Effect of free surface and strut on fins attached to a strut

Ocean Engineering 28 (2000) 159–177

Effect of free surface and strut on fins attachedto a strut

C.M. Lee a,*, I.R. Parkb, H.H. Chunb, S.J. Leea

a Advanced Fluids Engineering Research Center, University of Science and Technology, San 31,Hyoja-dong, Pohang 790-784, South Korea

b Pusan National University, Pusan, South Korea

Received 21 October 1998; accepted 6 January 1999

Abstract

An extensive experimental and computational investigation of the combined and separateeffects of free surface and body on the lift characteristics of a pair of fins attached to a strutand fin alone is conducted. The results reveal that the free-surface effect becomes significantwhen the depth of submergence to chord ratio (H/c) is less than three. The effect of the strutis also realized for shallower depth of submergence of the fins through free-surface deformationleading to a significant change in the incidence angle of the flow to the fins. The numericalresults based on the Higher Order Boundary Element Method with the linearized free-surfacecondition show good agreement with the experimental results for fin (foil) alone even at shal-low submergence, but some discrepancies appear for the fin attached to the strut at higherspeeds mostly due to the neglect of the nonlinear free-surface effect. 2000 Elsevier ScienceLtd. All rights reserved.

Keywords:Free surface; Fin; Struct; Wave elevation

1. Introduction

For high-speed ships to maintain stability and speed in harsh sea environments,some means of controlling the wave-excitation motions are necessary. Among vari-ous devices for motion control, fins have been recognized as being very effective inmaintaining the stability of submerged vehicles and air-flight projectiles. To a certain

* Corresponding author. Tel.:+82-562-279-5900; fax:+82-562-279-3199.E-mail address:[email protected] (C.M. Lee).

0029-8018/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S0029 -8018(99 )00063-3

160 C.M. Lee et al. / Ocean Engineering 28 (2000) 159–177

degree, the lifting characteristics of fins have been reasonably well predicted bysemi-empirical formulae shown, for instance, by Whicker and Fehlner (1958) andPitt et al. (1959). For the fins attached to surface ships, the lifting characteristics canbe significantly altered due to the effect of the free surface.

There have been numerous investigations on the effect of the free surface onsubmerged foils of large aspect ratio (Wu, 1954; Breslin, 1957; Nishiyama, 1966;Nakatake et al., 1988; Bai and Lee, 1992); however, similar investigations on finsattached to a body are scarce. Ohmatsu et al. (1983) investigated the lift character-istics of fins attached to a semi-submerged ship in calm water and wave conditions.Mori et al. (1988) and Qi and Mori (1990) carried out numerical and experimentalinvestigations on the resistance and flow characteristics of semi-submerged shipswith wings.

The aim of the present study is to validate the numerical method based on a HigherOrder Boundary Element Method (HOBEM) to predict the combined and separateeffects of free surface and rigid body on the lift characteristics on a pair of finsattached to a strut. The variables involved in the investigation are the ratio of depthof submergence to chord length of the fin (H/c), the angle of attack of the fin, andthe free-stream velocity.

Experimental investigation was carried out using a circulating water channel witha pair of fins of rectangular plane of aspect ratio 1.2 with the NACA-0015 foil shapeattached to a vertical strut of rectangular cross-section faired with elliptic ends. Thewave profiles were also measured by a laser light sheet technique to investigate thevariation of the free-surface elevation. The validation of the numerical method waschecked by comparing the numerical and experimental results.

2. Mathematical formulation

A right-handed Cartesian coordinate system is used with the origin located on thefree surfacez=0, the positivex-axis in the direction of the free stream, and thez-axis directed upward. The fluid is assumed to be inviscid, incompressible and irrot-ational, and the surface tension is neglected. With these assumptions, the velocitypotential F(x,y,z) is introduced and the governing equation in the fluid domainbecomes the Laplace equation

=2F50, in the fluid domain (1)

The velocity potentialF(x,y,z) can be decomposed into the uniform stream partand the velocity potentialf(x,y,z) disturbed by a body and the free surface as follows:

F(x,y,z)5U`x1f(x,y,z) (2)

Boundary conditions on each boundary surface to be satisfied are as follows:

Linearized free surface condition on the free surface:

161C.M. Lee et al. / Ocean Engineering 28 (2000) 159–177

∂f∂z

1U2

`

g∂2f∂x

50, onz50 (3)

whereg is the gravity.

Body boundary condition:

∂f∂n

52U`nx, on SB (4)

wherenx is the x-component of the unit normal vectorn which points out of thefluid domain.

Radiation condition at infinity:

=f→0 (5)

Kutta condition at the trailing edge (TE) of a foil:

|=f|,` (6)

Assumption of the wake surface:

n·(VU2VL)50 (7)

pU2pL50, onSW

where the wake surface is assumed as a flat plane. Subscript U and L are the uppersurface and lower surfaces of the wake, andV and p are velocity and pressure,respectively.

The velocity potentialf which is the solution of the Neumann–Kelvin problemcan be written as an integral equation by Green’s theorem:

C(x)f1E ESB1SF

f∂G∂n

ds1EESW

DfW

∂G∂n

ds1U2

`

g EESF

fxxG ds5EESB

∂f∂n

G ds (8)

wherex=(x,y,z) is the collocation point vector,DfW the potential jump on the wake,G the Green function=1/(4πr), r=|x2x|, x the location vector of the singularities, andC(x) the solid angle.

3. Numerical method

In general, Panel Methods have been widely used to calculate flows around anarbitrary body with singularity distribution on the boundary surfaces. By comparisonwith these Panel Methods, boundary surfaces and physical quantities in HOBEM arerepresented by curved elements of second order (or even higher) and a shape functionof the same order, respectively. The accuracy and numerical convergence of thecomputational results by HOBEM are thus expected to be improved. In the present


paper, the body boundary surfaces and physical quantities are represented by a 9-node Lagrangian shape function and, on the free surface, the geometry is representedby a 9-node Lagrangian element and the physical quantities by a bi-quadratic splinefunction that has the advantage of continuity in physical quantities across neighboringelements. The purpose of introducing the spline function into the calculation of thevelocity potentials on the free surface is to remove a numerical wave damping andto improve the wave dispersion property, which would lead to a better solution.Sclavounos and Nakos (1988) showed the advantage of this approach by comparingit with other schemes. They used their own radiation condition which will be men-tioned later.

Boo (1993) used a 16-node cubic order Lagrangian shape function on the freesurface. The order is higher than the bi-quadratic spline function, but it appears thata discontinuity in the physical quantities across the neighboring element could causea divergent solution. It was observed from the results in their paper that it wouldbe difficult to get a converged solution, especially for a shallowly submerged bodyand a surface-piercing body.

3.1. 9-node Lagrangian element method

Fig. 1a shows a local coordinates system for a boundary element and Fig. 1bshows a general 9-node element and a discontinuous element for an edge problemor singular point which has different boundaries. Boundary surfaces and physicalquantities are represented by a 9-node Lagrangian shape function as follows:

x(x,h)5O9j

Nj (x,h)xj ,

f(x,h)5O9j

Nj (x,h)fj , (9)

∂f∂n

(x,h)5O9j

Nj (x,h)∂f∂nj

whereNj(x,h) is the shape function.

3.2. Bi-quadratic spline method

The physical quantities on the free surface are represented by the bi-quadraticspline function as follows:

bs(x)551

2Dh2xSx+

32

DhxD2

, −32

Dhx#x#−12Dhx

1Dh2

xS−x2+

34Dh2

xD, −12Dhx#x#

12Dhx

12Dh2

xSx−

32

DhxD2

,12Dhx#x#

32

Dhx

6 (10)


Fig. 1. Description of numerical grids: (a) local coordinate system; (b) general 9-node Lagrangianelement and discontinuous element.

whereDhx is the length between the neighboring elements.The bi-quadratic spline function normalized on a mapped plane,21#x#1, is

as follows:

bs(x)55bs j−1(x)=

18(x+1)2

bs j (x)=14(−x2+3), −1#x#1

bs j+1(x)=18(−x+1)2

6 (11)

The velocity potential on the free surface is represented by the normalized bi-quadratic spline function as

f(x)5ONF

k51

Bks(x,h)jk (12)


whereSNFk51Bk

s(x,h)=S iS jbis(x)bj

s(h), i=k2NFY, k, k+NFY, j=k21, k, k+1 andjk isa weighting coefficient at the centroid of thek-th element. NF is the number of totalelements and NFY is the number of elements in they-direction on the free surface.

Then the following equation can be obtained from Eq. (8):

Cifi1ONB

e

O9j

EESe

B

∂G∂n

NjJ dx dh fj 1ONF

e

ONF

k

O9j

EESe

F

∂G∂n

BksJ dx dh jk

1ONW

e

O9j

E ESe

W

∂G∂n

NjJ dx dh DfWj2ONF

e

ONF

k

O9j

EESe

F

U2`

gG

∂2Bks

∂x2 J dx dh jk5

2ONB

e

O9j

EESe

B

GNj J dx dh∂f∂nj

(13)

whereJ is the Jacobian of the coordinate transformation. NB and NW are the numberof elements of the body and wake surface, respectively. The superscripte denotesone element of the surface.

A standard Gauss–Legendre quadrature is used for the evaluations of the integralequations and the singular integrals are regularized by introducing the nonlinear coor-dinates transformation technique by Telles (1987).

3.3. Radiation condition

Sclavounos and Nakos (1988) used the following equation at the upstream trunc-ation boundary for a radiation condition.

fx5fxx50 (14)

The above equation can be transformed intojm−1=jm=jm+1 at the upstream trunc-ation boundary for the bi-quadratic spline function method where the subscript isthe position of the neighboring three elements in the downstream direction. At thelateral boundary it can be imposed that the second derivatives in they-direction ofthe velocity potential may be zero.

3.4. Kutta condition

Discontinuous elements are used to avoid the evaluation of integrals in the planewhere three planes with different physical values meet. Then the Kutta condition isimposed at some points on the surface of a fin away from the trailing edge. A pressureKutta condition by Lee (1987) was employed in the present study and this has beenused efficiently for three-dimensional problems such as foil and propeller for theuniqueness and convergence of the solution.


4. Experimental method

The strut and the fin used in the present investigation are shown in Fig. 2. Thetype of strut for investigating the effect of the body is chosen for geometrical sim-plicity. The basic cross-section of the strut is a rectangle of width 500 mm, thickness50 mm and length 1300 mm. The fore and aft ends of the horizontal cross-sectionare faired by an ellipse of 2 to 1 ratio between the major and minor axes. The finis of NACA-0015 foil shape with a rectangular plane form of 100 mm in chord (c)and 120 mm in span. The material of the strut and fin is aluminum.

The fins can slide vertically along the groove of width 20 mm which is made inthe mid-width of the strut. Once the fins are positioned at the desired vertical position,the groove is filled flush with the strut surface with styrofoam and acrylic materials.The angle of attack of the fins can be adjusted by±20°. A load cell is attached ontop of the vertical axis at which the fins are attached with a horizontal rod. The loadcell is designed to measure the vertical force acting on the fins independently.

The experiment was carried out in a circulating water channel of test section of1 m in depth, 1 m in width and 4.53 m in length. The maximum flow velocity thatthe channel can generate is 2.2 m/s. Since the blockage ratio of the strut–fin structurewas about 5%, no correction was necessary for the blockage effect on the measure-ments according to Goldstein (1983).

Fig. 2. Fin and strut configuration (units: mm).


The experimental parameters in the investigation are the depth of submergence ofthe fins, i.e. the vertical distance from the calm-water surface to the center of themean chordline,H, the angle of attack of the fins,a, and the free stream velocity,U`. The definition of these variables is given in Fig. 3, and the values of the para-meters chosen in the experiment are as follows:

H/c: 0.5, 0.8, 1.0, 1.3, 1.6, 2.0, 2.5, 3.0, 4.0, 5.0

a in degrees:215,210,25, 0, 5, 10, 15, 20

Fr5U`/Îgc: 0.4, 0.6, 0.8, 1.0, 1.2, 1.5

wherec is the chord length of the fins. In the present case√gc<1 m/s and hencethe Froude number is identical withU` in m/s. Forc=2 m, Fr=1.5 corresponds toabout 19 knots. Thus, the Froude numbers considered in the present experiment arein the mid-speed range for high-speed ships of 25 to 35 knots cruising speed. Sucha choice is forced by the limitation of the test facility.

To investigate the lift characteristics of the fins alone, the pair of fins were joinedtogether and supported by a right-angled streamlined strut which is attached to themid-span of the trailing edge. Thus, the disturbance of the flow by the strut was mini-mized.

The measured quantities are the lift force of the fins,Lf, which is represented interms of the lift coefficient,CL=2Lf/(rU2

`A), wherer is the density of the water andA the plane area of the two fins, and the wave profiles along the side of the strutand on the extended centerline aft of the strut.

The longitudinal-plane section of the flow at the mid-span of the fin and theextended centerline of the strut aft of its trailing edge, as shown in Fig. 4a, wasvisualized using the laser light sheet technique to investigate the variation of thefree-surface elevation. Fig. 4b shows the schematic diagram of the experimental set-up for the flow visualization. In this experiment, a beam of a 4W Argon-ion laserwas irradiated onto the cylindrical lens immersed behind the fin, by which it wasspread into a 2-D laser light sheet. The wave patterns in the plane of the light sheetwere recorded using a Nikon F5 camera installed on an optical rail located outsidethe water channel. The visualized photographs were digitized with a scanner and the

Fig. 3. Definition of variables.


Fig. 4. Schematic diagram of experimental set-up: (a) top view (units: mm); (b) side view aty/c=20.85.

free-surface wave profiles were extracted from the photographic images with thehelp of digital processing techniques.

In order to check the reliability and usefulness of the photo-image technique, thewave heights at several locations in the extended centerline aft of the strut weremeasured by a capacitance-type wave height gauge and compared with those meas-ured by the present image processing technique. The comparison results shown inFig. 5 are in fairly good agreement with each other although two completely different

Fig. 5. Comparison of wave-elevation measurements between wave-height gauge and photo-image pro-cessing (Fr=1.2).


measuring systems were used. Therefore, the digital imaging processing techniquewas proven to be a useful and powerful tool for capturing the instantaneous waveprofile in the plane of the light sheet.

5. Results and discussions

The measured values of the lift coefficientCL of the fins attached to the strutversus the submergence ratio,H/c, for the angle of attack (a=0°,10°) for the Froudenumbers ranging from 0.4 to 1.5 are shown in Fig. 6. The lift force of a symmetricfoil in unbounded flow would be zero ata=0°. However, the lift coefficient of thefoils attached to the strut ata=0° in the presence of the free surface varies consider-ably with the free-stream velocity and the fin submergence depth due to the effectsof the free surface and the strut. The results show that the free-surface effect onCL

is greater for smaller depth of submergence and for greater free-stream velocity, asanticipated. The free-surface effect seems to increase significantly in the Froude num-ber range between 1.0 and 1.2 forH/c#3.0. The negativeCL is due to the fact thatthe strut-induced incidence angle of the flow to the fin is negative. However, theapparent reverse trend ofCL versusH/c at Fr=0.8 is interesting. Such a reversedtrend appears to be caused by the reversed angle of incidence of the flow and thisis discussed later in detail together with the wave profile comparisons.

The comparison of the experimental results with the computed results forCL ver-susH/c for the fins attached to the strut fora=0° is shown in Fig. 7. The results ofthe four representatives Froude numbersFr=0.4, 0.8, 1.2 and 1.5 are chosen forcomparison. Sufficient numbers of convergence tests on the numerical results withrespect to the computational domain and also element numbers are performed. It is

Fig. 6. Experimental results of lift coefficient versus submergence ratio for fins attached to strut: (a)a=0°; (b) a=10°.


Fig. 7. Comparison of experimental results with computed results forCL versusH/c for fins attachedto strut: (a)a=0°; (b) a=5°.

found that 286 elements for the half of the strut attached to the fins and 840 elementsfor the free surface of21#x/L#2 and 0#y/L#1 are sufficient to obtain a convergedsolution. The agreement between the two results atFr=0.4 and 0.8 is fairly goodwhile some discrepancy can be seen for the higher speeds. It is worthwhile notingthat the numerical result predicts well the trend of the positive increase inCL withthe decrease ofH/c at Fr=0.8.

Fig. 8 shows the computed and measuredCL versus angle of attack for the finsattached to the strut atH/c=1.0, 2.0 and 5.0. The computed and measuredCL forthe fins alone atH/c=1.0 and 3.0 is shown in Fig. 9. It can be seen in Fig. 9 thatthe shallower the depth of submergence and the greater the speed, the greater thediscrepancy between the computed and measured results. The lift curve slope forthe fins-alone case computed by the formula of Whicker and Fehlner (1958) is 2.76while the corresponding measured result shown in Fig. 8c is about 2.72. Thus, themeasured results ofCLa appear reasonable.

Fig. 10 shows the computed chordwise pressure distributions of the fin attachedto the strut atFr=0.8 for H/c=1.0 anda=5°. The numerical results show that theKutta condition at the TE for the fins with strut is well satisfied. Although the resultis not shown in this paper owing to the limited space, it was noted that the Kuttacondition at the TE (trailing edge) for the fin alone was well satisfied. It can be seenthat the present computational method with the linearized free surface condition canpredict well the lift force of a submerged fin alone at submergence ofH/c$1.0 wherethe nonlinear free surface effect seems to be negligible.

Therefore, it can be understood that the discrepancy between the computed andmeasured values at the higher speeds ofFr=1.2 and 1.5 is attributed to the strongnonlinear free-surface effect by the strut, but not by the fin. This understanding isalso supported by the fact that the discrepancy in the two results decreases asH/c


Fig. 8. Comparison of experimental results with computed results forCL versusa for fins attached tostrut: (a)H/c=1.0; (b) H/c=2.0; (c) H/c=5.0.

increases. As the depth of submergence increases, the effect of the strut on the liftof the fins diminishes as can be deduced by comparing the results shown in Figs. 7and 8. At the deepest submergence depth ofH/c=5.0 as shown in Fig. 8c,CL curvesfor all Froude numbers in the present experiment almost converge to one curve; theresults forFr=1.5 are presented as an example. It appears evident from the presentresults that the consideration of the nonlinear free surface effect for shallow depthsof submergence in the numerical scheme would improve the computational results.

In an unbounded fluid, it is commonly understood that the relation between thelift coefficient and the angle of attack for thin airfoils before the stall is linear. Thisis confirmed by the results shown in Fig. 9b for the fin alone case at a relatively


Fig. 9. Comparison of experimental results with computed results forCL versusa for fins alone: (a)H/c=1.0; (b) H/c=3.0.

Fig. 10. Computed chordwise pressure coefficient distribution on the fin with strut.

deep submergence ofH/c=3.0, as well as by the results shown in Fig. 8c for thefins attached to the strut at a deeper submergence ofH/c=5.0. However, when thefins are located close to the free surface, the behavior ofCL seems to be represented,as indicated in Fig. 8a and b, by an equation in the formCL=Co+C1a whereCo isa constant which depends on the free-stream velocity and depth of submergence andC1 is a pure constant. The constantCo, the value of which can be positive or negativedepending on the free stream speed and the submergence depth, is the measure ofthe free-surface deformation created by the strut. The results shown in Fig. 9a and


b are for the fins-alone case forH/c=1.0 and 3.0. Here, one can observe that thevalues of constantCo seem to be significantly smaller (almost approaching zero)than those for the case of fins with strut, indicating that the free-surface deformationgenerated by the fins is significantly smaller than that by the strut.

The effect of the strut on the lift force is shown in Fig. 11 forH/c=2.0 whereDCL is obtained by subtracting the lift coefficient of fins alone from that of the finsattached to the strut. The computed results show a linearly decreasing trend with theincrease ofa while the experimental results do not exhibit any definite trend. Thelarge discrepancies between the experimental and numerical results at the highestspeed ofFr=1.5 can be judged to be due mainly to the nonlinear free-surface effectsmentioned earlier.

From the wave-elevation results shown in Fig. 12, it can be deduced that the free-surface deformation generated by the strut apparently changes the incidence angleof the free stream to the fins, which changes with the free-stream velocity. In thepresent case, the strut-induced angle of incidence of the flow tends to be positivefor Fr=0.8 and negative for the other Froude numbers (Fr.1.0) as evidenced by themeasured wave profiles shown in Fig. 12a where the strut is located between20.5and 0.5 in thex-coordinate. TheCL shown in Fig. 6a supports the relation betweenthe angle of incidence created by the presence of the strut andCL. That is, forFr=0.8and H/c=1.0, CL.0, while at the other Froude numbersCL,0.

The presence of fins atH/c=0.5 appears to alter the wave profile slightly as shownin Fig. 12b compared to the wave profiles in Fig. 12a. Fig. 12b shows some ripplesat Fr=0.4, a slight reduction in the wave height and length atFr=0.8, and moreintensive wave breaking at the TE of the strut atFr=1.2. The wave profiles shownin Fig. 12c for the strut with fins forH/c=0.5, 0.8 and 1.0,a=0° and Fr=1.2 showvery few changes. Fig. 12d shows the effect of the angle of attack on the waveprofile. Ata=10° the wave breaking occurs before the TE of the strut, while it occurs

Fig. 11. Comparison of experimental results with computed results for strut effect onCL.


Fig. 12. Photo-image of wave elevation at20.5#x/L#0.5, y/c=20.85 and 0.5#x/L#1.5, y/c=0 for: (a)Froude numbers for strut alone; (b) Froude numbers atH/c=0.5 anda=0°; (c) foil depths atFr=1.2 anda=0°; (d) angles of attack atFr=1.2 andH/c=0.5.


Fig. 12. (continued)

right at the TE ata=0°. Thus, the presence of fins at the shallow depth of submerg-ence with a moderate angle of attack at higher speeds can disturb the free surfacesignificantly in the stern region of the strut.

The effect ofa on CL at H/c=0.5 andFr=1.2 is shown in Fig. 6a and b. The valueof CL for a=0° from Fig. 6a is20.4 while that fora=10° from Fig. 6b is about


0.0. One can deduce such results by observing the angle of incidence of flow at theleading edge of the fins in Fig. 12d fora=0° anda=10°.

Fig. 13 shows the computed and measured wave profiles at the longitudinal planeat y/c=20.85 for the strut alone and also for the strut with fins fora=0° at H/c=0.5and 1.0. The strut is located between20.5 and 0.5 in thex-coordinate. It can beclearly seen that the wave height due to the strut alone is slightly larger than thatdue to the strut with fins, which is also seen in Fig. 12a and b. For the case of thestrut with the fins, the wave height for the shallow fin submergence ofH/c=0.5is smaller than that for the deeper submergence ofH/c=1.0 due to the favorablewave interferences.

These results clearly indicate that when the fins attached to a ship’s hull approachthe free surface, they would definitely be influenced by the free-surface deformationcreated by the advance of the ship. Thus, the selection of the location of the controlfins for a high-speed ship should be carefully determined by considering the ship-generated wave contour alongside the ship.

6. Conclusions

The free-surface effect on the lift characteristics of the fins attached to a body isfound to be significant when the submergence depth of fins is less than three timesthe chord length. The dominant cause is the change in the angle of incidence of flowto the fins induced by the free-surface deformation caused by the strut. The waveelevations alongside the strut for the strut with fins are lower than those of the strut

Fig. 13. Comparison of experimental results with computed wave-elevation (z/c) for strut alone andstrut with fins atFr=0.8.


alone. Therefore, careful selection of the fin position relative to the body is necessaryin view of the lift and wave making resistance points.

The computational method based on the HOBEM appears to be useful in pre-dicting the quantitative lift and flow characteristics of fin alone submerged at thedepth ofH/c$1.0 and also of the fin attached to the strut up to moderately highspeed even at a relatively shallow submergence. Further improvements of the compu-tational scheme by taking into account the nonlinear free-surface effect will be neces-sary to improve the accuracy of prediction for the fin performance near the freesurface for high Froude numbers.

Acknowledgements

C.M. Lee and S.J. Lee express their gratitude for the supports of the Korea Scienceand Engineering Foundation and the Advanced Fluids Engineering Research Centerof Pohang University of Science and Technology (POSTECH). The assistance givenby POSTECH graduate students J.H. Choi and S.J. Kim is greatly appreciated.

References

Bai, K.J., Lee, H.K., 1992. A localized finite-element method for nonlinear free-surface wave problems.In: Proceedings of the Nineteenth Symposium on Naval Hydrodynamics, pp. 113–130.

Boo, S.Y., 1993. Application of higher order boundary element method to steady ship wave problem andtime domain simulation of nonlinear gravity waves. Ph.D. dissertation, Texas A&M University, USA,150 pp.

Breslin, J.P., 1957. Application of ship-wave theory to the hydrofoil of finite span. Journal of ShipResearch 1 (1), 27–55.

Goldstein, R.J., 1996. Fluid Mechanics Measurements, 2nd ed. Taylor & Francis, 712 pp.Lee, J.T., 1987. A potential-based panel method for the analysis of marine propellers in steady flow.

Ph.D. thesis, Department of Ocean Engineering, M.I.T., Cambridge, Mass., 150 pp.Mori, K.H., Hotta, T., Ebira, K., Qi, X., 1988. A study on semi-submergible high speed ship with wings—

its resistance characteristics and possibility. Naval Architecture and Ocean Engineering 27, 1–10.Nakatake, K., Kawagoe, T., Kataoka, K., Ando, J., 1988. Calculation of the hydrodynamic forces acting

on a hydrofoil. Transactions of the West-Japan Society of Naval Architects 76, 1–13.Nishiyama, T., 1966. Linearized steady theory of fully wetted hydrofoils. Advances in Hydroscience 3,

237–342.Ohmatsu, S., Yoshino, T., Yamamoto, T., Ishida, S., Nimura, T., Sugai, K., 1983. An experimental study

on the motion control of semi-submerged ships. Journal of the Society of Naval Architects of Japan152, 229–238.

Pitt, W.C., Nielsen, J.N., Kaatari, G.E., 1959. Lift and center pressure of wing–body–tail combinationsat subsonic, transonic and supersonic speeds. NACA report 1307.

Qi, X., Mori, K.H., 1990. A boundary element method for the numerical simulation of 3-D nonlinearwater waves created by a submerged lifting body. Journal of the Society of Naval Architects of Japan167, 25–34.

Sclavounos, P.D., Nakos, D.E., 1988. Stability analysis of panel methods for free-surface flows withforward speed. In: Proceedings of the Seventeenth Symposium on Naval Hydrodynamics, The Hague,The Netherlands, pp. 29–48.

Telles, J.C.F., 1987. A self-adaptive co-ordinate transformation for efficient numerical evaluation of gen-


eral boundary element integrals. International Journal for Numerical Methods in Engineering 24,959–973.

Whicker, L.F., Fehlner, L.F., 1958. Free-stream characteristics of a family of low aspect-ratio, all-move-able control surface for application to ship design. DTMB report 933, 119 pp.

Wu, Y.T., 1954. A theory for hydrofoils of finite span. Journal of Mathematics and Physics 33, 207–248.

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Effect of free surface and strut on fins attached to a strut