A theoretical prediction of the effect of a wall on the resistance of a fast ship shape in water of uniform depth

278

1. Introduction

There has been considerable interest in the effects of shallow and restricted water on ships over the years, particularly concerned with large ships moving at mo-dest speeds. A great deal of the more recent work has been concerned with the 'squat' phenomenon, such as for example Ref. 1, while other work, both theoretical and experimental, has studied the interaction between

the ship and a bank (Ref. 2) or between two ships passing (Refs. 3 and 4). Less attention has been paid to the effects on resistance of ships in restricted water be-cause they are likely to be moving slowly under such conditions and will usually therefore have adequate power available, although a method of estimating the added resistance, should this be needed, has been given by Schlicting and can be found in Ref. 5.

Since these larger ships are not moving very fast in restricted waters then they can be classified as moving at low sub-critical speeds in terms of the critical wave speed for a particular depth of water. This is given by the equation

V=(gd)½

where V is the speed of the critical wave, g is the gravi-tational acceleration and d is the depth of water. How-ever, many smaller ships, such as those which may be used as fast patrol boats or for similar duties, can

achieve speeds which may equal or even exceed the critical speed for a particular depth of water and are thus moving at high sub-critical or super-critical speeds. It therefore becomes important to be able to predict what effect may occur in different water depths and in restricted water at these higher speeds since there is evidence to show that the effects may be much larger than for the low sub-critical speeds.

Theoretical work which could be applied to these higher speeds was developed by Srettensky (Ref. 6)

for both shallow water and a channel and later by

Lunde (Ref. 7), Kirsch (Ref. 8) and others although

there has been little or no comparison with

experi-X) _{Department of Mechanical Engineering, The University of Liverpool,}

England.

A THEORETICAL PREDICTION OF THE EFFECT OF A WALL ON THE RESISTANCE OF A FAST SHIP SHAPE IN WATER OF UNIFORM

'*1ÑIsCHE UNIVERSITEIT

by _{Laboratorium voor}

A.J. Riche, J.L. Sproston* and A. Millward* Scheepshydromechardca Archief

Mekeiweg 2, 2628 CD Deift

Summary TeL: 015- 786873 - Faic 015-781838 A theoretical analysis has been developed to predict the resistance of a simplified ship shape in water of uniform restricted depth near a wall at high sub-critical and super-critical speeds. The results ofnumerical

cal-culations have been compared with experimental data for a fast round bilge displacement hull and have shown

good agreement.

mental data. On the experimental side some data was available on round bilge hulls (Refs. 9 and IO) and for one planing hull (Ref. 11). More recently these existing theories have been used to make comparisons with the available experimental data (Refs. 12 and 13) and with

newer data (Ref. 14) for shallow water.

It was noted however that no theoretical work was available for the particular case of a fast ship near a vertical bank or wall in shallow water. The present paper therefore develops the theory for a ship at high sub-critical and super-critical speeds in shallow water near a wall and includes a comparison with some new-ly available experimental data.

2. Theory

The analysis given below is concerned with the deri-vation of a simplified hull form moving parallel to a vertical wall in water of uniform restricted depth. The method is based on the work of Lunde (Ref. 7) and the following assumptions have been made:

The flow is inviscid, incompressible and irrotational

Wave heights at the free surface (z = O) are small compared with the wave lengths.

The motion is steady.

The flow can be represented to first order accuracy by a distribution of sources and sinks.

The velocity potential for a simple source of strength m in the proximity of a wall, at a position y = b, is given by (Ref. 15),

m m7r

= =

_{- f do f exp (K(fw - (z +f))}

_{}dK (1)}

r1 2ir o

together with its reflection in the wall,

m m

IT

f do f exp {K(iw1.(z+f))}dK (2)

r1 2ir-ir O where 2 2

r1x +y2+(z+f)2

x2+(y+2b)2+(z+f)2

= xcosü+ysino

wr

xcose+(y+2b)sino

and the velocity potential due to the presence of the solid boundary at a depth z = - d is similarly given by

m in

2

¡do

.Fexp K(iw1 (z+2d--[))}dK

where s represents the area containing the sources. For the case of a distribution involving variations in the x-direction only, then if all the sources are assumed to be on y = 0,

m

_{f a(h,0,f)ds.}

It is shown in Ref. 7 that the force experienced by a source strength m at a point in the fluid at which the velocity due to all other sources is u' is given by F = 4iTpmu'. Thus for a continuous distribution of sources

over a region of the xy plane

R = 4irp f f o'(h', 0,f')u' dh' df' (7)

where R is the wave resistance experienced by the body represented by u arid where u' isthex-component of fluid velocity of a source placed at the point (h', 0, f'). Thus substitution for u' by taking the derivative of

Eqn. 6 and integrating over the whole distribution gives

R = - 4pi ff o dxdz ¡f o'dhdf cosO do f

IT O

Kcosh(K(df))

[K+K0sec2O +ijisecû] cosh(Kd) [K - K0 sec2 O tanh(Kd) + i,tz secO I X cosh [K(d+z)] (1 +e2iKl sino )eK - d)dK

(8)

where now w = (xh) cosO +ysinû Defining N by

N= .7 cosO dûf

_ir

O

Kcosh(K(df))[K+K0sec2o +ip seco] (1 +e2bsm0)

cosh(Kd)[KK0sec2û tanh(Kd)+iMsecû]

X +ysino])dK ₍₉₎

it follows that the singularities of the integrand in Eqn. 9 are determined by the sign of secO in the denomin-ator so that it is necessary to limit the range of inte-gration in O to (0, ii/2) to ensure secO > 0.

For this purpose let (i = I - 4) be defined by R) =

(R + K0 sec2 O- Ip sec8)( I +e' ,lflO_)cos8 (xdi)cosG *yimO ea cosh(Rd)(k- K0sec2O tanh(kd)-ip secO)

(10) =

(Î + K0 sec2 -. Ip secO)(1+ 51116 )cosü (xl)cosO yiin8 I -Rd

cosh(kd)(R - K0sec2 O tanh(Rd) + Ip secO)

X

= -= - f dû f ex

_r2,

2ir -

p{K(iw1.(z+2df)) }dK

m m

In Ref. 7 for the case of water restricted in depth only, the expressions 4 and ₂ are required. A third potential function can be written, namely,

4)3 = F(û.K)cosh(K(z+d))exp(iKc1)

which satisfies the condition of zero normal velocity at the sea bed.

By substituting the sum of ₂ and 4)3 into the surface boundary condition

4)

+K4)

=0 where K0 "g/v2 (5)

an expression for F is determined. In order to keep in-tegrals determinate a frictional force, assumed to be proportional to velocity, is introduced which keeps the motion irrotational and leads to a solution involv-ing only those waves trailinvolv-ing aft. It can be assumed then that 4)3 now represents that infinite set of images of and ₂ due to the introduction of the free sur-face.

In the present case with the addition of a wall, a further potential function 4)3 representing the image of the above infinite set in the wall will need to be in-cluded, identical with 4). though with y replaced by (y + 2b). Combining Eqns. I to 4 together with and 4) obtained from Eqn. 5, the total 4)7 is

= + 2 + 4)3 + + 2 + 4)3.

=+++--

_{r1 r2 r1}

m cosh(K(df))[K+K0sec2û +iM seco]

-

L1

cosh( d)[KK0sec2O tanh(Kd)+iz secO X

X cosh[K(z+d)] (I +eKb50)eFK(_0')] do dK

(6) where

w = xcosû +ysinû

For the situation where the sources are a continuous distribution of density u, in must be replaced by the area integral

3(o,K) =

( +K0sec2O +zp sect )(1+e2be)csOeR[x.h)coe*ysrn8)_Îd

cosh(dXR - K0sec2 O tanh(Îd) + ip sect)

4(8,K) =

( +K sec2O - sect)( i + 2kb5mO )cost e [.x'h) coso +ystnoI

cosh(k'd)(Î - K0sec2 O tanh(d) - ¡psecO)

(13) where k is complex. Now the expression for N given in Eqn. 9 can be written as

A .. A

N

= f

_oc

KcoshK(df)coshK(d+z)

.(1+2+3+4)dRdO

(14) where imaginary parts are to be taken, and C is a suit-able contour.

After some algebraic manipulation, shown in detail in the Appendix. the expression forR (Eqo. 8) becomes R = - 8ir p f

(f2 +J2 )Kcos28K0sec28(1 + tarih(Kd))( i + cos(2Kb sinû))e Kd

cosh(Kd) K sinO where K0tanh(Kd) cosO = K

KK0tanh(Kd)

sinO -K

and where I. J are defined in the Appendix. Therefore,

R = - 8irpK0 7 j2 +J2)K2(1 +cos(2KbsinO))dK

0 _{cosh2 (Kd)(K - K0 tanh(Kd))1'}

(16) 3. Representation of a hull

In order to evaluate Eqn. 16 for the resistance a

suitable expression for the hull shape must be given. It was decided to use a simple representation of a hull, in the first instance, in order to obtain numerical values for the effect of a wall in water of uniform

depth and to make a comparison with some available experimental data (Ref. 14) for a fast round bilge dis-placement hull. The simplified hull shape chosen had parabolic water lines and parabolic cross-sections as shown in Figure 1 since earlier work (e.g. Ref. 13) had shown that there was reasonable agreement between the theory for this simplified hull shape and

experi-mental data for a round bilge displacement hull in shallow water of unrestricted width.

(12) dK (15) 5 6 35 3 2i 2 11 ½ is given by

-v ay

2ir ax where JK0tanh(Kd) C = (I V K

The coordinates of the hull are expressed in a non-di-mensional form by putting

x=!± zT;

y=

(20)

so that ay B an

òx L a

where the dimensionless equation for the simplified hull shape is

(1 _2)(J

.2) ₍₂₁₎ and

2(1

2) (22) Hence

I=i

2ir

i

(l (23)

uIIu-ai

L/B=8OO

T/L=OO5

Figure 1. Body sections of the mathematical hull form.

For convenience the hull is brought to rest by im-posing a velocity - y to the whole system, where V is

the forward velocity of the hull relative to a fixed co-ordinate system.

From Ref. 7 the source distribution of a hull of the form

y = N(x,z)

(17)

T

t-(18) In the expression for the resistance J O for a hull which is symmetrical fore and aft about the mid-ships section so that it is necessary only to evaluate the in-tegral

J = 5f ocoshK(dz)sin(Kxc)dxdz

(19)

9i9 BI8 73 7 6 5

B 2

After carrying out the integration it follows that

12 4a2B2

12

sinj- 1

1KLc _cos (KLc\ 12

K4L22c2lKLc \

2 J

(1 2

sinh(Kd)+----coshK(dT)+

\ K2T2! KT 2 2 + sinh(K(d - t)) (24) K2 r2

If the wave resistance in a finite depth of water near a wall is expressed in a non-dimensional form, following that suggested by Weinblum (Ref. 16), then

Rj,d =R Hence

R,d

7 (1 +cos(2Kbs)) LT2 K7'°= c2 cosh2 (Kd) [K- K0 tanh(Kd)Jt'z 2

smI .1 cos -

X J 2 ¡KLc\ KLc

2)

2jÇ

'<

j(l

2

'\sinh(Kd)+-- coshK(dT)+

K2 T2) KT + 2 sinh (K(d - T)) J2 dK (26) K2T2 where ., , 8pgBT Lir (25)

al about the mid-ship section and had parabolic water-lines and also parabolic cross-sections. The results of these calculations have been expressed as the ratio of the wave resistance in shallow water near a wall to the resistance in deep water of unrestricted width Rd/R and have been plotted against Froude number F,.

Walt offset b/L =035 5 s

t

4 o n L n 3 o V) V) w 2 n 0 1 2 Fraude number _,h

Figure 2. The variation of wave resistance ratio with water

depth and Froude number at one distance from the wall (b/L = 0.35). Wail offset b/L =058 ,- L/d=6 L/d=5 L/d=4 0 1 2 Fraude number Fflh

Figure 3. Tha variation of wave resistance ratio with water

depth and Froude number at one distance from the wall (b/L

0.58).

KK0 tanh(Kd)

s=

K

and

O if K0d < I or the positive root of K K0tanh(Kd) if K0d> 1

and the expression for the resistance can be evaluated numerically after evaluation near the lower limit using an analytical approximation. The corresponding equa-tions for the wave resistance in deep water followed those given by Kirsch (Ref. 8) which are based on the work of Michell (Ref. 1 7). The wave resistance in deep water of unrestricted width is:

B2T2

R0,=-pg

L _{o - 11½}

Jdy

where

lia

J = ff__e_0tsin(y)d.d

o o a and L 4. Discussion of results

Calculations were made to obtain the wave resist-ance of the simplified hull shape which was

symmetric-n82

5 Wall offset b/L=081 L/d =6 5 0 1 2 Froude number Fflh

Figure 4. The variation of wave resistance ratio with water

depth and Froude number at one distance from the wall (b/L = 0.81).

based on the undisturbed depth of water d. The graphs obtained are shown in Figures 2 to 4 for three different wall distances expressed as the ratio of the wall dis-tance to the hull length b/L. Each figure shows three curves corresponding to different depths of water ex-pressed as the ratio of hull length to water depth ratio

(Lid) of 4, 5 and 6. The results show the general

pattern obtained in previous work in shallow water of unrestricted width that there is an increase in the

re-b/L =035

b/L=058

b/L=081

Depth ratio L/d =4

sistance ratio at high sub-critical Froude numbers and a reduction in resistance ratio at super-critical Froude numbers. The diagrams illustrate clearly that both the increase in resistance ratio at sub-critical speeds and the reduction at super-critical speeds are a function of

hull length/water depth ratio Lid - the effect of the

shallow water becoming more pronounced as the

8 -= o n L. 5 4 C o V, Q)

5)2

> o b/L= 035 b/ L= 0 58 b/L= 081 Depth ratio L/d=5 b/L= 035 b/L =0 58 b/ L= 081

Depth ratio Lid = 6

Fraude number Fflh

Figure 6. The variation of wave resistance ratio with distance from a wall and Froude number in shallow water (L/d = 5).

1 2

Froude number Fflh

Figure 5. The variation of wave resistance ratio with distance Figure 7. The variation of wave resistance ratio with distance from a wall and Froude number in shallow water (L/d = 4). from a wall and Froude number in shallow water (Lid = 6).

o 1 2 Froude number Fflb O 2 5 8 4 o n L.

5)3

C o V, V, Q)

water depth decreases. It can also be seen that at higher Froude numbers the resistance ratios tend to one

showing that the wave resistance at very high super-critical speeds is effectively independent of water

depth.

A comparison of Figures 2 to 4, which are for wall offset ratios (b/L) of 0.35, 0.58 and 0.81, shows that as the hull is brought nearer to the wall both the in-crease in the resistance ratio at sub-critical speeds and the decrease at super-critical speeds become greater in magnitude. An alternative form of presentation is

given in Figures 5 to 7 where each diagram gives curves of resistance ratio at the three different wall offset

distances for one of the water depth ratios (L/d) se-parately. These diagrams show that the effect of prox-imity to the wall is to exaggerate the normal effect of shallow water both at sub-critical and super-critical speeds.

Figures 8 to 10 reproduce some experimental re-sults, which can be compared with Figures 2 to 4, for a round bilge hull (Model lOUA of the NPL) round bilge hull series, shown in Figure 11. measured in a towing tank (Ref. 14) for the same three wall offset distances. The curves, which give the ratio of residual resistance in shallow to deep water, are similar to the theoretical wave resistance curves with the main dif-ferences being in the region of the sub-critical resist-ance peak. Here the theory tends to predict a higher magnitude for the shallowest water depth (Lid = 6) and a lower magnitude of the peak for the other two water depths (Lid = 4 and 5). Away from the

resist-6

o

Walt offset biLO'35 -Lfd=6

L/dS

2

Froude number Fflh

Figure 8. The variation of residual resistance ratio with water

depth and Froude number at a distance from a wail (b/L 0.35).

(V 0 u,

ail

s 5-1 ₂ Froude number

Figure 9. The variation of residual resistance ratio with water

depth and Froude number at a distance from a wall (b/L = 0.58).

L/d =4

Walt offset biL 081 Lid 6 Wall offset b/L=058 L íd = 6 L/d=5 0 1 ₂ Froude number Fflh

Figure 10. The variation of residual resistance ratio with water depth and Froude number at a distance from a wall(b/L = 0.81).

284 C £

a

01 02 03 04 «6 08 10 20 30 'O «0 «0 ib WaU offset b/L

Figure 12. A comparison of the variation of maximum resist-ance ratio with distresist-ance from a wall in shallow water obtained from theoiy and experiment.

ance peak the agreement between the theory and the experimental data is much closer and shows the same trend for the effect of proximity to the wall.

The effect of distance from the wall is illustrated in a different way in Figure 12 which shows the maxi-mum resistance _{ratio, measured at the sub-critical}

resistance peak, plotted for each water depth ratio (Lid) against distance from the wall both for the

theoretical and experimental data noting that the in-formation for the wall offset value b/L 1.6 in the experiments was taken from results obtained in the centre of the towing tank where the model was equi-distant from two banks and is therefore not strictly

comparable although it seems likely that any error

caused by the second wall is likely to be small. It can be seen that the curves for the theoretical and experi-mental results have a similar shape indicating that the effect of the wall decreases very rapidly as the distance from the wall gets bigger and if the hull is more than approximately 1.5

hull lengths from the wall the

effect is negligible.

The calculations were extended to determine the effect of a wall on the same theoretical hull shape but in deep water. The results are shown in Figure 13 for

Theory L/d Experiment

.6

a

.5.

.4.

the same wall offset distances as the wave resistance divided by the hull weight plotted against the Froude number Fv based on the displacement. It can be seen that the effect of the wall is quite small, particularly when compared with the shallow water case close to the wall, and only results in a slight increase in the

resistance in the middle of the speed range, near a

Froude number of 1.5 approximately, for the closest wall offset (b/L = 0.35). There is also a very small re-duction in resistance shown for speeds above a Froude number of 2 although this is so small that it is unlikely to be of any practical significance. The corresponding curves for the measured residual resistance, again plotted as a ratio of the hull weight, against Froude number Ev, are shown in Figure 14 which is also taken from Ref. 14. It can be seen that there is good agree-ment with a similar result being measured in the expe-riments where the effect of the wall in deep water was also much less than in shallow water and also showed a small increase in residual resistance for the closest wall offset distance over the middle of the speedrange.

sa

0 1 2

Fraude number F5

Figure 14. The effect of distance from a wall on the residual

resistance in deep water.

The results shown in the present work can also be considered as _{an extension of the work given by}

Everest (Ref. 18) for a catamaran if the wall is assumed to act in the same manner as the centre-plane between a pair of hulls. Everest's work was for Froude numbers less than 0.5 approximately and for hull separations corresponding to wall offset distances (b/L) of less than 0.4 compared with Froude numbers greater than 0.3 and wall offset distances greater than 0.35 so that there is little overlap. The present work has shown a

more consistent pattern for the faster speeds and larger wall offset distances and none of the complicat-ed interaction effects obtaincomplicat-ed by Everest. This

sugg-ests that, although the spacing of catamaran hulls _can be optimised for a particular speed at the lower Froude

100 75 al a. Deep water -... w SO A b/L = Sa _{081 a} £ O 58 w 25 £ 035 100 75

I

a. a'

It.

50

I

_{Deep water} J' s k ₀₈₁b/L a 25 _$ A 0 58 035 .A'p 0 1 2 3 Fraude number F5

Figure 13. The effect of distance from a wall on the wave

re-sistance in deep water.

e

'V

'-4

a'

numbers, at the higher Froude numbers covered by the present work the effect of the spacing of the hulls follows a simpler pattern indicating that the wider the spacing the lower the added resistance, particularly in shallow water.

As has been suggested in previous work on the effect of shallow water on resistance it is thought that the differences between the theoretical and experimental results can be attributed to a number of factors:

1. The simplified hull form used in the theoretical cal-culations when compared with the real round bilge hull form used in the experiments.

The use of linearised wave theory which assumes small wave heights compared with wave lengths. This assumption is likely to be most in error in the region of the critical speed in shallow water where the major differences between theory and experi-ment have been observed.

The experimental results have shown that there is

a large change in both the trim and heave of the model hull in the region of the sub-critical resistance peak whereas the theory assumes that the hull is fixed in heave and trim.

The residual resistances obtained from the experi-ments, which have been compared with the theore-tical wave resistances, were deduced by subtracting the calculated frictional resistance from the mea-sured model resistance. This standard procedure makes use of the International Towing Tank Confe-rence formula (Ref. 5) to calculate the frictional resistance and is really intended for use in

unre-stricted water. It may well therefore be less accur-ate in restricted waccur-ater.

Even with these differences as listed above the agree-ment between the theory which has been developed in the present paper and the available experimental data is quite close and the theory could therefore be used

to estimate the effect of proximity to a wall on the

resistance of a ship in shallow water at high sub-criti-cal and super-critisub-criti-cal speeds with reasonable accuracy.

Conclusions

A theory has been developed using first order

lin-earised ship wave theory in order to calculate the effect of proximity to a wall on the resistance ofa simplified hull in shallow water at high sub-critical and super-critical speeds. The effect of the wall be-comes greater as the hull is brought closer to the wall. The theory has also been used to calculate the ef-fect of a wall on the sanie simplified hull shape in deep water. The results have shown that the effect of the

wall is much smaller than in shallow water but

re-sults in an increase in resistance over the middle part of the Froude number range.

The theoretical results have been compared with experimental data for a round bilge hull in both cases and have shown good agreement.

Acknowledgment

The authors wish to acknowledge the award of a

grant from the

Science and Engineering Research Council (Marine Technology Directorate) through Marinetech North West.

Notation

B beam of hull

depth Froude number

F7 displacement Froude number v/..,,/gV 1/3 K0 g/v2

L length of hull

R wave or residual resistance T draft of hull

W _{displacement (weight) of hull} b distance of wall from axis of hull

d water depth

f

depth of simple source - z-coordinate g acceleration due to gravity

h fore/aft position of source - x-coordinate

m strength of simple source r1, r2 moduli of position vectors s surface area

y velocity of hull

x,y,z

Cartesian coordinates (z positive up)

V displacement (volume) of hull

velocity potential

z/T non-dimensional z-coordinate

,(

, ) y/b/2 non-dimensional y-coordinate ji _{added friction parameter in equation (5)}

x/L/2 non-dimensional x-coordinate

p water density

u source density

integration from aft to forward Suffixes

R residual (resistance) W wave (resistance) d finite water depth

oo _{infinite water depth}

References

Dand, l.W. and Ferguson, A.M., 'The squat of full ships in shallow water', Transactions R.I.N.A., Vol. 115, pp. 237. 255, 1973.

Dand, I.W., 'On ship-bank interaction', Transactions R.I.N.A., Vol. 129, pp. 25-40, 1982.

Dand, I.W.. 'Ship-ship interaction in shallow water'. Proc.

11th Symposium Naval Hydrodynamics, pp. 637-653, 1976.

Tuck, E.O. and Newman, J.N., 'Hydrodynarnic interaction between ships', Proc. 10th Symposium Naval Hydrodynam-ics, pp. 35.70, 1974.

Comstock, J.P. (Ed.), 'Principles of Naval Architecture', S.N.A.M.E., 1967.

Srettensky, L.N., 'A theoretical investigation of wave resistance', (in Russian), Joukovsky Central Institute for

Aero-hydrodynamics, Report 317, 1937.

Lunde, J.K., 'On the linearised theory of wave resistance for displacement ships in steady and accelerated motion',

Trans. S.NA.M.E., Vol. 59, pp. 25-85 1951.

Kirsch, M., 'Shallow water and channel effects on wave resistance', Journal Ship Research, Vol. 10, pp. 164-181,

1966.

Sturtzel, W. and Graff, W., 'Systematic investigations of small ship shapes in shallow water in the sub- and

super-critical ranges', (in German). Forsch.Ber. Wirt. und Verk. Minist. N-Rhein-Westf., No. 617, 1958.

Sturtzel, W. and Graff, W., 'Investigation into the develop.

ment of optimum round bilge boat forms', (in German).

Forsch. Ber. Lands N-Rhein-Westf., No. 1137, 1963.

A1 = - acosO - IlsinO A2 = acosO 3sinO A3 = acosO + ¡lsinO A4 = - acosO - ilsin0 and

= tan(a/ß)

then the sign of A. for the various cases are given in the

following:

a>0

a<0

0877 i7O7r/2

t6ir/2

A1 <0

<0

>0

<0

A2 >0

<0

<0

>0

A3 >0

>0

<0

>0

A4 <0

>0

>0

>0

Appendix For the integration of the in equation (14) it is necessary on the sign of the argument in each of the exponential the original sources lie in the x, y plane) and also writing

Il. Toro, AI., 'Shallow water performance of a planing boat',

University Michigan, Dept. Naval. Architecture and Marine Engineering, Report 019, 1969.

Millward, A., 'The effect of shallow water on the resistance of a ship at high sub-critical und super-critical speeds',

Transactions R.LN.A., Vol. 124, pp. 175-181, 1982.

Millward, A., 'Shallow water and channel effects on ship

wave resistance at high sub-critical and super-critical

speeds', Transactions R.I.N.A., Vol. 125, pp. 163-170, 1983.

Millward, A. and Bevan, M.G., 'The behaviour of high speed

ship forms when operating in water restricted by a solid boundary', RINA. Paper W2, 1985.

Courant, R. and Hubert, D., 'Methods of mathematical

physics', Vol. II,Wiley and Sons, 1962.

Weinblum, G., 'Ein Verfahren zur Auswerting des Wellen-widerstandes vereinfachter Schiffsformen', Schiffstechnik, Vol. 3,pp. 278-287, 1956.

Michel, J.H., 'The wave resistance of a ship', Phil. Mag.,

Vol. 45,pp. 106-123, 1898.

Everest, J.T., 'Some research on the hydrodynamics of

catamarans and multi-hulled vessels in calm water', Trans-actions North East Coast Inst. Eng. Shipbuilders, Vol. 84, pp. 129.148, 1968.

to employ suitable contours. Figure 15, dependent factors. Writing a = (x - h), ß = 2b andy = O (since all of

(i') (iv)

Figure 15. Contours of integration in the K-plane for evaluation of Eqn. 14.

Using the above information and choosing an appropriate contour from Figure 15, the sum of the lì becomes, for a> O

ir/2 Am

Am

>Q =

f

f çte dmdû +

f

e 2 dmdü -f icos(KA2)dO A m 77/2 A m , A4m

- f

j e 2 _{drndû + f}

f qe

dmdû + f f Øe dmdo 770

00

00

ir/2 ir/2 mA

- f

liicos(KA)dû - f f qe

4dmdß 17 770 and for a < O -i

Am

ir/2 Am =

-

_{f'7 L, cos(A1 K) do - f f e dmdü + f f Øe} drndû 0

00

-110 A,m 77 77/2 n/2

-

A3K)dO

-

J I'cos(A,K) dû - f f e

' dmdû - f

icos( o -

00

0 -i Am 17/2

Am

T

- f f

e _{dmdû + f f} e dmdû

_J2

cos(A4K)dO

00

-710 11/2 mA

-f fØe

4dmdû o o where

m cosøcosm(d -f) cosm(d + z) (K0 sec2û cos(md) + msin(md)) cos(md) (m - K0sec2O tan(md))

and

2rr(K+K0sec2O)Kcoso coshK(dfl coshK(d+z)e

cosh(Kd)(1 K0dsec2O sech2(Kd))

Noting that A1 = A3 A2 = A4 and that if, for a < O (x - h) is replaced by - (x - h) so that only a> O need be considered then also (A1 _a>O = (A2 )1<O , (A2 = (A1 )a<O Then, if the terms in the above are added, their

sum is zero. This leaves the terms in i to be considered. Taking the first term in and expanding the cosine

gives

11/2

Vicos( KA2) dO= ir/2 ' [cos(Ka coso) cos(K3sinO)+ sin(Ka cosO) sin(K3sinO)] dû

Consider now the x, h integrals in the expression for R (equation (8) denoting by F, A the forward and after limits; then, since a> O

1= a dx a dh 17/2 [cos(Ka cosO) cos(Kßsinû)+ sin(Ka cosO) sin(KßsinO) do

A A 77

F F 17/2

= f a dh f a dx f

i [cos(Ka cosO) cos(Kßsinû) + sin(Ka cosO) sin(Kßsinû)} do

A h 17

In the last expression interchange x and h throughout

=!

o dx o dh 11/2 [cos(Ka cosA) cos(K(lsinû) - sin(Kcr cosA) sin(KsinO)] do

A A 77

Now adding and halving the sum of the first and last expression gives I, independent of the sign of a so

F F 71/2

1 f a dx f a dh f

' [cos(Ka cosA) cos(Kßsinû)} do

2A A 77

Performing a similar operation with the remaining expressions in i gives a total 'T of

F F 17/2

I

_T

2f odxf adh f

4[cos(Kacosû)cos(Kßsinû)] dû

A A o

Now the expression for the resistance can be written down as (remembering to add a similar expression for b = O) 17/2 Kcosû(K +K sec2û) coshK(d - f) coshK(d + z) cos[K(x - h) cosO]

R=l6pirfadxdzfadhdf f

x

S S ° _{cosh(Kd) [I K0dsec2O sech2(Kd)]}

Now expand the cosine term

cos[K(x - h) cosO] = cos(Kxcosû) cos(Khcosû) + sin(Kxcosû) sin(Khcosû) Then

ir/2 (J2 +J2) Kcosû(K+K sec2û) (I + cos(2Kbsinû)) e_K2

R=l6irp f

dû

cosh(Kd) (1 - Kdsec2û sech2(Kd)) where

J

=

ff

cTcos(Kx coso) cosh(K(d + z)) dxdz asin(Kx coso) cosh(K(d + z)) dxdz

In the above theory the singularity occurred at that value of K for which

K - K0 sec2 û tanh( Kd) = O

If K0 d> I then this has a positive root for K If K0d < 1 then K = Ois the root.

The above expression forR can now be converted to an integral over K. From above: dK - 2K0sec2û tanû tanh(Kd) do - K0dsec2û sech2(Kd) dK O

(I - K0dsec2û sech2(Kd)) dK = 2K0sec2û tanû tanh(Kd) dû Thus,

(J2 +J2) Kcosû(K+K sec2û) (1+ cos(2Kbsjnû)) e_'

RI6irp f

dK