The waves close behind a planing hull

w

Lab.

y.

Scheepsbouw!ie

Technische Fcsc

cu

Class Number

532.583,L-Report No. Aero,1992

-

Report 1fo. Aero.1992

November, 1944.

ROYAL ACT ESTAr,LSmT FABOU

The Waves Close Behind a Planing Hull by

-D. C.. MacPhail

and

.D.

1tre

M.A.P. Ref: Nil.

R.A.E. Ref: GJJ+.R/J140

£TJ1ARY

R. Reasons for enquiry

A number of cases have arisen in which large amplitude high speed porpoising of flying boat models seemed to be seriously

aggravated by inopportune striking of the.rear step on the surface of the trough behind the main stp. It would clearly be useful

to 1ow the shape of the trough in order to predict the likelihood of such

an

occurrence.

Range of investißation

After reviewing a number or the existing solutions for the transverse waves behind a planin surface, this report suggests that, for a no!mal flying boat, the diverging waves are more important than the transverse ones, except at very low speeds.

i approximate solution

is evolved

and compared with experiment

measurement s.

Conclus ions

It is shoun that the divergent wave system behind a planing hull is capable of accounting for the main features of the experimental trough observations. There are, however, a number of systematic

discrepancies which are attributed mainly to the pressure field around

the hull and. to the transverse wave formation, ipirical corrections

are introduced to compensate for this defect, and the resulting formulae are collected together in section 9 for practical use.

-2-10.1992

CONTENTS

L

Synfools 2, Introduction 3. enaral considerations

L

Filling of a depression in the water surface

5, Vaves ie to downash behind a hull.

6, Down4vash angles

Experimental measurements

Comparison of theoretical and experimental results

Ilse of the results

Effect of the afterbody

Conclusions

Suggestions for further work

References

114-. Circulation

.Apendix I. Gliding of a plate on the- surface of a liquid

Appendix II. Waves due to a travelling pressure point

Appendix III. Sources of error Figures

1. _Planing of a plate

2, Wave formation due to a single mong pressure rnint

3. Initial trough for flat bottom hull

14-. Flat bottom hull

With

bow wave

Vee bottom hull

Vee bottom hull with bow wave

7, Constant initial dovnwash dis tributi on

Triangular downwash distribution

Waves behind, a flat, bottom hull

iaves behind a flat bottom hull with bow wave

behind a vee bottom hull

aes behind a ve-e bottom hull with bow wave

15, Waves due to unifon dowìwash di,stribution

14.. Waves due to triangular downwash7distribution

Arrangement of the apparatus Experimental observations

Wake cross sections behind vee bottom hulls

fl H H I! T? H

Report No. Aero.1992

Symbols

= distance measured backwards from the hull step

y

= il to starboard of fore-and-ai't centre line

z upwards from undisturbed water level

elevation of water surface aoove undisturbed level

5 = draught at step

b = immersed beam t step for a draught 5 in. still water

a = inclination of keel to horizontal

= angle of deadrise

U forward speed of planing hull

u = absolute water velocity along OX

y = velocity along CY

w

= t, t, Oz

p U - Froude number based on immersed hull beam

X

= non-dimensional distance behind step

y L. = non-dimensional distance in an athwertships direction.

2. Introduction

When a normal two step flying boat is being taken off or landed,

it is invariably supported on both steps at low speeds; but, at higher speeds, it may plane on the forward or aft or both planing

bottoms. Many measurements of the water resistance of planing hulls have been made, and it has been shewn that the results for single

planing surfaces can be accurately and concisely specified. in terms of the attitude, draught, d.cadrise, and surface finish" 2, 5

a single planing bottom, the water pitching moment also can be internolated

relatively easily from a small nunber of experimental results; but,

when both.steps of a flying boat are in the water, neither the resistance

nor the pitching moment can be predicted with confidence. Under these

conditions the uncertainty arises because the after bottom runs not on

the.und.isturbed surface of the sea, but in the complicated trough loughd by the ± orward part of the hull.

In order to design the afteroody for adequate clearance from the water, it is necessary to know the details of the wave formation

bchind. the forebodyf; and it is the object of the present discussion

t outline an approximate method of estimating the dimensions of

.ero.i992

In the work which follows it has been expedient to me a

-number cf assumptions, sorne of which are not strictly

correct.

Lack of experimental data made some of these necessary,

end. others

were

required. in order to simplify the formulae to a

degree

when they would be useful in practice,

They were chosen with

a view to producing in the

final results that is believed to be

a reasonable balance

of deviations from the available experimental

data.

Although various possible refinements promised

to improve

the logic of the assunptions and the accuracy

of the results, they

have not been introduced at this stage

1ecause there seemed to be

a risk of complicating the calculatioiis too much for practical use

while still falling badly short of an exact

solution to the problem.

3. General considerations

The outstanding difficulties in the way of obtaIning a

theoretical

solution for the motion of water past a flying boat hull are due to

the markedly three dimensional nature of the flow

and to the free surface

which is distorbed. by gravity waves.

It seems probable that the

effect of gravity is insignificant near a fast-moving iill in comparison

with the relatively high pressures due to tL motion al the hull;

but

gravitational forces predominate further aft.

Although they d not

include all the import an phenomena, some of the two dimnsional so

lu-tions for flow past an infinitely wide piene form a suitable starting

point for a discussion of the three dirensìonal

problem.

A few such

solutions for the planing of a two dimensional plate are

mentioned.

briefly in Appendix I, and illustrated in Fig.l.

The transverse wave system which follows an

infinitely wide

planing plato is

of course, very different from the wave

formation

behind. a flying boat hull; but,as is tointed

out in the Appendix,

it may also differ greatly in height.

The failure of two

dThien-sional solutions is not surprising when one sees how far the flow

past a real hull departs from a two dimensional

rattern.

At an

incidence of f°

in still water the beam of a 250 deadrise hall is

equal to only

of the length of immersad keel.

2, 8

The simplest possible three dimension1 problii is to consider

the waves accompanying a single point of pressure

moving on the

surface of the see

This wave system is discussed very briefly

in Appendix II and. is illustrated. in Fig. 2.

Even this simple

representation of a flying boat hull leads t- a wave

configuration

which bears a remarkable resemblance to that following a moving ship

or flyin.

boat.

If the pressure distribution on the under side

of

a hull were 1iown in

full detail, it migitbe possible to reproduce

the correct wave system by adding together the waves caused. by a suitable

array of travelling pressure points.

Although such a solution would

be very useful for calculating the water resistance

íxt a flying boat,

it would have to be based on very accurate and complete data ii the

wave dimensions were to

'ce correlated with the incidence, de9rise,

ath draught of the wave-producing hull.

It is for this reaman that

it seems preferable at present -L-o attempt a solution which can be

fitted to the geometrical bourdary conditions

at the hull.

3.1 Basis of the

'esent Calculations

From visual observation of the waves

prod:uced b

fast boats

it con be

seen t1aat

as the speed is increased., the wavelength

of

the transverse waves increases and the

divergent waves encloae the

hull in a narrower and. narrower vec

For instance, Table I of

-4.-Report No. Aero.1992

Appendix

II shows that the crests of

tk

transverse ave

are about

2,000 feet apart at 60 knots,

_and,

_{at a point 50 feet}

_{behind the}

source of thu waves, the angle between the port and starboard.

divergent waves is only about l-°, This

mean

that the transverse waves

are

incapable of accounting for

the peaks in

_{the part of a}

wake

close to the hull. In fact,

tank

tests on flying boats

normally cover a

range

_{of speed extending above the velocity of}

even infinitely long waves.

_{It is therefore suggested. that tlr}

transverse wave system can be ignuredentirely for the present

investigation.

In a more accurate attac

_{on the problem it might}

need

to

be reintroduced..

Since the crests of tiie

divergent waves in the region of a

possible rear step strean nearly straight aft

at high speeds, it

may not be

too

seriou.s an

error

to suppose that the trough

fills

in entirely

from

_{the sides and hiat the wave system behaves}

_{as if}

it were two dimensional in transverse

_pianes.

_{It is proposed,}

there-fore, to assume that each transversc

_{slice of water, extending to}

infinity on each side and below,

_{conforms to}

_{the hape of the hull}

as

it passes the step and. has impressed _{on it a vertical velocity}

corresponding to the incidence of the hull.

_{The present pzblen}

is

to

find out how the trough fills

_{up under the influence of}

gravity.

The proposal outlined

in. the above

_{paragraph will introduce}

a

_number

_{of errors}

_of

which the following may be expected to be

the most important.

_{The transverse waves}

_{ire being ileglectod}

because of reasons which

_{are valid only at high speeds, at low}

speeds. this neglect may involve an.important error.

Also it

is being assumed that the behaviour of any transverse slice of

water is unaffected by the conditions further forward or aft,

and is controlled. entirely by gravity. _{This assumption, which}

has been made for simplicity, is equivalent to saying that fluid.

velocities and pressure gradients in

_{the direction of motion are}

not importent.

_{Since the}

_{esults of Fig. 1(a) and 1(b) show that}

the water surface behind

_{a plate flattens out quite rapidly}

_even

in the absence of gravity, it

_{is clearly not correct}

_{o ati bute}

the filling in of the trough

_{entirely to gravity.}

_These

_rors

boni so large as to

_{menace the accuracy of the proposed.}

_clution;

but it is suggested that they

will

turn out to be less important

than

the

_{customary neglect}

_{of the divergent}

_waves.

L.

The FilÏin

_{of a DeiDression in the 1ater}

Surface

Althouh the following clevebopnent _{chu be}

_found

_{in a number of}

places, the numeriral values which will

lter he used in this problem strain some _{r the initiai assumptions so seriously that}

it seems worth while collecting the assumptioni together by starting

with the equations of motion.

If

the motion is imagined to be

_{confined to planes at right}

angles to the x-axis, the equations

_{of motion are}

av

av

av

1 = p

ay

av

v w

a

i

_ap

=

-g

at

ay

_{az P}

4.1

or

O

O

'or irrotation1 motion

av aw az ay

-and. there is a velocity potential such that

=

ay

w =

itution of these relatiQns in ¿..l prod.uces the equation

- g

z 4

(y2 +

w2)

4, 2

p at

for an iìcompressible fluid..

If V2 and. w2 are neglected, accord.ing to the usual practice,

L.3

On the surface p = O so that the surface elevation is given by

1

-g

=0

On the understau&ig that the slope of the water surface is

ever'where negligible16 at az

=0

-

+g---Sz

The requirement for continuity is

2 2

=

2 2

3y

a

A solution of these ecuations is

Kz

-

sin ot. e cos Ny

Report No. Aero.1992

where A

anJ K

are still quite arbitrary Luid.

2

c

=gK

When t

0, 1.

5

becomes merely

=

A cos Ky

At t

O it is necessary to combine a nuber of waves like that of

¿7 in such a way as to form the desired. shae of depression.

This

can be done by means of the Fourierintegral1Ç

.An initial depression

or elevation of the form

=

_f(y)

will be given by

/

t=0

=

f

J

f(s) cosK (y

- o) de.

(

O -CD

co

Since the final solutiOn will certainly be symmetrical about the

fore-anüaft centre line fo

normal forms of flying boats, only the

coz Ky teìs in equation

4.

8 need

be

retained.

co

2 (*

t=o

=

f(0)

Ky cos KO dO. dJç

-o

'.0

To be correct for times other than t

=

0, this equation needs to be

modified in accordance with 4.5

CD

Thén - (coz t) f(o). cos Ky. cos KO, do, 0K 4,9

-

j

o Jo

It is now necessary to specify the shape of

the

trough at

the

step of

the

hull by

defining I(0).

In default of quantative experirnentaJ. information, sorne cpflcuìations

have

been made for a number of

arbitrarily

chosen trough shapes.

4.1 Troe by a Flat Bottom Hull

Por a flat bottom hull of draught

5

and beam

, as illustrated in

Fig.3, f(0) in equation 4. 9 has the ±ollowing

values

4.

6

4.8

S-7-f(o)

-8

for

f(e) = O for

2

2

8

The variables in the integral on

L, 1. 1 can very advantageously be made

way. The time t whieh has elapsed

the trough by the hull is related to

t It

is

conrit alsO to isolate

Froude number based on the hull beam.

i

2

ccs jjTc t. cos Ky. cos e

ä. dK

i

/-cas gK t. cos Ky. sin ¿1K

the right hand side of eation

non-a meusional in the following

since the initial fonning of the speed of the hj7U by

pitting

the group -- F, the

gb

In non-liniensional form

the

distance X of any point behind the Hill step is

x

- a

X

i

y

b F

Also n Kb. Then 4.. 1. 1 becanes

1

-

n

-

COS n X. sin . cas n Y. th

In

2

For brevity it is convenient to refer to the expression on the right

side as (X, Y) say; i.e. as the contribution of the initial height .of the water surface to the elevation at sorne point owii streom.

= i1

(x,

Y) 4.1.3

Since it has not been pos.sible to express E1 (X, y) in terms

of tabulated integrals, it has been evaluated by nrithrnetical summation, and its values are plotted in Pig.9. In view of the excessive labour involved and the doubts which have already been introduced into the results for other reasons, the range of integration has been taken from

n O to n

= 35.

4.t n

= 35

the factor in the integral makes the

n

high frequency harmonics so small that the accuracy is quite

Rcport No. Aero. 1992

The .curves given in Fig. 9 are plotted for constant values of

y = i. e. they are longitudinal sections through the wake, Some b

of the characteristics of the waves behind an. actual flying boat appear in this graDh. For instance the usual rising centre (or

leroachu) in the wake is seen to appear in the neighbourhood of

X=2.

4..2 _{Flat Bottom Hull With Bow Wave}

It is by no means clear that the initial conditions assumed in Fig,3 approach thos found in practice, In fact, a planing flat

plate usually makes quite a large bow wave which runs past the edges

and. streams aft. There is unfortunately no information available concerning either the shape and size of such a wave or the variationo

of the dimensions with Froude number. The assumption is therefore

maie in Fig.24- that all of the water thrown out in nicking the trough is heaped up in the f oxm of a conventionalized bow wave. This

asstnption is, of course, very crude, but is convenient for ari initial investigation. The subsequent motion can be found by suitable

superposition of system s of the form given in Fig. 9.

The initii boundary condition is

= - for

= 8 for

= o for

and the -result turns out to be

1 n - . _{003 jn L San - .. con n Y.}

di

2 8 1

-. con n n

112 is plotted in, Fig. 10.

4.3

Troih Made by aVeç Bottom Hull

Bere. .discussig the even wider varIety cf possible _{trough forms}

f _{a vee bottom hull it scemo bst to consider -the filling of a al&in}

vee trough like that of Fig.5, _{Here f(0) which appears in equation}

4.9.

takes the values

_

9--fl n

X. sin, conY. dn

2 2 4. 2.1 = 112 (x, y) .4.2.2

-

10

-s

-

20

for

o'-.o b

2

f(0)

= o

for

o7

This second condition is equivalent to saying that there is. no

bow

wave, and. might be based on an assumption that a vee bottom hull

makes its trough by throwing all of the water clear as spray.

2

Jo

cos

/i

t. (1-

).

cos Ky. ces KO. dO. dX

In non-dimensional

form

co

n

- (

J_.

cos

fr X. (1 - cos

) cas ny,

dii.

6 _J0 2

H3 (X, y).

This integran has been evaluated numerically ar. the

r.sult is

plotted in Fig. il.

4.4-

Vee Bottom Hull With Bow Wave

s a flying boat accelerates from

rest, the dyniic pressure

P U of the water soon becomes much greater

than the

static pressure

difference g P & corresponding for instance to the draught of the hull. In fact

b

U2

-

b

gP8

F

being

the Froude number. For this reason the bow Wave first

grows

in size as the

speed increases;

but, at very high

speeds,

the

effect

of gravity on

the flow

around

the

forebody may

be expected to be so

small that the

fluid motion approaches the

three dimensional ecuivalent

of that illustrated in Fig. i(a),

This variation of the inDortance

of

the bow

wave

with

speed

makes a

precise discussion of

the wake shape

difficult.

In section 2.

3.

has 'seen given a solution for a trough

without a bow wave; it is now proposed to

take the

other extreme in

which ail the water from the trough

is

formed into a

bow wave as in

28

ö(1 +2

-)

for

b

)

The appprìate valuos of f(o) are

for a point on the free water surface,

(ii.)

'O

2

or

O

(i

The resulting waves formation, which can be represented by a

suitable

superposition of waves of the shape shovn in Fig.5 is

given by

p:E

_¡

_f

2(1)

J 1

/2

:0

(

/_

x

n

-/ 2 +

J \

L

fcos

1T1

X(cos

n

1 2

_/

1+ 2

,I

Values of this expressionare charted in Fig. 12,

5.

Waves Due to the Dornwash Behind a Hull

The previous sections have been concerned with evolving solutions

corresponding to a nuner of assined initial disrlaceiuent conditions

at the hull step.

It is next necessarr to give the water a suitable

vertical velocity so tb.t it may stream smoothly off the aft end of the

hull bottom.

The relation to be satisfied at the swface o

the hull is

(U+u)tana_vt

w=O

5. 1

Bernoulli's

equation is

(Uu)2v2w2_2g"=U2

Report No. .&ero.1992

-_ _)

for

O L

(i.

2

On the fore-ath-aft centre line where y =

2 g

a - i

U _U2

on the centre line at the hull step. By comperison wIth equation.

_4,4.

it is seen that a velocity potential

2 = COS

\(rgx t. _{Cos Ky}

_5.4.

would. satisfy the equations and. produce a vertical velocity

- _{= B cos ¿K t. ces Ky} a

at the surface.

If the initial d.ownwash distribution is

at =

t=o

the c9-istribution at a later tie t is

2

(

(

-

-/

cos1Ç

t. f(e) ces Ky. cos KO. de.

dx

jo

5.1 Uniform Downwash

For a flat plate, the deadrise angle = O, and.

equations 5.1

and

5.2 yield. the following expression for the d.ownwash

velocity

at the trailing edgo. w /

2g

2 V sjn a U _{U2 u} 12

5.3

5.5

5.1. 1

+ 2 g sjn a U _U2

At speed.s so high that U2 ,> 2 g ?

w

Report No. Aero.1992 In accordance with the asiiiption elready made that V2 is negligible

in comparison

vrith 2 g

8 and that2 g 5 is much less then U2, this

reduces to

f(e)

= -Usina

2

..

-3:;.

Usina

sin a 5.1.2

This means that, to the extent that the above assurnptions are c'rcct,

the dowash

mealately behind a flat plate is constant. It will

be necessary to discuss the errors involved, in a later section and some

correction may

be needed to bring the results into agreement with

experimental results.

The values of f(e) to be substituted in .6 are

cc

= - -. u

sin a 1 i

sin

t. cos Ky.

sin K. dK

J

3/2 2

gK

In the non-dimensional iiota±ion used in section +, 1.

- 2

. sin

x.

sin . cos ny.

F. b. sin C n 3/2 2 5.l.L. is conveniently abbreviated to w1 (X, ) F. b. sin a for O ¿.O. 13 -b - .

oos gK t. cos Ky. cos KO. do. dK

5.

_!5

The integral

_w1(x,

Y) hás been cvaliiatea. numerically, nd a number of curves are plotted in Fig. 13.

f(e) o for b

2

2

Usinai

- cos

gK t. cos Ky. sin K. p 5.1.3

Equation

5.1

shows that, for points below the water surface,

the expression

analogous to

_5.

1. ₃ is

-sine

U

b

e

r-

--

_n.

cos n X. cos n

Y. sin

ctn 2 y

2.

=

sana U e

-

_co

X. sin sin n Y, th-i

5.

1.6

o

n

₂

ituld.

be possible to compare values of c'1culated from

this

U

relation vth experimental results

of

the type

described

in

reference

13;

but the technicue

of

reference 18 has the advantage that

it is

capableof finding the direction of' flow outside

the bondary

layer

on the surface of the planing plate.

5,2

Triangular Dovinwash Distribution

Equation 5.1 indicates

that

a larger deadrise

angle

¡3 tends

to reduce the dn'mwash near the sides of a vee bottom hull. In

default of detailed inÎorhation it is

proposed to assume

that

the

downwash decreases linearly from its rnadmum

value

at the centre

line

to

zero

at the sides of

'the h,ill.

This assumption is rather an

extreme on0,but it may be

useful

in combination with the

result

of

section 5.1 or it may serve 'to cOEnpensate for some of the other

defects ot the present investigatio It should be csidered a

rough

first guess because it arid an equation

similar to 5.1.6

are

very unlikely to satisfy equations

5.1

and. 5,2.

Then the f(0)

pearing in

5. 6

takes 'the following values

f (0)

= -

U s iri a (1 _{- _)} f o. _O

- o

f(e)

O for

2

The

resulting wave formation is given by

1

sin Jn X. (1 - cos ). cos n Y. dn

5/2 2

n

5.2.1

I

If the motion behind, the plate were

cor'ined

_{to planes at}

right angles

to OX, as ha already been supposed, the equation of cóntinuity would be

av

aw

-F. b. sin a

Values of this integral are given in Fig. 14-.

The corrspcnding transverse velocity distribution is

'b

r-= sin a _f . cas n X. (i - cos ) sin n Y. dn. 5.2. 3

u

_J

2

6, Downwash Angles

Athough

it seems

very unlikely that the present investigation

is complete enough to make possible an accurate calculation of the

argle of d.ownwash at the position of a rear step, it may 'be

usefulto set down the appropriite equatiois. Sections 6.1 -6.4. discuss the effects of the initial displacement of the w.ter

surface;

6.5

end 6.6 considet t1'e

results of initial dowiash.

6. J. Flat Bottom Hull (Fig.

3)

Displacement of the

water

surice, equation 4-.l.2

Icas q'

X. sin . ces n Y. dn

j n

2

o

au.b2;-/

sjnnX sinn. ccsnY.dn

6.i.i

ax

i.'..J

. '2'

is the fore-end-aft slope of the water surface, and a positive

ax

sign ïndicates an upwash. The integral on the right hand side uf

6. 1. 1 has not 'been

evaluated.

6. 2 Flat Bottom Hull With Bow

Wave (Fig.)+)

The displacement of the surface has 'been given in equation

2, 2. 1. Differentiation of 4. 2. 1 with respect to x yields the result.

00

2

... sin

L x.

sin

,.

cos n y dn

=

6.2.1

i n n

-

--. sin

X. sin - cos - Y, an

-

2 2

15

-5,2.2

r

6,3 Vee Bottom Hull (Fig;5)

The fore-and-aft slope of the water surface at some point

behind a

vee bottom hull running at zero incidence can be foimd.

by differentiating

equation 2 3,1 viithresoect to x,

The result is

'i

x

-5

-6.5

Surface Slope Due to Uniform Initiai Downwash

(Fig.7)

Equation 5,1.3 is the

result required.

In non-dimensional f orm

,co

= H

(X, Y).

See Fig. 9 for nurerica1 values.

6.6 Su.rface S1oe Due to Trianu1ar

Initial

Downwash Distribution

(Fig.8)

/

o

i. positive value indicates an

upwash.

6.2+

Vee Bottom Hull with Bov Wave (Fig. 6)

The f ouiae analogous to

6. 3. 1 can be derived from

i

/

r-\

2

'sin ¡_n

x\Ji_cos

3/2

I)

\

sin

3/2

i.

3/2

sìnX. (i

- 16

Report No.

From 5,2,1 it is found that

cosX. (1- cos)s cos nY.

2

H

= H3

(x, y)

-cosa). ces itYdri

6,3.1

2

- .lcos

₂

(i_cosa,). (cos

n

di

6.6.1

i

2

sine

ax

i

sin a

ax

oos

x. sin

cos n yi dn

6.5.1

17

-Report No. Acre. 1992

which is plotted in 'ig.l1.

Experiienta1 Measurents

Observations of the wave formation accompanying a

flying boat

'e Íequent1y described, and

det1ed measurements have sometimes

been made'9,

Since the present work was concened

meinly with the part

of the wake in the vicinity of a conventional

afcer step, mot of the

measurements described below vee made fairly close

to the model hulls

used.

Then these experLuents were started, the procedure was to tow in

the tsniç a

constant speeds a wedge-shaped model whose

&tìtude and

draught could be fixed.

A transverse row of pointed rods some

dis-tance behind the step was adjusted until each rod just touched the

surface of the trough behind the model.

The vertical displacement

of the water was found from a comparison

of the rod heights

with

their pasitions whentouching the water

at rest.

This technique

was rather incorvenient, however, for work at high speeds.

The

duration of a single runvas then so short

that it was usually necessury

to make several trips in order to map the cross

section of the trough at

a single fore and aft position

behiiid the hull.

The pointed rods were

abandoned, therefore, and. in their place was put a row of thin

stream-lined wires which could be allowed to cut through the water without

careful setting.

The measurements were made as follows,

Photographs

of the wires were taken by means of the arrangement

shown in Fig. l5,

and the actual trough dimensions were deduced from these photographs

(examples of which are given in Pig.16) by referring the

water surface

both in motion and at rest tb the black band

around e.ch wire.

A eries of wave measurements is given in Pigs. 17

and 18 for

comparison with the theory.

Th

examples of Fig.16 are included merely

as additional information.

Those observations were made too far

behind the step to be of very great interest from the

present point

of view,

It was found that, at model speeds below about 20 feet/second,

and in the. absence cf an afterbody, the trough was smooth

end

glossy so that the observations were easy to make,

At higher speeds,

however, the surface of the wake become rough and ill-defined so

the

results were less accurate,

Comparison, of the Theoretic1 and

cperimental Results for Vee

Bottom I-lulls

8.1

The present investigation ieglects'so many factors which may

have an inportant influence on. the wave formation that it

is not

clear what combination o1 the initial conditions of Pigs.5 -'8

should

lead to the best overall result,

In order to choose between the

initial displacemen conditions of Pigs. 5 and 6, calculations based

on both of thesassumptions, combined in

turn with the assumptions

of Figs.7 and 8, have been.00m arad with observations

made beh4nd hulls

planing at small incidenc

Under these circumstances the initial

ciownwash velocity was small and the initial shape of tke trough was the

dominating boundary condition.

This comparison led to the adoption

of the initial trough shape of Fig. 5 in preference to

that of Fig. 6.

From the 'next series of

in which the only variable ,was the

hull incidence, it was ascertained that, in conjunction with II3

the

function W1 (x,

) of Pigs. 7 and 13 was in good agreement with

x'oeriment for'points just behind the step, but consistently

Report No. Aen.l992

assuiñe

an

initial downwashdistriòutjon like that

_of

Fig.8.

_The

resulting contribution

to the

trough,, W2

Fi.14., was

found.

to

underestimate the depth of

the trough

near the sides and. to

over-estimate slightly that in the centre.

In

an

effort to improve

the

agreement with experiment, several

compromises between

Fig.7

and 8 were tested. In comparison with

the ex-oerimental data, however, all of these solutions were

found to

predict, too small

a

value for the longitudinal

curvature of the

water surface just

behind the step.

It

was finally concluded that

this deviation of the theory from the facts was due primarily to

the neglect of fluid velocities and pressure gradients in the OX

direction. Perkaps, also, the discrepancies have been unnecessarily enlarged by an error in visualizing the initial conditions for the

present calcuJ.,ations.

The checking and correcting

of these defects is likely to be

so laborious that it se s justifiabL. as an interim ineasure

to

use a small

arbitrary correction in

_{conjunction with W2, which agrees}

more closely rith experiment thn any of the othr

functions tested.

It is

proposed,

therefore, to use the value 0.9 U sin a for the

domash velocity at the keel. With this modification the best

avallable representation of the experimental results seems to "be

+ 0, 9

F. b.

W2 sin a

8.1.1

Values of

L

calculated from 8,1.1 set out in. Figs0 17

8

and 18 for comparison

with the experiments.

It is

seen that, except

for the case of

18 d, the

calculated and observed, trough depths

on the centre line agree to within about 18 of the draught at the

than step. The errors are much larger at the sides of the trough, but this is perhaps not too serious for work in connection with the

narrow pointed, rear steps now in common use. _{The results plotted}

in

Fig.

18cl have been included as an example of the poorest agreement

between the theory

and experiments that could be found..

Fortunately

the absolute errar implied would not in practice be

as serious as it

appears. For . flying boat with an afterboady length of x 35',

the wetted beam corresponding to

-i-

-

3.31 and F = 5.15 would. be

Fb

1. 9L,

and. for. a

_{dealrise of 25° the draught would be only 5j.}

The error in estimating the position of the wate'r surface would amount

to 3". On a

base length of

35'

a change of 3"

in the clearance between the after step and the trough could occur also

through

a change in

attitude of about

j°.

It is an extremely interesting fact that the theoretical curves describe

some of the most notable features

_{of the wave system with a} fair degree of accuracy. The rising centre or "roach" in the wake

is well defined., and the subsequent splitting cf

_the

ridge into.

two

divergent waves is illustrated "by the theoretical curves in Fig. 18

a, b, e, but the experimental observations do n9t show this detail as

clearly as it usually appears to the eye.

8.2 Com.arison of Theoretical and. x rimental Results

for Flat

Plates

_

18

-j

J2 it

as suggested.

that the bow wave in front of a p1±ng flate plate should be taken into account

when discussing the

Report No. Aera. 1992

therefore been aloptal for the following work, ani the initial dowuwash velocity has been taken to be 0.9 U sin a asin 9.1.1, but stribitd. as in Pig.7. The resulting expression is

=8H2+0.9FbW1 sin a

8.2.1

There are not a great many experimental rneasurnents with which

to compare values secured. from 8.2.1, but Sottorf has given in reference

1 some observations ma.e on the centre line of the trough behini a

flat plate. ig. 19 compares the experimental dnta of reference i

with the theoretical fomulae of reference l (ouoted. in section

1.2 of the present note) and with curves based on eQuation 8.2.1 above.

At a small incidence and. 1rge draught (Fig.19a), the present formula

is much closer than that of reference 3.4 to the experimental curve;

but, at larger incidences and smaller draughts as in Fig.19c, there is ittle to choose between the two formulae.

In a comparison like that made above, some uncertainty may be introduced by the fact that the "roach" bebind a flat plate is some-times exaggeratedto such an extent as to become a pLise of loose

spray rather than a wave. The calculations do not take account of

such a possibility. Subject to doiìots of tbis kind, equation 8.2.1 is recommended. as a suitaole basis for estimates of the wave profiles

behind. flat plateSa

9. Use of the Results

Por convenience jn use the present results are briefly simjmarised

below1 The fomulae should be considered. to be ap'oximations which may need to be modified as further data accumulate.

9.1 Flat Lottom Hulls

= ti + 0. 9 P. b. sin a

= elevation of water surface at . point distant x behind.

the step and y to port or starboard of the centre line.

8 = hull draught at the step.

hull beani at step.

incidence of hull bottom.

Froiic9e number U (Consistent units).

forward. velocity.

displace:ient of surface due to initial dmpresion, plotted in

Fl0,

displacement of surface due to dowuwash behind step, plotted.

in Fig. 13.

9.2 Vee Bottom Hulls

= 8H3+ O.9,P. b, W2. sina

= elevation of water surface at a point x behind the step and

y to rt or starboard. of cenere line.

19 b = a = F = U = H = 2 W1 =

8

draught at step.

For steps vdth a plan fonu of vee shape,

8

iaight be ten as the irmnersion of the keel at a

fore arid.

t station midway between the position of the step

at the

keel and that at the chine.

The results may be less accurate

for pointed than for straight steps.

b

beam corresponding to a still watr draught of 8

If the chines

are immersed, it seems probable that an

extended beam

b1

28 cot

should be used for a hull of deadrise angle

a =

incidence of hull bottom,

P

= Proude number

-

(Consistent units),

g b

U =

forward velocity.

H =

displacement of surface due to initial depression, Pig.11.

W2

displacement of surface due to downw h at step,

Fig.. lL.

9.3

Downwash Angles

The integrsLc appearing in some cf the dowash

equations of

section 6 were not evaluated because the probable accuracy

and

ii.portace

did. not seem to justify the labour.

If,.. however,

the inclination of the water surface at some place

behind the main

step is required, it can be found from the difference in elevation

at two points one slightly ahead of and one slightly behind. the

station in question.

10. Effect of the Afterbody

When the work described above was started, it was assumed that the

terbody would have no effect on the trough behind the forebody but

would plane on the surface of the trough as if on the surface of a wave

of the correct slope.

Since then, however, evidence has appeared

which

indicates a reduction in the pressi'e of the air in the sce

bween

the afterbody arid the surface of the trough behind the

forebody

Until further data are avtì1able, it is suggested that the effects of

these changes of' air pressure be Ignored.;

but It is clear that such

changes will produce a Ditching moment on n hull and

will also alter

the level of the trough.

The method of reference 6 seems to be

a suitable one for calculating

the latter effect.

II. Conclusions

The first conclusion which emerges from the above investigation

is that, daring the faster parts of the take-off and.

landing runs of

flying boats, the divergent waves make an important

contribution to

the motion of the water some distance behind. the

hull,

The shape of the

nearer part of the divergent wave

system has been calculated.

approxi-mately and. has been compared with ti-ia wave formation produced. by a

model hull.

The coelparison shows a number of systematic

discrepancies

which are attributed. to other ngencies

which bave not been considered

in the 'esent discussion.

The most important factors left out of

accourit are believed. to be the pressure field.

around a hull and the

transverse wave system.

Since, however, the approximate

calculation

of the divergent wave formation appears

to be capable of describing the

main feares of the experimental

observations, t he liberty

bas been

Report No. Aero..1992

taken of introducing a single empirical downwash correction factor

to compensate for some of the missions. This has not been done with any theoretical justification, but merely with the object of making the present results available as an approximate basis for

practical cculations.

Sugestioris for Further Work

Because of the difficulty of obtaining

as

analytical solution for three dinensional flow past a flying boat hull, it is suggested that an apprxiivation could be evolved by means of the analogy between

electrical end. rd.rodynainìcaì potentials. The idea might be, for instance, to place a non-conducting model of the under part of a hull in an electrolyte carrying on electric current, and to map the

current paths end strengths. The free surface of the trough behind. the hull (in the absence of gr.vity) could. be represented by me ans of

a rubber mcithrane stretched over a nueer of screws which would. ned to be adjusted until the membrane representnL a surface of constant

pressure. A lateral flow of current might have to be arranged under the forward part of the hull to rerresent the spray thrown out by a

real flying boat.

The present experiments have not included any work on the downwah

behind. hydrofoils. _{It would be useful to find out whether or not}

an appropriate adaption of the preaent fornulae irou1d serve as a

working basis for calculations. The method of reference 10 should. be applicable to motion at low speeds.

It is important to find a method for predicting the behaviour

of the trough behind a porpoising hull. _{The present work was}

meant to be the beginning of en investigation in this direction, end the apparatus for the eqerimental part of the work was made; but the whole progrsiruee had. to be abandoned in favour of more urgent problems.

References

No. Author Title

1 _{Sottorf Exerieents with gliding surfaces,}

_{s. 385,}

A.R.C.

₃₆₅₃

Perring and Eydrodynomic forces and moment-s on a sieiple Johnston planing surface re4 on a flying bcat hull.

R. &M. l6L.6 r

3 Gott Analysis of the force on seaplane tenR models

int hydrostatic pressure, hydròcîynomic pressure, and skin friction, _{Report No. B.A.]J1)11,}

Smith and A review of porpoizing instability of seaplanes

whit e _{M.A.E.2. Report No. H/os. 173, l9).}

5 Green Note on the gliding cf a plate on the surface of

a stream, Proc. Carnb. Phil. Soc.

32, p.4-8,

1936.

Lamb Hydrodynamics,

_1932.

p. lC.

Greenhill The theory of a stream line past a plane barrier.

R. & M. N-o. 19. _{Example 2), 1910.}

tachei: Appendices I - III Table i Fig.i 2 ₎ 3,4,5,6,7 and8.

9andlO

11 snd 12 ". 13 and. J24-t! 15 i6 " 17 ' 18 u ₁₉

R..E.

Report No. B.1678 (5213), May, 1941.

Drg. No. 15655s T? l5656s t! t! _15657s It t? 15658s .1! t! l5659s t? t!. 15660s Neg. No. 53623 Drg. No.

l5661s

t? t? _15662s t? ! 15663s 22 -i Author References (Contd.)

8 y1eigh The f or of standing waves on the surface of

runnin water. Proc. lend. Math. Soc. XV, p.69, 1883; Papers V. II p.258.

9 Lamb. Hydrodyncinics, 1932. p. 398.

10 Kelvin On stationary waves in flovng water.

Phil. Mag. )0(II p.517, 1886; Papers V1V p.282.

11 Wagner On the gliding of bodies on a water surface. Proceeding of the 4th International Congress for

Applied Mechanics. p.126, 193/4-; ReDort No. A.R. C. 1933, 5.2.52.

12 Pavienko On the theory of gliding. Proceedings of the 3rd. International Congress for Aplied Mechanics. V I. p.179, 1930.

Dursud Aerodynamic Theory, 193/4-, V VI p. 13/4-.

1/± _Shoemaker Tonic tests.of hat and V-bottom planing surfaces. N.A.C.A. Tech. Note No.509, 193/4-.

15 Lamb Hyrod.ynamics, 1932. i.4Y--.

16

Lamb Hydrodynamics, 1932. p. 365.

17 Lanb Hydrodynamics, 1932. p.384.

18 yard A new method of studying the f Jmw of the water along the bottom of a model of a flying boat

huÌ].

N..A..C.A. Tech. Note No. 7/4-9, 1940. 19 Bottomley Experiients with hLodels of seaplane floats.

R. & No. 365.

- 23

crt No, Aero.1992

Circulation:

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Aero (i), T/Á, P, S , S2, F, F/A, L, H(2), E, W,

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_No.6141R

i

Appendix I

Gliding of a Plate

on the Surface

of a Liquid

Liquid Free From Gravity

Fig.l(a) illustrates the two dimensional equivalent of a hull

planing

on

the surface of a liquid in the absence of gravity.

For

even this drastic simplification of a flying boat hull', the solution is very involved5; but, if the characteristic spray or jet in front

of theate is assumed. to be suppressed as in Fig.l(b)' on the ground.

that

it will not have very much ffect on the flow

behind

the plate,

the problem becomes much

sijnpler0,?,

Although

solutions of this kind. dc not represent the actual

state of affairs sufficiently completely to be useful' in estimating

the form of the trough behind a three dimensional 'hull, they show

that, even in the absence of ¿ravity, the watèr surface behind a

planing plate flattens out quite rapidly with increasing distance

froni

the trailing edge.

Liquid Sected to Gravit

If the pressure distribution on

the under side of a very wide

planing surface is measured experimentaflr or is estimated from the results-'

of references 5, 6 or 7, it is possible to calculate the shape of the wave profile behind

the body

by a method due to Rayleigh8'9.

A convenient formi of

his

result is

that

if the applied pressure

is

infinite at the origin and. zero everywhere else

in such a way that

pdc=P,

the surface elevation is given by

for x ' O,

-

2 iz sin . +

u2

i. e. on the downstream side, and

- 21 K +

u2

cosKx

_dK P cos K x u2 K +

u2

for x

O.

The effect of onyeSsure distribution

can

be found

by

superposition of the disturbances

2.1

and

2.2

due to concentrated

pressures.

p

is te density of the fluid, U is the forward speed of the

planing surface

and x

is distance measured downstream from an

arbitrary origin at rest with respect to the plate. A n'nbe of

investigations

based on

this

solution and similar work by Kelvin have been made in

t

_P

2. 1

r

Report No. I.ero,l992

more recent years11' 12

ai

are very

rief1y reviewed in reference 13.

In solutions of this kind the .ppray in front of the plate (Pig.la)

is suppressed by means of a sharp cambering of the foirard part of

the plate as in Fig. 1(c).

in the same way, any deviation of the

assuned pressure distribution noar the trailing edge frdL the correct

pattern involves a more or less marked camber near the aft end of the

plate.

Unfort'únately an error of this kind makes it difficult to correlate

accurately the wake shape and the clrauht and incidence of the plate.

In reference 13 these difficulties are avoided by supposing that the

water surface behind the plate can be represented by a trochoidal wave

satisfying the caditions that, at the trailing edge of the plate, the

water surface displacent and s'ope are equal to the draught end

mci-dence of the planing surface, i. e,

-8

at the trailing edge of the plate.

In order that the wave may travel

at the same speed as the hull, the

:ie1ength is given by

2

2t1J

g

For small incidences and high speeds the wave amplitude A

is

2

A.

IS

tan a

t,

Reference 13 states a nujiaber of 1iitations on the applicability

of these equations;

but, even within these limitations, calculations

based on

2.3

and

2,4

do

notbear comparison with visual

observations of the waves behind fast boats or flying boats.

For

instance, equation

2.4.

predicts that a planing surface travelling

at 60 iciots at an incidence of

5

would be followed by a train of

waves about 50 feet high from-'ough to crest.

Another set of relations satisfying the same s10

and dxaught

con-ditions at the trailing edge of the plate has been developed in reference

14-, and has been compared wtth experimental measurements for tiaree

dimen-clonai flow.

_{In the present notation the main result is}

r- ,

\\

:=3 -

/Lgtana

btan a

_2.5

V

Pb

L

/

where L is the 1'rdrodynamic lift and p is the density of the liquid under

the plate.

_{The derivation of this formula was based on the assumptions}

that all of the downwash has the same downward velocity

as it passes the

plate and. that the cross sectional area of vertically moving fluid is

constant for all distancbehind the step.

This mass of fluid is then

supposed to oscillate vertically under the action of gravity in such

a

way as to keep the sum of its potential plus kinetic energy

constant.

These assumptions ignore the dispersive properties of the

waves;

but,

as is shown in reference lL and Fig.l7 a

the end of this paper, the

agreement between equation

2. 5

end experiments is cuite g ood. at high

speeds.

- tan a

cx

2.3

-

26

-A?pendix II

Waves Due to a Travel1inz Pressure

Point

The wave system f brmed. on the surface of water by a travelling pressure point has 'been investigated aproximately15, arid the cal-culated

pattern

of

the

wave

crests

is reproduced in

Fig.2.

If the

speed

_{of the pressure point is U, the co-ordinates of points on}

the

wave crests

are

given by the two relations

2

x=n

_{(5cosç-cos3ç)}

2g

y=--n.

_{_J(sinçsin3)}

2g

where varies

from

O to -.

n can

_{tace the values 1, 2,}

_3.

corresponcHng to successive waves represented by the curves

_{shown in}

Fig.2. The position of the point (x1,

o),

_{for instance, is found}

byputtirgnland.=0.

Thenx12U2.

idea of the scale of a system of this

kind for speeds

comparable with

those experienced during the taking off and. landing of flying

_boats.

Because of the growth of the wave pattern with speed, a constant length "," would

appear

_{in Pig.2 to shrink at}

_high

_{speeds due to} the decrease in Por this reason

the

divergent waves at a

xi

distance

behind the pressure point

would

appear to stream more

nearly straight aft at high than at low speeds.

The third colunin

of Table i gives vhiúes of the angle between the port and.

starboard

divergent waves for which n

= 1, at

a distance of

₅₀

_{feet behind. O.}

27

-u

('roTs)

x1 (FEET)

IJTGL BETWEEN RT

DI1TERGT

WES O

FT' H2-IIND O.

10 56 63° 20 223

13

40 ₈₉₀

3.20

t

_0OO o

1.4.

80 4L.50

0,6°

100

_55D

ppcndix III

Sources of Error.

1lthouh it is rather clar:ìing to see the risks which have

been tskcn in :akin some of the assiziptions in the foregoing

onalysis, the fact that the results aro at all coarablo with

coo':ion experience means that soao of the errors are not os. serious

as they might SCOES.

Though rather toabus, it seems to be worth

while

ontionin

a nuiber of then.

The first three ass'ztions aro

usually

ado in vîve problems ord aro not peculiar to the present

iis cuss ion.

ssiption that the flowis irrotational, section !.

In the

rc5ion outsic].e the boundary loyer, this assution is probably safo

for veo bottoa hulls rwming at hih spes.

Por deeply Liiorscd

vc bottoi hulls and. for flat plates, howevcr, the motion in the

trough is ccrtaly not entirely free from ediLes.

Nc1cct of thsuaros of the iiturbco velocities.

This is

probably not iuportoxt at small incidonces but moy bu artly

responsible for the errors in the theoretical results for high

incidences.

_{Thu success of the method of reference 14. at hjrh}

incid erices rcinf o r ces this view,

(a)

ssution that the lc of the water surface is everhcre

nog1i.ible.

_{This ass'uption, cade in s:ctìon Lp..}

_{has boon}

violated in section 4.1 to the extent of moking part of the surface

vertical.'

In 1os

_{extre caseswhere the surface is not steep}

en?ugh to collapse

in

f OrO, the error

in

moJd.ng this ass'rrption moy

not be too important.

_{It should be pointoi cub that the solutions of}

references 8,10,

and 11 ossnc the sane thing,

Neglect of fluid vo1ocjtjs in the direction of motion of the

hull.

This is boliev

_{to b0 the inst serious defect of the present}

investigation.

The error from this source

con be expected to be

least importent at small incIdences and hi.h speeds.

¿.n attempt has

boon male to reluce this error under other conditions by treating

the flow close to the b1y

in

the manner of Pig.l (b) end that

further aft by the present method Cor a divergent Wave s3rsto, hut

the accuracy promised to be only

_{a li:ht improvement en that}

of sections 9.1 and 9.2.

NeCia et of the

_{transversa wave system.}

_{The error from this}

source is likely to be important whon the wavelength of the transverso

waves is comparable to the length of the hull.

_{Seo Appendix II.}

Errors in th initial conditions.

The initial boundary

conditions for the present

_{wave calculations have been based on}

visual data the dotail

of which are very incorçleta,

5PRAY

FREE 5UFAC.E

__1_'

-NO WP%VE

AI-4ED

U

. PLPNN OF A PLPTE. IN AB5ENC.E OF

RVTY

u

FIGS. I&2.

F1G.I

OEPRESION OF FREE

SUR FACE _{EHND PLAT E.}

COVER LAID ON FREE SURFACE TO 5UPPRE5S 5PRP/

u

b. SIMPLIFIED SEPRE.SENTArnON OF FI

I (o.) WITH NO SPRV

PRESSURE DISTRUTION

TRAVELLING PRESSURE. POINT

__,,#P0*"

DIVERE.NT WAVE. CRES

TRANSVERSE. WAVE. CRESTS

DEPRESSION SE.HIND

PL.rE.

TRANSVERSE WAVES BEHIND PLATE

C. PLANIN

OF A PLATE ON P LIQUID SUBJECTED TO

RA'JIT1.

PLATE. _{CAMEERED TO PRODUCE. ASSUMED PRE.5SURE. DISTRIBUTION.}

TWO DIMENSIONAL FLOW PAST PLANING

SURFACES

WAVE FORMATION DUE TO A SINGLE

MOVING PRESSURE

POiNT.

4

4-t

Y

i_i

¡k

ti

FiG3.

TÖUCI-4 MAOC

A

BOTTOM HULL.

(viEw or src FROM ASTERN.)

UNO4STURB.D LVL.

FI.4.

TRoUH MADE BY A PLAT

BOTTOM HULL WITH A

CON VENTIONAIJZZD

REPRE5NTATION OF ALAÇC

BOW WAVE UNNN CLO5

ALO NCS1O&..

flG.5.

TROLXH MADE BY A

VEE BOTTOM HL)L.L.

-- WATER FROM TROUGH

ASSUMED TO BE THROWN

OUT AS SPRAY.

FIG.6.

VEE BOTTOM HULL

wrr BOW WAVE.

CONSTANT DOWNWASH

DISTIBJTON

(vw FROM ASTE.RN)

F1c.8.

TRIANULP OOWNWASN

DI$TRIBL)TION.

FIGS.3-8

t i"

-I0

I0

-I0

AI'

a._

-__I.

-..._

_iSIUI

I

-( .si

'Yci7t 2 I

-- 7.

-__

!

Ra

Î_m

L-, FORM OF TROL)CH AT SltP.

.o

_a

s-

_EI'-'

mmii_a.

_U2UU

-..

1IU_...

.

'IPAIIPq'IJ

T

;___

JpI.

WAVES BEHIND A FLAT BOTTOM HULL WITH AND

WITHOUT A BOW WAVE.

FÎGS9 & to.

05 o

Io

H4 OO

IO

5HAP OF -rOucH AT STEP.

WAVES BEHIND A VEE BOTTOM HULL WITH AND WITHOUT A BOW WAVE.

FIGS. II&12.

FIG II.

FIG 12

$ 08

L_____

-Vi

-

Y',

..

-j____

I

-2O

J

o.

JL_-. Yso

SU

'(02

SRUIîUUdi

_

jr

.

.-).

.t

-'&'J.VA.

.:.O..ppp.

_

';::;::;::

WI O

-O5

w

w

-r:

. SIN X . J m. I&

WAVES DUE TO DOWNWASH BEHIND A HULL.

INITIAL DOWÑwASII DISTIJTIOt4

FIGS 13 & 14.

FiG 14.

IL!

::

?S0LIP_

i I_LIUNUII

os -. I O

11111

INIT AL DOWNWASH DSTIurnUM

I _

I_-k

_____ ---.

,

.'9,uuupppp

o 'e.

LENTICULAR

3CTION PRONG

5ECTION

TWICE FULL SIZE

CAMRP J

k1

MIRROR

t

\r

'-X

VIEW OF PON5 FOM PA

U

O. P1C.ASUiPG P.OJGS

AT R5T

b.

5PACIÑ I

\/IDTH

or

bLACK BAÑDS

O'S"

40

_,

_I5'/3Cc

flC.16

O:

5°

J: 15'/5C.0

80

U

ao/c

WAKC CROSS

SC1OÑS

HIND A VZ CTTOM

HUL.L

b

uIZ_..

X

9

îì-e.

_{O( G°}

_U=3O/5C.

ØL:G°

o

4°

u

o'/sce.

: 4°

u:2O'/sc.c.

c('S°

Ut3O,/CC.

ig°

U: t5'/C.

-04

JDISTURB

LEVL.-e

I

'6.

1

,

o

-I0

-10

-20

-I0

-vo

I-0

- 20

4O

-1O'

6-18Z

L)-/5CC.

P=34.7

-04-

-0-Z

O

0Z

04-I J I I

WAKE CROSS SECTIONS BEHIND VEE BOTTOM HULL$..

e EPERIP-lEÑtPL P0t.4TS.

G

F1G.17

Q-b

C

d

cY-6°

&5'

E,.u'I17

Ui.30/SC. F-ZO, h-035.

I

e

e

e

e

e

,_ßO ß-5

_{6-i rT}

U-ZO'/SEC.

F-,4G --063

0-z

o4-

o

cZ° .&=S"

6=0

'.J=36'/SC. F84

3

_o

s

3

_o

6

-10

-'O

o

- 5

6-Ii7'

U - zo/sc

F- 545

31 I I I

-06

-04-

-OZ.

O

0Z

04

O.

4

WAKE CROSS SECTIONS BEHIND VEE BOTTOM HULLS.

POINTi.

FIG. 18.

-04-

az.

O

oz

O4-

06

cI° 45" 6

I

7'

U-15'/SEC.

r-4g

- i9

C

d

UN0ISTUR.

LEVEL

c=4°

.&-5"

5-.117'

U_IS/SEC.

F-41

_{rb =}

-cB° 4-5" 6-' II7

U=15'j SEC.

F=4

-19

o

-IO

6.

05

1

»

I,, .i

rT19

-Uo.5

D

(J)..

co-4 9°

Z z

(n

cnÖ

cm.

Z-os

_1Iii

>-1 O.S

rTl0

0(4°

O

SOTTORFs MEASUREMENTS, REF. I

-O

CALCULATED IN REF. 14.

CALCULATED BY PRESENT METHOD.

-o-UNDISTUREED LEVEL.

-.

-o...

-c s

DISTANCE BEHIND TRAILIN

EDRE - FEET.

9-

o

EXPERIMENTAL, REF. I.

14. METHOD.

-0---- REF.

-f

PRESENT 3

4

FEET.

-8°

EXPERIMENTAL,REF. I.

14.

METHOD.

-- REF.

--- PRESENT

0

-

--2 3 4 FEET. o

__,

o