A study of hydrodynamics of three-dimensional planing surface

A study of hydrodynamics of three-dimensional

planing surface

Nan Xie*, Dracos Vassalos, Andrzej Jasionowski

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

Universities of Strathclyde and Glasgow, Henry Dyer Building, 100 Montrose Street, Glasgow G4 0LZ, UK

Received 6 August 2004; accepted 2 February 2005

Available online 2 April 2005

Abstract

The hydrodynamic problem of 3D planing surface is studied by a finite element approach. The

planing surface is represented by a number of pressure patches whose strengths are constant at each

element. The unknown pressure strength is obtained by using the free surface elevation condition

under the planing surface and Kutta condition at the transom stern. Previous studies indicate that,

when the constant pressure distribution method is used, the number of buttocks should be less than

five or six, otherwise the calculated pressure distribution will start to oscillate and even become

divergent. In the present study, after careful examination of the influence coefficients, it is found that

the accuracy of the influence coefficients matrix is very important to the convergence of the solution,

especially when the number of elements is relatively high. The oscillation of the pressure distribution

can be avoided by constant element method if the influence coefficients are sufficiently accurate. The

predicted results of the present paper with more number of buttocks are in good agreement with other

researchers’. It is concluded that the irregularity of the pressure distribution found in previous studies

is most likely caused by the low accuracy in their calculation of the influence coefficients, not by the

method itself.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Hydrodynamics; Planing; 3D; Pressure distribution; Numerical prediction

Ocean Engineering 32 (2005) 1539–1555

www.elsevier.com/locate/oceaneng

0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.oceaneng.2005.02.003

* Corresponding author.

E-mail addresses: [email protected] (N. Xie), [email protected] (D. Vassalos), a.jasionowski@

na-me.ac.uk (A. Jasionowski).

http://www.elsevier.com/locate/oceaneng

N. Xie et al. / Ocean Engineering 32 (2005) 1539–15551540

1. Introduction

The study of hydrodynamics of planing is important for high speed marine vessels. The

hydrodynamic forces on the hull and its appendages developed during forward motion can

be sufficient to support the craft and raise a substantial proportion of the hull out of the

water. The determination of the planing induced hydrodynamic force is an important part

in the analysis of vessel performance.

Early studies on planing problems were seen in the 1930s, but until the 1960s

theoretical investigations were restricted mainly to linearized two-dimensional planing

(Maruo, 1951). The three-dimensional planing problem was tackled in the 1960s, but

always with restrictions on either the planing speed or aspect ratio of the planing surface

(Maruo, 1967; Wang and Rispin, 1971; Shen and Ogilve, 1972; Tuck, 1975). Doctors

(1975) may be the first to study the three-dimensional planing without these restrictions. In

his approach, finite pressure elements were adopted to represent the wetted area of a

planing surface and in an iterative procedure the wetted area was adjusted to satisfy the

trailing-edge Kutta condition until it finally reached a constant, but the pressure

distributions thus obtained were seriously oscillatory. Wellicome and Jahangeer (1978)

and Tong (1989) prescribed the wetted area in advance, then calculated the pressure

distribution and shape of transom. The pressure oscillation, which Doctors found, was

avoided when the number of buttocks was not more than five or six, otherwise it would still

occur. The reason for such oscillation was believed to be the pressure discontinuities at the

side edges of the constant pressure element they employed. To avoid the pressure

oscillation problem, Cheng and Wellicome (1994) developed a pressure strip method, in

which a planing surface is represented by an assemblage of strips of transversely variable

pressure placed on the mean free surface. The pressure oscillatory problem was

successfully removed. More recently, Wang (2003) used linear pressure distribution over

each element with continuous pressure over the length of the planing surface. However,

the solution of the pressure distribution still had oscillatory behaviour, a similar wavy

behaviour of the inversed pressure solution on a flat plate can be also seen in the work of

Tuck et al. (2002).

The constant pressure approach provides relatively simple mathematical formulation

and therefore is easier to facilitate numerical implementations. The authors have applied

this method to the prediction of wash wave field of the Air-Lifted Catamaran (Xie et al.,

2004). The aim of the present paper is to re-examine the application of the ‘constant

pressure distribution’ method in the study of planing hydrodynamics. Within the frame of

linearized potential flow theory, the free surface elevation induced by the planing hull is

represented by pressure distribution over the wetted area. The wetted area of the planing

hull is discretized using a number of rectangular elements over which the pressure

strengths are constant. The induced free surface elevations within the wetted area are set to

be equal to the hull surface vertical coordinates. At the transom stern, the pressure is equal

to the atmospheric pressure, i.e. the planing pressure is zero in order to satisfy Kutta

condition. The free surface depression at the stern will be equal to the immersed depth of

the local trailing edge, which is also unknown. Solution of the linear algebraic equations

will be the pressure strengths on each element and the immersed depth of the trailing edge.

Previous studies indicated that when the number of buttocks was more than five or six,

N. Xie et al. / Ocean Engineering 32 (2005) 1539–1555 1541

the pressure would become oscillatory. Since the oscillatory pressure distribution on the

planing surface has not been supported by model test measurements, it can be concluded

that the pressure oscillation found in previous studies are caused by numerical

inaccuracies.

The elements of the matrix of the linear algebraic equations are the induced coefficients

between control points and the pressure elements. The pressure elements, representing the

planing hull, are deployed on the water surface forming a rectangular pressure patch which

is also moving at high speed, usually the element length based Froude Number

FrZU=ffiffiffiffigl

pO6:0, in which U is the craft speed, l is the length of the element, and g is the

acceleration due to gravity. Therefore, the free surface elevations (or induced coefficients)

are very small within the wetted area range. This requires high accuracy in the calculation

of the coefficients, especially in the cases when the number of elements is relatively large.

2. Mathematical formulation

The planing of a craft at a constant forward speed U on the otherwise calm water can be

equivalently transformed into a stationary planing craft, which is represented by a pressure

distribution, on the surface of a uniform incoming flow with speed U. It is also assumed

that the fluid is inviscid with an infinite depth, and the disturbance to the main stream

caused by the craft is small. The coordinate system oxyz is defined with xoy plane located

at the undisturbed free surface; the xoz plane is the symmetric about the central plane of

the craft; the oz axes is positive upwards and goes through the centre of trailing edge; the

ox axes is positive towards to the bow of the craft. A diagram of the coordinate system is

shown in Fig. 1.

The potential of disturbance should satisfy the following equations:

Laplace equation in the fluid domain:

4xx C4yy C4zz Z 0 (1)

On the free surface the velocity potential 4 and the surface elevation 2Z2(x,y) satisfy

the linearized kinematical and dynamical boundary conditions

U2x C4z Z 0; on z Z 0 (2)

pðx; yÞ

rCg2ðx; yÞKU4x Z 0 on z Z 0 (3)

U

x

y z

oplaning

surface

Fig. 1. Coordinate system.


on the bottom and far upstream condition

4z Z 0 z ZKN or x ZCN (4)

The boundary value problem (1)–(4) is solved by using Fourier transformation method,

and the following expression can be obtained (Wehausen and Laitone, 1960)

4ðx; y; zÞ Z1

p2rU

ððS

pðx;hÞdxdh

ðp=2

0dq sec q P:V:

ðN

0

k ekz

k Kk0 sec2qsin½kðx

KxÞcos q�cos½kðy KhÞsin q�dk Kk0

prU

!

ððS

pðx;hÞdxdh

ðp=2

0sec3 q ek0z sec2q cos½k0ðx KxÞsec q�cos½k0ðy

KhÞsec2q sin q�dq (5)

where S is the area on which the pressure is applied and k0Zg/U2 is the wave number. The

expression for free surface elevation 2(x,y) can be obtained by substituting 4(x,y,z) into the

free surface condition (3):

2ðx; yÞ Z1

p2rg

ððS

pðx;hÞdxdh

ðp=2

0P:V:

ðN

0

k2 cos½kðx KxÞcos q�

k Kk0 sec2q

!cos kðy KhÞsin q� �

dkdq Ck2

0

prg

ððS

pðx;hÞdxdh

ðp=2

0

sin½k0ðx KxÞsec q�

cos4q

!cos½k0ðy KhÞsec2q sin q�dq Kpðx; yÞ

rgð6Þ

In the present approach, rectangular pressure patches are adopted. Without losing

generality, let S be a rectangular area of length L and width B, which is divided into Nx

elements in length direction and Ny in the beam direction leading to a total of NTZNx!Ny

small rectangular elements. The free surface elevation at jth control point now becomes

2ðxj;yjÞZXNT

iZ1

pi

rg

1

p2

ððSi

dxdh

ðp=2

0P:V:

ðN

0

k2 cos½kðxj KxÞcos q�

k Kk0 sec2q

8><>:

!cos½kðyj KhÞsin q�dkdq

9=;C

XNT

iZ1

pi

rg

k20

p

ððSi

dxdh

ðp=2

0

sin½k0ðxj KxÞsec q�

cos4q

8><>:

!cos½k0ðyj KhÞsec2q sin q�dq

9=;K

pðxj;yjÞ

rgZ

XNT

iZ1

pi

rgDji j Z1;.;NT

ð7Þ


where Dji is the non-dimensional free surface elevation at field point (xj,yj) induced by the

rectangular pressure element of unit strength at (xi,hi) and can be expressed as (Kim and

Tsakonas, 1981)

Dðxj; yj; xi;hiÞ ZKdji K1

2p2

X8

mZ1

ðK1ÞmC1Re

ðp=2

0

expðik0 sec2qsmðqÞÞ

sin q cos q

!½E1ðik0 sec2qsmðqÞÞK ipð1 KsgnðsmÞÞ�dq

�ð8Þ

where dji is static free surface depression

dji Z

1 ðxj; yjÞ is inside Si

0:5 ðxj; yjÞ is on the boundary of Si

0:25 ðxj; yjÞ is on the corner of Si

0 ðxj; yjÞ is out of Si

8>>>><>>>>:

(9)

smðqÞ Z

ðxj Kxi KaiÞcos q C ðyj Khi KbiÞsin q m Z 1

ðxj Kxi KaiÞcos q C ðyj Khi CbiÞsin q m Z 2

ðxj Kxi CaiÞcos q C ðyj Khi CbiÞsin q m Z 3

ðxj Kxi CaiÞcos q C ðyj Khi KbiÞsin q m Z 4

ðxj Kxi KaiÞcos q K ðyj Khi CbiÞsin q m Z 5

ðxj Kxi KaiÞcos q K ðyj Khi KbiÞsin q m Z 6

ðxj Kxi CaiÞcos q K ðyj Khi KbiÞsin q m Z 7

ðxj Kxi CaiÞcos q K ðyj Khi CbiÞsin q m Z 8

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

(10)

ai, bi are half-length and width of ith pressure element, and E1(w) is the exponential

integral of complex argument, which can be defined as (Abramowtiz and Stegun, 1970)

E1ðwÞ Z

ðN

0

eKðtCwÞ

t Cwdt (11)

There is also a series expression for this integral

E1ðwÞ ZKg K lnðwÞKXN

nZ1

ðK1Þnwn

n$n!(12)

where gZ0.5772156649., which is also known as Euler’s constant.

3. Determination of pressure distribution on planing hull

A pressure element representation of a flat planing hull is shown in Fig. 2. Assuming

that there are Nx cuts in the length direction and Ny cuts in the beam direction, the free

surface elevation at jth control point can be obtained by adding the induced contributions

Spray rootignored

L

B

Fig. 2. Representation of a flat planning hull by pressure elements.


from all the pressure elements and expressed as

2j ZXNT

iZ1

Dji

pi

rgj Z 1;.;NT (13)

where jZ jy C ðjx K1ÞNy; iZ iyC ðix K1ÞNy; jy, ijZ1,.,Ny, jx,ixZ1,.,Nx and NTZNx!Ny. On the other hand, the body boundary condition requires

2j Z xjxtan t CHjy

j Z jy C ðjx K1ÞNy; jx Z 1;.;Nx; jy Z 1;.;Ny (14)

where xjxis the longitudinal distance from trailing edge, t is the trim angle and Hjy

is the

immersed depth of the trailing-edge for the jyth buttock.

In order to satisfy Kutta condition, zero pressure is assigned for the trailing-edge:

pi Z 0 i Z iy C ðix K1ÞNy; ix Z 1; iy Z 1;.;Ny (15)

Substituting (13) and (15) into (14), the following linear algebraic equations are obtained

XNT

iZ1

D̂jip̂i Z Ej (16)

where

p̂i Z

Hiy

B tan ti Z iy C ðix K1ÞNy; ix Z 1; iy Z 1;.;Ny

pi

Brg tan ti Z iy C ðix K1ÞNy; ix Z 2;.;Nx; iy Z 1;.;Ny

8><>: (17)

D̂ji Z

K1 jZjy Cðjx K1ÞNy; iZ iy Cðix K1ÞNy; ix Z1;jx Z1;.;Nx; jy Ziy; jy;iy Z1;.;Ny

0 jZjy Cðjx K1ÞNy; iZ iy Cðix K1ÞNy; ix Z1;jx Z1;.;Nx; jy siy; jy;iy Z1;.;Ny

Dji jZjy Cðjx K1ÞNy; iZ iy Cðix K1ÞNy; ix Z2;.;Nx; jx Z1;.;Nx; jy;iy Z1;.;Ny

8><>:

(18)

Ej Zxjx

Bj Z jy C ðjx K1ÞNy; jx Z 1;.;Nx; jy Z 1;.;Ny (19)


Solutions of (16) will be the trailing-edge profile and the pressure at each control point.

In the actual calculation, the surface symmetry is taken into account and only those control

points on one side of the hull will be used.

A number of non-dimensional parameters are used in this study which are defined as:

The non-dimensional pressure coefficient

cp Zp

0:5rU2 tan tZ

2

C2v

p̂ (20)

where Cv ZU=ffiffiffiffiffiffigB

pis the beam Froude number.

The non-dimensional lift force:

CL ZLift

0:5rU2BL tan tZ

PNTiZNyC1 pið4aibiÞ

0:5rU2BL tan t(21)

The non-dimensional location of pressure centre

LCP Z lCP=L (22)

which is measured from the trailing-edge.

4. Numerical methods

Most numerical work is for the evaluation of the induced coefficients Dji in (8) in which

singularities in the integrands and their highly oscillatory behaviour are two main

problems to be dealt with. It can be seen that the integrands in (8) have highly oscillatory

behaviour when q approaches p/2. A commonly used method is to make a variable change

of uZtan q to stretch the dimension in the fast oscillation area. However, numerical tests

indicate that the evaluation of the induced coefficients requires high accuracy in order to

obtain accurate pressure solution. The truncation of the integrals will depend on the vessel

speed, size of the pressure element and location of the control points. In the present study,

the following variable changes are adopted

u Zjgj

sin q

cos2qm Z 1; 4; 6; 7

jdjsin q

cos2qm Z 2; 3; 5; 8

8><>: (23)

where gZgðyjKhiKbiÞ=U2, dZgðyjKhiCbiÞ=U

2, gs0 and ds0. By doing this, the

integrals in (8) become

J1m Z

ðp=2

0


sin q cos qE1ðik0 sec2

qsmðqÞÞdq

Z

ðN

0

2u eitmðuÞ

4u2 Cg2 K jgjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4u2 Cg2

p E1ðitmÞdu m Z 1; 4; 6; 7 (24)


J2m ZKip

ðp=2

0


sin q cos q½1 KsgnðsmðqÞÞ�dq

ZKip

ðN

0

2u eitmðuÞ


p ½1 KsgnðtmðuÞÞ�du

m Z 1; 4; 6; 7

(25)

J1m Z

ðp=2

0


sin q cos qE1ðik0 sec2qsmðqÞÞdq

Z

ðN

0

2u eitmðuÞ

4u2 Cd2 K jdjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4u2 Cd2

p E1ðitmÞdu m Z 2; 3; 5; 8 (26)

J2m ZKip

ðp=2

0


sin q cos q½1 KsgnðsmðqÞÞ�dq

ZKip

ðN

0

2u eitmðuÞ


p ½1 KsgnðtmðuÞÞ�du m Z 2; 3; 5; 8 (27)

where

tmðuÞ

ffiffiffi2

pau

G1

Cg

jgju m Z 1ffiffiffi

2p

bu

G1

Cg

jgju m Z 4ffiffiffi

2p

au

G1

Kg

jgju m Z 6ffiffiffi

2p

bu

G1

Kg

jgju m Z 7

8>>>>>>>>>>><>>>>>>>>>>>:

(28)

tmðuÞ Z

ffiffiffi2

pau

G2

Cd

jdju m Z 2ffiffiffi

2p

bu

G2

Cd

jdju m Z 3ffiffiffi

2p

au

G2

Kd

jdju m Z 5ffiffiffi

2p

bu

G2

Kd

jdju m Z 8

8>>>>>>>>>>><>>>>>>>>>>>:

(29)

where G1 Zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijgj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4u2Cg2

pKg2

q, G2Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijdj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4u2Cd2

pKd2

q, aZgðxjKxiCaiÞ=U

2 and

bZgðxjKxiKaiÞ=U2. It can be seen that the highly oscillatory behaviour of the

integrands has been eliminated as u/N (i.e. q/p/2). In fact, tm(u)fGu as u/N from

(28) and (29).


The second problem in the numerical calculations is the convergence of the integrals in

(24)–(27). The integrals have logarithmic singularity at lower limit uZ0. In fact

limu/0

2u


p =1

uZ 1; lim

u/0

2u


p =1

uZ 1

(30)

This singularity can be removed by the following method:

J1m Z

ðN

0

2u eitmðuÞE1ðitmðuÞÞ


p Keitmð0ÞE1ðitmð0ÞÞ

u

" #du

C

ðN

0

eitmð0ÞE1ðitmð0ÞÞ

udu

m Z 1; 4; 6; 7

(31)

J2m ZKip

ðN

0

2u eitmðuÞ½1 KsgnðtmðuÞÞ�


p Keitmð0Þ½1 Ksgnðtmð0ÞÞ�

u

" #du

K ip

ðN

0

eitmð0Þ½1 Ksgnðtmð0ÞÞ�

udu

m Z 1; 4; 6; 7

(32)

Similar expressions can be derived for mZ2, 3, 5, 8. It can be shown that the second

integral in (31) and (32) have no contribution when they are summarised over all m. The

integrands in the first integral of (31) and (32) are regular at uZ0 (i.e. qZ0). The

convergence behaviours of (31) and (32) at their upper limits are analysed as follows

For large w, the following asymptotic expression can be used (Abramowitz and Stegun,

1970)

ew E1ðwÞ Z1

wK

1

w2C

2!

w3K

3!

w4C/ (33)

Therefore, the first integral in (31) is convergent. On the other hand, the integrand in the

first integral of (32) are oscillatory functions whose amplitudes monotonically decay as

m/N, thus the convergence can easily be achieved (similar to an alternating decay

series).

Another problem should be mentioned in the numerical calculation is that there are

logarithmic singularities in the integrands of (31). In fact, when

q Z q� Z tanK1 jxj KxiGaij

jyj KhiHbij

� �(34)

sm(q)Z0 for some of m 0s and therefore tm(u)Z0, the exponential function E1(itm) in

(31) will have logarithmic singularity, as can be seen in (12). Although theoretically this


singularity is integrable, in order to get high accurate results, proper numerical integration

scheme should be used.

For the cases of gZ0 or dZ0, one can leave those terms whose sm(q) are the same and

their contributions counteract each other, then the same procedure can be used.

5. Numerical results and discussions

The hydrodynamics of 3D flat planing surface is studied using the above method. The

wetted area is prescribed in advance, the pressure distribution and the transom profile is

determined by constant pressure element distribution method, in which the free surface

elevations at the control points are equal to the vertical coordinates of the local flat planing

surface.

The integrals in the induced coefficients expressions (31) and (32) are truncated at the

upper limits and calculated by using Simpson’s rule of integration. Since J1m usually

converges faster than J2m, the upper limits of integration of J2m should be larger than those

of J1m in the numerical calculations. Fig. 3 shows sample of the integrands after removing

the singularity at qZ0 in the integrals of the induced coefficients. In this example, the

integrand of J13 has a logarithmic singularity at uz4, as shown in Fig. 3a; the integration

around the singularity point will make major contributions to the integral. Therefore,

-0.60

-0.50

-0.40

-0.30

-0.20

-

0.00

0.10

0 2 4 6 8 10u

f1

m=8

m=3

-0.9

-0.6

-0.3

0.0

0.3

0.6

0.9

0 5 10 15 20u

f2

m=4

m=5

sample integrands in (31)

sample integrands in (30)

0.10

(a)

(b)

Fig. 3. Sample of integrands in the induced coefficients.


highly accurate scheme should be used. The integrand of J25 shows oscillatory decay

behaviour, as can be seen in Fig. 3b. A number of numerical tests have been carried out to

test the convergence of the integrals. It is found that the accuracy of the induced

coefficients plays a dominant role in the behaviour of the predicted pressure distributions.

If the induced coefficients (Dji in (13)) are not accurate enough, the pressure solution will

oscillate in a similar way to those found in previous studies (Tong, 1989; Doctors, 1975;

Wang, 2003; Tuck et al., 2002), even if the number of buttocks is three or four (not shown

here). The reason may be that the coefficients with low accuracy can damage the matrix of

unknown pressure equations and lead to erroneous solution. Table 1 shows an example of

the induced coefficients for a pressure element of b/aZ1.34, FrZU=ffiffiffiffigl

pZ9:04, where

lZ2a, a and b are half-length and beam of the element, respectively. A 3D view is also

shown in Fig. 4.

Numerical tests have also been carried out to study the effects of location of control

point on the solution of pressure distribution. It is found that placing the control point at

the centre of element is the best choice so far. The reason may be that the differences

between the flow patterns down stream and up stream of the pressure element are

significant due to the high element length based Froude Number (Fr), see Fig. 4. Another

reason may be that, in the present constant pressure distribution method, the ‘self-induced

coefficient’ is different from that of conventional source distribution method, in which the

self-induced coefficient is 2p (Newman, 1978). Fig. 5 shows an example of the self-

induced coefficients at the central line for a pressure element with length Froude Number

FrZ7.08 and b/aZ1.67, 1.00 and 0.56, respectively. It can be seen that the self-induced

coefficients vary significantly along the flow direction, especially for the element with

lower beam-length ratio. This corresponds to the cases in which higher number of buttocks

Table 1

Sample of the induced coefficients (b/aZ1.34, FrZ9.04, X, longitudinal distance from centre of the element

divided by length (2a); Y, lateral distance from the centre of element divided by beam (2b))

X Y

0 1 2 3 4

K7 K0.809781!10K1 0.267377!10K1 0.552860!10K2 0.246351!10K2 0.143168!10K2

K6 K0.697273!10K1 0.231254!10K1 0.481483!10K2 0.216792!10K2 0.126788!10K2

K5 K0.582469!10K1 0.194874!10K1 0.405869!10K2 0.185975!10K2 0.110599!10K2

K4 K0.465847!10K1 0.157250!10K1 0.338853!10K2 0.155573!10K2 0.952134!10K3

K3 K0.347012!10K1 0.119831!10K1 0.265249!10K2 0.128658!10K2 0.806927!10K3

K2 K0.227081!10K1 0.826154!10K2 0.201714!10K2 0.104593!10K2 0.676633!10K3

K1 K0.102581!10K1 0.485673!10K2 0.139680!10K2 0.829829!10K3 0.562147!10K3

0 0.367811!10K2 0.214768!10K2 0.988816!10K3 0.622272!10K3 0.464104!10K3

1 0.156091!10K2 0.944974!10K3 0.661455!10K3 0.476320!10K3 0.386360!10K3

2 0.758113!10K3 0.576489!10K3 0.459877!10K3 0.372690!10K3 0.322092!10K3

3 0.510338!10K3 0.374945!10K3 0.354668!10K3 0.312561!10K3 0.268896!10K3

4 0.417445!10K3 0.303526!10K3 0.240145!10K3 0.258015!10K3 0.226030!10K3

5 0.275688!10K3 0.231732!10K3 0.245897!10K3 0.207067!10K3 0.189273!10K3

6 0.170239!10K3 0.255575!10K3 0.182931!10K3 0.166518!10K3 0.161186!10K3

7 0.153567!10K3 0.182703!10K3 0.181568!10K3 0.149332!10K3 0.138721!10K3

Fig. 4. Three-dimensional view of the induced coefficients of a pressure element (X, longitudinal distance from

centre of the element divided by length (2a); Y, lateral distance from the centre of element divided by beam (2b),

fnZU=ffiffiffiffiffiffiffiffi2ag

p).


is employed. If the control points are not properly selected, the pressure solution could be

incorrect, even oscillatory. Fig. 6 shows a 3D view of the self-induced coefficients for an

element of b/aZ1.00 at FrZ7.08.

Fig. 7 shows comparison of pressure coefficients at the central line of a flat planing plate

of B/LZ0.5, FrZ2.24, where in the present method, nine buttocks were used (NyZ9). The

pressure distribution is free of oscillation and in good agreement with other researchers. It

can be seen that the present results have a better agreement with Cheng and Wellicome’s

and Wang and Rispin’s results than Tong’s, who also used the constant pressure element

distributions. The reason may be due to Tong’s number of buttocks is too few (less than 5).

Fig. 8 shows pressure distribution at central line for B/LZ1.0 and vessel Froude Number

FrZ2.24, the number of buttocks are 5 and 7, respectively, the present results are in

-0.020

-0.015

-0.010

-

0.000

0.005

0.010

-0.5 -0.3 -0.1 0.1 0.3 0.5X

D

b/a=1.67

b/a=1.00

b/a=0.56

0.005

fn=7.08

Fig. 5. Sample of pressure element self-induced coefficients at central line (X, longitudinal distance from centre of

the element divided by length (2a).

Fig. 6. Sample of self-induced coefficients of a element (X, longitudinal distance from centre of the element

divided by 2a, Y, lateral distance from the centre of the element divided by 2b).


good agreement with Wang et al.’s results and do not show any oscillation. Figs. 9 and 10

show pressure distribution for flat planing plate of B/LZ1.2 and FrZ1.5 for different

number of buttocks. The number of pressure strips is NxZ11. The convergence of the

present method can be seen from these figures. Fig. 11 is the three-dimensional

configuration of the pressure distribution on a flat planing hull of B/LZ1.2 and FrZ1.5.

The pressure distributions on the entire region do not show any oscillation, which validates

the present numerical method. Fig. 12 shows the calculated transom profile for B/LZ0.5,

FrZ2.24.

0.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0

Cp

Tong

Cheng&Wellicome

present

Wang&Rispin

B/L=0.5, Fn=2.24

x/L

Fig. 7. Central line pressure distribution (B/LZ0.5, FrZ2.24).

0.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8

Wang&Rispin

present, Ny=7present, Ny=5

X/L

CP

Fig. 8. Central line pressure distribution (B/LZ1.0, FrZ2.24).


6. Concluding remark

In the present paper, hydrodynamics of three-dimensional planing surface is studied

within linear potential theory. The planing surface is represented by a number of pressure

elements whose strengths are constant. The induced free surface elevations under

the planing surface is set to be the vertical coordinates of the planing surface, and also a

zero pressure condition, i.e. Kutta condition is applied at the trailing-edge of the planing

surface. A numerical method is proposed for the evaluation of the influence coefficient

matrix. The predicted pressure distributions using the present method are in good

agreement with other researchers’, which used different approaches. One of the important

findings of the present study is that the pressure distribution will not show oscillatory

behaviour using the constant pressure distribution method when the induced coefficients

0

1

2

3

4

5

6

-0.5 -0.3 -0.1 0.1 0.3 0.5y/B

Cp

Ny=11

Ny=9

Ny=7

Ny=5

Ny=3

B/L=1.2, Fn=1.5, Nx=11

Fig. 9. Pressure distribution at ixZ11 on flat planing hull (B/LZ1.2, FrZ1.50).

Fig. 11. Three-dimensional configuration of pressure distribution on flat planing hull (B/LZ1.2, FrZ1.50) (CLZ0.988, LCPZ0.721).

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.2 0.4 0.6 0.8 1.0x/L

CpB/L=1.2, Fn=1.5, Nx=11

4#, Ny=5

2#, Ny=9

1#, Ny=11

Fig. 10. Pressure distribution at central line of a flat planing hull (B/LZ1.2, FrZ1.50).


B/L=0.5, Fn=2.24

–2.0

–1.5

–1.0

–0.5

0.0–0.5 –0.3 –0.1 0.1 0.3 0.5y/B

H /

[B

tan

(tao

)]

Fig. 12. Predicted trailing-edge profile for flat planing hull (B/LZ05, FrZ2.24).


are evaluated with high accuracy. The irregularity of the pressure distribution in the

previous studies using the same method is most likely due to the inaccuracies in the

calculation of the pressure element induced coefficients, not the method itself.

Acknowledgements

The present study is partially supported by the 5th Framework EU project EFFISES, the

EU funding is grateful. The authors also thank Dr. X.M. Cheng and Professor L. Doctors

for their valuable discussions and reviewing on the manuscript.

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A study of hydrodynamics of three-dimensional planing surface