Experiments with planing surfaces

NATIONAL' ADVISORY COMMITTEE FOR AERONAUTICS

TECaNICAL MEMORANDUM NO. 681

ZRIMENTS 7ITH PLANING sui7Acrs*

By W. Sottorf

67th Repnrt of the Hamburg Naval Tank (H.S.7.A.)

Experiments with planing surfaces are fundamental

hy-drodynamic researches for the purpose of obtaining

the

most favorable forms for.planing boats, flying boats, ant

seaplane floats, with respect to water resistance

_and

sea-worthiness.

TESTS OF MODELS OF PLANING AND FLYING BOATS AND THEIR ANALYSIS

Figure 1 shows typical water-resistance curves for a

planing.boat and a flying boat.

The planing boat-has a resistance

_{curve which,}

be-cause of the constant weight of the boat, shows

_a

continu-ally increasing resistance with increasing steed.

_The

re-sistance increases approrimately q*adratically up to the

instant of the 'leginni:Ig of planing, and in

_{the planing}

pndition increasys with

_{a eower loss than}

_2.

The flying boat is lightened by the wing lift.

which

it,.:reases as the square of the speed.

_{The resistance}

curve.therefore first reaches a maximum value and

thou

.falls off to reach zero at the instant of

_{take-off: that}

is, when the wing lift equals the

gross weight.

The

fly-ing.beat can take off with

_{any load for which the sum of}

the water and air resistances remains

_{less than the}

pro-paler thrust,

aripanded treatment Of a paper read before thii -arincipal

meeting of the Gesellschaft der ?round

_{und Forderor dot}

Hamblrgischea'SChiffbau-Versuchsanstalt

_{G.m.b.H., UAY 25,}

1929.

_{From author's repriat of article}

_in

n7erft-Reede-rei-Hafen.0 NW: it. 1929; pp. 425-432.

I.

2 N.A. C.A. Mechrs .9.1. tle.morp.viAR, r,o., 66.1.

-Three classes of models

may.be.u,ke4in

this work;

Massive wood or paraffin models. These usually

are three to five times 841 heavy as the

.17.0tekk

correspond-ing to their

The excess weight illzt

bre balanced

by a weight suspended over a sheave R. (Fig. 2.) The

mass forces from oscillations it' the

direction of

motion,

which

affect the

resistance

measurements, are

therefore

three to five times

those

of a

model having

trUe mass

elm-, .

Those..from'vertical oscillations are five to

nine times as great.

Consequently, to

obtain unaffected

measurements,

Weightsitilar idodels_are generally used. Their

weight'inclusive of the

etaande

weight is G/X3.

Dynamically similar models are used,

ttpecially

for tests in waves, In order.to_give..4

rue reproduction

of the motion-of the full-site in pitching.

Their'model

weight is

and

their.masd_moment of

inertia

_T/X5

where

G = weight

and T-=

mass moment of inertia of

the

full scale, and

X = model scale.

Figure 2.ShOws.the test

arrangement A. The

weight-similar model is balanced about the C.g. of the whole

air-craft

r.:.nd suspended at the .c.g. by means of ,a fork., The

wing lift is produced bY'a'w4ight susPended over shSave

R. The model is tOwed'in the propeller thrust line.by.a

wire bridle, Which

reads forward to

the.resistance.dyna-mometer and aft'to i

small tension

weight.

Two gutdes

forward and aft of the model

hold it

in the direction of

motion.

The 'resistance ie

riaa-surei by means of a

spring,

while tho rise and the trim angle are read on

correspond-in.!; scales.

The fundamental test8. are runt made according to'..

Freud:9'7s law.,-*ith the model

free.

f*rim, at various

Con-stant slieede over the. 'speed

range up to take-off. The

'wing

lift

ia

suPplied

by the balance weights and it Corrected

during the

ruus according to the angle of attack of the

wings as determined from the trim angle.

However, with this method of towing it id

not

possi-ble to study two Tropossi-blems which

roquire dynamic

similari-ty of the .mAss.ea in

motion and massless lift:

1) The

testing, of

11. float

in acceleration

corres?odd-to the true starting regime

(tae-off),

_{This would be}

1.1%71 Dr1 T

luPDATA 1976

N.A.C.A. Technical Memorandum No. 661 3

ve-ey desirable in order to study its motions, since from experience, almost every float, beyond the hump, i.e., at

the start of planing, has a tendency to

pitch..

.Too much

importance is given to this pitching in tests made at con-stant speed, since then it naturally persists throughout

the run and gives poor readings.- In fact, when taking of'

the

aircraft quickly passes through the critical range ad

shortly thereafter

begins to plane, which results in smoot71

running.

An additional advantage of this method of testing would be that the whole of the resistance curve referred to above, together with the accelerating forces, would be obtained in one run.

2) _{The testing of a float in a seaway, both at}

con-stant speed and while being accelerated.. This presents

the same requirements.

The first problem can be stUdied in the circulAr:tank

of the Junkers Company. ',Me model is towed in

_a

circular

channel, suspended and guided by a hinged arm whose

verti-cal axis lies at the center of the channel. If the

was-pension point on the axis lies above the Attachment point on the model, then the vertical component of the centrip-etal force, which increases as the square of the r.p.m.,

is applied to the model. By changing the height of the

suspension point on the axis any wing lift can be obtained. The apparatus of the H.S.V.A. for towing dynamically similar models in a straight tening tank consists of a

combination of the resistance ody devrArped at the tank

for use on trial trips with the 'float test apparatus.

Since the resistance body has a water

resistance

increas-ing exactly

as

_{the square of the speed, it is oal;,-}

neces-sary to connect the pull of the resistance body to the

suspension wire of the float by a suitable linkage,

arrange-ment B, Figure 3. A lever and quadrant are interposed and

make it possible to

vary the

Point of attachment of the

pull from the resistance body. _{The lift}

corresponding to

get-away speed can be obtained by suitably locating the

point where the pull is applied to the lever. The

resist-ance body is a rather light cone, and the force required to accelerate it id so small in comparison to the total

force that it can be neglected, so that at each moment of

4 _{r.k.c.A. Technical Memorandum ro. 661}

-A -further development of the

jeer is

proposed in which

the slider to which the pulling

wire of

_{the resistance body}

is

taehed will

be moved.by a motor

'controlled by the

mod-el

itself in radii _{a manner as}

_to

_prodace

the lift

_corre

-sponding to the

_{exact angle of attack of the wings.}

,

The resistance body properly

should lie off to one

side of

the model

so that any

_{influence on the model may}

be avoided.

American

tests for the sma

pUrpose in. -Thich a

hydro-foil vas .arranged in the

water undOr

the

-model to give t'ne

corresponding lift

led, among other things,- to _{errors due}

to interference

with the model.

'Full information on

the-nodel'is

given only by tests

at

different

constant

trim ancles,

since

the

_resistance is

dependent on the trim

angle and _{further. important} re-sistane'effeCt appears which is dependent on the trim

an-gle and is produced by

the

water' flowing under the

_step

and vatting the:.afterbody.

By moving a sliding weight (fig. 2) the

_necessary

mo-ment is applie& to the model

during the run to maintain

the.

desired

angle of trim.

It is the' task of the airplane designer

to construct

the float or boat so that as far as possible it will give

the

most favorable angle of attack

_{for the rings of the}

airplane and

so that the,Control surfaces can apply the

proper moments, in order that the ret-away

may

take place

at the 'Most favorable -angle

of attack for the -particular

speed range.

The economy of the flying boat is increased by

lower-ing

the maximum

resistance.

This gives an

_{increase in the}

useful load, which means an

Increase

_{in fuel capacity or}

an increase

in range. 71th it is also obtained _{a shorter}

tine of take-off, much desired

on account of the heavy

.structural loads while tcl,:ing off in

a-seaway.

By making

changes in the planing'bottom,

changes in the position and

height

of the steps, changes

_{in.thel.afterbody with regard}

to

the spray, or by

_{tehe Construction of a new boat}

with

different principal

dimensions, etc.', It is possible

simple comparison to

deermino.the relatively most

favor-able model.

;77777777-7777k

irOPOATAL;MMIMIMW.11,7717ft

0."iiffig..`

N.A.C.A. Technical Memorandum Fo. 661 5

.Yigure 4 shows the comparison of systematic tests of

'a flying boat with

five

different bottoms. There are

im-portant differences in the resistances of the individual

designs: Bottom form No. 1, for instance, has two

condi-tions-of maximum resistance; traveling in smooth water the water does not break away from the step with the result

that the resistance steadily rises. rnsticking can be

.accomplished by jarring the model.- The model then rises on the step and the resistance falls to the lower branch

of the curved In practice this boat would be able to get

-off only in rough water.

This method of developing 's. type is still followed

generally. The tests nevertheless remain unsatisfactory

because they are derived from a form which, lacking

previ-ous fundamental knowledge, has been developed by the

de-signer principally by feeling or instinct, and a variation of all the design elements Which have a bearing on the prob-lem must be neglected, primarily because of the great

length of time required for the tests and the

correspond-ingly high costs. Consequently the information Which is

obtained regarding the effects of changes is often deCep-tivey.e.g., a change in the. bottom form of tad forebody

1141.4t be favoiM71-71-1 itself but because the afterbody is

not suitable for use with it an apparent failure might :be obtained because of the effects of the spray.

A recognition of this fact lod.the firm of Rohrbach to cenduct at the.Eamburg tank the first systematic tests

with different bottom forms of forobody alone. The

after-body -was 'separated by a vortical cut at the step, while

the road remained equal to tho total load. In those tests

the great increase in resistance due to the wake running

back along the afterbedy appeared plainly. For instance,

the combination of no of the completely investigated

fora-bodies with a certain afterbody gave a 30 per cent increase in resistance because the afterbody, although Of itself out of the water, was wetted for its entire length by a

stream of water 'coming out from under the step. A

suita-ble deflecting device reduced the resistance to almost that of the forebody alone.

Models of similar type but with

different loadings

and .get-;away speeds are compared according to Figure 5. ,

In this the ordinates are the planing numbers = W/A, n

where W is ti.e resistance, A the dynamic lift, an the

1

a N.A.C.A. 'Technical Memorandum

No. 361

v _{_ observed isneed of model;}

abscissas are -- = speed ratio

vo _{take-off speed of model.}

-By making numerous tests with seaplane models of similar Loading and the same scale, it is probably possible to de-termine empirically an envelope for the numerous curves which would represent the curve for the best model.

However,' the question of how near we are to the

theo-retically best model is still unsolved. _{The test results}

from work done for different private concerns must be

treated as confidential, consequently material for compar-ison is restricted.

II. SYSTELIATIC TESTS WITE PLANING SURFACES

As early as 1924 tests with box-shaped bodies _wero

undertaken at the Hamburg tank by _{Eng. Popp as e}

basis for planing boat construction, in order to _determine

the planing number for flat and. 7-bottom forms _{and for}

different trim angles end loads. These

experiments. coUld

hot be carried out to the proposed extent because of lack

of funds.

Lbeut a year and a half ago the test program which is

described in more detail later on was set-up because _{of the}

pressing necessity for.a

basis

for analyzing 'flying boats.

The first results of this program are presented-here.

The following considerations lead immediately to the

choice of a plane rectangular plate for the fundamental investigations, as being the planing surface with presum-ably the best planing number.

Tangential and normal forces act on the under side

of

a plate which is moving through a fluid at rest while the

upper side remains under constant atmospheric _pressure.

It is assumed- that the Water flows away freely from _the

bounding edges of the bottom surface..

In the case of a frictionless fluid the tangential of

friction forces equal zero. From Figure Za it. is seen

that the resultant of t:le normal force N for trim angle

a

gives. 7 = A tan

a

as a minimum resistance.

From Figure Sb it IS Seen that, assuming the addition

of the friction force T, the resistance becomes

carnal.1711.1.1k

r.A.c.A. Todinical hemerandum

ro.

361 7

U = k tan

a +

. as, in accore.nnce -rith the

condi-cos

a

tins of the tests, the lift

A

in virs;:td

to

-)et constant.

If one considers the cross section of a 7-bottom plate

1

with piano inclined surfaces and assumes for !implicit7

that the trim angle is small an:i say

be neglected,

_the

nor-force

on ono side, according to Figure 6c is =

1 Al

or N -A , For the flat plate sin

(1

= 1.

sin

fa'

sin

(15.) \2,

1.2 kj2/

hence P = A. nth increasing deadrise, 7 and the lost

comnoneat Nv both increase: as well as the wetted

sur-face, if. the constant lift A is maintained, as a result

of

7whit

the total resistance of the 7-bottom, and also of the curved bottom, exceeds that of the flat plate.

The Test Program

The systematic study of the plates was conducted ps

follows:

For a given plate, the influence of trim anfle _a

on planing number c = 7/1 was determined by towing at

constant speed and load, but with each test run at a dif- j.

ferent trimming moment.

Tho affect of increasing load Was determined

re-Deanne tho first tests at

higher

_loadings.

The cffom of increasing speed was found by

re-'seating

the first grout of tests at higher speeds. _It

was assumed that the lift increased es the _{square of the}

speed.

After the relations for one plate _{were kuorn,}

tests were made for comparison with one or two other

ries of tests, in order to determine the effe t _of

vary-ing breadth of plate (variation of aspect ratio) _by

tow-ing relatively wider plates at constant loads and constant

speeds.

The conversion of the model results _{to full}

alto

by Fronde's method showed good agreement almost to the

maximum resistance, however, _{as tho size of the models}

was reduced an increasing discrepancy appeared, _{as Eernann}

(\Ad floss pointed out as a result of testsrwith modals at

f;707111

B _{T.A.C.A. Technical liomoranium}

_{:so. 651}

varions scales carried oat at the

(ne.

_7.)

Al-t-acol'ca -an to the .laxinum resistance the hydro- _{and aero-.}

dynathc lifts are small, and co:levy:ter-fly _{there is no}

lir-ference etweet a flying boat, a planing bent, ani _a

dis-placemeat boat, thereafter, with increased hydrodynamic

lift, the planing condition is.reached, _{and it is}

necessa-ry to obtrin a conversion formula of the form

r

_{w x3}

-f C).;. The first term expresses the conversion of the

toe

ronistence neasured as Inc reeistance, according _to

Prodels law, while the second ox3resses tho _difference

in frictIon, according to Reytolis lew, as a correction

!or 3See1e Effect." By testing plates of different widths,

cntlinel above undor (4), the law of conversion _{for the}

planing condition can be found. The scales

of

the models

aro derived from the relations of the ?late widths to each

other. Cn fee basis of ?rondo's law tho tests _were

con-inated a: corresponding speeds and lords and _{the results}

1)'

to determine the second tern of the conversion

for-mnla.

This method has the advantage over tests with _actual

hull noels in

nil.:

the effects of spray which would cause

the results to lose their general validity, are avoided.

3) In the same way the influence of different

tot-tom forms, deadrisu, curvature, etc., arc to be

deter-rimed by coh-arative teets, in which, in place of _flat

plates test floats are used with the same over-all

dimen-sions but with different bottom forms.

Apparatus

Preliminary tests were run with ea flat glass plate to

determine the forms of the wetted surfaces. In the

appa-ratus describe& hereafter, the angle of attack and _the

length below the still water level were determined by

reading the change of draft (fore and aft) from the zero

reading (plate level on the water !surface). The impact

preeeure of the water on the plate increases the true

wet-ted sur.:ace. _{By lookiag through the glass plate the}

wet-tel length is easily determined by reading the impact

wa-ter line on a scale on the rlate. Tor a -elate having a

with of 0.3 meter the dynemic water line is a flat arc

of about 10 mm midordinate for all speeds, angles, and

loads. Accordingly, it was nermiosible in

the succeeding

experiments to use an aluminum plate about 6 mm thick, in

DTYPDATA 1976

semssandeslie

;.A.C.A. Tochnicel :_icmorandue 17o. 5i1 9

which a flags plata 50 rem wide had teen fitted nt the

quarter point. A mean value of the wetted length was read

terough this small glass plate.

The Tlate was pin-jointod to two vertical rods, each of which was guided at both top and bottom of its guide

frame b; three ball-bearing rollers. (?ig. 8.) Ihe guide

frames were supported at their c.g. axis on knife-edres. Their 7eirht was supported from tae carriage while the

rods formed a part of tae load on the plate. It the level

of the axle of each guide frame a stool wire was secured

to a short cross aember on tho guide rod. Tho

counter-woLfhts ncro attached to those wires and suspended ovor

sheaves RI and R2. Tho loading neights woro placed on

sealopans on the guide rods. By charging tho loading

701hts or counterweights any desired loading of the plate could be obtained, while by shifting tho loading weights or counterwoights from one rod to the other any =anent

could "ea obtained. Rise and trim angle wore determined Iny

gra;hical record of the change of riso of each guide) rod

oa a drum.

The resistance dynamometer consists of an equal armed

/ever su7;orted freely on two knife-edges. A coarse

bal-ance -Teleht and the tension of a snring, which is

varied

electric

motor controlled by two contacts, hold the

resistance in balance. The extension of the spring

regis-ters on et drum so that an accurate mean resistance is

ob-tained. The time and travel of the carriage are registered

at the same time to determine its spool. By this

arrange-ment the

weighing lever, and hence the forward cuido rod

are izaintained in the vertical position. The slight

angu-larity of the after guide rod at large angles of trim is

ta2:en care of by a correction.

The Test Procedure

When the carriage is not in motion ring stops on the guide rods hold the nlate so that the ',slain; edge is above

water. Marin:: the accelerating rnu the ;late gradnally

assumes, of itself, the depth corresnonding to the

prede-termined load and the angle of attack corresponding to

t:.0

°redetermined noment, so that on reaching constant speed measurements can be taken.

Goailmer,

'case

4 ,

PCIA1.11=1

r.A.C.A. Technical Liemorantiani.7o. 661

Tests on the

.3t Tide flat

Plate

The 13 As and speeds chosen a:pear it the following table.

DC load is Increased as the square of the s7el, and

accord-A

agly the Dm? coefficientn

v3 - a

b

remains constant.

V 2 . 2 be=

in :an

coefficient

P -b- v2

X 3.216

715

415

275

113 60 50 42 33 25 / 011 4 4 2

Snot in meters per second

4

g

_9

kg

92051/11;1.925

610 4 1.419

430

i 1.052

450 ::; 0.522

475 :7;

0.433

550:o.,

0.373

550

3.325

630 "1 0.263

i

770 -..j

0.232 / t . -A 22.3

ti

141

(

/ling `1#

In.-le

1

ILoteth

1:otoatt

irotted

below

Resist

...

in

I an D a-I

li

lemPth

i

the

la ,

water

nce

e

at:a7k

..p., mk g 1

&brat

-.4c

surface

in g

trail-,

tug

1

:fig.14)

i

4 mis,

Loading case I;

tp

Monett

coot-

lp

ficient

jr

J

_t,

29.404e -1.i.C.7 :1

21 4.653C.751

15.05

itt

0,3

6.

03

7.98 0.312 16 / .0..:, 320

5.72

4.92

'J '0.C52

14, C.941

4.33 t 1/4.330

354 fr 0.'190

I

lo Ar

II

X 0.218

9"

Et

32

45.2

X 0.218

13

'

27

I 48 " IV

x 0218

13 "

36

It I !

Spead

7 =

L. /if

11

770

; )1 2

IZ

465

; )2 2

43

325

tker 4 3

₁₆₀

5

.C5

25

100

5 .17 6

34

95

7 J.:C.7

21 70

8 W 9

6 SO 9

10

,4

nr

45 = 4 kg

0.25

C.5

1.0

I:n

1

:fig.14)

10

3-9 3 4 5 4.' 3 5 1 7 7 8 8 5 9 22

it

54 7 8 9 0

a

5 6 ,1IPT71411°A 1071 855 705 565 495 395 320 255

la

/ 00- 2 57 755 !Ip 3. 1 748 1 4 13 545 3 5 25 395 4 6 34 278 1 15 7 31 215 16 7 58 195 7 : 3 170 A i 8 11 18 140 1

Speed v = 4

-.Is, :caking

_case

i

nis. Loading case I;

1 ateigfaci2 A = 12 kg 10.55

-r

0.301 9.36

t

0.345 8,20 ' 0.682 7.09 0.593 5.83 ₁ 0.729 4.82 0.726 4.39 :) 0.740 A = 16

kg

e.zs

7.61 6.31 5.02 4.12 A = 9 kg 32.90,'4 23.10,d 13,3o 0

9.00,

e.45,r, 7.02 r 0.587 0.327 0..362 0,579 0.586 0.754 3.709 0,798 :.105 D.787 0,789 III; 750 1630 '5.212

jc

1545 5.512 491 1710 40E 1805 4.159 30.5 1980 _3.460 234 2150 2.332 171 2350 2.578

Loading ease IV;

792 362

22501

9.540 2420: 7.371 499 1 2525111 6.609 359 1 2920, ! 5.249 . 270 1 324000, 4.315 59'5 / 1C70 "3.821 14.60 ' ' 0.337 675 1140. 33.335 14.70 .01- 0.535 472 0 1105. '3.112 330 1100 -'2.423 215 1130, ,1.714 11.90 .40 0.708 3.29.1: C.741 5.55 -, 0.751 e _s. Iis

te

L 155 1180 .1.387 5..31 _0.798 140 1220 _1.244 4.75 - 0.785 110 1350',- 1.075 4.12.

4

0.717 1 84 I 1600 , 0.847 3.24 . -0.730 Speed. v = 5. 6 910 6 7 785 it 7 49 615 9 31 475

i's

10 54 385

re

Speed.

211 22 710 3.t 1 490 4.13 8 240 4.4-, 37 4.1. 45 220 177 150 655 1710 4.863 433 1410-s 3.406 188 1105. 1,954

174

1050fl 1.458

130

1C20.-j 1.245

104

1100

1.035

ot

N.A.C.A. Technical 1-6emorandu..4 ro. 561 11

11.11

lo.

Angle of attack

ao

!Length

7etted I below Length ! the water In mm !

surface

In

cm I

(fig.14)

*Resist

ance

In g

Moment

U in

mkg

about

trail-ing

edge

Moment coef- lp ft dent

it

(fig.14)

03

A b

SI-er'i v = 4 m/s. Ioliinp case :I;

A = 8 kg

2 6 5.

10

(fig.14,

4.828

2 3 6 2

o.

2

21, 36

3

3.3t 47 4

4.; 10

5

4t15

6 5.., 8 7 _{5.v- 31} 8

5.40

9 6-0 5ti

0

54

1

9-' 58

2 9.";,34

;772.7.7:77471

'Length below the water surface 1 in mm

(fig.14)

Speed =

6 mic,

Loading case /; A = 9 k4

deg.min. 5.,

64'

9 7.11 0- 8.1 1

9):

14006..vg 915.322 810 .170

780.31

580 -Pt

515 Jr

490 4'2

318 -"a

235 Ulf

1821.gcr

163/4-Speed v = 6

mis. Loading

_{case II;}

?"if 4

43320,2, 14,350

2880' ":

2320.04 2770,11 3

2595..w.-42630

.4 132644E2 ml 838 7S-C

0 739 3./

2 705 .!

4 507 444, ail 448 501

/0 420 -An

2: 250 fir

84 168 II.

csi-117 4 lit

Cat201 1 C Moment I Moment Resist- II in I coat-ance of . mkg I ficient about in kg I trail-lp (fig.14) tug A b C3 edge

1125n'0.840

I

5.70,4 0.856

1210ny 0.733

1300.( +; 0.503

1380r6 0.562

1575,',3 0.497

4.97,1

s.848

c.a3.3

3.81.1 0.945

0.973

1 1

A = 18_4

e s.s.I r 3 Fr Me .4 7, .2 L

24.4

0.533

11.587

19.70

;

0.597

10.833

18.45)1 0.704

9.930

16.90)01 0.724_

I

6.329

_{10.78.,A 0.748}

7.03

12.00!- i

C,.752

2330..

5.531

11.17.c44 0.730

1

'2710.c.

4.518

7.671 0.737

2670ow 3,080

_5.23±

0,785

1 1

3050., li 2.431

_{4.13:"Ii 0.740}

317qeer.

2.181

3.72. 444 0,733

Speed v =

6 mis,

Loading case III; A =

27 kg

rirst.ffic

2.1:141

1695

11590 t;34147C0

26.370 1

19.95M

I

0.570

,

'talc 31

3c 34

1310

°z1205 161

320.4/ 22.609 1 17.C7M1, 1

_0,532

4.:; 22

1095

7 994"

/ 4130.PA 20.030 I 15.15.k ;I 3 374

5: 13

937

/4 832 r n

4/2.0,4 17.531

i 13.22,:.3. 0,393

5.Y. 26

873

.33-e 716 2

44140"AI 15.478 .

1Z..45.1411 .3.321

5.- 24

682

?-3 587"t 4230.-

_{13.3.31 I 10,32.A 0.709}

43

640

R71

545731

4280.0c° 13.142 1

9,93.--f

n.717

7.

25

538

u 442"i 1 4360 ,,stg. 10.834 II

8.19.1',

t.723

8st

3

478

My, 380 5i 4o 4540,

_{3.438 I}

_{7,13. "}

_.:.723

9Y 12

390

445 28720 il 4770.,,s.

7.391

5,58 "1 3.728

9:048

346

r, 252 in

4950

6.697

6,,08 0E1 0.709

.174 ,

,

3

-49

108 57 95

47

82

35

62 53 56

N.s.c.s.

Tad: nical Mernorar_dum _71:1. 3.33. 12

7.-5 66 4:-/ 58 72- 44 442r 30 zi.1'

pi

28

a's 3 4 5 8 7 8 9 1. 2 3

Angle

of

attack

Wetted

length

in

(fig.14)

0 14: 7 1

7-In2;le un

of

attack

a°

7 8 9 1 3 4

Speed

t. 3.1:t52

4:- 19

5).

5 5..

35

8,. 16

3.. 43 3. 57

7'. 43

8.', 11

a): 45

-

r33

5-

10 C 8

t

10'71. 2 Steed Wetted length

it

4

r.A.c.A. Technicel Memorandum No. 551

in mm surface

(fig.14)

I in mm

(f

ig.14)

v =

mis,

Loading case IV;

a'. Length Moment below Resist in the ance mkg water 7 about in

kg

trail-ing edge

= B

miss

Loading case I; A

Moment

coef-ficient IL. A b CB

= 36 kg

= 16 kg

lp

V

-(fig.14),

,c't<

13

CS:bq0

etc.;110

_4. a SC; /V I

Jr

4.

e

2.- 32 ;.--810

67r 750 /40*4313c0thi:

_10.187

_38.90.117:f

_0.779

2 It 42

-1620

1222 565

71$74t900,41

8.011

SO . 60.14

0.801

3 r 24 p-401

c7o ;540 r.00.22260,,A

5.093

19.45 ecv!

0.810

43

f275

g 217

"44442060.41

3.538

13.90;

_0.821

4 35

M1178

54 118

// 1820rc

2.200

8.42..ge

0.781

5 0

a43

_lad

Bg'qP

1750A

_1.848

_0.802

b.

5.

5 '

7__ 13 51

57

57

mr3145 7,7125

iv 38

..2'b65

k7

94 22- /18 5Clut.

l31

_78a

193 0. ent, Ia/iC

43 4

21eq,,:,

20

2470mPs

1.859

1.534

1.217

1.076

7.10 or'

0.796

6.05.0T 0.827

4.66. Gt

0,837

4.11.0" 0.833

91 12

Jot,72 MI 2890,.ig

1.319

3.89 .0,g

0.870

Sneed v

8 mb,

Loading case

II;

A

.4 41

= 32 kg

.2.44792 1,31

Sil 709 /2

45150

I

19.092

18.25

0,748

5:"

3 640

Lye 560 n 04870?

15.508

14.834/-

0.780

5'

G...' 31 19

/.? 507

7 350

_kr7

Lx.425 4

275

as4780,1[ 47 60 o'

12.539

8.348

12.08.4:

7.98 'II

0.774

0,796

37

7.

57

20

/ -4,306.an258 V3

frig

236/

1923485

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7.294

6.302

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0.764

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1535

#1/JI17/

7 14355/ 115870

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1385 -P71427 125539 2/5790:3'. 30.750

13.08.V1 0.812

1200 2i5.:710752:f

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28.350

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1120

99314 '65670. I 328.510

_{11.25.04 0.550}

985

8504

v8143° 943

820/6

oo570..1i24.680

_{05960,4 22.710}

10.50.,M4 0.388

9.63.0 0.574

923 /675

cut,

_coo

830/s

046010..; 22.724

_9.63

_0.674

745 stis

312

52497

590 JOS' 97-

56421

445250 0 1 19,521

_{8.28.V: 0.681}

'

43300.e3 )18.020

7.63.U3 0.678

523 .1; 2,1

_{4984L 0/6490.,1 316.052}

_6.81.nJ

_0.7C1

2.94:

509 'to,

_{38537 40760/ 113.338}

_{5.67.1)4 0.712}

433

_.if-ffrv7/

3432

o7010

411.883

_{5.05-;41 0,702}

6-B. 9- 0-1 2 3 4 5 6 7- a- 9- 0-

2-0.

le

4-5 0 3 21 v = 7 7 A

1-tarsal;

11

.A.C.A. Technical Memorandum

Ito.

661

Speed v

= 8

14

3-deg.min.

7;3;20

./.77248

895179

P/C

case II;

A

6,307

= 32 kg

oid

6.02

et s.41,7C : '

0.750

4-

7 t' 39

- VI 57

.-744238115202 84 168 7orc,

es 130 tzzs

4980.clh'_5070411

5.270

6.336

6.02.°'';'

5.04.r-

0.750

0.751

' 7 ' 6.471 25 9.1.140

.0-190

150

60 120 me"

Al 82 Uri 5230.04,

558001,

4.323

3.376

4.42.t!-!

3.51;1.4

0748

0.753

1_

Speed

v = 8 tais.

Loading case III; A

= 48

kg

8-

&I, 38

96

'' 59

1 .46738 2.4'625 31C 632 IV fr.o 521 676

7880

Lti

798tk

26.017

22.479

41.

11.05

9.557:f

0.726

0.741

. yie'a

7.0 10

.0527

F

41 523 cent

7920..); 22.483

9.55

Itt

0.73£

S.1 II

/...3438

112 365 ni f

8220.01,1 16.810

7.20"

0,738

8-41: 2.5

0434:0

ye* 329

837Orm. 14.008

5.94 .1-1

0.737

Speed v

= 9.5

mis, Loading case I; A

= 22,6 kg

t::

e.

(5, )3. 2.2:150 1226680

1305 610 150 r 4166

1 12.167!

32

,90''inl 0,790

4-

all 10

/17422

in 357 liti. 4 353004

8.135

22 .00.t '3

0.820

)5- 3.13 52

.**295.

pc 233/30 i. 2790.-.1

5.159

14.06.14/:

0.802

)6-

4X 45

4a99

67 134 4-44 I 2600pb

f

3.801

10.30, m 0.813

)7-

5.-t13

.5417C

55 110

(1,Ls 266QrsNr

3.104

8.39 .4 0.901

)8- 6-33 20 .3.1110

z7

54 72

2910"1:

_2.039

5.52/14 0.813

M

19- _73h124

-At 80

_it,i.

28 /ft.

1

3120.k

1.263

:

3.41.04 0,693

4-Speed

v = 9,5 mis,

Loading

ease II;

=

45.2eigi

2-4M29

5,0, 5./3 7

50

-.73820 J119328

/47490

367735/5

272 545 7:64 2o3 407 <in-9 4/ 6980

6730.0'

6680.o

28.015

22.319

16.550

18.98.1

15.11.14

11.22.w.

0.749

0.755

0.763

P, 3- 6.1^ 42 /.0,5-315

//6232te4i 6 680050 10.834

7.3(1.4-f 0.754

4- 733 44 .F-06242 8+168 'To

7070,:9

8.146

a.,

1"

Length Eoment Moment

an

ro.

Angle

of

attack Wetted length

it

below the water

Rest

st-ance

I; U in mkg aboat t

coef-ficient

lp

_it

ao

In ma

(fig.14)

surface1 in mm

(fig.14)

in

hg

trail-edge

(fig.14)

CB

m/s, Loading

. A

ing

A

r.A.c:A:

Technical Uemorandum /To. 661 15

In Figure are entered:

The measured resistances less the

air

drag

deter-mined by towing the horizontal plate just above the water.

The curves of for resistance TF = A tan

a.

The curves of frictional resistance TR,

com-muted according to Prandtlts friction formula for a

turn-lent botaidary layer with laminar approach. .

7R = 2

-

m2 F cf

in which P = don.ity

-g

= mean velocity of the water relative to

the plate

-= measured wetted surface = b

If T, wetted length cf = 0.073 1600 v/ R = Reynolds :umber

= Tr_

The ourvea.

(Tp +

TR) . .

The mean reduction in speed of the water

rela-tive to the plate vu = v - vm in per cent of the towing

peed v, as determined by pressure measuree,ents.

A complete agreement of the measured values with the

curves

Mr

+ TR) appears at lower loads, while with

it-creE.sing loads at

sehll

angles of attack an

increasing

difference appears. This difference is explained in part

by the gradual appearance at small enact; of attack of spilling over the edge, and in part by the increasing ef-fect of the edges on the establishing of the boundary

la:-er and consequently on the coefficient of friction cf

with inCreasing ratio of lib.

Figure 10 presents tho curves of planing number

e =

16 _{N.A.t.A. Technical iiiemorandunare. 651}

.

-diagram show the change of planing number with _increasing

seed at.. constant load coefficient for loadings _{I and II.}

A notable variatl.on is apparent drily for v F

_{4-131.7P.S. at}

small angles, wnere a condition of laminar boundary _layer

predominates.

The lower part of the diagram gives the _planing

num-bers determined for the higher velocities for _{the four}

cases of loading I to IV. _{At large angles of attack with}

decreasing friction We, ' the curve of resistance W

ap-proaches the curve of rorm resistance 73, = A tan

a

asymp-totically, from which the planing number, tan

a,

is

a)-rived. as the asymptote. _{The minimum planing number lies}

between 40 and 60. Toward smaller angles the planing

num-ber increases with the rapidly increasing wetted _surface.

Toward larger angles the planing number increases rapidly

with the form resistance as a function of _tan

_a.

_The

lowest value is 0.114, the resistance is therefore about

119 the lift. Further tests with lower loads are proposed

Ifor t'ae determination of the curve of optimum planing num-'. biers.

The results indicate that with increasing load, _{as a}

result of tho greater wetted length in relation to _the

breadth, the planing number becomes worse,

The Pressure and Velocity Distribution _{on the Plate}

For the study of the pressure distribution the plates

were fitted with 85 holes of 2 mm diameter. _{These were}

fitted on one side only, at the center line, at the

quar-ter beam, and at 4 mm from the edge, as well as at _a

num-ber of intermediate points. From each hole a connecting

tube led to a glass manometer tube secured on a panel at

rigat angles.to the plate. he tubes were connected

to-gether at tho upper ends by a cross tube, by which tho

wa-ter in the connecting tubes could be sucked up

simultane-ously and colored. For each tost the elate was secured

to the two guide rode in the proper position to giro _tho

desired load and angle of attack. _{Men constant speed}

was reached, air was allowed to enter the cross tube, _so

that the water in each tube Stood at the height

corre-s7onding to the pressure, and could be photographed _or

marked.

The pressure distributton was investigated for v = 6

Tochnicel Uomorndurto. 551 17

and A = 18 kg for 4°, 50, and 80 angle of attack.

(Fig. 11.)

At the.leadin3 edge of the wetted surface, where Li-pact and change in direction of the water occurs, the

pressure rises immediately to a maxi:num, and

eetickly

Arors

off. At small angles it roaches zero at the trailing

edge. Toward the sides it falls off only a little

com-pared to the pressure at the center line. At the s1i6s

the water escapes as a

jet on

account of the sudden

pres-sure drop. From the trailing edge the -rater continues on

it

the direction given

by the

plate until a jet shoots up

.from the rebound of the water displaced by the plate. This jet is the Principal cause of the groat increase of resistance in flying boats when the water touches a poor-ly formed afterbody.

From the pressure

distribution it is

deduced that,

for instance, at 40, the mean pressure, with respect to

the 7ho1e wetted surface, is smaller

than with a wider

7ith the same load whore the 'after part of the

ires-sure range is elisging. But if the wider -.late has the

greater mean pressure the wetter-Surface,

T.ET'conseouent-ly the friction and total resistance, is smaller. Tests

with

plates of various widths should clear up this

ques-tion experimentally.

Bernoulli's equation makes it possible to determine the velocity distribution on the ?late from the pressure

distriblItio%, as shown

in Figure

11. The volocit7

along the elate length is also plotted against the plate

width, from which as the mean of

the velocity

crone

sec-tion, the speed

vm

is obtained as an average speed over

the entire

Plate. For tho computation of the frictional

resistance

7,

the moan reduction in spood vu = v rn

as a per cent

of the towing speed

v,

is shown In 71.7nr.1

9 as a function of the trim

angle

a.

Separation of

the Resistances

The total resistance is first divided into frictional

resistance WR, and form resistance

!4.

This last is to

be divided into the induced

resistance

j,

analogous to the

induced resisence in the wine theory, and the wave

resist-ance 7w, which

is due to the motion of a body moving in a

boundary between two media. As Wit is readily determine with

1

S.A.P.A.

m.p.s. plato :teen

-""1"...11m"..".".1"IbrAILTT100^1AT A 10'71AI hm 4°

0.238 m

0.617 kg

n ^_{P V}

_{hm b}

According to equation (I) hm and Wi are determined

for tIle three cases investigated, as

1.8

Late'CI.A.

Technical

Memorand.wa Fo. 661

sufficient exactness, we obtain 77( after determining

According to the law of momentum tho newly created

momentum per second or the momentum P F1 v w is equal to

the

lift In

this

P =

"fig = 'onsity, Fl is the cross

section of

tho mass

affected, v is the horizontal speed

of

the plate, and

w

is the

downward velocity of the mass

considered. For simplicity lot-a

rectangular section

.10

be assumed over the width b of tho plato, which because

of the relatively small

reduction of

pressure toward tho

side appears Permissible. Furtho-, lot there be introduced

for the determination

of

the downward

valocitv

w the

av-°rage velocity vm over the entire pinto, then wm vm

sin a. If one calls the mean height of he affected mass

hm then we have

P hm b v wm = A

(i)

The kinetic energy created per second is equal to the

work of the induced

resistance; that is,

W m2

P hL

b v

= 7; v

0.162 m

0.126 m

0.906 kg

1_ 1.15

kg

Figure 12 shows the constituent parts Of the separate

resistances, which at

the miniMum

resistance form

appro71-mately equal

parts of the tdhl.

In

contrast to

the wing

theory the share of the frictional resistance in the totL1

resistance is eat, and on account of the

unknown wetted

length 11

it cannot be

determined theoretically with

accu-racy; on the other hand,

there enters

also a

notably

lart:e

from

which

A2

=

(2)

Y.A.O:A. Technical Llomorandum IT,. 551

1:5

7ave-making resistance which can'lic determined

mathemat-ically only by the expenditure of very

much time,

so that

only systorlatic research can 7./..ouco the necessary bass

for the predotor:aination of the tesistanco, -.7;oont, and

angle of attack of any gliding body.

Figuro 13 shows the nomont coefficient C- =

- b

plotted against a. Tho nomont is referred to the trailing

cage of the

plato. With increasing speeds the moment c-Irve

for a

single

loading

condition approaches a

boundary curve

which corresponds approximately to the curve for

v

9.5

mis. At large angles

the

curves approach a'cOmmen

as-tote. The introduction of

the wetted length

instead

of

Cm

gives a Troup of carves of

the same character. The separation of the curves is therefore detnrminod by thc chancing of the pressures with a -cower not equal to 2,

an

yet 7roator than

2 at low spocds, which falls to 2 at

the

boundary

curve.

For this reason the

prel.otorminod

lift, which varies as

the square, is reached at a

shorter

wetted length

at the lower speeds.

Further toots r.ro

planned for tho

confirmation of those

first results.

In Figure 14 the position

of the cantor of nressuro

in plotted as

_1-0/1'

in

which lp

is the

distance

of the

center of pressure from the after

edge

and. lt is the

wet-ted length. All the points of the same loading case fall

on the same slightly curved line, that is

the ratio of

lp/11

is approximately constant.

With

increasing load

the value of /p/i.. decreases.

On the basis of results so far,

since the intensity

of loading corresponds approximately to that of the

fly-ing boat, Figure

4, a minimum lino can be drawn in

that

figure which shows that with

the

best dosins, whore more

value is attached to low rosistance than to the best

sea-going qualities, the minimum resistance of tho

flat elate

is still exceeded by 20 to

40 per cent.

The most important result of these tests

to date Nay be said to be that it is possible to make comparisons

be-tween the later

tests

of the program and one or two series

of suitably related tests, which will ivo the work a

nrac-tical

application.

A 1 Ors

irk

0 1 2 3 4 E 6 7 8 speed. of mold in mis

Fig. 1 Resistance curve of a planing boat with a disllicemnt of oP kg (blow) and of a flyin:: boat

with a distlacemmt of 12 kg (above). Fig.

eiTTI.rt a /Soft }hi

ST 4 I-AT 2 I-

/

-9 11-C

123

S 789

L. 40 co 6 I-ca 6 0 2! 7 = Vo= 2 Conditions Spucd of modal.

Getaway sroed of model.

Single float I " II icTwin float. 1 .4 .6 Planing number c G - T W = Resistance

G = Weida

of aircraft T = Lift from wings

5 Comparison of F seadine floats. 220

tint

P2,10. 9 10 1 v 0

N.A.C.A. Technical Memorandum No. 661. Pigs. 1,4,5

5

It

0 3 ca-4 n4 , 0 _I. 0 1 4

5 518

Sped of model

in mis.

Fig. 4 Resistances of flying boat with five different forms of bottom. Displacement = 18 kg.

0

-3 Bottor_ I / II 6 0

1-3 /.1\1:

/

I\

-_____________.__ , I , , m -Li R , g-.

.--Spoed recoeder'1

!;-1-1

Rise

I ,-1 1 .. I P -"!i.

C

'

ri\

1:-I ! ; -i : . 11 I S

,

Balancing reweightt__I

/I

'Weighing spring

---- ... c

L ;

.' o il '' .0, TriggIcan ent al

ti;;Wing lift

W (---

LA

l''Slidine; weight

..4,1 ! t

Resistance dynamometer

I / V ---... !

it i

'

___

. a,Center of gravity of I. t .11

system.

1 ! I -Trim .

-i

Coarse weight _./Jangle ;

!..,,.."irs/

. Hi- .7

Tension

weight

\ /

\ . i Vi 1i

ci'

,,,. ; -:-..-1, 1 I!Guide 11.'

Retstance

-I !I' --La .

Fig.2 Diagram showing test arrenrement A for flying boats.

C

N.A.C.A. Technical Memorandum No.661 Fig.3 stance

4-1"-faralPsOnt

long --Lever .-Slider Trim

ngle

//CI

/

----Resist;;;iM

body

/Piga

Diagram showing test arrangement B for towing dynamically

similar models.

ct,

Ni

t

_(a)

(b)

2

Fig.6

(a) Forces on the flat plate in a frictionless fluid.

11 II ft 11 11 11

viscous fluid.

11 it

with deadrise at mall

angles of

trim.

2 2 0 3 (b) (0) (c)

N.A.C.A. Technical Memorandum No.661 _Figs.7,10 a,v = b,v = c.v = d,v 9.5 6 EST7PDATA 197;11 I

a

.15k

4

taA h

r

0)1

\

C.

-.10

I I IC -C51 i G2 I 4 6 E. Trim, a.r.,;le,o, !....certf-- ..:1-Planing number I 411- e fen

Leading

...e,...e----i Case II.

--x2ianlm Planing Number E for Loading

r,...,settf

c-tse I.

"7.-1C

Fig.1C Curves of the Planing; nuabers, e .

Mead Planing Number c for

Loading Cases

I IV.

350r

-$

wor

n

r '

1.,\

1.81:4 2E>01- ' -

1:2

2C0 I-

iii

':`

...:

'

N7,--- - 1:1

_-N.,. 4.0; 150 1CJF 50 L -A-ek 5 1C 15 Speed

in mis

Fig.? Resistances of similar floats tested at scales of 1:1, 1:2.

1:4and 12, converted to the resistance of a float '-iavi_scr. a displacement of one ton.

;.,Load cazo. IV; op=1 0.218 g,Load caso Lat.

11 "

III;c3=0.75 A

0.218 h, " 0.218 ,

II;c3=C.5

I;cB=C.2,) 2

-.1

Record of spring tension / Resistance dynamometer I

\--1 ! / Coarie weight

I

-e:rs.

17

'Speed recorder wasoisvI 1 ; Rosisterca /

rounter-r

'weights L.; L_ \i --- -Loading \

fiti

_\

lit

\I

II

. )

--Rise and trim

Fig.

Diagram showing the arrangement used in tests of planing surfaces.

weights

vs Sas

I \

3

--Firm resistance Wr=4A tan -1,

Frictional resistance WH=25 vg7

(leTR)

°Measured

resistances

Mean reduction in

Breed over

the

r2C

surface vii=v-vm

lb in per

cent of

1. e ,A

36 kg

rot

= 27 "

0.0,,3

(-1

)02

1600

cf =

R v114

-= Mean speed over surface,

0.0,,3

(-1

)02

1600

cf =

R v114

-= Mean speed over surface,

F = Area 3 f

wetted surface.

F = Area 3 f wetted

surface. -

T-1,--, 1 7-Liv =

9.i mi

t

t

1--f

04.7

.

1 1

7

1 IA = 4.512 ic.

V I 8

is

contuft_

1.

--- -

Ift

-r1)1 r

Ar

--t-,

ti. 4 __ -- -- ---1---1 I 1`,.3. " 1 1

-

6 1

4

i I

/

-1---7-1-1-I

,

o) iig I +a 5 "! tp to 1 r4 -

T-1,--, 1 7-Liv =

9.i mi

t

t

1--f

04.7

.

1 1

7

1 IA = 4.512 ic.

V I 8

is

contuft_

1.

- If-

t

-r1)1 r

Ar

--t-,

ti. 4 __ -- -- ---1---1 I 1`,.3. " 1 1

-

6 1

4

i I

/

-1---7-1-1-I

,

o) iig I +a 5 "! tp to 1 r4 99 88 aa .

- j

. I I ;kt---f:-.. t ' 1 I

/

S , I 1

'

f

.

t, n 2 4 6 4

10 12 I

2 4 6 8

10 12 J

2 4 6 A 1 '

12 0

2 4 6 8

10 12

Trim angle,,,..

Fig.')

Resistance of the 0 .2 in wide

flat planing

surfqce,referred to trim angle 1.

;a1100P1s+f

I

4 . 1 1

r i

,04.4tiat V / t-"i'll".16 ---s.a...i. M.:-. b,A c,A

= 16 kg

= 12 "

3 to = 4 " It If

r,A =

hbA = v

6mis constant

--9 = = v = constant

/

.22. Cave

N.A.C.A. Technical Memorandum No.662 Piga'

PITT tin+A PCP7C

Width of surface

r1

a,Locations of pressure

orifices.

5.57 Wsmean snood over

Mean speed

Along the the whole surface.

plate. Speed distribution! II ifto -- 11.--_...44 _{11=245 mm}

t....,,..____rc_t_iii"

1..- mr441' rs___L =175 ran

1X

A

co d

₄₉

,x/

44/ /,'

0

ac-

rii/ e,Side(4 mm in from the edge.)

5

"

f,1/4 beam.

300 g,Center line

I/1

....

. 41 Pressure distribution for 8° trim 400. fq projected over the trailing edge.

to _itt

''

lif 1-500. . 11 a. 60C

4

Fig.11 Speed and rressure distribution over a wide flat Planing surface. Speed of towing, v = 6 mis Lend, A = 18 kg

(Continued on next two pages)

V-4 V M -0.072 V ',Center line .14 beam u,Side

Y.A.C.A. Technical Memorandum Na.661 Fig.11

(Continuation)

Width of plate _{75,81 mis mean speed}

-46r-

.. ...;,_,_:___,.,_ __I__ IL over the whole

,

-b L _al/

- surface.

Mean speed alourthe plate.

0.1,Locations of pressure orifices.

v - v n1=C .C32 to61.

1

---0. t It/4 - ---=--,--. , v 51

ne

11

1 a,Side , b,1/4 beam ic,Center

line

Speed distribution

lc

-d, Si e,1/4 beam f,Center line

Pressure distribution for 6° trim. (Continuation of Fico,11) 1,'=450 mm 1 =370

Dt1P3ATA

.

f

30C1--N.A.C.A. Technical Memorandum No.661

agJecations of

Width

_of'

.ate

.21.6

131-'

Mean speed along

the plate.

0 +, r) 200 or4 A 300 400 1.5:b 5._ .10 1. I,

a,Side

b,1/4 beam

c,Center line

Speed distribution

I =

Prussure distribution for

(Conclusion of Fig.11)

vm

= C.013

40 trim.

Fic.11

(Conclusion)

pressure orifices.

,-5.92 m/s moan speed

over the whole surface

C 6 100 -1. = EOC -v V

c-"RiS

Frs.:be-a

N.A.C.A. Technical Memorandum Na. S61 Figs. 12,14

--Total resistance AS measured.

1.0 C.8 0.6 0.4 0.2 Wi = Atana-0 2 4 6 8 Trim angle,

a

Fig.

a2 Division of the resistance for

V = 6 mis and A = 18 kg. 10

k..Point about which mcments are taken. I w Corrsc tion

w, - -,'"'

-t-

_11171111111

k,

r

* 1111 - - - w

Mks,

IIIIIII--Loa..

, cas II

Mit

cask ri III .. It tt , IV 2 4 6 10 Trim angle, a

Fig. 14 Postiltion of the cantor of pressure.

raVAN SI. /6/7 IP 111 a A 3 2 1 =WF

30

10 ;0. -A I 71 N

\

\

\ P

I _._ \

9.5 rile

4.7.iiiir. --I

Leading

ease I

1 I

-,

-\IN

\\ \

-\

\

8Y`9'5

1/4, N.

.;-. N..

\

-"

II

" .1II " IV 1

---

-.. N _ III - ..-c.---_-_-_ b --z--- :,-;_i -4. ---.&---A;L:--- ---.&---A;L:---a -.&---A;L:--- t-:--- --,... [:--34" - -.. ---. ...I - z --- - ": -. -;-.._ -- : [ I

1-ar_...±._

, -"1-1. 10 9 7 3 4

6

6

Trim angle, e

?/g. 12 :Moment coefficient cm -againt trim angle al

I

1