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REPORT 1355

A THEORETICAL AND EXPERIMENTAL STUDY OF

PLANING SURFACES INCLUDING EFFECTS OF

CROSS SECTION AND PLAN FORM

By CHARLES L. SHUFORD. Jr.

Langley Aeronautical Laboratory Langley Field, Va.

gt&g

.. W'çe.ncjen

HO L L A N D. TECHNISCHE !JN!VERS!TEIT

La5cator;um voor

Scheepshyc!romechanjca Arch ief Meke!weg 2, 2523CD Deift Tel.: 015.786873 - Fax 015-781830 I

(2)

By CHARLES LSHUFORD,Jr.

NATIQNAL ADVISORY COMMITTEE

FOR AERONAUTICS

CROSS.SECTION AND PLAN FORM

For 1e by the Superintendent o! Documents, U. S. Government Printing Office, Washington 25. D. C. eariy 8ubscription. 10; foreign, $13;

aingle copy price vanee according to nize Price 45 cento

'N 6?

51355

(3)

National Advisory Committee for Aeronautics

Head quarters, 151S H Street NW., Wa8hington 5, D. C.

Created by Act of Congress approved March 3, 1915, for the supervision and direction of the scientific study of the problems of flight (U. S. Code, title 50, sec. 151). Its membership was increased from 12 to 15 by act approved March 2, 1929, and to 17 by act approved May 25, 1948. The members are appointed by the President and serve as such without compensation.

JAMES H. DoOL.n-rx..E, Sc. D., Vice President, Shell OilCompany, Chairman LEONARD CARMICHAEL, Ph. D., Secretary, SmithsonianInetitution, Vice Chairman

ALLEN V. ASrIN, Ph. D., Director, National Bureau of Standards. PRESTON R. BASSETT, D. Sc.

DETLEY W. BEoNI, Ph. D., President, Rockefeller Institute for Medical Research.

FREDERICE C. CRAWFORD, Sc. D., Chairman of the Board,

Thompson Products, Inc.

WILLIAM V. DAVIS, JR., Vice Admiral, United States Navy,

Deputy Chief of Naval Operations (Air).

PAtTL D. Foovs, Ph. D., Assistant Secretary of Defense,

Re-search and Engineering.

WELLINGTON T. HINES, Rear Admiral, United States Navy, Assistant Chief for Procurement, Bureau of Aeronautics. JEROME C. HtNSAEER, Sc. D., Massachusetts Institute of

Technology.

CHARLES J. MCCARTHY, S. B., Chairman of the Board, Chance Vought Aircraft, Inc.

DONALD L. Ptrr, Lieutenant General, United States Air Force, Deputy Chief of Staff, Development.

JAMES T. PYLE, A. B., Administrator of Civil Aeronautics. Fs&NcIs W. REICHELDERFER, Sc. D., Chief, United States

Weather Bureau.

EDWARD V. RICEENBACEER, Sc. D., Chathnan ofthe Board,

Eastern Air Lines, Inc.

Lotis S. ROTHSCHILD, Ph. B., Under Secretary of Commerce for Transportation.

THOMAS D. WHITE, General, United States Air Force, Chief of Staff.

HENRY J. E. REID, D. Eng., Director, Langley AeronauticalLaboratory, Langley Field, Va. SMITH J. DEFB.ANCE, D. Eng., Director, Ames Aeronautical Laboratory, Moffett Field, Calif. EDWARD R. SH.&RP. Sc. D., Director, Lewis Flight Propulsion Laboratory, Cleveland. Ohio

WALTER C. WILLIAMS, B. S., Chief, High-Speed Flight Station, Edwards, Calif.

Htoa L. DRTDEN, PR. D., Director JOHN F. VICTORY, LL. D., &ecutive Secretary

JOHN W. CROWLEY, JR., B. S., Associate Director for Research

(4)

REPORT 1355

A THEORETICAL AND EXPERIMENTAL STUDY OF

PLANING SURFACES INCLUDING EFFECTS

OF CROSS SECTION AND PLAN FORM'

By CHARLES L Snueoao, Jr.

SUMMARY

A summary is given of the background and present status of the pure-planing theory for rectangular flat plates and V-bottom

surfaces. The equations reviewed are compared with

experi-ment. In order to extend the range of available planing data, the principal planing characteristics for models having sharp chines hace been obtained for a rectangular fiat and two V-bottom surfaces having constant angles of dead rise of 200 and

400. Planing data were also obtained for fiat-plate surfaces

with very slightly rounded chines for which decreased lift and

drag coefficients are obtained.

A revision of the theory presented in NACA Technical Note 3233 is presented for the rectangular fiat plate. The revised

theory bases the aerodynamic suction effects on the total lift rather than solely on the linear component. Also a crosxfiow drag coefficient which is dependent on the shape of the chines was found from experiment to be constant for a giren immersed cross section; however, for surfaces, such as those having horizontal chine fiare or vertical chine strips, the crossftow drag coecient is constant only for the chine-immersed condi-tion. The theory is extended to include triangular fiat plates planing with base forward and V-shaped prismatic surfaces

having a constant angle of dead rise, horizontal chins fiare, or vertical chine strips. A method is also presented for estimating

the center of pressure for surfaces having either rectangularor

triangular plan form. The results calculated by the proposed theory have been correlated only with the data consideredto be

pure planing; however, for conditions not considered pure

plan-ing, a method is given for estimating the effects of buoyancy.

The agreement between the results calculated by the proposed theory and the experimental data is, in general, good for

calculation,.s of pure-planing lift and center-of-pressure location

for fiat plate, V-bottom, and related planing surfaces.

LN TRODUCTION

Recent developments in water-based aircraft have resulted in configurations utilizing planing surfaces operating at angles of trim, length-beam ratio, and Froude number beyond those for which most of the available planing theories

were correlated with experimental data. In reference i a preliminary review of these theories for a pure-planing rectangular fiat plate was made to determine whether available planing theories were adequate in estimating the

Supenetei SAcA Tchnii oe i535 by CharIe L. Sbuíord. Jr.. 1957. 459901-58

planing lift in these extended ranges. In addition to this review, a modification and addition to existing theory which

is useful in predicting the lift and center of pressure for pure-planing rectangular fiat plates was presented.

The review in reference i indicated there were no data. available in the extended ranges of combined high trim and high length-beam ratios; consequently, the principal planing characteristics for models having sharp chines have been obtained in these extended ranges for a rectangular flat and two V-bottom surfaces. It was also noted in reference i that there was a difference in the lift coefficients obtained from various experimental investigations; therefore, data have been obtained for rectangular flat-plate surfaces having very slightly rounded chines to determine the influence of slight differences in construction at the point of flow separa-tion on the lift coefficient.

The review of existing theories and data has been extended

to include those applicable to V-bottom surfaces. The

theory presented in reference i for estimating the lift and center-of-pressure location of a pure-planing rectangular fiat plate has been revised and extended to include triangular flat plates planing with base forward and V-shaped prismatic

surfaces having a constant angle of dead rise, horizontal chine fiare, or vertical chine strips. Since water-based aircraft operate at low Froude numbers as well as high Froude numbers, an approximate method hasalso been pre.-sented for estimating the effect of buoyancyon lift coefficient.

SYMBOLS

aspect ratio,

r

ratio of maximum beam to overall length (see

fig. 40)

beam of planing surface, ft

D drag coefficient based on square of beam, crossfiow drag coefficient

crossflow drag coefficient for a cross section having an effective angle of dead rise of 00 drag coefficient based on principal wetted area,

D qS

induced drag coefficient, CL,S tanr

skin-friction coefficient, CD.3CL

tan r

I A A, b CD,0 CD3 (CD.,)ß.-o .3 CDI Cf

(5)

lift coefficient

lift coefficient due to buoyancv,- (see eq. (31))

L lift coefficient based on square of beam,

-lift coefficient based on principal wetted area,

L

lift coefficient due to buoyancy based on total wedge-shaped volumetric displacement of the planing surface, qS---' (see eqs. (28) to (30)) speed coefficient or Froude number,

.gb

drag of planing surface, lb

dead-rise function (applied only to crossflow term, see fig. 2)

acceleration due to gravity, 32.2 ft/sec2 dead-rise function (applied only to linear term.

see fig. 1)

lift of planing surface, lb lift due to buoyancy, lb

lift due to buoyancy based on total wedge-shaped volumetric displacement of the plan-ing surface, lb

length of planing surface, ft chine wetted length, ft keel wetted length, ft

mean wetted length (distance from aft end of planing surface to tite mean of the heavy spray line). ft

center-of-pressure location (measured forward of trailing edge), ft

nondiirensional cent er-of-pressure location normal force, lb

free-stream dvtiamic pressure, pF2, lbi'sq ft Reynolds number,

principal wetted area (bounded by trailing edge. chiites, and heavy spray line), sq ft horizontal velocity, ft/sec

angle of dead rise, radians unless otherwise stated

effective angle of dead rise (angle between a

straight line drawn from keel to the chines and the horizontal), radians unless otherwise stated

basic angle of dead rise (angle between V-shaped portion of model and a horizontal line perpendicular to keel), radians unless other-wise stated

mass density of water, slugs/eu ft

trim (angle between planing bottom and hoii-zontal), radians unless otherwise stated kinematic viscosity, sq ft/sec

script:

. 9 used to indicate various terms in equations for

lift coefficient

R EPORT 1355NATIONAL ADVISOR Y COM_MITTEE FOR .&EItONALT1CS

REVIEW OF EXISTING PLANING-LIFT THEORY In reference 1 the pure-planing lift equations for rectangu-lar flat plates presented in references 2 to il were reviewed

and compared with experiment. In addition to lift theories

for rectangular flat plates, the present review considers V-shaped surfaces havinga constant angle of dead rise and

V-shaped surfaces having horizontal chine flare.

Since publication of reference 1, Farshing (ref. 12) pre-sented a cubic equation for the lifton rectangular flat plates derived from a consideration of deflected mass and based on an effective angle of attack. The equation hasthe form C'L3+ [(2.292 1 .57lA)-2.379A]c2+

[2A+4+ (6.283A 4 .584)71 ÜL-6.283A7=0 (1)

However, the lift coefficient obtainedfrom equation (I) was multiplied by an empirical factor to get better agreement with experimental data; thus,

CL.S C1.E (2) where + A \

=l.359tanh (---l--)

(2° r 18°)

(3)

( 1 +A)+(r l80

) tanh

(lS0r300)

E=l.359arilì

84 90.53 , (4) and r is measured in degrees.

P. R. Crewe of Sanders-Roe Ltd. (British) in

correspond-ence with the Langley Laboratory proposed an equation for

rectangular flat plates and a V-shaped surface having a basic angle of dead rise of 20° and horizontal chine flare that had a linear term with a forni analogous to airfoillifting-surface

theory. This equation, based on the data of Kaprvan arti

Weinstein (ref. 13), is ('L.s sin r (OS 7' E

s1

(l)+

l+/1+()

T 2 sui r-1' siI1 r] (5) '. here B==2.67 (A <2.0) 13=3.0 (A >2.0)

arid ß, is the basic angle of dead rise in radians for a mode! having horizontal chine flare.

In reference 14, Korvin-Kroukovsky, Savitskv, and

Lehman proposed an equation forrectangular flat plates and

V-shaped surfaces having a constant angle of dead rise that

was derived primarily on the basis of the data of Sottorf (ref. 15) and Sambraus (ref. 16). This formula can be

vritten as ('L,s=0.012(57.3r)4°30.006o(5e.35)A [0.012(57.37)7 I]O.8 (6)

vi

ß,.) 'e

(6)

Locke (ref. 17) proposed that the lift characteristics of

rectangular flat plates and V-shaped surfaces having a constant angle of dead rise can be presented by a power

function of the form

CL.s=0.5 (i_y) Kr'

(7)

where K and depend only on aspect ratio and are obtained

from curves given in reference 17.

Schnitzer (ref. 18) presented an equation forrectangular

flat plates and V-shaped surfaces which was derived from a

consideration of two-dimensional deflected mass and was modified for three-dimensional flow by Pabst's empirical aspect-ratio correction factor (ref. 19) and Bobyleff's flow

coefficients presented in reference 20. The equation can be

written in the form:

(r/

\

CL.S=Acos r tan

B(!tanr

(8)

The term , which is dependent on aspect ratio, and the term B. which is dependent on angle of dead rise, are given in references 19 and 20, respectively. For the case of a flat-plate planing surface, equation (8) reduces to

In reference 21, Brown presented empirical equations based on deflected-mass considerations for rectangular flat lates arid V-shaped surfaces having a constant angle of Jead rise. The equations for a flat plate can be written in :he form:

2w

-

(I,.,b)

(10)

cot +w+(2 cot

md

.s=(1.67 sin r+0.09) sin r COS

2w b i,, 3cot

-or a surface having a constant angle of dead rise,

(I,b)

A THEORETICAL AND EXPERIMtNTAL STtDY

where

'k. cot r tan $ (14)

which is defined as the critical keel wetted length. For surfaces having a constant angle of dead rise and a transverse step, the critical keel wetted length is defined as the keel wetted length at which the stiliwater line passes through

the rearmost point of the chine. For the flat plate the value of the critical keel wetted length was assumed, after analysis of experimental data, to be equal to the beam.

PROPOSED THEORY

LIFT

In reference i an equation for the lift on a rectangular flat plate was developed from a consideration of linear and nonlinear components of lift (an approach generally used in low-aspect-ratio and slender-body airfoil theory).

In the

present report this equation is revised and extended to include V-bottom surfaces. The equation is divided into three parts: (1) a reasonably accurate approximation to the linear components of lift is made; (2) a method for calculating the crossflow effects is presented: and (3) an estimation of the aerodynamic leading-edge suction is made.

Linear term.The linear term is determined in

refer-ence i from a consideration of the lifting-line theory and is given by

OF PLANING StRFACES 3

0.5wAr

li- A

This relation gives the linear component of lift on a

pure-planing flat plate.

In references 3 and 18, a dead-rise function was determined

from a consideration of an iterative solution made by Wagner (ref. 2) for the impact force on a V-bottom surface

immersing with a constant vertical velocity. The dead-rise

function can be written

K($)=(I_$)2(ta; $)2

This dead-rise function (developed for application to equa-tions derived from virtual mass concepts) does not correlate

well with experiment when applied to equation (15) for angles of dead rise above approximately 250. Therefore, another dead-rise function 1sin $ which correlates well with experiment up to angles of dead rise of 50° is used; thus,

0.5wAr

1-f-A (1 sinß,)

This expression is for the linear component of lift on

rec-tangular flat and V-bottom planing surfaces. A comparison

of the (lead-rise function 1sin$ with the (lead-rise function based on Wagner's solution is given in figure 1.

Crossflow effects.For a simple theoretical

considera-tion of the crossflow effects, the velocity component perpen-dicular to the surface of a flat plate is assumed to be of the magnitude V sin r. The flow is projected into components (16) cottß sin3r(1 sin r) cos r

(1 lk,)

(12)

nd

sin r+0.09] sin r cos

0.9 sin r(1 sin r) cos3r (l ike,) (13)

7w3A

(7)

pendicular to and parallel to the planing surface, and the g force associated with the flow perpendicular to the

Ining surface is calculated. The normal force on a flat

ite, therefore, is

N=CD., S(Vsin )2

en

CL 3 CDC sin cos r (17)

lift coefficient due to crossuioweffects, and is proportional

sln2r. This relation is the concept presented for airfoils by tz in reference 22. The crossflow drag coefficient

CO3, d in this elementary derivation of the crossflow term was umed in reference i to be one-halfthe value GD,C=2 gen-Jiv used for aerodynamic surfaces. The value of CDC çnown to vary with the shape of the cross section and to be isitive to local shape at the edges. Since the theoretical ermination of these effects is very difficult and the simple ;es which have been solved have not correlated with ex-riment, the analysis of suitableexperiments will generally vide the easiest and most accurate method of determining For the case of the V-bottom the theoretical effect of dead e is given by Bobyleff in reference 20 fora bent lamina, the ,tion of which consists of two equal straight lines forming

angle. Bobvleff's flow coefficient, which

can be approxi-ted by cos ß. (see fig. 2), represents the ratio of the

result-t pressure on a V-boresult-tresult-tom result-to result-tharesult-t experienced by a flat

te of the same beam in normal flow; thus,

CL.4=(CD,C)$,..O Sjrì2r cos r cos ß (18)

ich is the crossflow component of lift.

uction component of lift.An airfoil has a suction corn-ient of lift due to the large negativepressures produced

the flow around the leading edgeof the airfoil; however, a planing surface where there is no flow around the lead-edge, this suction does not appear. In the strictest sense suction component of lift should be based only on the ar term (see ref. 1); however, comparison of experiment h theory indicates that betteragreement is obtained if the

tion component of lift is based on CL,2+CL4. Therefore,

lift is less than that predictedby equations (16) and (18) an amount

CLS== (CL.i+ CL,4) sin2r (19)

otal lift.The total lift

on pure-planing surfaces

can be ined from the sum of the linear component of lift (eq. ) and the crossflow effects (eq. (18)) minus the suction ponant of lift (eq. (19)); thus, bycombining terms

REPORT 135 5NATJOAL ADVISORYCOMMITTEE FOR AERONAVTICS

L 1+A

cos'r(l_sinß)]+

[(CD.Ç)OC_o Cos3r sin2r cosß}

= C+ (,

re

0.5TA,'

1+A cos2r (1sin ß)

} (20)

(21)

and

CL .? (ÜD.C)51_O CoS'r sinrr cos (22) For equation (20) to predict adequately the lift on triangu-lar surfaces planing with base forward, it hasbeen necessary to define the aspect ratio as the ratio ofmaximum beam to overall length; that is, A,=b/1.

APPLICATION 07 tiri ThEORY

In order to use equation (20) to predict the lift of planing surfaces, only the determination of the proper value of CD. is required. Values of CD,C for various chine configurations for which experimental data are available are presented in

figure 3. For a given model CD.0 did not vary with trim or

length-beam ratio. Also it can be seen that, as long as the angle of dead rise was constant for

the entire beam, (,.

did not vary with the angle of deadrise.

Rectangular fiat and V-bottom surfaces having a constant

angle of dead rise.The crossflow drag coefficient for the sharp-chine models was determined from tests (from ref. 23 and data presented in thepresent report) to be 4/3. This value is two-thirds the value given for a two-dimensional flat-plate airfoil; thus, from equation (20)

0.5vAr

l±A

costr(1sin ß)+ sin2r cos3rcos ß

The relative magnitudes of the total lift (eq. (23)), the total lift before removal of lift due to leading-edge suction (eq. (16) plus eq. (18) with C».=) and the crossflowterm (eq. (22) with cD.C=)1sshown in figure 4 for surfaces hav-ing angles of dead rise of 0°, 20°, and 40°.

Horizontal chine flareThe total lift on a pure-planing V-shaped prismatic surface with horizontal chine flare similar to the models shown in figure 5 can be determined from equa-tion (20). The crossflow drag coefficients CD.0 determined from data presented in references 13, 24, and 25 are given in figure 3.

Vertical chine strips.The total lift on a pure-planing V-shaped prismatic surface with vertical chine strips similar to the models shown in figure 6 can be determined from equation (20). The crossflow drag coefficients determined from the data presented in references 25 and 26 are given in figure 3.

Triangular fiat plate.The total lift on a pure-planing triangular flat plate planing with base forward can be est i-mated from equation (23) if the aspect ratio is defined as the ratio of the maximum beam to the overall length or

thus,

0.5TA,r

C3

1+A, COS2r+COS'r sin2r CE?CTER OF PRESSURE

The center of pressure on a planing surface may be deter-mined from the lift coefficientsgiven by equations (21) and

(22) and by estimating the location of the ('enter of pressure of these two components of the total lift coefficient for a

given planing-surface plan form.

(8)

A THEORETICAL AND EXPERIMENTAL STUDY OF PLANING SuRFACES 5

Rectangular plan form.The center of pressure of the component of lift given by equation (21) is assumed to be located at seven-eighths of the mean wetted length from the trailing edge of the planing surface. This location is between the three-quarter-chord position generally assumed in lifting-line theory and the position obtained from the prediction of no lift behind the section maximum width for low-aspect-ratio airfoils (ref. 27).

The center of pressure for the lift due to crossflow effects is generally assumed to be located at the center of the area in airfoil theory. Therefore, the center of pressure for the component of lift given by equation (22) is assumed to be located at the center of the mean wetted length; thus,

7 1

(1\

L. 5+('L.?

'\l ),r.Ae CLS

where CL.ß is given by equation (21), C'L.7 is given by equa-tion (22), and CL.S is given by equaequa-tion (20).

Triangular plan form.The center of pressure of the

component of lift given by the first term on the right-hand side of equation (24) is assumed to be located at the mean of the heav spray line which is approximately the section of maximum wetted width.

The center of pressure for the component of lift given by the second term on the right-hand side of equation (24) (that is. the crossflow term) is assumed to be at approxi-mately the center of the wetted area; thus,

(

CL.S+ CL.9

\1,,. )e.io CLS

and is the center-of-pressure location for triangular flat plates planing with base forward. The value of CL.S is

determined from equation (24) where CL,$ and are given

by the first and second terms on the right-hand side of equa-tion (24), respectively.

COMPARISON or PROPOSED AND PREVIO(IS PLANING FORMULAS A comparison of the values of lift coefficient (plotted against trim for constant length-beam ratio) calculated from the proposed theor (eq. (20)) and from previous summarized planing formulas is given in figures 7 to 10 and an index to the comparison is given in the following table:

Value of C0.. of 43- (9ee eq. )) uaed uniesa othwtae noted.

Lift coeicient were not p otted since the resulte depended on the airfoil data oued. "VaJue of Co.. of 1.59 uued In equation J)).

In figure 11 the values of lift coefficient (plotted against mean-wetted-lengthbeam ratio for constant trim) cal-culated from the proposed theory (eq. (23)) and planing formulas as presented in references 14, 17, 18, and 21 are

compared with the data of the present report (see tables i(a), II, and III) and references 23 and 28 for models having angles of dead rise of 0° (fig. Il(a)), 20° (fig. 11(b)), and 40° (fig. 11(c)). Only the theories that apply to both flat-plate and V-shaped surfaces have been compared in figure11. It can be seen from figures il(a) to 11(c) thatnone of the planing formulas presented in references 14, 17, 18, and 21 are adequate for estimating the lift coefficients for either flat-plate or V-bottom planing surfaces, whereas the lift co-efficients calculated from the equation proposed in the present

report (eq. (23)) agree very well with experiment. The equation presented in reference 12 (eq. (2)), however, gives a good approximation of the lift coefficient for a flat plate. (See fig. 7(d).)

EXPERIMENTAL INVESTIGATION DESCRIPTION OF MODELS

The models used for this investigation had a beam of 4 inches and a length of 36 inches. The models shown in figure 12 for a flat plate and surfaces having angles of dead rise of 20° and 40° were constructed of brass and are the same models investigated in references 23 and 28.

Addi-tional flat-plate models that had sharp chines, 4-inch-radius

chines, and 16-inch-radius chines were constructed of plastic.

(See fig. 13.) The model with the 34-inch-radius chines was made by rounding the chines on the sharp-chine model after the tests with the sharp-chine model had been completed. The plastic models were backed with a %-inch reinforcing steel plate.

APPARATUS AND PROCEDURF

The experimental investigation was made with the main towing carriage in Langley tank no. 2 and existing strain-gage balances which independently measured the lift, drag, and moment. The lift and drag were measured with the balances capable of measuring: (1) 600 pounds of lift and 250 pounds of drag, and (2) 1.000 pounds of lift and 600 pounds of drag. The moment was measured about an arbitrary point above the model. The tests were made with the wind and spray shield installed, as shown in figure 14, unless otherwise indicated.

The wetted areas were determined from underwater photo-.

graphs made with a 70-millimeter camera mounted in a waterproof box located at the bottom of the tank. The camera and high-speed flash lamps were set off by the action

of the carriage interrupting a photoelectric beam. The wetted length was obtained from markings on the bottom of

the models. In order to assure a very smooth bottom, the

markings on the brass models were erased except in the region of the heavy spray line. (See fig. 15.) The plastic models had markings each inch for the full length of the models.

The force measurements were made at constant speeds for

fixed angles of trim. The change in trim due to structural

deflection caused by the lift and drag forces on the model was obtained during the calibration of the balances and the trim of the model was adjusted accordingly before each run. Slight adjustments to lift and resistance to correct the data

Con6guratIc

Equation IJ)) compared with planing foe.

mulas presented In rient vaIueLift coelfi-presented In

figure-Referen Equation

Rectangular flat plate

4.5.6,andl 8, 9, and 10 11 (7). (9). (10). and (11) (2). (5). and (6) 7(a) 7(b) 7(c) 7(d) 7(e)

V-shaped surface having

a conotant angle of dead rise of se

(6) and (7)

(8). (12), and(13) 8(a)8th)

V-shaped surface having ac000tant angle of dead

clue of 40

(6) and (7)

(8), (12), and (13) 9(a)9(b)

V-shaped surface having an angle of dead rise of J)and horizontal chine flare .=l6°)

(9)

REPORT 135 5NATIONAL ADVISORYCOaLMXTTEE FOR AERONAUT!CS ie desired trim were made after completion of tests for

tases where the forces or center-of-pressure locationwere rent from the values used to estimate the trim due to :tural deflection. The change in trim due to structurai ction did not exceed 0.2° for most conditions although few cases changes up to 0.6° occurred.

ie aerodynamic forces on the model and towing gear found to be negligible when the windscreen was used. aerodvnamc tares were subtracted from the data when

-ind screen was not used.

ie accuracy of the quantities measured are believed to ithin the following limits:

l,ft/sev

±0.20

e forces were converted to coefficient form by using a tured value of density of 1.942 slugs/cu ft. The kine-c viskine-cosity measured during the tests varied from xiO-° sq ft/sec to 1.80X10 sq ft/sec.

RESULTS AND DISCUSSION

GENERAL

ìe lift coefficient, resistance coefficient, ratio of wetted .h to beam, ratio of center-of-pressure location to mean d length, speed coefficient, and kinematic viscosity are

'nted at given trims in tables I to III for all models.

lift and drag coefficients are ex-pressed both in terms of quare of the beam and in terms of principal wetted area. arp ehiiies.The lift coefficients and center-of-pressure ion for the sharp-chine models are considered in the )fl "Comparison of Theory and Experiment for Lift."

e resistance data for the sharp-chine brass models having .ant angles of dead rise of 0°, 20°, and 40° are presented

ure 16 as plots of the variation of drag coefficient and induced drag coefficient CD. (which is equal to ;an r) with mean-wetted-length--beam ratios for given The difference between the solid and dashed lines sents the friction drag. (Since the data were obtained )eeds above the critical speed of wave propagation for foot-deep tank, there is no wave drag due to transverse

included; however, there may be some drag due to or other causes included in this difference.) At high and low length-beam ratios the induced drag exceeds )tal drag and indicates an apparent negative friction (This result was previouslyreported in ref. 23.) The te of forward spray is large at high trims and appears to

ihigh forward velocity with respect to the model. The

'e velocity of tite model in the region of forward spray Dre is effectively reversed (see fig.

17) so that the

n drag due to this spray acts in a direction opposite t of the drag in the principal wetted area and thereby

s the total drag.

Therefore, at low length-beam

where the friction drag is small, this negative friction ue to forward spray may cause a negative friction force L trims.

variation o! with trim for the models

having

sharp chines and constant angles of dead rise of 0°, 20°, and 40° is given in figure 18. At a trim of 12°, the value of approximately constant for all length-beam ratios for the models having constant angles of dead rise of 0°, 20°, or

40°. At high trims, however, the values of for the

flat-plate model increase with increase inlength-beam ratio. are approximately constant for a given trim for a model hawing a constant angle of dead rise of 20°, and decrease with an in-crease in length-beam ratio for a model having a constant

angle of dead rise of 40°. The value of for the flat-plate

model decreases with increase in trim at low length-beam ratios and increases with increase in trim at high length-beam ratios; however, the value of decreases with in-crease in trim for all length-beam ratios forthe models hay-ing constant angles of dead rise of 20° and 40°.

Wind screen and spray shieldThe lift coefficient for the flat-plate model with wind screen and sprat shield

removed (aerodynamic taressubtracted) was approimatelv

the same as the lift coefficient obtained when the windscreen

and spray shield were used. (See fig. 19.) At a trim of 12°

the drag coefficient for the flat-plate model with the wind screen removed was approximately the same as the drag coefficient obtained with the wind screen installed (see fig. 20): however, for a trim of 18° the drag coefficient of the flat

plate with the wind screen removed was less than that obtained when the wind screen was used even before the aerodynamic tares were subtracted. The value of the dif-ference is in the wrong direction to be explained by the aero-dynamic tares. (The aerodynamic tares subtracted were less than the difference in fig. 20.) The variation of tue

center-of-pressure location with mean length-beam ratio on

the flat-plate model was approximately the same for data taken with and without the wind screen and spray shield

installed. (See fig. 21.)

SpeedThe effect of speed at high trims (24°) is show:i in figures 22 to 24. The variation of 1it coefficient, drag coefficient, and center-of-pressurelocation is a pproxima t ely the same for speeds of 30 and 60 feet per sec9nd for 4-iiL('h-beam prismatic models having constant angles of dead rise of 0°, 20°, and 40°; therefore, there was apparently no speed effect for this range of speeds.

Rounded chines.The effect of 34-inc1i-radius and .-inch-radius chines on the lift coefficient, drag coefficient,

center-of-pressure location, skin-frictioncoefficient, and 11f t-drag ratio of a 4-inch-beam rectangular flat plate is shown in

figures 25 to 29. Rounding the sharp chines of the fiat-plate model to radii of % inch and inch resulted in a de-crease in lift and drag coefficients; however, tile center-of-pressure location, skin-friction coefficients, and lift-drag ratios remained approximately the same. A decrease in lift of approxmatelv 5 and 9percent resulted from rounding the sharp chines to a radii of inch and 3 inch, respectively. (See fig. 25.) A decrease in lift for a small rounding of the chines was also observed by Perry (ref. 29).

The variation of skin-friction coefficient with Reynolds

Ib ±5.0 tance, Ib ±3.0 roing moment, ft-lb ±3. 0 ed length, ft ±0. 01 deg ±0. 15

(10)

A THEORETICAL AND EXPERIMENTAL STtTDY OF PLANING SURFACES 7 number for a trim of 8° is presented in figure 28 for a

flat-plate model having sharp chines and 3-inch-radius chines. The agreement between the data and the Schoenherr

turbu-lent-flow line indicates that, at low trims and high Reynolds numbers, the drag can be calculated with reasonable accu-racy from

CD.s=Cf+CL.s tan r (27)

where ( is determined from the Schoenherr turbulent flow

line. (See ref. 30.) The lift-drag ratios at high trims are influenced little by the chine condition; however, at low trims (8°) the lift-drag ratios for the sharp-chine modelsare slightly higher than those for models having rounded chines.

(See fig. 29.)

Pure planing.The experimental data were considered as pure planing if the lift coefficient due to buoyancy based on the total wedge-shaped volumetric displacement of the

planing surface CL. did not exceed a given value. The lift coefficient due to buoyancy was calculated from the wedge-shaped volumetric displacement of the planing surface below the level water surface given by

,_ i

= sin 2r

for rectangular flat plates and

cLVOI(,+,

2[!sin

2r+(21+1k) tan s] (29)

for rectangular surfaces having dead rise and

CL;.ol:=:s1n 2r

il

(30)

for triangular flat plates with straight leading edge and

pointed trailing edge.

The allowable lift coefficient due to buoyancy ('Lvj, as

determined from equations (28) to (30), was arbitrarily selected as 0.01 at a trim of 16°. The maximum allowable

lift coefficient due to buoyancy C'L,VO: for other trims was

determined by drawing a straight line from zero trim (and zero lift coefficient due to buoyancy ('L.yoi) through the value 0.01 at a trim of 16° on a curve of the variation of lift

coefficient with trim. For the flat-plate data the maximum

allowable lift coefficient due to buoyancy ('L.VL selected by this method at a trim of 2° varied from 16 percent of the predicted lift coefficient at a length_beam ratio of 8 to3.3

percent of predicted lift coefficient (eq. (23)) at a length-beam ratio of 3. These values decreased with increasing trim so that at 30° they would vary from 6.6 percent at a

ength-beam ratio of 8 to 3.0 percent at a length-beam -atio of . The permissible lift coefficient for surfaces having

lead rise is, in general, a slightly greater percentage of the

Dredicted lift coefficient than the values given for the -ectangular flat plate.

499O1 58.-2

(28)

Buoyancy.The experimental lift coefficients given in reference 31 less the lift coefficients calculated from equation (20) with CO3=i.15 plotted against the lift coefficient due

to buoyancy (Lvj calculated from equation (28)

are plotted in figure 30. Since equation (20) with C'.=i.l5 is approximately the pure-planing lift for the model inves-tigated in reference 31 (see fig. 32 (c)), the subtraction of this value from the experimental lift coefficients should indicate the amount of lift due to buoyancy present in the data. Only values of the difference between the experi-mental lift coefficient and the calculated lift coefficient

greater than 0.01 are considered since, for small differences between experimental and calculated values, this method is not considered to be sufficiently accurate to determine the lift coefficient due to buoyancy present in the experi-mental data; however, this method should give reasonably accurate indications of the lift coefficient due to buoyancy present in the experimental data for the cases where the lift coefficient due to buoyancy is large. Figure 30 shows that the magnitude of the lift coefficient due to buoyancy for different speeds is approximately one-half the lift due to buoyancy based on the total wedge-shaped volumetric dis-placement computed by equation (28); therefore, a rough empirical approximation of the increase in lift coefficient

due to buoyancy can be calculated with reasonable accuracy

from

CL.BCL.V.

(r8°; C3)

(31)

where C.01 is given in equations (28) to (30). For low

trims (4°) a lift coefficient due to buoyancy greater than that given by equation (31) is required to account for the additional lift coefficient due to buoyancy as indicated by the flagged symbols in figure 30.

COMPARISON O THEORY ANO EXPERIMENT

LiftOnly the experimental data indicated as pure planing by the method discussed in the preceding section are

con-sidered for the comparison with theory. Also, the data considered are only for the chine-immersed condition. The theory is applicable to the non-chine-immersed condition; however, for surfaces having other than a constant angle of dead rise such as those having horizontal chine flare or vertical chine strips, the shape of the cross section varies, and, therefore, the crossflow drag coefficient would not be the same value as that determined for the chine-immersed

condition. The values were calculated from the proposed

theory as if there were no non-chine-immersed conditions. For the non-chine-immersed condition, the lift coefficient for a surface having a constant angle of dead rise is approx-imately the value determined at the instant of chine immer-sion and is a constant for a given trim and angle of dead rise. (The length-beam ratio is approximatelya constant value for all non-chine-immersed conditions for a given trim and angle of dead rise.)

In order to simplify the comparison, the data are sum-marized in the following table:

(11)

REPORT 1353NATIONAL ADVISORY COMMiTTEE FOR AERONAUTICS

some of the experimental data that were obtained with oden models (for example, see ref. 31) were lower than values predicted by the proposed theory; this difference is )ught to be due to the influence of the local shape at the

es (slightly rounded or roughenedchines).

Fhe effects of Reynolds number, scale, and nonuniform ne radii on Cß have not been determined because of the ited data available.

rhe lift on various pure-planing surfaces with rectangular triangular plan forms similar to those considered can be

mated by changing the value of the crosaflow drag fficient for a given configuration.

Values of the

sflow drag coefficient should be determined from tests;

7ever, reasonably accurate approximations that are satis-ory for engineering calculations can probably be made

fig. 3) that will approximate the pure-planing lift for aces similar to those considered herein.

or planing surfaces that vary considerably from those ;idered herein, only data for a given angle of trim and

The values of Cri, for eguatlon (25) were determined Irons ltgru-e 3.

aspect ratio (for a given effective angle of dead rise) are required to determine the value ofCr from equation (20). (The experimental values of lift coefficient, trim, aspect ratio, and effective angle of dead rise are substituted into equation (20), which is then solved for the value of Since the value of C». is a constant for s given planing-surface cross section, the lift coefficient for wide ranges of trim and aspect ratio can then be estimated. If values of are obtained for two or more effective angles of dead rise for a given type of planingsurface, the value of C'DC for similar surfaces having a different effective angle of dead rise can be estimated by interpolation. Therefore, in order to calculate the lift coefficient from equation (20) for wide ranges of trim, length-beam ratio, and effective angle of dead rise for a given family of planing surfaces, only a very few test points are required.

Center-of-pressure location.A comparisonof theory and experiment for the center of pressure is given in the following table:

Condgural ion Description of mode) used

Data 1,0 be

esmpared-Data presented

at figure Remarks

Equation of

present paper Experimental dataof reference-4-Inch-beam brass model

31(a) (Agreement good eacept at trims above

approtimateiy

Sharp chine plastic model

31(b) 30 at large Iength.beani ratios 4.jnch-beam brass model

I 32 31 (cl Agreement good

Various models 12, 32, 31.33.34. 16,15,

and 25 31 (d) lo 31 (k) Agreement good; some. differences withmodels wooden

4-inch-beam plastic' model with i I Present paper 32(a) and 32(b)

Cts., reduced to 1.15 and 1.31) Rectangular flat plate (ta-inch-radius chines and (4,. I

I

Inch-radius cirineo (30)

J 1

Models used In reference dala had either slightly

rounded or roughened chines and reduced valuesof

Wooden model same model used

in both ref. 31 and 35) 31, 32, and 33 32(c) Io 32 (e)

Co., resulted. In case of reference3,5 the chineshad greater chine radius or roughness ana result of went-lu use; further reduction An Co., resulted. With angie of desti r-13e nl 2)' 1

28 and 25 33 (Agreement good except for trims above apprssl.

With angle of dead rise of 40' I 28 and 25

34 t tetiY 311' Basic V-surface

I

i

Agreement good for length-beam ratios abovebelow this value the ezperlmentai data laded3,0; to

With ang of dead rie of 30'

36 35 chow the usual Ineasein

Ct as 1,/b decreased.

Slmliareffeet slightly evident in 0g. 34(b) forß-40'.

V-surface with horizon, WithrIse 0116'aneffective angle of dead ) (20)

13 and 23 36 Agreement good

tal chine flare With an effective angie of dead (Co.. value from f

rise of 32'4' j fig. 3) 24 and 25 37 Agreement good

V-surface with vertical Withrise of 15'33'an effective angle of dead 1 (36)

28 311 Agreement good

chine strip With an effective angle of dead l(Co., value from {

rIse of Sl'SY'

J fig. 3) 26 36 Agreement good

'I'rlanguiar flat

(base forward) plate Wooden surfaces (see Bg. 40( (24) 31 and o.npub(ished

tank no.2 data 41 Agreement good up to trims of 16'. Values tower than those at trim of 30'; chines may be slightly rounded since they are made of wood.

Configuration Deserlptioo of modet used

Data to be compared-I)ata pee3enved In 0gw-e--Remar8s Equation of present paper (.) Experimental data of

reference-Rectangular flat plate 4-inch-beam brass model

(251 23, 31, and 25 42 Good agreement

Basic Vsurlare W 11h angle of dead nne of 20'With angle of dessi rise 5(40'

Withangleofdeadriseollo' (231 i23) (231 25 and 28 25 and 28 36 43 44 45 Good agreement Good agreement Good agreement V-surface wllh horizontal chine fiat-e

With an effective angle of

dead rise 0116'

With an effective angle of

dead rise of 32'47' (251 (23) 13 and 25 24 and 25 46 47 Good agreement Good agreement

V-surface with vertical

chine strips

With an effective angle of

dead rise si l5°33'

With an effective angle of

dead rise of 31'59' (23) (231 26 26 48 49 Good agreement Good agreement Triangular plan form Wooden surfaces (sec fig. 40)

(24- 3! and unpublished

(12)

A THEORETICAL AND EXPERIMENTAL STUDY OF PLANING SURFACES 9

The principal planing characteristics for models have been obtained in extended ranges of trim and length-beam ratio for a rectangular flat plate and two V-bottom surfaces; therefore, force approximations for water-based aircraft can be made in these extended ranges with more confidence. The data obtained for rectangular-flat-plate surfaces having very slightly rounded chines indicated that slight differences in construction at the point of flow separation can result in

decreased lift and drag coefficients obtained for a given flat-plate configuration; however, the center-of-pressure location, skin-friction coefficients, and lift-drag ratios remained approx-imately the same for the trims tested (8° to 18°). These data showed that slight differences in construction at the point of flow separation were probably the reason for the differences

in experimental data obtained for a given configuration

by various experimenters.

The proposed theory appears to predict with engineering accuracy the lift and center-of-pressure location of

rectangu-Shuford, Charles L.. Jr.: A Review of Planing Theory and

Experi-ment \Vith a Theoretical Study of Pure-Planing Lift of Ree-tarigular Flat Plates. NACA TN 3233, 1954.

Wagner, Herbert: Phenomena A.ssociated With Impacts and Sliding on Liquid Surfaces. S-302. British ARC., July 14, 1936.

Mayo, Wilbur L.: Analysis and Modification of Theory for Impac t

of Seaplanes on Water. NACA Rep. 810, 145. (Supersedes NACA TN 1008.)

Sokolov. N. A.: Hydrodvnamic Properties of Planing Surfaces and Flying Boats. NACA TM 1246. 1950.

Sedov. L.: Scale Effect and Optimum Relations for Sea Surface Planing. NACA TM 1097, 1947.

Perelinuter, A.: On the Determination of the Take-Off Characteris-tics of a Seaplane. NACA TM 863, 1938.

Sottorf, W.: Analysis of Experimental Investigations of the Planing Process on the Surface of Water. NACA TM 1061, 1944.

S. Perring, W. G .A., and Johnston, L.: Hydrodvnamic Forces and Moments on a Simple Planing Surface and on a Flying Boat

Hull. R. & M. No. 1646, British ARC., 1935.

9. Korvin-Kroukovsky, B. V.: Lift of Planing Surfaces. Jour. .&ero.

Sci. (Readers' Forum). voI. 17, no. 9, Sept. 1950, pp. 597-599. lO. Siler, William: Lift and Moment of Flat Rectangular Low Aspect

Ratio Lifting Surfaces. Tech. Memo. No. 96, Exp. Towing

Tank. Stevens Inst. Tech., 1949.

Perry, Byrne: The Effect of Aspect Ratio on the Lift of Flat Planing Surfaces. Rep. No. E-24.5 (Contract N6onr-24424, Project

NR 234-001), Hydrod. Lab., C.I.T., Sept. 1952.

Farshing, Donald David, Jr.: The Lift Coefficient of Flat Planing Surfaces. M. S. Thesis, Stevens Inst. Tech.. 1955.

Kapryan, Walter J.. and Weinstein, Irving: The Planing

Charac-teristics of a Surface Having a Basic Angle of Dead Rise of 20° and Horizontal Chine Flare. NACA TN 2804, 1952.

L4. Korvin.Kroukovsky, B. V., Savitsky, Daniel, and Lehman,

William F.: Wetted Area and Center of Pressure of Planing

Surfaces. Preprint No. 244, SMF. Fund Paper, Inst. Aero. Sci. (Rep. No. 360, Project No. NR062-012, Office Naval Res., Exp. Towing Tank. Stevens Lost. Tech., Aug. 1949.)

Sottorf, W.: Experiments With Planing Surfaces. NACA TM 661, 1932.

Sambraus, A.: Planing-Surface Tests at Large Froude

Numbers-Airfoil Comparison. NACA TM 848, 1938.

(7. Locke, F. W. S., Jr.: An Empirical Study of Low Aspect Ratio

Lifting Surfaces With Particular Regard to Planing Craft. Jour. Aero. Sci.. vol. 16, no. 3, Mar. 1949, pp. 184-188.

CONCLUDING REMARKS

REFERENCES

lar flat plates, triangular flat plates planing with base for-ward, and V-shaped surfaces having a constant angle of dead rise, horizontal chine flare, or vertical chine strips. A reasonably accurate approximation can probably be made for the crossflow drag coefficient of a given model that will result in satisfactory engineering calculations of lift and center of pressure for pure-planing surfaces similar to those considered in the present report. Also, the proposed theory (which can be applied to both the chine-immersed and the non-chine-immersed condition) together with the method for approximating the lift coefficient due to buoyancy gives s reasonably accurate method for estimating the lift charac-teristics of planing surfaces for a wide range of conditions.

LANGLEY AER0N.tUTtcAr. LABORATORY,

N.tTIONAL Arsvisoa COMMITTEE FOR AERONAUTICS, LANGLEY FIELD, V.&., Nocember 23, 19.56.

Schnitzer, Emanuel: Theory and Procedure for Determining Loads

and Motions in Chine-Immersed Hvdrodynamic Impacts of

Prismatic Bodies. NACA Rep. 1152, 1953. (Supersedes NACA TN 2813.)

Pabst, Wilhelm: Landing Impact of Seaplanes. NACA TM 624, 1931.

Lamb, Horace: Hydrodynamics. Sixth ed., Cambridge [niv. Press, 1932.

Brown, P. Ward: An Empirical Analysis of the Planing

Charac-teristics of Rectangular Flat-Plates and Wedges. Hydrod. Note No. 47, Short Bros. & Harland Ltd. (Belfast), Sept. 1954. Betz, A. : Applied Airfoil Theory. Airfoils or Wings of Finite

Span. Vol. IV of Aerodynamic Theory, div. J, Ch. lIt, sec. 7, W. F. Durand, ed., Julius Springer (Berlin), 1935 (reprinted by Durand Reprinting Committee, 1943), pp. 69-72.

Weinstein, Irving, and Kaprvan, Walter J.: The High-Speed Planing Characteristics of a Rectangular Flat Plate Over a Vide

Range of Trim and Wetted Length. NACA TN 2981, 1953. Blanchard, flysse J.: The Planing Characteristics of a Surface

Having a Basic Angle of Dead Rise of 40° and Horizontal Chine Flare. NACA TN 2842, 1952.

Kapryan, Walter J., and Boyd, George M., Jr.: Hvdrodynamic Pressure Distributions Obtained During a Planing

Investiga-tion of Five Related Prismatic Surfaces. NACA TN 3477, 1955. Kapryan, Walter J., and Boyd. George M., Jr.: The Effect of Vertical Chine Strips on the Planing Characteristics of V-Shaped Prismatic Surfaces Having Angles of Dead Rise of 20° and 40°. NACA TN 3052, 1953.

Jones, Robert T.: Properties of Low-Aspect-Ratio Pointed Wings

at Speeds Below and Above the Speed of Sound. NACA Rep 835, 1946. (Supersedes NACA TN 1032.)

Chambliss, Derrill B., and Boyd, George M., Jr.: The Planing Characteristics of Two V-Shaped Prismatic Surfaces Having

Angles of Dead Rise of 20° and 40°. NACA TN 2876, 1953.

Perry, Byrne: Lift Measurements on Small-Scale Flat Planing Surfaces. Rep. No. E-2410 (Contract N6onr-24424, Projeet NR 234-001). Hydrod. Lab., CIT., Apr. 1954.

Davidson, Kenneth S. M.: Resistance and Powering. Detailed

ConsiderationsSkin Friction. Vol. II of Principles of Naval

Architecture, ch. II. pt. 2, see. 7, Henry E. Rossell and Lawrence

B. Chapman, eds., Soc. Naval Arch. and Marine Eng., 1939,

(13)

Io

1ridhri, h'iiis't i I ...:iid \Ir( h .10)111 IL: I'l:iiiiiig ('Inir:ieh'r-¡.tic- of Turi'' iir'f:u. I'pri'.)'Ill1)Ik of Ils' rn-)I l"orril14, N M'A ItsI 1.1003. 1/119.

\Icflrid'. Ellis E.: An Exj.lrilIll'l)l;ll lnv.slig:ilioii of thi' Sc:1l' fl'hii ion.. for ih.. Iriipliiging '\sit r Sprsiv (.ii.'rsitid II ¡t

I'I;tii-rig iirfae.'. NM'.\ TN 3615. I93i.

Slio.'irisikr, J:tiiii.. M.: T:iiik Tvst ofFl:ii SISO) V-flot I 0111 l'Isining irrf:ig.. NM A TN 50/I. I ¶434.

TAl5l.1 I

EXPERIMENTAL I'LANING DATA OBTAINEDFOR A

RECTANGULAR FLAT PLATE

(a) 11 r,s" nio,1,'I I,avin sharp dunes

TrIriÌ. Cr Is St si :31 1. 533 2.151 Sr.'s 4.82 11.64) 334. 1:3' 1.7(1 1$. I'S 2.91 114.19 3.3(2 IS 114.19 4.914 114.22 5.79 18. IR 7.30) 19:13 7,74 39, 13 19,153 16.1(1 I.-b 1. 64 374 4. 6.834 1.82 2.97 3.813 ',IM; 5.8.3 7.05 7.841 2.04) 2.80 2.92 4. 12 5, 1)' 0.04 7.16 7.18 8.03

.LI. .s'r.s.i 11/secI,, Ca., C,.. 1,R I. ¿Idi II. 7945 1.911X10-' 2.74 .744 . IM) 3.77, .7(4 3.7$ 4.57 .4)147 1.7$ 9.41$ .41344 3.76 (1.4)2i .652 1.07 7,714 .4(12 1.76 1.84 3.03) 2.442 313 3.54) 7.07 7.94 1.92 3.07 ro3141 4.111 14.4s 7.04 8.02 8.1(1 2.02 2.82 2.02 4.34 5.72 41:07 7.20' 7.22 8.05 SI 2.117 3.19 14 3.24 (1344 31.21) 7. 12 14.1$ 3.447 4. 33 3.21 .3(73 .1)78 3(14 4133 41.42 4135 ('Ill 4)40) 41533 4(1(1.

IIEpoIfl' J 35 5-NATIOXAL AI)\'17O1lY ('OMMITr EE F1111 AF.111)NAI'TI('S

0.4161 087 104) 1241 .141 .1541 .171 104 140 175 .213 .234 .2411 .281 34tO .220 24(3 287 .332 SSS 385 431 .444 .3)8.) 307 .379 .441 .455 5305 53.2 .1)10 0. 2545 .375 453 .324 4142 4144 0141) 314(t) 7354 $91411 .802 9513 4811 .4156 .744) .853 1.025 1.335 1.143 1.2743 3.2443 .582 7(44 .780 864 1,111 1.249 1,425 1.429 1.510 0037 0.174 .032' .1314 .029 .121 (13311 j914 .0211 .103 .023 .0435 .4172' .09(4 .057 .214 .4449 .671 .344.5 .158 .042 .1441 .4140 .3314 .4137 12$ .0344 .123 .04)5 .255 .072 .219 .0357 .200 .0041.19(1 .059 .172 .0541, .164 055 .1413 .0111, .159 .0S5. .199 .133 .29! 167 .27:1 .142 .2157 .0342:234 J .083 .213 042, .2031 07(1 .1931 06 190 .076 J .1811 .472 53); .29s .2.52 .2191 .237 229 30(1 19,274 34', 1.3)0 1.02 737 1,41:4 I 054 ) 330 415 .521) 314.25 2,4)5 2)64 2.12 .4102 lILI ' .493, .851 ; .2491 .44)7 P1.22"3,112 :1))). 3.13 .4547 1.41:4 I '3.127, .2!) .3.4 ('.33 4.24 4371434; 5149 1.345 j $341 1.4741 .3(10 .373 ¡'.32 5.57 S.44 7.5! .455 1,54'. 1.1135 1.744 . 15) .323) 41447 )I,2 i.31'4530 .8); 1,331 1.348,1(418; .102 .7315 Is.).') 7,445 7.t2 l91

.1,8'.

'2.177 ...81 1.314 3.2'; 334 .53); illS 03 :1.4813 1.74$ .1145 .331. '.1.10 4.2 432 4.30 .31(9 I'S) ' .1.2131.417' .192 :35%

9)0 SILl :4I')'3.34; 111.3 I'S) 135 1.111 20,1 .3214

114.22(:( 2.48,.1.7 (.0532.10; !.11iI2.464 '(74SIrS 1.11:4i.r.s .23.53.1 ' .1311'.3112 .303..371 .3416.341

I". I:) 322 3 142 :4.33; 3(12 1,341 I .671 ' 1,2341 .2(7 .375. l'SS 4.22 4.37 4.42 .032 1.34'.

J 1.124 1.0101 .257 .54416

t'. 2" 5. :4'. 384' '.5.54 .1413.. t .34. i 1.7131 I. 1(3% .210 .33!

II

II

II... I ' .4.04

3-I. I.i,t'k', I". \\'. S., Jr. r 1'.'t.. lii si l'mi iolt),tl; I'l;iiii;gSurfac',' To

I).'),'rrl)ill.'

lii'

IIl('l'liti))li (If I'l;ltiillg

..N\\ .\f'l/

J)I. J''j. I (1/II',, 13ir....d'I.. I)''. ¡¶4-IS.

35. ('liristop)o'r, K,'lila'IIl 55'.: EIF.'et of l;sdIow 55;iI.r n, iii, Hvdro-(I\'l);i)lIir (Ii;lr;u'Ii'I'i,.tit'14 91,1 si J"l;iI-lIoIIonl I'l;lI,iIig Ñlrtslce.

NAIS TN 3642, I/ISO.

36, Sjlring.tiill, (.'org.' JI...zind Ss'r,, Clifford ¡,...Jr.:

Tu.' I'Isiliillg ('lia roc-I .'rjst¡4'% of s, V-l1;i1H'l) ¡1riSIilat le ÑIt'fsil'i' \\'iI h 54)

Do-grvl% J).'od Ifl'.'. Il.'p. ¶42(1. ¡)avid 55'. Taylor Modo) 1388111,

Nave DIpl . , Ful t. I ¶455.

TABLE I.-ColIIiIItIed

EXPERIMENTAL PLANING DATA OBTAINED FOR A

RECTANGULAR FLAT PLATE

(bi Br,iss iflo,lel Iisiv Ing SII;Ir(1 cli ries; no u lad IdCrn

sIiri cI)in,'s: no and .'.er'.'ri or SIrIS' shit:

JI I...2s 'I Is.114 32 '15.13 1.3.4 359 1.4(2 2. 711 2 75 2 78 3.IISIZ4.o4:3.st II SII 1,04X1101 . 71151, 1.4.4 ..ss 1.34 0.01141 0.2,91 ((,145(1 I)114) .45539; .31314 '432 .III .10!. . .4(12 '1:57 .1113 j 113 22I 15.19 4.743.63 ,4.13)1J 4 442 .673 1.34 .1211' 53$ 1323 .I ii) 5.814 ,.5812 .494' 1.14 .154.5 r .395 16,51

I,,) I'lasI k' rr,oil.'I IIs)s')r1S slisri' chitas'

114.4(1 1.24 3.29 1.31 (1.735 1.714X130' (4.1623.5 11.1942 I)1(45. li,Jl1,lS 133.101 2.4825212..54j 2i 1.73. .(E373:.2Itisi ."Il "54.37 j I(i) :3.98 3.4)2I3.454 .711.' 1.71. ((4415 ' .78)4 .1)13%:(17(4,) S; 18.411 Iilr45 . 4.17 ' 3 525 4.3.4 , 5.30'5.54 1 3.31; .712: 1.73. 71)8I 4 75 .115213 .23.11 "117 .41622 .4L*3'2 1.31647 13)53', ((57$

I 114.11. III4) 7 .551'S..57Ii. Ill 4; 43 J 6 457.51. 7111$ii42 i 7%I.

'

((('.71.4)749 . 34)91574)' f499,9l()44 :.4"2'.(494

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: ' Is) .17113 ;.4184314 '(44,, lls"l 11;,..';1444i191.184; (74 17s .34)495 .4516' .41.3.334

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.l'i14 I 114.19 9191.(tlII 7.1)5 .44.)'Jt.. l J 3)4411 LiSo' 337 l'il' .I.ri.1i, C. t. 1_ e. s'i li 'sec C,,.3 ('L., Cr,,s C91.s , '("g

T

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(14)

iii;

. -(,llI

1111421

EXI'EItIMENTAL I'LANIN(; DATA OIITAINLI) Folt A

RECTA NG ILAR FLAT PLA1'E

(, ¡'Ils) II motivi I,:lviujz '.,-ii,cI,-r.sdiw. (11111,9

(1) lIaslic 010(1, lì"Ing .1,-i1)(h$9Ul ¡4

TABLE II

EXPERIMENTAL DATA OBT.;INED FOR A PLANING

S1JRFACE HAVING A 200 ANGLE OF DEAD RISE

TABLE III

EXPEIIIMENTAI. DATA OBTAINED FOIt A PLANING

SURFACE HAVING 40° ANGLE OF DEAD RISE r . 1 . : I l'o .6 KS) .4 .6 G 2e) .4 70 80 90 i'r,in , r,.l.'g (?i I

¡

/_'jj: i,

¡

i ..

:

y. SI) rl'4''4 Cl.. (':.. (,, . .. ' ('i. .8 I

IS, 35 I. 7(8 I. 54 1.541 I). 74 I I. OTXI)(' (I. *145)5 I), 21)5(4 o. ((47 o. 11941)

IS. (3 2 .541.. '44 2(12 7)4.5 I 17 1)7111 3443 .44_494 .133)) IS. 19 3.444 3. (,4 3. 4141 . 7(04 I.17 ((972 . 43122 (4297 11141 II 18.19 4,79 4,144 4,447 .085 .07 .11141 .1)4115 .1)144 .11)32 ¡14,16 5.51.2 5 .2(42.. '4* (491 1,117 .12)41 .53191 .44232 ((971 (8.07 tiSI ¿4.92 ¿4.9.5 .9113 4,07 .1.4.9) .9.5(7 ((.124 .691(1 18,07 7.60 7.68 7.014 .9.141 1,07 .11812 .67(,4 .0217 I 18.28 2.48) 2(145 2.114 .7418 ((42 .1615 .45483 .11769 .31375 18.19 2.84 2,94) 2.113 (444 1,07 .2)949 (112)0) (1713 .21314 (8. (9 3, 79 3. 184 3. (4(1 . 672 I. 147 . 2.41) . 7439 . 11)153 . 1937 IM 18. IO 4. 92 4. 98 5. (1) . 640 1. (.7 . 2945 (((142(4 , 44.54342 . 1734 18.13 5,89 5.4144 2.97 (45 1,07 .331'4) .9789 14.2)19 .1648 Is, (9 7, (II 7. ((6 7. (#4 . 444 1. (17 , 3441 I I. 12111 . (15.54 . 15(45 (8.314 7,7f, 7.91 7.144 .1.37 1.417 .41447 1.2)4.145 .0.537 .1542 28. 13 I. (t4 1.68 1.5$ O. 712 1. 78X1591 (I (172 0, 1(41.14 (1,01112 '0(1)l,14 (8.13 2,13 2,19 2.2)1 .7111 l.7s (4112)4'.18:2(1 .1)117 .111433 s (14,11)(8,11) 3,5(41,50 3,514 4.54 3.584 .54'. 73(4 72m) (,7p, 1.744 (447 ..23414 (42)4 .91 24 (1123(III) .0.14714(1)1.4.) (7.86 1r38 5.413 5.45 .7(li 1.75 (414941 .2948 (41449 0.1443 17.93 (1,49 6.54 (1,54; 74(1 I,7 (91.4 .3257 .)ltl() .0498 18.00 7.53 7.35 7.441 *494 1.7$ (1713 3487 .18494 .04(41 (8.27 1.04 1.69 1.71 .71)7 1.5)) ,)L55S .2(131) .14.431) .1556 18.27 2.62 2.98 2.764 .704 1.50 41715 ' .3371 (42741 125$ 19.27 3.7.5 3.1144 3.83 I'M.) 1.91) (COl .4)1441 1)14.14 .1074 Il ¡9.33 4,64 4,69 1.71 .4489 1.75 14111' .47.94 .023)' .1)6)9 18.205,71 5,72.14.75 .6714 (.714 ,I114.'.5405.()224 .09313 18. 13 6. 76 0, 82 6. 444 . (174 1. 78 . 1453 I . 14941 . 0243 .4)977 18. 13 7.84 7.86 7.89 .l.11.5 1.20 , 1051 ..4812(1 .02114 .33643 IS. 13 I. 75 1.49) 1.1)3 .6941 1.79 . ¡4)13 ' 441)6 (4943 .2481 18.13 2.74 2,744 2.82 .993 1.63 .44187.98)3 (1743 .211)4 ¡8.13 4,01 4,06 4.634 .1446) 1.78 .21445 .7420 .(6i114 .1832 18 18.13 5.02 8.49.4 5.09 .4.4)4 (.79 I7I.8.'S .31573 .1683

Is. 20 6. 22 (I. l'i 9.. 2(4 .3141 I. 7$ . 33544 :.97s41 (4241 .14412

(8.13 6,90 (595 (,.99 .642 1.79 311744 1,48117 .0525 .1549 IS. 13 7. 91 7. 461 7. 99 . 4141 1. 79 . 44419 I, ((477 . (4241 . 14417 TrITo, r. ll11: Cr 1, . 1, b 14 6 (j,, "I '15., Cp., CL., C,..4 Cc.a 19.31 1.6$ I 44' 212 (17)49 I. 7.;xli'-1 (1.4121 4), 220 0.4)27 II. 124 1$. 13 2.16 . 2.344 2.82 '.74 .4(93 .275 .0271 (III 114 19 2.29222 2.74 72.14 (.7.1 .063 .277 .027 ' .1111 12 IS. Ill 1$. 19 3.45 3 1 2.81 4.27 4.14 4.68 ('449 (1446) .093 .314.3 ($441 .41)3 .11231 .1981 .1991

IS. 19 2.49 8.711 2.91 II(S) 1.71

III

.44i (49141 .1492

IS. 13 ¿4.29 ...r 34' (1.73 1184 .425 .19.4 .0.41)i .4)76 18.29 746 (14.2 .449 534 .02).) .076 IS. 47 1.494 1.1114 74(44 .1244 .3191 .04171 .205 18.31 2.79 2.14" 3113 1.73 .174 .5(4 .140), .477 16.29 3,1441 3143 4,116 (1413 1.74 .241 .112)4 .054 . 1444 Is 19. 13 4.72 193 (163 1.71 .22.1 .712 .4)59 .133 IS. 14 5.7(4 293 (.43 1.71 .29)1 .5.44) .011) .144

IS. i (1. (40 9, ;i. li. 91 .1134 1.71 3)' 944 .1149 .1314

1$. 16 7.91 794 9.( 92(1 1.71 .373 4(144 .017: .131 19.31 2.1122.122.21 .11441 1.711 .237 .44)14 .121 .2(414 19.25 3.44 3. ¡4 3. 2)) .11415 1.117 3441 .771 .111 .240 18. II'. 3.64 4.113 4. II' (49 I. 72 .423 .931 .IUS .2.14 18.13 4.16) 4.440 .90 .1195 1.72 4(49 1.1)12 .6)1: .221 24 1$. 13 Is. 19 Ii. 2(1 (1 143 Ii. 44 7. (16 7, II 7.24 .11911 .1129 I. (17 I. .6)44 1.314 .147) 441 .41)14;.11)14 .207.2)4.4 19.31 7. $41 141 9. (II .1114 4. 34 .7344 .350 .4)42 .144) V. III 3.11 2.49 3.20 11M 1.07 .314 .7714 .1044. .214 8.11 4. II. I 9 4.27' . I.17 .431 49)6 .1114 .24)

9.117 517 2 11 2.33 .6)4 III I lIS ¡III .2(44

114.12)' III) 1.12 4.51 ....191 1(13 .4(42 .224 .204 .:s;l 18.19 2.32 2.5' 1 43 . (.21 1.143 .143 .77) .457 .324 18.19 3.3$ 3.41 3.144 .6114) 1.62 .41(45 (.020 (70 .2447 :10 137 18.31 3.26 3.42 3:041 .647 1.2). 4.27 431 4.39 .112)1 1.2)1 .5i'.724 1.2(15.91(3 .177 .216) (117 .27)4 18.31 5.35 42 5414 .1114 1.54 .4453 .172 (.13

III

1$. 2! ...II. 11 .184 I. 54 1,14.35 I 70;) I!.' .2444 114.23 7.21 1 7:12 .14(9 1.34 1.119. 1,576 .429 254. 18.13 1.54 I. '2 (.4)6 .1412 1.113 .4481 .41)44 .222 .3714 19.16 2.41 2,41 1.414 .034! 1.113 .549' .54)6 .1444 .553 34 14.31 15.22 3.14; 3,7' 2,75 ..;;.: 1.54', 1.6). 4:) 4.74 ItlS .2)1 141) 1.144 .2_42 :4244 .1494 .1:12 III .3144 18.31 .14.141 5.9. 5)94 I 1.194 I 5(2 .211 .2491 11141 9.42 9.16 )L2)I 441 1,24 :147 1.907 .2117 .2914 ¡4,32 7.31.' 7,4(1 7,484 ...4 1 4494 Trio', (, ' 1!. t, 4199 3, .7 14 i. I I ,41))111.4 ('II., ' CL., i C',1.,m C'i. i 15.4)7 I

(4.148 (.14 1.4144 0,73(1 1. 72X ((I-' 0.1(2.14 (I. Ill) (1.14112 0.0145 15.117 1.114) 2.14 2.118 .6914 1.72 .041 .107 .1)19 .1)714 114.11112.91 3,3. 3,j : .653 1.72 .0(9) .2214 (11M (94. 12 19.13 3,14.14 IS.)); 4.9); 4.41; II. .141 4.4(5 .4431) (tOn ....34 1.721.72 ((72 .273 .1)114 .331 .0)6.017 1112 (4434

IS. IC' 2. ((2 (i. 41 6.92 i (117

1.444 .102 .314; ((IO .44.145 16.24 9.90 7.51 8.06 .1422 1.69 .120 .444 .016 .44.55 1$. 23 1. 14 1. 4. 1.81 1.72 .674 .219 (114) . 14$ 11913i 2.2(4 2.1$ 2.36) (1414 1.1.3 .119 .339 ((46 . 13) 19.19 3.44) 3,74 4.01 (44g 1. 72 .12(1 .456 .042 .12.) 114 114.19 4.32 4,9.14 4.98 034 1.67 .19!. .53'S .042 (IO 19.13 2.42 .14.74 6.114 .617 .294 .626 (9444 .14(9 19.25 9.34) Il )17 6,99 14 .253 .7(94 1(38 .105 (14. (6 7.4(1 8.07 005 1.149 .390 .761 (Osi .098 18.13 1.71 1,141 2.101 .1433 1.63 .170 .371 .9149 . 194 11113121W 248' 2.99 .03*) 1.72 32 I .1134 184 19.162.91 1.11$ 4.24 .6114 1.72 .314 ' .673 .678 . (67 18.1614.57 4.I9 .611 1.72 .3.146 .7(1) 075 lOO (5.114 :5,74 5,91 (III i .6(4 1.67 .45(1 .953 . OlI 24 i 19,1312.64 6.111 6.111 (1(6 ¡.72 .4413 962 .1(91 15.114 7,04 7.24 .3(46 1.1144 .5(44 I (Wi .071 .1.58 (9.114' 7.61 7.71. 7.9(1 .4842 1.4144 .253 1151 ((74 14$ 14(1 264 2)44 2.99 .C9l'2 1.67 .233 .512 .4914 163 9.13.3.78 3,1)4 417 I .611 1.67 .3(14' .0814 .11*0 173 9.69 4.81 5)42 5.2)) .601 2.97 .402 .4214 .080 .1(S) IS. 13 I. IS 1.17 1.39 .627 ¡.63 44,14 .311) 1411 .244) 1111111 2.04 2.44 2.25 .412.4 1.63 .2)45i .505 . 138 .2241 11(13 3.65 3.44 3.94 .61)4 1.614 .424 .155 . 2'Jfl 46.13 4.0.14 4.43 4.23 .4415 1.113 .545' .9)414 .132 .3140 16.31 4.4(5 5.111 2.22 .507 1.5$ .626 I(.6112 . 125 .2)91 11125 j18.19 (14.111 (14.07 9. 22 .587 I. 5$ . 7314 1.2.94 . 124 .2)14 7.47 71(3 7.58 .1479 1.544 ' .4(11 I1.493 . 121 . (97 18.07 1.29 1.37 (.48 (.63 .255 .377 .186 275 18.22'2.3$ 2,4" 2.55 1.6)4 . .44 1 .619 .181 . 2444 19.11313.10 3,114 3.24 ¡.413 .47 : .791) .173 . 254) 34 . 18.19 4.20 4,2.2 4.25 1.5$ «1(9 .993 .184 19.332.32 5.33 fr39 SSS 1.5(4 .683 1.263 .165 .236 (8.341 9.35 44.12 (4.4*) .353 1.1.8 .1(91 ' 1,44)1 .162 .224) 114. 3(1 7.16 7.244 7,25 .251 1.49* 1.181 1.41.58 .164 .3181

A TUEORETICAL AND EXPEHIMENTAL STUIÏV OF PLANING SURFACEI 11

O IO 20 30 40 50 60

ß9, deg

F'Ir.I'68: I.-C(1I)III:lris'lii (If (1l:I(1-ri»i' fuIlIlioli ppIi,'d Lo IiIIe8r OrnI wit Ii d :,4l-r,.'(' f111111 I issed ou \aglI4rs vork

IC' 30 40 50 GO

70 80 90

ße. 09g

Fir L'ni: 2.-('oll;Il:ri»llll of (Il'1l-ri93' f1111141 loll :Ip4lie(l to crosflov tenti

(15)

12 L4 C L2 o 1.0 o L) .8 .8 in C .8 o .4 2.2 2.0 (.8 (.6

REPORT 1335NATIONAL AD\1SOHY CO?t1,T tiTEE FOR AERONAUTICS

V-shaped surface having vertical chine Strips (rets. 25 and 26)

V-shoped surfoce having horizontal chine flore (rets. 13, 24, ond 25)

Rectangular flat, triangular flat, and V-shaped prismatic surfaces having a constont ongle of dead rise (rets. 23, 28, 36, and dota of present report)

Rectongulor flot with 1/64-inch-rodius chnes

(4-inch-beam model)

Rectangular flat with 1/16-inch-radius chines

(4-inch-beam model)

FICURE 3.Variatic.iìof cro'flosv drug eot'ffi-iiiit for var)oui types of pluming ritirfaces. .8 in C 4 .8 in .4 C : o .8 o 4 (b) Dead rise, 20°. Flat-RE4Cotitintied.

(e) Dead rise, 4110. F'ii;IRE 4.C'oiietiidi'd

-due (eq Proposed Proposed Crossflow ta ((6) + term theory, theory leading-edge

total tif? (eq (23)) before removal at lift

suction effects

eq ((8) with C0 4x3) (eq. (22) with Co, 4x3)

R.

R

Rl

u-j

-f6

--1_ I

.---

(b) Proposed Proposed due to )eq (16) Crassf low + term theory, theory leading-edge

total lift (eq. (23)) before removal of lift

Suction effects eq (IS) with CO3r

(eq. (22) with C0 4/3)

n-.

'H

lifl I I irr I

_---H'1

Proposed theory , total lift (eq (23)) Proposed theory before removal of lift

due to leading-edge Suction effects

(eq (IS) + eq. ((8) with Co, 4x3) Crossf low term (eq. (22) with C0 4/3)

-- -3--11m.

I

I

-

---

I

--

----

(o) 50 20 30 ß, deg 0 8 '6 24 32 0 Trim, r, deg i_il l(:ti p(:itr'.

1"iiut-its: 4.IleLil iii iiu.g,i)t titi' of (-OtiiIii)iiilit' tif itOX)Si'(l t lti-urv.

6 32 8 24 (6 24 32 0 8 16 24 32 0 8 Trim, T, deg 24 32 16 8 16 24 32 0 8 Trim, r, deg

(16)

.821"---(b)

4.000'

4.000"

A THEORETICAL AND EXPEHIM};NTAI. STI'I)Y OF PLANING SUItFACES

90"

¡.288'

(a f Fffective ¡uigk' of dead rie. 1 ti. (See ref.1 :3.

(b Effr"etivt angle of di»ad rie. :l247. (See ref. 24.)

EtCURI: 3.Cro's set'tioli of ,.tu'face.,

li:iiii

liorizuitlal climi flare.

(0) 2.000" 2.000' 4(25' (b) 4.125' .390'

Effective angle of dead rise, 15°33. (See ref. 26.)

Effective angle of dead rise. 3l°59'. (See ref. 26.)

FIGL-RE 6.Cross section of surfaces having vertical chine strips.

13

Proposed theory (eq. (23))

Sotcolo, (ref 4) Sedov (ref 5) Perelmuler (ref. 6) Sottorf (ref 7)

-iuurruiuuu

__

-,

0 4 8 12

6 0

4 8 2 Trim, r, deç

(:i I Prnjsed theory and ref.'r('taet's 4 to 7.

I:ii.Hp. ".---'arial nit uf lift. coeflititit. vjt h trim for rectangular flat-plat. l)it. formulas.

2.40" rod. 20 6" .574. .154"

--.30 20 C t) u o o u .20

jo

(17)

a) o -J .6 .4 .2

Proposed theory (eq (23) J Perring arid Johnston (ref 8) Korviri-Xroukovsky (rel. 9) Suer (ref 0)

(h) Proposed theory sud references 8 to 10. FIGURE7.Continued.

.6

.4

.2

Proposed theory (eq. (23))

rew

Farshirrig (eq.(2))

Korvin-Krouhovsky, Sarsutsky, and Lefurnon (eq. (6))

-r

-i.

Id)

Proposed theory (eq (23)) Schrrutzer (eq (9)) Locke (eq (7)

Brown (eqs (IO) arid (II))

-(c)

Proposed theory (eq (23)) Sford (ref. I)

--I

I I

mm

.iiiiii

(e)

14 REPORT i 33NAT1ONAL ADVISORY CÙ3IMITTEE FOR AERONArTI('S

O 8 6 24 32 0 8 6 24 32

Trum, r,

(d) Proposed theory sud references 12, 14, and Crewe's equatiout (eq. (5)).

FIGURE 7.Continued.

8 1G 24 32 0 8 f6 24 32

Trim, r, deq

(u') Proposed theory auud r('fureulce 1.

FIGURE 7.Con('lud,'d.

0 8 6 24 32 0 8 6 24

Trim, r, deq

(e) i'ropoed t lueorv and ref,'reuiees 17, 18, and 21.

FIGURE7.Continued. .6 .4 'ri L .2 C a) E O a) o 'i .4 -J .2

(18)

.2

45ie,(iI-5g---A THEORETIC45ie,(iI-5g---AL 45ie,(iI-5g---AND EXPERIMENT45ie,(iI-5g---AL STUDY OF PL45ie,(iI-5g---ANING SURF45ie,(iI-5g---ACES

Proposed theory (eq (23))

Schnitzer (eq. (8)) Brown (eqs (12) and (13))

(b) Propu-ed theory and references 18 and 21:

FIGURE SConcluded. (b) 8 16 24 32 0 8 6 24 32 Trim, 7, deq .2 15

Proposed theory (eq. (23)) Schnitzer (eq. (8)) Brown (eqs. (12) ord (13))

(b) lB 24 32 0 8 16 2 32 Trim, deg Proposed theory Locke (eq (7)) Korvin- Kroukovsky,

IL7

(eq (23))

Sovitsky, and Lehman (eq (6))

-

-U

WIUU

a va aau

RAU

rn_

u

IL

3I

_a.

I ()

Proposed theory (eq (23))

Locke (eq. (7)) Korwn-Kroukovsky,Sovttsky,ond (6))

-

Letwron (eq

au__

_

__ia

I_iI_

_-

r

up

___,_u_________

u______

u.

u.

_____u

0 8 6 24 32 0 8 6 24 32 Trim i deg

(a) Proposed theory and references 14 and 17.

FIOrRE 9.Variation of lift coefficient with trim for a surface having an angle of dead rise of 400.

0 8 6 24 32 0 B IB 24 32

Trim, T, deg

(a) Proposed theory and references 14 and 17.

FIGURE 8.Variation of lift coefficient with trim for a surface having an angle of dead rise of 200.

(h l'ropooed theory atid references 18 and 21. FIGURE 9.Concluded.

.6

Cytaty

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