A Kutta condition for ship seakeeping computations with a Rankine Panel Method

A K u t t a Condition for Ship Seakeeping Computations

with a Ranlcine Panel Method

V o l k e r B e r t r a m , T U Hamburg-Harburg^ G e r h a r d D . T h i a r t , Univ. Stellenbosch^

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

The ship (especially including the rudder) can be considered as a vertical f o i l of very short span. For a steady yaw angle, i.e. a typical manoeuvring application, one w o u l d certainly enforce some k i n d of K u t t a condition i n a p o t e n t i a l flow computation. However, f o r the corre-sponding periodic m o t i o n i n seakeeping, even mentioning of the K u t t a c o n d i t i o n is rarely f o u n d for 3-d methods. Wu (1994) includes the K u t t a condition to compute the r a d i a t i o n problem for an oscillating surface-piercing plate at f o r w a r d speed using a Green f u n c t i o n method. Zou

(1995) solves the same problem using a Rankine panel method. T h e only apphcations t o ships

at f o r w a r d speed are an extended abstract by Zou (1994) for ^ Rankine panel m e t h o d and

Schellin and Rathje (1995) for a Green f u n c t i o n method. However, Zou solves only the

ra-d i a t i o n problem anra-d neglects the steara-dy ra-disturbance potential, anra-d Schellin anra-d R a t h j e have some inconsistency i n f u l f i l l i n g the free-surface condition. For the related p r o b l e m of fluttering vibrations of airfoils i n aerospace engineering, enforcing a I ^ u t t a c o n d i t i o n appears to be stan-dard procedure. I t is unclear i f o m i t t i n g the K u t t a condition is based on some physical insight about the negligible effects or oversight. I t is of no importance as long as the applications are l i m i t e d to head or following waves, i.e. the most frequent test cases. B u t of course, for prac-tical requirements a method must be applicable for a l l wave directions. We shall numerically investigate the effect of the K u t t a c o n d i t i o n and hope to contribute this way to a clarification of its necessity.

2. P h y s i c a l m o d e l

We consider a ship w i t h average speed /7 i n a regular wave of small a m p l i t u d e h. Bertram

(1996) , Bertram and Thiart (1998) presented a linearized 3-d Rankine panel m e t h o d i n c l u d i n g

a l l forward-speed effects. T h e same m e t h o d is now extended to include the K u t t a c o n d i t i o n . We w i l l therefore o m i t other details of the m e t h o d and focus here only on the K u t t a c o n d i t i o n and the associated dipole elements applied to compute the motions i n oblique waves.

The p r o b l e m is f o r m u l a t e d i n right-handed Cartesian coordinate systems. T h e i n e r t i a l

Oxyz system moves u n i f o r m l y w i t h velocity U. x points f o r w a r d , z downwards. T h e angle of

encounter / i between body and incident wave is defined such t h a t n = 180° denotes head sea and fi = 90° sea f r o m starboard.

The ship has 6 degrees of freedom f o r r i g i d body m o t i o n expressed i n the m o t i o n vector ü = { u i , U 2 , « 3 } ^ and the r o t a t i o n a l m o t i o n vector d = { u 4 , u g , u e } ^ . A p e r t u r b a t i o n f o r m u l a t i o n for the p o t e n t i a l is used o m i t t i n g higher-order terms:

is the p a r t of the p o t e n t i a l independent of the wave amplitude h. I t is the s o l u t i o n o f the steady wave-resistance problem, (j)^^^ is p r o p o r t i o n a l to h and accounts (Hnearized) for the contributions of the seaway.

. ^ ( i ) = i ? e ( ^ W ( a ; , y , z ) e ' - ^ * ) (2)

The symbol " denotes a complex a m p l i t u d e . We is the encounter frequency. T h e harmonic potential (j)^^^ is decomposed i n t o the p o t e n t i a l o f t h e incident wave the d i f f r a c t i o n p o t e n t i a l

^Lammersieth 90, D-22305 Hamburg, bertram@schiffbau.uni-hamburg.d400.de ^Dept. Mech. Eng., 7600 Stellenbosch, South Africa

(p'^, and 6 r a d i a t i o n potentials:

<^(') = < ^ ' + r + E f « i (3) t"=i

I t is convenient to divide cf)^ and into symmetrical and antisymmetrical parts to take ad-vantage of the (usual) geometrical symmetry:

9 {x,y,z) = -I- (4)

(j)<i = / . « + ^''.« (5)

T h e K u t t a condition requires t h a t at the t r a i l i n g edge the pressures are equal on b o t h sides. I f the pressure is decomposed into a symmetric and an antisymmetric part, t h e n this is equivalent to requiring zero antisymmetric pressure:

^'•^ = -piiue^)''" + V<^(°)V<^*'°) = 0 (6)

where 4>^'°' is an antisymmetric unsteady pressure, i — 1,3 and 5 for the three antisymmetric

radiation modes and i = d for the antisymmetric d i f f r a c t i o n part.

The f o u r antisymmetric potentials are solved i n a Rankine panel m e t h o d w h i c h uses first-order panels (plane and constant strength) on the h u l l (up to a height above the average wetted surface) and around the ship. I n a d d i t i o n , a dipole d i s t r i b u t i o n inside the h u l l and t r a i l i n g behind the h u l l is used for the antisymmetric cases. T h e theoretical details of the dipole dis-t r i b u dis-t i o n are discussed i n dis-the appendix. T h e K u dis-t dis-t a condidis-tion (6) is f u l f i l l e d adis-t dis-the lasdis-t column of collocation points at the ship stern. A corresponding number of semi-infinite dipole strips is used on the centerplane of the ship. T h e dipole strips start amidships and have the same height as the corresponding last panel on the stern. T h e simultaneous f u l f i f i m e n t of no-penetration condition and K u t t a condition on the collocation points poses no numerical p r o b l e m , as already demonstrated by Zou (1994). T h e four systems of equations f o r the a n t i s y m m e t r i c a l potentials share the same coefficient m a t r i x w i t h only the r.h.s. being different. A l l f o u r cases are solved simultaneously using Gauss elimination. T h e subsequent computations leading t o the motions are then straight-forward.

3. R e s u l t s for S-175 c o n t a i n e r s h i p

T h e S-175 containership using the same discretisation as i n previous computations, Bertram

and Thiart (1998), was computed f o r Proude number F „ = 0.275. The results f o r the symmetric

motions are of course not affected by the K u t t a condition.

F i g . 1 shows the result for p = 150°, F i g . 2 for /.i = 6 0 ° . T h e results f o r heave and p i t c h show excellent agreement w i t h experiments. T h i s is typical f o r a l l wave lengths a n d angles of encounter investigated. No experimental results are available for surge f o r these angles of encounter, b u t the tendency is similar as f o r head waves i n previous computations, Bertram

and Thiart (1998).

For IJ, = 150° the K u t t a c o n d i t i o n has only significant effects for long waves i n the region where resonance occurs. Here the K u t t a condition simulates to some extent the effect o f viscous d a m p i n g and reduces drastically (by factors between 2 and 4) the a n t i s y m m e t r i c motions. However, f o r r o l l additional viscous effects (those t h a t would be apparent also at zero speed) reduce i n reality the motions even more. For yaw and sway, no experimental data are here available, b u t we expect that autopilots i n experiments w i l l also prevent the very large predicted m o t i o n amplitudes of the computations.

For / i = 60° some short-comings of the present method become apparent. Yaw is predicted w i t h good accuracy over most of the range of frequencies, b u t for low frequencies (long waves) the computations overpredict motions strongly, even i f a K u t t a c o n d i t i o n is enforced. We suspect t h a t this is due t o an autopilot used i n experiments which restores motions for low frequencies, b u t this assumption would require f u r t h e r investigation. R o l l motions show only satisfactory agreement between computations and experiments. However, the measured r o l l motions appear less plausible f o r high frequencies t h a n the computations. Sway shows strong scatter of results. A p p a r e n t l y the K u t t a condition makes things worse here.

A conclusive improvement of results due to the K u t t a c o n d i t i o n could not be demonstrated. The improvements for the narrow region of resonance are drastic, b u t ad hoc solutions h m i t i n g the response amplitude operators to m a x i m u m values t a k i n g f r o m experiments for similar ships may work almost as well i n practice, while being a lot simpler. Sway motions for low encounter frequencies ( i n f o l l o w i n g oblique waves) require special treatment. T h i s w i l l be subject to f u r -ther research.

R e f e r e n c e s

BERTRAM, V, (1996), A 3-d Rankine panel method to compute added resistance of ships, IfS-Report 566, Univ, Hamburg

BERTRAM, V.; T H I A R T , G.D. (1998), Fully three-dimensional ship seakeeping computations with a

surge-corrected Rankine panel method, J. Marine Science Technology 3/2

SCHELLIN, T.E.; RATHJE, H. (1995), A panel method applied to hydrodynamic analysis of twin-hull

ships, FAST'95, Travemiinde

W U , G.X. (1994), Wave radiation by an oscillating surface-piercing plate at forward speed, Int. Shipb. Progr. 41, pp.179-190

ZOU, Z.J. (1995), A 3-d numerical solution for a surface-piercing plate oscillating at forward speed, 10th Workshop Water Waves and Floating Bodies, Oxford

ZOU, Z.J. (1994), A 3-d panel method for the radiation problem with forward speed, 9th Workshop Water Waves and Floating Bodies, K u j u

SÖDING, H. (1975), Springing of ships, ESS-Report 7, Univ. of Hannover

1.0-T , o o q, 0 >-3 4 uJLJ^ -90 H O o o o 1.(1 \Ü2\/h 0.5 •] è o o o + ,-1- + f 41 4 uJhfg 0^ -90- * é 41 | « 3 | M 1,0-0.5 H 2.0 H

1.0-112

!p-Fig. 1: Response amplitude operators for S-175, p, = 150°, Fn = 0.275; experiment, -|- R P M witiiout K u t t a condition, o R P M with Kutta condition

\\ü,\lh 1.0 4 O O O 1.0 4 0.5--1 n 1 2 3 4 w^/LJ^ 90¬ 0¬ -90-904 0¬ -90-3 4 wJhlg + * 1.0 0.5 \uz\lh O • S ° 90 0 -90 1.0 0.5 1 2 3 4 u,s/L/g |M5 kh "O 3 4 uj^/YTTa 90-1 0¬ -90-G.O 5.0 4.0 3.0 2.0 1.0 90 O -90 4 kh 1.0 4 0.5 1W6 /c/i 90 O -90 ~i n 3 4 w-^/L/g ? + O 1 2 3 4 Uy/L/^ e O * -^ è f

Fig. 2: Response amplitude operators for S-175, fj, = 60°, = 0.275; experiment, + R P M without K u t t a condition, o R P M with K u t t a condition

Appendix: Velocity Potential for a Rectangular Vortex Panel w i t h Harmonically

Oscillat-ing Strength

The oscillating ship creates a vorticity. The problem is similar to that of an oscillating airfoil. The circulation is assumed constant within the ship. Behind the ship, vorticity is shed downstreams with ship speed U. Then:

u f ) ^ { x , z , t ) = 0. ( A l )

7 is the vortex density, i.e. the strength distribution for a continuous vortex sheet. The following

distribution fulfills condition ( A l ) :

^ix,z,t) = Re {%{z) • e'(<^"/t')(^-=^".) . e'^'') for x < Xa (A2)

where 7a is the vorticity density at the trailing edge Xa (stern of the ship). We continue the vortex sheet inside the ship at the symmetry plane y = 0, assuming like Zou (1994) a constant vorticity density:

7(.T, z, t) = Re {% {z) • e''^'*) for a;,, < a; < Xf (A3) Xf is the leading edge (forward stem of the ship). This vorticity density is spatially constant within the

ship.

A vortex distribution is equivalent to a dipole distribution i f the vortex density 7 and the dipole density m are coupled by:

dm / i ,\

The potential of an equivalent semi-infinite strip of dipoles is then obtained by integration. This potential is given (except for a so far arbitrary 'strength' constant) by:

^x,y,z)=Re [ ƒ J miO^ dC e'^A = Re {tp{x,y, z) • e'^'') (A5)

with r = ^/{x -^y +y' + {z-Cy and

" ^ ( 0 - [ (1 - ei(-=/c^)(«-<-)) -f {xj - Xa) for - 00 < e < Xa ^ '

I t is convenient to write ip as:

<p{x, y, z) = [yh + f i h + hh + hyh] (A7)

where h =

fj

^d^dC (A9)

The velocity components and higher derivatives are then derived by differentiation of $, which reduced to differentiation of <p:

V $ = Re ( - 7 „ • • e'"=*) with

<^a; = yhx hhm + hhx + Uvhx

tpy = Ii+yliy + hhy + hhy + ƒ4 (^4 + yhy) = yhz + fihz + f s h z + liyhz

^xx = yhxx - 2 / 2 x + l2l2xx + hhxx + fiyhxx (Pyy = 2hy + yhyy + f i h y y + Szhyy + h{'i-hy + yhyy)

Vxy = hx + y h x y - hy + f2l2xy + f s h x y + U i l i x + y h x y )

<Pyz = h z + y h y z + f 2 l 2 y z + h h y z + M h z + y h y z )

H>xz = Vhxz - hz

+

+

+

The integral I\ and its derivatives are:

'{{z - Zg) + rao)((z - Zu) + r/u)" {{z - Zo) + rfo){{z - Zu)+rau).

where rao - ^/{x - i v ) ^ + y'^ + {z - ZoY, = y^(a; - a;/)^ + j/2 + - x „ ) 2 , etc,

a; — So a; — a; ƒ x — xj Jl = In can be (A15) (A16) (A17) (A18) (A19) (A20) (A21) (A22) (A23) (A24) ' 1 ; / [z- Zo + r.

Taoiz - Zo+Tao) Tfu{z - Zu + rJu) rfo{z - Zo + rfo) Tau{z - Zu + ran)

y V y y rfu{z - Zu+r/u) rfo{z - Zo + r/o) {Z-Zu + Tau)

1 1 1 — + Llxx — Llxy nyz -llxz

rlojz - Zo+ Tgo) - {x - Xa)'^{z - Zp + 2rao) rlo{z - Zo + Vao)^

'r)uiz - Zu+Tfu) - (a; - Xf)'^{z - Zu + 2rfu)

+

T)O{Z -ZQ-V Tjo) - (x - x f f { z - z„ + 2r/o) '••)okz- Zo-^TfoY

rlujz -Zu+ Tau) - j x - Xa)'^(z - Z^ + 2rau) rluiz - Zu + rau)^

I'lojz -Zo + rao) -y'^jz- Zo + 2rao) rlo{z- Zo+rao)'^

+

T)U{Z

-ZU+ Tfu) - y'^iz - + 2 r / u )

Zu + Tfu)'^

r%{z- Zo+Tfo) -y'^(z-Zo + 2rfo) Tjoiz- Zo +rfo)'^

''auiz ~ Zu + Tau) -y'^{z-Zu+ 2rau)

+

rlu(z - Zu+Tau)^

(x - Xa)y{z - Zo + 2rao) _ {x-Xf)y{z-Zu + 2rju) rL{z -Zo+ Tao)"^ rj^{z -Zu+ Tfu)^ {x~Xf)y{z- Zo + 2rfo) ^ {x - Xa)y{z -Zu + 2rau)

r}oiz - Zo+Tfo)'^ rluiz - Zu+Tau)'^ jy y ^ y ^ y

/ft3 7»3 ,p3 ij>3

' ao ' fu ' fo ' a u

X "^fl ^ f ^ ' ƒ ^ ^Ci

ao ' f o

The integral I2 and its derivatives are;

'{Z - Zo){x-Xf) I2 = arctan

-arctan {Z-Zu){x-Xf)

- arctan [z - Zo)ix - Xa)

yrfu + arctan yVao {Z - Zu){x - Xa) yi'a (A25) hx = hy =

y{z - Zo) y{z- Zo)

'•fo{{x-Xf)^-Vy-') rao{{x-Xa)^+y^) y{z - Zu) _|_ y{z - Zu)

rfu{{x -Xf)'^ + rau{(x - Xa)'^ + y2)

1 -{x-Xf){z-Zo) + {X - Xa){z - Zo) + {X-Xf){z-Zu) -{x - Xa){z - Zu) y j x - x f )

rfo{(x -Xf)'^ + y 2 ) Tfoiiz - ZoY + y^)

1 1

+ •

Taoiix-XaY^y-") r „ „ ( ( z - Z„)2 + y^)

1 1

+

r/ii((a; - a;/)2 + J/2) Tfu{{z - ZuY ^y"^)

1 1

+

VTau{{x - XaY + 2/^) Taui^Z - ZuY + y^) y{x - Xa)

Tfoiiz-Zo)^+y-') raoiiz-Zo)^ + y^) y j x - X f ) yjx - Xg)

z — z.

^•^•yy

rfuiiz-zu)^ + y^) Tauiiz - zuf+y'')

{x-Xf)yiz-Zo)iir%-iz-Zo)-') (a; - a;,)j;(^ - ^ „ ) ( 3 r L - (

jx - Xf)yiz - Zu)i^u - j z - Zu?) jx - Xa)yiz - Zu)i2.rlu - (z - Zu)^)

r)uii^~^fy+v'Y Tlui{x-Xa)^+yT ( x - x f ) y i z - Zo) Ho {x - Xa)yiz - Zo) rlui{x-xa)'+y^)^ 3r%-iz-Zo)' 3r%-ix-xf)'' 3 r L - ( ^ - ^ o ) ^ , 3rlo-ix-Xa)'-. ( ( a ; - 3rlo-ix-Xa)'-. T „ ) 2 + 2 / 2 ) 2 + ( ( ^ _ ^ ^ ) 2 + y 2 ) 2

(g; - Xf)yiz - Zu) ï ^ r ^ - j z - Z u ) ' 3r%-ix-Xf)^

r% [{ix-Xf)^+y^)-^ ^ ( ( ^ _ ^ J 2 + j / 2 ) 2

{x - Xa)yiz - Zu) \ 2>rlu - iz - Zuf 3r^„ - jx - x^ 2 1

hxy -'••iyz Z - ZQ 1'ao Z - Zu Tfu ^ Z - Z u Tau X — Xf T f o X - Xc au \" ^ui . au { X x - X a Y ^ W y iiz-ZuY^y-'f 1 y 2 ( 3 ^ 2 ^ _ ( ^ _ ^ ^ ) 2 ) -(a; - a;/)2 + 1/2 r2^(-(a; _ a.^)2 + ^^2^2 1 ^2(3^2^ _ ( ^ _ _ , ^ ) 2 j

Xx-XaY + y-" rloiix-Xaf-Vy-^Y.

1 V\^r)^ - j z - z u f )

ix-XfY^y"^ rj^iix-Xf)^ +y^)^

1 y'i3rlu-iz-zu)')

{x-XaY+y' rluiix-xaY+y'Y

1 yH3r%-ix-Xf)^)'

' ao

X y fo v-^ ^fJ J

iz-ZoY + y' " r%iiz-ZoY + y'Y_

1 ,/^3rlo-ix-Xa)^]

iz-zoY + y^ rU(z-ZoY + y^Y

X — X _Xf +-y^{3r%-ix-Xfr) (z-ZuY+y^ r%i(z~ZuY+yY

1 y^3rlu-{x-xary

The integral h and its derivatives are:

h — arctan —arctan {z-Zo){x - Xa) yrao [Z - Za){x - Xa) — arctan {Z - Zo) yva '{z - Zu)' + arctan '{z - Zu)' . y . (A26)

y{z - Zo) y{z

-

raoiix-Xa)^+y^) rauiix-xa)^+y^)

1

hy = -{X - Xa){z - Zo)

+ {x - Xa)iz - Zu) Z - Zo

Vaoiix - Xa)^ + y^) + ra„{{z - Zo? + 2/^) J

1 1

+

1

+

Tau{{x-Xay ^y-") Tau{{z - Zu? + y^)

Z- Zu

hz =

{z-ZoY^-y•' {z-zu)'' + y-'

y{x - Xg) y { x - X a )

+

2y{z - Zo) 2y{z - Zu)

( { z - Z o ) ' + y ^ ) ' ^ i{z-zu)^ + y^)^ Srlu-jz-zu)' 3rlu-{x-xa)' {{x-XaY + y')^^ { i z - Z u ) ^ + y T hxy — hyz Z - ZQ Tao Z — Z y^{irlo-{z-ZoY [ x - x a Y + y ' ' rlo{{x-xaY + y''Y. 1 y^2,rlu-{z-Zu)-') X Xn ' { x - X a Y + y ^ rluii^-XaY + y'Y 1 yH^rlo~ix-Xa)Y Tao X - Xa {z-ZoY+y' rlo{(z-ZoY+y^Y 1 y^3rlu-{x-XaY ' au hxz — — X z - z u Y + y ' rlu({z-zuY + y'Y j z - Z o Y - y ' j z - Z u Y - y ' { { z - z o Y + y'Y { { z - z u Y + y^Y y ' ao 'au

The integral h and its derivatives are:

Xa

T, = [ pi'^'i/" Z-ZQ Z-Zu J [ r o { { x - i Y + y^) ru{{x~iY + y^)

—OQ

(A27)

where To = ^/{x - + + {z - Zof and r „ = sj{x - ^ y''^ {z - z^Y.

/

[z - Zu){x - QjSrl - {z - ZuY) _ {z - Zo){x - OiSrl - jz - ZQY) rliix-^Y + y^Y

rli{x-0^ + y^Y

jz - zu)y{3rl ~ { z - ZuY) (z - Zo)y{3rl - jz - ZQY)

rl{{x-0'+y^Y rl{{x-0^+y^Y 1 1 0 ' U J df i-Axx —

- I

riiix - 0^ + ' / ) rliix - 0' + y'Y Tuiix - 0' + y'Y 8{x -

0'

-iz-zo)\ I ' r ' t : ^ . . ^ ' ^ ' - ' ^ - ^ ^

rUi^-O' + y') riiix-O' + y'Y roHx-O' + y^Y

2 > 82/2 [Z - Zu) — oo -{z - Zo) y I e'"^(/"ix-0 [iz-Zo)(^

rl-3y^ , 2{rl-2y')

Tliix-O' + y') ri{{x-0'+y'Y ru{{x-0' + y^Y 82/2

rl - 3 2 / ' ^ 2{rl

-2y^

rUi^-O'+y') rl{{x-0^ + y'Y roi{x-0'+y^Y 4 ( 3 r g - ( . - . „ n 3

^Ti{{x-CY + y'Y rU{x-0^ + y^) 3 ) / 4 ( 3 r 2 - ( . - . . n E n

/

The integral w i t h respect to f in the expressions for U and its derivatives are evaluated numerically following the modified Simpson's method of Söding (1975):

Xi+2h dx ^ikxi , 2 i M / 0 - 5 / i - 2 / 2 + 1.5/3 V , kh

' y '

+

where A = /(a;i), h = fi^i + h), h = f{xi + 2h), A ^ / ^ fi - 2h + /s.

The integration interval is truncated 0.1 x (a,^ - Xa) up- and downstream of the the field point x. The wake "panel" is subdivided into a suitable number of "sub-panels" depending on the minimum distance of the field point to the wake.

The Fortran source code for the elements is available upon request by email.