Numerical calculation of free-surface potential flow around a ship using the modified Rankime source panel method

(1)

Available online at www.sciencedirect.com

ScienceDirect

E L S E V I E R _{Ocean Engineering 35 (2008) 536-544}

E N G I N E E R I N G

www.elsevier.com/Iocate/oceaneng

Numerical calculation of free-surface potential flow around a ship using

the modified Rankine source panel method

Md. Shahjada Tarafder'^'*, Kazuo Suzuki^

''Departweni of Namil Architecture cmd Marine Engineering, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh ^Faculty of Engineering, Yokohama National University, Yokohama, Japan

Received 28 May 2007; accepted 20 November 2007 Available online 22 November 2007

Abstract

A modified Rankine source panel method is presented for solving a Hnearized free-surface flow problem with respect to the double-body potential. The method of solution is based on the distribution of Rankine sources on the hull as well as its image and on the free surface. A n iterative algorithm is used for determining the free surface and wave resistance using upstream finite difference operator. A verification o f numerical modeling is made using the Wigley hull and the validity o f the computer program is examined by comparing the details of wave profiles and wave making resistance with Series 60 model.

Keywords: Modified rankine source method; Iteration process; Resistance; Wave pattern

1. Introduction

M a n y o f problems f a c i n g engineers involve such difficulties as a nonlinear governing equation and nonlinear b o u n d a r y conditions at k n o w n or u n k n o w n b o u n d -aries. The determination o f surface ship waves is a nonlinear p r o b l e m wherein the governing equation is linear but there is a nonlinear free-surface c o n d i t i o n imposed at an u n k n o w n boundary. I n practice the solution is approximated using numerical techniques, analytical tech-niques and combinations o f b o t h . Foremost among the analytical techniques is the systematic m e t h o d o f perturba-tions i n terms o f a small parameter. Problems o f this type are generally treated after the boundary c o n d i t i o n o n the free surface is linearized.

G a d d (1976) computed the wave-making potential flow based on Rankine sources distributed over a double model o f the h u l l and free surface. A non-linear f o r m o f free-surface condition is satisfied together w i t h the exact h u l l boundary condition. Dawson (1977) presented a numerical method i n calculating the linear wave resistance, w h i c h was a

*Corresponding author.

m a j o r development i n numerical analysis i n wave resistance problems. The general approach to solve the wave-resistance problem before Dawson was based on the concept o f systematic perturbation. The perturbahon potential that perturbs the k n o w n basic flow potential should be a small quantity compared to the basic flow potential. The basic flow used i n thin-ship theory is the u n i f o r m stream flow, whereas i n Dawson's method, the basic flow is the double-body flow, i n w h i c h the free surface is treated as a w a l l (indeed a zeroth-order Bernoulli wave) and the solution is sought below the water level around the h u l l .

X i a and Larsson (1981) developed a m e t h o d f o r investigating hulls and keels o f sailing yachts at an angle o f attack. The m e t h o d is considered to be an extension o f Hess panel m e t h o d f o r l i f t i n g , unbounded p o t e n t i a l flows to include the effect o f the free surface, w h i c h may play an i m p o r t a n t role i n the o p t i m i z a t i o n o f the design.

M u s k e r (1988) used R a n k i n e source m e t h o d t o predict the ship wave resistance. N o n - l i n e a r f o r m s o f the b o u n d a r y conditions were used and considerable care was taken to model the associated velocity gradients accurately. Calisal et a l . (1991) presented a numerical ship-wave-resistance f o r m u l a t i o n based on c o n t r o l v o l u m e analysis w i t h an iterative procedure w h i c h accommodates non-linear

(2)

M. Shahjada Tarafder, K. Suzuki / Ocean Emjineering 35 (2008) 536-544

freesurface conditions, parallel t o Gadd's (1976) f o r m u l a -t i o n . He ra-tionalized -the procedure already i n use and estabUshed the limits o f the approximations and inherent assumptions and presented the numerical results f o r the flow about a two-dimensional f o i l beneath the free surface. Scragg and T a l c o t t (1991) presented a m e t h o d o f solving the free-surface ship wave p r o b l e m satisfying Dawson's double-body linearization o f the free-surface boundary c o n d i t i o n , w h i c h employs distributed Havelock singula-rities o n b o t h the h u l l surface and on the free surface. The m e t h o d combined the superior aspects o f R a n k i n e -D a w s o n methods i n the calculation o f near field waves and the f a r field superiority o f the Havelock method.

Raven (1994) used the r a p i d (raised-panel iterative Dawson) approach f o r solving the f u l l y non-Hnear wave-resistance p r o b l e m . This m e t h o d used an iterative proce-dure based o n Rankine panel m e t h o d similar to that o f D a w s o n . The non-linear solution was f o u n d t o be realistic and very accurate numerically.

X i n m i n and X i u h e n g (1996) studied the canal bank effects on the h y d r o d y n a m i c forces and wave patterns o f ship hulls by the R a n k i n e source method. Parabolic curve panels were adopted to m o d e l the h u l l surface and a body fitted grid was chosen to divide the local free surface. The l i f t i n g effect was simulated by a vortex surface placed o n the l o n g i t u d i n a l central plane o f the ship and vortex filaments t r a i l i n g f o r m the whole vortex surface.

Bruzzone et al. (2000) carried out an experimental and numerical investigation based o n Rankine source m e t h o d on various configurations o f Wigley t r i m a r a n . Resistance, t r i m , sinkage and wave profiles along l o n g i t u d i n a l cuts have been measured f o r each c o n d i t i o n .

M i a o a n d X i a (2003) used a m o d i f i e d numerical approach based o n Dawson's m e t h o d t o study the lateral force, yaw moment and wave patterns o f a ship traveling along the central fine o f a rectangular channel. The free-surface b o u n d a r y c o n d i t i o n is linearized i n terms o f the double-body solutions and the R a n k i n e sources are distributed over b o t h the vessel and channel boundary surfaces to solve the ship wave p r o b l e m .

M i l l w a r d et a l . (2003) developed a numerical m e t h o d based o n p o t e n t i a l flow theory using R a n k i n e sources f o r predicting the wave resistance o f a high-speed ship w i t h a transom stern. I n order to m o d e l the flow a r o u n d the transom stern o f a fast h u l l , special b o u n d a r y conditions are applied to the transom and to the p o r d o n o f the free-surface downstream o f the stern.

T h e a i m o f the paper is to compute the wave m a k i n g resistance o f a ship m o v i n g w i t h a constant speed i n calm water by the m o d i f i e d R a n k i n e source method. A n iterative a l g o r i t h m is used f o r determining the free surface and wave resistance and Dawson's upstream finite difference opera-tor is used to satisfy the r a d i a d o n c o n d i t i o n . A v e r i f i c a d o n o f numerical modeling is made using Wigley h u l l and v a l i d i t y o f the computer p r o g r a m is examined by compar-ing the details o f wave profiles and wave m a k i n g resistance w i t h Series 60 model.

537

2. Mathematical modeling of the problem

L e t us consider a ship i n a stream o f u n i f o r m flow w i t h a velocity U as shown i n F i g . 1. The z-axis is vertically upwards and the y-axis extends to starboard. The origin o f the co-ordinate system is located i n an undisturbed free surface at amidship. The t o t a l velocity p o t e n t i a l 4> is the sum o f the double-body velocity p o t e n t i a l (P and the perturbed velocity potential q) representing the effect o f free surface.

(j) = ^ + (p. (1)

N o w the p r o b l e m f o r a ship can be constructed by specifying the Laplace equation

V ' - ( ^ + (p) = 0 (2) w i t h the f o f l o w i n g b o u n d a r y conditions:

(a) H u l l boundary conditions: The n o r m a l velocity c o m -ponent o n the h u l l surface must be zero.

V ( $ + cp) • 11 = 0, (3) where h = n j + n j + ihk represents a n o r m a l t o the

h u l l surface.

(b) Free-surface c o n d i t i o n : The velocity p o t e n t i a l needs to satisfy the dynamic and the kinematic conditions on the free surface

0C + ^ V < / . . V ( ^ = ^ C / 2 o n z = i:{x,y). (4) ' / ' . v G + = 0 on

z = i{x,y).

(5) E l i m i n a t i n g C f r o m Eqs. (4) and (5) i < ^ . , ( V ^ • V ( ^ ) , + ^ (l>y{V(l> • V<i))y + CJ4,,_ = 0 on Z = C(A-,J')-F i n a l l y i t is necessary t o impose a r a d i a t i o n c o n d i t i o n to ensure that the free-surface waves vanish upstream o f the disturbance.

X

n

(3)

538 _{M. Shahjada Tarafder, K. Suzuki / Ocean Engineering 35 (2008) 536-544}

3. Linearization of free-surface condition

The free-surface c o n d i t i o n E q . (6) is nonlinear i n and should be satisfied o n the free surface, w h i c h is u n k n o w n and can be linearized about the double-body solution by neglecting the non-linear terms o f (p. The p e r t u r b a t i o n p o t e n t i a l (p is assumed to be small relative to the double-body potential <P. The double-body potential corresponds to the l i m i t i n g s o l u t i o n as the Fronde number goes to zero f o r w h i c h case the free surface acts as a reflection plane. A reflection boundary c o n d i t i o n is to be applied on the p o s i t i o n o f the undisturbed free surface.

(7) (8) (9) = 0 o n z = 0. F r o m E q . (6) \ <I>M + <t>} + 4>% + \ 4>yi4>l + + + g4>z = o. Substituting E q . (1) i n E q . (8) we o b t a i n + \ { ^ + ( P ) y { { ' ^ + <P)1 + (<P + (pfy + ( < ^ + (P)]} + + cp\ = 0.

A p p l y i n g E q . (7) and neglecting the non-linear terms o f (p the free-surface c o n d i t i o n E q . (9) can be linearized about the double-body s o l u t i o n ^ as

+ \ < P M + ^% + \ + <f])y 0

on z = 0. (10)

N o w f o r any f u n c t i o n F{x,y) = 0

^.v^.v + ^yFy = <f,F,,

where the subscript / denotes the d i f f e r e n t i a t i o n along a streamline o f double-body potential cP o n the symmetry panel z = 0. Thus the E q . (10) can be w r i t t e n as

\ ^i(^j)i + H^i(Pi)i + ^ (Pi{<Pj)i + cj(p, = 0 o n z = 0.

S i m p l i f y i n g the above equation we o b t a i n ,

$j(Pl, + 2$i$ii(p, + gcp, = on z = 0.

(11)

(12)

4. Method of solution for free-surface problem

The velocity potentials <P and (p are expressed by R a n k i n e sources d i s t r i b u t e d o n the surface o f double body

5B and the undisturbed free surface Sp, respectively,

<P(x,y,z)=Ux- JJdBj dS, (p(x,y,z) = - I J f f F ^ d S -

JJAOBJ,

dS, (13) (14) where '•' = ] / { x - i f + ( } ' - i l f + z ' - .

The p e r t u r b a t i o n velocity potential cp o f E q . (14) is composed o f the velocity potential due to free surface and the velocity potential due to the effect o f free surface on the h u l l , respectively. I n the present numerical scheme, Dawson's f o r m u l a t i o n f o r the doublebody flow is m o d -ified by i n c o r p o r a t i n g the second t e r m o f the r i g h t - h a n d side o f E q . (14).

The flow past a double b o d y is obtained by a numerical solution o f the b o u n d a r y value p r o b l e m subject t o the N e u m a n n boundary c o n d i t i o n o n the double-body h u l l . N o w f r o m Eqs. (3) and (13) we obtain

dS •« = 0.

I n order to get a n approximate s o l u t i o n o f the above equation the double-body h u l l surface SB is discretized i n t o a number o f NB panels and the strengths o f the sources CTB are assumed constant at the center o f the panels. So the above equation at the /th panel becomes

rr

/1 \

-2naB{i) + J 2 j J ^B(/")Ö;^ f ; j dS = n(i) • U.

This equation is a two-dimensional F r e d h o l m integral equation o f the second k i n d over the surface ^ B - A f t e r simplification the above equation can be w r i t t e n as

NB - InaBiO + ^ (TB(/') 7=1 SB «(/) • u, ' d z j j r SB

^B

- 271(78(0 + Yl '^s(j)[ih-iV.xij + nyiVyij + n,iV,ij] 7=1

= n(i)-U, i=]-NB. (15)

The double-body s o l u t i o n is obtained after calculating the velocity components F^, Vy and f r o m Eqs. (27), (28)

(4)

M. Shalijadd Tarafder, K. Suzuki / Ocean Engineering 35 (2008) 536-544 539

and (29) o f Hess-Smith (1964). T h i s double-body solution is an a p p r o x i m a t e solution t o the free-surface p r o b l e m i n the l i m i t i n g case o f zero F r o u d e number w h e n the r i g i d w a l l linearizes the free surface. A f t e r getting the double-b o d y velocity potential (P f r o m E q . (13), the doudouble-ble-double-body streamlines are traced on the mean water level. These streamlines w i l l not penetrate the h u l l surface and are used to set up a free-surface g r i d .

The free-surface b o i m d a r y c o n d i t i o n given i n E q . (12) involves the gradient o f the velocity p o t e n t i a l along a streamwise direction designated by 1 and d i f f e r e n t i a t i o n is carried o u t a l o n g the corresponding double-body stream-lines. T h e streamwise velocity o n the free surface is computed by

= 'P.v +

(p

N o t e that this d i f f e r e n t i a t i o n scheme approximates the free-surface flow direction by the double-body flow direction. I f the free surface is discretized i n t o a number o f A'^F panels, the derivatives cp/ and cpn i n E q . (12) at the /th panel o n the free surface can be expressed as

(plii) = ö-F(/')iF(y) + Y ^CB(J)LBW), . / = 1 ( / ) = J2 ffF(/)CLF(//) + J2 AC^B(/)CLB(//), (16) (17)

- I I

Lp(ij) • <P.v ^ ^ 2 8A-V'-SB

I I Jq

~ I I \ j ^1 +

^2

dyV

SF JV-1 dS dS, (18) a n d A ' - l CLF(J7) = Y " "'j^' (19)

i n w h i c h e„ is an A^-point upstream finite difference operator given i n A p p e n d i x . The vertical velocity c o m p o -nent on the free surface is

(Pz =

-Inapif), i=J,

0, i^j.

(20)

Substituting Eqs. (16), (17) and (20) into E q . (12), the system o f linear equation f o r ap and AOB can be obtained as

•Np WB

' ^ ' ( 0 Y ^ F O ' ) C ^ F ( V ) + Y ^^^BOICLBUJ)

•Np iVB

+ 2<f/(/)3>//(/) Y '^F(/')iF(//') + Y ^('B(J)LBW)

- 2ngcTp(i) = -<P]ii)<P„{i).

Rearranging the above equation we get

A'F

^B

Y '^F(/')^F(y) -t- Y '^'^B(/VB(y)

- Ingapii) = 5 ( / ) , / = 1 - A ^ F ,

where

ABiiJ) = 'Pj(i)CLBW) + 2$imii(i)LB{ij),

AFW) = ^jiOCLpW) + 2<P/(/)(P,/(/)LF(//), B{i) = -^jmiiii). Substituting E q . (14) i n t o E q . (3) we get A'F Y ^F<J)VFW) + Y A f ^ B ( / ) F B ( / / ) = 0,

i = NF+\-NB+NF,

where VBW) • JJ 9n \r dS, 9

f l

SB VFW) •• dS, 7. +'h'^,{7. _•dz\i8

n

(21) (22) (23) (24) dS,

Ih" ('•'.

(25)

A n iterative m e t h o d is used t o o b t a i n the s o l u t i o n o f Eqs. (21) a n d (24). R e w r i t i n g E q . (21) as

A'F ^ B

Y ^F(J)AFW) - 27:gcTFi{) = B(i) - Y A T B ( / - ) ^ B ( / / ) . (26)

j=i

y=i

F o r an i n i t i a l value, the source d i s t r i b u t i o n over the h u l l surface can be a p p r o x i m a t e d by the double-body solution as

(5)

540 M. Shahjada Tarajder, K. Suzuki / Ocean Engineering 35 (2008) 536-544

F o r seeking tlie first a p p r o x i m a t i o n o f we substitute Eq. (27) i n E q . (26)

Np

E 4"0')^F(//-) - Inga^PU) = B{i). (28)

The solution ff^;'' f r o m E q . (28) is substituted i n E q . (24) and we o b t a i n

(29)

Np A^B

Ê

4"(/')^F(V)

+ £ A 4 ' ) ( 0 F B ( / / - ) = 0.

A f t e r getting the first approximated solution A u g ' f r o m Eq. (29) the second a p p r o x i m a t i o n o f ap is obtained f r o m Eq. (26) as

Ê of m m - 2ngaf(i) = B(i) - ^ Aa^^^{j)AB(Jj).

J=l 7=1

(30) Substituting the s o l u t i o n af^ f r o m E q . (30) i n E q . (24) we get the second a p p r o x i m a t i o n o f A(TB as

A'p A'B

E ' ^ F ' W ^ F ( y ) + E ^B('7) = 0. (31) 7=1 7 = 1

This procedure repeats u n t i l Eqs. (21) and (24) are satisfied to a given error tolerance ( £ = 1 x 10"'').

5. Calculation of wave making resistance

T h e pressure on the h u l l surface can be calculated f r o m the p e r t u r b a t i o n velocity p o t e n t i a l by using a hnearized version o f the B e r n o u l l i equation that is consistent w i t h the linearized D a w s o n freesurface b o u n d -ary c o n d i t i o n

P + pgz + \py<i>y(l> =Poo+\pU^,

P-Poo=\pU' - pgz - ^ / J V ( ^ + cp) • V ( ( P + (p),

P-Po -p\u^-2gz-<^l-^]-<!>l

- 2$,.(?>, - 2^y(py - 2$,(p, .

N o w the pressure co-efficient

P-P

( l / 2 ) p f / 2 U 5= p[ C/ 2- 2 i y z - 3> 2 _ ^ 2 _ ^ 2

-2<I>,-(p, - 2<Py(py - 20,cp, .

Assuming that the pressure is constant w i t h i n a h u l l surface panel, the wave resistance can be determined by

A'B/2

C w =

-\/2U^L- 2 J2 Cp(/)«.v,-A5,-, (32)

where A^,- is the area o f the h u l l surface panel and w.^,- is the A--component o f the u n i t n o r m a l o n a surface panel. The wave profile can be obtained f r o m E q . (4) as

C(A-,y) = ^ j i u ' - ' ^ l - - 2 < f - 2<Py(py). (33)

6. Results and discussion

The numerical a l g o r i t h m o u t l i n e d i n the preceding sections has been applied t o the p r e d i c t i o n o f wave-making resistance and wave p r o f i l e f o r t w o h u l l models at a wide range o f Froude numbers. The wave drag is computed by

J 2

Cal. (Fixed) Exp. (Fixed)

Fig. 3. Wave-making resistance o f the Wigley hull.

(6)

M. Shahjada Tarafder, K. Suzuki / Ocean Engineering 35 (2008) 536-544 541 0.02 0.00 Cal. (Fixed) IHHi (Fixed) G © -1.0 -0.5 0.02 0.00 -1.0 0.02 0.00 -0.02 -1.0 0.02 0.00 -0.02 0.0 2X/L 0.5 1.0 Cai. (Fixed) IHHI (Fixed) -0.5 0.0 2x/L 0.5 1.0 Cal. (Fixed) IHHI (Fixed) -0.5 0.0 2X/L 0.5 1.0 Cal. (Fixed)

-• •

• IHHI (Fixed)

•

o /

• • •

- l 1 1

•

1 -1.0 -0.5 0.0 2x/L 0.5 1.0

Fig. 4. Wave profile of the Wigley hull at various speed, (a) Wave profile at F„ = 0.25, (b) wave profile at F„ = 0.267, (c) wave profile at F„ = 0.289 and (d) wave profile at F„ = 0.316.

integrating the pressure over the wetted surface. The first case considered is the w e l l - k n o w n Wigley h u l l model defined by the analytical f o r m u l a

4x

1 - ^

where L, B and T are the length, breadth and d r a f t o f the ship, respectively, at still water. The characteristic dimen-sions o f the Wigley h u l l are BjL = OA,T/L = 0.0625 and C B = 0.444. F i g . 2 shows the panel arrangement o f the Wigley h u l l . Since the h u l l is symmetric, one h a l f o f the c o m p u t a t i o n a l d o m a i n is used i n the numerical treatment. The h a l f o f the h u l l surface is discretized by 36 x 8 panels and a free-surface d o m a i n ( - 2 < . Y / L ^ 7 and 0^y/L^2) is discretized by 9 0 x 1 6 panels, respectively. I n order to satisfy the r a d i a t i o n c o n d i t i o n the second derivative o f the velocity potential along the double-body streamline is calculated by backward (upstream) finite difference opera-t i o n . A opera-t opera-the foremosopera-t a n d rearmosopera-t c o n opera-t r o l poinopera-ts o n opera-the free surface t w o p o i n t operator is used and three and f o u r -p o i n t o-perators are used o n the remaining c o n t r o l -points o f the free surface.

I n F i g . 3 the computed wave m a k i n g co-efficient o f the Wigley h u l l w i t h fixed sinkage and t r i m is compared w i t h its experimental results carried out by Shearer and Cross (1965)

10

s Ü

0.1

Fig. 6. Wave making resistance o f the Series 60 hull.

(7)

542 M. Shahjada Tarafder, K. Suzuki / Ocean Engineering 35 (2008) 536-544

and the agreement is quite satisfactory. F i g . 4 shows a comparison o f computed and measured wave profile at various speed o f the h u l l i n fixed sinkage and t r i m c o n d i t i o n . The phases are almost same but the m a i n difference is f o u n d at the first crest o f the b o w and at t r o u g h and after that the difference is insignificant. These differences are Ukely to have been caused by the f o l l o w i n g reasons: the wave profiles are taken f r o m the free-surface elevations at the panels next t o the body, n o t at the actual h u l l surface, which resulted i n some error especially near

the bow. A n o t h e r i m p o r t a n t fact is that the wave profile near the b o w region is strongly influenced by the nonlinear terms and the linearized free-surface c o n d i t i o n may not simulate the exact b o u n d a r y c o n d h i o n properly.

The second case considered here is the Series 60, C B = 0.6 h u l l m o d e l . The h a l f o f the hull surface as weU as the associated free surface is discretized by 40 x 10 and 90 X 16 panels, respectively, as shown i n F i g . 5. I n F i g . 6, the calculated wave-making resistance o f the Series 60 h u l l w i t h fixed sinkage and t r i m is compared w i t h the

Fig. 7. Wave profile o f the Series 60 hull at various speeds, (a) Wave profile at F„ = 0.18, (b) wave profile at F„ = 0.22, (c) wave profile at F„ (d) wave profile at F„ = 0.28 and (e) wave profile at F„ = 0.30, (f) wave profile at F„ = 0.32, (g) wave profile at F„ = 0.34.

(8)

M. Shahjada Tarafder, K. Suzuki / Ocean Emjineering 35 (2008) 536-544 543

experimental measurements conducted at the University o f T o k y o ( U T ) and I s h i k a w a j i m a - H a r i m a Heavy Industries Co., L t d . ( I H H I ) .

The computed wave profile o f Series 60 h u l l i n fixed sinkage and t r i m c o n d i t i o n are compared w i t h the experimental measurements conducted at the Ship Re-search Institute ( S R I ) (see Takeshi et al., 1987) i n F i g . 7

10

Fig. 8. Effect of the size of the free-surface panels on the wave-making coefficient o f the Series 60 hull.

and some discrepancies are f o u n d at the first crest o f the bow. The d i f f i c u l t y i n this c o m p u t a t i o n lies i n the discretization o f the b o d y surface. This h u l l consists o f a flat b o t t o m ; parallel middle body and stern section w i t h large curvature.

Since the numerical results are dependent o n the panel d i s t r i b u t i o n , p a r t i c u l a r l y on the free surface, calculations are also p e r f o r m e d w i t h 70 x 16, 75 x 22 and 80 x 27 panels on one-half o f the free surface keeping all other conditions (such as d o m a i n size, number o f the panel o n the h u l l surface and upstream finite difference operator) the same. The numerical results that are plotted i n F i g . 8 are f o u n d to be converged.

F i g . 9 shows a comparison o f wave pattern a r o u n d the Series 60 h u l l at various speeds. A s can be seen i n these figures, the diverging waves are radiating f r o m the b o w together w i t h transverse wave f o l l o w i n g behind the stern o f the ship and they l o o k very similar to the wave pattern i n deep water.

7. Conclusions

The paper presents the m o d i f i e d Rankine source panel m e t h o d f o r solving the steady free-surface ship wave p r o b l e m using double-body linearization o f the free-surface

(9)

5-'l4 M. Shahjada Tarafder. K. Suzuki / Ocean Eiujineerimj 35 (2008) 536-544

b o u n d a r y c o n d i t i o n . The f o l l o w i n g conclusions can be d r a w n f r o m the present numerical analysis:

(1) The present m e t h o d c o u l d be an efficient t o o l f o r evaluating the flow field, wave pattern and wave resistance f o r practical ship f o r m s .

(2) The agreement between calculated and measured wave m a k i n g resistance is quite satisfactory b u t the m a i n difference between computed and measured wave p r o f i l e is f o u n d at the first crest o f the b o w and at the t r o u g h .

(3) The calculated results depend to a certain extent o n the discretization o f the h u l l and the free surface. Similar panel arrangement should therefore be used i f relative merits o f d i f f e r e n t c o m p e t i n g ship designs are t o be j u d g e d .

Appendix. A . scheme to ensure wave propagation

I n order to satisfy the r a d i a t i o n c o n d i t i o n the second derivative o f the velocity p o t e n t i a l a l o n g the double-body streamline is calculated by b a c k w a r d (upstream) finite difference operator. The derivative o f a f u n c t i o n J{x,y,z) o n the free surface along the streamhne d i r e c t i o n / w i t h respect t o the g l o b a l co-ordinate system {x,y,z) can be expressed as d / - - f , { i j ) ^ ^ y>j

=fxiij),

$2..-f-(p2..

--fyiU), dx dy

= fy(U) =

d^ dx di] dx' dfii,J)d^ ^ d / ( / , ; ) d > , d ^ dy dt] dy'

The derivatives o f the f u n c d o n s are calculated b y using the one-sided xipstream finite difference operator

d x d ^ ( x i i + l ) - x { i ) x ( 0 - . x ( / - l ) ( 3 A ( / ) - 4 A ( / - l ) - | - A - ( ; - 2 )

2 (1

1A(/)

-

18A((

- 1)

- I - 9A(/

- 2) -

2A-(/

- 3))

a =

2), (/ = 3),

0'

= 4), d ^ d£ j < / ) - X / - i ) ( 3 X 0 - 4 X / - l ) + J ' ( / - 2 )

2

(113'(0 - 18J'(/ - I) + 9yii - 2) - 2y(i - 3))

0 = 1 ) ,

a =

2),

('•=3),

('• = 4), .f i+\J f i j f i j ~ fi-\J ( l l / , . , . - 1 8 / , _ i , . + 9 / , _ ; , . - 2 / , _ 3 , . ) )

( ' = 1),

('• = 2), (/ = 3),

(/ = 4).

S i m i l a r l y , the derivatives ( d x / d j ; ) , (dj'/d?;) and (d//d//) can be obtained. The relationships among the derivatives o f the co-ordinate systems {x,y,z) and (^,);,Q o n the free-surface are given by 9A-

~ IJ| 8>;'

9A ~ | / | dl'

9 ^ _ _ l _ 9 x ^ _ J _ Ë : ^ ;

93'"171 9//' 9 ^ ~ M Ï 9 ^ '

| / | =

9A

9J

9A"

9^

9 j 9 i y ~ 9 ^ 8 ^ '

References

Bruzzone, D . , Cassella, P., Zotti, I . , 2000. Resistance components o f a Wigley trimaran: experimental and numerical investigation. I n : Proceedings of the Fourth International Conference on Hydrody-namics, pp. 115-120.

Calisal, S.M., Goren, O., Okan, B., 1991. On an iterative solution f o r nonlinear wave calculations. Journal of Ship Research 35 ( I ) , 1-14. Dawson, C.W., 1977. A practical computer method for solving ship-wave problems. I n : Proceedings of Second International Conference on Numerical Ship Hydrodynamics, pp. 30-38.

Gadd, G.E., 1976. A method of computing the flow and Surface wave pattern around full forms. Transactions o f the Royal Institute o f Naval Architects 118, 207-218.

Hess, A.J., Smith, A . M . O . , 1964. Calculation o f non-lifting potendal flow about arbitrary three-dimensional bodies. Journal of Ship Research, 22-44.

Miao, Q., X i a , J., 2003. Numerical study o f bank effects on a ship traveling in a channel. I n : Eighth International Conference on Numerical Ship Hydrodynamics.

Millward, A . , Nicolaou, D . , Rigby, S.G., 2003. Numerical modeling o f the water flow around a fast ship with a transom stern. International Journal o f Maritime Engineering, Royal Institution o f Naval Architects, 181-194.

Musker, A.J., 1988. A panel method for predicting ship wave resistance. In: Seventeenth Symposium on Naval Hydrodynamics, pp. 143-149. Raven, H.C., 1994. Nonlinear ship wave calculations using the rapid

method. I n : Sixth International Conference on Numerical Ship Hydrodynamics, Iowa City.

Scragg, C , Talcott, J., 1991. Numerical solution o f the Dawson free surface problem using Havelock singularities. I n : Eighteenth Sympo-sium on Naval Hydrodynamics, pp. 259-271.

Shearer, J.R., Cross, J.J., 1965. The experimental determination o f the components of ship resistance for a mathematical model. Transactions of I N A 107.

Takeshi, H . , Hino, T., Hinatsu, M . , Tsukada, Y . , Fujisawa, J., 1987. I T T C Cooperative Experiments on a Series 60 Model at Ship Research Institute-Flow Measurements and Resistance Test, pp. 10-13. Xia, F., Larsson, L . , 1981. A calculating method for the lifting potential

flow around Yawed, surface piercing 3-D bodies. I n : Sixteenth Symposium on Naval Hydrodynamics, pp. 583-595.

Xinmin, X . , Xiuheng, W., 1996. A study on maneuvering hydrodynamic forces acting on 3d ship hulls with free surface effect in restricted water. International Shipbuilding Progress 43 (433), 48-69.

(10)