ARTICLE IN PRESS

0029-8018/$ - se

doi:10.1016/j.oc

�CorrespondiE-mail addre

d.vassalos@na-

Ocean Engineering 34 (2007) 1257–1264

www.elsevier.com/locate/oceaneng

Performance analysis of 3D hydrofoil under free surface

Nan Xie�, Dracos Vassalos

The Ship Stability Research Centre, Department of Naval Architecture and Marine Engineering, Universities of Glasgow and Strathclyde,

Henry Dyer Building, 100 Montrose Street, Glasgow G4 0LZ, UK

Received 25 January 2006; accepted 30 May 2006

Available online 20 September 2006

Abstract

The purpose of the present paper is to develop a potential-based panel method for determining the steady potential flow about three-

dimensional hydrofoil under free surface. The method uses constant-strength doublets and source density distribution over the foil body

surface and thereby Dirichlet-type boundary condition is used instead of Neumann-type condition. On the undisturbed free surface

source density is used to meet the free surface condition that is linearised in terms of double-body model approach and is discretised by a

one-side, upstream, four-point finite difference operator. After solving the doublets on the foil and sources on the free surface, the

numerical results of pressure, lift and resistance coefficients and also wave profiles can then be calculated for different Froude number

and depth of submergence to demonstrate the influence of free surface and aspect ratio effects on performance of the hydrofoil.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: 3D hydrofoil; Double model; Free surface; Potential flow

1. Introduction

Hydrofoils are widely used in ships and marine vehicles.The analysis of performance of hydrofoil is one ofimportant subjects. When submergence of hydrofoilbecomes small, effect of free surface should be considered,including evaluations of free surface profile, pressuredistribution, lift and resistance as well.

Most of early studies for this problem are on 2Dhydrofoil with free surface. Parkin et al. (1956) carried outmodel tests on symmetric Joukowski section with 12%thickness. Giesing and Smith (1967) adopt Kelvin source-type complex potential to satisfy Neumann-type bodyboundary condition and linearised free surface condition tosolve 2D hydrofoil potential flow. Yeung and Bouger(1977) used hybrid integral equation method, while Bai(1978) and Bai and Han (1994) adopt localised finiteelement method to solve 2D hydrofoil potential flow withlinear and non-linear free surface conditions. Plotkin andKennel (1984) applied a second-order theory to 2D flat

e front matter r 2006 Elsevier Ltd. All rights reserved.

eaneng.2006.05.008

ng author.

sses: nan.xie@na-me.ac.uk (N. Xie),

me.ac.uk (D. Vassalos).

plane problem, source/vortices are distributed on foilsurface and the free surface condition is kept to secondorder. More recently, Kouh et al. (2002) analysedperformance of 2D hydrofoil under free surface. Theydistribute source on undisturbed free surface and doubleton foil and wake surface. Dirichlet-type body boundarycondition is used instead of Neumann-type boundarycondition, the free surface condition is linearised by freestream potential.In the analysis of 3D lifting body with free surface, Lee

and Joo (1996) used a mixed source and doublet distribu-tion on body surface and source distribution on freesurface to calculate wave-making resistance of catamaran.Dirichlet-type body boundary condition is used on thebody surface. In their formulations, source strength onbody surface is set as the component of incoming flowvelocity on body surface normal direction, the inducedvelocity of source distribution on free surface is notincluded, the body boundary condition is thus not exactlysatisfied. They strongly recommended checking normalvelocity component on body surface.Larsson and Janson (1999) developed a 3D panel

method for yacht potential flow simulation. In theirmethod, source and doublet are distributed on the lifting

ARTICLE IN PRESS

Fig. 1. Coordinate system.

N. Xie, D. Vassalos / Ocean Engineering 34 (2007) 1257–12641258

part of the yacht. The use of Neumann boundary condition(normal velocity equals to zero) leads number of equationsless than the unknown strengths of source and doublet.In order to make the problem closure, the lifting bodysurface is divided into strips, essentially, parallel withthe undisturbed flow direction. At each strip, the doubletstrength is assumed constant span wise and varies linearlywith the arc length from the trailing edge of the pressureside around the nose, back to the trailing edge on thesuction side. Behind the trailing edge, several wakepanels are added along which the doublet strength isconstant. Kutta condition is satisfied by prescribing adirection of the flow immediately behind the trailingedge, where the velocity vector is assumed to be in thebisector plane. Numerically, this is accomplished byspecifying the normal to the surface and setting the velocityin this direction to be zero. This turns out to be exactlythe same condition as the hull surface condition, so Kuttaequations are of exactly the same form as the hullcondition. In this way, the problem is closure. Butproblem arises in calculating the induced velocity ofthe doublet on the lifting body surface and Kutta panels.In order to avoid this problem, some researchers distributethe doublet on the central plane of the lifting body, see,for example, Nakatake et al. (1990) and Zou andSoding (1994); while Chen and Liu (2005) distribute thedoublet on a sub-surface inside the body (de-singularitymethod).

In the present paper, a potential flow-based panelmethod is developed to analyse 3D hydrofoil under freesurface. The free surface boundary condition is linearisedwith double body flow potential. The total velocitypotential is split into double-body flow potential anddisturbance flow potential. On the body surface, Dirichlet-type boundary condition is applied by the mixed sourceand doublet distribution in which body boundarycondition is also satisfied. A doublet distribution is alsodeployed on the wake surface. There is source distributionon the free surface. The free surface condition is discretisedby the one-side, upstream, four-point finite differenceoperator. After solving the doublets on the foil andsource on the free surface, the numerical results ofpressure, lift, wave-making resistance and wave surfaceelevations can then be calculated at Froude number, depthof submergence and aspect ratio to demonstrate theinfluence of free surface effect on performance of thehydrofoil.

2. Mathematical formulations

Potential flow theory will be used in the present study.This means that the fluid is ideal and incompressible andthe flow is irrotational. A right-hand coordinate systemo� xyz is assumed, located on the foil advancing atforward speed U, oxy plane is on the undisturbed watersurface; oz-axis is positive upward and through the centreof trailing edge of the foil; see Fig. 1. The velocity potential,

Fðx; y; zÞ, satisfies the following conditions:

r2F ¼ 0, (1)

gqFqzþ

1

2rF � r rF � rFð Þ ¼ 0 on z ¼ B, (2)

qFqn¼ 0 on SB, (3)

rFj jo1 on TE of foil; (4)

Fðx; y; zÞ ! �xU far away upstream; (5)

where (4) is the Kutta condition. The free surface problemformulated above is nonlinear, due to the free surfaceboundary condition and the unknown position of thecorresponding boundary. The fully non-linear problem canbe solved iteratively, or solved with a linearised free surfacecondition. Dawson (1977) suggested the double bodyflow as the base flow to linearise the free surface condition.In the present study, Dawson’s method is used. It isassumed that the velocity potential consists of the principalflow potential, fðx; y; zÞ, and disturbance flow potential,jðx; y; zÞ:

Fðx; y; zÞ ¼ fðx; y; zÞ þ jðx; y; zÞ. (6)

The double-body flow potential satisfies the followingconditions:

r2f ¼ 0, (7)

qfqn¼ 0 on SB, (8)

rf�� ��o1 TE of foil; (9)

f!�xU far away upstream: (10)

It is also assumed that the disturbance potential is muchless than the principal flow potential. With the linearisedfree surface condition (Bertram, 1999; Xie and Vassalos,2005), the disturbance flow potential is the solution of the

ARTICLE IN PRESSN. Xie, D. Vassalos / Ocean Engineering 34 (2007) 1257–1264 1259

following boundary value problem:

r2j ¼ 0, (11)

f2l jl

� �lþ gjz � fzzfljl ¼ �f

2l fll �

12fzz U2 � f2

l

� �z ¼ 0,

(12)

qjqn¼ 0 on SB, (13)

rj�� ��o1 TE of foil; (14)

j ¼ 0 far away upstream; (15)

where SB is the foil surface, and ð�Þl stands derivative alongstreamline of the double-body flow on z ¼ 0 plane.

The double-body potential consists of free stream flowpotential and flow due to presence of the foil:

fðx; y; zÞ ¼ f1ðx; y; zÞ þ fBðx; y; zÞ, (16)

where f1 is free stream velocity potential:

f1 ¼ �xU , (17)

and velocity potential of the perturbation flow due topresence of the foil is obtained by using Green’s secondidentity (Newman, 1977):

fBðx; y; zÞ ¼1

4p

ZZSB

1

r

qfB

qn� fB

qqn

1

r

� �� �ds

�1

4p

ZZSw

fw

qqn

1

r

� �� �ds. ð18Þ

The integration on the wake surface, Sw, is added due toKutta condition should be satisfied. Substituting bodyboundary condition (8) into (18):

fBðx; y; zÞ ¼1

4p

ZZSB

nx

rU � fB

qqn

1

r

� �� �ds

�1

4p

ZZSw

fw

qqn

1

r

� �� �ds. ð19Þ

The disturbance flow potential consists of potentials due todisturbance from foil and free surface

jðx; y; zÞ ¼ jBðx; y; zÞ þ jFðx; y; zÞ, (20)

where the mixed doublet and source distribution on foil isused:

jBðx; y; zÞ ¼1

4p

ZZSB

1

r

qjB

qn� jB

qqn

1

r

� �� �ds

�1

4p

ZZSw

jw

qqn

1

r

� �� �ds ð21Þ

and source distribution on free surface:

jFðx; y; zÞ ¼

ZZSF

sFr

ds, (22)

where SF is the mean free surface ( i.e., z ¼ 0). By using thefoil body boundary condition (13), (21) becomes

jBðx; y; zÞ ¼1

4p

ZZSB

�1

r

qjF

qn� jB

qqn

1

r

� �� �ds

�1

4p

ZZSw

jw

qqn

1

r

� �� �ds. ð23Þ

3. Numerical method

When the field point, p, is on the foil surface, the doublebody flow potential (19) becomes

2pfBðpÞ þ

ZZSB�S�

fBðqÞqqn

1

rpq

� �dsq

þ

ZZSw

fwðqÞqqn

1

rpq

� �dsq ¼ U

ZZSB

nxðqÞ

rpq

dsq; p 2 SB,

ð24Þ

where Se is a small part of foil surface surrounding fieldpoint p. Eq. (24) is the integral equation for the unknownvelocity potential (doublet strength) on foil surface, and issolved by Hess and Smith method (Hess and Smith, 1964;Hess, 1972). The foil body surface is divided into a numberof panels in chord and spanwise directions, on which thestrengths of source/doublet are constant. At the trailingedge, Morino type of Kutta condition is applied (Morinoand Kuo, 1974):

fw ¼ fþ � f� on Sw, (25)

where f+ is strength of the doublet on suction side oftrailing edge of the foil; while f� is strength of the doubleton pressure side of trailing edge of the foil. The unknownson the wake surface can be represented by those (velocitypotential) of the panels near trailing edge on the same stripof the foil surface. The geometric symmetry of the foil istaken into account to reduce the unknowns. One equationis obtained at the null point of each panel:

XNB

j¼1

Ai;jfB;j ¼ E0;i; i ¼ 1; . . . ;NB, (26)

where the influence coefficients:

Ai;j ¼

ZZDSB;j

qqnj

1

rij

� �dsþ di;j

ZZSw;jy

qqnj

1

rij

� �ds, (27)

E0;i ¼ UXNB

k¼1

nk

ZZDSB;k

ds

rk;i, (28)

where di,j is the difference coefficients near the trailing edge,Sw;jy

stands jyth strip on the wake surface. Velocitypotentials on trailing edge potential are determined interms of values on the null point of the near panels on the

ARTICLE IN PRESSN. Xie, D. Vassalos / Ocean Engineering 34 (2007) 1257–12641260

same strip by the one-side three-point finite differencescheme:

f� ¼ C1fB;1 þ C2fB;2, (29)

fþ ¼ CKfB;K þ CK�1fB;K�1, (30)

where fB,1 is the velocity potential of the first panel of thestrip on foil (pressure side), fB,K is the velocity potential ofthe last panel of the strip on foil (pressure side) and Ci arecoefficients of the one-side difference scheme depending onthe distances among the three points.

Once the velocity potential (doublet strength, solution of(26)) on the foil surface is obtained, velocity distributionof the double-body flow in the fluid domain can becalculated as

~vðpÞ ¼ rfðpÞ ¼ rf1 þ rfB. (31)

The streamline on the undisturbed free surface can becalculated with (31), and a Runge–Kutta scheme is used forthis integrating process.

From (23), when the field points are on foil surface, theintegration equation for the disturbance flow is obtained as

2pjBðpÞ þ

ZZSB�S�

fBðqÞqqn

1

rpq

� �dsq

þ

ZZSw

fwðqÞqqn

1

rpq

� �dsq

þ

ZZSB

1

rpq

qjF

qndSq ¼ 0; p 2 SB. ð32Þ

The last term in (32) is the effect of free surface sourcedistribution on foil. By Morino’s Kutta condition:

jw ¼ jþ � j� on Sw. (33)

Discretising (32) and making use of (22), the followingequations are obtained:

XNB

j¼1

Ai;jjB;j þXNF

j¼1

Qi;jsF;j ¼ 0; i ¼ 1; . . . ;NB, (34)

where [Ai,j] are the same as in (28) except the integrating isover under water foil surface (without mirror body aboveundisturbed water surface) and

Qi;j ¼XNB

k¼1

~nk � r

ZZDSF;j

ds

rk;j

0B@

1CA ZZ

DSB;k

ds

ri;k

0B@

1CA; j ¼ 1; . . . ;NF.

(35)

In order to discretise free surface condition (12), the oneside, four-point upstream finite difference scheme is usedalong each streamline (Dawson, 1977):

f l

� �i� aif i þ bif i�1 þ gif i�2 þ dif i�3, (36)

where f is a physical quantity of the flow field, ai, bi, gi, di

are finite difference coefficients determined by thestreamline arc length coordinates (from up-stream to

downstream) li�3; li�2; li�1; li. The expressions of ai, bi, gi,di can be found, for example, in Bertram (1999).The free surface condition becomes

ai f2l;ijl;i

� �þ bi f2

l;i�1jl;i�1

� �þ gi f2

l;i�2jl;i�2

� �þ di f2

l;i�3jl;i�3

� �þ gjz;i � fzz;ifl;ijl;i ¼ Ei,

i ¼ 1; . . . ;NF, ð37Þ

where

Ei ¼ � aif2l;ifl;i � bif

2l;ifl;i�1 � gif

2l;ifl;i�2

� dif2l;ifl;i�3 � 0:5fzz;i U2 � f2

l;i

� �ð38Þ

and

fl;i ¼~li � rf ¼

qfqx

lx;i þqfqy

ly;i. (39)

Here ~li is the direction of streamline of double body flowon z ¼ 0. The velocity of the disturbance flow is expressedas the strength of doublet on the foil surface and sourcestrength on free surface, thus equations for the unknownsare obtained asXNB

j¼1

Pi;jjB;j þXNF

j¼1

Ri;jsF;j ¼ Ei; i ¼ 1; . . . ;NF, (40)

where

Pi;j ¼ aif2l;iV

Bli;j þ bif

2l;i�1VBl

i�1;j þ gif2l;i�2VBl

i�2;j

þ dif2l;i�3VBl

i�3;j þ gVBzi;j � fzz;ifl;iV

Bli;j ; j ¼ 1; . . . ;NB,

ð41Þ

Ri;j ¼ aif2l;iV

Fli;j þ bif

2l;i�1V

Fli�1;j þ gif

2l;i�2VFl

i�2;j

þ dif2l;i�3VFl

i�3;j þ gVFzi;j � fzz;ifl;iV

Fli;j ; j ¼ 1; . . . ;NF.

ð42Þ

The second-order derivative of the double-body flow (fzz)is also calculated by the four-point difference scheme. Inthe actual calculation, the source strengths of the first threepanels on each streamline are assumed to be zero.The influence coefficients in (41) and (42) are calculated

in the following way:

VBli;j ¼ �

1

4p~li � r

ZZDSB;j

qqnj

1

ri;j

� � ds

�di;j

4p~li � r

ZZSw;jy

qqnj

1

ri;j

� � ds; j ¼ 1; . . . ;NB,

ð43Þ

VFli;j ¼

~li � r

ZZDSF;j

1

ri;j

� �ds�

1

4p

XNB

k¼1

~nk � r

ZZDSF;j

1

rk;j

� �ds

0B@

1CA

� ~li � r

ZZDSB;k

1

ri;k

� �ds

8><>:

9>=>;; j ¼ 1; . . . ;NF, ð44Þ

ARTICLE IN PRESS

NACA4412, 5degh/c=1, Fc=1.0

-1.0

-0.5

0.0

0.5

1.0

-4 -3 -2 -1 0 1 2 3

x/c

2gz/

U2

present

2D, Kouh et al

Fig. 2. Comparison of wave profile at the central plane of NACA4412 foil

with 2D calculation at Fc ¼ 1.0, a ¼ 51, h/c ¼ 1 and AR ¼ 6.

N. Xie, D. Vassalos / Ocean Engineering 34 (2007) 1257–1264 1261

VBzi;j ¼ �

1

4p~k � r

ZZDSB;j

qqnj

1

ri;j

� � ds

�di;j

4p~k � r

ZZSw;jy

qqnj

1

ri;j

� � ds; j ¼ 1; . . . ;NB,

ð45Þ

VFzi;j ¼

~k � r

ZZDSF;j

1

ri;j

� �ds�

1

4p

XNB

k¼1

~nk � r

ZZDSF;j

1

rk;j

� �ds

0B@

1CA

� ~k � r

ZZDSB;k

1

ri;k

� �ds

8><>:

9>=>;; j ¼ 1; . . . ;NF. ð46Þ

Strengths of doublet and source for disturbance flow willbe the solution of the following equations:

A Q

P R

jB

sF

" #¼

0

E

. (47)

An iterative approach is adopted to solve (47) (Xie andVassalos, 2005).

Velocity distribution on foil surface:

~V ¼ Vn~nþ V e1~e1 þ V e2~e2 ¼ Ve1~e1 þ Ve2~e2

¼ VB;e1~e1 þ VB;e2~e2 þ rjF þ rf1� �

�~e1� �

~e1

þ rjF þrf1� �

�~e2� �

~e2, ð48Þ

where ~e1 and ~e2 are unit vectors on the foil surface, whichform a right-hand system with surface normal, ~n. The firsttwo terms on RHS of (48) are calculated numerically by adifference scheme:

VB;e1

� �i;j¼

qðfB þ jBÞ

qe1

� �i;j

¼ a fB þ jB

� �i�1;j

þ b fB þ jB

� �i;jþ g fB þ jB

� �iþ1;j

, ð49Þ

VB;e2

� �i;j¼

qðfB þ jBÞ

qe2

� �i;j

¼ aðfB þ jBÞi;j�1

þ bðfB þ jBÞi;j þ cðfB þ jBÞi;jþ1, ð50Þ

where a, b, g and a, b, c are difference coefficients. Theremaining terms in (48) are calculated analytically. Pressuredistribution on foil surface is

CpBðx; y; zÞ ¼ 1�V

U

� �2

¼ 1�V 2

e1 þ V2e2

U2ðx; y; zÞ 2 SB.

(51)

Wave-making resistance of the foil:

Rw ¼ �1

2rU2

ZZSB

cpBðx; y; zÞnx ds. (52)

Lift of the foil:

L ¼1

2rU2

ZZSB

cpBðx; y; zÞnz ds. (53)

The non-dimensional wave resistance and lift coefficientare defined as

Cw ¼Rw

0:5rU2cl, (54)

CL ¼L

0:5rU2cl, (55)

where c is chord length and l is span of the foil. Freesurface elevation is calculated as

Bðx; yÞ ¼1

2gU2 � rF � rF� �

ðx; yÞ 2 SF, (56)

where the total velocity potential is

Fðx; y; zÞ ¼ f1 þ fB þ jB þ jF. (57)

4. Numerical results and discussions

NACA 4412 foil section is selected for the calculation.The present results are compared with those of 2D of otherresearchers due to the availability of the data. Fig. 2 is thecomparison of wave profile at the central plane of the foilwith aspect ratio (AR) ¼ 6, immersion h/c ¼ 1, chordlength Froude number Fc ¼ 1.0 and angle of attack a ¼ 51,the agreement is satisfactory. Comparisons of lift andwave-making resistance coefficients are shown in Figs. 3and 4. Fig. 5 shows pressure distribution on the centralsection of the foil with AR ¼ 10. Good agreement isachieved with the 2D results. These comparisons show theapplicability of the present approach.Fig. 6 shows an example of the doublet (velocity

potential) distribution on the foil surface. In this case ofnumerical calculations, the foil surface is divided into 12strips in span wise, the first strip is near the central plane,while the 12th strip is located at the end of foil. The doubletis plotted against non-dimensional arc length along strips.It can be seen that, the curves basically consists of two

ARTICLE IN PRESS

NACA4412, h/c=1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.4 0.8 1.2 1.6 2.0

Fc

CL

present

2D - Yeung

Fig. 3. Comparison of lift force at the central plane of NACA4412 foil

with 2D calculation at Fc ¼ 1.0, a ¼ 51, h/c ¼ 1 and AR ¼ 6.

NACA4412, h/c=1.0

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.4 0.8 1.2 1.6 2.0

Fc

Cw

present

2D - Yeung

Fig. 4. Comparison of wave-making resistance at the central plane of

NACA4412 foil with 2D calculation at Fc ¼ 1.0, a ¼ 51, h/c ¼ 1 and

AR ¼ 6.

NACA4412, h/c=1, Fc=1

-1.5

-1.2

-0.9

-0.6

-0.3

0.0

0.3

0.6

0.9

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x/c

Cp

present

2D - Yeung et al

Fig. 5. Comparison of pressure distribution at central section with 2D

results for NACA4412 foil at a ¼ 51 and AR ¼ 10.

NACA4412, h/c=1, Fc=1, 5deg.

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0

arc length

doub

let s

tren

gth strip No.1

strip No.6

strip No.12

Fig. 6. Sample of doublet strength on the foil surface for NACA4412 at

AR ¼ 6.

Fig. 7. Sample of doublet strength distribution on foil surface for

NACA4412 at AR ¼ 4.

Fig. 8. Pressure distribution for NACA4412 foil at Fc ¼ 1, a ¼ 51, h/c ¼ 1

and AR ¼ 6.

N. Xie, D. Vassalos / Ocean Engineering 34 (2007) 1257–12641262

parts (on each of pressure and suction side), each of whichis approximately linear, however, the entire distribution oneach strip is not linear, i.e., not in proportion to the arclength. Figs. 7–9 show samples of doublet strength,pressure distribution on the foil and wave pattern,respectively. Figs. 10 and 11 show lift and wave-makingresistance varies with foil immersion depth for chordFroude numbers 0.7, 1.0 and 1.5, respectively, it can beseen that the effect of immersion on the hydrodynamicperformance is significant when the foil is located near free

surface. Fig. 12 shows lift for foils with three aspect ratiosof AR ¼ 4, 5, 6, respectively, the immersion depth ish/c ¼ 1. The lift forces decrease as the aspect ratiodeceases.

5. Concluding remarks

In the present paper, a potential-based panel method isdeveloped to predict performance of three-dimensional(3D) hydrofoil under free surface. Comparisons with other

ARTICLE IN PRESS

Fig. 9. Sample of wave pattern (NACA4412, Fc ¼ 1.0, a ¼ 51, h/c ¼ 1).

NACA4412, AR=6, attack=5deg

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1 3 542

h/c

CL

Fc=0.7Fc=1.0Fc=1.5

Fig. 10. Lift force varies with immersion depth.

NACA4412, AR=6, attack=5deg

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.001 3

h/c

Cw

Fc=0.7Fc=1.0Fc=1.5

2 4 5

Fig. 11. Wave-making resistance varies with immersion depth.

NACA4412, h/c=1, attack=5deg

0.8

0.6

0.4

0.2

0.0

1.0

1.2

0.3 0.6 0.9 1.2 1.5

Fc

CL

AR=4

AR=5AR=6

Fig. 12. Lift coefficient for foils with three aspect ratios.

N. Xie, D. Vassalos / Ocean Engineering 34 (2007) 1257–1264 1263

published results show the applicability of the presentapproach on the analysis of hydrodynamic performance of3D lifting body under free surface. The Rankine sourcedistribution method and Dawson’s double-body flowapproach enable the present method having the flexibilityto be extended to handle 3D lifting body penetrating freesurface (i.e., catamaran and trimaran) and combined hull–foil case (i.e., high-speed craft with ride control hydrofoil).These will be subjected to further publications.

References

Bai, K.J., 1978. A localized finite-element method for two-dimensional

steady potential flows with a free surface. Journal of Ship Research 22,

216–230.

Bai, K.J., Han, J.H., 1994. A localized finite element method for the non-

linear steady waves due to a two-dimensional hydrofoil. Journal of

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