propeller lifing-surface corrections"correction factors for camber, ideal angle due to loading,...

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HydrodynamicsSection Report No. Hy - 11 .November 1968

Propeller

Lifing-Surface Corrections

BY

WM. B. MORGAN,

VLADIMIR SILOVIC,, and

STEPHEN B. DENNY

4 ''

CL EA R INGHOUSEI "J 1 & -Cn' ) %

Sprng-id o 21~51

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Hy-1 PROHASKA, C. W. Anal -is of Ship Model Experimentsand Frediction of Ship Performance 5.00(Second printing)

Hy-2 PROHASKA, C. W. Trial Trip Analysis for Six Sister Ships 6.00

Hy-3 SILOVIC, V. A Five Hole Spherical Pitot Tube for 6.00Three Dimensional Wake Measurements

Hy-4 STROM-TEJSEN, J. The HyA ALGOL-Programme for Analysi " C-. ' J41 h, "t 6,00

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Hy-S ABKOWITZ, M. A. Lectures on Ship Hydrodynamics - I 20,00

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special reference to a Conversionto Purse-Seiners

Hy-10 WAGNER SMITT, L. The Reversed Spiral Test. 10,00A Note on Bech's Spiral Test .

and some Unexpected Resultsof its Applications to Coasters

Hy-11 WM. B. MORGAN, Propeller 25,00VLADIMIR SILOVIC, and Lifting.Surface CorrectionsSTEPHEN B. DENNY

-!- :~

Propeller Lifting-Surface Corrections

by

WM. B. MORGAN,

VLADIMIR SILOVIC, and

STEPHEN B. DENNY

~ HYDRO- OG AERODYNAMISK LABORATORIUMLYNGRY DENMARK

TABLE -OF CONTENTS

Introduction ................................. IThearetical BackgTround .......................... 3Presentaiion of the Series ..................................... -8Experim-enta1 Checks on the Theory ............... ...... .. .20.

Coniclusions ......................................... 27

Acknowledigments . .......... ........ ..................................... 28 -

References .................................................... 28

PREFACE

During the summer of 1967 Dr. Win. B. Morgan, Naval Architect, Hydro-mechanic- Laboratory, Naval Ship Research and Development Center, Washing-ton, D. C., spent three months at the Hydro- and Aerodynamics Laboratory.Although his stay was short it become very fruitful. Besides giving lectures on

Propeller Design at the Technical University of Denmark he found time to pre-

pare in co-operation with Dr. V. Silovi;. Head of Propeller Section at HyA, thepresent paper on Propeller Lifting-Surface Corrections, which was finalis$'ed afterDr. Morgan's return to the N.S.R.D.C. with the further assistance of Mr, Ste-phen B. Denny, mathematician, N.S.R.D.C.

The Hydro- and Aerodynamics Laboratory is grateful to the Naval Ship Re-search and Development Center for having made this co-operation possible, alsoto the Applied Mathematics Laboratory of N.S.R.D.C., and to the NorthernEurope University Computing Centre, Lyngby, Denmark, for assistance in thenumerical calculations.

PROHASKAProfessor, Dr. tcchn.Director, Hydro- and

Acrodynamisk Laboratory

THE SOCIETY OF NAVAL ARCPITECTS AND MARINE ENGINEERS74 Trinity Place, New York, N. Y. 10006 no. 10Advance copy of /per to I prnimnd at the AmaI Meai&, New York, N. Y., No, sber 13-16, 1968

to Propeller Lifting-Surface Corrections

By Wn. B. Morgan,' Member, Vladimir SilovIC,2 Member, and Stephen B. Denny,3 Visitor

"Correction factors for camber, ideal angle due to loading, and ideal angle due tothickness, which ere based on propeller lifting surface theory, are presented for aseries of propellers. Th;s series consists of optimum free-running propellers withchordwise loadings the same as an HACA a = 0.8 mean line and with NACA-66chordwise thicknss distributions. The results of the calculations show that the three-dimensional camber and ideal angle are generally greater than the two-dimensionalcamber and ideal angle at the same lift coefficient. The correction factors increasewith increasing expanded a'ea ratio, and those for camber aid ideal angle due toloading decrease with increasing number of blades. Thickness,, in general, induces apositive angle to the flow, which necessitates a correction to the blade pitch. Thisideal angle is largest near the blade root and decreases to negligible values towardthe blade tip and increases with increasing number of blades. Skew induces aninflow angle, necessitating a pitch change which is positive toward the blade rootand negative toward the blade tip.

Introduction Ludwieg and Ginzel [31 (19.44). More recently,

ThE trend in ships has been toward increasing because of the numerical evaluations possiblespeed, size, and horsepower. As a result, there with high-speed computers, emphasis has beenis an increasing demand for the design of propel- placed on more sophisticated mathematical modelslers which are efficient and yet produce minimum than hitherto possible; e.g., Sparenberg 141, Coxcavitation and induced vibration. Such a task [51, Pien [6), and Kerwin 17].requires a knowledge of the flow field in which a A complete review of the various lifting-propeller operates and an accurate determination surface theories will not be given here since ade-of the flow over the propeller blade surfaces. quate reviews for work through 1964 are given by

Much effort has been devoted recently to the Vu [81, Isay 191, and Lerbs et al. [101. Recentdevelopment of a more sophisticated propeller work not mentioned in these three references .,etheory. This theory has proceeded from the studies by Cheng 111, 121, Kerwin and Leopoldsimple momentum concepts of Rankine [1]1 1131, Nelson 114, 15], Sulmont [16, 171, Nishi-(1865) to the lifting-line model of Goldstein 121 yama and Sasajima 1181, Malavard and Sulmont(1929) and, finally, to the lifting-surface model of 1191, and Murray [201. Murray develops the

lifting-surface theory of both contrarotating and'Naval Architect, Hydromeclhanics Laboratory, Naval conventional propellers.

Slip Research and Development Center, Washington. In general, the investigators have dealt with theI).C. same basic mathematical model but have used

Naval Architect, Hydro-og Aerodynamisk Labora- different assumptions to obtain numerical solu-torium, Lyngby. Denmark.

Mathematician. Hydromcichanics Laboratory, Naval tions. The propeller-blade boundary conditionsShip Research and Development Center. Washington. ae linearized and the blade and its helical wake1). C.

Numbers in brackets designate References at end of arc replaced by vortex systems. Assumptionsmpar. are made for the chordwise and spanwise loading

For I)resentation at the Annual Meeting, New York.N. V., November 1I-16, 198, of Tt SOCIETY OP NAVAL on the blade and a complicated singular integralARCIIZTICTS AND MARINK ENGINEEaRS. is derived from which the necessary distortion

to the blade can be calculated to obtain the pre- Because of the linearized boundary conditions,scribed loading. In addition to determining load- camber and ideal angle corrections, which areing effects, Kerwin and Leopold [21], Nelson independent of tle magnitude of the propeller[141, and Murray 120] also considered the effect loading [10], can be derived by taking a ratio ofof blade thickness, in the linearized sense, and the three-dimensional camber and ideal anglefound that it contributed significantly to the re- to the two-dimensional camber and ideal angle,qtired blade distortion. respectively. These correction factors are, of

For an unskewed propeller blade, the effect of course, dependent upon the chordwise and span-loading on the blade shape is to require both a wise load (or pitch) distributions, number oflarger camber and a larger ideal angle than would blades, and blade area and shape. Similarly, abe re uired its two-dimensional flow in order to thickness correction factor can be derived whichproduce the same lift. The principal effect of is independent of the magnitude of the thicknessthickness is to distort the flow such that an in- but is, of course, dependent upon the chordwisecrease in angle of the blade is required to maintain and spanwise thickness distributions.the desired loading. Likewise, the principal Since ",ting-surface correction factors can beeffect of skew is to necessitate a blade angle change derived which are independent of the magnitudebut to require little change in camber. of the propeller loading and thickness, a system-

Nomenclature

A R - blade expanded area r = radial cordinate nondimensionahized by,* propeller disk area propeller radius

BTF = blade-thickness fract.on rh - dimensinless hub radiusC(r) - coefficient of blade outline (r, 9, z) = dimenionless cylindrical coordinates

coefficient 23rG(r) VD T = propeller thrustC Vf I (r, X) = half-thickness ordinate

Cma - two-dimensional maxinum mean-line or- t,,(r) = maximum-thickness ordinatedinate for CL = 1.0 U(r) = resultant induced velocity from lifting-line

PD theoryCp -. power coefficient, P U.0 = velocity induced by loading normal to blade-

;;A0 section chord2 T U.#r) = velocity induced by thickne-ss normal to

On, - thrust coefficient, blade-section chordP UT(r) = tangential induced velocity; V = ship speed

c - section chord VA(r) = speed of advance at a given radiusD = propeller diameter V,(P) = induced velocity vectorf - two-dimensional camber ordinate V,(r) = resultant section inflow velocity

-f, .2 , maximum two-dimensional camber ordinate w(r) = circumferential mean-wake coefficient at afA(r, x..) - maximum camber ordinate given radius

Jr(r, x) = chordwise camber oidinate (x, y, z) = dimensionless cartesian coordinatesft,(r. x,) - camber ordinate induced by thickness V, = chordwise section abscissa nondimensional-j(r, Mr) = chardwise camber normalized by maximum ized by chord

camber .Y,.. - chordwise position of maximum camberG(r) = dimensionless circulation from lifting-line Z = number of blades

theory, I'(r)/TDV a,(r) = section ideal angle of attackG,(r, 9) - dimensionless circulation over lifting sur- a, = two-dimensional ideal angle of attack for

face, r(r, 9)/I)D 17 CL = H)G1 (r) - dimensionless circulation of helical free at(r) = angle-of-attack correction from thickness

vortices $,(r) = hydrodynamic pitch angleGf(r, 0) - dimensionless circulation of helical free A, - induced advance coefficient - r tan 0,

vortices over lifting surfaces VkM(r) = camber correction = apparent advance coefficient,kt(r) - angle-of-attack correction for thickness a(r, ) = source strengthk.(r) - ideal angle-of-attack correction O(r) = angular position of blade-section leading

n - revolutions per unit time edgePI) = propeller power 0. = ikew at blade tip, degP. - hydrodynamic pitch 01(r) = angular position of blade-section trailingR - propeller radius edge

2 Propeller Lifting-Surface Corrections

atic series of these correction factors can be The following sections of the paper will reviewcalculated for design use. By varying such briefly the theory used for the calculations and)arameters as blade number, blade area, blade outline the assumptions and limitations. Results

skew. and hydrodynamic pitch angle, the effect of the calculations will be discussed. Some corn-of these parameters for a practical blade shape parisons between experiment and theory willcan be ascertained and a better understanding of be presented and conclusions will be drawn fromthe flow over the blade will result. These cor- the apparent trends as to the applicability of therection factors can be utilized in propeller design theory.to obtain the blade pitch and camber withoutempirical adjustments. Design methods--devel-oped over the years-based on empirical correc- Theoretical Backgroundtion factors and on correction factors from simpli-fied mathematical models should be regarded as Statement of the Problemoutnoded.5 In addition to providing the de- The problem can be stated as follows: "For asigner with coefficients with which he can design given number, loading, area, skew, chord distri-propellers by modern theoretical methods, these bution, and thickness distribution, determinecorrection factors can be used for making qualita- the required blade-section camber and idealtive checks of lifting-surface computer calcula- angle." This problem is similar to the inversetions, provided that the prepeller geometry and problem of airfoil theory, except that the thick-loading do not differ radically front the series. ness is specified.

Lifting-surface correction factors are presentedfor four, five, and six blades, for blade area ratios Assumptionsfrom 0.35 to 1.15, for hydrodynamic pitch ratiosfrom 0.4 to 2.0, and for three blade skews. The The brief theoretical background presentedcalculations are for optimum" free-running pro- here will serve only to bring out the salient fea-pellers and the results are similar to those pre- tures of the procedures used. Details of thesented by Lerbs [101 but with two essential dif- theories involved are available in referencesferences. One is that the assumed chordwise [121 and [211.load distribution is that of the NACA a = 0.8 In the mathematical model for loading, amean line: i.e., constant loading from the leading distribution of bound vortices is assumed to coveredge to 0.8 chord and then a constant slope to the blades, and free vortices are shed from thesezero at the trailing edge. The second difference bound vortices downstream along helical paths.is the inclusion of a correction for thickness. For thickness, a network of sources and sinks isExperimental results have shown that these assumed to be distributed over the blades.differences are significant in applying lifting- The following assumptions based on this mathe-surface theory to practical propeller designs. matical model are generally made:The thickness correction must be included if the I The fluid is inviscid and incompressible.propeller is to have sufficient pitch, and the NACA 2 The free-stream velocity is axisymmetricit = O.S mean line is a mean line which achieves, and steady, allowing the propeller to be wake-al)proximately, its theoretical lift in viscous flow. adapted.For example, the NACA a = 1.0 mean line 3 Each propeller blade is replaced by a dis-achieves only 74 percent of its theoretical lift tribution of bound vortices for loading effects,in a real fluid, and it is not feasible in the present and sources and sinks for thickness effects. Thelifting-surface theory to account for viscous circulation is distributed in both the chordwiseeffects on lift. In analogy with two-dimensional and spanwise directions. It follows front vortexresults, tihe NACA a = 0.8 load distribution theory that free vortices are shed from the boundrequires an ideal angle for the mean tint to operate vortices andl, in a coordinate system which ro-at shock-free entry. rates with the propeller, these free vortices form

a general helical surface behind the propeller.4 Each of the free vortices has a constant

Reference 1221 is an example of an outmoded niethod.In this particular reference the camber correction factors k, diameter and a constant pitch in the downstreamand k2 are replaced by the caniber correction factor k,. direction, but the pitch may vary in the radialand the pitch correction procedure is replaced by the ideal direction. This means that effects of slipstreamangle due to loading and thickness.

I The word optimiumi used here means the propeller contraction and centrifugal force on the shape ofhas a constant hydrodynamic pitch. Strictly speaking, the free-vortex sheets are ignored.such a propeller would be optimum only when lightly S The boundary conditions on the blade areloaded, when operating in an inviscid fluid, and if it had aninfinite number of blades. linearized, which implies that the lifting surface

Propeller Lifting-Surface Corrections 3

2C2ai

p I

¥-- ( -U(

pitch. This assumption is similar to the linear- Fig. 2 Coordinate system

ized theory of two-dimensional airfoils where theboundary condition is not satisfied on the profile sultant inflow velocity to the blade section, Ubut on the profile chord. Also, the linearization is the resultant induced %elocitv from lifting-lineenables separation of the loading and thickness theory, and U,, is the induced velocity normaleffects. to the blade-section chord. The angle , shown

6 The pitch of the blade and of the trailing in Fig. I is the hydrodynamic pitch angle.vortex sheets is the hydrodynamic pitch obtained Both a car,.-zian coordinate system (x, y, z) andfrom lifting-line theory. Within the context a cylindrical system (r, 0, z) will be used (seeof the linearized boundary conditions, this is not Fig. 2). The bound circulation will be assumedan assumption--except for skewed propellers- to have the strength G,(r) such thatsince the lifting-line theory does not account for

se.el) G,(r, O)dO = G(r) (2)7 The hub is assumed to be small enough J06()

that it is not necessary to satisfy the hub boundarycondition. The coordinates 01(r) and 0,(r) define the angu-

8 Blade rake is not considered. lar position of the blade-section leading and trail-The assumptions listed apply to moderately ing edges, respectively, and G(r) is the non-

loaded propeller theory. Assumption 4 would dimensionalized bound circulation as determinedhave to be removed for heavily loaded propellers, from lifting-line theory [231. Since G,(r, 0)and an additional assumption would have to be may vary in both the radial and chordwisemade for lightly loaded propellers; namely, that directions, a free vortex may be shed at eachthe effect of the induced velocities on the pitch of point on the blade surface and will trail behindthe vortex sheets is negligible, along a helix of pitch 7rr tan #, (tan 01 from lifting-

line theory).Loading It follows from vortex theory that the strengths

of the helical free vortices are equal to the negativeFig. I shows the velocity diagram for a pro- of the radial rate of change of the bound circula-

peller blade section in the absence of thickness. tion at the point the vortex is shed. If theFrom this figure and within the concepts of linear- strength of each helical free vortex is assumed toized theory, the boundary condition at each sec- be Gf(r) behind the trailing edge, thention is

aEa + G(r) = - r dr (3)

af ,() + (r , x ,) - (r) ( ) d r(o () x, =1T,1r) l ,

With this equation and equation (2), the strengthswheref, is the camber along the chord x,, a, is the of the free vortices shed from the blade are foundideal angle of attack (a, -- tanai), V, is the re- to be

4 Propeller Lifing-Surfcce Correcfions

p

d I) (r, 0)d~dr f -Gr) (S -) dr( )GAO) = ~r -)1d At," )liltin - .

S G,(r.) ddr G,(r. el) is identical to the velocity induced at a lifting0#(,) Or line by the trailing vortex system. Tile velocity

d +, dO ( induced by the lifting surface itself isdr d+ G,(r, ,) - dr (4)dv,(P,) _ ff I;,r ) X r),

The first term on the right corresponds to the i7 - 2 - J " 0) -- "-integral of the n~ee vortex strengths from the lead- 2;, ,(

ing edge to the trailing edge, which results from a + f f , i,,(r, 0)radial change in the bound vortices. The two "ifting line ".

remaining terms on the right are strengths of the f1( ,G rO ,t Sx dr

free vortices shed along the blade outline. It + ()

follows from this equation that, within the lifting _ d r S X d')surface, the free vortices have strength JJad (r) - )dr (7)

G,(r. 0) = -G,(r, 0) - dr This equation gives the velocity induced by a

ha d single blade at the point P. The total velocity, (r, )dOodr, 0, > 0 > Ot (5) induced by all the blades at the point P on one

0o r of the blades is obtained by sunmming this equa-tion with respect to the blade position in, of which

From the Biot-Savart law, the induced velocity S is a function. The velocity V,(P) includesV, at any point P on the lifting surface is found to the radial induced velocity as well as the axialbe and tangential velocities. The radial velocity

iound Vo.rtices does not appear it, the boundary condition,Y, _(P) 1 __, l0)f (S_Xd dO equation (1), and can be neglected. Then, the

i- - JJ, -3 nornmal velocity isfree vortices V _ ,f,, .o~¢,,V, V, U

G ~ , O S X l (11 n . u ,, . V ,

and the equation which must be evaluated inwhere A, is the area of the lifting surface, :1A2 is eetermining vflx, is

the area of the helical surface behind the trailingedge of the blade, I' is the freestream velocity / ( f L,(r, .,) U~r)or ship speed. S is the vector distance fr,'i a , + . . ...point Q on the helical surface to the point P () V; rIalso on the helical surface, d/ is the elementary I n- , G,(,.O) S X dvector tangent to the vortex line, and S = ISI. 2 I.- A. S3 )

The second term in this equation can be expressed Z, + ""' (r. Q,,, 0(rO) (r,0"in two parts by equations (3) and (5), and the +2 o , . Orquantity -r. JA f

dl 3 d(fr) (SXdl) d X S X-d) dr _ ff (1((r) (S X dl)12 ff , drk SSrA r S(8)

is now added to and subtracted from this equation;.13 is the area between the trailing edge and a where Z is the number of blades. The bladegenerating line along which a lifting line would be camber is given byplaced. A new coordinate system is introduced f r, 0) _(r)

such that the point P always lies on the lifting fp(r, x,) + xa(r) I I=- 1. diesurface' 16 1. This ineans that the integral 0)CL'

I For this change in coordinate system, the lifting lineiray deviate from the blade surface at radial pmints other To determine at, this integral is evaluated fromthan P when the propeller is not of -onstant pitch. The the leading edge to the trailing edge and f' iseffect on the results is probably small if the pitch does notdeviate too much from a constant value, taken to be zero.


Trhe equation for the slope of the camber line, = _ 1equation (.)), avoids integration limits of in- k.(r) = aI.CG

finity, but the double integrals are difficult to fotrU ((r)levaluate because the integrands are singular X - (r, x) - - dx (13)when S becoms zero. To facilitate the numerical f [ I'l V, 1evaluation of these singular integrals, Pien 16 where at,.. is the two-dimensional ideal angle ofand Cheng 1121 made a number of simplifications: attack for C, = 1.0 for the NACA a = 0.8 mean

I The radial bound circulation distribution line.G(r) and the hydrodynamic pitch ratio distribu-tion are expanded in a half-range cosine series Thicknesswith a finite number of terms. Blade thickness effects are determined by

2 The singular points of the integrands are introducing a source-sink system distributed overisolated and the integrals are evaluated numeri- the lifting surface [211. The induced velocitycally. In the region of the singular point, the field is given byradial integration is done analytically by assumingthat pitch and circulation are constant. V(P) 0)

1 For calculating the blade mean line, the I V tlkn,* -rSPS G(r)

induced normal velocity is expressed as a powerseries. X H,(P, r0, 0.)j0 dOdr (14)

The sinmplifications are numerical approxima- &

tions and not assumptions made in the theory. where H, is the velocity induced at the point PThe precise accuracy of the evaluations is difficult by a unit source located at point r, 0; i.e.to ascertain. Z

Correction factors for camber and ideal angle H, F, gradclue to loading which are independent of the M-1 \ T47Smagnitude of propeller loading can be derived by and where dx, is the element of blade chord, andnormalizing the three-dimensional camber and a(r, 0) is the source strength. It is assumed thatideal angle by the two-dimensional section values, the source strength distribution is known and isWith regard to camber, only the correction factors that derived by the usual linear approximationsfor the naximun chordwise caniber will be formed from airfoil theory; i.e.in this way. Front equation (9), then

f~~~(r,~1 o (rO) ,t(ro, V,)k ,(r) = f .. = .. . rV x ,

" r'x) U( -x a(r) X, V cos 1, ox,x [ L, ) dx J -

(10) where l(ro, a,) is the thickness distributed alongthe chord x,, Ur is the tangential induced velocity,

where x,,,,, is the chordwise position of maximun and X, is the apparent advance ratio of thecamber, Cl, is the blade-section lift coefficient, propeller; i.e., V/TrnD.and f,., is the two-dimensional maximum camber As for the effect of loading, only the axial andand equal to tangential induced velocity components are

required from equation (14). If the change inf, .a. = Cn....CL (i1) camber along the chord x, clue to thickness is

fp,(r, x,), and the ideal angle induced by thicknessThe function C,. is the maximumn mean-line is a,(r), the linearized boundary condition on theordinate for CL = 1.0 in two-dimensional flow, blade isFor other chordwise positions, the three-dimen-sional canmbers will be normalized by the niaxi- a,(r) + -)-"- (-,(r, x,) (6)

fum camxber; i.e., ax, 1,

r, .v) r(12) where U., is the induced velocity normal to thef(r, X111(1 section chord from equation (14) and whereat(r) = tal a(r). Equation (16) is similar to

Correction factors for the ideal angle due to equation (1), and the change in curvature clueloading call be formed in the same way as the to thickness is found by integrating equationcamber correction factor. (16) in the nanner of equation (9).


To facilitate evaluating the integrals of equa- cited in Fortran statements 201 to 204 of Sub-tion (11), Kerwin and Leopold [211 used a dis- routine SUB3 should rtad 0.9999 and 0.0001tribution of quadrilateral source elements over the respectively, rather than 0.99 and 0.01. Thisblades. This approach is similar to that of Hess oversight results in considerable differences inand Smith 1211 for potential flow about arbitrary calculated camber values when compared tobo'lies, except that the elements are placed along results of the former program [11] for constantthe chord rather than over the surface. There- chordwise loading cases.fore, the source-sink distribution is not continuous 2 Initial input to the Cheng program is thebut is made up of discrete elements. If a sufli- designation of a fan angle and grid spacing whichcient number of elements are used to describe the govern the location and number of control points.sorface, the accuracy should be good. The fan angle input must contain the projected

Calculations have shown that thickness in- view of the propeller blade, and the grid spacingduces an inflow angle and a camber change but in that fan should be as fine as possible for thethat camber alteration is small 1211 except for very greatest accuracy in flow calculation at the con-small pitch ratios. This flow distortion requires trol points. The total number of chordwisean increase in blade angle (ideal angle) to main- stations on either side of the control point, how-tain the desired loading. Correction factors ever, cannot exceed 90. Ia order to meet thesefor this ideal angle can be made independent of restrictions, the angle and grid spacing should bethe magnitude of thickness by dividing the angle carefully checked before each propeller calcula-(in radians) by the blade thickness fraction for tion. and particular care should be taken forwhich the calculations were made; e.g. large blade areas and highly skewed propellers.

U x1v No automatic checks exist in the program tok,(r) f - (r, (17) assume this task.

11 J 11r 3 Most critical of the input parameters is the

where BTF is the blade thickness fraction. A specification of chord-wise coordinates at whichcamber correction due to thickness can be formed the camber and induced velocities will be cal-in a sinilar manner, but since the correction is culated. Considerable differences in calculatedsmall it will be ignored. It should be noted from inducedl velocities appear for relatively small

equation (17) that k,(r) is also a function of the abscissa changes along the chord near the leading

propeller loading since V, is a function of the and trailing edges. The inclusion of these changesloading. This effect is small, however, and no in the resulting integrals yield rather small per-correction will be made in the results which are cent changes in cambei but relatively large per-

presented. cent changes in ideal angle. The series dis-cussed in this paper was calculated with chord-

Problems in Numerical Evaluation wise coordinates specified from .5.0 to 95.0-percentchord. Questionable ideal angle and camber

Because of their complexity, the expressions corrections arose occasionally for small blade areasarising in the lifting surface calculation procedure and/or Ioi% pitch ratios. A change of chordwisedescribed in this paper were necessarily evaluated coordinates to a range from 2.0 to 8 0 percentby digital computation. Thi, resulted in re- chord improved upon the erratic results in thesepeated numerical operations to obtain discrete regions, but, for consistency in the series data,values of the functions involved and the occur- these calculated values are not presented. Itrence of inaccuracies, % hich were dependent upon should be noted also that the questionable datathe characteristic,, of the functions themselves were obvious only in comparisons to other dataand were not always predictable by the program in the series, alone they perhaps would haveuser. gone unnoticed. Independent propeller calcula-

The Cheng comluter program, described in tions should be checked against the series pre-reference 1121, offers many input and output data sented in this paper or other similar designs ifoptions which are helpful in describing ipropellers possible.with special geometries and loadings. This 4 Lifting-surface correction data are givenflexibility, however, can lead to inaccuracies in herein only at radial positions r/R = 0.3 throughthe calculations. Points which merit special r/R = 0.9 for 0.2R hub propellers. The lifting-attention in both the operation of the program surface calcul;iions were made also at r/R = 0 25and the analysis of the results are: and r/R = 0.95. However, the result- were

I The comluter program, as listed in Appen- questionable which is probably due to a com-dix 13 of reference [121, must be modified slightly bination of numerical difficulties involving theto give correct results. The convergence criteria singularities at the blade hub and tip.


- - - Since the calculations are based on similar pro-: ', ] cedures, the reason for the difference bef ween the

-- - results of Lerbs and Cheng is not known. Thedifference toward the blade tip is probably causedby different numerical techniques, and the differ-ence toward the root is mainly due to the lifting-

- t' line calculations; i.e., induction factors versus

- -i Goldstein factors.For a further comparison of lifting-surface

calculations, the camber correction factor for a

chordwise load distribution corresponding to a

NACA a = 0.8 mean line is also plotted in Fig. 3.- - It is interesting to noui that the camber correction

factor for this mean line is everywhere smaller

- than the corresponding factor for the constantload distribution. The three-dimensional camberis higher, however, since C,, is approximately

Fig. 3 Comparison of camber correction factors for -3 percent larger for the a = 0.8 than for the athree blades, i, -1.0472, and Ax/Ao - 0.75 .0 pen lrer1.0 meanl line.

Lifting-surface corrections were calculated for

propellers with four. five, and six blades, withPr$*tefkm of "9 Saw$ expanded area ratios from 0.35 to 1.15, with

hydrodynamic pitch ratios of 0.4 to 2.0. and withProcef-rn a symmetrical outline and skew angles of 7,

The propellers for the series were free-running 14, and 21 deg. The skew angle 0e is defined as the

propellers of constant hydrodynamic pitch with angle between two straight lines in the projected

a hub diameter of 0.2 of the propeller diameter. plane--one from the shaft centerline throughLifting-line calculations were from a computer the midchord at the root section and the otherprogram based on the induction-factor methodof Lerbs [251. With loading and pitch distribu-tions available from the lifting-line calculations, Table I Ordinates for NACA 66 (Mod) Thicknesslifting-surface calculations for loading were made Distribution and NACA a = 0.8 Camber Distribution

using the program developed by Cheng [12] and Thickness Camberfor thickness using the program developed by Station, Ordinate, Ordinate,

Kerwin and Leopold [21]. These have been x, 0. f/ffa

combined into one program at NSRDC for run- 0 0 0

ning on the IBM-7090 computer. 0.005 0.0665 0 043

Because of the complicated nature of the numer- 0.0075 0.0812 0.0595

ical procedures, it is difficult to state the accuracy 0.025 0.1466 0.1580

of the results. To this end, calculations were 0.05 0.2066 0.2712

made and compared in Fig. 3 to those of Cox 0.075 0.2525 0.3657

[5], Kerwin [13], and Lerbs et al. [10] for a 0.1 0.2907 0.4482

three-bladed propeller with an induced advance 0.15 0.3521 0.58690.2 0.4000 0.6993

coefficient s ,f of 0.3333, expanded area ratio of 0.25 0.4363 0.7905

0.75, and a constant chordwise load distribution. 0.3 0.4637 0.8635

The camber correction factors, k(r), calculated 0.35 0.4832 0.9202

using results of the Cheng method are within 1 0.4 0.4952 0.96150.45 0.5 0 9881

percent of those derived from the Lerbs method 0.5 0.4962 1 0for radii between 0.4 and 0.8 but deviate consider- 0.55 0 4846 0.9971

ably at the 0.3 and 0.9 radii. Both the Cox and 0.6 0.4653 0.9786

the Kerwin nethods give correction factors which 0.65 0.4383 0.9434

are, in general, higher than those obtained by the 0.75 0.36125 0.8121

Lerbs and the Cheng methods. These compari- 0.8 0.3110 0.7027

sons are consistent with the way Cox and Kerwin 0.85 0.2532 0 5425

arrive at their camber correction factors [10]. 0.9 0.1877 0.3580.95 0.1143 0.1713

The induced advance coefficient X, is related to the 0.975 0.748 0.0823

hydrodynamic pitch ratio by r, - r tan#, = (P/ID),. 1.0 0.0333 0


Table 2 Distribution of Blade Chord LMRAIOLoI9

r C(r) StE

0,2 16ow 8-N0.3 1 80820 4 1 9648 NSK0 5 2. oI,67

('7 23208 2 1719

0,9 1 hV31095 1 ,62

1E\

Table 3 Skew Distribution for 14 it Skew and U E.

irX, of 1.2Skew R

0 2 003 0 00370 4 0 01480. 5 0 03360.6 0 0604 Fig. 4 Expanded blade outlines, Ar/Ao 0.75, Z - 5,0 7 0.0957 with symmetrical blade outline and 14-deg skew0.8 0 14020.9 0 194911) 0,2616 half-thickness ordinates as normialized by the

maximum thickness, Howver, the lifting-sur-Table 4 Calculated Propellers with Symmetrical face correction for thickness would be expected

Blade Outline for Four, Five, and Six Blades to be essentially independent of the chordwisethickness distribution, assuming, of course, that

7xi 0.4 0.8 1.2 1.6 2.0 the shapes are reasonable.As 0.35 0.35 0.35 0.35 0.35 The spanwise distribution of thickness was

5 0. i 0.55 0.55 assumed to be linear with respect to radius and0.75 0.75 0.75 0.75 0.75 was given by the following equation:0 95 0.95 0.95 0 95 0.951.15 1 15 1. 15 1.1.5 15 15a

= (BTF - 0.003)(1 - r) + 0.003 (18)D

fromn the shaft centerline through the inidchord where BTF is the blade thickness fraction. I jisat the blade tip. choice was based on an examination of a numberof propellers of current design.

The blade outlines chosen were slightly widertoward the tip than the Troost outline and were

Choice of Parameters given byGeometric properties of the propeller series were

chosen to be consistent with present-day practice. c/D = ( [ ' _(19)All the propellers were designed to have the same Z ( .,I(chordwise load distribution as that of the NACAa = 0.8 mean line, i.e., a constant chordwise load where C(r) is given in Table 2 and (.I E/Ao) is thefrom the blade leading edge to 0.8 of the chord, expanded area ratio. This is a nathematicaland a constant slope to zero at the trailing edge. outline given by Cox [51 with his constantsThis nean line was chosen because viscous effects o = 0.732 and I = 0.7, and is essentially theon its lift are small [261, and hence the potential same outline as given by Schoenherr [28] with hissolution closely approximates the true physical constants c = 0.A and n = 0.3. An expandedflow. The naxiniun ordinate of this mean line blade for al .1 :/.10 of 0.75 and five blades isfor a CL = 1.0, C,,.,, is 0.06790, and the two- shown in Fig. 4.dimensional ideal angle of attack for CL = 1.0, The radial distribution of skew was chosen soac,,, is 1.54 (leg 1261. The chordwise thickness that the blade-section niid-chord line followed adistribution chosen was the NACA-66 section, circular arc in the expanded plane, as shown inTMB modification [271, which has a desirable Figure 4, and was calculated fron the followingpressure distribution. Table I indicates the equation:


P? %

-T- -1 -

S .," .".n .A... - . .....~f... t ....... - ........V~

, N ...... ~ J. _.. A -0 t1 A ... . . -- , ,.~ ~e ,

I V.

.~- -A *iW~ -I-.

N, ---- -------'

. . . . . . . . ...... . .;. . . . . .. Z :- ' ,_.__ .

A. , i J ",- i-'.. .. ..... - ....... snN N

ske R,' - 0' /f', - (r -02)% a.qn .--is the ske anl . As n a le, ske

"-I a.=O in the exp nded pln as lcua ted rom th-

A , - , .~flnt0 - N 4'& nu 0 4,

=:ttl 06 --

K~ ;z

10Prpele . CrrctionsNJ ~ ~ ~ - z' -' tc'N 2gfCU.4

1- . . .. .. 1. 1,09~ i- IT!~2 T!I

i'~ ~ i'z I ~ ' N . 9199 .-- ,:( 'f1N A-0

. . .- . .. - - - -

E2 1 :

R-'R. -- 0.2)-i h x anddpane ascluae frm ti

R, +..A.

~,ThelY pvAropelle&rs * whc weec44alclte orti

ske ti sytematic seris Q C ar litdi ale4,ofe

coo to fatr fo camber ideal anl du olaig

10Pople Lifin-Sufc Coretin

- -myiw -q......... ..*, ........ r..9.

A- 7 - r [ --- ---_

.. . r-. I w*i' ..

. - , ....

0 - --------, - - ---- .. -- ,4

" - " -i- I. i"":' ;; , --- * ,--,

r..,"... ",.:.0...... . .......... .. h.....

Z ----- .. . . ..

.. . .........V ....

- -.-- ----- ---- -----

- - -T%4 -:

o~~2 2, '2 - , -*

anld ideal angle (lie to thicknessas determlined by radius are ,shownl in Figs. 5 throughl 1:3 for a

equations (10), {l13), and (17). respectively, are representative part of the data. Figs. 5,.6, and 7listed in TFable 5 for these propellers. These show the correction factors for a five-bladeddata, in general, are direct computer outputs ,and propeller ,xith a 7r, of 1.2 for the range of ex-

are not cross-faired. The data should not be panded area ratios investigated. Figs. , 9, andregarded as more accurate than to the s econd 10 show the correction factors for a five-bladeddecimal place even though three places are given. propeller wVith .1E/.1 = 0.75 for the range ofPlots of the correction factors versus the propeller induced advance coefficients investigated. Fi-

Propeller Lifting-Surface Correctiors l1l

.:p.~ ~~ .. .. ... . .... , Tn".!*::• . ... . ....--- - ..

- - - --- ,

III

- -- - -

.- OO..N f......... C **.l ___ _. -* *

.11

-= M

- - - - - - - - -

I '!,. ........ .. • N

-. ---- - - -

j-7 ..... ----- - -------

S...NA' , .. e4ACN. ,.. €; € C € 6 "3. oo o . . .

.'... '' .: O.A N."$ , .. r '~~ ,,, ,*O'~ . O•O*N

J. . ... ;;

nally, Figs. 11, 12, and 13 show the correction with the various parameters is small. The datafactors for rX, = 1.2 and A/Aa for four, five, given in this table are diret computer output and,and six blades. obviously, there are discrepancies at some radii.

In addition to these results, the chordwise These radii are marked in the table. Thedistributions of camber, as normalized by the questionable data have been carefully checkedniaximnim camber, equation (12), are listed in and no apparent error could be found. TheTable 6. Only the normalized coordinates for discrepancies are probably linked to the difficultypart of the series are presented since the variation encountered in the calculations for low pitch


iI

! -

II

, • . . .. . ...... - ...... f.. . .......

..... ... .. Vi ... .. ... :': 't'; .'.1"

.. .. . .... .. . .... ... fl..h ... 0 . ................

1c, , - r ,,,,. ,.,...0 ,,,,, ,,,,,,, 0o ~ ~ ~ 0 0rtIO N Ot o"fl ot-

o - ' 54" C N O f t( 1 [0t - V.. I.' O t'.lt,flO ', 0 fl ,0 'Vflfle,,t -. 07,)'

-- ~ .. (0(~0n.(tC*r 0 o~ .. toN o .....3wnc'....::-8

... .... ......... . ......... ..... .. ..... - .......

',o .... t . t .... r................. r ................

Propeller~ .'iftiSuface Corrctitons 13-t,..0.~.tx

* ) .. 00 -~-, O...t.13 .01 -t )-0. O,70 .'II01 OEIf~0

0

...... g~~... 1. z S .. 0 U I sf.51 - 0 lr ot rlh.N o

* -0V~ Ol flVC- 0 r,-qtO~I.;N-3n.0 oN -Mtmt0,0.

CO, WOfl 1 0.0 U IC. . .,. .oe . 11cr,.~ - -0I0 r - a.... ..... .. O~t&I f.. . ...-qVt O A0 w f..O0 . 14 M N O O t .100

I- 0.0.utS~0Vl0,10I~t0 ~ ~ ~*0C0501.IS0tS 1Cd-

010.rop lle Li)( O.I 0 ftg-Sur0t~face V VI) Coretin 134N ~ ae0V .

.. .........

*;O - f ,

.......... %O .. e.-V

~,.qfl VI ~.b~lt.p~ ~ ... . .~ N..~... . . .C .. ~ .I. ...

...........

x s On N r o9 O l..q c .... 0

Nm" 10tW --- I, 1 .1 .4 O.Jn.00.1 co 0 - . Cffl.. 1 1.

.n....... ii9 9 .91 .1 O~l~,~%n, f. .9 OT~t. , ,......

~ OO~ny)N O ............l

U-. 0-a~ta - 0.1.w0 1 o- on~~a- OC ,a qd~f~. snO .....qo O w. .... Q - ~ 3 oa . . o

- - f........... ~ AO

0 qi,. V?& t . 0..~ 9 0?4f2 -00 ............. t...0... .?--

Ofl.,fl 0, fl ~ O Or~tA.0...,..... C) 9 l . t...fl.., ....

N 0 0vq.eO~a.-.-.,o ,. Et.C %~~.0CtIJ0.)3*9 .... O.... .. .

AS

14~9 g9 C Propeller9 Liftng-Srf00 C flrfl.%

* ~eO400A 0 "O

*n :o -.

00 0- CO0C.o.a

~ ................

C; nr0400* 40f 000

E 4

Waa,- 00 m n.. ... . ...-

1 - o - ..V ~ ~ o~ r~ r~ o , '. .. . ... .. c . .

On=o..Gn - 2 0 0000'fl0= =t-..cc" .a -.

VO~'tC0Of.0 0. C ..........O~t .. C, ~ .c.. coo.-

*0

O~ t O .C. )O flC q~ - . .. t....... ... .. ;? .- ,

.' - tw w.n .o .1 ,11. 0- C...o........o. ).0AQ OIdC4

C; .... 0..... I . ...C.ro. . .. no c-w-,o n0 qs. ,

..... IiC.flfo..S00

in An,..~ 11-% i li

On In ......... bI0C-I.10V~aO0a C ~ ~ 0 V 0C.0 ~ -w.rr,,co.

PrpleItln.uraeCretos1

oS o, Os 06 07 ii o Fig. 8 Camber correction factor, h (r). for five bladesand A/A A 0.75

Fig. S Camber correction factor for five blades and- 1.2 SCI1 II III.... II1IE

[ -r -03- , °' 04 z5 0 07 of 09

*1 04 05 0 07 1 Of Fig. 9 Cam orrection factor k(j for idala iled e oladns

for five blades and A/Ao = 0.75

' '--""wise data should not be regarded as more accurate' " -- - than to the second decimal place.

I 0: 04 t 06 0, 0 0, A comparison of the chordwise dlistribution ofFig. 6 Correction factor for ideal angle due to loading camber with the two-dimensional value from

for five blades and xX = 1.2 reference [261 is shown in Table 7for a typical.:propeller. The three-dimensional wd~oues are flat-

ter near the leading edge and fuller toward the- - .-- - trailing edge of the blade, and~ the dlifferencesF,[ I decrease slightly toward the blade tip for this

- ,1- -,I propeller. As can be seen from Table 6, there is a- - r,-., tendency for the coordinates to become flatter

toward the leading edge with increasing induced- _ _ a - - - -fridvance coefficient and with increasing expanded

,$ area ratio. Also, the ordinates become slightly

0 35fuller toward the trailing edge for increasing in-

- - -- - - duced advance coefficient, but show little change0 - - - - - with area ratio. These data show only slight

0 006variation in the spanwise direction, however,

-'" --- and these dlifferences are essentially within theo , accuracy of the calculations. This leads to theo o. o o7 o, interesting supposition that the nornmlized cam:-

Fig. 7 Correction factor for ideal angle due to thickness ber ordinates are almost independent of theirfor five blades, r) , = 1.2 and BTF = 0.4 spanwise position.

.16 Propeller lifting-Surface Corrections

. Z-5 AE A*. CI 12 ci,'s ,

Fig. 10 Correction factor for ideal angle due to thick-ness for five blades and Az/Ac© - 0.75

Fig. 12 Correction factor for ideal angle due to loadingfor~r? = 1.2 and AE/,9 - 0.75

o , 4 5 _ I ...... .. -

t-0~

Fig. 11 Camber correction factor for 2r,, -= 1.2 andA.A 0 0.75 Fig. 13 Correction factor for ideal angle due to thick-

ness for rX = 1.2 and ArI/A = 0.75

It is quite apparent from Table 5 and Figs..5, 6.and 7 that the effect of expanded area ratio idea! angle due to loading increase x ith increasingdloninates the correction factors. The correc- induced advance coefficient, except possibly neartion factors all increase with expanded area ratio the hub and tip of the blades, but decrease forfor a given number of blades and pitch ratio. increasing number of blades, i.e., for the rangeIn general, the correction factors for camber and investigated. The factors for camber and ideal

Table 7 Comparison of Cbordwise Distributions of Camber for Five Blades,7rX = 1.2, and A/Ao = 0.75

Clnord Distribution Distribution DistributionlPosition, 2-Dimentsional for for for

x, Distribution r = 0.3 r = 0.6 r = 0.9

0 0 0 0 00.025 0 159 0 121 0. 132 0. 1330.05 0 271 0 230 0.243 0 243

0.1 0).448 0 410 0.424 0.4310.2 0 699 O.675 0 687 0.698

0 3 0i 864 0 8.56 0.860'X 0 8710.4 0 962 0.961 0.963 0 9,650F5 1 (X) 1 (XX) 1 00) 1.000

0 6 0 9179 0 9)77 0 979 !) 9730.7 0 889 0 882 0.889 0.8780 8 0 703 0 682 0.69 0.682

0 9 0. 359 0).364 0 366 0. 3650.95 0 171 0 177 0 177 0i.176)0.975 0 082 0 085 0.086 0 082

1t0 0 0 0 0

Propeller iftin.Sur face Corrections 17

~~ ,~ \ N ~TRMOT- -/I I I-COO/ SERIES VC,

-I _ ROOSTS I I

/ 03 04 OS 04 07 08 09

Fig. 1 5 Comparison of camber correction factors forthree blade outlines of Fig. 14

Fig. 14 Comparison of three blade outlines for five. SE-- Ro -' ' IES ''bladed propeller with expanded area ratio of 0.75 T*I

angle are greater than unity, except for the higher -numbers of blades at low induced advance co- _0eflicients. This means that the three-dimensionalvalues are, in general, greater than the two- Fig. 16 Comparison of correction factors for idealdimensional values. In most cases, the thickness angle due to loading for three blade outlines of Fig. 14induces an ideal angle which is largest near theblade root and decreases to negligible values to- _

ward the blade tip. In general, this angle de-creases with an increasing induced advance .S--.. H--

coefficient but increases with increasing numberof blades. IROOST

To show the effect of blade outline, spanwise Coxloading, and spanwise thickness distribution onthe correction factors, propellers were calculatedin addition to those shown in Table 4. All Oo0 A 05 0 0 7 01 09

calculations were for a five-bladed lropeller withan induced advance coefficient 7rX , of 1.0 and a Fig. 17 Comparison of correction factors for idealblade area ratio of 0.75. Two blade outlines, in angle due to thickness for three blade outlines of Fig. 14addition to the one of the series, were investigated:one was the Cox Type 1 and the other was aTroost outline. Fig. 14 compares these outlines, tip to show any such effect. The trends of theand Figs. 15--17 show the correction factors for correction factors with the blade outlines suggestthe two additional outlines and the series outline, that correction factors could be approximatedThe correction factors generally follow the blade fron the series for an arbitrary outline by usingchord distribution, i.e., the blade outline which is an "equivalent" area ratio at each radius. Thenarrowest near the root and widest toward the "equivalent" area ratio is defined as the expandedtip) (Cox Type 1) has the smallest correction blade area ratio for a propeller of the series whichfactors near the root and the largest near the tip. has the same chord at the particular radius inCorrection factors for the Troost outline, which is question as the arbitrary outline. Correctionthe widest near the root and the narrowest toward factors derived fron the series with "equivalent"the tip, show opposite trends. This is true for all outlines for the Cox Type I and Troost bladethree correction factors, except that the ideal shapes were found to be reasonably close to theangles due to thickness are too small near the calculated values, but not in every case were

1 8 Propeller Lifting-Surface Corrections

~ WAEE ADAPTED

SstolE

ESTMIATED

0 _T I

fig. 18 Comparison of ideal angle correction factorsI-for radial variation in thickness 03 04 0S 06 07 ON 09

- -- - - - Fig. 20 Comparison of camber correction factors for- - -- ~ Ithree pitch distributions of Fig. 19

REDUCED LOADINGAT TI SERIES PROPELLER

091, WAXI ADAPTED 20- - REDUCE ID LOAD-h* AT TIP7

- 7-

03 0S 04 Its 06 07 of 09 10-- 03 04 05 06 07 01 09

Fig. 19 Comparison of three pitch distributions for Fig. 21 Comparioofcretnfaosfridlfive-bladed propeller with expanded area ratio of 0.75 anl due to laisng forree chsrbtion s fatr ofa

Fig. 19

they better than the series data without, use of an11equivalent'" blade area. Data presenoted byCox 15 1 show similar trendls.

Calculations were also made using a nonlinear -SERIES PROPELLER

thickness dlistribution. andI the tabulated thick- ~ I iniess values for this variation andI the series are -Shownvi in TIable S. Thei relation between the [jthickness corrections for these two dijstributions is ~ Ishowno in F'i,. IS for a five-bladed propeller with 03 0 A 05 016 C7 a 09

an inducedl advance coefficient ,rX1 of 1.0 andl ab~lade area ratio of (0.75. A curve of the correction Fig. 22 Cornparison of correction factors for idealfactors a-; estinmatedl from the series dtla is also angle due to thickness for three pitch distributions ofshown in this figuire. Thle estimation was madle Fig. 19

by multiplying the series correction factor by aratio of the local thickness givenl in, Table 8. Asinight he expected, the estimation gives a cor- Table 8 Comparison of Radial Distributionsrectioli whoich is too low. However, if the cor- of Thicknessrectioo factors were calculated on the basis of an ~m ~average of the series and the estimating pro.- ew ~ ~ !

cetlure used, the estimated correction factors 0 2 0 (9326 (1 0S3216wouild have been extremely close. 0 3 () 0289 1) 02761

Th'le effects of two additional loading variations 04 0 0252 1) 02339)0 5 0:21115 11 0)1939

were also investigated. 'rhese loading variations 0 (6 M N)7 1) 01572resulted in a dlistribution of p~itch which was not 0 7 0 0141 1) if1229

01 8 0) 0104 II (X89constant. One p~ropeller was a "~ake-adapted0 1)( 0 M74 00.592poropeller and the other had a redlucedh loadinog I 0) 0) 0MM0 1) WPM


Table 9 Cakulated Propellers with Skewr ,,0.40.81.2 - - 2.0-

A, 0.35 0.75 1 15 0.35 0.75 1.15 0.35 0.75 1.1.5 0.35 0.75 1.1.5

7 7 7 7 7 7 7 7 7 70'. (leg 14 14 14 14 14 14 14 14 14 14 14 14

21 21 21 21 21 21 21 21 21 21

at the tip. The resulting pitch distributions are - ---- --

shown in Fig. 19 along with the wake distribution - -

for the wake-adapted propeller. The relationbetween the correction factors for these two loaddistributions, as compared to the series results, - - _-

is shown in Figs. 20, 21, and 22. It is quite ap-parent that the radial load distribution has asignificant effect on the camber and ideal angle Ndue to loading correction factors. These cor- K,- - \-rection factors for the propeller with the reduced -

loading at the tip are largest toward the bladeroot and smallest toward the blade tip. The .2Ccorrection factors for the wake-adapted propellerare close to the series results, probably becausethe radial load distribution is much the same. An .30 a 0 9 0? 01 -iattempt was made to estimate the correctionfactors from Figs. 8, 9, and 10 by enteritg the Fig. 23 Ideal angle correction factor induced by skew,figures at the Xj of each radius for the wake- Z- 5, X,- 1.2, and Ar/Aa - 0.75adapted propeller and that with reduced loadingat the tip. Although the values of the correctionfactors estimated in this manner were, in general, The dominant effect of skew is on the ideal anglecloser to the calculated values, there was not a correction due to loading k.. Fig. 23 shows thesignificant improvement over using the series cor- ideal angle correction factor induced by skew forrection factors directly. The result of the analysis this series versus the propeller radius for Z = 5,shows clearly that the effect of the radial load ,rX = 1.2, and AN/Ao = 0.75. To obtain thisdistribution is important. factor, the ideal angle correction factor for the

symmetrical outline was subtracted from theSries of Propel/ers With Skew correction factor for the skewed outlines. This

The propellers which were calculated with skew figure shows that skew effect is significant andare listed in Table 9. Correction factors for induces a positive angle toward the blade rootcamber, ideal angle due to loading, and ideal and a negative angle toward the blade tip. Thisangle due to thicknes, as determined by equations effect has been found by many others [101.(10), (13), and (17), respectively, are listed in The chordwise distributions of camber, asTable 10 for these propellers. As stated for the normalized by the maximumn camber, equationsymmetrical blade outline, the data are not cross- (12), are listed in Table II. Only data for part offaired and should not be regarded as more accurate the series are presented, as the results are verythan to the second decimal )lace. close to the results obtained for the symmetrical

Tables 5 and 101 show that the camber correc- outline. The trends shown by the data for thetion factor k, is almost independent of skew but skewed outline are the same as for the symmetricalthat it does tend to increase slightly toward the outline.blade tip above the value for the symmetricaloutline for increasing skew. This effect increases Expermental Checks on the Theoryslightly with increasing blade area and with in- Using lifting-surface theory, several propelerscreasing induced advance coefficient. The ideal have been designed andl tested at NSRDC.angle correction due to thickness tends to be None of these has been taken specifically fromslightly smaller than the symmetrical outline the series, but the computer programs used invalue. Otherwise, the relationship beotween this making the lifting-surface calculations were thecorrection factor and skew shows no distinct trend. same as used for the series or were older versions


Table 10 Correction -adors for Skewed Propellers, Z 4

.%-0.4 !AS* 0.1 1. Ao-7 187

4.jAt/A, - 0.36

r 0.3 -.' 2.64 37Is~ A7 8 243 1.7;3 160.4 3 7 10574 316 :151A t 1. 1 U 1.797 3O0 " WS 1' 1.0 USfIi I.3a 1.731, .1203

0 .. '.-. 3.1 .' 3S f3 1.1V4 1.4 . 2

0.8 3:s. 31I .23? .7-~2 1.33 37 111 t134'

*CA 0.7IS

1:.3 36-) ' ~ 7- 3.~ .5 .41, .634 M.?9 .793 2.34 270.4 IF K, .2, 1 3.x 2.304 *35, 3.3 3 ;!.31. .2 :2t t.4 51 2.424 .20.,0.5 3v, 147 3.j3 2.077 z2z ;.34 237 .in 3 ~454 2.315 ISO!0.6 '. :5" 1 37S97 .S1, 3.33, 1.231. .11^ 1.24.4Z6 ? .354 -1 3.11 . .349'

0. .2 7 .7 .7)S .11.2 3.570. 3 .113 .17' 11.6-7 2.02 .3K7i

0. 1. 3 4 .. .4 . 21 j6A 5 1.S5 .9 1 M I4 US3 .415;

* 1.1 2.22 S .74 1 4 17 2347 o.71 .2

0.4 :.7: :10 2.: No37 1397 2 .3 *0:.g 1.' 1' -4~ 1.1-4 .'2 1.35292 .0

0.7 ..'- ; 7 .334 3.7 NI :4; 2.437 2.270 2I39

0. ".ei .. . .23 !. .. 41 .3071 2.9 .005

06g 1.A, 013 l;t 27- 2 -)

r* 0.3 3*~ 2.041 .11 1I3S 2.1$3 .17Z 1.175 2.194 Wts~ 1~.7.4 2.274 .1l60.4 1 1.033 AS1 1.042 2.C&2 .4S I *1W4 Z.203 1l34t a .2 2.=3 :'I2Z0.5 1.3 1.775 m7 134S3 3.9!0 .114' 1.05 2.05 .12r.11 1.3:.1 a? 2.244 330.6 1_247 1.571 .": I . lC3 1.774 .OSS "..2 3.0%3 '37Ss 11 :213 2.039 :19"70.7 3.314 3..3 .11

31 7 1.30 054 I.20U 1.545 .03'r I2S 1.590 0374

0.8 1..3 .23 '5 .011, -1.296 3.305 :C,342 I .3co 3.34S .7 '.32S 1.013% .0540.9 1.MS m57 417337 3. 04 1.5. -. 123 :036, -,3 .1 .:30 .0471

AtA,*0.75- -

*0. z.3 7.33. 1:5 3.40 MCC0 .543 1.0 .!7 .3551) t.i7r 2.904 .2360.4 1.3 .. 2 *3 233 .63 .1 3r 1.327 2.7e, -,71,1 6 3.44 1.220.5 ..,:'7! -4 1.211) 2.54 .2SS 1.347 1.!01 !513 1:462 2:.949 .31740.6 12.33 3 .11, l32 2.3C2 .132 1.4.1 Z:45 : 1 133 1l.2Z 2.709 .344,0.7 It7 I.:, z .- :t 3.739 I.4 . 30;'Ilo 3.-1 2.~9 ̂ 2l2 ' I s' .644 2.263 .352;0.8 2.. 337 .4 .713 1.019 .067 . 3.*3 1.1 f .734 .417 CF5,;0.9 . .17, .. !1 1- .64^ .01!, 2.37, .373 .i5- is$37 .328 AU;

A,/%0 - 1.15

4. .71 .^7. .'' . .$?4 .$7 2-; 3.4?4 -j 40' ',T52) 3.413 .234'0.4 1.77- 2.927 .W 1.74., 3.723 .40' ' .%2 3.7 .W 4 j 376 .70 .2 40.5 1." ;.Z .4, 1.7U6 3..'A .34 1-.37 3:376 :V93 I.3 .7 370.6 1.-' 2.c7 13 -. 3.2 .23 ' .. 30 .23 I 12.42 3 .368 .167,

0.6 1. .132 1 Z..14 I.3!' .303 'i :3.412 : .3) 13 5 2.s, 2.37 .1302,0.9~j'- i: .3 Z4~323 3413 03 3.379 323 .39 ~3.342 .1" .'034

6S-210. At1%A - 0.35 j~ .r -0.3 1.' 334 .7 .37E a.497 ~ .41 2.503 .30

0.4 1.~ 23. 2. A05 2.323I *313! jl3 2.7S5 .117~0.5 3. 2.0155 . .022 2.403 .33 , I 3.I3U, 2.5.3 .1310.6 3. S. 2.S77 31 .151 2 9.317 I07 1.315- 2.425 .,.D,0.7 1.~- l7 .733. 3.237 37 9 345' It 272 1.967 .0710.8 1.S 51 2.-M. .:17 1.355 1.6 CS 43 1 7.3s4 1.330 03C50.9 I.V . .1 5z C . -. 5434.07.) ii .&Z0 -3.098 C0311

*0.3 1.113 3.23? .7 1 .340 3.377 . 1 .629 3.207 .3 1.768 3.256 :17310.4 3.2 3.1' .4'2 3.237 -3.2 .349 1.336 3.435 Zr'? t 31 .473 1.6 .70:5 .57 74 .3 2 .949 .239' .360 3.355 .224t 1.474 z.-176 .79

06 1.332 2.394 .3K6 3.2347 2.679 W,3 1:344E 2.3 .344 5.43 3.137 .460.7 .31 2.32!^ .,72 1.519 2.247 .31.0' 1.GO2 2.35 .313 3.478 2.53 .301

0.9 2.2 ~ .33 .4' .33 .40: Z.45 -.531C 2.6 -.74 .01.

AE/A,, 1.15

r.03 73 4.264 .332.351 3.761 '.399 1 2.321 3.733 23410'4 3 .77. 4.4$7 *7331 1.22 4.395 .3451 1.984 4:358 .201

0:.j 322 3.704 .51' 1.37 3.340 .21 I77011 53: .57 SIM0 .331 3E 3.540 .4.2.-6 34703 1

07 1 3.~ Z.549 .321 1:'154 3.033 .157, 2.231 3.314 .33'0.8 '.147 2.022 G0'21 2.535 2.333 .1071 " 2.553 2:.199 30 20.9~ " .1 304 .2 3.441 .114 -07V U13c -. 34 .078


Table 10 (cont) Correction Factors for Skewed Propellers, Z =5

' -0.3 , 1.1-S le.47 2; U.;155'- 4 ''.. 3:57Z .270 1:050 1. 7 .11951.167 3.111 5 .4S .155

S.172 tt 1 .232 .93 os . 347 *C71

AL/A. 0.75 __. '-V

"74 2.01s .:i 0 3 .6 2.143 .322 1.:".35 .33 3; 1 2.:.a .331

64$O V.5 -2 1. 1:21- ''.'11 .2I

3.1 1 ~ .2 1;,S7 .47; 1.334 . 4 .02, 1.6:3 1.501 104Z

~'~ 0.3, 13.9 1.5~7 3 .124 .3 12.304 C571 4 .33 -044 3 .95; X.;7 I.7

0.27 2.2 .5 39 334 ~ .3 70 .2.6 2.0 23 338 333 .3170.7 ~13& 143 .9 .3 .3 34 .3 3,S-4 .34 23951 M.I .

0.9 . .34'; .01 2.2 A53 .S .75 .3 07 ') -.03 .3S7

1~03 5 s3.- 3.A45 3.2S3 a.373 3.7 3.23 2.1 .

41 0: A.3 .430. i337 1.727 SO 3 i32' .~ .3 .47 .33 Z. 163 2.07 .14*1

_.__t. _365e; __________ 13 27 3I2.:I?

0.7 3.M !1.73 .3, .6 1.37m 1 3 4 .21 1 Cs's Z.' 1 .1 M,

0.3 .Z .st G3 1 C 1.67 Ic- ,f 1.724 .115 3.13 2.W3 .30.7 1:,)>; 1.7.0 CA "7 15- & I lis 2.333 .309 1.2:6 2.33 3

1.0 .37 Ito$3 1.20 00.9~~.3 1.i 3S ~ ~ ,.:9 .3423 3.313 r,52

r 03 .' 27,7 1 its7 2.93 .73 is 155 2.661 .352 1.1 2.7211 .0.4,4 1 4V30 2.500 637 1.7 2:r S5 .33- 1.25i W.26 .434 133247 .30.5~ SS4. 2.424 _3 133, 2.200 .347) 1.241 2.549 :3U .37 3.30 .32%70.611.)21) 155 M;k 3 7 2.3 .2 1.2 2.72 24334 30 ~ .0

0.7 1 .1 1.93. 1 .3 .304 M33 3.-SI 4.72 W31 153 333 .0.8 13. 01 .013 I S 3.%-S .055 2.07; 3.3704 174 349 .0

0.9 1.4 42 3~I1 4 -. 6 03 1.3 .4 .07 .0 97 .

r- 0:3 5.4 .2 3.37 1 '~2. 3.113 .61 .2.354 3.13^ .39042 1.6 5.3C z 511

0.6 3,4, .2 ~ ~ -~ .3 3.137 .332 S.739 4. 4V. .3120. , * 10.34 -;"5s v . .3 .30 .3 .7 3.154 .390

0.4 -. 0 --. C . .3s .09227 3 -4 qs30

22 Prpele I.i-IiOs 2u^.71e.Correc0io15

Table 10 (cont) Corrodkon Factors for ;.;k*wod Propellers, Z 6

k kt

0, - 7'. ACIA, - 0.3s

r - 0.3 1.1U I.Ov! .36, '.4360.4 1.7sl 1.4i: .; .220 -. 245

J.=i- GOC, ; .22- "J.081,0.5 1.355 .123,0.4 .773 1.21.12 .07 TO,?, Ij .1.02 1.06 .149a ? R24 - I.Zzt .0?1 X.107, 1. 02, A.34E.,0:8 IMI 1.343 .02

1.210.04 1.*"- .067, '.160 1.02,3- " 07S

0.9 V.217_1__-__ .00- VOMO 4.236 _149 4.423, _ -CW

PtIA, - 0.75

1.4?4 C5 -00 1.4 ;3 2.14 1 b4 AJ27' 1. 33 -- 2.037"-ON 21:1311 77, -1:1700 1.122 2.03 '.72S :1.213 2.04, Z;7 1 =* UM; ;' .463,

Srz 1445 :Z 6 I.M ;4?9 'I 175 1033 .-.43A' 11.295 24" , ".1 17*0: S6 Oss 1.4cl 53

2, 1.647 -1.195 1 '.297, 1 .'2770.7 1.052 1.327 :11 , 1.1,01 .171 1.,2!3 .3.604 .1911 1.365 'PI.775 .1990.6 9 ^X.9 37. '.122 1-.427, .141343, ';lot 1.443, 1,373*17 _ . -1.0540.9 1N3,5 11:3164 .0 1:M 'M OSI .1.440, .091, Nso III

A. 1.1

0.3 2451 3469 1.69 2.1 7 2..57 .4 ?C 2_. 2W 2.546 .1271.290 1.777' 2.136

0.4 1.447 1.645, 2.7M .711 .499O-S 1.211 2.303 1.537 2.465 .1 55 U.Sak 2. 7;, .4220.6 1.113 943 550 Z I a 2.297' .412 1-681- 2.452'. .335,

:219 627, *,2434 ;V7 I_.7Z 2.2%tl' 457NO 1:10 1 579o 1 435 1.474 .12$ ,A.839, 1.936, 1.902 UM

1.432 1.90* .01OP3 2;41S2 Y.599 ' 24 2.475 1' .1 90!_ " :I49

14 is

r - 0 1 I.IU 1.?11 .372 1.101 I.S72 1 ;302.952 1.740 1.010 1.815

0.5 I.W'- I.C33 TOM .197 3.034 143 4z -2.0:4, .2031. *12 1. 5 1.030 -. 123 I.C61 7,32- 1 3

0.7 1.44Z ION 1.433 .07a 1.133 4C, 1 1 4 1.632 .1050.9 t.I.T 1.1.1 1.10 1.074 .049 11164 1 C651 1:213 1.065 ".0720.9 1 . 31 4 .551 1.322 .025 1.331.- , -:IAZ -- , .044:U .499, - -. 5GA _4351

AjjA - 0.75

r - 0.3 1 1.474 2. 41_ 1.4121 1.127 2.530:I 1:03C .7 4 1 '630' 2-ii?

0:4 _2 3, -1.223 2.5Z3 1.34715 :,,4 2.412 714 ,.3w , U125 .43S

0 - I -. 1 1 :1^4 ?:2,7 A71 I ISI 2.4 2 'I.J05- 2.704- .353,1:15C :4z, I 1 _ ,.6 .- , , - 3 431,

0 . 3r 747 21 r . 21 :29 1:1ci 2,211 .297. X.30 2.507, .2720.7 1 731 .172 1.3^o 14551 .172 J-392 .2494 1230.8 1.247 1 :3,99 1:30A -. 101 1.4G I.= -. 123 :11.530. I M -. 133

.455 Ta -1.577 .24S 497. seo. M2, to,,

0 J2.3SI 3.371 1. 7 115 3.172 1.22S 2.149 3.053 .870 1416' 2J20, .5153.4910 1.315 J93 I.SO 3.35S 4737. 1.5,26' 2.934 '.46S

0: 11:41 555 3 SE3

211 2.820 ."5 11:449 2:911 .745 1.557 3.012 OFS 1.471, 2.SC4 .4000.6 1.113 2.339 .587 1.37? 2.574 .512 1.550 2.741 .452 1.490 2.657 AM0.7 1.140 I.M? .309 I-el 2.173 .316 1.636 2.30 .295' '1;.590 2.34C .2500.8 1.4313 1.397 .139 1.744 1.659 IS2 I.C71 1.762 1.168 1:791 4;6% .1?70.9 t %432 1.240 .114 2.494 .902 .131 2.496 014 129 1 334 ' 132

210, At1% - 0.35

0 3 W4,0:4 m ZIS , " - -# : S, I - It-' 1 1.12S 2.167 .442 GISS, - 2.157 . .315

1.033 2.- , ;_ 332 .324 1.103 2.425 .26Z

0.5 .9 1 1.13, 1.049 2.205 .227 1,112- 2.375 2070 6 1.01; 1.7F7 71 OS3 .15F t.Ift, 2.24S 11521. IM 4 .1050:7 1..57 4 1: 56

4 :157 11.29 1 1 .246 -.0110: 89 1 ULF AS SI? 1.423 -.291_ .044 1.624 . -1.002 .039

0 p.n I EACIA - 0.75 't 9,

r - 0.3 .174 Z.S5. 1., Z 1.463 2.779 ].^.1) 1.527 2 Sol 763 .451,

0.1 .944 LOI ."I I.IZ9 2.9S7 .710 1.232 M5, :55G ..432

- _497 .4 s 1.195 2495 .42) .357O.S .913 2_71 .4 5 I.M I0.6 06 Z.C9- .?1, 1.117 2.409 .2Z9 1.227 2.630 .294 42710.7 1.0$2 1.-41 .1-3 1.231 2.045 .10 1.525 2.127 .1?6 190

0.8 1.:16 1.4-,7 Vit t.433 1.374 .028 1.513 1.390 .119, .1280.9 ;07& 1.943 -.443 .093 .101

Atl/A

r - 0 3 '326 3.502 1.698 2.149 3.236, A27 2.261; _3'.Ij4 -5640:4 .447 4.077 1.275 Y.02 3.375 70a ww 34966, 3130 5 .211 3.304 .894 t.572 3.01 :55S 1. 7 1 O' '3 7590.1 .113 2.741 .56 1.566 13.165 .414 T.712 , 3:53s, 3440.7 .1*0 2.104 .314 1.653 2.627 U4 111.903 '3.049,- '.2540.8 .435, 1.474 .147 1.210 1.814 .10 2.010, IOW .1790.9 .423 .617 mt 2.W _-.CU .127 2.599"'.. , -;395:, -;134


a V .~N OO&(. . 4 .ge

"-, mo or t. m *

Ofl~flCS .t.. . .. . . . . . . . .. v., .1. .

'4 - "I

.... .' .. ... ti . ....... ... ...-.

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

....b1~ C ......... S .C. . . .... .... ~. ~C. -

ov 1- sC- , 1MC-- C 41C'S-

72 00

ID0 C O C . Nc ~ . O . . IOO C t O* C C S o .. I.OC . . I'I ......

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

0: =7. .VO ~ O V.~f. ltlj (~I~.V

..........

q u m . .,. .- . . .

'-4 00 10C200 O)CS~r55 N~ .5..1s

24~~~NC Prpete Lfin-Srfc Crrdi

..... ..... ........ .1.......... . ~ .

vt n ,.. . . 9 VIln..1.9i . . . . . . . .

I i - i " i i

* -*-

.1 C:C It .:. ! CC

0 .... ........... ...............

." ." .."" .......... .. ..

-I -'C - C"

.. ,,, f1N% CflO 0%bO~41'fdW h

M .N"- . . ...- . . . . . . . . . . ... .f " .

d 0 o- . ... ....oo I oc, .. ... e . t- ...... 0.0

01 0-

....... ..... .... 0 0 " -

. .t e. .- % 'O_, • . . . ... ...a c c o ~ n ~ f~ . I 4 1. . . v n ~ l n c ~ n , ' ll

0--O

o :..o ... ........... -.............. ... ..............I---.-- , -- ~ - - -

............. _ § I~ 1. . .. . . . . . ... .. . . . . . . . . . . . . _

Propeller,0.,-%c Lict ong.u ce~ ,torrections 25~'t~0.ff1 %fIf

00-,0 gg.Of 0 .2.1z .~.a "00. f??I gz . AfO.(n-O G, 0..t'nt1 0,0

C; C .... 3'ocb o. ) o1f0b. o.~cfs ... .. ...0f4 f% . ' f:

Q ~ ~ ~ ~ O ', 011%'4'0 0 1

0

0 . * 0~... f.0.0.t ..... O- .r.. ........- '111-.~c 0. O.. . . . . .

I-X

0Prope.fO.. 1 ,fler Litg-Surfce Corrhc4% tions 25~~CffO%.t~lfCf*.

;-.~ ~~~~~~~~~~~~~~ ....- +', ... ... ; .. . . ..." . . . . •. . .. 2. .. .....I ... ........ ...... . -io, . .. . . . . . , .... ..... ..

INC " - .. .. .. ...........

S .... .............. ..........

VI. IN-

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

- ' I ... ..... . .

~m _.,-

4 Ofl....... ............ ! O........ ....... .... '. . S.,:'::;~

V .. ........ .. ..

o_4 ,,., ... .. .....g.4~. ,.-+:4. -. -._-4

" . . "....... ....... ........ . ..

,4 • ::...;.." ''% : = . .C4O.,S0U4'

44. .:-;:.. .4... ,.4 • -- .

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

4 0.4 -. 4,

S 4 ................ .


Table 12 Comparison Between Theory and Experiment for aConstant Thrust Loading

Experi- Eperi-P/D at mental X, mental C,

Propeller Z :l/AO 0.7R BTF Ci Design X, Design C,

A 5 1.318 1.473 0.054 0.566 1.013 0.976B '3 0.606 1.077 0.040 .A563 0.W99O 1.025C 3 O.601 1,08. 0.080 0.563 0.987 0.989D 3 0.303 1.086 0.057 0.570 0.995 1.016F 3 1.212 1.073 0.028 0.5 19 1.014 0.965G 5 1.480 1.503 0.036 0.361 1.008 0.971

of the programs. Table 12 compares theoretical but the camber correction is negligible except forand experimental values for six of these propellers, low pitches.designated A, B, C, D, F, and G; all used the 4 A correction factor can be formed for theNACA a = 0.8 mean line and had no skew. In ideal angle induced by thickness which is in-general, the theoretical values are within the dependent of the magnitude of thickness butaccuracy of the experiments and within the accu- dependent on the thickness distribution.racy with which the blade-section viscous drag 5 Correction factors for camber and ideal anglecan be chosen. due to loading are largest near the blade tip and

There appears to be a trend for wide-bladed smallest at 0.4 and 0.5 radii. They increase withpropellers A, F, and G to be slightly over-pitched, increasing expanded area ratio and, generally,Also, the thick propeller C tended to be slightly with increasing induced advance coefficient,under-pitched as compared to the propeller of but they decrease with increasing number ofstandard thickness, B. It is possible that both blades.of these differences could arise from determination 6 The ideal angle due to thickness is largestof the section viscous drag from airfoil data. near the blade root and decreases to negligiblePropellers A, F, and G have blades with thickness- values toward the blade tip. In general, thischord ratios lower than those for which experi- angle decreases with increasing induced advancemental data are available, and propeller B has coefficient but increases with increasing bladeblades with thickness-chord ratios which are number.higher than common. A further discussion of 7 The chordwise distribution of camber ismost of these propellers will be found in reference somewhat flatter toward the blade leading edge[29]. and, in general, fuller toward the trailing edge asConclusions compared with the two-dimensional values. The

spanwise change in the camber chordwise dis-From the numerous calculations made and the tribution is small. In practice, the use of the

series data presented, a number of conclusions two-diniensional distribution is probably reason-can be drawn with reference to lifting-surface able.corrections for propellers. Many of these con- 8 The shape of the blade outline has a signif-clusions, of course, have been made previously icant effect on the correction factors for camberby other investigators. and ideal angle due to loading.

I Use of a realistic chordwise load similar to 9 The spanwise load distribution has a signif-that for an NACA a = 0.8 mean line results in icant effect on the correction factors for camberboth all induced camber and ideal angle. The and ideal angle due to loading.camber correction factors for this load distribution 10 Skew has little effect on the camber andare somewhat less than for the constant-load mean ideal angle due to thickness but has a large effectline, but, in general, they are greater than unity, on the ideal angle due to loading. Skew in-i.e., greater than the two-dimensional camber for duces a positive angle near the blade root and athe same lift. The ideal angle correction factor negative angle toward the blade tip.due to loading is, of course, zero for the constant 11 Experimental results indicate that, for aload. chordwise load distribution corresponding to that

2 Correction factors call be formed for the of the NACA a = 0.8 mean line, thc use of lifting-camber and ideal angle (cue to loading which arc surface corrections gives propellers which, inindependent of the magnitude of propeller loading, general, meet their predicted performance within

3 Thickness induces an angle and a camber, the accuracy of the experinents and tile accuracy


with which the blade-section viscous drag can be 7 J. E. Kerwin. "The Solution of Propellerchosen. Lifting-Surface Problems by Vortex Lattice

12 Correction factors derived herein should Methods," Massachusetts Institute of Tech-replace those calculated by less sophisticated nology, Department of Naval Architecture andmethods; for example, those in reference j221. Marine Engineering, June 1961.

13 The numerical evaluation of the compli- 8 T. Y. Wu, "Some Recent Developments incated theoretical equations may lead to solutions Propeller Theory," Schiffstechnik, vol. 12, no. 60,which are erroneous, and calculations must be I9(65, pp. 1-11.carefully checked. 9 W. H. Isay. Propeller Theorie-Hydro-

dynamische Problems, Springer Verlag Publishers,Acknowledgments 1964.

The authors appreciate the assistance of many 10 H. V. Lerbs, W. Alef, and K. Albrecht,staff members of the Naval Ship Research and "Numerical Analyses of Propeller Lifting SurfaceDevelopment Center and the Hydro-og Aero- Theory" (Numerische Auswertungen zur Theoriedynamisk Laboratorium, Lyngby, Denmark. der Tragenden Flache von Propellern). JahrbuchSpecial mention goes to C. W. Prohaska of Hydro- Schiffbautechnische Gesellschaft, vol. 5s. 1964,og and to W. E. Cummins and the directorate of pp. 295-318, David Taylor Model Basin Trans-NSRDC for their generous support; to J. B. lation 330.Hadler of NSRDC and H. W. Lerbs of Ham- 11 H. M. Cheng, "Hydrodynamic Aspect ofburgischen Schiffbau-Versuchsanstalt for their Propeller Design Based on Lifting Surface Theory:encouragement; to H. M. Cheng, E. B. Caster, Part I-Uniform Chordwise Load Distribution,"and E. Harley for their valuable assistance on the David Taylor Model Basin Report 1802, 1964.various computer programs; to G. G. Cox for 12 H. M. Cheng, "Hydrodynanic Aspect ofreviewing the manuscript; and to Mrs. Shirley Propeller Design Based on Lifting Surface Theory:Childers, Miss Danielle Dubas, and A. A. Canpo Part II-Arbitrary Chordwise Load Distribu-for preparation of the manuscript. Also, a tion," David Taylor Model Basin Report 1803.paper of this type would not have been possible 1965.without the services and cooperation of the 13 J. E. Kerwin and R. Leopold. "'A DesignNorthern European University Computing Center, Theory for Subcavitating Propellers." TRANS.Lyngby, Denmark, and the Applied Mathematics

Laboratory of NSRDC in making the lengthy SNAME, vol. 72, 1964, pp. 294-335.

computer calculations. 14 D. M. Nelson, "A Lifting Surface Pro-peller Design Method for High Speed Computers,"

R~emncos NAVWEPS Report 8442, NOTS TP 3399, 1964.15 D. M. Nelson, "Numerical Results from

1 W. J. M. Rankine, "On the Mechanical the NOTS Lifting Surface Propeller DesignPrinciples of the Action of Propellers," Trans. Method," NAVWEPS Report 8772. NOTS TPINA, vol. 6, 1865, pp. 13-39. 3856, 1965.

2 S. Goldstein, "On the Vortex Theory of 16 P. Suhnont, "Theoretical Study on MarineScrew Propellers," Proceedings of The Royal Society, Propeller Performance Using the Rheo-ElectricalLondon, England, Series A, vol. 63. 1929, pp. Analogy Technique," Bureau d'Analyse et de440-465. Recherche Appliqu~es, Report on ONR Contract

3 H. Ludwieg and I. Ginzel. "On the Theory Nonr(G)000!!-61, 191.of Screws with Wide Blades" (Zur Theorie derBreitblattschraube), Aerodynamische Versuchsan- 17 P. Suhnont, "Analog Study of Somestall, Gottingen, Germany, Report 44/A/0S, 1944. Problems Relative to Lifting Surfaces in a Heli-

4 J. A. Sparenberg, "Application of Lifting coidal Movement," Bureau ('Analyse et de

Surface Theory to Ship Screws," International Recherche Appliqu~es, Report on ONR ContractShipbulding Progress, vol. 7, no. 67, 1960, pp. No. N62558-2997, 1962.99-106. 1S T. Nishiyania and T. Sasajina. "Induced

5 G. G. Cox, "Corrections to the Camber of Curve-Flow Effect in the Lifting Surface TheoryConstant Pitch Propellers," RINA, vol. 10:3, of the Widely Bladed Propellers," Journal of1961, pp. 227-243. Ship Research, vol. I1, no. 1, 1967, pp. 61-70.

6 P. C. Pien, "The Calculation of Marine 19 L. Malavard and P. Sulmont, "ApplicationPropellers Based on Lifting-Surface Theory," of the Lifting Foil Theory Solved by RheoclectricJournal of Ship Research, vol. 5, no. 2, 1961, pp. Analogy to the Calculation and Design of Sub-1-14. and Supercavitating Propellers," Bureau (I'An-


alyse et de Recherche Appliqu~es, Report on ONR Inc. Report E. S. 40622, Long Beach, Calif., 1962.Contract N62558-4998, 1967. 25 H. W. Lerbs, "Moderately Loaded Pro-

20 M. T. Murray, "Propeller Design and peilers with a Finite Number of Blades and anAnalysis by Lifting Surface Theory," Inter- Arbitrary Distribution of Circulation," TRANS.national Shipbuilding Progress, vol. 14, no. 160, SNAME, vol. 60, 1952, pp. 73-117.1967, pp. 433-451. 26 I. H. Abbott, A. E. von Doenhoff, and

21 J. E. Kerwin and R. Leopold, "Pro- L. S. Stivers, Jr., "Summary of Airfoil Data,"pelter Incidence Correction due to Blade Thick- NACA Report 824, 1945.ness," Journal of Ship Research, vol. 7, no. 2, 27 T. Brockett, "Minimun Pressure En-1963, pp. 1-6. velopes for Modified NACA-66 Sections with

22 M. K. Eckhardt and W. B. Morgan, "A NACA a = 0.8 Camber and BuShips Type I andPropeller Design Method," TRANS. SNAME, Type II .Sections," David Taylor Model Basinvol. 63, 1955, pp. 325-374. Report 1780, 1966.

23 W. B. Morgan and J. W. Wrench, Jr., 28 K. E. Schoenherr, "Propulsion and Pro-"Some Computational Aspects of Propeller De- pellers," Principles oj Naval Architecture, vol. 2,sign," Methods in Mathematical Physics, vol. 4, edited by H. E. Rossell and L. B. Chapman,Academic Press Inc., New York, N. Y., 1965, SNAME, New York, 1939, p. 157.pp. 301-331. 29 S. B. Denny, "Cavitation and Open Water

24 J. L. Hess and A. M. 0. Smith, "Calcula- Performance of a Series of Propellers Designedtion of Nonlifting Potential Flow about Arbitrary by Lifting-Surface Methods," NSRDC ReportThree-Dimensional Bodies," Douglas Aircraft Co. (in preparation).

Propeller Lifting.Surface Corrections 29

propeller lifing-surface corrections"correction factors for camber, ideal angle due to loading,...

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