The forces on an aircushion vehicle executing an unsteady motion

ARCHIEF

ab.

_{Technische Hogeschool}

vt

c eeps ouw

unde

School of Mechanical and Industrial Engineering

Deift

University of New South Wales

Sydney, Australia

May

1972

DOC UMtN I 1IE

c4'6Q

TI FORCES

ON AN AIRCUSHION VEHICLE

CUTING AN

UNSTEADY MOTION

by

Lawrence J. Doctors

Prepared for Presentation at the Ninth Symposium on Naval Hydrodynamics

Paris, France August

1972

:jbluotheek van d

Onderafdeinr

r'-

-epsbouwkunde

sc e kogeschoo,

DOCUMEN FATIE

22

DATUM:

a etcr.

i

ABSTRACT

This paper treats the theoretical problem of an air-cushion vehicle

(ACV) travelling over water of uniform ±inite or infinite depth, with an arbitrary motion. The ACV is modelled by a pressure distribution applied to the free surface of an inviscid. incompressible fluid and the boundary conditions on the free surface are linearized.

Numerical results are presented firstly for accelerated motion from

rest. In deep water, the hump condition is delayed to a higher Froude

number, while in finite depth, the hump resistance is appreciably

reduced - even by moderate levels of acceleration.

The effect of tank walls on these results when carrying out mode].

tests is next examined. The side walls of the tank alter the resistance by less than one per cent in accelerated motion, if the tank width is greater than four times the model beam.

Finally, calculations of the side force acting on a yawing ACV are

presented. For super-hump speeds, the side force is of the same order as the wave resistance and. favorably aids the turn. It is also shown that the steady-state forces are realized when the craft has travelled 'a distance of less than two vehicle lengths after a manoeuver.

a = half length of craft 1/2F2

half beam of craft

width of tank

velocity of craft

water depth

Froude number =

depth Froude number = c/V acceleration due to gravity

circular wave number = +

_u2

fundamental wave number =

g/c2

length of tank

indices for lonitudinal and transverse wavenubers in a tank

cushion pressure measured in the moving reference frame xyz = cushion pressure measured in the stationary frame

jz

p0

= nominal cushion pressure

P,Q = functions defined by Eq. (26) or (148)

wave resistance

wave resistance coefficient defined by Eq. (35) distance travelled by craft

induced side force, or area of pressure distribution

side force coefficient, analogous to R

time

induced longitudinal and transverse wavenumbers

R = R s = S = Sc = t = w,u = A = b = B = = d = F = Fd = g = k = = L = n,m = p =

W weight of craft

x,y,z = coordinate system travelling with craft, but not rotating with it

= longitudinal cushion pressure fall-off parameter

= transverse cushion pressure fall-off parameter

= /gk.tarth(kd)

= yaw angle of vehicle relative to x axis (see Eq. (147))

= free surface elevation

e = wave direction with respect to the x axis

= longitudinal coordinate in the stationary reference frame p = water density

= initial position of model in the tank

-r = dummy time variable

= velocity potential in the stationary reference frame, such that

the velocity is its positive derivative

SUPERSCRIPTS

*

= variable referred to axis system moving and rotating with craft I _{= dummy variable}

BACKGROUIW

Havelock

(1909, 1914

and

1926)

was the first to study the theoretical problem of the wave resistance of a pressure distribution. His inter-est lay in a desire .to represent the disturbance from a ship. As a re-sult, the pressure distributions that he chose to analyze were very smooth and were not typical of the pressure underneath an air-cushion vehicle

(A cv). However, later on, Havelock (1932) derived the genera). expression for a pressure distribution travelling at a constant speed of advance. In this paper, he also showed that, under the assumption of a sm dis-turbance, the action of the pressure was equivalent to a source distri-bution on the undisturbed free surface. The relation was:

- p ax

(i)

where and p are the source intensity and pressure at the same point,

c is the velocity in the x direction, p is the density of the fluid

g is the acceleration due to gravity.

Lu.nde (1951a) extended the theoretical treatment to include the case

of an arbitrary distribution moving over finite depth. Numerical cal-culations which are directly applicable to the ACV have been carried out

by other workers. For example, Newman and. Poole (1962) considered the two cases of a constant pressure acting over a rectangular area, and over an elliptical area. The most striking feature of their results is the

very strong interaction between the bow and stern wave systems. The

resistance curves displayed a series of humps and hollows - particularly

for the rectangular distribution (where the interaction would be greater).

A hump is produced when the bow and stern wave systems are in phase and

combine to give a trailing wave of a maximum height. A hollow occurs when the two wave systems are out of phase by half a wavelength so that

the combined amplitude is a minimum.

The interference effects are found to be stronger for large beam to

length ratios, as would be expected from this argument, since the wave

motion becomes more nearly two-dimensional. The humps are found to occur

at Froudé numbers given approximately by

F=l//(2n_1)7rfornl,2,3'

(2)

and. the hollows occur approximately when

F = l/V for n = 1, 2., 3 (3)

In the limit of infinite beam (a two-dimensional pressure band), the

humps and hollows are given precisely by Eqs (2). and (3). This result

was found. by Lamb (1932).

In water of finite depth, the main, or "last", hump (n = 1) is shif-ted. to a lower Froudé number., and for sufficiently shalloW water,, occurs at a depth Froude number, Fd , equal to unity . (that is, at the depth critical speed). Similar calculations were given by Barratt

(1965).

Newman and Poole also considered the effect of a restricted waterway,

such as a canal. In such a case, the wave pattern is constituted from wavelets of discrete frequencies only, which can exist in the tank,

where-as in laterally unrestricted water a continuous distribution of frequencies

exists. As the speed is increased past the critical speed, the transverse wave, component can no longer occur, and as a result, there is a

d-iscontin-in the data.

uity in the wave resistance curve. This Sudden drop in resistance is given by

3W2 2pgBd2

where W

is the weight of the ACV, B is the width of the tank and d is the water depth.

Havelock (1922) also presented some resti.lts for a very smooth pressure distribution moving over water of finite depth. These., too, showed the

shift of the main hump and the increase in its magnitude in shallower

water. Havelock's curves displayed only the main hump. The secondary (.n = 2) and other humps did not occur because of his choice of pressure distribution.

In recent years, a number of experiments have been carried out in

order. to check the above-mentioned theoretical results. These have been

performed in particular by Everest Cl966a, 1966b,

l966c

and 1967) and

Hogben (1966). The fundamental question pointed out in these papers is the resolution of the total drag on the ACV into its components. Apart

from the wave resistance, the forces acting on the craft are aerodynamic drag and wetting drag.

I these experiments, the total drag was usually measured with a

dynamometer. The aerodynamic drag was then estimated from the drag coefficient on the model, and the momentum drag was obtained from the

mass flow into the cushiOn. The agreement between theory and experiment

was found to he best at speeds greater than hump. At lover speeds,

nan-linear and viscous effects become important and there was a large scatter

To avoid the troublesome wetting drag, Everest (1966a) attempted to

eliminate it using a thin polythene sheet floating on the water surface. The resistance breakdown is further discussed by Hogben (1966). The

ex-periments only indicated the presence of the first two (n = 1, 2) and possibly three (n = 3) humps. Hogben

_(.1965)

showed that this result

fitted in with the idea that the maximum ratio of wave height to length

is about one seventh; That is, wave breaking prevents the occurrence

of the additional humps.

Further experimental work by Everest, Willis and Hogben (1968 and

1969)

dealt with an ACV at an arbitrary angle of yaw. This problem was also studied numerically by Murthy (1970). In the experiments, the wave

resistance was measured directly from a wave pattern survey, using the

transverse cut method. There was considerably less scatter in the data using this method, since the rather doubtful techniq.ue of estimating the wetting drag was eliminated. Indeed, the agreement was found to be

somewhat better, particularly for the lower of the two cushion pressures tested.

In an attempt to get better agreement with experiments at lower speeds, Doctors and Sharma

_{(.1970 and.1972)}

used a pressure distribution which

essentially acted on a rectangulararea but had a controlled amount of smoothing - both in the longitudinal (x axis) and in the transverse

(y axis) direction. The distribution used was:

p(x,y) = p0 [tanh{a(x + a)} - tath{c(x -

a)}}

where p0 is the nominal, cushion pressure, and a and b are the

nom-inal half-length and half-beam respectively. The rate of pressure fall-of at the edges is determined by the parameters and . This

func-tion is shown in Fig.].. As a special case, ct ,

+

is equivalent

to a uniform pressure acting on a rectangular area 2a2b.

In practice, of course., the pressure at the edge of an ACV does fall

off at a finite rate, corresponding approximately to a = a O. Nevertheless, it was found that only by selecting ca = a = 5 , could

the humps in the resistance curve above the third be essentially

elimin-ated. Thus clearly, viscosity and non-linearity are important at low

speeds. Some of these calculations arereviewed in thispaper.

Another problem of practical interest is the wave resistance during

accelerated motion, as one is frequently concerned with the ability of a

heavily-laden craft to overcome the hump resistance in order to reach the

cruising speed.

The problem of accelerated motion for a ship has been treated by

Sretensky (1939), Lunde (195lb, l953a and. 1953b) and Shebalov (1966.). Wehausen (l961) made numerics.], calculations for a particular motion of a

ship model starting from rest. His results consisted of asymptotic

expiessions valid for large values of the time after a steady speed was

obtained.

Djachenko (1966) derived an expression for the resistance of an

arbitrary pressure distribution moving with a general acceleration

pat-tern in deep water. He also presented some results for atwo-d.imensional

distribution.

Doctors and Sharma found that the main effect of accelerated motion

finite depth to reduce the magnitude of the hump quite significantly. These results are partially presented in this paper.

PRESENT WORX

The basic theory for the wave resistance of a time varying pressure

distribution will first be given. Then the results will be appl-ied to

the case of an ACV executing rectilinear motion in a horizontally

unrestricted region.

The work will then be extended to the case of an ACV moving along

the centerline of a rectangular channel. From theBe calculations it is

possible to determine the effect of the tank walls on the wave resistance.

Finally, the case of a yawing ACVwilJ. be examined. In particular,

the induced side force acting on the craft will be determined, so that

BASIC TIORY

PROBLEM STATEMENT

We represent the ACV by a pressure distribution p(x,y,t) acting on

the free surface, and travelling with the speed of the craft. Two right-handed coordinate systems (reference frames) will be used, and are shown

in Fig. 2. A third coordinate system that rotates with the craft during a yawing motion will be introduced later. The axis system jz is fixed in space, and the system xyz moves with the craft, z being vertically upwards while x and are in the direction of the rectilinear motion.

The relation between the coordinates is then given by

x = - s(t)

=

-

J

c() dT

(6)

where s is the distance that the craft has moved. The pressure in S

the stationary reference frame is denoted by p (,y,t). The velocity

potential in the stationary frame, (such that the velocity is its

positive derivative), satisfies the Laplace equation, so that

v24-o

.

(7)

The kinematic boundary condition on the free surface requires that a particle of fluid. on the surface remains on it (for example, see Stoker

(1957)),

so that

8

;.

[

-

(,it)]

= 0

,

where is the elevation of the free surface. Now we have the

sub-stantia]. derivative:

D_3

a a

at34)yay 'zz'

so that the exact kinematic condition becomes

- 4)y _y] z= - t

=

(8)

The linearized kinematic condition is Obtained by dropping the second

order terms, and then writing the remaining terms as a Taylor expansion

about. the point z = 0 . After dropping the higher order terms again,

we obtain simply

[4)z]

z0 -

t

= 0

The dynamic condition on the surface the Bernoulli equation

-in the stationary frame is

[4) +

+ + 4)z2)]

z= + + g = f

where f is an arbitrary function of time, which may be put to zero

without loss of generality. Eq.

(io)

is now linearized to give

(9)

and the Laplace transform pair:

[ctt1

_{z0 +}

+ g =

The combined free surface condition is obtained from Eqs (9) and

(ii) by eliniinating

r

1

is

(12)

The final boundary condition needed states that there should be no

flux through the water bed:

[]

z-d =

L13).

THE POTENTIAL

The solution of this set of equations can be obtained by an

applic-ation of the double Fourier transform pair:

FCw,u) =

J

J

exp{-i(wx + uy)}

and. f(t) 10 L{f(t)} = f(t) exp(-q.t) dt 0 6+i -i--L{f(t)} exp(qt) dt

= 2Tri

I

being a positive constant.

The Fourier transform of Eq..

(7)

is first taken, giving c

-k2=0

,

zz

where is the Fourier transform of , and k2 = w2

+ U2

The solution of Eq.. (16) subject to the transformed bed condition,

Eq.. (13), is

(w,u;z,t) = A(w,u;t).cosh{k(z + a)} . (18) Eq.. (18) is now substituted into the Fourier transform of the free

surface condition, Eq.. (12), giving

Att + y2A =

1

sech(kd) where P is the Fourier. transform of and

= gk'tanh(kd)

(2 +

2) L{A} = - sech(kd) [q L{P(w,u;t)} -

P(wu;O)]

The inverse Laplace transform is taken, using the convolution theorem on

the first term:

t (w,u;z,t) - p.cosh(kd) [_

J

P(w,u;t).cos{y(t - t) dt coshk(z + d)} [ 0 sin(yt) P(wu;O)] I

We express P by means of the Fourier transform, and then the inverse Fourier transfom is teken:

1

(,y,z,t) -

-Jj

dS' J dT J dw J

pS(t yt T)

cosh{k(z + d)} cosh(kd) 0 00 cos{V'gktanh(kd)'(t - T)}exp{i(w( - + u

y

-00 00 + dS' dw du

pS(t

ye ) cosh{k(z + a)} cosh(kd.) S 00 -00 sin(/gk.tanh(kd).t)

exp{i(w( - + u(y - y'))} , (21)

/gk.tanli(ka)

where PS =

pS(tyt1)

_{, defined over the area} St _{, while}

12

Eq. (21) is the potential for an arbitrary time-varying pressure

distribution starting at t = 0 . Thus the solution for the general

motion of an ACV is obtained. I-n the following sections, we shall

con-sider special motions of a pressure distribution which is non-time varying with respect to axes rigidly attached to the vehicle.

RECTILINEAR MOTION IN HORIZONTALLY UNRESTRICTED WATER

THE POTENTIAL

We now consider motion of a craft starting from rest at t = 0

The expression for the potential, Eq. (21), may be simplified by partial integration of the five-fold integral with respect to T

t

J

sin{y(t- T)}exp{i(w(

- ') + u(y yt))) (22)

I

The pressure distribution, , as measureä. in the stationary reference

frame is a function of time. it is related to the pressure in the moving frame, p , by the following eq.uation:

p5(,y,t) = p(x,y)

= p( -.s(t),y) . (23)

THE WAVE RESISTANCE

The resistance of the pressure distribution is defined as the longi-tudinal component of the force acting on the free surface, and is there-fore given by

J

dw

J

du cosh{k(z +a)} CO sh (kd)

(,y,z,t)

= - w2p

Jj

dS' Si ₀

R=-i

g S 114 R(t) =

J

ps(y,t)

dy . (214) S

Substituting Eq.. (ii), we obtain

.exp{i(w(x - x' + s(t) - s()) + u

d.y

The real part of the integrand is now expanded. Then it is simplified by invoking properties of even and odd functions. The final result is:

t

y-y'))}

cos{/gk'tanh(kd)(t - t)}.cos{w(s(t) - s(t))}, (25)

The second term in this expression contributes nothing to the integral providing the pressure drops to zero at = . The result for d) ,

Eq.. (22), is now used. If one expresses the pressure in terms of the

moving frame by Eq. (23), then the wave resistance becomes:

t =

JJ p as if ptdS' J

c() dt

J dw J

duw2.cos{y(t

-S 0 - -R

= 22pg

J

c(t) dT

J dw J

du.w2 p2 .,-Q2)s 0 0

-C

Iw2dwldu

1iii2pg j J 0 -where

=JJ(x)

+ dx dy (26) S

The range of the u integration in Eq. (25) may be halved for a pressure symmetric about the x axis.

Eq. (25) is similar to that for a thin ship obtained by Lunde (1951b).

His formula included an additidnal integral which was simpiy proportional

to the instantaneous acceleration. This extra term is zero if the

sin-gularity distribution (Eq. (1)) lies on the free surface - as for a

pressure distribution.

The steady-state wave resistance can be derived from Eq. (25) by

allowing the velocity of the craft to be constant for a long time. If

the velocity is suddenly established at a value c , then one obtains

and

u=ksin0 ,

p2 _{+ Q2)}

15ih11

+ wc)t}sin{(Y - wc)t}

L

-y+wc

Ywc

As t - , the oscillations in the integrand increase so that there is

only a contribution from the second term, and this occurs when

y - wc = 0 . (27)

This is the relationship between the transverse and longitudinal wave

numbers for free waves travelling at the speed of the craft. The analysis is simplified if we use polar coordinates:

w = k cos 0

R

.Q(w,u) = 0

where k

is the circular wave number and 0 is the wave direction. The limit process is carried out for a similar case by Havelock (1958), and the final result is

-O w/2

1 1 1

k3cos8

2irpg

J + J

1- kod..sec2Osech2(kd)

-ir/2 _Oi

{P2(k cos 0,k sin 0) +.Q2(k cos O,k sin e)} dO , _(29).

in which k0 = g/c2 (30)

and k is the non-zero solution of Eq. .(2T), that is, of

k - kosec20.tanh(kd) = 0 (31)

The lower limit for 0 is taken as 0 , the smallest positive value of

0 satisfying Eq. (31) for a real k . It is given by:

= 0 for k0d > 1 (subcritical speed)

(32)

= arccoa

,&

for k0d < 1 (supercritical speed)

RESULTS

Some results previously published (Figs. 3 to T) are now.presented

to show some of the effects of the choice of pressure distribution given

by Eq. (5). For this choice., it was shown that

Tr°sin(av) ¶.Sjfl(bu) P(w,u) _{= P0 a.sinh(1w/ci5°.siñh(7ru/28)}

while the weight supported by the pressure is just

Wl4p0

ab

For convenience the wave resistance is expressed as a dimensionless coefficient:

R =-"

C

_{W Po}

Fig.. 3a shows the wave resistance of a distribution with a beam to length ratio of 1/2. The variable used for the abscissa is A = i/2F2

This has the effect of expanding the horizontal scale at low Froude numbers.

Curve 1, with ca = a = , displays the unrealistic low-speed oscillations

which are characteristic of the sharp-edged distribution and were referred

to previously. It is seen, that with increasing degrees of smoothing (smaller values of a.a, and a), the low-speed humps and hollows may be

eliminated. The case with cw = a

_{= 5}

results in only about three humps, more in keeping with experiments. Fig. 3b presents results for

finite depth water for three different distributions. The chief dif-ference now is that the main hump is shifted to the right and. occurs near the critical depth Froude number. It is seen that Curve 2 has smoothing

applied only at the bow and stern equivalent to a sidewall ACV. The

result is similar to the case for smoothing all around, showing that the

wave pattern is essentially produced by the fore and aft portions of the

cushion and not the sides. The resistance in the region of the main

hump is hardly affected by the smoothing.

The result of varying the depth of water is displayed in Fig. .

18

occurs in each case at a depth Froude number slightly less than unity. The location of the other humps is also affected, but to a lesser degree.

Beam to length ratio is varied in Fig.

5.

The general effect of

increasing the beam is to increase the maxima and to decrease the minima

in the wave resistance curve. This is due to the transverse waves

assuming greater importance as the two-dimensional case is approached. A secondary effect is a shift in the locations of the oscillations to the right, so that in the limit of infinite beam, they occur at Froude

numbers given precisely by Eqs (2) and (3).

We now turn to the effect of constant levels of acceleration of the craft from rest. Fig. 6a applies to a smooth (aa

= 5)

two-dimensional pressure band moving over eep water. A general displacement

of the oscillations to higher Froud.e numbers occurs. This shift is greater for the higher acceleration. In addition, most of the low-speed

oscillatiOns apparent in steady-state motion do not occur in accelerated

motion. The resistance of a smooth band over finite depth water is shown in Fig. 6b. Here the reduction of the peaks is even more dramatic than in deep water. More striking, however, is that for this and for all

other two-dimensional cases studied, the wave resistance becomes negative beyond a certain velocity in finite depth. The resistance then asymp-totically approaches zero. (The ordinate in this figure is plotted on

an

arsinh scale.) The depth Froud.e number at which the negative peak

resistance occurs in shallow water has been found to be

Fd = 1 + 2

hc/gd

(36)

The resistance of an accelerating three-dimensional pressure dis-tribution is shown in Fig. Ta (deep water) and Fig. Tb (finite depth).

In deep water, the wave resistance shows similar, but less marked, effects due to acceleration as does the ôorresponding two-dimensional case (Fig. 6a). In finite depth, there is again a strong reduction in the main peak as well as an elimination of nearly a.0 the low-speed oscillations. However,

there is no region of negative wave resistance - thus indicating the

20

RECTILINEAR MOTION IN A TANK

TEE POTENTIAL

We now consider the problem of an ACV moving along the centerline

of a rectangular tank of length L and width B . The initial distance at t = 0 between the starting end of the tank and. the coordinate axes

xyz fixed to the model is taken as a . This problem is crucial to the

testing of models, as one. must know the effect of tank walls. For in-stance, during steady motion in an infinitely long tank, Newman and Poole

showed that the effect of tank width in the neighborhood of. unit depth Froude number to be importance (see Eq. (14)).

We utilize Eq. (22) for the potential in a horizontally unbounded

region, and satisfy. the additional condition of rio flux through the four

tank walls, by employing a system of image ACVs as shown in Fig.

8.

We consider first only the array of distributions on the tank centerline,

and. later on apply the boundary condition on the tank sidewalls. The

potentials for the individual distributions, , are related to the

primary potential, , as follows:

(y,z,t) =

- nL,y,z,t) for n even, = + (n + l)L - 2a,y,z,t) for n odd,.

We add to (-l) (i) to (-2) (2) to

,and soon.

This only alters the exponential factor in Eq.. (22), which now becomes: factor = 2 exp{i(v(- a - ') + u(y - y'))}cos{w(a + )} exp(2inwL)

The integral with respect to w of this factor in Eq. (22) can be

simplified using the Poisson summation formula to give

=

- 2pL

dS'

sin{y(t - r)}

si.n{y(t

-

T)}

1mn

where _urn = 2TTE/B ,

k

2=w2+u2,

mu n m

du

cosh{k(z +

cósh(kd)

cos{w

(a +

)}'exp{i(w (-

a -

') +

u(y

-n n

We now satisfy the condition on the side walls of the tank by

in-cluding the image ACVs on lines parallel to the tank centerline. The

procedure is similar to that just carried out, and if we assume that the

pressure distribution is symmetric about the x axis, then

t cosh{k (z + d)} --

if dS'

j dT El

P5C'Y't)

cosh(ka)

S

(38)

+ )}.exp{i(w(_

a

') +

u(y - y'))}

,

(39)

where w

=irn/L,

(37)

k 2 = w 2

+ u2

n

where

22

v.2=gk tanh(k

d)

'mn mn mn

and

= 1/2, = 1 for xi >

THE WAVE RESISTANCE

The method of obtaining the wave. drag is the same as in the previous

sectior. and utilizes Eqs (2I)

and (38).

After some algebra, one obtains:

t

R

= pgBL I C(T)

dt

xi

e

m Vn

2.cos&k

mn

tarih(k d.).(t

mn

-J n=O

0

{cos(w (s(t) - s())) -

cos(w (s(t)

+ s(t) + 2a))} +

_Q2

_{{eos(w(s(t) - S(T).)) + cos(w (s(t) + s(T) + 2a))}}

+2 P sin(w(s(t) + s(T) + 2a))] ,

(12)

P

P(v,u)

xi

m

and

=

Q(v,u)

it is clear that the fluid motiOn in the

tank consists only of wavelets

whose wavenumbers are

given by Eqs

(37)

and (39), and that in the

limit

of

L

and

B

+

,

the result for a longitudinally and laterally

unbounded region is recovered.

The terms containing

a

are due to

reflections

off

the starting end of the tank, and as

a --

, they

The wave resistance for steady motion in an endless tank may be

obtained from Eq.

₍₂₉₎

by setting up a1laterally disposed array of images. The result, derived by Newman and. Poole, in the present notation, is

k 2.tanh(k d.) {p 2 + Q 2}

rn in

.m

pgB m 2k _- kotarth(kmd) - kmkodsech2(kmd) in which U is given by Eq. (39) and W by

k 2 = 2 +

The circular wave number, k , is the solution of

m

k

-kktanh(kd)=U2

m

in0

in

m

(The value of k when in = 0 is distinct from, and generally not equal to,. k0 , the flmdamenta]. wave number,)

RESULTS

The wave resistance of a smoothed rectangular distribution moving

in a tank is shown in Fig.

9.

In deep water (Fig. 9a), it is seen that

the effect of the walls is small for B/a = 2 . For B/a > 14 (tank

width greater than four times model width), the resistance coefficient

differs from the infinite width value by less than 0.0]. . It may be poin-ted out here that for the special case of B/a = 1 , that is, the tank width equal to the nominal beaiii of the model' the pressure carries approximately

7 of the weight of theACV beyond the tank walls However, it can be shown that this case is mathematically equivalent to a two-dimensional

2I

pressure band spanning the width of the channel.

In finite depth (Fig. 9b) the influence of the tank walls in the

region of unit depth Froude number is considerably greater, as was shown

by Newman and Poole. The drop in wave resistance (Eq. (fl)) at the cri-tica.]. speed does not depend on smoothing. Even when B/a = 61 , so that

the tank width is sixty-four times the model beam, there is a discontinuity

in resistance. coefficient of 0.188 . Thus steady-state experiments in this speed range are difficult.

The effect of side walls of an endless tank on the wave resistance

of an accelerating ACV is displayed in Fig. 10. Two different levels of acceleration in both deep water and finite depth were calculated. In all cases the wave resistance is a smooth function of the tank width.

For the low-speed range, increasing tank width generally decreases the

wave resistance. On the other hand, this trend is reversed for high

speeds (greater than the hump speed').

The case of infinite tank width is not plotted, in order to avoid

confusion with the case of B/a = , with whidh it is almost identical.

This difference in wave resistance coefficient for the cases calculated

is less than 0.01 , so that one might consider that a tank width equal

to four times the model beam to be essentially infinite.

Even in finite depth there is no sudden change in resistance as the

model accelerates through the critical depth Froude number. (A depth

Froude number of unity is passed when tv7 = l4.14 if /g = 0.05 ,

and when

t,Q = T.OT

if /g = 0.3. .) This sharply contrasts the

case of steady motion, in which the

drop

or discontinuity in wave

resistance coefficient when d/a = 0.5 and B/a = is 3.0

in finite depth, and thus only the former is shown, in Fig. 11. The case of an infinitely wide tank is presented in Fig. ha for a/a = 1, 2 and In the region near t 0 , there is a slight increase in the resistance when a/a = 1 only. Incidentally, when a/a = 1 , part of the pressure "extends" beyond the starting end wall, so one must expect some interference. When a/a = 2 , the clearance from the starting end.

wall is half a craft length and there is no noticeable interference. The two curves for the finite values of a were calculated for a

tank.length L/a = 20 . There is no perceptible effect fromthe far end wall until the model "passes" through its image - as indicated by one or two oscillations in the curves near

t/7 =

20 .

The case of B/a = 1 (that is, a two-dimensional pressure band) is

shown in Fig. hib. For the case of no nominal separation of the craft from the starting end wall at t = 0 , there is now a slightly greater effect on the wave resistance.

26

FORCES ON A YAWING ACV

THE POTENTLAL

We now consider the special case of an ACV travelling for a long

time in the longitudinal or x direction. The craft is either fixed in a steady yaw position, or it starts a yawing motion after initial transients have died away. We may therefore use Eq. (21) for the

po-tential, and drop the second term which will approach zero as t

-THE FORCES

The wave resistance is defined by Eq. and the side force by

s(t)

=

JJ

p5(,y,t)

y d dy .

(5)

Thus the side force is the positive force to port (the y direction) required to hold the craft on a straight course.

The analysis for the two forces now continues, as in the case for

rectilinear unyawed motion in horizontally unrestricted water. The forces are: t =

-J

d

JJ

p(x,y,t) dS

JJ

(x',y',t) dS' d.w

du ()'y.

0 S St

And after some simplification: t

R_

1

S - 22pg

J

dt

J dv J

du (w).yssin{y(t,_ t)} 0 0 -- PQ').cos{w(s(t) - s(t))} + (pp' + QQ').sin{w(s(t) -in which P = P(w,u,t) Q = Q(w,u,t) P' = P(w,u,t) and Qt = Q(w,u,t)

It is convenient to calculate the P and Q functions using an

**

axis system x y z that rotates with the craft rather than the xyz system, in which the x axis lies in the direction of motion. This is illustrated in Fig. 12. The yaw angle c(t) is taken positive for

* clockwise rotation of the craft, when looking down on it.

If w

and

*

u are the induced wavenumbers relative to these craft axes, then

w*(t)

_{= w cos{(t)} - u}

_{sin{(t)} = k cos{e +}

*

(T)

and u (t) = w sin{(t)} + u cos{c(.t)} = k sin{e + cCt)}

Then it maybe shown that

P(w,u,t) I * * * cos * * * * * * Q(w,u,t) = p (x ,y ) .

(w x +

U y )

dx dy sin (18) S analogous to Eq. (26).

(6)

28

For the pressure distribution given by Eq. (5), it immediately follows

from Eq. (33) that

P(w,u,t)

= p0

rr.sin(aw*)

irsin(bu)

csinh(rrw

/2)

°sinh(yru

/2)

and.

Q(v,u,t) = 0

We now consider a craft travelling at a constant velocity at a

fixed azigle of yaw from time

-T

to

0 _,.

and then allowed to yaw up to

time

t

.

The

t

integral in Eq. (h6) for just the first phase of the

motion is

I =

sin{y(t

-

t)}.[(QPt PQ') cos{wcCt

-+ (PP' -+ QQ') sin{wc(t -

di

=

_{{Q(w,u,-t) P(w,u,-0)}

_{- P(w,u,t) Q(w,u,-0)}.}

.rcos{(y+wc)t}

cos{(y+wc)(t+T)}

cos{(y-we)t}

cos{(y-wc)(t+T)}

L y+wc

-

y+wc

y-wc

y-wc

+

{P(w,u,t) P(w,u,-.0) + Q(w,u,t) Q(w,u,-0)}'

.rsin{(y+wc)t}

sin{(y+wc)(t+T)}

sin{(y-.wc)t}

sin{(y-wc)(t+T)}

L y+wc y+wc

y-wc

y-wc

We consider first the case when

t = 0

and

T

(that is, a

steady state).

The four terms containing the cosine factors, and the

first and third sine factors are zero. The fourth sine term is the only one that gives a non-zero result in the wu. integral of Eq. (I6) as T + . The stead'-state forces may be obtained in the same manner

as the limit of Eq. (25) for large time:

R_ 1

S - 2rrpg

.{P2(k cos O,k sin e) + .Q2(k cos 0,k sin O)} dO , (so)

where k ,

k0 and 01 are given by Eqs (30), (31) and. (32).

If we now assume that the ACV starts yawing at t = 0 , then as

T - , the second and fourth cosine terms, and the second sine term contribute nothing to the wu integral in Eq. (16). The expression for the forces after t = 0 becomes:

-0

ff/2

k.(''?5

0)

sinO

R 1

S - 2pg

[

_J2 +

J

]

1 - k0d.sec20.seh2(kd) 0

.{P(w,u,t) P(w,u,-0) + Q(w,u,t) Q(w,u,-Q)} d8

+ l.ff2pg

J

dwj du (W).y.{Q(w,u,t) P(w,u,-0) - P(w,u,t) Q(w,u,-0)}.

.rcos{(y + wc)t} + cos{(y - wc)t} L

y+wc

y-WC

(cont'd over) k3.(COS 8) sin 0 1 k0d.sec2O.sech2(kd)

1 + 0

22pg

J

di

J

0 0

.[Qwut)

+ {P(w,u,t) du (W)

P(w,u,t) P(v,u,t) Q(w,u,t)} 30

(w).y.{P(wu,t) P(w,u,-0) + Q(w,u,t) Q(w,u,-O)}

.rsin{(y + wc)t} sin{(y wc)t}

L

y+wc

y-WC

cos{wCs(t)

RESULTS

The (steady-state) wave resistance of a yawed ACV is shown in

Fig. 13. Fig. 13a indicates the marked effect of smoothing the pressure fall-off on a rectangular cushion, for a Eroud.e nber of unity. This is accentuated for yaw angles in the neighborhood of 100 and

85°.

The peaks would seem to be caused by interference between short wavelets

as short wave components are not produced by a smoothed distribution.

The slopes of the curves are zero at yaw angles of 00 900 as required by symmetry.

The variation of wave resistance of a smoothed distribution with yaw

angle for a series of different Froude numbers is displayed in Fig. 13b. P(w,u,'r) + Q(w,u,t) Q(w,u,t)} sin{w(s(t)

At super-hump speeds, yawing the vehicle increases the effective Froude

number so that the. resistance drops a little. On the other tiand, yawing at a sub-hump speed (for example, F = Q.) can bring the craft

onto the hump (at constant speed of advance), and thereby increase the resistance.

The wave-induced side force is shown in Fig. l. It is non-dimensionalized in the same mariner as the wave resistance in Eq. (35). The effect of smoothing on side force (Fig. J)4a) is seen to be even more vivid than on resistance (Fig. 13a). Increase in sharpness has a considerable effect on the side force for very small, or for very large,

yaw angles - even at this relatively high speed. At the same Froude

number, the effect of sharpness on unyawed wave resistance (Fig. 3a) was

considerably less. The linear theory predicts a peak dimensionless side

force of

2.63

in contrast to a dimensionless wave resistance of

_0.73

at zero yaw angle. It seems that nonlinear and viscous effects would

preclude the development of such large side forces in practice. Different Froude numbers are considered in Fig. llb.. The side force (for aa = a = 5) is seen to be positive for superhump speeds, and therefore favorable during a coordinated turn. It reaches a

inaxi-mum at a yaw angle of about

300

. Thus there is an optimum sideslip

angle for generating the maximum side force. For subhuxnp speeds, there is a range of yaw angle in which the side force is negative.

Unsteady yawing motion is now considered. The side force for

dif-ferent rates of constant rotational speed after travelling at zero yaw

angle for a long time is presented in Fig. 15. The abscissa is the yaw angle, and is proportional to the time after the initiation of the

32

the available side force. However, as trpica]. average yaw rates are in the vicinity of 50 per unit time, it is clear that the unsteady in-fluence is of secondary importance. The side force q,ualitatively fol-lows the same trends at the two speeds considered, namely F = 0.6

(Fig. iSa) and F = 1.0 (Fig. 15b).

Finally, in Fig. 16, a manoeuver is studied, in which the yaw angle

is instantaneously öhanged from zero to 50, 100, 15° and 20°. The distance the ACV must travel before the steady-state side force is

achieved, is slightly greater for larger manoeuvers. Nevertheless, this effect is small Almost the full steady-state side force is generated

after the vehicle has moved one craft length at F = 0.6. (Fig.-16a), and after 1.25 craft lengths at F = 1.0 (Fig. 16b);

A favorable side force is developed inmiediately after this sudden yaw manoeuver, and then increases slowly at first. It may be shown that for a. small jump in yaw angle, the initially generated side force is just one half of the final steady-state side force. This feature is evident in the curves, particularly for the smaller manoeuvers.

33

CONCLUDING REMARKS

PRESENT WORK

Turning firstly to the. case of Rectilinear Motion in a Tank, it is

clear that the problem of interference from the side walls during

accelerated motion in finite depth water is considerably less than that during steady motion. Model tests under such unsteady conditions would be much easier to perform as a tank width equal to four times the model

beam essentially simulates the laterallyunrestricted case.

With. regard to the yawing ACV, the. great dependence. of side force at superhump speeds on smoothing was an unexpected result. So much so,

that it would be unrealistic to model the pressure under the craft with

a sharp distribution. Even assuming practical values of ca = a (which has a neglig-ibJ.e effect on unyawed wave resistance) reduces the maximum predicted induced side force by almost one half. A study of

the expression for the steady-state. forces, Eq. (5.0), reveals that this

difference is due to the high frequency oscillations in the integral

for 0 just less than ir/2 . The effect is worst for a yawed sharp

distribution when the oscillations decay very slowly and is further

emphasized in the integral for side force which contains a sin e factor, rather than the integral for wave resistance which contains a. cos 0 factor. A particularly large number of subdivisions in the

integration is therefore required under these conditions. This probably

explains the small discrepancies found at small non-zero yaw angles

and yaw angles just below 9Q0 , when attempting to verify the theoreti-cal wave resistance theoreti-calculated by Murthy

₍₁₉₇₀₎

and Everest

_(1969).

31

In practice, these high frequency wavelets probably break due to excessive theoretical steepness, and other practical effects such as cushion air flow.

The induced side force has nevertheless been found to be

signifi-cant, being of similar magnitude to the wave resistance. It clearly plays a role in the control of ACVs. This force has been experienced by drivers of air-cushion vehicles, who usually refer to it as "keel

effect1'.

During a trpical manoeuver, it has been found that the induced

side force is almost equal to the steady-state value at the sazae instantaneous yaw angle.

FUTURE WORK

It would be interesting to verify some of the above-mentioned

theoretical results by experimefit. In particular, one would like to

know how accurately the induced side force is predicted - or what the

equivalent smoothing wOuld be. Such an experiment would have to take

into account aerodynamic and momentum hide forces as veil as skirt

con-tact, which might be significant.

Numerical work can be extended in various areas. Further test cases, including the effect of finite depth might be examined. Incidentally, many manoeuvers are carried out in finite depth near the terminals. This aspect is therefore important.

Possibilities for theoretical work include an investigation into

the yawing moment acting on the vehicle about the vertical axis. Some experiments by Everest indicated that the craft is generally stable in yaw.

I

AC0WLGEI'ENTS

The writer is grateful to the Office of Naval Research, Washington

for their support of part of this work under Contract No.

NO0011-67-A-0l8l-00l8 Task No. NE 062-li.20, which was carried out during 1969

and

1970 in the Department of Naval Architecture and Marine

Engin-eering at the University of Michigan in Ann Arbor, Michigan. This work is briefly covered in the section on Rectilinear Motion in

Hori-zontally Unrestricted Water. For a more detailed account, the reader

is referred to Doctors and Sharma

₍₁₉₇₀

and 1972).

The section on Rectilinear Motion in a Tank represents some

cal-culations performed for research supported by the Australian Research

Grants Committee during 1972.

The writer also wishes to acknowledge valuable suggestions

per-taining to this paper made by Professor P.T. Fink, Dean of the Faculty

36

BIBLIOGRAPHY

Barratt, M.J.: "The Wave Drag of a Hovercraft", J. Fluid Mechanics,

22, Part 1, pp 39 - 7

(1965)

Djachenko, V.K.: "The Wave Resistance of a Surface Pressure Distribution in Unsteady Motion", Proc. Leningrad Shipbuilding Inst. (Hydrodynamics and Theory of Ships Division). English Translation: Dept. Naval Architecture and Marine Engineering, University of Michigan, Ann

Arbor, Michigan, Report 11i, 12 pp

₍₁₉₆₆₎

Doctors, L.J. and Sharma, S.D.: "The Wave Resistance of an Air-Cushion

Vehicle in Accelerated Motion", Dept. Naval Architecture and Marine

Engineering, University of Michigan, Ann Arbor, Michigan, Report

_99,

1O pp + 92 figs (1970)

Doctors, L.J. and Sharma, S.D.: "The Wave Resistance of an Air-Cushion Vehicle in Steady and Accelerated Motion", accepted for publication

by 3. Ship Research (1972)

Everest, J.T.: "The Calm Water Performance of a Rectangular Hovercraft",

National Physical Laboratory (Ship Division), Report 72, 12 _{pp +}

₂₉

figs

₍₁₉₆₆₎

-Everest, J.T.: "Shallow Water Wave Drag of a Rectangular Hovercraft", Ibid., Report

79, 8

_pp

+ 19

figs

₍₁₉₆₆₎

Everest, J.T-.:. "Measurements of the Wave Pattern Resistance of a Rec-tangular Hovercraft", Ibid., Teôh.

_{Mem.l17 (1966)}

Everest, J.T. and Hogben, N.: "Research on Hovercraft over Calm Water", Trans. Royal Inst. Naval Architects, pp 311 -

326 (l967)

Everest, J.T. and , E.C.: "Experiments on the Skirted Hovercraft

Running at Angles of Yaw with Special Attention to Wave Drag",

National Physical Laboratory (Ship Division), Report 119,

_{8 pp +}

15

figs

₍₁₉₆₈₎

Everest, J.T. and Hogben, N.: "A Theoretical and Experimental Study of the Wavefliaking of Hovercraft of Arbitrary Planform and Angle

or

Yaw", Trans. Royal Inst. Naval Architects, 111, pp 311.3

_{- 365 (1969)}

Havelock, T.H.: "The Wave-Making Resistance of Ships: A Theoretical and Practical Analysis", Proc. Royal Soc. London, Series A,

82,

pp 276 - 300 (1909)

Havelock, T.H.: "Ship Resistance.: The Wave-Making Properties of Certain Travelling Pressure Disturbances", Ibid.,

_{pp 1489 - 11.99}

_(19114) Havelock, T.H.: "The Effect of Shallow Water on Wave Resistance", Ibid.,

100, pp 1499

- 505 (1922)

Havelock, T.H.; "Some Aspects of the Theory of Ship Waves and Wave Resistance", Trans. North-East Coast Inst. Engineers and

Ship-builders,

142, pp 71 - 83 (1926)

Havelock, T.H.: "The Theory of Wave Resjstance", Proc. Royal Soc. London, Series A,

138, pp 339 - 311.8 (1932)

Havelock, T.N.: "The Effect of Speed of Advance upon the Damping of Heave and Pitch", Trans. Royal Inst Naval Architects, 100,

pp

131 -

_{135 (1958)}

Hogben, N.: "Wave Resistance of Steep Two-Dimensional Waves", National

Physical Laboratory (Ship Division), Report

_{55, 9 pp + 5}

figs

₍₁₉₆₅₎

Hogben, N.: "An Investigation of Hovercraft Wavèmaking", J.

38

Lamb, H.: Hydrodynamics,New York, Dover Pubs.,

738 pp (1945).

Orig-inally Cambridge., Cambrid.gë University Press

₍₁₉₃₂₎

Lund.e, J.K..: "On the Linearized Theory of Wave Resistance for a Pressure

Distribution Moving at 'Constant Speed of Advance on the Surface of

Deep or Shallow Water", Skipsmodeiltanken, Norges Tekniske H$gskole,

Trondheim, Medd.

8, 48

pp, in English

₍₁₉₅₁₎

Lund.e, J.K.: "On the Linearized Theory of Wave Resistance for

Displace-ment Ships in Steady and Accelerated Motion", Trans. Soc. Naval

Architects and Marine Engineers, ,

pp 25 - 8

(1951)

Lunde, J.K.: "The Linearized Theory of Wave Resistance and its Applic-ation to Ship-Shaped Bodies in Motion on the Surface of a Deep,

Previously Undisturbed Flü.id", Skipsmodelltanken, Norges Tekniske

Hgsko1e, Trondheiin, Medeflelse

.23 (.1953).

Translation: Soc.

Naval Architects and Marine Engineers, Tech. and Research Bulletin

1 - 8, TO pp (1957)

Lunde, J.K.: "A Note

on.

the Linearized Theory of Wave Resistance for Accelerated Motion", Skipsmodeltanken, Norges Tekuiske ff$gskole,

Trondheim, Medellelse

27, 14 pp (1953)

Murthy, T.K.S.:

"The Wave Resistance of a Drifting Hovercraft", Hovering

Craft and Hydrofoil,

Q, pp 20 - 24 (1970)

Newman, J.N. and Poole, F.AP.: "The Wave Resistance of a Moving Pressure

Distribution in a Canal", Schiffstecbnik, ,

pp 21 - 26,

in English

Shebalov, A.N.: "Theory of Ship Wave Resistance for Unsteady Motion in Still Water", Proc. Leningrad Shipbuilding Inst. (Iiydromechanics and Theory of Ships Division). English Translation: Dept. Naval Arch-itectu.re and Marine Engineering, University of Michigan, Ann

Arbor, Michigan, Report

6i,

i1 pp

(1966)

Sretensky, L.N.: "On the Theory of Wave Resistance", Trudy Tsentral. Aero-Gidrodinam. Inst.,

3L8, 28

pp, in Russian

₍₂₉₃₉₎

Stoker, J.J.: "Water Waves", I of Pure and. Applied Mathematics, Inter-science Publishers Inc., New York,

₅₆₇

_pp

_C1957)

Wehausen, J.V.: "Effect of the Initial Acceleration upon the Wave

LIST OF FIGURES

Pressure Distribution Used

The Two Coordinate Systems

Wave Resistance for Different Amounts of Smoothing

Deep Water

Finite Depth

Wave Resistance for Different Depths

5. Wave Resistance for Different Beam to Length Ratios

6.

Unsteady Two-Dimensional Wave Resistance

Deep Water

Finite Depth

7. Unsteady Three-Dimensional Wave Resistance

Deep Water

Finite Depth

8. Image System Used to Represent Tank Walls

9.

Wave Resistance in an Endless Tank

Deep Water

Finite Depth

10. Unsteady Wave Resistance for Different Widths or an Endless Tank

Deep Water, */g = 0.05 Deep Water, /g = 0.1 Finite Depth, c/g = 0.05 Finite Depth, /g = 0.1

II. Unsteadk Wave ResiStance for Different Locations of Tank EndsIn:

Finite Depth

Infinitely Wide Tank

Two-Dimensional Pressure Band

12. Axis System Fixed to Craft

13. Wave Resistance in Deep Water while Yawed

For Different Amounts of Smoothing

For Different Froude Numbers

14. Side Force in Deep Water while Yawed For Different Amounts of Smoothing

For Different Froude Numbers

15. Unsteady Side Force in Deep Water while Yawing

(a)

F0.6

(b)

F=i.O.

16 Unsteady Side Force in Deep Water After a Step Change in Yaw Angle

(a)

F0.6

(b)

F1.O

2.0

1.5

Curve 3:

R

I

't

c*a=8a=1O

C I

1.0

0.5

0

I 2 4 6 I I

8

10

12

CUrve 1:

ca =

=

Curve 2:

cta =

= 20

Curve 4:

c*a=a= 5

b/a = 0.5

d/a=

U

14

Fig. 3 Wave Resistance for Different Anounts of Snothing,

(a)

Deep

Water

R

C

1.5

1.0

6

8

10

1/2F2

Fig. 3 (cxnt.)

(b) Finite Depth

12

1.4

16

0.5

-0

2 0

3.5

Curvel: d/a=..

Curve 2:

d/a = 1

Curve 3: Wa = 0.5

Curve 4:

d/a = 0.25

b/a = 0.5

= 5

8a = 5

0 1 2 3 4 5 6 7

1/2F2

1.5 1.0 0.5 Curve 1:

b/a=

Curve 2: b/a=1

Curve 3:

b/a=O.5

Curve 4: b/a = 0.25 1/2F2

Fig. 5 Wave Resistance for Different Beani to Length Ratios

2.0 1.5 1.0 0.5

curvel: c/g=0

Curve 2: /g = 0.05 Curve 3: /g = 0.1 Curve 4:

/g=0.2

1/2F2

Fig. 6 Unsteady Two-Dizinsiona1 Wave Resistance, (a) Deep Water

10

R C 2

-1

1

Wa

= 0.5

c*a

= 5

Curve 1:

= 0

Curve 2:

= 0.05

Curve 3:

= 0.1

Curve 4:

= 0.2

b/a =

2 3 4 5

Fig. 6 (cont.)

(b) Finite Depth

V I

I

I I I

R C

1.0

0.8

0.6

0.4

0.2

b/a = 0.5

d/a =

= 5

/1

8a = 5

Curvel:

/g=0

Curve2:

/g=0.05

Curve

3:

/g=0.1

0 1 2 3 4 5 6 7

1.5

1.0

0.5

Curvel: Q/g0

Curve 2:

/g = 0.05

Cuxve.3:

/g=0.1

b/a

0.5

3/a = 0.5

= 5

= 5

0

1

2

3.

4

5

6. 7

Fig. 8

Image System Used to Represent Tank Walls

-(E_:

r

y

x

1.5

1.0

0.5

Curve 1:

B/a=

Curve 2:

B/a=2

Curve 3:

B/a = 1

b/a =0.5

d/a =

aa = 5

3a

= 5.

1/2F2

Fig. 9

Wave Resistance in an

Endless Tank,

(a) Deep Water

R

1/2F2

Fig. 9 (cent.)

(b). Finite Depth

1.5 1.0 0.5 Curve 1: Curve 2: Curve 3: Curve 4: b/a = 0.5 d/a = co = 5 = 5 = 0.05 I

I

.1

1 I 0

5.

10 15 20 t/

Fig. 10 Unsteady Wave Resistance for Different Widths of an Endless Tank, (a) Deep Water and c/g = 0.05

2.0

Curvel: B/a4

Cuzve2: J3/a=2

Curve3: B/a=4/3.

Curve 4: B/a=1

b/a = 0.5

d/a

= Oo

c.a

= 5

= 5

= 0.1

1.5

1.0

0.5

1

I

6 8

10

12

14

Fig. 10 (cent.)

(b) Deep Water and

/g = 0.1

I

16

18

20

2.0

-1.5

1.0

0.5

--0.5

0

I

5

I

10

15

I

20

Fig. 10 (oont.)

(c) Finite Depth and

c/g = 0.05

Curve 1:

B/a=4

Cuxve2: B/a=2

Curve 3:

B/a = 4/3

Curve 4: B/a=1

b/a = 0.5

d/a = 0.5

cia

= 5

a =5

= 0.05

25

30

R

C

2.0

1.5

1.0

0.5

Curve 1: B/a=4

Curve 2,:

B/a=2

Curve 3:

B/a = 4/3

Curve 4:

B/a=1

b/a = 0.5

d/a = 0.5

= 5

= 5

= 0.1

I I I I I I

I

_!

2

4

6

8

10

12

14

16

t4

Fig. 10 (cont.)

Cd) Finite Depth and

/g = 0.1

-0.5

0

I

R

C

1.0

-0.5 -0.5 Curie 1: cu/a=co

Curve2: cr/a=2

Curve3: cr/a=1

b/a = 0.5 d/a = 0.5 cza = 5 = 5 = 0.1 B/a = L/a = 20 I I

I

I. I 1 I 0 2 4 6 8 10 12 14 16 18 20

2.0

-1.0

0

I

2 4 6 8

10

12

14

16.

18

20

t/

1.0

0.8

0.6

0.4

0.2

-0

Curve 1: aa=Ba=

Curve 2: ca=a=40

Curve 3: ca=a=20

Curve 4: aa=a=I0

Curve 5:

ca =

= 5

b/a = 0.5

d/a =

F

=1

I I II I I I I I

00

_10°

_20°

_30°

_40°

_50°

_60°

_70°

800

900

C

R

C

Curve 3: F=O.6

Curve 4: F=O.7

Curve 5: F=O.8

Curve 6: F=O.9

Curve 7:

F=1.O

C

3.0

2.0

-0°

100

20°

30°

40°

50°

600

Curve 1: aa=a= °

Curve 2: cza=a=4O

Curve 3: cza=a=20

Curve 4: c*a=a=1O

Curve 5: aa=a= 5

b/a = 0.5

Wa =

F

=1

70°

800

Fig. 14

Side Force in Deep Water while Yawed,

(a) For Different nounts of Sixcothing

0.2 -0.2

Cuivel: F=0.4

Curve 2: F=0.5

3

Curve 3:

F=0..6

::

a =a=5

b/a = 0.5 d/a = I 00 100 200 _30° 40° 50° 60° 700 80° 90° £

Fig. 14 (cont.) (b) For Different Frote. Ninbers

I I -I I I I I

0.6

-0.7

0.6

0.5

0.4

0.3

0.2

-Curve 1:

=

0°

Curre 2: h=

50

Curve 3: 4= io

Curve 4:

=

20°

b/a = 0.5

d/a =

cxa

= 5

= 5

F

=06

I I I

Fig. 15

Unsteady Side Force in Deep Water while Yawing,

(a) F = 0.6

0.6

0.5

0.4

0.3

0.2

0.1

-Curve 1: c=

c=15°

Curve 2:

Curve 3:

Curve4:

b/a = 0.5

d/a =

= 5

= 5

F

=0.6

0 I ,1 II I I 0

0.25

0.5

0.75

1.0

1.25

1.25

s/2a

0

1.25

1.5

1.0

0.25.

0.5

0.75

s/2a

Fig. 16 '(cant.,)

(b)F = 1

0