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sECURITY INrOR0ATIO.4

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WEXPERIMENTAL TOwINO TANK

STEVENS I|STITUTE OF TECHNOLOGY

HOBOKEN, NEW JEH'SEY

4

I -I ,

LIFT AND DRAG OF HYDROFOILS:APPL!CATIOS OF THEORY

f- . TO

EXPERIMENTAL RESULTS

. by

8. 1'. Koryin-Krookovakyand

Robert J. fernich

PREPARED UNDER

CONTRACT Nest 26301U.S. NAVY

OFFICF OF NAVAL RESCARCN

f (I.T.T. PROJECT DZ1407)

, , Mey 1952 Report NS._________ I. : .or w o

2r

m . . ... .. .. ..

!FC. IT tM:FORMATION

.J

EXPERIPOENTAL TOWIO TANK

STEVENS 1I',lTUTE OF TEC:4HC'.Ot YHOBOKEN. NEW jr--EY

LIFT AND DRAG OF HYDROFOILS:

Ai'PLICAfION OV tiW.rURYTO

EXPERIMENTAL RESULTS

by

R. V. Korion .Kroakovskyend

Roabrt J. Wernick

PREPARED UNIER

CONTRACT Nes 263.01U.S. NAVY

rffICf Of NAVAL RESEARCH

(I.T.T. PROJECT OZt407)

May 1952 Report No. 438

l ~ -A;

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~P-436

SECURITY IN ORA'AIO"'TABLE OF CONTENTS

Page

Sumn-ary ............................................................ 1

Iztroditntion ....................................................... 3

Symbols . ........................................................... 5

Hydrofoil Theory ................................................... 7

Induced Diag .................................................... 7Critical Velocity ............................................... ILWave-Making Drag Theories ....................................... 11Helation between Lift Coefficient and Angle of Attack ........... 19

Analysis of Experimental Resultt .................................. 21

Description of Models and Sources of Test Data .................. 21Petermination of Profile Drag ................................. 21

Model I& .................................................. 21Model Ib ............................................. 23

Model II ..................................................... 24

Rosults and Discussion ............................................. 25

Variation of Profile Drag with Lift Coefficient and Depth ....... 25Dependence of Profile Drag on Reynolds Number ................... 26Slope of Lift Coefficient with Angle of Attack .................. 26

Concluding Re srk3 ................................................. 28

Ceferences ................................................. 30

Tables I- III ....... ...................................... 32

Figures 1 -18

4f

U R IJRITY INFORMATIOn

NOW_.777,77 110M r

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ACNOLECEEN

This~~~~ ~ ~ ~ ~ reor has' benpeaedb h tafo h

Tehnoreogy, hin onetren byi thfe Sa of Nahe

Research Contract No. Nonr 263-01.

P.

P-438P-438

A hydrofoil %ovir'g at a given angle of ..ztack near the surface of afluid cxvericncea a resistance which can be broken down into the follow.Iirg coponets. a wave-naking drag aused by bound vorticea in proximi~ty

to the surface. an induced V-1 caused by t-ailing vortices resultin8, !roma finite span, and a profile drag consisting of the frictional and eddy-making drag of the hydrofoil. *

It is found in the present repert that the induced drag of a hydro-foil of aspect ratio .6 (the smallest investigated) can be computed with

sufficicat ac tricy by a procedure similar to that used for airfoils in !

1Ue several methods for computing the wave-making drag of a hydro-foil in media of finite or infinite depth are cimpared and discussed. Inshallow water, there is a maxisam velocity for the propagation of waves,termed the critical velocity. If the hydrofoil moves at a supercritical-elocity. the waves caused by the bound vortices cannot follow, and wave-

* making drag is nil. Jowever. in the transition range near the criticalvelocity, the rat& at which wave-making drag decreases with velocity fromi

a finite value to zero is not given by existing theories. In the presentreport, as empirical correction factor depetding on the ratio of velocityto 'critical velocity is applied to the equation for the twe-dimmsismal

wave-makizkq drag of a flat plate in s. medima of infinite depth in ergerto approximat* the decrease of thi. eve-making drag in a medium of fimit*depth. It the actual operaticn of a hydrofoil in open water, this care-rectiom facter become dagligible.( f-

11ave-making drag and induced drag are subtracted from experimentaltotal drag to give values of profile drag. At. exasinatin of the result-isS profile drag curves and those obtained directly from N.A.C.A. wind-tuanel teste shows:

(1) very close agreemnt when the hydrofoil and airfoil sectionswere tested at the sam Reynolds numers,

(2) similar form and reasoable magnitude when the hydrofoil wastested at lower Reynolds tumers.

Since the wave-aking drag and induced drag of a hydrofoil are in-iipendent of scale effect, it is possible, given the profila drag at a

JRTY INFORMATION

...........................

-2 - C'iFI 6E ., -At

suitable Reynolds number. to find the fu. -cale total dreg at that

Reynolds number.

It is also shown that the slope of t t lift coefficient with angleo'f attack is a function of submergence, and Sacreases wi.n depth towarda zheoretical infinite-depth value dependir, or Lite aspect ratio. "

.

I.I

p

L

I.

'COKF2OE#tifAL

, 4'e :' 25

R-438 1-438CONFIOENTIAL -3- 3

! ..

INTRODUCTION I'

This report is a study of the ep2lictt. na of the theories of induced

doinwash Angle and of waeo-aking resistance to the analysis of expari-

twintal lift end drag data oa hydrofoils. A deeply-suboerged hydrofoil act s

At au airfoil in an infinite -idiun. i.e., the effect of the freet ss -ce

is oeg - .tble. In the case of a foil of infinite span. there is no inducedlownwash usgle and thus no induced drag. Therefore, the only obiervable

resistance is tb- :.rofile or seOtice d'iA. Wfwever. in the case of a deep-ly-sutergad hyl:ofoil of finit's ,.T i.railhng vortices occur, causing a

downwash enale ved thereby mo d r'z."hG ,,lips of the ctrve of lift coef-

ficient vs. agl. of vttt, t i-.4v4,, dotuvail angle will produce in-

du~ced (trviling tort*%.r,, to ,'e. :* tnal drag is the aum of the

prafite and induLA ,f' cimapmeati.

i foil ::w!-..ej ue.r the suLf.ee proeuces e.#oo vbbih, in turn, pro-

vide added icrese.-:s of velocity to .he flow art'-±" it, $urther cz-.ifying

the lift and drset. Wdifivatioas causei 11. the pruxi.) of the bound vor-

tices to thf, Nt:ace occu" esether the p is finite or infinite. Thire

are zeveral m',ok# avilable for e,luti'u the wsaa-ak no {bound vo-

*..A) Vi14. VtV:4 (-kSftrf.ue 1) t'sprezan . the fail e-; *k z b-

WAiWt ami #*.-4!d tlae ,.wtica of tha lift -.o thu- :vtc;tioa &a 44the Sa) -L C .'i& &"th, i.e., tbe Kugta-Joukcaski ,rp:,ticn. .otchin

2;er ,n,.es'-wseit - f-il of; a mzsale vortex filamnt, but

considers znirsl. ut the frt onice _-the relati i?*..p ovsvea thelift end tLa oWnlotaic . sch ati Lsm vraLtiev ( fersaco 3), in their

study of the free surfate effects, repreaent the fc.il by a lifting surface,

i.e., a surface ot di ecotinulty on wLlch !:,nrcos ar.c vortices of varying

strengt are placed. Thee three sa.kch ore gcovsred and discussed here-

Is shallm water, thor is a mximm velocity for the propagation ofwaves, tormd the eritlui v61.ity. If a hydrofcil moves with a super-Critical Vloesity, the wev eassed by the boud vortices eset follwo,

sd weven .aing rinvoee is oll. 5,vir, in Oh transitiom rep osearthe eritel velocity, the rote of decrease of the wave-me'hig drag rom[a finite vs)** to sew is sot givsa b- ,nistlin .,ories, Therefore, an

empirical relationehip is developed in this report for the value -f wave.

making drag in the treaition rasae between suberiti cl and supercriticalvelocities.

CONIOENTVI .M ..

•~~ A -"d ,

nM

4-

.4-. CONFIDENTIAL,

It should be amphasi ed th.t the varishing of the wave-maeing re-

aistance at supercritical velocities is eisentielly a result of Inborn-tory conditions; it would ocur in full-scale only in shaliow water, The ,

evaluation of the ,ave-mak.n, drag of a full-acale hydrofoil under nor- , "

m l conditions of opration therefore depands on the theory developa*

for -foil in a meiua of irest depth. In the preseot analysis of the

shllw-water data obtain ,d A. a towing L nk, the equations resultingfroi deep-water theory 4re oW iied to tike into account the depth of

the medim.

In this refort, the term 'wave-making dreg' is applied, in lifting

line theory, to the effects of the bound vortes is. proximity to the free

surface (or, ia lifting surface theory, to the effect& of the distribu-

ticn of bound vertices). Tese effects are dotoraied for the infinite

span, i.e., for two-dimnsional flo, with the fotmitios of transverse

aves. 7he term induced drag' is applied, followrg aeronautical usae,

to the inerew t of drag caused by a finite span, i.e., cauzed by the

trailing vortices. Thm trailing vortices also form w ais, but of oblique

form.

The above nomenclature differs from that recently used by Vavra in ; IReference 1. The terminology of 'bound vortex drag" and 'trailing vortex

dra* is used by UYas ad Lsye, end in leference I Vavra groups both

of tb'se untler the name of 'sismsking drag,' which is defined as in-

eluding all resistance in txcesa o1 the drag of a foil in an infinitetoo-dimesa-cnael medium, i.e., profile dres. In Vara's notativa, then,

the drag ef the trailin vortices i3 divided into two parts:

() the induced drag of the finite aspect ratio foil in s. in-

finite nedium,

an additional increment due to the effect ai the surface in-cluded A part of the expression for bound vortex drag.

It was felt by the present authors that this terninolo y might lead tosam confusion, sad so en term '"ve-making' and *induced' dreg as

dfiwsd in the prceding peragaph ere used in this report.

CO4F IDEW I AL !

. I

R-438CON7 I DENTr I At. -- 5 -

SYMBOLS

Il. various ;erms ard coefficients tived "; this report are ta

Gen "erl Symbols

S - acce!etation of gravity, 32 17 £t./sec.v - kinema.tic visc.-sity

p " mass density

Mkrodynauic and Hydrodynsaic S,,,'tz*,

A - aspi t ratio

As a equivalent aspect ratio

b - apen of tb hydrofoil, ft.

to a equivalent srant ft.

Cs , induced drag coefficient

, profile dras coefficient

C," total drag coefficientCD - wave-making drag coefficient

C" a three-dimensional lift -.oef.icientC* - two-dimensional lift coefficient

e w chord length, ft. X1i(s) - exponential integral, f (f/)dW

F - Froude number, Vi../, baaed on the chord as charasceristiclength

N - tank water depth,. ft.

h w hydrofoil depth f aubmrsioro ft.

I. - total lift force, lb.L, " lift force per unit. spen, lb./ft.

N - "unk spen factor (o* weqtic (2))PI, f.s Qj, Qj " coei- icienta (see equations (25)-(28))

Re - Reynolus nbatzr, Ve/'S - plaufore area, ft. t •1

V horitoatal velocity of the foil, ft./saec.

-e " critical velocity, ft./se.

r a dowmassh velocity, ft./sqeo 'a- sle of attack, radiaaa

e - e.tion angle af attack, i.e., angle of attack of a fo.l ofinfinite span, radians

41. asigle of attack at sare lift, radians

CONFIDENTIAL

, ., , : ... /m - ..,+.++'" I ' ,m,'-'" ,. . :;,,+ ,.m .. .......,,+,,+:J .,,, ,,,,17. ,7 .-A", +. ,,.+,-,,.

5m _ _

CMFNIC'ENTIAL

S*planfcrm factor (aeo Page 10)r-circuzlation i ie-downwsal e fl , radians

aaform of Frctd- iiumber, 0 2 1t.

e a form of Froude nuaber, 2Sh/V', based on vubwtrgenrek as char-ecteriatic length

aMunk interfere% fasctor (see page 8)

CONFIDENTIA

CO'NF'DENTIAL -7- },

HYDROFOIL THEORY

110M~ DRAG

There are several developments of exprcsions for trailing vortexdrat. ladinirov (feference 4) "ves a sketch) treatment hised e adingle horseshoe vortex, and Varre and Mayor have treated the problem

(Referencel) in the case of the baud vortex line of varying strength*d a cosplew trailia 8 vortex sheet. This latter treatmer. livas adub! inateipal which is soluble in closed form for ciy s few limitingcases (so*0 oquiioos (19)-(22) of the presentt report;). Th nlleessary

numerical integretia for the more general cases has not yet bea can-platt. There is also a study being oad at the E.T.T. im which the of-fect if the trailing vorticas i, . ee , , "At the results are notyet available. Coosequently. x- chi* uietier .f the report, a sinpli-fied a*ipression is use4 whicW '.-..,ere the surlace disturbae, caused

by the ,railing vortices to 6. ,1gligible. Which is actually the condi-tio4 at high speeds This permits the application of x technique widelywesd ia aerodynamics.

The induced dosMekSh a0ge of airfoils is determined b' intepet.ing the vertical velocities induced by the trailing vortices. I% thepartioular case of an elliptic lift distributim, it my be shown thatthe downuask velocity is uniform alo% the open and can be computed bysAmentum theory, using as the vitual wass the me of a cylinder offluid having a diameter equal to the spen of the airfoil (Sketch A be-Ice).

tme su"ArO

AIRFOIL IN AN PLANING FOIL

1I4FINITE MED IUI

SKETCH A SKETc I •

i:,

~1NPIENTIA

( 1 111 JiTI 'I,

_______________________i

~~~~~-~"M 5'&?' t-S .~' SP1W' , S34tk'.v Wn1%C f~t e. ,WRAt.

-4.38

V

For body meving neat the 9*sr surface. the system Can bO Completedby its ref lection from the surface. Thus. for a body planing as the surf&-.*of the water. good raulta are obtained by taking )Waf tL3 cylinder (SketchB. wige 7).A the virtual aw (PBeferenoes 6 and 6).

In the present derivation of the exprossica for the induced drvg to. Iafficient. C84l, an itage &par 'Sketch C, pope 9) is semed at a 4i*,euco h(equial to the mubmergence of the wtuavl toil) above the undisturbed surfaoseUse induced drag is then developed by a procedure similar to Oat for a hi.plaew in an ifiite medium. In biplane theory (Reference ?), the induceddreg is fm-uA by determining the disaneien of a moooplamo eqtsivloat tothe bipis-u' tsbtch D. page 9) ad then proceieding by mosoplan., OAW.ee

711e equivalent monoplane spa 6~ it. given by

N is the &m pan factor, ad is defined as

sheree-La/L&, the ratio of the lifts of ewA spen

is a De/8 the ratio of the soan lengthsw a the Mwk ieterfoerone faro.

For the hydrfoil, iC 1, ad th eprosiew for beceaes

2()

Ike iaterfaeroce fectior ie determined by numerical integration of the doublelsaeal found ont pape 2? of %feresee 7. It is shown plotted spinet anextended range of W/6l (where A' is the distance between the tw, opan,equa to l2A) as Figure 1. For the hydrefoil of rectangular plenform, theeuivsleat aspect ratio, As. is given by

As Mb (4)

CONIDCSTIAL

F7.77'

R-438 R-430CO)NFIDENTIAL -9-

By using the sme method s in Fleference 6. the lift is written od

L*C .evspvvj ()

sAere J a nfb /8. the shaded area in Sketch D belo.

IMAGE FOIL

HYROOL - ~ I~AC

ATUAL. AND IMAGE FOILS UIA f FL

SKETCH C SKETCH D I~ -

Solving equation (5) for the downwash velocity, w, gives

'sc& (6)2J

- ~

The downwesh angle at the lifting line is e * /2V, Therefors.

a SCL

The induced drag coefficient is given omy

1hes the foil hes a rectangular pleafen, S be @$.A. Sahtitatiag for3 and J1 in equation (0) gives

Fine equation (4). A* N '6812&@ * '4/2 for the rectauplsa foil. There-fare,

CONFIDENTIAL R

R-43810 CONFMIDENTI AL

In tcrms of the interference factor.

C =1 ,* C s . 1 ).

• ,

Equation (11) us7 be obtained in a different way. Equation (28) o rpege 248 of Reference 7 give% the induced drag of a biplane as the -- ,dced drag of each span plus the interference effects of each span on theother, i.e.. the induced drag. D, is

2 Le 1L6 L

HQever, in the cci~e of the hydrofoil, only the induced drag of a singleactual span and the interference effect of an assumd inae span are com-'sidered. Thu

D i D- +. D 2. - . ; t L ' 1 2,L' £ (12

Therefore.

Dividing b7 (p/2)V'S and sebetitutim. C(p/2) VS for L give

=' ---L- C m.)G

Ca V'AI

which is equation (11). I r4

Equations (10) gad (11) hold for the elliptic foil; for the rectos-plar foil, the induced drag is obtained by mltiplying the induced dragcoefficient for the elliptic foil by (1 + S), shere 8 is a pleaform fee-top. tbue, eqmatimos (10) and (11) becon

0 ,,I ( $)( F)CI (140),;:

Th factors a sad 8 are small, so that for low aspect ratioe, equation

ANtFIODIN IAL

I-

R-433 3CONFIDENTIAL U

(14.) may be written as

...SL (14b)

Values of 8 oyer an exteaded range of A are abstracted from Reference Sanid plotted on Figure ?.

CRITICAL VELOCITY

In Reference 9, Vavra shows analytically that in shallow etr,

e* g., a towing tank, two entirely different flow regimes exist. Mse forr <~i aiua h.-other for V > -47s where Tis thedepth of the water Vin the tank. V - H is termed the critical velo4...ty, which is the I '

maximm velocity of waves in shallow water. [.At superritical velocities, the free trearverse waves caused by

the bound vortices cannot follow at the speed of the moel. a"d the aur- I Lface disturbance taken the form of a solitary wave traveliag with themodel. Us"r is so wave train in the wake. and coesequsetly se energyloss, and no component of wave-making drag. noe tosts reported is nsf-ereace 10 were all made at velocities above the criticwl, and in theanalysis of the data made in ti~e present repct (se Figures 11.sand 12).the exclusi.a of a wave..akisz drag gives results which are closelychecked by wied-tuanel tests (Paferenka 11).

The data of Referoe 12 il-d velocities ranging from olsbotiti-Cal to eupercritical. ad the problea aisesa s to how abruptly the on"e.making dreg CeAse" to exist at the critical velocity: whether there is ajump discontinuity, or a smooth decrase in the aset of energy tress-witted to the wake as IF increases toward the critical.

WAVE-MPING DA THECRIES fUa Irferesce 1. yvara gives the following equation for the two-

dimensional wave-makial dreg. Drt Of a Single verteX line is 0 udlenM Of

infinite depth: (5

L

where the symbols ane thern give ma pag 5W . 1k oempare this eqUtLes

COW. I I IA

R 7 , --. 1 E

J2 -" CONFIDENTIAL

-pith the expression obtained by Kotchin in Reference 2.

Dip a " " 0 "-h (16)VS

and states that sith the circulation obtained by the Kutta-Joukowski equa-

tion.

r - 4 (1?)4

equations (IS) and (16) are identical. However. equation (17) applies only

to au infinite medium. Kotchin gives the equetiou for L, in the case of a

semi-infinite medium (infinite depth, free surface) as

L, -oy - + Us~() 1e)

where IS(x) a J.j,/u)d,. Equation (18e) is thus seen to be the Kutta-Joukowski lift times a factor varying with depth, velocity. and c€rculs-tie:

IrL - OrrL * I - .(h (18b)

The actual error involved is wing equtios (17) in a semi-infinite medium- - will be a mlight oen at low values Qf V/. h eceaainq to cere at

Yl-fsk-f I.S?, and increasing with -'/14 thereafter. I

-In Bliforeee IS, Vow,. gives an exprssioe for the weve-making dragof a vortex line of finite sen, .moving in a fluid of finite depth.with varying circulatios around the spes (lift variible slong the open):

- WIa ) (19)P

A hlois. %ei is estsis is abo tomeou pp foslsOe IS. bheas voe.o dee I. ealed#a is i$b ,.sei sss of skis fe $# ed is 091.vss.. 1, is I ssd r.s

COWu IOWETI AL

. .. . .-

* . -. . " *-; -°

where[

2L In-h,41 U (4,i'/b')' 0i

awan exponent depending on the distribution of lift along thespan (for uniform distributionj a *1; Lot elliptic distri-bution. a ~

o 2~ (t~ - 11 )/411

H (I'land H,0 Hnkel functions. 4Closed solutions of equation (19) are possible only in three limitingCases:

(a) Very small values of 0:

(2) 2)~20

(b) Very large values of 0:

L (2 1)

whene

(L111) a ame value of LIS Wlowg t00 spes,

'square of the sam vale of L, *long the sem.

The quotient (i1"T/(r)' is unity for uniform lift distribution, sod 1.0for elliptic lift distribution.

(c) Very small values of X:

2L'_ flail * (6'/4h'2)1l (2

* - CONFIDENTIAL

' 7f le ,"

-;J4 -CONMIEWIAL

If the wave-making drag is obtained from equation (19). the inducedjdrag, Dj, is taken in FReference X tot that for a foil in an infinite medium,because surface effects are assumed ti. be ac...unted for by the followingterm in equation (19):

It will be noted that this term in identical to the interference factor a,

given in l'eference 7 which is discussed ourlice in the present report on

for the 'lift and wawe-mshing drag of various bodies submerged in a two-

dimensional semi-infinite medium. for a hydrofoil. the I.rcb!an is Lo firdthe characteristic fuction of flow having a given velocity V nd a given

line representing the foil. Cy placing em this line sources of intensityfq(*) do ad vorbices of intesity y(s) do which give the amme flow patternas that actually occurring shout a this foil, tho desired discontinuity of

- - the normal and taugeatial velocities is obtained. The boundary conditionsde to the free surface are impoeed on the am of the sources and vortices.from the flow patters thus obtairad, expressions for the lift and dreg atsay angle ei stu.ck are forulated. Actually. in Reference 3, the derive-

(Jtiem is carried through only foi.. this flat plate at an angh. of attack

Sop chord e. ad sulbmergence h. In the present report, thi actual hydro-foil is assumed to he equivalent to thoi this flat plate. lb. zero angle ofattack for asymmetric foils is assumed to he the angle of sero lift. of 0

lb. expressions for two-dimensional lift and wave-making drag as do-. :~"oloped in Referene 3 are as follow:?

L, v1feVla (P1 - sos (23)

'4;:P,~ +

2 9(25) .

CONFIDENTIAL

.8. .

. . . . -. . _ _ .._ .. .. ... ...

R-438 -438

CONFIDENTIAL -15-

---I V (26)2h2)t 8K K I

Q'O t 2h - 14 2h - (

horeUh• /V2

P1, Pg. Q1, ad Q, are shows plotted aspinet x at various subuirgaces emFigures 3. 4. 5. and 6.

ihere is a certain amonat of coafusion in the referesce material te- O-- *grdiag the above expressions. In Dftfresee 3, as additicoal coeeficiest

of 2 is feud in equation (24). It is emitted here aice it is a" found

in tfoeroese 4. chitk quotes the Keldys" ad Lavrativ results, or is

Wberefte 14. which developa the eM expMesisas idepeudeatly. Ok theother hug. Vlad, i resce 4, eppret0ly cestd kin elf-

ties an hi.her spoes sA deeper submergesces, since be wrote the vew':mking dre exa re ,ics. as

witting teJ follwing tem of equation& (27) am (211:

Soe n 1/64. is of sok mgligibln mnitude that it may be mitted at oil

but the vary leset speeds. 24(0.3125) 0 0, but i(h,) rapidly bomes ap-rmiable at vales of A, above sad below 0.3725. It is easily mss thet the

edero tom will be of suffisimt sies to he isclded t moderate speeds

sod ehemepeee, Therefore. in the psit mopet, the complete eapres-

,, / ¢ONVIEIINTI AL

* /

7, .. .7

• . . ... .,-/, ,

• _ o/

4'-

!7~

a R

5"f7-M.,7 T,1 7MS ",.A

.18- COFIOENtIAL

*ions. i.e., equatione (2?) and (28), (23) used.4

In coefficient forn, equations3) n 2)bcm

C 2mIP -Q -a*Q) ( 30)

Any comprison of the theoretical resuls for a finite-san hydrofoilin a finite-depth madium obtained loy the formalas of Vswra end those UseIdin the subsequen~t &aalyaix mest neceusarily be made on the basis of tha..u

that Vavra uses the induced drag of the foil is an infinite aiediu. a"d c-r-

recta for all suface effectsi he wavemaking dragtom whas itoe a--

presajona applied to experimntal data is the present reporttaeio c

count a surface effect is botis the induced and wave-making drag term.

In order to compare the two ajtheds. take the particular case of ahydrofnil nder the following conditions: A -20, e - 2.5 is., 8 - 5.P. ft.,S- es chord, sand beam-depth ratio. 0 a 20.

Conaider firnt Varra's method. ine A is large. equauion (21) is OP-plicables for determining Cs. A uniform diatribution of lift along the span

is assumed. The wawea-paking drag par unit spas is thani *

or 0- , #(32a)

or, is coefficient fern,

Z_:A6L (32b')

Use induced dreg coefficient for a feil in en -.nfinite nedium is (.)~~

Thi semd method am the K.Udyeeh-Lavrastiev focmula. equauion (31).'medified by a facter to tahe into account the fact that the mdium is of

C3UIDENTIAL ,

5_____________________________________

cot~gggNR-M R-4W8CWFI~ff AL -

finite depth. This factor in discussed fartber os peis 22. 1he sb.Ilow.water wsve-ahing drag coefficient, Cff, is given by

where V. d~ the critical -locity. 'The induaced drag is fouind by eqms- ;ties (14a). The am of the wave-making ad induced drag coefficients inshallow water then became

Use results of equations (33) and (35) are plotted against CL~ anas Figure 7. It will be noted that equation (33.) consistently gives higlierresults in this cese.

For a semi-infinite medium, i.e., tack depth 8 very large, it willhe sees, that is equations (35),

There are three equations for the two-dimensional wave-meklux dreg*of the hydrofoil in a semi-infinit. fluid: Vavra's (Squatios (15)), gotekin's(Nquaties (16)). and that of leldyse.4 ead Lavrentiev (equation 31)) Iscoefficient form, eqnaties (15i beeses.

cot ~ (36)

In order to pat squation (16) ibts coefficient form, it is heoessrY tofind am, expression far the sarmulation, r'. in a assignsammoradu (Ibefer-aece 15), Gibbs ad Con. Ibe. sugested that the valm of r arun a this&ymtrial foil A& as Iinite medion be'used to sppgrivats the circula-ties a-.vm a hydrofoil. Pres Wsereace 1. pages, 198-M,4 this eirselationis famd to be meV. bheiatis watefo rin evietion (m6)gives

CONFIDBDITAL

7' - .4 , =kle<

4' '4 't~

-18- CONFIDENTIAL

Aeplacing r by m.eV in *qution (186) results in

whr x a (/4h (g/'e4i?3 Saititatiug equatiot (38) into (3?)gives

C a- ,(39)

In order to put equW tie (31) into the &am fouw, Gquatice (30) is re-

Sinhetituting for a@ in equation (31) yields [

Ca lm - Gov, (40)

Raqutioss (36). (39). mid (40). which are all in the fore (Cj/4K)I1(where J a 1.2.3). are compared in Figure 6. where Ij is plotted againstred amber, YArsjt. at various aeptbs.

At the dempeaw submegssces, it is am that the three eqntioas areis close agreement. Namever, equstioms, (36) ad (39) bu..ams loes and lessaccurate as the submargemov dermose, since they ane heeid on values oflift derived from the circlatiae armmd A single vetex in an infinite

madium, ad melect the considerable effeit of the prosimity to the freesurface an the circlatiem. 0a pap 38 of Refrence 14, Kotchis giOf" sVexpresseion fco r which* takes into accont the free suface, bet it is coe-

sidered tee compicated to evaluate for the perves of this report. Let [

it suffice to say, therefore, that equatios (36) ad (39) give good re-sults at the deeper suumrgemsce In a medium of grest depth.

CONFIDSIAL J

- * TW O

CONFIDENTIAL-

Equati'mn (40). which considers a number of sources inl vortices dii.tributed alon-t a line 91 discontinuity, with the free surface as a bettr. arycondition. Maeines the circulation by the requiicment of a-finia. 'relorityat t -- trailing ed9re. It also ha.n its limitations in proximity to the our-fact., since at sz.& troude nuxbors at ahallow subcmergences, it results in&

negative save-mtking dreg, *hit.. is physical'y impossible. However. atshallower submergence* (about 1.5. to 0.75c): the results obtained usingeqiation (40) are moe reliable then those obtained using equatimse (56)and (39).

RELATICt4 OFTWEEN LIFT C(XFFICIENT AND AWAXL OF ATTACKIIf the usual pattern of deriving the lift coefficiet slope used 3a

airfoil theory (PBsference ?) is followed, the angle oi attack of a foil of. *finite spa with raspect to the undisturbed surface, a, cast be def!;aisd asI

aMe+a(41)L Jwhere

as th agle of attack exclusive of the induced velocity compooset, .3.e.0, the angle of attack at infinite apect ratio, for the samslift coefficient,

c - angle due to the vertical velocity induced at the lifting lieas the result @f lite spen.

Differentiating with respect to the li ft coefficient, C'J, gi, .d

.a.5+ - e(42)

For a deeply-euberged hydru~oIl da#/dC. a 1/2w, end, from equities (?), .:

Si.

As the asborgesce increesbe, J approaches w62/4, 0?,

(0 we (43)

Consequetly.

CONFIDEWT!AL

A-438-20- CONFiDENTIAL

and

rhereforo,

(dc\ A -

To bbtain the slope of the lift coefficient with angle. of attack atinfinite spsect ratio, equation (42) is raarragpd:

Use term 4ia/dCs my be obtained fres .periuestal data. and from eqa.

for the elliptic feil. 'Fro'- W;'atift (14a). *54

do 1!;U +(MI+i0

for the reatageer fell.

CMIMALI

t I-1~~~.,s 19"MPW MsI"JK

CONFIDENTI AL -21- i

ANALYSIS OF EXPEnIMENTAt RESULTS

DLWtPTIION OF MELS AND .4RUcE OF TEST DATA 1I%* "ss reposted in Wefrence 16 were couducted at he I.T.T. tm

wooden atl of N.A.C.A. 641-A412 section, bereisaitr termd Model Ta(chord lenuth - 2% in.; aspect ratio a 20). The velocity on was from I4.85 to 9.70 ft./see., and the --- imm'lift coefficient mm 0.62. oefer.once 10 reports the tests conducted at the N.A.C.A. on a stainles steel

model of N.A.C.A. 641 -A412 ectim, hereinafter trend Model lb (chordlength - I in.; aspect ratio - 10). The velocity remoo was frm 15 to 3S

ft./sec., and tote maxim lift coefficient Was just Over 0.50. Peorms" I12 and 17 report the results of tests mde at the E.T.T. en a symtricalaluminum modoil of N.A.C.A. 0012 section, hereinafter termed Model II(chord length aS in.; aspect ratio - 6). The velocity rnag as fem 3.5 Ito 16.0 ft./sec., and the test were run at two water depths, 5.33 ft. end3.75 ft., to give a variation of the critical velocity. The maxim lift Icoefficient "a 0.68. I

DOTV4INTION OF FArILE DRAG

NOBEL Is. Tt angle of attack (4"), lift coefficient (C) ad total (drag coefficient (Cs)OC reported for the-hydrofoil in Reference 16 aregiten in colima 4. , nd 6, respectively. of Table I. The Car value ofthe foil ar plotted against C12 in Figure 9, and least squara cvrves oftU form Cpr'u Cjt' 6 are diawn throg the points for eack .tode m-,ber and smerge ce.I .

Since the toits of Reference 16 sen all made at subcritical velo-

itie, it was originally seaned that the profile drag, Cop, eeuld be ,

ritten as

co, * C, j - c oC (49)

where

C'.ieesed lfm" os% to *s. p00466 r. . £11 rw -oo ser 4 sl 8- to &I *1eseek @ad so"4 da0 *eo 4bbi04 ges emo bi. fqnat pep S *I bpeteos it.

1CM64 lot se e 0.4 etesed .effe is $be Poes 01 s o sets eeer*SPis4 see6"69d fee fi .I. bforge.. M,.

CONFIDENTIAL;5;'"

-4Z A"

- 22 ~COViI DENT IAL

(1U + 00 + 01C92 , from equa tion (14.)

fhathe results thus obtaimea ware plotted agafast -C ,'LQ E, rmeqain(1 it was sees thatCpdecreased with lift. (7his was also observed when equation (49) was

applied to Mode 11 data at subcritical velocities.) flowever, is the wind.tomlteats repvted is Resference 11, the profile drag invariably shared

as iscesioase with- lift for every Reynolds nmber. It was assumed, therefore.that this WwAa.d hold tm. to. the nrum of Playnolds numbers considered isPlefoer... 16. Equation (14a) was derived with the assumptiom, that the scr-face disturbance is thlgili.1is assuption is. ohows to be reasonablefrem the analysis of the results for "oIes b and It at high (suprcriti-eel) Freud. nmbrs, where me wave-emking drag exist., sizes the-profiledrug is fond to increase slightly with lift, as expected. Equation (31)was derived for a to-dimensional foil in a semi-infinite mmdiim, whereasthe data analysed are obtained from tests on a three-dimniml foil ina medium of finite depth. It was assumed, therefore. that altowh both'i .

eqation. are is erroe at low Fradesombr. the results of equation (31) Icontribunte by far the preater part of the error. It was decided to applyas empirical correction fees.. to tk& wave-making ireg torm is equation(40) to tat* isto account the fact teat there is a critic'al wave-smn I-

velocity. The factor decided nup" mo l - Wyf* 13, where re is the criti-Cal. velocity. re a .41. Thme appliCation of this factor~ given a smoothtransition to a wavrmeaing drag of zero at V a V., indiceting a steady4wea.a of sho aasunt of emargy transited by the bound vertex as thevelocity app obe the critical. In the &%beritical rag. the expression ~for go then becme

cot S, l -(V/A)]fl a! - (50)Cop ~ S, Co" S

At suporcritisal. veocitsica x1 .

It shoald be meted that the use of amns formula, equation (33),wi1 livs value. of Cp, which, decrease with lift, Big"e the C9 + Cog re-cults are higher them these obtained by the see of equation (35)

CONW I DOITI AL 4

) ~R-438 -4CONFIDENTIAL -23- 2

The ropjlta of applying eplatioa (50) to the *A-o1 IS date are gives

in Table 1: column 7 lioti. values of Coi + C&W3 , crt.O *:lUM A $hooo Gb.

tained by subtracting column 7 from C,,. (colw. 6). 0a Figure 10, C., is

plc.ted against C.' and a iesa square* curve of the foru Cop aCj13 + b

is drawn thrcugh the points to CLt -0

Model I& is an asymmetrical modal, and theroforeo its point of mini-SMM profile drag doss not occur at C5 0, as indicated by the form of the

leasnt squares equation adopted. To obtain a parabolic curve eith a imin

point at CLI 00 the least squares equation should be of the formCo (eC, + ) The point oaf minimum Co, Loeid fat Lhis foil is Wafr-

Saco 11 ts at L'*0.04. Above this Yalu* of CL the datum points throughwhich the two curves (i.e., C,, a GCL 1 + 6, and Cap -a~ L + 01) mst be

draws can fit either curvc, closely. But since it is easier to compete the

coefficients is the equation Cap a eCL' I 6 this is the form sed. Toe

least squares curve bel-w CL I* 0.04 is shown dotted on Figure 10.

006EL 1b. Poference 10 givesn faired plots of CL MAd CO~ vs. velocity

at various angles of eatat and submegencon. Values of CL sad Co. were

taken off these plats at Foode numbers corresponding, insofar as possible.

/ to those reported is 11efaremee 16. Mwe drag and angle of attack were cor-

reted for ground effect, and strut tares es were Subtracted from the total

dreg coefficient. Ike lift and drag mwe then croes-fairea Lay plotting them

a piat depth of submersion and angle of attack. C,, and Co. were them ead

edfatthe subnergesces tested'at the E..T. an MoAel Ia. The corrected and

erea-faired values of aso C., and i.pt are gives in columns 4. 5. and E.respectively, of Table 11. Mwe C#, values are plotted against CL' iSo FJIgu

11. mad least squareaucurvee of the form Co. + bC * so draws throughthe points for ech Frsude nomber sod depth of submersion.

Since the tests reported in Refereace 10 wore all rum at supercriticalveloeitie. co, is found by

wbieC is obtained. as before, from equation (14.).

0 to....I ab. toot *..d.e~dem 06 e P.A.c.A. 66 -41S *"#(*a.. Mob d194t trm 0heP.AC.A. 44f

1 . o t d.,.. is *611 L Reb"..11 s $e stailtee 64P oem,

ee 040 Me@~ ~ goo *g644 free tit~we 1 00f blerego. If.

IMN7IOENTIAL

1N

................. *

1-438-24- CONFIDENTI AL

The results of applying equation (51) to the Model lb data are given

in Table I: values of Cg, are listed in column 7 and Crp in coluam 8. Fig-

ure 12 shows Coo plotted against C~t , with le*.t squares curves of the form

5o - eeL 2 b drawn throulh the points to C. - 0. This form ia adoptedfor the sae reasons given above for Model Is. The minimum profile dra5 is

alto assumed to occur at the '--e point as frr Model Ia, i.e., atCLso C.04.

Below this point, the least squares curve is shown as a dotted line.

MOREL It. Columns 4. 5, and 6 of Table III list the values of a, CL,

and Cal, reported in References 12 and 17. Figure 13 shows Co. plotted

asinst CL, with least squares curves of the form Cr - sct + 6 dranthrough tba pei ta at the various Fronds numbers and submergences.

The teats reported in References 12 aud 17 range it velocity from tutsubcritical to the supercritical range, and the profile drag is obtained

from equations (50) and (51). The results of the analysis are given inTable III: cclumn 7 shows Cei + Coo' (at suprcritical velocities. Coo'00)

and colum I lists values of Cap found by subtracting column 7 from column

6 (Cpr). Coo is plotted against CLS on Figure 14, with least squares curves

of the form Cop e&C + 6 drawn through the points. Midel II is a syme-

trical foil. snd the minimm profile drag point occurs, therefore, atC'L'C O. *0

* C eoeeed fee eposed elrfeee I0 th. veeeee V.9.01.

C.,eoosed too osfed offet ed sent $*to is 0he pro***& top"%I. t toe*. get. se does@.e" Ise the seas of Asteteeee s ted It $ed at e*eee" to be 1/3 of great as obee gl9ees

aFlge. 1"1 1...... 1I.

CONFIDENTIAL

R438 j -438CONFIDENTIAL -25- 25-

RESULTS AND DISCUSSION

VARIATI OF FROFILE DRAG WITH LIFT COEFFICIENiT AND DEPTH

Figures 10. 12. and 14 show the C#;, results at various velocitiesfor Models Is, 1b, and II. rt,. actively. Theaes profile drag points ateobtained from Car data at various submergence*. Since profile drag is thedrag in an infinite medium, it o3iould show. so dependence an the depth ofsubmersion of the hydrofoil. It is seen that. almost invariably, the C&Ppoints fall close to the least square* lines. with only a normal "antof scatter. lTb- only exceptions are in the case of Model Is at low speeds(Figure 10), which my he explained by the fact that there is a largespread in the C11, data. However, on Figure i2. it is sees that there ispractically so scatter of the Cp1 points, which were obtained from cross-faired 0o. values. This indicates that if all the Cr values had booncross-faired, the scatter in the Csr values of Figures 10 and 14 would

have been greatly reduced.

With the adoption of the enpitical correction to the wave-makingdrag, it is also seen. on Figures 10 and 14, that there is a small posi-tive slope to the least square lines drawn through the Cpp points ateach Froudoe somber. i.e., the Cp, increases siightly with lift Thus, atlower A* (below the critical velocity), the use of the correct&on factorgives results which are reasonable whokn comperod with those oko.aisaedfrom wind-tunnel teots at higher A-!.

Figure 12 shmws the variation of Cop with lift within the sra ofReynolds numers repotted for the wistd-tuaael t4sta of Reference 11. Thesmooth surface ad standard roughness C#V values of Reference 11 are alsoplotted on Figure 12. Comparison shows that the Cpp values obtained byapplying eqUation (51) to the Car from the towing-teak tests of Dleference10 are slightly higher tha the smooth surface Cpp of the wind-tmeltesta and well below the standard roughs~s Co,,. However, as is mentionedin Defersoce, 10, the surface of the hydrofoil model was slightly pittedduring the tests by the salt water in the task, and therefore ms prob-ably somewhat rogher than the smooth airfoil tested in the wind-tunnel.

A very isterotiag anod illuminating result in obtained in the caseof the Model 11 data at V a 12.0 ft./soc. (Figure 14). The Ca,~ data fromwhich those profile dreg points were obtained were taken at two submer-

CONFIDENTIAL

-26- CONFIDENTIAL

gences (h - 0.75c and 1.50c) and three tak depths (H - 3.75 ft., 5.33 ftand 5.83 it.). Although at N - 5.33 ft. and It a 5.83 ft., the velocity issubcritical and equation (50) applies, and a' i - 3.75 ft. the velocity issu.-ercritical and equation (1) applies, it may be seen that the points obtaised in each case fall very close to the least squares line dra-n throu h

3l of them,

DEPElDENCE OF FPCF ILE DPAG ON REYNOLfl6 NU4ER

The valueA of Minimum Up. for the three hydrofoils considered areplotted against Reynolds number on Figure 15. (For Models re and 1b, theVinimum Cop ac .urs at CL' - 0.04; for Model II. the ainirw. Cp is at

C 0. ) he following curves are also plotted for comperison:

(a) Blasius laninar line(b) Prsndtl transitiom line

(c) Scioonherr man line(d) Smooth surface C., for N.A.C.A. 641-412 and 14. C A. 0012 sC-

(o) Standard roughness Co for N.A.C.A. 641-412 and N.A.C.A. 0012fations.

The slope of the Model I& minimi C9p, points is greater the that ofthe BIasiu laminar line, but the geseral shape of the curve indicate that

-; the flow was lsminesr, with oametioa causing a high drag.

Ssectioe sape of Models Ta and lb m desived to give laminarf11w eve at high Reynelds numbers, under favorable couditios. Euniza- I '

tics of the o, el I data am Figure U indicates that the flow s prob. .•

ably at least partly raimler at the lower Reynolds mmbers. As A# in-ere"", the fall very ClO" to the transitie line.

The lime das through the Model 11 points is exactly parallel tothe Schenerr me liet, which indicates that the floew around the foilwas turbelent, Bwever, the line is displsod @lightly above the [Schoberw line. This my be de to toughnee of the model surface,eparstiom of WOO flow, or a embsatift of both.

5M OF LIFT C IrIDT WITH A.E Of ATIACK

The nles of attack (corrected for ground effect) and lift cooffi-cieste from the tests of leforec*es 10, 12, 16, and 17 are give, in

CONFIDENTIAL

... . .,v-l I un L n - ...... - - - --,

. //I

CCtEFIfTIN.27

colmm 4 amd S. respecti'vely. of Tables I -I1II. U hs. are plotted witha as abscissa and CL as ordinate as Figures 16. 17. Ad He ford"Is Ua,1b, amd 11 respectively, end least *squares WaXes of the form C, asme+aft drows through the points. Also ohms. as thoe. figures is t~ie theotgti-Cal lift coefficient at infinite depth, (C),, wkich is obtaine" by in-togpating mmueim (45): j ;t

where 9 is a coeat atendled by aat zerlift, *s For Modelslahug

1b,.g a *-4.0** ed!"; for *4111 q .0 rediam.

-~~ - A examination of joh corem as Figares 16, 1?, and 11 ohmw thatthe slope, iscres., with depth toward the thensetical vale obtained ffts

* ~equation(2)

Y-2

-~~~ *s~o .m as Wo si of Ibism - U tan, %s 131). Is is ese ses a ern me wade Wa Obs-O"soems of eagle of 8as6e6. a. lie see s eO.83. a"a inesw* Is she famsa see66.

'I I. if '

...... Ile

CON4CLUDING REZdA1KS

It should be esi-iie to apply the exPresvions given in this reportto the predictiom of the lift and drag of a iull-acale hyirofoil. In the

- prev.ou0S sections, theories of induced P~M wea-making resistance, modi-fied for laboratory conditions, were applied to towing-task data. The re-sults, obtained in regard to profile drag are consistent with wind-tunnelresults, and ia the cases where direct compariso is possible, they arefoe"d to agree very closely with wind-tunnel results. It therefore appears

that the initial hypothesis that th3 total drag of a hydrofoil may be di-

verified. and can be accepted ts a physical fact.

Induced drag is a fnction of aspect ratio, depth-to-span rutio, sadlift coefficient; wave-waking drag depends on Frond. amber, depth-to-chordratio, and iift coefficient (circulation). Therefore, the Frowde Law ofComparison holds, i.e., these components of drag are independent of scaleeffect.

Thus, f 3 a full-scale hydrofoil. equtin 1%14a) on page 10 my beused to predict induced drag.

anud equatioa (31) on page 16 to predict wave-waking dra.

S.. Figures S a"d 6 for Talmes of Q, and Q1.

The pre~ilt drag., however, is a function of the foil shape saidflaynlds number. A full-asos hydrofoil having a chord length of say 10 ft.my Operate, for instance, at a dssined A@ 1% S x 10'. and c 5 ay oe de-terned from wind-tunel toots of the foll at this rPsynelds n~mr.

PIs the special case of a hydrofoil in shallow eater (which is thecondition under which towing-task toots are conduct'et), it must be re-meubered that the wave-aking dreg component vanishes at velocities he-youd the critical. Thes procedures followed in analyzing tho test data in

CMWFID4IAL

-. __ - ___ 1.I ~

R-438 1 -436

th present report ~y be comaidered analogous to thos. used whoa reducing Iraw wind-tunnel date to readily applicable for... I

'I* -- I.

S <

I.

~ )* / 1.1-/ ,

I ~ ~N

CCSUIDUITIN.

- I. / . -. *, -.

.1' - S.'[I .--. -/

- . *1 * /

R-4,3

-30 - CONFIDENTIAL

REFERENCES

1. Vsvra. M.N. Review -if ave Drag Theories. The Hydrofoil Corp.. , IAnnapolis, Md., Technical Memoraudum No. HM-13. April 1951.CCPN10SNTIAL

2. Kotchin. N.E.; Kibel. J.A.; and Rose, N.D. Theoretical ffydr mchs" c&Chapter VIII. Arc. 19, Mosccc. 1948. (Translation by B.V- Korv--iKroukovwky, unpublished.)

S. Keldyach, M.V.. and Lavrentiev. M.A. On the Notion of an AereoilIhder tse Surface of a fHeey Flui. i.e.. £ Liquid. Science Trans-latio Service. Cambridge. Mass.. SIS-7S. November 1949.

4. Vladimirov, A. Apprximate Hydrodynaeic Calculation of a ydrefois ofFinite Span. ZAII hport No. 311; Englisb Traslation. AdmiraltyDocument 1W53280/D, Iky 1946.

S. Wager, N. Plening of Wtercreaf. N.A.C.A. TM No. 1139. April 1948.

6. Korvia-Kroakovsky. B.V. Lift of Planing Surfeces. Journal of theastitit. of Aeremau tic t Sciences. Vol. 1T. N. 9. September 1950.

7. Von Mis. R. Theory of Flight. First Edition, McGee- Hill, NewYork,1945.

8. Nydrfei I#eaeearch Craft for Off ice of Newl I eearch. Dath Iron Works

Corp.. by Gibbe a"d Cox. Inc., Technical Report N 1. 4,uuuuaITIAL

9. Vam. M.H. Few* Dreg of Suice. ged Feil in Shollox e ter. The Hydro-foil Corp., An"polis Md.. Technical Memoradm No. W3, Movebeu r1950. CoS:ZUUsIL

10. Madlin, K.L; emen, J.A.; and MaGeho., J.H. Tank Test# at Subeavits-ties Speed@&! an Aspect-ettie-1Q Hydrofoil eith a Single Strut.N.A.C.A. T No. t91K4.. CoOPlssv9uua.

11. Loftin. LK.. and Sith, N.A. Aeredynamic Characteristics of 15 N. .CA.Seetise of Seven Reynolds Nunere from 0.7 X I0 te 0.0 x Joe.N.A.C.A. I 1945. October 1949. 1

12. Sutherland, I.H. Letter Repert om Test# of $1 Cherd Aspect Ratio By-drefoil Mael. aperimentoal Teming Tank ipert No. 414, Jume 1951.

13. Vart M.. Move Drag- of Sub erged Foils. ne Hydrofoil Corp..Anpolis, i. Mae NO. W.-, October 4. 1950. ceuvzSINyzai

CONF ID01TIAL

f .

CON PE NTIAL 31-

14. Kotchin. N.z. (4 hthe fae-.Mkr Resisatance and Lift of ao iea Sob.aerged in fater. Transactio..A of the (^nference on the Thozy of

Wave Resistance, U.S.S.R., Moscow. 1i37; Trenalatins by A.L(T)

Air Ministry, B.T.P. No. 666, March 1938, The Society of ?hvalArchitects and Marine Engineers Techaical and Research BulletinNo. 1-. August 1951.

15. Michel, W.H. Comparison of wave Dreg Theories. Gibb a d Cox. Inc.,Design Meorandum, March 1, 1951. CONVIN tIAL

16. &Atherland, W.H. Letter Report on Istension of Single and TdenFail Toia of 2%' Chord Aapcct Ratio 20 fydrofoil. ExperisemtalTowing Tank Report No. 410, June 1951. COXY1SUNTIAL

' 17. Kaplan. P. Letter Report um Extension of Single and Fanerie FoilTests of 2%1 Chord Aspect Ratio 20 Hydrofoil and S' Chord Aepset

Ratio $ Hydrofoil. Experimntal Towing Tank Report No. 428,Novober 1951. CONUItMNTIAL

16. Sutherlamd, 1.11 Exploate"eT Model Tets for Engineering Design of aHydrofoiI Vossel. Experimntel Towing Tank Report No. 407. Jamory1951. CONFIDINTZAL

C E

'i:.

CONF OEN I A

.. . . .... . I I II I I I I I I I I I II I! 11r / / ) -

• ' V

; o* //2

,// "

,10

-. 32-CCNFIOENT AL

TABLE I

EXPE11INTA~L AND TNHORETICAL RESULTS '

(Experimeatal Or"g COefflcleats Are Corrected for Strvt Toro and ke 4a Effect):j

4y) () (1) (q) (: 6 (7) a

SlOIMMLENM a 2* ses. TAK WATR DEPTH 5.96 FT. DPh1203

0 160 .6"1.0362.01 . MSO.11 .0001 .14

2 1 40 .0201, .08 .020a

sag ig .46 2.00.1 .24 .91 .0? 06

al 0.01 .241 .0206 . 604 .0216I2.61 .401 .6201 .0041 .0746

S401 .26 .6ill .0190 ..371~44 6.0a1i :1 i *Ii

~~~1~4.62 .30 .09 .4104 .

S20 1.80 1.4, o .01 .27 .3 .660 .0193:. 411 INN2 .60136 .03'12

C.IL .55: .0833 .0140 .93.2 0.0 1.44 IS .201 .42 .60211 40

111 IN "62:1 *."" I..1 36 .040 1 O

cI0 E6?I'*in. TAWK WATER DEPTH4 5.96 FT. CKPT14 1.5 4mg

1.6? 4.06 0.6 01 0 .13 .0111 .04s .040 16is .2 02 1

2.1 130 I .01 .213 .@46 .092 002.01 .84 .0160 .0ft .01094.01 .556 .6460 .016? .021#

2.0 .30 1.44 2 10 .0 .276 .0200 .004 .01624.01 S .0261 .01"0 .0200

4.6 .40 .11 .0173 .01"22.26 0.56 1.6 91 0.0 ISO 0:a"288 ::T .0083 .0150

8.01 .219 .02S? .6004 .6161

0.1 .106 -- AMo 44 .1 . .08to .01726I.0178 4 .1331

1.46_ __.__1 .1- -#6 .0084_ .1 _ _ __ _

4.6 .. 0 C3$./1 0

.S /.0aI$ 90 .8 $9 63 05

/ /s

P-439CONFIDENTIAL -33-

TABLE 11

EXPERIENKTAL AND TNEORETICAL REIVUSTN.A.C.A. #it 1A02 (MODI!L 10)

(Experimental Dral Coefficients Are Corrected for Strut Tore and green Effect)

Avg Re a a. I CD* 1~ 4 1 fC C & I OV/48c ft./*"o. I Veil' 11,. 11 1i (9 -

CH ~ i.TN WTRDPTH 8 6.00 FT. DEPTH I Cwwo3

5.4 1.2 0.953 xS 1 0 .226 .0103 .024 ."IT0.5 .4S OlS .034 0019

1.0 '1 O .0, 11,3 .00t .001.5 .344 .0143 .0a0 .01122.0 .311 .04 .00ts .4414s

2.5 .422 .14: .31 002.0 INS2 .01,96 .01C .041111

2.14 11.34 1.10 led0 0 .226 .6095 .0026 .40120.5 .23 .11 0 $'* 034 I0M

I. It J4 .012is .0041 .04352. .301 .0152 .0015 .00112.5 .422 .1 .00391 .01,163.0 .462 .01s0 .01"3 .401

0.4, 31.50 :.346 186 4 .Zif .00194 .0026 .00710.5 211 .014 .010 .01110

1.0 .105 .18 .04's .0101.5 .344 .0112 .11041 .04112.0 .38, .0141 * 0415 e4012.5 .022 .0165 .0491 0143.0 .442 .0153 .00 01114

5.1 2.51.50 160 .2*4 .0094 .0026 "To19.4 .265 .0205 .0086 .1611

1 0 :05 011? .0640 .0$ft#44 .. 110 .06661 .0049

J, SI .4S2 .0163 .8091 .00123.0 32 .0145 .01 .0412

4.411 31.11 1.0 10' 0 .f4 .0 .0024 .04160.5 is$5 .0105 .0086 .10611

.5 .44 .01301 940"1 .04502.0 .3@8 .8145 "00S .012. 42 .0142 .01 .0012

30 .41.7 .0101 .0169 .00123. 3.1 14 0 .224 .006 .0024 380181.54 25.0 2.1 a 1 30 .011 .040"it4

1. 344 .0180 .11111 .0%482 .143 11M9 .00s

2 __.____ 0 .4:2 .020:1 .0109 .012

044W U"NSI IN5 TANK WATER 09PTH 6.0D rT. DEPTHIN 1.043

2.4 15.02 0.05 10 .234 .0101 .024 .441I i:, @01 03 6011

p2.0 .41 1,3, :004: .003

- I - _________05 .8205 .0111 .0002

3* ese to wompied offset

CONFIDENTIAL

7,7-,--~ -T--

K- r4.' P2l

R- -v - - ---v

Ico +. .4611 .61o 0

I 1 .663 .17W6e6.01I3.6 .4"8 .6106 .0111 .0410

4.64 U."4 1.66 I4P 4 .S34 .6166 .0096 Woe

1.5 .43 6 1 .669ws1.4 .6 .642 .6648.46

*. ."1 16 60 66

0.41 U..356 10J .0009 .034.0

1 .86 .01" .01 06,*1.0 1.0 631 I 6 .841 .6066 .606

Ile An7 .610 ."14

is.1 .0.66."O A. 36." .6106 op"9.06

1 .6 SAS1 fieL42.8 M44 .160 .1.0.6 A"1 :01700 61 .4"16

LOS ..8316 0$6 .64t .400 ow1?

0S .6:18 flit0 .46 .69

I I4..0 88.0 ..0 11 ."416 .061s

2.0 . .8130}18:1. .40? .11 60

_ _ _ a6." I.If so0$10

A.$ 4440 .017 sm.a4j

COPW I HT I AL

COFDE7A -3-

TABL 11(Cntd.

V/SION ARC LAW R-3 /V d4t8

04060D LENGTH 0 a In. TANK WTtCWPT014 0 6.00 9T. OEPIN 1.8 CNOR(J

0.5 .30s .01"21 60331 .0,1.6 .363 .6115 .040 .01s8I .6 .4#6 .6161 ."Is5 .0016

1. .44?9 .6 Z .03 6

6 .46 SO."0 1.668 lo 16 .350 .!19s .0026 .4615

1.6 .343 .0131 .064t .04

'.2K .0 .405 .0111 .6S1 .0016,\S .6 ."4? .69 .0091 .6616

.0 6 .463 9 .016 i0 .606

0.5 .51 .060 .05 .0016

3. .01 .1151 .0015 .0016

3. .41 .0160 .0001 .40183.0 .46 .0166 ft"6 .*0No

*Ce16dtee pesad off***

cow I DOET IN.

"AtON. CO1.ET P.RA

TABL IN

TA L j I I*+ I02

. .020 1 .111 01L11

1.09 lsono f.og Coffeas Are Co0.01 e for0 .014? rem Go-WMot

1.5" ol 3 .4 TAN a0 -001 .633 r.1100211.2 68# 2.6 0 0.* -. 160 .011

-4.0 -. 048 .0114+4.000 .14 .02

44.010 *.11 .0113.0' .131 .0166

0 66.04 .01214.10 4.149 -out4.01* .113 .012

0.05 .116 .01140.8 "81 .0154

+.60 60 .24f. .08 .511 .0104

6 -. 805 .0214

44.00' 4.144 .0111.04 .414 .927$

5.4 1.16 6.4. as£' -m.01 .. 6 ow 0106 .000160140 -.064 .0111 .14"0 .2

- 4.016 +164 .0114 4440 .00066.94 .418 .01144 .0460 .iI11

14.146 15.60 4.11 g 10 V 40 .0s" wdd0 .04.0

-. 1 - 16 .f" .6".11

aSm 55001 a 6 on. T~ms eam1 a 5.3 PT. mmW 0.73 00

'.1is alla Zu

6.6 .14 .05

:.6 4106 .0420=04 is' .is0 ap

0w" loeot poved 611..s

COWFIOENTIANLTh.

.....................................

TABLEt II (C, 'd

030W LENOm *sm. TAMS oDET *5.3 DeT"

a'.-.91 ..011

W' *.;At .0149 9'

6.33' .oo11 .at .019' 032

2.2 oo 1.010 .01 .0131 .4416 .03

#a $"2 46 S0"5 TP

4.04 w3l.00 .0144 .013"a. .994 .44H3 .044 .31

.44 .09 .01 .30 .013?0.3' os' .01, .4141 oslt

4.06' .444 .4444 .40 03

6.6 -. 64 .Uf:.03 04*8.U .0 4 m$ 4 .4.033

* oaft~ .01:30.01 JOS .0333 .4411 .4061

4.4 ."1 .0160 .01m3 .0330

4404 . :3? .544 sm3 .slit

0.04 no6 .01I$ a"!9 .001.10 od0 .64'.o 60 .3W1

4 .018 .4416 .004I

8440 -1 .0341 46 Its0O .00 2 .058 .039

.030 04 .053

040 on' .4o0 .42000 10,11'U.s, .84 .443 ."Is4 .811404" :4 * 06 .030Z =4 .03140.041 111.91 .0224 103 e.s0ese e se

11.6 -48 00-."-' -. 00 .01 .10114.1

44,4.4.6. : S" fist ."I. 41210

44

CC:IOEETA* lgeL

-6 044%'- 134 4

R"4

__ __ _ __ __ _ __ _ __ __ _ __ __ _ __ _ _""A_ __ _ jk"' rip ,

CONFIVENTIAL

TASLK III (Conmod.)

Re.S'- ft.ous Cs/ c COf.e~ 4F

-F - [ _-- - - .on.03 -.844 .0)1 .0414 0.0049

5.14 10.1 4.4 P 69 .801 .E.3 010

StI"* - -

t .946 .0101 .104 Sam00 .34 .0338 .i ~ .9

,14.03 .0338 .0)04

.0109 .0)144.4 8.0 411 SS0 1:~ 30' Uzi 94 .0)) 181

.8.0' -.146tsto 4o a .0134 .01 .093..00 . A.,44 .004) .0M0

Os -08, .040 o0w0 .0ON .34 01 .146

* .040 .041 0.044.01 . :00 :60 044

8.01' . ~ .0000 ."1# .044. vs .0 1 .0162 .00 .406

06 s1Auoi IN g. Tam paiw 0 3.75 FT. DEPTH * 0. CH40§

8.59 0.00 4.149 got 0 I-.095 .0AM.0* ~ o " .0 .09 0 0* ~ W I .0 .04M 09

on .0 .0000 0 .0"Isso Ii 004 0 08.4 111 .084 .011 .014.00 ~~.010 008 .014.61 .014 .00 .0110

*o -. On 933 0 08* .00 .0111 09,11

8.00 *.ii .088 .033 .01:64 ft 84 .041 .004f I0

4.0411 061 .804.01:4091 .81 04 1 .041.080.35 .419 -Not .1 .0140

4.0 .898 .0153.1 4 .9114 0 84 .141 . 0" Is .111

4.0 ."44 .041 .109.180.0 .866 .38 .0)0 .18

Nf~ .40 .IS, .0100 .0

4.0? .809 .0111 .6018 .01181

U.0 .10is .814 .0110 08 .411*106)' .100 .018? .0381 .14 . 8 4V9 . 4 1. 1 8 .0 1 1 5

'~ hZ94 * TAN. WIN oC3 S. 13S V. o(1 3.50 CH~O$

#.40 O 0 046e~ .1119j4.00 am14 .34..ilI__ :: on, Is;4

* ~ ~ ~ ~ ~ a a~~P~f 1.Sm .18 to.

CONV iDfTIAL

R-438 -4-4CONFDENTAL -9- -9

TABLE III (Cont'd.

A-$. Y. Re. , ~ c c co

icdhoL~w~h 5 N. *1 DEPTH a 5.33 FT. DEPTH 1-5.0 CHORDS

6.4 21 -.012 .62 0 ss

6.1 *.63.2 Is :1 .0142 S.03.454.61 .272 .019S 64 01

3.63 63 4.07166 1.6? 59 646s ~5 s

6 .1 .0131 1 024.63 .32 .016) .05 .01246.66 .566 .466 .6262 .0150

5.39 12.0 4.16 IS I6 A"66 .6219 6.61111

4.04 .29 .612 .6I3NS126.0 .421 .410 .0176 .6131

6.66 .623 .6411 .62?? .61345.4 151 .45 III6 .664 .6110 0 .0116

.309 .031 .606S .613394.24 16.66 $.1I 1.6t.24 61

6 .92 .6136 0 .61264.64 +.343 .6319 .60463 .066

040, LV'YN In. TANK DEPTH S.U3PF. OEMT 1.90 CN06

.6 .10 .12,1 .64 I613. 46

4.09 4216 .04Ms 96*S .64.64 .439 .4.06 .0196 .3c

1.00 $. 4 .?S e l 1060 -6 9 .0136 1 .612 ,6.6 41 13 .041 .0$1# .03

6 -. 04 .646 0 .6120:0-.60 .6286 It .0219..1 424 .016 .0019 .015?

4.0 .29 .190 .0! .6134.06 .359 .014 .64?8 .0120.64 .432 .ss6 .6119 .021

4.6 .414 ."257 .466 .611121.64 1.1 86 .40 .20 .26 62

6.40 4.566 vu45 .6:Is .03513.3 646 3.1 1 80 -60 .622 :01, .012?

0~ 0 .62,: .624.06 .256 .0214 1",3 .611S.60 .553 .0no.#5 .0144

6.01 1 .51 .01, .19 .6116

0 .64 .113 0 .011

4.06 .14 .619 65. .61136.6 .1st .0591 .6346 .0161

*Cwems4d I"ee ea efI...

C~OIENTIAL

ot,.

J OA k A, kw.

TABLEM MI (CoeC'4)

006B ENSW 5m.T"U wpTW 3. 83 FT. cam * .50 CHO"$*1

120 13.02 S .~ .61~ 1 011

3.1 10 .4131 .0414 .611?303, .36 .01?0 .06? .6119

6.02 .4*1 .01 else1 .0146

.2 .64 ."It0 .02

77.~ 7K MT-7R M77M 7,7"77

CON FIDENTIAL

t. -- ... .. f

rT Tt T7*I 7

AA

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rr ''r

7I- + . r Tfi7*r7T

7{t fT.."7!

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- CONFIDENTIAL

- -. -, - - - - - - --

R-436

CONFIDENTIAL

I. I' I1 -~4...4

1 1. 7 -

/_-.. . .... i

171-.* I .I I.

71........... .... ... .

.. .. | "*,

06 .. .. ............ .... . . . ... .... ... . ..

I JOI N I E I°

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CONFIDENTIAL

I-Z

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R-438CONFIDENTIAL

4-4.I

mt-t..±4+5 4 74 F %413...4~__ rJ2i1

'a~~~~ FdF~ - '~ ~ ~FV.,

A, .-. -t ..... i ~ .

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R 4 84 3 8

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I 4*~. GON~tDENTIALI a* 1

*:J4~.P~j.IOU M~ WV AIG R~:I~~-~~*1*i 2 *-*7.2--4-1-T

2IN.'MEDIU~i,,,.~ DSTI7

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CONF£?~IC E Cj J7..

R-438

CONFIDENTIAL

i ~ i~~...1............

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CONFIDENTIAL

R-438

CONFIDENTIAL

7 CO0,PUTEDINb V40; -M b 4 '"""~-LA

- ~ ' WAE 1 XPLRIPALIL0A D -

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R-438

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R-438

CONFIDENTIAL

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7.. -14 AT, w~l6 II'1'rA8 R

--4d

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.04 Di

U -. 71 Ut7. ~

1: -__

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CONFIDENTIAL

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CONFI DENTIAL

-4 X

T-H.

V- 1-, opt&

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4./

CONFIDENTIALMIN IV

1 U

IR-436R

CONFIDENTIAL

V.. ... .... .-

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CONFIDENTIAL

I~a I tio "-MA•S " "

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(CONFiMENTIAL

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CONFIDENTIAL

~13t, ~ ~ ' ' V

s' 'k *

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CONFI DENTIAL x-3 H__1

-T 1It .

1±

iIV~u5 -~ A~ -:1, ~L!J~ 01: L 7

it±! iRL*~;:~ ,

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v 5. F.7sa IFist*

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CONFI DENTIAL

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CONFIDENTIAL

*~ ~ td L rF'T tuc1~ I f Nf~4 Iuq o1 y4,wGI

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