o d -ies d u r i n g the v e r t i c a l i m p a c t onto the l i q u i d f r e e surface. N u m e r i c a l results w e r e c o m p a r e d w i t h those obtained f r o m W A M I T ® . Good a g r e e m e n t was f o u n d .

The m a i n advantage o f the proposed v a r i a t i o n a l m e t h o d is related to the f a c t t h a t the a d d e d mass coefficients are d e t e r m i n e d w i t h a second-order e r r o r i n the energy n o r m . This t u r n s c o m p u t a t i o n fast and so enables the c a l c u l a t i o n o f the added mass tensor at each i n stant o f time. Moreover, the f o r m u l a t i o n o f the m e t h o d is q u i t e g e n -eral and its i m p l e m e n t a t i o n is extensible to i n c l u d e any c o n v e n i e n t t h r e e d i m e n s i o n a l case. Further w o r k s are supposed to consider c o n -vex g e o m e t r i e s and t h e free surface elevation at the v i c i n i t y o f the i m p a c t i n g body.

A c k n o w l e d g e m e n t s

W e a c k n o w l e d g e FAPESP, the State o f Sao Paulo Research Founda-tion, f o r t h e PhD scholarships no. 2 0 1 0 / 0 7 0 0 8 - 9 , f i r s t author, a n d no. 0 4 / 0 4 6 1 1 - 5 , t h i r d author, and a CNPq, Research Grant no. 3 0 3 8 3 8 / 2008-6. Thanks also to Dr. Marcos Donato Ferreira, Petiobras, f o r s u p p o r t i n g research o n theoretical o f f s h o r e h y d r o d y n a m i c s , and Dr. A l e x a n d r e N . Simos f o r discussions o n the subject. The a u t h o r s w i s h to t h a n k the r e v i e w e r s f o r t h e i r c a r e f u l reading and valuable sugges-tions.

Appendix A. T h e v a r i a t i o n a l m e t h o d applied to 3D potential flows over bodies i n u n b o u n d e d fluid

The f o r m u l a t i o n o f t h e v a r i a t i o n a l m e t h o d presented here is essen-tially taken f r o m Pesce a n d Simos [ 2 9 ] . The f o l l o w i n g equations are satisfied f o r p o t e n t i a l p r o b l e m s i n incompressible u n b o u n d e d fluids.^

v 2 0 = O

V 0 • n = U • n = Lf„, o n SB V 0 ^ 0 (as 1 / r w i t h

( A l )

w h e r e U = U ( t ) is the v e l o c i t y of the t r a n s l a t i n g body and SB its surface. Let i/f (r) be any square i n t e g r a b l e f u n c t i o n i n the s e m i - n o r m '

1/2

( A 2 )

This class o f f u n c t i o n s is a Sobolev space, M^^\v). D e f i n i n g the f u n c -t i o n a l

( A 3 )

the k i n e t i c energy associated to the p o t e n t i a l f u n c t i o n 0 is s i m p l y g i v e n by

T = l p G ( 0 , < / . ) , (A.4)

w h e r e p is the density o f the fluid.

Taking the Laplacian o f cj) (see Eq. ( A . l ) ) , m u l t i p l y i n g i t by i n -t e g r a -t i n g i n -the fluid d o m a i n and u s i n g -t h e divergence -t h e o r e m and the boundary c o n d i t i o n s (see ( A l ) ) , the r e s u l t is a w e a k e q u a t i o n f o r the p o t e n t i a l p r o b l e m :

( A 5 )

w h e r e V [ i f ) = }^J{l„dS.

Note that, i f tjr satisfies Laplace's e q u a t i o n and the f a r f i e l d c o n d i -t i o n as i n Eq. ( A . l ) , -t h e n G{4&g-t;, f ) = fsg'^4&g-t; • nifr dS.

The Lagrangian can be p u t i n the f o r m :

Lict>)=^-pG{(l>,,j>)-pV{4>). ( A 6 )

such t h a t the s t a t i o n a r i t y o f L{(p) leads to Eq. (A.5) and vice versa. Let 0(r) be a n u m e r i c a l a p p r o x i m a t i o n o f cf>(^T). Eq. (A.5) can be solved i n a finite d i m e n s i o n a l sub-space o f finite energy spanned by a l i n e a r l y i n d e p e n d e n t set o f t r i a l f u n c t i o n s { T / r ) ; j = 1 N}. The t r i a l f u n c t i o n s r j ( r ) satisfy Laplace's e q u a t i o n and the f a r - f i e l d c o n d i t i o n (see Eq. ( A . l ) ) . W r i t i n g ,

( A 7 )

the w e a k e q u a t i o n changes to a linear algebraic system i n the u n -k n o w n coefficients {qj;j= 1 , . . . , N}, i.e.

G q = V

G = [G{TiJj)]

V = (V/(T,)}

( A S )

The s o l u t i o n o f Eq. (A.8) corresponds t o a s t a t i o n a r y p o i n t o f the Lagrangian (see Eq. (A.6)), and the o b t a i n e d r e s u l t is t h e best a p p r o x -i m a t -i o n -i n the f-in-ite sub-space spanned b y {rj(r); j = 1 , . . . , N}.

The g e n e r a l i z a t i o n o f the v a r i a t i o n a l m e t h o d f o r a l l six degrees o f f r e e d o m can be seen i n Pesce and Simos [ 2 9 ] .

•5 The free surface boundary condition 0 = 0 is the infinite frequency asymptotic limit of the usual oscillating floating body problem.

Flavia M. Santos et al. /Applied Ocean Research 39 (2012) 75-82 81

Fig. 13. Sketch of the displaced dipole at the plane z = 0.

Fig. 14. Sketch of the generic a-ring.

A p p e n d i x B. T r i a l f u n c t i o n s

I n t h e present study, dipole, rings o f discrete dipoles and v o r t e x rings are e m p l o y e d as t r i a l f u n c t i o n s . ' ' I n p o l a r c y l i n d r i c a l coordinates, t h e y can be, respectively, w r i t t e n as

Dipole^ w h e r e , !O = 2JT / , = 0 ; 'n = — — f n - 2 (B.5) 4>{r,z\a)-. 1 / a2 3 / 2 ( B . l ) Rings of dipoles

The v e l o c i t y p o t e n t i a l o f the i - t h d i p o l e displaced ( i n f ) at the plane z = 0 w i t h respect to the o r i g i n o f the c o o r d i n a t e s y s t e m (see Fig. 13 ) is g i v e n by

<Pi[r"i.z";a) = •ii2 ,„2

3 / 2

(B.2)

and the p o t e n t i a l o f the ring o f dipoles ( w i t h radius f , see Fig. 1 3 ) can be w r i t t e n as a s u m o f dipoles displaced w i t h respect to t h e o r i g i n (see Eq. (B.2)) 9. This leads to

. ( r " , z " ; a ) = ^ 0 ( ( r ; ' , 2 " ; a ) ,

i = l

(B.3)

w h e r e is the n u m b e r o f dipoles t h a t represents the discrete r i n g . In Eqs. (B.2) and (B.3), r'/^ =r^+ r'^ - 2 r r ' cos(i^ - tp'j) and z" = z.

The rings o f discrete dipoles w h i c h are used i n this w o r k are p o s i -tioned at the p l a n e z = 0, w i t h radius r" and <p- = 2:7r(f - l ) / n d .

Circular vortex rings^"

' Other elementary singularities, as source nngs, may also be used. 8 See Lamb [33].

^ Vertical dipole positioned at 0". See Pesce and Simos [29].

u"+' / „ ( r , z ; R ) = / „ ( a ) = / — 2^n+3/2 d i i , a^{r,z:R): R2 (B.6) (B.7)

The v o r t e x rings w h i c h are used i n this w o r k are p o s i t i o n e d at t h e plane z = 0, w i t h radius R. I t f o l l o w s that, f o r n = 0:

/ o ( r , z ; i ? ) = / o ( a ) = l r

Vl

(B.8)

For n > 1, the d e f i n i t e i n t e g r a l i n Eq. (B.6) can be w r i t t e n i n t e r m s o f h y p e r g e o m e t r i c f u n c t i o n s (see [ 2 9 ] ) i n the f o r m :

f„{r,z;R) = fn{cx)-- 1 1

(B.9) (2 + n) {a2 + ^)''+V2

F (3/2 -F n, 1; 2 4- n / 2 ; 1

/ ( n - CT^^) ; all o > 0

The v o r t e x rings used i n t h i s paper are c o m p l e t e d a n d circular, i.e. the aperture angle is a = 2;r, see Fig. 14. The n u m b e r o f t e r m s used i n the series expansions is 960 < n < 1018. Note t h a t to c o n -s t r u c t the matrice-s [ £ ( 1 , , Tj )J b y n u m e r i c a l l y c a l c u l a t i n g the i n t e g r a l g i v e n i n G{^. i / f ) = / s g V 0 • ni/f dS the g r a d i e n t o f the t r i a l f u n c t i o n s so becomes necessary. This is the reason f o r the c o n s i d e r a t i o n o f h y -p e r g e o m e t r i c f u n c t i o n s . T h e i r recursive relations o n d e r i v a t i v e s are s t r a i g h t f o r w a r d . Further details can be f o u n d i n [ 2 9 ] .

Appendix C. A d d e d m a s s r e s u l t s u s i n g o t h e r t r i a l f u n c t i o n s

A s i m p l e n u m e r i c a l e x p e r i m e n t , w h i c h f o l l o w s f r o m the consider-a t i o n o f consider-a v e r t i c consider-a l d i p o l e consider-a n d consider-a single v o r t e x r i n g consider-at s y m m e t r y plconsider-ane z = 0, was s y s t e m a t i c a l l y c a r r i e d o u t f o r d i f f e r e n t values o f f *. Such an e x p e r i m e n t a i m e d at i n v e s t i g a t i n g h o w precise can be the n u m e r -ical results w i t h the use o f a single v o r t e x ring. O p t i m a l values f o r the v o r t e x ring radius, R, at each f *, w e r e thus o b t a i n e d t h r o u g h the

etc parameter, see Eq. ( 2 5 ) . This is s h o w n i n Tables 2 and 3, w h e r e

the She and e^^ parameters are presented as a f u n c t i o n o f f *for b o t h geometries.

It can be seen i n Tables 2 and 3 t h a t the a g r e e m e n t b e t w e e n t h e n u m e r i c a l results is good, as o n l y t w o t r i a l f u n c t i o n s w e r e used, i.e. a

82 Flavia M. Santos et al. / Applied Ocean Research 39 (2012) 75-82

Table 2 ;

Numerical results for a sphere as a function of f using just two trial functions (dipole and vortex ring).

f R/rc IW; (WAMIT) M£ (VM) Rbc (%) 0.05 0.827 0.0089 0.0090 14.1567 1.2958 0.10 0.880 0.0230 0.0222 12.4549 3.5169 0.15 0.905 0.0393 0.0385 14.5313 2.0235 0.20 0.921 0.0572 0.0561 13.2230 1.8072 0.25 0.941 0.0750 0.0758 10.2013 1.0639 0.30 0.944 0.0917 0.0920 5.6063 0.2705 0.40 0.954 0.1250 0.1244 7.8814 0.4966 0.60 0.980 0.1833 0.1819 3.4683 0.7737 Table 3

Numerical results for an oblate spheroid {b/a = 0.6) as a function of if * using just two trial functions (dipole and vortex ring).

R/tc Mj (WAMIT) M£ (VM) etc (%) PAfiC%)

0.05 0.740 0.0319 0.0253 48.2692 20.7868 0.10 0.840 0.0812 0.0739 26.7426 9.0147 0.15 0.865 0.1345 0.1214 17.6676 9.7864 0.30 0.910 0.2853 0.2730 12.1889 4.3386 0.45 0.911 0.3930 0.3801 10.2604 3.2889 0.60 0.924 0.4539 0.4295 10.4454 5.3683

d i p o l e and a single v o r t e x ring. A systematic i n c l u s i o n o f inner e q u i -spaced v o r t e x rings can be done to i m p r o v e the added mass results.

References

[1] Von Karman T. The impact on seaplane floats during landing. NACA/NASA Langley Research Center; (Technical Note no. 321 ]; 1929.

| 2 | Wagner H, Landing of seaplanes. NACA/NASA Langley Research Center; [Tech-nical Memorandum no. 622]; 1931.

[3] Dobrovol'skaya ZN. On some problems of similarity flow of fluid with a free surface. Journal of Fluid Mechanics. 1969;36:805-829.

[4| B Molin, R Cointe, E Fontaine. On energy arguments applied to the slamming force. In: l i s t international workshop on water waves and floating bodies, Hamburg, Germany; 1996.

[5| Mei X, Liu Y, Yen D. On the water impact of general two-dimensional sections. Applied Ocean Research. 1999;21(1):1-15.

[6] Chuang SL. Theoretical investigations on slamming of cone-shaped bodies. Jour-nal of Ship Research. 1969;13(4):276-283.

[7| MilohT. On the Initial stage slamming of a rigid sphere in vertical water entry. Applied Ocean Research. 1991 ;13(l):43-48.

[8] MilohT. On the oblique water-entry problem of a rigid sphere, journal of Engi-neering Mathematics. 1999:25(11 ):77-92.

[91 M Shiffman, DC Spencer. The force of impact on a sphere striking a water surface. Applied Mathematics Panel Report 42 IRAMG-NYU no. 105; 1945.

[10] Shiffman M, Spencer DC. The force of impact on a cone striking a water surface (vertical entry). Communications on Pure and Applied Mathematics. 1951;4(4):379-418.

[11] Scolan Y-M, Korobkin AA. Three-dimensional theory of water impact. Part 1. inverse Wagner problem, journal of Fluid Mechanics. 2001 ;440:293-326. [12] Zhao R, Faltinsen OM. Water entry of two-dimensional bodies, journal of Fluid

Mechanics. 1993;246:593-612.

[13] Zhao R, Faltinsen OM, Aarsnes JV. Water entry of arbitrary two-dimensional sections with and without flow separation. In: 21 st Naval Hydrodynamics Sym-posium, Trondheim, Norway; 1996.

]14] Takagi K. Numerical evaluation of three-dimensional water impact by the displacement potential formulation. Journal of Engineering Mathematics. 2004;48(3-4):339-352.

[15] Peseux B, Cornet L, Donguy B. Hydrodynamic impact; numerical and experi-mental investigations. Journal of Fluids and Structures. 2005;21:277-303. [16] Sun H, Faltinsen OM. Water impact of circular cylinders and cylindrical shells.

Applied Ocean Research. 2006;28:299-311.

117) Wu GX. Numencal simulation of water entry of twin wedges. Journal of Fluids and Structures. 2006:22:99-108.

[18] Xu GD, Duan WY, Wu GX. Numencal simulation of oblique water entry of an asymmetrical wedge. Ocean Engineenng. 2008;35:1597-1603.

[19] Aranha JAP, Pesce CP. A variational method for water wave radiation and diffrac-tion problems, journal of Fluid Mechanics. 1989;204:135-157.

120] Cao Y, Schultz WW, Beck RF. Three-dimensional desingularized boundary inte-gral methods for potential problems. International Journal for Numerical Meth-ods in Fluids. 1991;12:785-803.

[21] Y-M Scolan. Some numerical aspects of nonlinear free surface motions by a Method of Fundamental Solutions. In: 25th international workshop on water waves and floaring bodies, Harbin, China; 2010.

122] Scolan Y-M. Some aspects of the flip-through phenomenon: a numerical study based on the desingularized technique, journal of Fluids and Structures. 2010:26:918-953.

]23] NewmanJN. Marine hydrodynamics. The MIT Press; 402 pp.; 1978. ]24 j Falrinsen OM, Chezhian M. Ageneralized Wagner method for three-dimensional

slamming, journal of Ship Research. 2005;49(4):279-287.

[25] Korobkin AA. Analytical models of water impacL European Journal of Applied Mathematics. 2004; 15:821 -838.

[26] S Malenica, AA Korobkin. Some aspects of slamming calcularions in seakeeping. In: 9th international conference in numerical ship hydrodynamics, Michigan, USA; 2007.

]27] AA Korobkin, S Malenica. Modified Logvinovich model for hydrodynamic loads on asymmetric contours entering water. In: 20th international workshop on water waves and floating bodies, Longyearbyen, Noi-way; 2005.

[28] L Casetta, CP Pesce. The proper definition of the added mass for the water enti7 problem. In: 21st internarional workshop on water waves and floating bodies, Loughborough, United Kingdom; 2006.

[29] Pesce CP, Simos AN. A family of vortex rings and a variarional application to po-tential flows around three-dimensional bodies, journal of the Brazilian Society ofMechanical Sciences and Engineering 2008;XXX{2);116-131.

[30] CP Pesce. Estudo do comportamento de corpos flutuantes em ondas: um enfoque variacional e aplicaijoes da teoria do corpo esbelto. PhD thesis. University of Sao Paulo; 1988 [in Portuguese].

]31 ] Faltinsen OM. Sea loads on ships and offshore structures. Cambridge University Press; Cambridge; 1990.

132] L Casetta. Investigafoes teóricas sobre o problema de impacto hidrodinamico. MSc dissertation. University of Sao Paulo; 2004 [in Portuguesel.