### xperixnental Towing Tank

### Sbevens Institute of Technology

### Hoboken, New Jersey

THE EFFECT 0F SPEED AND SIZE

ON THE PERFORLiANCE 0F A MODEL SHIP PROPELLER
AT OPTTh1UM _{FFICIENCY}

B, V. Korvin-Kroukovsky

### Presented at the March

16, 1951### Meeting

### of the

Section of Mathematics### and

Engineeringof the New York Acaderr of Sciences

Prepared under

### Contract No. N6orir-2L7O7

### U.S. Navy

Office of

### Naval Research

### (E.T.T. Project No

_{DF1321)}

TABLE OF CONTENTS
PAGE
Summary _{i}
Introduction _{i}
Experimental Daba _{2}
Iiethod of Analysis

Analysis of the Blade Section Drag .. _{8}

Lift of the Blade Sections _{IO}

Concluding Remarks _{11}

References _{12}

Table I: _{Model Propeller ?erforniance at V/nD of}

Maximum Efficiency .. .

### i1

Table II Ordinates of the Propeller Section at

_{OThR}

### ...

_{15}

Figures . ... _{....} _{16}

UMIAhY

The torque and thrust coefficients of three model propellers of the

motorship SAN FR.NCISCO, tested independently in open water in three tow-ing tanks) are analyzed by the methods of circulation theory of propellers

to obtain the lift and drag characteristics of a typical blade section.

The drag coefficients are found to vary with Reynolds number in a manner

indicating that a turbulent boundary layer condition edsts even at low

Rrnolds numbers This may be attributed to the turbulence stimulation

caused by the vibration connected with the high rotational speed of small

propellers. The lift coefficient is lower than can be expected from the

propeller and airfoil theories, This can be attributed to the reduction

in absolute value of zero lifte angle, and corresponds to a similar

reduc-tion in the case of wind-tunnel tests of airfoils at very low Rrnolds

numbers.

The analysis and some of the tests were conducted at the ßxperinen't

Towing Tank of Stevens Institute cf Technolor) under the auspices of the

Office of Nava].. Research,

INTRODUCTION

A naval architect is forced to depend on test results obtained on

rrdel ships and model propellers for the prediction of ship speed, for the

specification of the power to be provided, and for the specification of

speed of rotation of the propeller shafts of the full-s:ze vessel, In

this connection) he is always confronted with the question as to how accurately full-scale performance can be predicted from model data, or more specifically) as to what kind of data and methods can be used to make an accurate full-scale prediction, A part of this problem will be

examined in this paper --- that of the performance of the propeller itself)

not considering the propeller-hull interaction, the so-called

free-water performance of propellers0 The problem will be further limited to

a study of one particular propeller design, tested independently in three

difL'erent laboratories,,

Kempf1 and van Lammeren2 conducted tests on model propellers of dif

f erent sizes and at different speeds, and presented data on the effect of

TM-102

TM-102

### -2-.

these parameters on the propeller thrust torque and efficiency. However,

an examination of the changes in the over-all characteristics of the tested propellers often results in a confusing pictire, and sheds little

light on the methods of extrapolation to full size, and on the limitation

of such methods. In particular, it often leads to an arbitrary and

errone-ous statement of the minimum practical size of the model, as well as to

the unwarranted assumption that a model of a certain larger size will well

represent the characteristics of the full-size prototype. A more useful

and reliable guide can be obtained .t'rom an analysis of the model propeller

performance by applying the methods of existing well-developed propeller

theory to assist in reducing the test data to the lift and drag

character-istics of an isolated blade section. This procedure can vary in scope and complexity, depending on the nature of the experimental data, and on the

completeness of the results desired. In this paper', a simple method is used to obtain the lift and drag characteristics of a single section

located at 7i% of the maximum blade radius, and at an advance ratio V/nD

at which the maximum propeller efficiency is obtained.

The work is part of the research on self-propulsion tests of ship

models done at the Experimental Towing Tank of Stevens Institute of

Tech-nolor, Hoboken, N.J., under contract with the Office of Naval Research,

Task Order VII.

EXPERIUENTAL DATA

The propeller design chosen for this analysis is that of the

motor-ship SAN FRéNCISCO which was initially investigated by Kempf1 in Hamburg.

Recently, a model of the same size was made from Kexnpf's drawings, and was

tested at the David Taylor Model Basin in Carderock, Md. Also, a small

model of this same propeller was made and tested at the Experimental Towing

Tank of Stevens Institute of Technolor. The E.T.T. model was copied' from the DTMB model by a pantograph attachment to a lathe. The geometric

di-mensions, test speeds, and resulting performance of the tested propellers

are given in Table I. A typical plot of the data is given on Figure 1.

* This paper, illustrated with slides, was prepared for presentation at

the meeting of the section of h&athematics and Engineering of the New York

### T-l02

-3-This customary way of presenting the data consists of plotting the measured

thrust T and torque Q in the form of nondirnensional thrust and torque

Kq coefficients (as defined on the chart) against the ratio of the

veloc-ity V through undisturbed water to the product of the revolution per

second n and the propeller diameter D (Le0, J V/nD). Each series of

tests was conducted at approximately the same RPI, with J varying with

the velocity of advance V from zero to the speed at whioh the thrust

vanishes. As can be observed from Figure 1, there is a certain scatter

of the data, and it is difficult to distinguish by eye the relatively

small effect of the speed of rotation as shown by different coding of

the four series of tests shown0 In order to make the comparison as

re-liable as possible, a definite method of fairing by calculation. was

adopted, and was applied to all test data, without any regard to the

visual fairing performed by the original investigators. In the case of

the thrust coefficient Kt, it is observed that the curve is practically a

straight line in the range from J

### = o55

to J at which Kt becomes zero. Furthermore, it is noted that in this range of J, the slope dK/dJ of theline is about uniform in all cases, and that any error in the estimation of the slope would not be important in vier of the location of the finally

desired points at J 0.80, which is abou: in the middle of the range

chosen0 The equation of this part of the thrust line then becomes:

Kt = A - C'5O J (1)

The values of the coefficient A were computed for all observed test values

of in the range

### 0.55

< J < max, and were averaged separately for eachseries of tests0 The final f aired value of for J of maximum efficiency

### (0,75

or### o,8o)

was then computed by Equation (1), using the averagedval-ues of A, and is given in the fifth column of Table I.

A similar process was used for the torque coefficient K; except

in this case, the plot of Figure 1 is represented by the parabolic arc

Kq

### = c - o

j2The constant C was computed for each test value of Kq averaged for

each series, and the value of Kq for J of madmum efficiency was computed

on the basis of this average, and is given in the sixth column of Table I.

TM-102

-b--of Lairing were eliminated, and each value -b--of Kt or Kq at J -b--of maximum

ef-ficiency was found as an average of a number of test points. In the

seventh and eighth columns of Table I, these data are converted to another form of nondimensional coefficients, and which are more convenient

for use in the contemplated analysis. The efficiency n

_{= TIP}

is given
in the last column. In view of the process of fairing used, these

effi-ciencies do not always correspond to bhe ones reported by the original

in-vestigators.

LETHOD OF ANkLYSIS

An analysis of the data will be made utilizing the modern theory of

propellers, known as the circulation theoxy Although this theory is

dif-ficult in its original derivation, it is ectremely simple in its

applica-tion. Figure 2 shows a four-bladed propeller with a helicoidal vortex

sheet trailing from the edge of each blade. Betz3 has shown that the

opti-mum efficiency of a propeller is obtained when the velocity distribution

in the wake is such that these helicoidal vortex sheets trail aft without change of form, Le., as if they were rigid-material sheets. Prandtl3

made the first approximate calculation of the velocity distribution needed

to satisfy this condition, and Goldstein developed the final rigorous

solution.

A specified velocity distribution in the wake implies a specified distribution of the thrust along the blade; this is referred to as

Goldstein's (or Betz') thrust distribution, Strictly speaking, the

cir-culation theory applies only to propellers designed to provide this thrust

distribution, and only to their performance at the speed of maximum eff

i-ciency. In practice, it is found, however, that the maximum efficiency

is affected very little by a certain amount of deviation of thrust

distri-bution from the optimum, which thus implies a certain range of deviation

of the ratio V/riD from the optimum Little error will be involved,

there-fore, in the inverted application of the theory to the analysis of a given

propeller, in assuming that the propeller operating at maximum efficiency

conforms to the ideal thrust distribution. With this assumption, the

analysis can proceed following the elementary Rarikine-Froude momentum

the mean velocity r a computed by Goldstein0

First, reference is made to Figure 3, which gives the velocity and

force vector diagrams at the propeller blade, and also the velocity vector

diagram at the surface of the helicoidal sheet at an infinite distance aft

of the propeller, where the flow in the wake becomes steady0 The

hori-zontal vector C*Jr is the tangential velocity of a blade element due to the

velocity of' rotation; the vector V is the velocity of advance; and U is

the resultant velocity of a blade element with respect to the undisturbed

fluid. The reaction to the lift force L is an induced velocity vector

w/2 produced at the blade, so that the actual velocity of the water with

respect to the blade is shown by the vector U. The angles of the vectors

U' and U with the plane of the propeller are designated by and

,

re-spectively0 At the propeller blade, the vector U is tangent to the

heli-coidal vortex sheet, which by definition floats with the fluid, Le0, moves normal to itself with the induced velocity

_{w/20}

At an infinite
distance aft of the propeller, the induced velocity increases to w, and,
since its axial and tangential components u and u0 increase in the
same proportion, the helicoid rotates as it moves aft without change of

shape0 In making this statement it is assumed that the diameter of the

helicoid remains equal to the propeller diameter, or in other words, that

the contraction of the wake is neglected0 This assumption is permissible

for moderately loaded propellers used in common practice.

The movement of the helicoidal surfaces is caused by the induced

fluid velocity w, and conversely it can be said that movement of the helicoid causes the velocity w0 Since a perfect fluid is considered,

and the forces of friction do not exist, the velocity of the helicoidal

sheet along its surface will cause no motion in the fluid This being

the case, it is convenient to replace the vortex sheet velocity w in a

direction normal to itself by the velocity of' displacement W along the

direction z parallel to the propeller axis0 In either case, the fluid

will have the velocity w with components u _{and u0, but now these are}

expressed in terms of W as

= w

### u

### = Wcos

### sinÇ..

### (2)

TM-l02

### --r1-1o2

-6-Furthermore, the velocity in the wake is not uniform, and it was shown by Prandtl3 and Go1dstein that the fluid velocity is smaller in the space

between the helicoidal sheets than at the surface, The mean axial

com-ponent Uzm of the fluid velocity is, therefore,

### u

### =Ku

where the constant K was Origiflally computed by Goldstein, and then the

computations were ecended by Lock5 and Kramer6; they are used here as

reproduced by Theodorsen7.

Now the Rankine-Froude axial momentum relations can be rritten,

Consider the area contained between radii r and r + dr:

### 2,irdr

### (3)

### The fluid mass flowing through the above area is:

### 2iP (V

_{+}

### u)rdr

,### ()

where P is the mass density of the

### fluid.

The### change of momentum of the

### fluid mass which is equal to the

thrust increment dT is:dT =

### 2tP (V

+### -u)urdr

### (s)

Expressing mean velocity u in terms of displacement V of helicoids gives

U a KW

### cas2

.(6)

The thrust increment in terms of' W is:

dT =

### 2p

### [VwK co21$.

### (WKcos2.)2]rdr

. (7)Let r

### xR

arid divide bydT = li

### [(K

+### (K

2

)2

### ]x

dx . (8)The thrust can be obtained now by integrating dT (x) from x O to

### x a

1, Le., over the length cf the blade. A short cut is available, how-ever, in that, in practice, the distribution of the thrust along the bladeis found to be suí'ficiently uniform for different propellers so that the

integral can be evaluated on the assumption that

### (dT/dx) atx= a

T = I dT(x)

c _{(9)}

C. i constant

I

On this basis, the performance of a propeller is often judged by

the performance of a single section at x = O7 or x = O75 In the present

case, the analysis is based on x = 07li, since a section was shown on

Kempfts drawing at r = 100 mm, or r = 0071R0 The interpolation of the

8 9 10

data given for this type of analysis by Lock , Driggs', Hartman , and

11

Lerbs leads, in the present case, to an estimate of the constant as

being

### l63,

so that:T

### l63

[(K cos2### ç.) + (K

cos2i

The ideal efficiency of a propeller _{l} is defined as

i tank

- i+w/2v

### tan.

and the power disk loading coefficient is

P

### =T /77.

C.

_{i}

C._{i}

In the Rankine-Froude linear momentum theozy, the velocity W in

Equation (li) above was understood to be the true axial velocity of the

uniform wake, and the only energy loss was considered to be in connection

with the increment of the axial velocity0

In the circulation theory, W is understood to be the fictitious

9displacement velocity" of helicoids, greater than the mean axial velocity

Uzm and as such, it includes the effects of energy losses of velocity in

tangential and radial components, as well as in the axial component0

The subscript i in the above designates the ideal efficiency, i.e., the efficiency obtained by neglecting the profile drag of blade sections.

The values of Tc, _{Pc19} and

### r.

are uniquely determined by the displace-ment velocity ratio### w/v.

On Figures L and### 5,

the upper curves designatedE = O represent plots of the ideal efficiency _{77} vs thnist disk loading

coefficient T0 or power disk loading coefficient co The displacement

velocity ratio

### w/v

is indicated by the dotted parametric lines0 This curveof ideal efficiency agrees closely with similar (and greater in slope)

curves published by Kramer6 and Lerbs12.

)2J

TM-102

-7-(10)

Ti-1O2

### -8-.

As shown on Figure

_{3,}

the effect of the profile drag D is to reduce
the thrust and to increase the torque of a blade element. Bienen and
### l3

von LarInan have shown that the integrated effect for the entire blade

length can be put into the forni of a correction to the ideal thnist and

torque coefficients, so that the actual coefficients become:

T = T

### (1- 2C tanp)

C C. (13)### i

### P =P (1+.a)

### (Th)

C C. i owhere _{=} CU/CL, i.e., the ratio of drag coefficient to lift coefficient,

and tan = tan

.

at r = R (which means at the tip of the blade)

The blade efficiency

b is:

i - 26tan

71b 2 £

,

+ 3 tan
and the total efficiency _{t} is

= b

(16)

Parametric curves of the total efficiency T are plotted on Figures

L and 5 for a number of values of the drag/lift ratio E

Finally, the lift coefficient of a blade element is given asfllS:

C - - li sink. tan('.

### -L0

i### r1

(17)where

K = a coefficient related to K, and is tabulated by Lock in

Refer-ence

### 5,

O = the solidity defined as Nb/2 n r, where N = the number of blades, and b = the chord of a blade at radius r.

ANALYSIS OF T1 BLADE SECTION DEG

The analysis of the propeller performance now becomes very simple

The observed efficiency taken from Table I is plotted vs. T and on

CD/CL for the blade element are read directly from the relation of

the plotted point to the parametric lines w/v = const and L = conste

With w/v known, tan

.

is determined as

### tan.

(i+ tan

Tank is known, since tan= \1/cr. With and known, the lift

coef-ficient is computed by Equation (17) and the drag coefcoef-ficient C is comrn.

puted as _{£CL.}

The drag coefficients CD found in this way are plotted vs Reynolds

number on Figure 6 for the three tested propellers All ten test condi tions are plotted and compared with three standard friction resistance

curves. Although there is a considerable scatter of data, there is a c1ear

ly defined tendency to follow an average line, essential]y parallel t'o the

lines expressing the frictional drag for the turbulent condition of the

boundary layer. The mean profile drag of the section at OThR of the

propeller design under consideration is found to be equal to two tes the

friction drag of a flat plate having the same projected area-j Since thi

factor of 2 is unexpected] high, the wir:d=tunnel test data of an old design

airfoil RAF6, tested in a very turbulent type wind tunnel at low Reynolds

number, are plotted on Figure 6 for comparison. The two sets of data agree

as to the order of magnitude, and, in fact, the direction of: the plotted

curve indicates a turbulent condition of the boundary 1ayer. This

eondi-tion for an airfoil is probably brought about j the turbulent condition

of the wind tunnel, and by the type of airfoil section possessing a harsh

curvature

### at the leading edge.

### The propeller blade section is described

by the ordinates given in Table II, and appears to be of the Joukows1r type

with circular arc median

### line.

Then tested in the towing tanks, each bladeencountered undisturbed water as free of turbulence as could be expect-edo

It was surprising, as well as gr&tïfying, therefore, to find the turbulent

boundary layer type of resistance curve, gratifying because it permits a

convenient extrapolation to air value of the Rrncids number,

### including

full sizeIn looking for reasons to explain the occurrence of a turbulent

### T-I32

- lo -.

vanced that the turbulence is probably stimulated by the vibration of' the

rapidly rotating propeller model.

A test was made in the cundition corresponding to Test

_{L,}

_{in which}the model propeller was made to operate in the turbulent wake of a suries of round rods towed in front of the propeller.

_{The relation of T}

_{and}

in this test remained the same as in Test ), indicating that added

turbu-lence had no effect on the drag. This fact can be taken as supporting

evidence that the boundary layer was already turbulent.

LIFT OF TF BLADE SECTIONS

The lift coefficient was computed by 'quation (17), and was corrected

to represent the section as it would be tested in the wind tunnel by

apply-ing ttcurvature correction" as developed by HiliTh. No wind-tunnel data

for the section in question were available, but the lift characteristics

to be expected can be estimated by the computational methods known from

the theory of airfoils. The lift coefficient as tested was found to be

very much lower than indicated by this estimate. Again a comparison was

made with the low Reynolds number tests of an PLAF airfoil, and it became

apparent that the low value of the lift coefficient is due to the decrease

in the absolute value of the angle of zero lift at small Reynolds numbers.

The angles of zero lift as computed from the test data are shown on Figure

7, where the asymptotic value for the larger Reynolds number is also shown. The mode of approach to this asymptotic value is, however, not known. It

appears that smaller, more rapidly rotating propellers have a smaller

de-ficiency in the angle of zero lift than larger propellers compared at the

2

sanie Reynolds numbers. This checks with the data of van Lanimeren , where

at a given Reynolds number, smaller propellers showed higher efficiency

than the 1argcr ones.

The propeller used for Tests 9 and 10 shows an abnormally large de-ficiency in the angle of zero lift. It also shows the best efficiency

V/nD at a value of J =

### 0.75

instead of at J = 0.60 which was common forthe other tested propellers, indicating a reduced strength of circulation.

The finish of the blades appeared to be unusually rough, but no data are

### resulting change in the angle of zero lift.

CONCLUDING REMAEKS

### The analysis of the test dat.a for model propellers on the basis of

### the circulation theory explains their performance in terms of the lift

### and drag characteristics of the blade sections.

### The drag characteristics

### follow the turbulent boundary layer law of skin frictions so that

### ex-trapolation of drag coefficients from model to full scale should present

### no difficulty. This conclusion refers only to the specific blade section

### analyzed in this paper.

### There is evidence in the van Lamineren

### tests

### (not yet analyzed in detail) that a mixed laminar-turbulent flow exists

### in small propeller models with the low drag type of blade sections of a

### more modern design.

### The attempts

### the

### T.T. and van Lammeren to use

### an artificial turbulence stimulation are inconclusive since the models

### tested were already (unknowingly) in a developed tu bulent condition.

### The most. serious dïscrepancy is found in the values of the angle

### of zero lift., which will require further investigation.

### Tentatively

### it appears that in order to give the saine performance

### smaller model

### propellers should be made with a degree or two larger pitch angle than

### the full-size prototypes. Again this appears to apply to the particular

### propeller analyzed.

### In an example given by

### Lerbs11,

### a good agreement

### of lift coefficient was obtained or a very thin ogival section at

### some-what greater Reynolds number.

### The range of Reynolds numbers between the 'sxnall' and fllargett

### models is rather narrow, and even the 'large't models cannot be assumed

### to represent directly the full-size performance.

### If this type of analysis had been applied to the propeller when it

### was designed

### it vould have clearly indicated the possibility of greatly

### improved performance by reducing the profile drag losses, preferably

### by-using a more favorable blade section, or by reducing the number of blades

### from four to three with a corresponding increase of pitch (and a resultant

### increase of the angle of attack and of the lift coefficïent).

TM-102

-TM-iD 2

12

-REFERENCES

Kempf, Gunther: "A Study of Ship Performance in Smooth and Rough

Water," Transactions of The Society of Naval Architects and Marine Engineers, Vol, W4, 1936, pp. l9-227.

Van Lamrneren, Ir. W.P.A.: _{"Propulsion Scale Effect,"} _{Transactions}

of the North-East-Coast Institution of Engineers and Shipbuilders,

1939/RO, pp. ll-12.

Prandtl, L. and Betz, A.: "Vier Abhandlengen zur rdrodynamik and

Aerodynamik," Gottingen 1927; Edwards Brothers Inc. Ann Arbor,

Michigan, 19b3.

). Goldstein, Sydney: "On the Vortex Theory of Screw Propellers," Proceedings of the Royal Society, 1929, pp.

_{W4o-I65,}

Lock, C.N.N. and Teatman, D.: "Tables for Use in an Improved Method of Airscrew Theory Ca1culations" ARC (British) R. & M. No. 1671

October l93I. (Also included in the bound volume 1935-36.)

Kramer, K.N.: "The Induced Efficiency of Optimum Propellers Having

a Finite Number of Blades," NACA T,M, No, 88h, January 1939.

Theodorsen, Theodore: "The Theory of Propellers, I Determination

of the Circulation Function and the Mass Coefficient for Dual

Rotating Propellers," NACA Report No. 775, _{19W4.}

Lock, C.N.N.: "Graphical Method of Calculating Performance of

Air-screws," ARC (British) R. & M. No0 _{1675, 193h.}

Driggs, I.H.: "Simplified Propeller Calculations" Journal of the

Aeronautical Sciences Vol. 5, No. 9, July 1938, pp. 337-3W4.

10, Hartman, Edwin P. and Feldman, Lewi.s "Aerodynamic Problems in the

Design of Efficient Propellers," NACA Wartime Report L-753,

August l9Li2. (Formerly ACR, Aug, 19L2.)

li. Lerbs, Hermann W.: "On the Effects of Scale and Roughness on Free-Running Propellers," Journal of the American Society of Naval

Engineers, Vol.

### 63,

No. 1, February 1951, pp. 58-9L,Lerbs, Hermann W.: "An Approximate Theory of Heavily Loaded Free-Running Propellers in the Optimum Condition," Paper presented at

the annual meeting of the Society of Naval Architects and Marine Engineers on November 9 and 10, 1950. (To be published in

Trans-actions of the Society of Naval Architects and Marine Engineers,

Vol. 58.)

Bienen, Th. and von Kimn, Th,: "Zur Theory der Luftsehrauben,"

Zeitschrift des Vereines Deutschdr Ingenieure, Vol. 68, Nos. L8

1h. Hill, J.G.: "The Design of Propellers," Transactions of the Society of Naval Architects and 1arine Engineers, Vol.

### 6, 19)48, pp.

1b3-192.### 15.

Kane, J.R.: Discussion of "The Design of Propellers," by J03, Hill,### Vol.

### 6, 19)48,

### p. 172.

TM-102

-TABLE I

MODEL PROPELLER PEFOI(ANCE AT

V/nD OF MAXIMUM EFFICICY

Designation

No.

Tested By; Prop. Designation; Diameter; j

V/riD at Maximum RPM -60n Reynolds No. at r-0.714RxlO7 K T K T T r C 3 2 t

### q Pn2D5

C 2### 2pn

Efficiency 1 Experimental Taxing 800 0.145 0.105 0.O21.9 0.1418 0.776 0.573Tank, Stevens

In-2

stitute of Tech- nology;

1000 0.56 0.116 0.0262 0.1462 0.819 0.5614 3 1200 0.67 0.120 0.0271 0.1478 0.8147 0.5614

### b

Model No. 52; 1308 0.73 0.123 0.02614 0.1489 0.825 0.593 D - 0.323 ft.; 5 j 0.80 il400 0.78 0.1214 0.0256 0.14914 o.800 0.618 6 Kempf'; 201.4 0.86 0.102 0.0225 0.1407 0.702 0.580 7 Model No. 35142; 363 1.53 0.109 0.0217 0.1433 0.678 0.638 D - 0.885 ft.; 8 j = 0.80 792 3.3 0.1214 0.02146 0.1492 0.767 0.6145 9 David Taylor 175 0.80 0.106 0.0208 0.1478 0.785 0.609 Model Basi-ri; 10 500 2.27 0.126 0.02314 0.570 0.8814 o.6)*14 Model No. 22114; D - 0.885 ft.; J 0.75TABLE II

ORDINATiS 0F T ?ROPELLER 3ECION AT

### 0.7LR

(Scaled from Kempi' ts Áodel

### Drawing,

Station### at r = 13On)

Thi-102 -

### 15

-Distance (,### in % of

### chord

Ordinates y,### in % of

chordLower 'Jpper

### Oedian

Line Half ThicknessO O D O D 1.25

### -o.6h

1.38 0.37 1.01 2.5 i.o6 2.32### 0.63

### 1.67

5### -iJ8

3.W1### 0.98

### 2J6

### 7.5

### -1.72

.l8 1.23 2.95 10_{-1.90}

_{1..67}

_{1.L9}

_{3.39}

15 -1,90 ### 5.87

1.99### 3.89

20### -.1.80

_{6.1i5}

2.33 I,l3
30 -1.00 ### 7,2!

### 3.12

### b.12

### -0.13

### 7iO

### 3.6k

### 3.70

So 0.27 7.25### 3.76

### 3,50

### 6o

### 0.53

### 6.60

### 3.57

3.02 70### o,)8

### 5.55

3.01 2.5I 60 o.i6 t,,.02 2.09 1.93 90### -0.05

2.22 1.09 100_{o}o o

Radius Trailing Edge =

### CHARACTERISTIC CURVES

### "SAN FRANCISCO" 4- BLADED VARYING PITCH PROPELLER

### ( STEVENS PROPELLER NO. 52)

### (CONSTRUCTION & REPAIR DWG. NO. C- 989)

### FIG. i.

### RIGHT- HAND ROTATION

### MODEL DIAM.,FT.

### 0.323

### MODEL PITCH, FT.

### 0.332

### AT 0.7 RADIUS

IDEV.### AREA RATIOzO.309

### PROPELLER IMMERSED 0.323 FT., 00% d.

### i-1400 R.P.M.

### o 1200 R.P.M.

D### 1000 RPM.

### 800 R.PM.

### .7

### .9

### LO

### 8

### 2

### 3

### .4

### .5

### .6

AXIAL INDUCED VELOCITY W W COS ß

AXIAL COMPONENT OF INDUCED VELOCITY U W COS2

TANGENTIAL COMPONENT OF INDUCED VELOCITY u8: W COS Sin

VELOCITY DIAGRAM

### AT INFINITY

DISPLACEMENT VELOCIT'! W OF HELICOIDAL SURFACES dT### ---- -

dT D. dT-AdT AdQ I### L_i

dQ dQ+ AdaFIG. 2: DIAGRAM SHOWING HELICOIDAL VORTEX SHEETS

IN THE WAKE OF A PROPELLER

rr

2

### VELOCITY DIAGRAM AT PROPELLER BLADE

### FIG. 3

T M-102

-17-TM-IO2

### -Is-FIG. 4: MAXIMUM EFFICIENCY VS. Tc Pc, AND C FOR J

### .80

### .80

### .95

.90 .85 u### o

### Z70

w### o

### u-w.65

### .55

### .50

O .1 .2 . .4 .5 .6### T/i/2r17R2V2

.70 .65### .60

.55### .50

.3 .4 .5 .6 .7### .8

.9 1.0 1.1### P/I/2f1TR2V3

### .95

### .90

85### .80

.75### .95

### .90

### .85

i,### .75

### g-

### >-o

### z

### LU.70

### o

### u-w

### .65

### .60

### .55

### FIG. 5

### MAXIMUM EFFICIENCY

### vs. Pc AND E

### FOR J=75

### .50

### 0

.1### 2

### .3

### .4

### .5

### .6

### .7

### .8

### .9

### 1.0

### LI

### ¡.2

### IC: T/I/2r1TR2V2

T M-102### -'9-.50

### .80

### -o

### g-.10

_{.09 .08}

### D

### .07

### o03

### 025

### 007

LZ### »Q09

### .008

### .007

### .006

### IO

### 1.5

### 2.0

### 3.0

### 4.0 5.0 6.0

### 8.0 IO

### 1.5

### 2.0

### 3.0

### 4.0 5.0 6.0

### ao iø'

### REYNOLDS NUMBER, Re

### Ub/t)

### -f

### o-H

### ___

### ____

### FtG6

### DRAG COEFFIC$ENT

.T### IR VS. REYNOLDS

### ----F--fl__.4

### PROflLE

### SECTiONS

### NUMBEP

### -HH----1

### &_4__ _J

### OF BLADE

### RAF

ft.._{6}

### F OL1

### i

### -1:

J r### PRANDTL- (ARA FPCTQN

### :

### j

### MEAN JNE: T

### = log,0

### eCf

### Ti

### L_BLAS;1S LAAR LE.

### 2 L32'

### t

-j

### SCHOEMEPR

### O.242//(Jf/2

### o

LU### DO

### o

LU### >2

### I-

### o

UJ### u-

u

### -u-J--4

### -5

### -6

### FIG. 7

### ANGLE OF ZERO LIFT a0 PLOTTED VS. REYNOLDS NUMBER

2 3