Ship Technology Research/Schiffstechnik

J o u r n a l f o r R e s e a r c h i n S h i p b u i l d i n g a n d R e l a t e d S u b j e c t s

SHIP TECHNOLOGY RESEARCH/SCHIFFSTECHNIK was founded by K. Wendel in 1952. It is edited by H. Söding and V. Bertram in collaboration with experts from universities and model basins in Berlin, Duisburg and Hamburg, from

Germa-nischer Lloyd and other research-oriented organizations in Germany.

Papers and discussions proposed for publication should be sent to Prof. Dr.-lng. H . Söding, Institut fiir Schiffbau, Lammer-sieth 90, 22305 Hamburg, Germany

Vol. 42 No. 2-April 1995

TECHNISCHE UNIVERSITEIT

U b o r a t o r i u m voor Scheepshydromechanlca

Archlsf

Broder Hinriclisen Mekelweg 2, 2628 C D Delft B e n d i n g of D o u b l e W a l l P a n e l s w i t h o n e F r e e E d g e Teb015-786873^ Fax! 015--781838

Ship Technology Research 42 (1995), 55-67

T h e a n a l y t i c a l solution of the d i f f e r e n t i a l equation f o r rectangular o r t h o t r o p i c plates is applied t o double w a l l panels w i t h t w o opposite sides s i m p l y supported, the t h i r d edge free and the f o u r t h edge b u i l t i n or s i m p l y supported. T h e pressure load varies l i n e a r l y i n the d i r e c t i o n of t h e opposite s i m p l y supported edges while i t is constant i n the other d i r e c t o n . T h e results are presented as design curves i n the same f o r m as Schade's diagrams.

K e y w o r d s : o r t h o t r o p i c plate, double w a l l panel, stiffened plate, L e v y ' s m e t h o d , deflection, b e n d i n g m o m e n t

M a l i c k B a and M i c h e l G u i l b a u d

A F a s t M e t h o d of E v a l u a t i o n f o r t h e T r a n s l a t i n g a n d P u l s a t i n g G r e e n ' s F u n c t i o n Ship Technology Research 42 (1995), 68-80

T h e c o m p u t a t i o n s of the unsteady ( h a r m o n i c ) Green's f u n c t i o n w i t h f o r w a r d speed involve n u m e r i c a l difficulties and require h i g h c o m p u t a t i o n a l t i m e . A new fast and precise m e t h o d of c a l c u l a t i o n is presented, valid regardless of the values f o r frequency and f o r w a r d speed. T h e integrations over the complex exponential i n t e g r a l use a R u n g e - K u t t a f o u r t h order scheme w i t h a stepsize c o n t r o l . Calculations are compared w i t h other methods, and the hnearized free surface b o u n d a r y c o n d i t i o n is satisfied w i t h good accuracy. Examples of free surface elevations above a submerged p u l s a t i n g and t r a n s l a t i n g source are presented.

K e y w o r d s : C F D , Green's f u n c t i o n , f o r w a r d speed, free surface, seakeeping, source, panel m e t h o d

F a r i t G . A v h a d i e v and D i m i t r i j V . M a k l a k o v A T h e o r y o f P r e s s u r e E n v e l o p e s f o r H y d r o f o i l s

Ship Technology Research 42(1995), 81-102

For t w o - d i m e n s i o n a l p o t e n t i a l flows, a theory of pressure envelopes f o r h y d r o f o i l s is developed. T h e t h e o r y predicts whether or not an a r b i t r a r y given f u n c t i o n can be realized as the pressure envelope f o r a certain closed p r o f i l e . Exact lower bounds of the pressure envelopes f o r s y m m e t -r i c a l h y d -r o f o i l s a-re de-rived and a new a n a l y t i c a l m e t h o d of h y d -r o f o i l design is developed. T h e m e t h o d produces a profile shape w i t h exactly the specified pressure envelope.

K e y w o r d s : p o t e n t i a l flow, pressure, h y d r o f o i l , c a v i t a t i o n , profile

Kostas Spyrou

S u r f - R i d i n g , Y a w I n s t a b i l i t y a n d L a r g e H e e l i n g o f S h i p s i n F o l l o w i n g / Q u a r t e r i n g W a v e s

Ship Technology Research 42 (1995), 103-112

States of s u r f - r i d i n g are i d e n t i f i e d f o r one wave l e n g t h and a r b i t r a r i l y selected vessel heading. T h e y represent the s t a t i o n a r y solutions of the coupled, non-Mnear system of m o t i o n equations. S t a b i h t y analysis w i t h simultaneous consideration of the l o n g i t u d i n a l and l a t e r a l directions on the wave is used t o i d e n t i f y the areas of stable and unstable s u r f - r i d i n g . O s c i l l a t o r y s u r f - r i d i n g can exist i n q u a r t e r i n g waves. S t u d y of transient behaviour indicates specific mechanisms w h i c h can be responsible f o r the development of considerable heel, even w h e n the vessel's metacentric height is kept h i g h .

K e y w o r d s : ship m o t i o n , wave, d y n a m i c stability, s u r f - r i d i n g , broaching, surging, yaw

Verlag:

Schiffahrts-Verlag „Hansa" C. Schroedter & Co. (GmbH & Co KG) Striepenweg 31,21147 Hamburg, Postfach 92 06 55,21136 Hamburg Telefon: (040) 7 97 13 - 02, Telefax: (040) 7 97 13 - 208,

Telegr.-Adr.: Hansapress Scliriftleitung:

Prof Dr.-lng. H. Söding, Dr.-lng. V, Bertram

Institut für Schiflbau, Lammersieth 90,22305 Hamburg Aborniementsveraaltung;

Marcus Thies Tel. (040) 7 97 13 - 220 Vertriebsleiter; Hans-Focko Koehler Tel. (040) 7 97 13 - 321

Nachdruck, auch auszugsweise, nur mit Genehmigung des Veriages. Die SCHIFFSTECHNIK erscheint viermal jahriich. Ahonnemenfspreise: Inland: jahrMch DM 192,20 inkl. Mehr\vertsteuer und Versandkosten; Ausland: jahrlich DM 192,00 zuzilglich Versandkosten. Der Abonnementspreis ist im voraus fallig. Zahlbar innerhalb 30 Tagen nach Rechnungseingang. Einzelpreis: DM 51,00 inkl. Mehrwertsteuer, inkl. Versandkosten. Abonnementskündi-gungen sind nur schriftUch mit einer Frist von 6 Wochen zum Ende eines Kalenderjahres beim Veriag möglich.

Höhere Gewalt entbmdet den Veriag von jeder LieferungsverpfMchtung. -Erfilllungsort und Gerichtsstand Hamburg.

Gesamthenitellung: Hans Kock, Buch- und Offsetdruck GmbH, Bielefeld ISSN 0937-7255

Publishen

Schiffahrts-Veriag "Hansa" C. Schroedter & Co. (GmbH & Co KG) Striepenweg 31,21147 Hamburg, Postfach 92 06 55,21136 Hamburg Telefon: (040) 7 97 13 - 02, Telefax: (040) 7 97 13 - 208,

Telegraphic address: Hansapress Editon

Prof Dr.-lng. H. Söding, Dr.-lng. V. Bertram

Institut für Schiflbau, Lammersieth 90, D-22305 Hamburg Subscriptiondepartmenfc

Marcus Thies Tel. (040) 7 97 13 - 220

Sales Director Hans-Focko Koehler Tel. (040) 7 97 13 - 321

All rights reserved. Reprints prohibited without permission of the pubUsher. SHIP TECHNOLOGY RESEARCH is issued quarterly. Subscription price: D M 192,00 per year + mailing cost, to be payed in advance 30 days after re-ceipt of invoice. Price of smgle copies: D M 51,00 (mailing included). Cancella-tion of subscripCancella-tions at Ihe end of a year only by written notice to the publish-er 6 weeks in advance.

in case of force majeure the pubUsher is freed from delivery. Court competency and place of conttact: Hamburg.

Production: Hans Kock, Buch- und Oflsetdruck GmbH, Bielefeld

Bending of Double W a l l Panels w i t h one Free Edge

B r o d e r H i n r i c h s e n , I n s t i t u t f i i r Schiffbau^

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

D o u b l e w a l l panels can be analyzed by small-deflection o r t h o t r o p i c t h i n plate theory. Schade

(1938,1940) was a pioneer i n a p p l y i n g this m e t l i o d t o double-wall ship panels such as

dou-ble b o t t o m s . A l t h o u g h more accurate computer-based analysis methods are availadou-ble today,

Schade's (1941) design curves are a simple and economic m e t h o d t o estimate the expected

deflection and stress level. Here these design curves are extended t o panels w i t h t w o opposite sides s i m p l y s u p p o r t e d , the t h i r d edge free, and the f o u r t h edge b u i l t i n or s i m p l y s u p p o r t e d . 2. O r t h o t r o p i c P l a t e T h e o r y

T h e o r t h o t r o p i c plate theory is a generahzation of the theory f o r isotropic plates w i t h l a t e r a l dimensions large and deflections under l a t e r a l loads small relative t o the thickness. F i r s t we consider a homogenous plate of constant thickness w i t h elastic properties of the m a t e r i a l different i n t w o o r t h o g o n a l directions i n the plate's plane. L a t e r we wiU a p p l y the o b t a i n e d results t o double w a l l panels. T h e stress-strain equations f o r an o r t h o t r o p i c m a t e r i a l f o r plane stress are:

(^xx (^yy ^ ^yy

„

'^^^ fr - 111 - ^ rr (1)

S y m m e t r y of the s t r a i n energy yields v^Ey = UyE^.

We assume t h a t linear elements perpendicular t o the m i d d l e plane of the plate before b e n d i n g r e m a i n s t r a i g h t and n o r m a l t o the deflection surface of the plate after bending.

e^^ = -ziD\^^ £yy = -ziD^yy e^y = - zw^^y (2)

T h e corresponding stress components, f r o m eq. (1) are:

E E z

CTxx = 1 — i^xx + yyeyy) = " z ^ (^'^1- + ^yW\yy)

1 - V^Vy 1 - Vx^y

^-^i£yy + '^x£xx) = ^'^^ iw\yy + l^x'Wl^:,) (3)

1 - ly^J^y 1 - ^xi^y

Gxy — 2Gxy£xy — Gxyjxy — ^G^yZW^^y

W i t h these expressions, the bending and t w i s t i n g moment intensities per u n i t l e n g t h are

h/2 m^^ = j a^^z dz = -K,J,w\^^ \ ^yW^yy) -hji rrlyy = J CTyyZ dZ = -|- l^xW^^x) -h/2 h/2

m^y = ƒ T^yZ dz = -2K^yiD\^y -h/2

where the b e n d i n g rigidities i i ' ^ . and Ky and the torsional r i g i d i t y K^^y axe constants:

E.-h^ Eyh^ _G,yh^

^ ^ ^ = 1 2 ( 1 3 ^ ^ ^ Ï 2 ( Ï 3 ^ ^ ^ 2

^Univ. Hamburg, Lammersieth 90, D22305 Hamburg, Germany, in cooperation w i t h Haye Hinrichsen, F U -Berlin

F u r t h e r we i n t r o d u c e t h e "effective t o r s i o n a l stiffness":

^ 6 12(1 - Va:l'y) ^ > T h e s t r a i n energy V of t h e plate f o r hnear elastic behavior is:

U=IJ

dx dy

J

dz {s,,a,, + SyyCiyy + 2 £ , , a , , ) (7)

G -h/2

Expressing t h e strains i n terms o f t h e displacement field of the m i d p l a n e we o b t a i n :

U = \ j dx dy {iQwl^ + Ky^l^ + AK^yW^^ + {2H - AK,y)w\,,w\y\ (8) G

We derive t h e d i f f e r e n t i a l equation of the deflection surface and the a p p r o p r i a t e b o u n d a r y conditions using the v i r t u a l displacement m e t h o d t o get a well-conditioned b o u n d a r y p r o b l e m . T h e d e r i v a t i o n corresponds t o t h a t of Landau and Lifschitz (1989) f o r the i s o t r o p i c p l a t e . For convenience we d i v i d e the expression f o r s t r a i n energy i n t w o parts:

^7 = C/i + 2L{,yU2 ( 9 ) w i t h

G

^2 = ƒ dx dy {w\^^ - w^,,w\yy)

G

A p p l y i n g t h e p r i n c i p l e of v i r t u a l displacements, we assume t h a t an i n f i n i t e l y small v a r i a t i o n èw of t h e deflections w of t h e plate is produced. T h e n the corresponding change i n s t r a i n energy is:

= ƒ + KyW^yyèW^yy + H {W^,jW\yy + W\yy6W\,,)) ( ü )

G

^^2 = j dx dy {2w\^y6w^^y - w\^Jw^yy - w^yy 8iü\^^) (12) G

= j dx dy {d,{W^,y8W\y - W\yy8lü\,) ^ ( | , , IÜ | , - | t ü | , ) ) G

I n t e g r a t i n g t w i c e by parts we have 6Ui = Li - L2 + L3 w i t h :

^1 = ƒ {dxiL{,W^,jW\, + HW^yySW^,) + dyili'yW^yyêWly + H W \, ^ W\y))

^2 - j dxdy {dx{KxW\^^Jw +Hw\^yyèiü) + dy{L{yW\yyy6iü +Hiü\^^y6w)) (14)

G

^3 = j d x d y 6 w { L { X w + L(yd^yW + 2Hdldlw) ( I 5 )

G

T h e double i n t e g r a t i o n i n / i , and èU^ can be replaced by a simple hne i n t e g r a l u s i n g t h e divergence t h e o r e m :

dx dyV - Ip = j ) dln-ijj ( i g ) G dG

dG is t h e b o u n d a r y of t h e d o m a i n G w i t h the n o r m a l vector u a n d the t a n g e n t i a l vector / : n = Therefore we get: h = f J dG h ^ f _J dG 6U2 J dG (18)

dl {n:,{w\^y6w\y - wiyyêwi^) + ny{wi^y6w\^ - w\^Jwiy))

T h e p a r t i a l derivatives d^^ and dy can be s u b s t i t u t e d b y dn and d f .

dn = n • V = n^d^ + Uydy d^ = n^dn - Uydi dl = l-V = n^dy-nyd^ dy = Uydn + n^di

(19)

W i t h especially ^ ^ i ^ ; = njw\n - nySw\i and Sw\y = ny6w\n + njw\i we o b t a i n :

h = j dl6w\n(nl{K^w\^^^-Hw\yy)-Vnl{KyW\yy +Hw\^S) (20) dG

+ j dl èw\i n^riy (^{H - Kx)w\^x - {H - Ky)w\yy^ dG

SU2 = j dl8w\n{^'nxnyW\^y-nlw\^^-nlw\y^ (21) dG

+ j ) dl 6w\i [{nl - nl)w\^y - n^ny{w\^^ - w\

I n t e g r a t i n g b y parts t h e terms of the t y p e dl 8w\i... using the f a c t t h a t b o u n d a r y t e r m s i n l i n e integrals along a closed contour vanish, we can express the first v a r i a t i o n of t h e s t r a i n energy i n t h e f o r m :

6U = j dxdy8w(^K^dtw + Kyd^w + 2Hdldltu) (22)

G

+ j> dl8w < dl n^Uy (^{K^: - H)w\^:^ - {Ky - H)w\yy^ - n^{KxW\^^j; + Hw\^yy) dG ^ ^

,2 „2^

-ny{KyW\yyy ^ Hw\^^y) "f 2K^y dl {n^ny{wi^^ - iDiyy) - {n^ - ny)iu\^y

+ i dl 8w\n \nl{K^w\^^ + Hw\yy) + nl{KyW\yy + Ew\^^) J

dG

A n elastic b o d y is i n t l i e state of e q u i l i b r i u m , i f SU is equal t o the w o r k done by the e x t e r n a l forces d u r i n g the assumed v i r t u a l displacement:

SU = J dx dySwp{x,y) (23) G

Using the f u n d a m e n t a l l e m m a of the calculus of variations, the d o m a i n i n t e g r a l provides t h e d i f f e r e n t i a l equation f o r o r t h o t r o p i c plates:

dlw + Ky 9 > + 2E dldlw = pix, y) (24)

As the second result of the v a r i a t i o n a l process, we get t w o sets of b o u n d a r y c o n d i t i o n s , n a m e l y the n a t u r a l and the k i n e m a t i c b o u n d a r y conditions. For rectangular plate, the f o l l o w i n g con-d i t i o n s m u s t be m e t : A l o n g boundaries x = const.: either | K^w^^^^ + {H + 2K^y)w\^yy either I K^w\^^ + {H - 2Kxy)w or \yy A l o n g boundaries y = const.: e i t h e r \ K^w\yyy + (H + 2K^y)wi^^y} = 0 either { / i ' ^ ^\yy ~ 2K^y^wyr^ ] = 0 or or or Ul is prescribed (25) w\j. is prescribed (26) w is prescribed (27) w^y is prescribed (28)

T h e s o l u t i o n of (24) involves t w o nondimensional parameters, the v i r t u a l side r a t i o p and the t o r s i o n coefficient T?, which are already k n o w n f r o m Schade (1938). (a and b are the l a t e r a l dimensions of the plate.)

P = a I k , b \ T. y V H (29)

These parameters are sufficient t o describe plates w i t h o u t free edges since i n this case there is no exphcit infiuence of the torsional r i g i d i t y K^^y. For the considered case of free edges K^y enters the s o l u t i o n and we need t o i n t r o d u c e the "mean Poisson's r a t i o " as a new n o n d i m e n s i o n a l parameter:

H - 2K,..

0 <T <r) T =

^/KJi\ = ^JVxVy (30)

r = 0 means no infiuence of the Poisson's r a t i o , r = 77 means vanishing t o r s i o n a l r i g i d i t y .

3. S o l u t i o n o f p l a t e e q u a t i o n

y b

U s i n g the m e t h o d of single f o u r i e r series, we take the solution of (24) i n the f o r m

':{x,y)= Uk{y) sinnkX (31) fc=l,3,..

where ^fc = -Kk/a. Each t e r m of series (31) satisfies the b o u n d a r y conditions along the t w o opposite s i m p l y supported sides. W i t h

p{y) = p{y)l Y T^"^^^-'^" k=l,3,...

we get a f o u r t h order o r d i n a r y d i f f e r e n t i a l equation:

K.l^t^kiy) + Kyd'yUkiy) - 2H iildluuiy) = ^ P{y) (33)

We assume t h a t p{y) is a linear f u n c t i o n : p ( y ) = Po + PiV/b. T h e p a r t i c u l a r s o l u t i o n of (33) is:

i"'"'(a--,2/) = K?/) E ^^^^

fc=l,3,.. ^

A useful f o r m u l a t i o n f o r the variety of homogenous solutions is:

4 - i y ) = - j ^ A , e ' ^ ' ' y (35)

T h e roots of the corresponding characteristic equation K^o + KyX^ - 2H X"^ = 0 are:

/

\ i / 2

Ai,2 =

-^{H +

J h ^

- K J ( y ) (36) \Ky J

\ 1/2

1

A3,4= \^-^^iH-^H^-K.Ky)j

A c c o r d i n g t o the sign of the d i s c r i m i n a n t we have t o consider the f o l l o w i n g three cases:

a) > KJCy

A l l roots of eq. (36) are real.

H - - > 2 / ) - 7 ^ E ' ^ ^ E ^ / = . - P ( ^ ' ^ ^ » ) + K ^ / ) ) (37)

;c=i,3,...

V i = i

/

T h e b o u n d a r y conditions at the edges y = const, determine the coefficients

A,^k-Y S . , A , , = B, (7 = 1 , . . . 4) (38)

i = i

W i t h w = -Kb I a the m a t r i x S is: S l , , = uX.ke^^^^'iKyX'^-H ^2K,y) (39) 52,, = e^^^''{KyX]-H + 2K,y) Ss,, = 1

r KyXj -H + 2K^y case 1

T h e vector of the r i g h t h a n d side of eq. (38) is constant: Bl = {H + 2K,y)pi E3 = -po I {H - 2K^y)pQ e a s e l -pi case 2 (40) T h e s o l u t i o n is of the f o r m M = ^ / ( A i , A 3 : A, d / ( A 2 , A 4 ) A3 = ^ / ( A 3 , A i ) Aa = ^ / ( A 4 , A 2 ) (41) (42)

where d is p r o p o r t i o n a l t o the determinant of the equation system and f{u,v) is s t i l l t o be developed. W i t h the abbreviations ü = v? - and v = v"^ - v^e f i n d t h a t :

For Case 1:

f{u,v) = po fo{u,v) + Pl fi{u,v)

fo{u,v) = vk v?üv -I- [u^ - v^)uvx cosh uvk

- ve~'^^'' [uv"^ sinh Lovk + vv? cosh Lovk)

fi(u,v) = ( M ^ - v'^) (uvurck cosh u>vk + v?v + v"^

LO sinh uvk d{u, v) = 2fc (u^ - v'^)[vu'^ sinh wu/c cosh uvk - uv'^ cosh uuk sinh wt;/c

For Case 2:

fo{u,v) = üwA; 72i;(;/^ + ö e " ' ^ " ' ' ) sinh a;t;/fc

-j-wf i}(i^^ + V cosh wüfc) e"'^"^ - ü{v + cosh LOvk)

fi{u, v) = vujkv^ [vu sinh uvk + -«(u e "^"^ - iZ cosh wufc)

+ « « ( t ; e""^"^ - - ü) sinh wu/c

y 6"'^"'= + («2 + v)v\ cosh wu/t - - ' ^ (43) (44) (45) (46) (47) (48)

+

d{u,v) = 2ojk

^[i^x + (u^ + v)e'

2uüvv - uv{v? + v'^) cosh uuk cosh tovk

+ (ïi^ü^ + u^iZ^) sinh wuA; sinh ujvk

(49)

b ) < J^J^^

T h e f o u r roots A i . . . A 4 are complex conjugate pairs. T h e previous results are s t i l l v a l i d . c) = KJ{y

T h i s case includes the isotropic plate. T h e characteristic equation has t w o double roots:

T h e s o l u t i o n f o r the f i r s t t w o cases does not h o l d since the d e t e r m i n a n t d{u,v) vanishes. W e have t o m o d i f y the f u n d a m e n t a l s o l u t i o n (37):

U s i n g t l i e previously defined f u n c t i o n f{u,v) and d(xi,v), the coefhcient A , are:

(dl - 2d^d, + dl)f{u,v] {dl-2dud. + d^,)diu,v)

2 { d u - d , ) f { u , v )

(52)

--v=±\

We o m i t a presentation of the extensive expression f o r Aj. 4. A p p l i c a t i o n to d o u b l e w a l l p a n e l s

A double w a l l panel consists of b o t t o m and top p l a t i n g and transverse o r / a n d l o n g i t u d i n a l webs.

A l t h o u g h i t is composed of isotropic m a t e r i a l , the b e n d i n g stiffness i n l o n g i t u d i n a l d i r e c t i o n is o f t e n different f r o m t h a t i n transverse d i r e c t i o n . I f the webs are u n i f o r m , numerous and closely spaced, they can be "smeared" over the p l a t i n g , thus allowing the entire panel t o be t r e a t e d as a single o r t h o t r o p i c plate. Therefore the derivation of the governing d i f f e r e n t i a l e q u a t i o n follows the same steps as f o r ( 2 4 ) . T h e first step is t o derive an expression f o r the s t r a i n energy of the panel. T h i s was already done b y Schade (1938). Since we assume t h a t the webs s i m p l y act as technical beams, t h e i r s t r a i n energy is:

a 0

Us = ]^E j dx j dy '«'f^^ + iwy w

(53)

T h e u n i t moments of i n e r t i a iyjx and i^iy are the moments of i n e r t i a of the webs d i v i d e d b y t h e i r distance. T h e y are t a k e n about the respective n e u t r a l plane of the o r t h o t r o p i c p l a t e , n o t a b o u t the n e u t r a l axis of the web alone. As i n plate theory, the stresses and strains i n the t o p p l a t i i r g are:

(54) Ttx W\xx ety = rty wiyy

^ixy — dxU + dy CTtx = ^ r-CTtx = <7ty = ^ (i <7ty = '^txy — E '^txy — 2(1 + u) \xy 'rtxW\xx + yny W\yy —^{nx + ny)w\xy

rtx and rty are the distances of the t o p p l a t i n g f r o m the n e u t r a l axis of b e n d i n g i n x- and

d i r e c t i o n respectively. The d e t e r m i n a t i o n of stress and s t r a i n for the b o t t o m p l a t i n g is carried out by s u b s t i t u t i n g and rty by - r ^ ^ and -rby. The s t r a i n energy of the p l a t i n g is:

E

0 0

-f- 2U {ttrtxTty + hrbxHy) W\xx W\yy 1 - V + [tt {rtx + rtyY + tb {rbx + n y f ] ,2 \xy (55) I n t r o d u c i n g the expressions ^ix 'Ifxy 1 - / . 2 1 1 -l _ ^ 2 {ttrtxTty + tbrbxTby) •

w i t h ix = ijx + i-wx and iy = ijy + i^„y, the t o t a l s t r a i n energy is: a b

U E

dx

J

dy

; 0

1-u

^^^\xx + h^lyy + '^vifxyW\xxW\yy + - ^ { i f x + i j y + '^ijxy)W\^y (57)

B y comparison w i t h (8) we get:

Kx = Eix K y ^ E i y Kxy = ^ _{E I f x + I j y} 2i _ixy

( 1 - V ) { i j x + i j y ) + 2(1 + U) ijxy

(56)

(58)

These equations f a c i l i t a t e the f o r m u l a t i o n of the differential equation w i t h a p p r o p r i a t e b o u n d -ary c o n d i t i o n f o r double w a l l panels. The useful a p p r o x i m a t i o n s = rty and rbx = rby give:

I xy { l - t , ) E i f x y H = E i f x y (59)

,,y = 0, and The b o u n d a r y conditions f o r e.g. y = const, are w = 0 or iy w\yyy + {2 ~ v)ifxy w\

iu\y = 0 or iy w\yy - f u i f x y w\xx = 0.

A n o t h e r i m p o r t a n t result f r o m the above a p p r o x i m a t i o n s is a constant r e l a t i o n between r and 7/, namely t = urj. v is the Poission r a t i o o f t h e isotropic m a t e r i a l . W i t h this constant r e l a t i o n , the elastic behavior of double w a l l panels depends only on rj and p.

5. D e s i g n C u r v e s

A p p e n d i x A presents tlie solutions f o r double w a l l panels w i t h v = 0.3. T h e deflections and stresses are a l l expressed i n terms of the l o a d i n g , dimensions of the s t r u c t u r e and the Young's m o d u l u s , m u l t i p h e d by coefhcients K which are f u n c t i o n s only of the curve parameters r] and the v i r t u a l side r a t i o p as abscissa. T h e diagrams are only v a h d f o r the case of r = 0.3?]. A change of t h e mean Poisson's r a t i o r changes noticeably the evaluated results.

T h e diagrams include the isotropic plate w i t h = 0.3 and 77 = 1. T h i s case is discussed by Timoshenko (1959). A comprehensive investigation especially of the e x t r e m cases p ^ 0 and p ^ 00 is presented. I f the l e n g t h b of the opposite simply supported edges of a u n i f o r m l y loaded i s o t r o p i c plate is very large compared t o a, the deflection of the free edge is the same as t h a t of a u n i f o r m l y loaded and simply supported s t r i p of the l e n g t h a m u l t i p h e d b y t h e constant f a c t o r (3 - i / j ( l + + u). For another extreme case, when a is very large compared

t o b, the m a x i m u m deflection of the plate is either the same as f o r a u n i f o r m l y loaded s t r i p of the l e n g t h b b u i l t i n at one end and free at the other, or, i f three edges of the plate are s i m p l y supported, the behavior corresponds w i t h St. Venant's t o r s i o n theory.

6. E x a m p l e

To demonstrate the m e t h o d and also t o v e r i f y i t s accuracy, we now use is t o analyze a doclc gate. A d e t a i l o f t h e v e r t i c a l section o f t h e structure is shown below. T h e s t r u c t u r a l dimensions and parameters are:

W i d t h : a = 20.0 m H o r i z o n t a l webs: t-wx = 8 m m = 1.5 m {bm/b)x = 1.1 Height: b = 7.5 m V e r t i c a l webs: = 8 m m Sy = 2.0 m {bm/b)x = 0.87 H o r i z o n t a l stiffeners Top and b o t t o m p l a t i n g Distance between platings

H P 100 X J t = 8 m m h = l.2m

(8)

800 1 all stiffeners HP 100x8 ^ la W ^

Based on these values the o r t h o t r o p i c plate parameters are:

E ]

uoo

Q = b

V

T h e gate is loaded b y h y d r o s t a t i c pressure w i t h the m a x i m u m :

Po = g - g - L y = 1 • 9.81 • 7.5 l < N / m ^

= 7843 cm^ iy = 5937 c m ' i f x y = 5414 cm''

F r o m Figs. 7, 9 and 11 of A p p e n d i x A :

iv = 0.013 "

1 ^

= 15.4 m m E j l x t y

r . = ! ^ = 0 . 1 2 ^ = 43 N / m m ^

h x 2 ^ ^ ; ^ '

r , = ! ^ = 0.056 ^ = 23 N / m m ^

A f i n i t e element analysis w i t h a detailed m o d e l i n c l u d i n g a l l s t r u c t u r a l members ( p l a t i n g , webs w i t h c u t o u t s , stiffeners) provides the f o l l o w i n g values:

w = 16.9 m = 44 N / m m ^ Ty = 28 N / m m ^

The m a i n reason f o r the d e v i a t i o n i n p r e d i c t i o n of deflection and stresses using o r t h o t r o p i c plate t h e o r y is the neglect of shear d i s t o r t i o n of the webs. For a comprehensive i n v e s t i g a t i o n of the role of shear i n b e n d i n g of double w a l l panels see Hughes (1988).

R e f e r e n c e s

L A N D A U , L . D . ; LIFSCHITZ, E . M . (1989), Lehrbuch der Theoreiischen Physik, Akademie-Verlag Berlin T I M O S H E N K O , S.; W O I N O W S K Y - K R I E G E R , S. (1959), Theory of plates and shells, McGraw-Hih SCHADE, H.A. (1938), Theory of ship bottom structure, Trans. SNAME 46

SCHADE, H.A. (1940), The orthogonally stiffened plate under uniform lateral load, J. A p p l . Mech. 7 SCHADE, H.A. (1941), Design curves for cross-stiffenend plating, Trans. S N A M E 49

A p p e n d i x A : D e s i g n C u r v e s f o r D o u b l e W a l l P a n e l s w i t h u = 0.3

Note: m^.^ is the bending moment per unit length about the axis perpendicular to the free

edge;

rriyy

is the bending moment per unit length about the axis parallel to the free edge.

-T3 O 0) S s

1

SP

a 1-^ 11 II "o faO a CD o s g 0) O ft a j3 1^

A Fast M e t h o d of Evaluation

for the Translating and Pulsating Green's Function

M a l i c k B a , E . N . S . M . A . Chasseneuil d u P o i t o u - France^, a n d M i c h e l G u i l b a u d , C . E . A . T . U n i v e r s i t é de Poitiers - France^

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

T h e seakeeping of m a r i n e structures, which is a m a j o r p r o b l e m i n h y d r o d y n a m i c s , has been w i d e l y s t u d i e d i n the last t w e n t y years. T h e solution can be considered as satisfactory f o r offshore p l a t f o r m s w i t h o u t f o r w a r d speed or f o r ships w i t h low t o moderate f o r w a r d speeds, b u t n o t f o r ships w i t h h i g h speeds. T h i s d i f f i c u l t p r o b l e m can be solved by b o u n d a r y i n t e g r a l equations using a Green's f u n c t i o n adapted t o the pecuhar p r o b l e m . A t the b e g i n n i n g of t h e 80's, m a n y authors have studied t h e h a r m o n i c Green's f u n c t i o n as Bougis (1980), Inglis and

Price (1980) or Guevel and Bougis (1982) b u t , due t o h m i t e d c o m p u t e r resources, i t s use i n

seakeeping codes w i t h f o r w a r d speed was almost impossible. T o s i m p h f y these codes, some a p p r o x i m a t e f o r m u l a t i o n s of the hnearized free surface b o u n d a r y c o n d i t i o n were proposed b y

Guevel and Grekas (1981), Grekas (1981), Huijsmans and Hermans (1985), Delhommeau and al. (1991), b u t correct results have been o b t a i n e d only at low f r e q u e n c y a n d l o w f o r w a r d speed.

T h e p r o b l e m can also be solved using Rankine singularities, b u t the m a i n d i f f i c u l t i e s are due t o the n u m e r i c a l satisfaction of the r a d i a t i o n c o n d i t i o n and t o the increase of the order of t h e complex system t o solve. Following the a b s o r p t i o n c o n d i t i o n chosen f o r u p s t r e a m waves, codes are efficient either at l o w values (less t h a n 0.25) o f t h e non-dimensional parameter r = uU/g (w frequency, U f o r w a r d speed and g g r a v i t a t i o n a l acceleration) where u p s t r e a m influence exists,

Maisonneuve et al. (1993) or f o r higher values, Nakos and Sclavounos (1990a,b).

M o r e recently, w i t h t h e increase of computer performance, new studies of the unsteady Green's f u n c t i o n have been achieved, Wu and Eatock Taylor (1987), Bougis and Coudray

(1991), Squires and Wilson (1992), Chan (1989). Coudray and Le Guen (1992) have compared

t h e wave p a t t e r n s generated by a submerged source f o r various s i m p l i f l c a t i o n s of the hnearized Neumann-ICelvin b o u n d a r y c o n d i t i o n at low value of r = 0.128.

For c o m p u t a t i o n s of seakeeping at any Froude number and frequency, i t is necessary t o use K e l v i n singularities and t o get a fast and efficient m e t h o d t o c o m p u t e the unsteady Green's f u n c t i o n . T h e development of such a m e t h o d , not only f o r the f u n c t i o n b u t also f o r i t s first and second derivatives, is presented here, vahd f o r any value of r . T h e m e t h o d is i n t e n d e d t o be i n t r o d u c e d l a t e r i n seakeeping codes.

2 T h e h a r m o n i c G r e e n ' s f u n c t i o n w i t h f o r w a r d s p e e d

^ A coordinate system Oxyz is defined m o v i n g at constant speed U, w i t h x p o i n t i n g i n t h e d i r e c t i o n of f o r w a r d speed and z positive upwards f r o m the mean surface. H a r m o n i c m o t i o n at circular frequency u) is assumed, w i t h an iirviscid and incompressible fiuid; the free surface extends t o i n f i n i t y . T h e surface tension is assumed t o be neghgible w i t h respect t o i n e r t i a forces. For weak amphtudes a n d mean camber of the free surface elevations, the p e r t u r b a t i o n velocity p o t e n t i a l (p satisfies the classical N e u m a n n - K e l v i n b o u n d a r y c o n d i t i o n . P a r t i c u l a r l y , the hnearized free-surface b o u n d a r y c o n d i t i o n is given by

, , 2 Ö ' G dG . dG

^ J ^ - ^ ' ^ U ^ - ^ (^ + 9 j ^ = 0 on z = 0. (1) G is the t r a n s l a t i n g and p u l s a t i n g Green's f u n c t i o n . Following Bougis (1980), t h e t o t a l p o t e n t i a l

$ IS searched under the f o r m

= $ * ( M ) c o s w i + $ * * ( M ) s m w < .

W i t h these assumptions, t w o Green's f u n c t i o n s Gc and Gs appear, related by

Gs IdGc

'OJ dt

So, only the f u n c t i o n Gc is to be computed; Gc is defined by

COSOJt Gc

\MM

j- + gc{M,M'j) = Go + Gr + G2.

M{x,y,z) is the f i e l d point and M'{x',y',z') the source p o i n t . For convenience, G is s p h t t e d

i n 3 different f u n c t i o n s dehned by

Go{M,M',t) cos(wt)

-tujt

^y{X-X>f+{Y-Y'y + (Z-Z')^ ^ / { x - x ' f + ( y - y r + ( z + z ' )

/2 Kl [gi (KiO + ffl iK\e)] - K2 [ f f l {K2O + g l {K2e)]

-iwt A/ 1 + 4 r cos 0 •Zi\g2(Z!,i)+g2(Z^i')]-Zi[g2{Zit)+g2{Zii')]^Q V4TCOSe-l G 2 ( M , M ' , i ) = — :g3lgi(^30+gl(^3^')]-^4[53(^4^)+33(^4g')] V 4T COSÖ —1 , ^ - i ; : * f W 2 JC3[g3(A'30+53(A-3g')1-A-4[gi(A-40+5i(A-4^')]^g . + ^ J ö c + a c \/1-4TCOSÖ (2) 'R is f o r real p a r t , w i t h e = Z + Z ' + i [ ( X - X ' ) COSÖ + (Y - Y ' ) s i n ö ] ; = Z + Z ' + i [ ( X - X ' ) cos e-{Y - Y') s i n ö ] ;

6 = a ö = ö,) ;

i'^ = i'{e = ec) ;

x = ^ ; y = | ; z = t ; X ' = | ; ; Y' = i ; Z ' - - z _to

T h e non-dimensional numbers used are w = 0J^/Q~g ; ï = i \ / f f / 4 , the Froude n u m b e r i^n = Vj^fglo and the parameter r = ^ = wi^n; ^0 is a reference l e n g t h . X i t o X 4 and Z 3 , are f u n c t i o n s of r and F „ , Bougis (1980). T h e expressions f o r Go and G i present no c o m p u t a t i o n a l d i f f i c u l t i e s . I n G 2 , the h m i t s of i n t e g r a t i o n are given by

i f r < 0 . 2 5 , f c = 0^ = = 0;

i f 0.25 < r < 0 . 5 , = cos^^ ( ^ ) ,B'c = ^, i n f i n i t e l y small, i f r > 0.5, Öc = c o s - i ( i ) , 9'^ = c o s - i ( ^ ) , i n f i n i t e l y small.

Functions ffi , ...,to ffa are the m o d i f i e d complex i n t e g r a l f u n c t i o n s defined by

ffi(0 = e « £ i ( 0 i f 0 < a r g ( O < 2 7 r

ff2

(0 =

e^Ei

(0

i f - < arg

(0

< TT

w i t h

£ i ( 0 =

Ei{0-2i7r i f

S ( O < 0 .

T h e complex exponential i n t e g r a l f u n c t i o n Ei is

^ i ( 0 =

i r ^ - l ^ d t

i f - 7 r < a r g ( 0 < 7 r ^ i ( 0 = A ° ° S ^ r f ^ i f 3?(O>0. T l i e hrst derivatives are g f - G „ ( M , M ' , t ) = ! ï # i { ( A ' - . Y ' ) H ( y - r ' ) 2 + ( ^ + ^ ' ) ' } ^ { { X - X ' y + {Y-Y')^ + (Z-Z')^}i 3 G l = - V 3 ? < i e - « ^ ' T.— G2 = —^3? dxj Tril r7r/2 p K f [gi{KiO+i-iy+'ai{K\e)]-Ki g\(A2g)+(-l)-'+^g-i(A'2^')] JO ^ V l + 4 r c o s ö " ' ^ ' re', ,n^l[MZ30H-iy+'MZ3i')]-zl[g2{Zii)+{-iy+'g2{Zie)] j „ , g-to^T f ö c - a c , D ^ I [ g l ( ^ 3 0 + ( - l ) ^ + ^ g i ( Z 3 r ) ] - ^ | [ g 3 ( ^ 4 0 + ( - l H + ^5-3(^4g')] \ / 4T COSÖ—1 27r7t-2(l-,) ('e-f^<:^c + ( _ i ) j + l e A ' c ? i (W VT sin

p/2

^Al[53(A-3a + ( - l ) ^ + '53(A'3gO]-A-|[gi(K-40+(-l)J + ' g - i ( A - , e ' ) ] \ / 1 - 4T I dO (3) For the second derivatives, we have

^ G „ ( M , M ' , i ) = ^ { + 3 (

[{X-X')^ + {Y-Y'y + {Z+Z'y]2 [(X-X')2 + {Y-Y'y + {Z~Z'yy. Al

[{X-X')^ + {Y-Y'y + {Z-Z')-^]I [{X-X')2 + (Y-Y')2 + {Z+Z'y]i ) }

di ' j 'T^o \ ''0 \/1+4= ^ S f J

| e - « ; ^ r / 2

^2

A t [gl (A'l O+gj (A-i^')]TCOS - A t

l3i

0

(A'20+5l

{K2i')\ g - i a ; T , ; ^ 2 ^3'[g2(g30+g2(Z3^')]-Z,3[g2(Z4g)+g2(Z3g')] ''0 V'4T COSÖ-1 ^^-iZPt röc-«c „-p2^|[5i(^30+gH^3r)]-Z,^[g3(Z4g)+g3(Z4^')]^p >/4T COS 0 - 1 ^ g - i a . i ^ 2 2 7 r A 3 ( l - 0 ( e-f'''<:«':+e-f''=f^ (4) \ / T sin 0c ' \ i r.-3r w i t h i f j = 1 i f i = 2 i f J = 3

fT/2

p 2 A | [ g b ( A - 3 0 + g 3 ( A - 3 ^ ' ) ] - A | [gi (A-40+gi ( A 4 g ' ) ] a-i = .-c, yl = A l = X - X ' , 5 = i c o s ö , = i cos 0^;

X2^y, A = A l = y - y , B = i sin 0, B^ = i sin 0^; X3 = z, A = Z + Z', Al = Z - Z', 5 = 1.

3 N u m e r i c a l i n t e g r a t i o n o f t h e G r e e n ' s f u n c t i o n

T h e m a i n d i f h c u l t y i n the c o m p u t a t i o n of the f u n c t i o n G is the calculation of the integrals over variable 0 i n (2) f o r G'l and G 2 . The f u n c t i o n s to be integrated are sometimes q u i t e oscillating, r e q u i r i n g very small i n t e g r a t i o n steps to o b t a i n a desired accuracy and consequently a h i g h c o m p u t a t i o n a l t i m e . F u r t h e r m o r e , oscillations are not over the f u l l range of values of 6 b u t sometimes only over a s m a l l p a r t of i t , excluding the use of classical methods of i n t e g r a t i o n .

A m e t h o d of resolution of d i f f e r e n t i a l equations, Press and al. (1986), c o m p u t e d the integrals. Let Y be t h e i n t e g r a l over the f u n c t i o n F{e) t o be computed

Y = r F{d)de.

T h e p r o b l e m is replaced by the c o m p u t a t i o n of the f u n c t i o n Hiö) defined by its derivative

'^EM^F{e) w i t h i f ( ö i ) = 0. dO

W i t h these assumptions, the i n t e g r a l is given by Y = H {62) • T h e solution of an o r d i n a r y d i f f e r e n t i a l equation is generally obtained using an i t e r a t i v e process

H{e + de) = Hie) +

^ ^ 4 0 ^ + 0

{de')

de is the step of i n t e g r a t i o n . These steps depend on the desired h i g h accuracy, r e q u i r i n g m a n y

cahs of the f u n c t i o n t o integrate. To prevent a drastic increase of C P U t i m e , the f u n c t i o n s t o be i n t e g r a t e d are i n t e r p o l a t e d by polynomials of f o u r t h order. A stepsize c o n t r o l determines the local step l e n g t h b y e x t r a p o l a t i o n o f t h e previous ones w i t h respect t o the local slope o f t h e i n t e g r a n d a n d o f t h e corresponding accuracy. FinaUy, a m i n i m u m step, chosen t o 10 ^ r a d i a n , is defined t o take i n t o account the t r u n c a t i o n errors. C o m p u t a t i o n s are done w i t h double precision. W i t h this m e t h o d of c o m p u t a t i o n , i t is not necessary t o have different processes i n the t w o ranges r > 0.25 or r < 0.25, i n spite of the existence of a s i n g u l a r i t y of the i n t e g r a n d f o r the last range. C o m p u t a t i o n s can be done as close t o r = 0.25 as required, only t h i s value must be excluded.

T h e integrals are i m p r o p e r f o r Ö = 7 r / 2 or e^. Close t o 0 = 7 r / 2 , the i n t e g r a n d m u s t be replaced b y an equivalent f o r 7r/2 - I O - * < 0 < ir/2. T h e f u n c t i o n £ i ( r ) is c o m p u t e d using a development t o the f o u r t h order t o keep an accuracy of 0 . 0 1 % .

A r o u n d 0 = 6»^, an equivalent of the f u n c t i o n f o r - a e < Ö < + « c is c o m p u t e d , t a k i n g care of choosing smaU enough t o o b t a i n a good accuracy (a^ ~ l O ' ^ ) . A computer code has been w r i t t e n t o calculate the Green's f u n c t i o n and i t s first and second derivatives w i t h a prescribed error. T h e m e t h o d of Telste and Noblesse (1986) computed EI{T). T h e various f u n c t i o n s are c o m p u t e d w i t h an accuracy of 0.01% w i t h low C P U t i m e , even on a simple personal computer ( t i m e is d i v i d e d by about 120 w i t h respect t o classical i n t e g r a t i o n ) .

4 N u m e r i c a l r e s u l t s a n d c o m p a r i s o n w i t h o t h e r m e t h o d s 4.1 T h e G r e e n ' s f u n c t i o n a n d its d e r i v a t i v e s

To check the results of the c o m p u t a t i o n s , the Green's f u n c t i o n is p l o t t e d i n the f o r m used by

Wu and Eatock Taylor (1987); F i g . 1 shows the real p a r t of f u n c t i o n G and i t s first derivatives

i n comparison w i t h steady results for the h m i t i n g steady case (w ^ 0 ) . F i g . 2 shows real and i m a g i n a r y p a r t of f u n c t i o n s for r > 0.25 {U = 2 m / s and = 1.4 r a d / s ) compared t o results of Inglis and Price (1980). These authors decomposed the integrals i n ( 2 ) , w i t h very osciUating integrands i n v o l v i n g the exponential f u n c t i o n , i n t o a p r i n c i p a l value i n t e g r a l and a residual t e r m ; the p r i n c i p a l value i n t e g r a l is t h e n evaluated along a ray i n the complex plane, so the i m a g i n a r y p a r t of the argument of the exponential f u n c t i o n is zero. T h e osciUatory behaviour of the i n t e g r a n d is therefore avoided. T h e values of the f u n c t i o n s are given on t h e free surface, on a r a d i a l hne m a k i n g the angle a = t a n - ^ O - S ) w i t h x axis; the source is l o c a t e d at p o s i t i o n (0,0,-1). T l i e figures sliow the precision of our c o m p u t a t i o n s . Figs. 3 t o 5 show simUar calculations f o r the second derivatives. F i g . 3 gives the influence of an increasing speed

(1 < U < 2 m / s ) at constant frequency, LO = 1.4 r a d / s . W i t h increasing velocities ( r < .25)

ripples appear downstream of the source, w i t h amphtudes m a x i m u m j u s t f o r r shghtly less t h a n 0.25. For the highest velocity, i n the supercritical region, oscillations occur at a m u c h lower wave l e n g t h . Figs. 4 and 5 i l l u s t r a t e the infiuence of an increasing frequency at U = I and 5 m / s , respectively, w i t h same values of r . T w o frequencies correspond t o either side of the c r i t i c a l p o i n t b u t the second derivatives are less sensitive here t h a n the f u n c t i o n or i t s first derivations (Wu and Eatock Taylor 1987). F i g . 5 shows an increase of the amphtudes of the d o w n s t r e a m and u p s t r e a m ripples on derivatives w i t h respect io y OT z when r is s h g h t l y lower t h a n 0.25. T h e influence of the frequency is weaker at the highest velocity, p a r t i c u l a r l y on the real p a r t s .

C o m p u t a t i o n t i m e is about 0.1s f o r b o t h real and i m a g i n a r y parts of the Green's f u n c t i o n , f r o m 0.17 t o 0.2s f o r the first derivatives and f r o m 0.25 t o 0.4s f o r the second derivatives o n an A L L I A N T F X 4 0 w i t h 4 processors.

4.2 F r e e s u r f a c e c o m p u t a t i o n s

A n o t h e r means t o check the validity of the results is t o compute the l e f t side of the free surface b o u n d a r y c o n d i t i o n (1) and t o compare the results w i t h zero as devised by Squires and

Wilson (1992). A f t e r separating real 3? and i m a g i n a r y parts 9 i n ( 1 ) , we o b t a i n ' f tir + t2T + i s r + tir = « r

1

hi + hi

+ ^3; +

tii = tti

where i i , = §ft(C

/20)

and

tu

= ö ( C

/ 2 0 ) ,

and s i m i l a r l y f o r the other terms of ( 1 ) . T h e percentage error on the real and i m a g i n a r y parts of the free surface b o u n d a r y c o n d i t i o n are given as

% error i n the real p a r t of b o u n d a r y c o n d i t i o n = ^""I'^'-l % error i n the i m a g i n a r y p a r t of b.c. = ioo\a,\

Results i n Table I show t h a t this c o n d i t i o n is satisfied t o a h i g h order of accuracy ( b e t t e r t h a n 0.01%).

Table I . Free surface b o u n d a r y c o n d i t i o n errors source(0,0,-2), field(0.707,0.707,0), F = 0 . 6

r Real p a r t _{I m a g i n a r y p a r t ö percent error 3? percent error} 0.1 9.4652E-05 -3.5763E-05 8.2549E-03 1.9817E-03 0.2 2.5272E-05 -2.9564E-05 6.5960E-04 1.0461E-03 0.3 7.0095E-05 -9.2030E-05 2.4975E-03 6.1451E-03 0.4 6.4373E-05 4.1366E-05 2.5170E-03 3.1684E-03 0.5 -2.3842E-06 1.0848E-05 9.7471E-05 8.5377E-04 0.6 2.3842E-07 1.3590E-05 9.9899E-06 1.0615E-03 0.7 2.8610E-06 1.6212E-05 1.2160E-04 1.2435E-03 0.8 8.1062E-06 1.8358E-05 3.4737E-04 1.3801E-03 0.9 -1.1444E-05 2.7776E-05 4.9250E-04 2.0496E-03 1.0 -8.3447E-06 3.6120E-05 3.5969E-04 2.6231E-03

Once the Green's f u n c t i o n and first derivatives are c o m p u t e d , i t is easy t o compute t h e free surface elevations f o r a submerged source (0,0,-1). F i g . 6 shows various wave patterns of an oscillating and t r a n s l a t i n g source f o r different f o r w a r d speeds w i t h Froude number r a n g i n g

f r o m Fn = 0.24 t o 0.72 at t i m e t = Os. For r sliglitly greater t l i a n 0.25, some circular waves f o u n d f o r r < 0.25 disappear, g i v i n g an almost f l a t free surface. F i g . 7 shows patterns f o r the same values of r b u t obtained w i t h F „ = 0 . 6 4 at various frequencies. T h i s g r a p h shows more clearly the disappearance of upstream waves f o r r > 0.25. T h e downstream waves are t h e r e m a i n i n g parts of the concentric waves due t o the p u l s a t i o n . For r > 0.3, these waves appear o n l y i n a d o w n s t r e a m sector, w i t h decreasing angle f o r increasing r . For large r , short waves appear w i t h i n the K e l v i n angle. T h e shapes obtained agree well w i t h those of Chang (1977) and Noblesse and Hendrix (1992), at various values of r r a n g i n g f r o m 0.2 t o 4, f o r the fax field wave patterns of a ship advancing i n a t r a i n of regular waves.

5 R e f e r e n c e s

BOUGIS, J. (1980), Etude de la diffraction-radiation dans le cas d'un flotteur indéformable animé d'une Vitesse moyenne constante et solliché par une houle sinusoïdale de faible amplitude, Ph.D. thesis, E.N.S.M., Nantes

BOUGIS, J. and COUDRAY, T . (1991), Méthodes rapides de calcul des fonctions de Green des problèmes de Neumann-Kelvin et de diffraction-radiation avec vitesse d'avance, 3rd J. Hydrody-namiques, Grenoble

C H A N , C S . (1989), Green function in the theory of unsteady forward motion. Report NAOE-89-22, University of Glasgow

C H A N G , M . (1977), Computations of three-dimensional ship motions with forward speed, 2nd I n t . Conf. Num. Ship Hydrod.

C O U D R A Y , T . and L E GUEN, J.F. (1992), Vahdation of a 3-D sea-keeping software, 4th C A D M O , Madrid

D E L H O M M E A U , G , A L E S S A N D R I N I , B . and V I L L E G E R , F. (1991), Comparaison de différentes approximations du problème de diffraction-radiation avec vitesse d'avance, 3rd J. Hydrodynamiques, Grenoble

GREKAS, A . (1981), Contribution a l'étude théorique et expérimentale des efforts du second ordre et du comportement dynamique d'une structure marine sohichée par une houle régulière et un courant, Ph.D. thesis, E.N.S.M., Nantes

G U E V E L , P. and GREKAS, A . (1981), Le théorème de Lagally générahsé et ses apphcations en hydro-dynamique navale, Bulletin de l ' A T M A , Paris

G U E V E L , P. and BOUGIS, J. (1982), Ship motions with forward speed in inhnite depth, Int. Ship. Prog. 29

HUIJSMANS, R . H . M . and HERMANS, A . J . (1985), A fast algorithm for computations of 3-D ship motions at moderate forward speed, 4th Int. Conf. Num. Ship Hydrodyn., Washington

INGLIS, R.B. and PRICE, W . G . (1980), Calculations o f t h e velocity potential of a translating, pulsating source, R.I.N.A.

M A I S O N N E U V E , J.J., F E R R A N T , P. and D E H O M M E A U , G. (1993), Diffraction-radiation avec vitesse d'avance par une m é t h o d e de singularités de Rankine, 4. J. Hydrodynamiques

NAKOS, D.E. and SCLAVOUNOS, P.D. (1990a), Ship motions by a three dimensional Rankine panel method, 18th Symp. Naval Hydrodyn., Ann Arbour

NAKOS, D.E. and SCLAVOUNOS, P.D. (1990b), On steady and unsteady ship wave patterns, J. Fluid Mech. 215

NOBLESSE, F. and H E N D R I X , D. (1992), On the theory of potential flow about a ship advancing in waves, J. Ship Research 36,1

T E L S T E , J. and NOBLESSE, F. (1986), Numerical evaluation of the Green function of water-wave radiation and diffraction, J. Ship Research 30,2

PRESS, W . H . , F L A N N E R Y , B.P., T E U K O L V S K Y , S.A. and V E T T E R L I N G , W . T . (1986), Numerical recipes, Cambridge University Press

SQUIRES, M . A . and W I L S O N , P.A. (1992), A n investigation into tlie translating, pulsating Green's function for ship motions prediction, 4th C A D M O , Madrid

W U , G.X. and E A T O C K T A Y L O R , R. (1987), A Green's function form for ship motions at forward speed, I n t . Ship. Prog. 34

4.0 2.0

.•I •

1 • ]

S

i

j

, x / • 1

• 1 •

1

• 1 • '

( ^ 1

r

\ i I ; \ ;

\

j V ^

1 1 1—i~i—1 1 1 r 1

_{' 1 • 1 ^ . 1 . 1 ^}

4.0 ao zo -e -4 0 -8 -4 3.0

J-1

' I ' I ' I 2.0-1 ^ -to -2.0

f T T 7 •

a'' V w 4 8

' • I • ' • I • • 3.0

[•2J) rio 0. ' I ' I ' I ' I ' I ' I ' I ' I ' I -6 -4 0 4 8 R r--10

- Inglis and Price X Present method 0.4

I

--0.4 H

• ' 1

/ i

^ } • i /

' ' ' ' ' ' I l l l —

JK 1

1

1 1 1 1 1

1 ]

1— •0.8 0.4 h-0.4 0 3.0 to.

I

0,. -2.0--4 _ 1 _ 0 4 8

1 . 1 ,

* .'

_{1 /}

-t—f

I

X -« -4 0

n

4 8 •3.0 ZO '10 a -to -2.0 U = 2 m / s , LO = .005, r = .0001, F = .64, z' = - 1 . 0 0 Figure 1 T h e zero frequency h m i t

4.0 SJO 10

l f

- J

l

,\

i

If ^

i ^

\ ^

K 'r

\ /

^ V ^

'i , ,-( 10 0. •to ao-10^

' > 1 i ' ',

'/ y v^, 1 i ao 2.0 O R O R - Inglis and Price

X Present method -8 0.8-^ 0.4 -0.4 _1 1—l_J—1—1—1—'—'—

r-

A

r^-

ii

-a4 O R 3.0

i

" .10 -2.0 - « • 4 0

I

\ I

V

I ' l ' l ' l ' l

•* -4 4 8 I I 1 . I

I ' l ' l ' l ' l

4 S ao !L0 to &-to -ZJ

,, . 1 , , , 1 ,

. 1 . 1 \ \ \ \ \ \ -\

-Lililihlil

i

>

~ '

a -to O R ao I . o l -to -iO -B -4 I. I I

I

I I

t t

-6

1 I

V

' l ' l

-4 4 S

• I • ' • I ' '

l ' l I

ao t-zo to 0. ^-to O .fi 0.4I -^ -0.4 -a -4 I I

' I '

I O 4

I

I 1

A A

y

\f

li

I '

I

' I ' I

O R

I '

I

' I '

I 4 e 0.4 -0.8 -e . . 1 , 1 2D Ik -2.0 0 4 0

I

I I I

I •

1 '^ A / \

I

'• ' it i

I

_{I ' l l}

_{, 1 11}

» ^ V I 11

^ •* -4 \4

' I '

I '

1

' I O 4 B 2.0 U = 2 m / s , w = 1.4, r = .29, F = .64, 2 ' = - 1 . 0 0

tv » •e \ C i CV 2 . O OJ

M

li

—'—'—'—1—1—1—1—i—1 ; 1 ij

-4

O R 7.5 5.0 -5.0 H cv 5.0 2.SA

0.-3

'-i -2.5 €

-5.0--8 -4

1 , 1 . 1 . 1 . 1 .

3 4

, 1 , 1 .

8

. 1

A

i\ 1

1—1—1—r—1—1—1—1—1—1— i 1 i

-4

O .R 5.0 E-2.5 -2.5 -5.0

-8

7.6 cv \ -—• e 0. -2.5 -5.0 • —4 1 1 1 1 1 1—1 1 1—

•-

. . - ^

—1—1—1—1—1 r—1—1—1—I—

f

O R 7.5 5.0 2.5 0. l:-2.5 H-5.0 5.0 c? 2.5 \ 0.-cv •ö -2.5^ -5.0-4 1 1 i 1 1 1 , 1 e-5.0 O R 7.5-4 cv CV § 0.. ö -2.5 ^ -5.0

. 1

1

1

1

1

1

1

1

1

1^

ll

A A / N / | L / / V// . 1 , 1 , 1 , 1 , , i\

• i\

\\V .

W l i ï ]

\j \i

y 1

' 1

' 1 '

1

1

1

1

-8 -4

C R 1

1

1

1

1

1

1

1

1

1

4 8

•5.0 2.5 0. 5.0 H a 5 t N t i 5r' O-cv >S -2.54

-5.0--8

I I

I

-4

I I

I

i /

Y'-'i \

i ƒ i

' 1

; I 1

i

U = 0 . 5 0 0 = 1.400 0.071 F I >

I

-8

0.160

\ \

iV'i

4

I I

I

5.0 2.5 -2.5 -5.0 O R x ' = y ' = 0 , z' U = 1.000 Oi = 1.400 T = 0.143 F = -1.000 -1.000 U 1.500 co = 1.400 T = 0.214 F 0.319 x ' = y ' = 0 , z' 0.479 x ' = y ' = 0 , z' = -1,000 0.639 x ' = y ' = 0 , z' = -1.000 U = 2 . 0 0 0 co = 1.400 T = 0.285 F

-8 -6 15.0-H tJ \ . U Oi B CD -10.0

1 , 1 , 1 . 1 , 1 ,

" A

. 1 . 1 . 1 , 1 , 1

A\ A

A : V V i ! | 3—1—1—1—1—1—1—1—1—1—V r

'V

T-—I

1 1 1 . 1 , , . 1

15.0 5.0 0. -10.0 -8 0 R 10.0^ H tv ta -5.0 -10.0 ft / / \ / \ A i \ / \

i

;

\\ \l

i l l i V • V V \ i U l i 'j : v..^ /

\ i

\\

\ i

V A -0 R -8 -10.0 1 . 1 , 1 , 1 . 1 . , 1 , 1 . 1 . 1 . 1 i ^

1

1

1

1

1

1

1 ' 1 '

f ^

. 1

1

1

1

1

1 1 1

1

0 R 10.0-tJ \ C i t i 5.0-4 -5.0-3 -10.0

I l l l l l l l l

4 8 I . I . I •10.0 •5.0 t-5.0

I ' l ' l ' l ' l

-4 0 4 8 R -10.0 -4 -8 '^^ N t J w.o 5.0 -10.0 1

A

J ^ A /

p

\i \i \\

Y

—1—1 1

1

1

1

1 I 1

1 li.' \ .'• \ /'

y /

• ' 1

1

1

1

1 ' 1 ' 1

15.0 0. -5.0 • -10.0 10.0 H t j cs 0. -5.0^ -10.0 t i 1

i

i i

_{i i}

f \

'i i / ': ! \

V

V V Ipfi

n/ / \ / -^ .

eio.o ^5.0 -5.0 -10.0 R R u = 1.000 CO = 0 . 0 4 9 T = 0 . 0 0 5 F = 0.319 = 0 , ?' = -1.000 u = 1.000 CO = 2.403 T = 0.245 F = 0.319 X' = Y ' = 0 , z' = -1.000 u = 1.000 CO = 2.550 T 0.260 F = 0.319 X' = Y ' = 0 , z' = -1.000 u = 1.000 CO = 5.003 r = 0.510 F = 0.319 X' = Y ' = 0 , z' = -1.000

U = 5 . 0 0 0 O) = 0.010 T = 0 . 0 0 5 F = 1.596 x ' = y ' = 0 , z' = -1.000 U = 5 . 0 0 0 o) = 0 . 4 8 0 T = 0.245 F = 1.596 x ' = y ' = 0 , z' = -1.000 U = 5 . 0 0 0 o> = 0.510 T = 0.260 F = 1,596 x ' = y ' = 0 , z' = -1.000 U = 5 . 0 0 0 CO = 1.000 T = 0.510 F = 1.596 x ' = y ' = 0 , z' = -1.000

A T h e o r y of Pressure Envelopes for Hydrofoils^

F a r i t G . A v h a d i e v a n d D i m i t r i j V . M a k l a l c o v , K a z a n State University^

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

I n h y d r o f o i l t h e o r y a pressure envelope means the f u n c t i o n

F{a) = -Cpminia) = 2

where Cpmin is the coefficient of the m i n i m u m pressure p ^ i n o n the h y d r o f o i l surface, a is t h e angle of a t t a c k ; p^,p, and Voo are the pressure at i n f i n i t y , density of the fluid, and t h e v e l o c i t y at i n f i n i t y , respectively. T h e f u n c t i o n F(a) is one of the m a i n characteristics of h y d r o f o i l s , which allows the c a v i t a t i o n - f r e e incidence range t o be predicted i n advance. T h e classical c o n d i t i o n of n o n c a v i t a t i n g fiow imphes t h a t the pressure p must be greater t h a n the vapour pressure py everywhere i n the fiuid, e.g. Knapp et al. (1970). I n terms of F{a) t h i s c o n d i t i o n can be w r i t t e n as _ «

Q is t h e vapour c a v i t a t i o n number.

I n a seaway, t h e c a v i t a t i o n number Q and changes i n angles of a t t a c k (the l a t t e r can be caused by a sea state or by control devices of incidence variations) depend on the c r a f t ' s speed. T h u s according t o c r a f t o p e r a t i n g requirements various types of pressure envelopes can be desired to operate i n a seaway w i t h o u t the danger of c a v i t a t i o n . T h e desired envelope can be obtained by the technique presented by Eppler (1990). T h i s technique employs a procedure of c o n f o r m a l m a p p i n g and consists i n specifying a velocity d i s t r i b u t i o n i n d i f f e r e n t segments of t h e p a r a m e t r i c circle f o r different angles of a t t a c k . Except on the first and last segments near the t r a i h n g edge, the velocities at the corresponding angles of attack are specified as constants, whose values are u n k n o w n and must be determined i n solving. T h e technique developed o r i g i n a l l y f o r airfoils works also well f o r h y d r o f o i l s , Eppler and Shen (1979), (1981), b u t pressure envelopes and t h e i r s u i t a b i h t y f o r c r a f t o p e r a t i n g requirements can be o b t a i n e d and checked o n l y after the design procedure. Due t o this inconvenience an i t e r a t i v e process complements the design m e t h o d .

Our approach consists i n finding a w i n g section shape t h a t generates exactly a specified pressure envelope. I n t h i s w o r k we present:

1. A m a t h e m a t i c a l description of the set of aU pressure envelope f u n c t i o n s . 2. A n a n a l y t i c a l m e t h o d of h y d r o f o d design w i t h a given pressure envelope.

3. The exact lower bounds of pressure envelopes f o r h y d r o f o i l s being s y m m e t r i c a l w i t h respect t o t h e chord.

4. A comparison of our m e t h o d and results w i t h those of Eppler and Shen (1979).

2. M a t h e m a t i c a l f o r m u l a t i o n

We consider a t w o - d i m e n s i o n a l p o t e n t i a l flow of an ideal incompressible fluid over a single proflle i n the z-plane. L e t z = z{t) be the c o n f o r m a l m a p p i n g o f t h e d o m a i n exterior t o the u n i t circle i n the p a r a m e t r i c f-plane onto the the flow region i n the 2:-plane. T h e correspondence of points is: z{oo) = 0 0 , 2 ( 1 ) = zt, where zt is the complex coordinate of the p r o f i l e t r a i h n g edge, F i g . l . T h e m a p p i n g z = z{t) matches i n one-to-one manner the points on the p a r a m e t r i c circle and the points on the profile. Let 7 be a polar angle i n the t-plane, a be an angle of

iThis work was supported by the Russian Foundation of Basic Research under grants N9A - 01 - 01763 and 7V93 - 01 - 17552.

a t t a c k relative t o the zero-lift hne. We denote by 1 ^ ( 7 , a ) the velocity d i s t r i b u t i o n along the p a r a m e t r i c circle at the angle of a t t a c k a. T h e velocity at i n f i n i t y is taken t o be u n i t y .

Z e r o - l i f t line

(b) (t)

/ A m ^

F i g . l : (a) Physical 2:-plane; ( b ) Parametric ^ p l a n e

For the f u n c t i o n ^ ( 7 , 0 ) the equation holds, Lighthill (1945), Eppler (1990):

V{l,a) _ V{l,ai)

| c o s ( 7 / 2 - a ) | | c o s ( 7 / 2 - a i ) | (3) I f F ( 7 , q ) is k n o w n at a certain angle of attack a, the p r o f i l e shape can be constructed by the p r o f i l e design theory, e.g. Lighthill (1945), Eppler (1990). T h e velocity d i s t r i b u t i o n cannot be specified a r b i t r a r i l y b u t must satisfy the conditions of solvabiUty:

l o g F ( 7 , 0 ) f i 7 = 0 TT

e ' ^ l o g F ( 7 , 0 ) ( i 7 = 0

(4)

(5) C o n d i t i o n (4) provides t h a t the velocity at i n f i n i t y is equal t o u n i t y , c o n d i t i o n (5) is necessary t o o b t a i n a m e a n i n g f u l closed profile. Let us i n t r o d u c e a f u n c t i o n

f { a ) = m a x V ( 7 , a ) ₍₆₎

T h i s f u n c t i o n w i l l be an envelope of the f a m i l y of the f u n c t i o n s 1 ^ ( 7 , a ) , i f 7 is taken as a parameter of the f a m i l y and a is taken as a variable. We have the same f o r the pressure envelope f u n c t i o n . T h u s / ( a ) can be called a velocity envelope. B e r n o u l l i ' s i n t e g r a l relates the velocity envelope f { a ) t o the pressure envelope F{a) by

f{a) = y i + F{a) ₍₇₎

T h e 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 p r o b l e m of h y d r o f o i l design w i t h a given pressure envelope consists i n the f o l l o w i n g .

P r o b l e m 1. L e t F{a) be an a r b i t r a r y given f u n c t i o n . F i n d aU the f u n c t i o n s F ( 7 , a ) t h a t satisfy the conditions (3) t o (7)

P r o b l e m 1 does not have a solution f o r each f u n c t i o n F{a), e.g. i t is impossible t o take

F{a) < 0 at a certain a because t h a t leads to f{a) < 1, which contradicts the m a x i m u m

modulus p r i n c i p l e f o r a n a l y t i c f u n c t i o n s . T h i s reasoning gives rise to P r o b l e m 2 .

P r o b l e m 2. F i n d conditions, t h a t must be imposed on the f u n c t i o n F{a) t o p r o v i d e t h a t P r o b l e m 1 is solvable.

3. F o r m a l s o l u t i o n to P r o b l e m 1

Let F(cx) be an a r b i t r a r y given f u n c t i o n . I f t i i i s f u n c t i o n is the pressure envelope f o r a c e r t a i n p r o f i l e , t h e n the velocity envelope can be f o u n d f r o m eq. ( 7 ) . T h u s , i t does n o t m a t t e r whether F{a) or f { a ) are given. For m a t h e m a t i c a l f o r m u l a t i o n s , i t is more convenient t o deal w i t h velocity envelope f u n c t i o n s . So we shall w r i t e our m a t h e m a t i c a l results i n terms of / ( a ) , keeping i n m i n d t h a t the r e f o r m u l a t i o n s i n terms of F{a) are always possible.

F r o m (3) follows t h a t

y ( 7 , a ) = | c o s ( 7 / 2 - a ) | f f ( 7 ) i ^ j ^ ( 7 ) is a f u n c t i o n independent of a. Let 7 ( a ) be a p o i n t where the f u n c t i o n V{-/,a) achieves

its m a x i m u m value at the fixed angle of a t t a c k a. We i n t r o d u c e t w o f u n c t i o n s

q{a) = 2 _{a + arctan} ƒ ' ( " ) (9)

ƒ ( « )

( T ( 7 , a ) = l o g / ( a ) - l o g l c o s ( 7 / 2 - a ) | - l o g 5 ( 7 ) ( 1 ^ ) Since / ( a ) = V[jia),a] > V{^,a) f o r a r b i t r a r y 7 and a, the f u n c t i o n ( 7 ( 7 , 0 ) of the t w o

variables 7 and a achieves its m i n i m u m at 7 = 7 ( a ) - We deduce f r o m here t h a t da/da - 0 at 7 = 7 ( a ) t o conclude after d i f f e r e n t i a t i o n

§ 4 = t a n [ 7 ( a ) / 2 - a ] (11)

ƒ ( " ) Eqs. (9),(11) y i e l d

7 ( a ) = g ( a )

So, the polar angles of the points on the p a r a m e t r i c circle, where the m a x i m u m s of velocity occur, can be f o u n d f r o m the simple f o r m u l a s ( 9 ) , ( 1 2 ) .

A t the p o i n t 7 = 7 ( a ) we have

y [ 7 ( a ) , a ] = / ( a ) (13) T h e r e f o r e , i f q - ^ l ) is a f u n c t i o n inverse t o ^ ( a ) , t h e n according t o (3) the velocity d i s t r i b u t i o n

at the angle o f a t t a c k /3 can be w r i t t e n as

F ( 7 , / 3 ) = M ( 7 , / 9 ) (14) w i t h M ( 7 , / 9 ) = / ( a ) c o s ( 7 / 2 - / 3 ) a = q-Hl) (1^) c o s ( 7 / 2 - a)

Let now f { a ) be a given f u n c t i o n of a i n a certain range a i < a < 0 5 . F r o m ( 9 ) , (12) we caii determine t h e l o c a t i o n of the m a x i m u m velocities on the p a r a m e t r i c circle. F r o m ( 1 4 ) , (15) the values of velocities at these points can be f o u n d f o r aU angles of a t t a c k . So, eqs. ( 9 ) , ( 1 2 ) , (14), (15) give, generally speaking, a s o l u t i o n t o P r o b l e m 1, b u t t h i s s o l u t i o n is o n l y f o r m a l :

1. There is no guarantee t h a t the whole p a r a m e t r i c circle w i l l be covered by the points o b t a i n e d b y means of ( 1 2 ) . T h e r e f o r e the f o r m a l a p p h c a t i o n of eqs. ( 9 ) , ( 1 2 ) , ( 1 4 ) , (15) can give rise t o some arcs on the p a r a m e t r i c circle where t h e velocity d i s t r i b u t i o n is u n d e t e r m i n e d .

2. There is no guarantee t h a t the points obtained by eq. (12) w i l l not lie u p o n one another, w h i c h w i l l result i n o v e r d e t e r m i n a t i o n of the p r o b l e m , since the f u n c t i o n V{-i,a) w i l l be m a n y - v a l u e d .